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

Core: VLSM Equivocators Fixed Equivocation

Section sec_equivocators_fixed_equivocations_vlsm.

Context
  {message : Type}
  `{finite.Finite index}
  (IM : index -> VLSM message)
  `{forall i : index, HasBeenSentCapability (IM i)}
  (equivocator_IM := equivocator_IM IM)
  (equivocating : set index)
  .

Definition of fixed-equivocation for states of the composition of equivocators.

Definition state_has_fixed_equivocation
  (s : composite_state equivocator_IM)
  : Prop
  :=
  equivocating_indices IM (enum index) s ⊆ equivocating.

Definition equivocators_fixed_equivocations_constraint
  (l : composite_label equivocator_IM)
  (som : composite_state equivocator_IM * option message)
  (som' := composite_transition equivocator_IM l som)
  : Prop
  := equivocators_no_equivocations_constraint IM l som
  /\ state_has_fixed_equivocation (fst som').

If the starting state_has_fixed_equivocation and the transition is made on an equivocating index, then the resulting state_has_fixed_equivocation.

Lemma equivocating_transition_preserves_fixed_equivocation l s om s' om'
  (Ht : composite_transition equivocator_IM l (s, om) = (s', om'))
  (Hs : state_has_fixed_equivocation s)
  (Heqv : projT1 l ∈ equivocating)
  : state_has_fixed_equivocation s'.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
l: composite_label equivocator_IM
s: composite_state equivocator_IM
om: option message
s': composite_state equivocator_IM
om': option message
Ht: composite_transition equivocator_IM l (s, om) =
(s', om')
Hs: state_has_fixed_equivocation s
Heqv: projT1 l ∈ equivocating
state_has_fixed_equivocation s'
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
l: composite_label equivocator_IM
s: composite_state equivocator_IM
om: option message
s': composite_state equivocator_IM
om': option message
Ht: composite_transition equivocator_IM l (s, om) =
(s', om')
Hs: state_has_fixed_equivocation s
Heqv: projT1 l ∈ equivocating
state_has_fixed_equivocation s'
  intros i Hi.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
l: composite_label equivocator_IM
s: composite_state equivocator_IM
om: option message
s': composite_state equivocator_IM
om': option message
Ht: composite_transition equivocator_IM l (s, om) =
(s', om')
Hs: state_has_fixed_equivocation s
Heqv: projT1 l ∈ equivocating
i: index
Hi: i ∈ equivocating_indices IM (enum index) s'
i ∈ equivocating
  destruct (decide (i = projT1 l)); subst; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
l: composite_label equivocator_IM
s: composite_state equivocator_IM
om: option message
s': composite_state equivocator_IM
om': option message
Ht: composite_transition equivocator_IM l (s, om) =
(s', om')
Hs: state_has_fixed_equivocation s
Heqv: projT1 l ∈ equivocating
i: index
Hi: i ∈ equivocating_indices IM (enum index) s'
n: i ≠ projT1 l
i ∈ equivocating
  destruct l; cbn in *.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
x: index
l: EquivocatorLabel (IM x)
s: composite_state equivocator_IM
om: option message
s': composite_state equivocator_IM
om': option message
Ht: (let (si', om') :=
   equivocator_transition (IM x) l (s x, om) in
 (state_update equivocator_IM s x si', om')) =
(s', om')
Hs: state_has_fixed_equivocation s
Heqv: x ∈ equivocating
i: index
Hi: i ∈ equivocating_indices IM (enum index) s'
n: i ≠ x
i ∈ equivocating
  destruct (equivocator_transition _ _ _).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
x: index
l: EquivocatorLabel (IM x)
s: composite_state equivocator_IM
om: option message
s': composite_state equivocator_IM
om': option message
e: equivocator_state (IM x)
o: option message
Ht: (state_update equivocator_IM s x e, o) =
(s', om')
Hs: state_has_fixed_equivocation s
Heqv: x ∈ equivocating
i: index
Hi: i ∈ equivocating_indices IM (enum index) s'
n: i ≠ x
i ∈ equivocating
  inversion Ht; subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
x: index
l: EquivocatorLabel (IM x)
s: composite_state equivocator_IM
om, om': option message
e: equivocator_state (IM x)
Ht: (state_update equivocator_IM s x e, om') =
(state_update equivocator_IM s x e, om')
Hs: state_has_fixed_equivocation s
Heqv: x ∈ equivocating
i: index
Hi: i
∈ equivocating_indices IM 
    (enum index)
    (state_update equivocator_IM s x e)
n: i ≠ x
i ∈ equivocating
  apply elem_of_list_filter, proj1 in Hi.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
x: index
l: EquivocatorLabel (IM x)
s: composite_state equivocator_IM
om, om': option message
e: equivocator_state (IM x)
Ht: (state_update equivocator_IM s x e, om') =
(state_update equivocator_IM s x e, om')
Hs: state_has_fixed_equivocation s
Heqv: x ∈ equivocating
i: index
Hi: is_equivocating_state (IM i)
  (state_update equivocator_IM s x e i)
n: i ≠ x
i ∈ equivocating
  state_update_simpl.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
x: index
l: EquivocatorLabel (IM x)
s: composite_state equivocator_IM
om, om': option message
e: equivocator_state (IM x)
Ht: (state_update equivocator_IM s x e, om') =
(state_update equivocator_IM s x e, om')
Hs: state_has_fixed_equivocation s
Heqv: x ∈ equivocating
i: index
Hi: is_equivocating_state (IM i) (s i)
n: i ≠ x
i ∈ equivocating
  by apply Hs, elem_of_list_filter; split; [| apply elem_of_enum].
Qed.

If the starting state_has_fixed_equivocation and the transition is made on the original copy, then the resulting state_has_fixed_equivocation.

Lemma zero_descriptor_transition_preserves_fixed_equivocation l s om s' om'
  (Ht : composite_transition equivocator_IM l (s, om) = (s', om'))
  (Hs : state_has_fixed_equivocation s)
  li
  (Hzero : projT2 l = ContinueWith 0 li)
  : state_has_fixed_equivocation s'.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
l: composite_label equivocator_IM
s: composite_state equivocator_IM
om: option message
s': composite_state equivocator_IM
om': option message
Ht: composite_transition equivocator_IM l (s, om) =
(s', om')
Hs: state_has_fixed_equivocation s
li: label (IM (projT1 l))
Hzero: projT2 l = ContinueWith 0 li
state_has_fixed_equivocation s'
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
l: composite_label equivocator_IM
s: composite_state equivocator_IM
om: option message
s': composite_state equivocator_IM
om': option message
Ht: composite_transition equivocator_IM l (s, om) =
(s', om')
Hs: state_has_fixed_equivocation s
li: label (IM (projT1 l))
Hzero: projT2 l = ContinueWith 0 li
state_has_fixed_equivocation s'
  intros i Hi.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
l: composite_label equivocator_IM
s: composite_state equivocator_IM
om: option message
s': composite_state equivocator_IM
om': option message
Ht: composite_transition equivocator_IM l (s, om) =
(s', om')
Hs: state_has_fixed_equivocation s
li: label (IM (projT1 l))
Hzero: projT2 l = ContinueWith 0 li
i: index
Hi: i ∈ equivocating_indices IM (enum index) s'
i ∈ equivocating apply Hs.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
l: composite_label equivocator_IM
s: composite_state equivocator_IM
om: option message
s': composite_state equivocator_IM
om': option message
Ht: composite_transition equivocator_IM l (s, om) =
(s', om')
Hs: state_has_fixed_equivocation s
li: label (IM (projT1 l))
Hzero: projT2 l = ContinueWith 0 li
i: index
Hi: i ∈ equivocating_indices IM (enum index) s'
i ∈ equivocating_indices IM (enum index) s
  apply elem_of_list_filter, proj1 in Hi.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
l: composite_label equivocator_IM
s: composite_state equivocator_IM
om: option message
s': composite_state equivocator_IM
om': option message
Ht: composite_transition equivocator_IM l (s, om) =
(s', om')
Hs: state_has_fixed_equivocation s
li: label (IM (projT1 l))
Hzero: projT2 l = ContinueWith 0 li
i: index
Hi: is_equivocating_state (IM i) (s' i)
i ∈ equivocating_indices IM (enum index) s
  apply elem_of_list_filter.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
l: composite_label equivocator_IM
s: composite_state equivocator_IM
om: option message
s': composite_state equivocator_IM
om': option message
Ht: composite_transition equivocator_IM l (s, om) =
(s', om')
Hs: state_has_fixed_equivocation s
li: label (IM (projT1 l))
Hzero: projT2 l = ContinueWith 0 li
i: index
Hi: is_equivocating_state (IM i) (s' i)
is_equivocating_state (IM i) (s i) ∧ i ∈ enum index
  split; [| by apply elem_of_enum].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
l: composite_label equivocator_IM
s: composite_state equivocator_IM
om: option message
s': composite_state equivocator_IM
om': option message
Ht: composite_transition equivocator_IM l (s, om) =
(s', om')
Hs: state_has_fixed_equivocation s
li: label (IM (projT1 l))
Hzero: projT2 l = ContinueWith 0 li
i: index
Hi: is_equivocating_state (IM i) (s' i)
is_equivocating_state (IM i) (s i)
  cbn in Ht.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
l: composite_label equivocator_IM
s: composite_state equivocator_IM
om: option message
s': composite_state equivocator_IM
om': option message
Ht: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition (IM i) li (s i, om) in
 (state_update equivocator_IM s i si', om')) =
(s', om')
Hs: state_has_fixed_equivocation s
li: label (IM (projT1 l))
Hzero: projT2 l = ContinueWith 0 li
i: index
Hi: is_equivocating_state (IM i) (s' i)
is_equivocating_state (IM i) (s i)
  destruct l as (eqv, leqv).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
eqv: index
leqv: label (equivocator_IM eqv)
s: composite_state equivocator_IM
om: option message
s': composite_state equivocator_IM
om': option message
Ht: (let (si', om') :=
   equivocator_transition 
     (IM eqv) leqv (s eqv, om) in
 (state_update equivocator_IM s eqv si', om')) =
(s', om')
Hs: state_has_fixed_equivocation s
li: label (IM (projT1 (existT eqv leqv)))
Hzero: projT2 (existT eqv leqv) = ContinueWith 0 li
i: index
Hi: is_equivocating_state (IM i) (s' i)
is_equivocating_state (IM i) (s i)
  destruct (equivocator_transition _ _ _) as (si', _om') eqn: Hti.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
eqv: index
leqv: label (equivocator_IM eqv)
s: composite_state equivocator_IM
om: option message
s': composite_state equivocator_IM
om': option message
si': equivocator_state (IM eqv)
_om': option message
Hti: equivocator_transition (IM eqv) leqv (s eqv, om) =
(si', _om')
Ht: (state_update equivocator_IM s eqv si', _om') =
(s', om')
Hs: state_has_fixed_equivocation s
li: label (IM (projT1 (existT eqv leqv)))
Hzero: projT2 (existT eqv leqv) = ContinueWith 0 li
i: index
Hi: is_equivocating_state (IM i) (s' i)
is_equivocating_state (IM i) (s i)
  inversion Ht.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
eqv: index
leqv: label (equivocator_IM eqv)
s: composite_state equivocator_IM
om: option message
s': composite_state equivocator_IM
om': option message
si': equivocator_state (IM eqv)
_om': option message
Hti: equivocator_transition (IM eqv) leqv (s eqv, om) =
(si', _om')
Ht: (state_update equivocator_IM s eqv si', _om') =
(s', om')
Hs: state_has_fixed_equivocation s
li: label (IM (projT1 (existT eqv leqv)))
Hzero: projT2 (existT eqv leqv) = ContinueWith 0 li
i: index
Hi: is_equivocating_state (IM i) (s' i)
H2: state_update equivocator_IM s eqv si' = s'
H3: _om' = om'
is_equivocating_state (IM i) (s i) subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
eqv: index
leqv: label (equivocator_IM eqv)
s: composite_state equivocator_IM
om, om': option message
si': equivocator_state (IM eqv)
Ht: (state_update equivocator_IM s eqv si', om') =
(state_update equivocator_IM s eqv si', om')
Hti: equivocator_transition (IM eqv) leqv (s eqv, om) =
(si', om')
Hs: state_has_fixed_equivocation s
li: label (IM (projT1 (existT eqv leqv)))
Hzero: projT2 (existT eqv leqv) = ContinueWith 0 li
i: index
Hi: is_equivocating_state (IM i)
  (state_update equivocator_IM s eqv si' i)
is_equivocating_state (IM i) (s i) clear Ht.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
eqv: index
leqv: label (equivocator_IM eqv)
s: composite_state equivocator_IM
om, om': option message
si': equivocator_state (IM eqv)
Hti: equivocator_transition (IM eqv) leqv (s eqv, om) =
(si', om')
Hs: state_has_fixed_equivocation s
li: label (IM (projT1 (existT eqv leqv)))
Hzero: projT2 (existT eqv leqv) = ContinueWith 0 li
i: index
Hi: is_equivocating_state (IM i)
  (state_update equivocator_IM s eqv si' i)
is_equivocating_state (IM i) (s i)
  destruct (decide (i = eqv)); subst; state_update_simpl; [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
eqv: index
leqv: label (equivocator_IM eqv)
s: composite_state equivocator_IM
om, om': option message
si': equivocator_state (IM eqv)
Hti: equivocator_transition (IM eqv) leqv (s eqv, om) =
(si', om')
Hs: state_has_fixed_equivocation s
li: label (IM (projT1 (existT eqv leqv)))
Hzero: projT2 (existT eqv leqv) = ContinueWith 0 li
Hi: is_equivocating_state (IM eqv) si'
is_equivocating_state (IM eqv) (s eqv)
  by apply (zero_descriptor_transition_reflects_equivocating_state (IM eqv) _ _ _ _ _ Hti _ Hzero Hi).
Qed.

If a future state has fixed equivocation, then so must the current state.

Lemma in_futures_reflects_fixed_equivocation
  (s1 s2 : composite_state equivocator_IM)
  (Hfutures :
    in_futures (preloaded_with_all_messages_vlsm (free_composite_vlsm equivocator_IM)) s1 s2)
  : state_has_fixed_equivocation s2 -> state_has_fixed_equivocation s1.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s1, s2: composite_state equivocator_IM
Hfutures: in_futures
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM))
  s1 s2
state_has_fixed_equivocation s2
→ state_has_fixed_equivocation s1
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s1, s2: composite_state equivocator_IM
Hfutures: in_futures
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM))
  s1 s2
state_has_fixed_equivocation s2
→ state_has_fixed_equivocation s1
  destruct Hfutures as [tr Htr].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s1, s2: composite_state equivocator_IM
tr: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM)) s1 s2
  tr
state_has_fixed_equivocation s2
→ state_has_fixed_equivocation s1
  induction Htr; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s, f: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM))
tl: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM)) s f
  tl
s': state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM))
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM)) l
  (s', iom) (s, oom)
IHHtr: state_has_fixed_equivocation f
→ state_has_fixed_equivocation s
state_has_fixed_equivocation f
→ state_has_fixed_equivocation s'
  intros Hf.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s, f: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM))
tl: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM)) s f
  tl
s': state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM))
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM)) l
  (s', iom) (s, oom)
IHHtr: state_has_fixed_equivocation f
→ state_has_fixed_equivocation s
Hf: state_has_fixed_equivocation f
state_has_fixed_equivocation s' specialize (IHHtr Hf).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s, f: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM))
tl: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM)) s f
  tl
s': state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM))
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM)) l
  (s', iom) (s, oom)
IHHtr: state_has_fixed_equivocation s
Hf: state_has_fixed_equivocation f
state_has_fixed_equivocation s' revert IHHtr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s, f: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM))
tl: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM)) s f
  tl
s': state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM))
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM)) l
  (s', iom) (s, oom)
Hf: state_has_fixed_equivocation f
state_has_fixed_equivocation s
→ state_has_fixed_equivocation s'
  apply transitivity.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s, f: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM))
tl: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM)) s f
  tl
s': state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM))
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM)) l
  (s', iom) (s, oom)
Hf: state_has_fixed_equivocation f
equivocating_indices IM (enum index) s'
⊆ equivocating_indices IM (enum index) s
  destruct Ht as [_ Ht].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s, f: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM))
tl: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM)) s f
  tl
s': state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM))
Ht: transition l (s', iom) = (s, oom)
Hf: state_has_fixed_equivocation f
equivocating_indices IM (enum index) s'
⊆ equivocating_indices IM (enum index) s
  revert Ht.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s, f: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM))
tl: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM)) s f
  tl
s': state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM))
Hf: state_has_fixed_equivocation f
transition l (s', iom) = (s, oom)
→ equivocating_indices IM (enum index) s'
  ⊆ equivocating_indices IM (enum index) s
  by rapply @equivocators_transition_preserves_equivocating_indices.
Qed.

Composition of equivocators with no message equivocation and a fixed set of machines allowed to state-equivocate.

Definition equivocators_fixed_equivocations_vlsm
  : VLSM message
  :=
  composite_vlsm equivocator_IM equivocators_fixed_equivocations_constraint.

Lemma not_equivocating_index_has_singleton_state
  (s : composite_state equivocator_IM)
  (Hs : valid_state_prop equivocators_fixed_equivocations_vlsm s)
  (i : index)
  (Hi : i ∉ equivocating)
  : is_singleton_state (IM i) (s i).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s: composite_state equivocator_IM
Hs: valid_state_prop
  equivocators_fixed_equivocations_vlsm s
i: index
Hi: i ∉ equivocating
is_singleton_state (IM i) (s i)
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s: composite_state equivocator_IM
Hs: valid_state_prop
  equivocators_fixed_equivocations_vlsm s
i: index
Hi: i ∉ equivocating
is_singleton_state (IM i) (s i)
  apply valid_state_has_trace in Hs as (is & tr & Htr & His).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s: composite_state equivocator_IM
is: state equivocators_fixed_equivocations_vlsm
tr: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm is s tr
His: initial_state_prop is
i: index
Hi: i ∉ equivocating
is_singleton_state (IM i) (s i)
  assert (is_singleton_state (IM i) (is i)) by apply His; clear His.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s: composite_state equivocator_IM
is: state equivocators_fixed_equivocations_vlsm
tr: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm is s tr
i: index
Hi: i ∉ equivocating
H1: is_singleton_state (IM i) (is i)
is_singleton_state (IM i) (s i)
  induction Htr; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s, f: state equivocators_fixed_equivocations_vlsm
tl: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm s f tl
s': state equivocators_fixed_equivocations_vlsm
iom, oom: option message
l: label equivocators_fixed_equivocations_vlsm
Ht: input_valid_transition
  equivocators_fixed_equivocations_vlsm l
  (s', iom) (s, oom)
i: index
Hi: i ∉ equivocating
H1: is_singleton_state (IM i) (s' i)
IHHtr: is_singleton_state (IM i) (s i)
→ is_singleton_state (IM i) (f i)
is_singleton_state (IM i) (f i)
  apply IHHtr; clear IHHtr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s, f: state equivocators_fixed_equivocations_vlsm
tl: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm s f tl
s': state equivocators_fixed_equivocations_vlsm
iom, oom: option message
l: label equivocators_fixed_equivocations_vlsm
Ht: input_valid_transition
  equivocators_fixed_equivocations_vlsm l
  (s', iom) (s, oom)
i: index
Hi: i ∉ equivocating
H1: is_singleton_state (IM i) (s' i)
is_singleton_state (IM i) (s i)
  destruct Ht as [(_ & _ & _ & _ & Hfixed) Ht].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s, f: state equivocators_fixed_equivocations_vlsm
tl: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm s f tl
s': state equivocators_fixed_equivocations_vlsm
iom, oom: option message
l: label equivocators_fixed_equivocations_vlsm
Hfixed: state_has_fixed_equivocation
  (composite_transition equivocator_IM l
     (s', iom)).1
Ht: transition l (s', iom) = (s, oom)
i: index
Hi: i ∉ equivocating
H1: is_singleton_state (IM i) (s' i)
is_singleton_state (IM i) (s i)
  cbn in *; rewrite Ht in Hfixed; clear Ht; cbn in *.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s, f: composite_state equivocator_IM
tl: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm s f tl
s': composite_state equivocator_IM
iom, oom: option message
l: composite_label equivocator_IM
Hfixed: state_has_fixed_equivocation s
i: index
Hi: i ∉ equivocating
H1: is_singleton_state (IM i) (s' i)
is_singleton_state (IM i) (s i)
  destruct (decide (equivocator_state_n (s i) = 1)); [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s, f: composite_state equivocator_IM
tl: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm s f tl
s': composite_state equivocator_IM
iom, oom: option message
l: composite_label equivocator_IM
Hfixed: state_has_fixed_equivocation s
i: index
Hi: i ∉ equivocating
H1: is_singleton_state (IM i) (s' i)
n: equivocator_state_n (s i) ≠ 1
is_singleton_state (IM i) (s i)
  elim Hi.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s, f: composite_state equivocator_IM
tl: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm s f tl
s': composite_state equivocator_IM
iom, oom: option message
l: composite_label equivocator_IM
Hfixed: state_has_fixed_equivocation s
i: index
Hi: i ∉ equivocating
H1: is_singleton_state (IM i) (s' i)
n: equivocator_state_n (s i) ≠ 1
i ∈ equivocating apply Hfixed.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s, f: composite_state equivocator_IM
tl: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm s f tl
s': composite_state equivocator_IM
iom, oom: option message
l: composite_label equivocator_IM
Hfixed: state_has_fixed_equivocation s
i: index
Hi: i ∉ equivocating
H1: is_singleton_state (IM i) (s' i)
n: equivocator_state_n (s i) ≠ 1
i ∈ equivocating_indices IM (enum index) s
  by apply elem_of_list_filter; split; [| apply elem_of_enum].
Qed.

Lemma valid_state_has_fixed_equivocation
  (s : composite_state equivocator_IM)
  (Hs : valid_state_prop equivocators_fixed_equivocations_vlsm s)
  : state_has_fixed_equivocation s.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s: composite_state equivocator_IM
Hs: valid_state_prop
  equivocators_fixed_equivocations_vlsm s
state_has_fixed_equivocation s
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s: composite_state equivocator_IM
Hs: valid_state_prop
  equivocators_fixed_equivocations_vlsm s
state_has_fixed_equivocation s
  apply valid_state_has_trace in Hs.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s: composite_state equivocator_IM
Hs: ∃ (is : state
          equivocators_fixed_equivocations_vlsm) 
  (tr : list transition_item),
  finite_valid_trace_init_to
    equivocators_fixed_equivocations_vlsm is s tr
state_has_fixed_equivocation s
  destruct Hs as [is [tr [Htr Hinit]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s: composite_state equivocator_IM
is: state equivocators_fixed_equivocations_vlsm
tr: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm is s tr
Hinit: initial_state_prop is
state_has_fixed_equivocation s
  assert (state_has_fixed_equivocation is) as His.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s: composite_state equivocator_IM
is: state equivocators_fixed_equivocations_vlsm
tr: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm is s tr
Hinit: initial_state_prop is
state_has_fixed_equivocation is
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s: composite_state equivocator_IM
is: state equivocators_fixed_equivocations_vlsm
tr: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm is s tr
Hinit: initial_state_prop is
His: state_has_fixed_equivocation is
state_has_fixed_equivocation s {message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s: composite_state equivocator_IM
is: state equivocators_fixed_equivocations_vlsm
tr: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm is s tr
Hinit: initial_state_prop is
state_has_fixed_equivocation is
    intros i Hin.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s: composite_state equivocator_IM
is: state equivocators_fixed_equivocations_vlsm
tr: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm is s tr
Hinit: initial_state_prop is
i: index
Hin: i ∈ equivocating_indices IM (enum index) is
i ∈ equivocating
    apply elem_of_list_filter in Hin.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s: composite_state equivocator_IM
is: state equivocators_fixed_equivocations_vlsm
tr: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm is s tr
Hinit: initial_state_prop is
i: index
Hin: is_equivocating_state (IM i) (is i)
∧ i ∈ enum index
i ∈ equivocating
    destruct Hin as [His Hin].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s: composite_state equivocator_IM
is: state equivocators_fixed_equivocations_vlsm
tr: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm is s tr
Hinit: initial_state_prop is
i: index
His: is_equivocating_state (IM i) (is i)
Hin: i ∈ enum index
i ∈ equivocating
    unfold is_equivocating_state in His.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s: composite_state equivocator_IM
is: state equivocators_fixed_equivocations_vlsm
tr: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm is s tr
Hinit: initial_state_prop is
i: index
His: ¬ is_singleton_state (IM i) (is i)
Hin: i ∈ enum index
i ∈ equivocating
    contradict His.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s: composite_state equivocator_IM
is: state equivocators_fixed_equivocations_vlsm
tr: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm is s tr
Hinit: initial_state_prop is
i: index
Hin: i ∈ enum index
is_singleton_state (IM i) (is i)
    by apply Hinit.
  }message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s: composite_state equivocator_IM
is: state equivocators_fixed_equivocations_vlsm
tr: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm is s tr
Hinit: initial_state_prop is
His: state_has_fixed_equivocation is
state_has_fixed_equivocation s
  clear Hinit.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s: composite_state equivocator_IM
is: state equivocators_fixed_equivocations_vlsm
tr: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm is s tr
His: state_has_fixed_equivocation is
state_has_fixed_equivocation s
  induction Htr; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s, f: state equivocators_fixed_equivocations_vlsm
tl: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm s f tl
s': state equivocators_fixed_equivocations_vlsm
iom, oom: option message
l: label equivocators_fixed_equivocations_vlsm
Ht: input_valid_transition
  equivocators_fixed_equivocations_vlsm l
  (s', iom) (s, oom)
His: state_has_fixed_equivocation s'
IHHtr: state_has_fixed_equivocation s
→ state_has_fixed_equivocation f
state_has_fixed_equivocation f
  apply IHHtr; clear IHHtr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s, f: state equivocators_fixed_equivocations_vlsm
tl: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm s f tl
s': state equivocators_fixed_equivocations_vlsm
iom, oom: option message
l: label equivocators_fixed_equivocations_vlsm
Ht: input_valid_transition
  equivocators_fixed_equivocations_vlsm l
  (s', iom) (s, oom)
His: state_has_fixed_equivocation s'
state_has_fixed_equivocation s
  destruct Ht as [(_ & _ & _ & _ & Hv) Ht].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
s, f: state equivocators_fixed_equivocations_vlsm
tl: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm s f tl
s': state equivocators_fixed_equivocations_vlsm
iom, oom: option message
l: label equivocators_fixed_equivocations_vlsm
Hv: state_has_fixed_equivocation
  (composite_transition equivocator_IM l
     (s', iom)).1
Ht: transition l (s', iom) = (s, oom)
His: state_has_fixed_equivocation s'
state_has_fixed_equivocation s
  by setoid_rewrite Ht in Hv.
Qed.

Inclusion in the free composition.

Lemma equivocators_fixed_equivocations_vlsm_incl_free
  : VLSM_incl equivocators_fixed_equivocations_vlsm (free_composite_vlsm equivocator_IM).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
VLSM_incl equivocators_fixed_equivocations_vlsm
  (free_composite_vlsm equivocator_IM)
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
VLSM_incl equivocators_fixed_equivocations_vlsm
  (free_composite_vlsm equivocator_IM)
  by apply VLSM_incl_constrained_vlsm.
Qed.

Inclusion into the preloaded free composition.

Lemma equivocators_fixed_equivocations_vlsm_incl_PreFree :
  VLSM_incl equivocators_fixed_equivocations_vlsm
    (preloaded_with_all_messages_vlsm (free_composite_vlsm equivocator_IM)).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
VLSM_incl equivocators_fixed_equivocations_vlsm
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM))
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
VLSM_incl equivocators_fixed_equivocations_vlsm
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM))
  by apply constrained_preloaded_incl.
Qed.

Inclusion of preloaded machine into the preloaded free composition.

Lemma preloaded_equivocators_fixed_equivocations_vlsm_incl_free :
  VLSM_incl
    (preloaded_with_all_messages_vlsm equivocators_fixed_equivocations_vlsm)
    (preloaded_with_all_messages_vlsm (free_composite_vlsm equivocator_IM)).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
VLSM_incl
  (preloaded_with_all_messages_vlsm
     equivocators_fixed_equivocations_vlsm)
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM))
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
VLSM_incl
  (preloaded_with_all_messages_vlsm
     equivocators_fixed_equivocations_vlsm)
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM))
  by apply basic_VLSM_incl_preloaded; [intro | inversion 1 | intro].
Qed.

Inclusion in the composition of equivocators with no message equivocation (no restriction on state equivocation).

Lemma equivocators_fixed_equivocations_vlsm_incl_no_equivocations
  : VLSM_incl equivocators_fixed_equivocations_vlsm (equivocators_no_equivocations_vlsm IM).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
VLSM_incl equivocators_fixed_equivocations_vlsm
  (equivocators_no_equivocations_vlsm IM)
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
VLSM_incl equivocators_fixed_equivocations_vlsm
  (equivocators_no_equivocations_vlsm 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)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating: set index
input_valid_constraint_subsumption
  (free_composite_vlsm
     (EquivocatorsComposition.equivocator_IM IM))
  equivocators_fixed_equivocations_constraint
  (equivocators_no_equivocations_constraint IM)
  by intros l [s om] Hv; apply Hv.
Qed.

End sec_equivocators_fixed_equivocations_vlsm.

Lemma equivocators_fixed_equivocations_vlsm_subset_incl
  {message : Type}
  `{finite.Finite index}
  (IM : index -> VLSM message)
  `{forall i : index, HasBeenSentCapability (IM i)}
  (equivocator_IM := equivocator_IM IM)
  (equivocating1 equivocating2 : set index)
  (Hincl : equivocating1 ⊆ equivocating2) :
  VLSM_incl
    (equivocators_fixed_equivocations_vlsm IM equivocating1)
    (equivocators_fixed_equivocations_vlsm IM equivocating2).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating1, equivocating2: set index
Hincl: equivocating1 ⊆ equivocating2
VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     equivocating1)
  (equivocators_fixed_equivocations_vlsm IM
     equivocating2)
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating1, equivocating2: set index
Hincl: equivocating1 ⊆ equivocating2
VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     equivocating1)
  (equivocators_fixed_equivocations_vlsm IM
     equivocating2)
  apply constraint_subsumption_incl.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating1, equivocating2: set index
Hincl: equivocating1 ⊆ equivocating2
input_valid_constraint_subsumption
  (free_composite_vlsm
     (EquivocatorsComposition.equivocator_IM IM))
  (equivocators_fixed_equivocations_constraint IM
     equivocating1)
  (equivocators_fixed_equivocations_constraint IM
     equivocating2)
  apply preloaded_constraint_subsumption_stronger,
    strong_constraint_subsumption_strongest.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating1, equivocating2: set index
Hincl: equivocating1 ⊆ equivocating2
strong_constraint_subsumption
  (free_composite_vlsm
     (EquivocatorsComposition.equivocator_IM IM))
  (equivocators_fixed_equivocations_constraint IM
     equivocating1)
  (equivocators_fixed_equivocations_constraint IM
     equivocating2)
  intros l (_s, om) [Hmsg Hfixed].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating1, equivocating2: set index
Hincl: equivocating1 ⊆ equivocating2
l: label
  (free_composite_vlsm
     (EquivocatorsComposition.equivocator_IM IM))
_s: state
  (free_composite_vlsm
     (EquivocatorsComposition.equivocator_IM IM))
om: option message
Hmsg: equivocators_no_equivocations_constraint IM l
  (_s, om)
Hfixed: state_has_fixed_equivocation IM equivocating1
  (composite_transition
     (EquivocatorsComposition.equivocator_IM
        IM) l (_s, om)).1
equivocators_fixed_equivocations_constraint IM
  equivocating2 l (_s, om)
  split; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating1, equivocating2: set index
Hincl: equivocating1 ⊆ equivocating2
l: label
  (free_composite_vlsm
     (EquivocatorsComposition.equivocator_IM IM))
_s: state
  (free_composite_vlsm
     (EquivocatorsComposition.equivocator_IM IM))
om: option message
Hmsg: equivocators_no_equivocations_constraint IM l
  (_s, om)
Hfixed: state_has_fixed_equivocation IM equivocating1
  (composite_transition
     (EquivocatorsComposition.equivocator_IM
        IM) l (_s, om)).1
state_has_fixed_equivocation IM equivocating2
  (composite_transition
     (EquivocatorsComposition.equivocator_IM IM) l
     (_s, om)).1
  unfold state_has_fixed_equivocation.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= EquivocatorsComposition.equivocator_IM
  IM: index → VLSM message
equivocating1, equivocating2: set index
Hincl: equivocating1 ⊆ equivocating2
l: label
  (free_composite_vlsm
     (EquivocatorsComposition.equivocator_IM IM))
_s: state
  (free_composite_vlsm
     (EquivocatorsComposition.equivocator_IM IM))
om: option message
Hmsg: equivocators_no_equivocations_constraint IM l
  (_s, om)
Hfixed: state_has_fixed_equivocation IM equivocating1
  (composite_transition
     (EquivocatorsComposition.equivocator_IM
        IM) l (_s, om)).1
equivocating_indices IM (enum index)
  (composite_transition
     (EquivocatorsComposition.equivocator_IM IM) l
     (_s, om)).1 ⊆ equivocating2
  by set_solver.
Qed.

Section sec_fixed_equivocation_with_fullnode.

This section instantiates the full_node_condition_for_admissible_equivocators by choosing the admissible indices to be the fixed set of components allowed to equivocate, and then shows that this constraint is stronger than the fixed_equivocation_constraint.

Context setting the stage for, and instantiating the full_node_condition_for_admissible_equivocators. It requires that the components have has_been_sent and has_been_received capabilities, that the number of components is finite.

Additionally to the above we require that equality on messages is decidable and that the set of VLSMs allowed to equivocate is non-empty.

Context
  `{EqDecision message}
  `{FinSet index Ci}
  `{finite.Finite index}
  (IM : index -> VLSM message)
  `{forall i : index, HasBeenSentCapability (IM i)}
  `{forall i : index, HasBeenReceivedCapability (IM i)}
  (equivocating : Ci)
  .

The full_node_constraint_alt is stronger than the fixed_equivocation_constraint.

Lemma fixed_equivocation_constraint_subsumption_alt :
  strong_constraint_subsumption (free_composite_vlsm IM)
    (full_node_condition_for_admissible_equivocators_alt IM (fun _ i => i ∈ equivocating))
    (fixed_equivocation_constraint IM equivocating).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
strong_constraint_subsumption (free_composite_vlsm IM)
  (full_node_condition_for_admissible_equivocators_alt
     IM
     (λ (_ : composite_state IM) (i : index),
        i ∈ equivocating))
  (fixed_equivocation_constraint IM equivocating)
Proof.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
strong_constraint_subsumption (free_composite_vlsm IM)
  (full_node_condition_for_admissible_equivocators_alt
     IM
     (λ (_ : composite_state IM) (i : index),
        i ∈ equivocating))
  (fixed_equivocation_constraint IM equivocating)
  intros l (s, [m |]) Hc ; [| done].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
l: label (free_composite_vlsm IM)
s: state (free_composite_vlsm IM)
m: message
Hc: full_node_condition_for_admissible_equivocators_alt
  IM
  (λ (_ : composite_state IM) (i : index),
     i ∈ equivocating) l (
  s, Some m)
fixed_equivocation_constraint IM equivocating l
  (s, Some m)
  destruct Hc as [Hno_equiv | [i [Hi Hm]]]; [by left |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
l: label (free_composite_vlsm IM)
s: state (free_composite_vlsm IM)
m: message
i: index
Hi: i ∈ equivocating
Hm: node_generated_without_further_equivocation_alt
  IM s m i
fixed_equivocation_constraint IM equivocating l
  (s, Some m)
  unfold node_generated_without_further_equivocation_alt in Hm.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
l: label (free_composite_vlsm IM)
s: state (free_composite_vlsm IM)
m: message
i: index
Hi: i ∈ equivocating
Hm: can_emit
  (preloaded_vlsm (IM i)
     (composite_has_been_directly_observed IM s))
  m
fixed_equivocation_constraint IM equivocating l
  (s, Some m)
  right.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
l: label (free_composite_vlsm IM)
s: state (free_composite_vlsm IM)
m: message
i: index
Hi: i ∈ equivocating
Hm: can_emit
  (preloaded_vlsm (IM i)
     (composite_has_been_directly_observed IM s))
  m
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating s) m
  apply elem_of_elements in Hi.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
l: label (free_composite_vlsm IM)
s: state (free_composite_vlsm IM)
m: message
i: index
Hi: i ∈ elements equivocating
Hm: can_emit
  (preloaded_vlsm (IM i)
     (composite_has_been_directly_observed IM s))
  m
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating s) m
  eapply can_emit_composite_free_lift with (j := dexist i Hi); [| done].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
l: label (free_composite_vlsm IM)
s: state (free_composite_vlsm IM)
m: message
i: index
Hi: i ∈ elements equivocating
Hm: can_emit
  (preloaded_vlsm (IM i)
     (composite_has_been_directly_observed IM s))
  m
∀ m : message,
  composite_has_been_directly_observed IM s m
  → composite_has_been_directly_observed IM s m
  by eauto.
Qed.

If all components have the cannot_resend_message_stepwise_property, then the full node condition is stronger than the fixed_equivocation_constraint (by reduction to the result above).

Lemma fixed_equivocation_constraint_subsumption
  (Hno_resend : forall i : index, cannot_resend_message_stepwise_prop (IM i))
  : preloaded_constraint_subsumption (free_composite_vlsm IM)
      (full_node_condition_for_admissible_equivocators IM (fun _ i => i ∈ equivocating))
      (fixed_equivocation_constraint IM equivocating).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
Hno_resend: ∀ i : index,
  cannot_resend_message_stepwise_prop
    (IM i)
preloaded_constraint_subsumption
  (free_composite_vlsm IM)
  (full_node_condition_for_admissible_equivocators IM
     (λ (_ : composite_state IM) (i : index),
        i ∈ equivocating))
  (fixed_equivocation_constraint IM equivocating)
Proof.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
Hno_resend: ∀ i : index,
  cannot_resend_message_stepwise_prop
    (IM i)
preloaded_constraint_subsumption
  (free_composite_vlsm IM)
  (full_node_condition_for_admissible_equivocators IM
     (λ (_ : composite_state IM) (i : index),
        i ∈ equivocating))
  (fixed_equivocation_constraint IM equivocating)
  intros l (s, om) Hv.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
Hno_resend: ∀ i : index,
  cannot_resend_message_stepwise_prop
    (IM i)
l: label (free_composite_vlsm IM)
s: state
  (preloaded_with_all_messages_vlsm
     (constrained_vlsm 
        (free_composite_vlsm IM)
        (full_node_condition_for_admissible_equivocators
           IM
           (λ (_ : composite_state IM) (i : index),
              i ∈ equivocating))))
om: option message
Hv: input_constrained
  (constrained_vlsm (free_composite_vlsm IM)
     (full_node_condition_for_admissible_equivocators
        IM
        (λ (_ : composite_state IM) (i : index),
           i ∈ equivocating))) l (
  s, om)
fixed_equivocation_constraint IM equivocating l
  (s, om)
  apply fixed_equivocation_constraint_subsumption_alt.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
Hno_resend: ∀ i : index,
  cannot_resend_message_stepwise_prop
    (IM i)
l: label (free_composite_vlsm IM)
s: state
  (preloaded_with_all_messages_vlsm
     (constrained_vlsm 
        (free_composite_vlsm IM)
        (full_node_condition_for_admissible_equivocators
           IM
           (λ (_ : composite_state IM) (i : index),
              i ∈ equivocating))))
om: option message
Hv: input_constrained
  (constrained_vlsm (free_composite_vlsm IM)
     (full_node_condition_for_admissible_equivocators
        IM
        (λ (_ : composite_state IM) (i : index),
           i ∈ equivocating))) l (
  s, om)
full_node_condition_for_admissible_equivocators_alt IM
  (λ (_ : composite_state IM) (i : index),
     i ∈ equivocating) l (s, om)
  by eapply full_node_condition_for_admissible_equivocators_subsumption.
Qed.

End sec_fixed_equivocation_with_fullnode.

Section sec_from_equivocators_to_components.

From composition of equivocators to composition of simple components

In this section we show that the projection of valid_traces of a composition of equivocators with a fixed equivocation constraint are valid_traces for the composition with a similar fixed equivocation constraint.

Context
  `{EqDecision message}
  {index : Type}
  `{FinSet index Ci}
  `{finite.Finite index}
  (IM : index -> VLSM message)
  `{forall i : index, HasBeenSentCapability (IM i)}
  `{forall i : index, HasBeenReceivedCapability (IM i)}
  (equivocating : Ci)
  (XE : VLSM message := equivocators_fixed_equivocations_vlsm IM (elements equivocating))
  (X : VLSM message := fixed_equivocation_vlsm_composition IM equivocating)
  (FreeE : VLSM message := free_composite_vlsm (equivocator_IM IM))
  (Hdec_init : forall i, decidable_initial_messages_prop (IM i))
  (Free := free_composite_vlsm IM)
  .

proper_equivocator_descriptors strengthen for fixed-equivocation. We require that if the index is not equivocating, than the corresponding descriptor is a zero_descriptor.

Definition proper_fixed_equivocator_descriptors
  (eqv_descriptors : equivocator_descriptors IM)
  (s : state (free_composite_vlsm (equivocator_IM IM)))
  : Prop
  := proper_equivocator_descriptors IM eqv_descriptors s /\
    forall i, i ∉ equivocating -> eqv_descriptors i = Existing 0.

not_equivocating_equivocator_descriptors satisfy the proper_fixed_equivocator_descriptors property.

Lemma not_equivocating_equivocator_descriptors_proper_fixed
  (s : composite_state (equivocator_IM IM))
  (Hs : valid_state_prop XE s)
  (eqv_descriptors : equivocator_descriptors IM)
  (Heqv_descriptors : not_equivocating_equivocator_descriptors IM eqv_descriptors s)
  : proper_fixed_equivocator_descriptors eqv_descriptors s.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
Hs: valid_state_prop XE s
eqv_descriptors: equivocator_descriptors IM
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors s
proper_fixed_equivocator_descriptors eqv_descriptors s
Proof.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
Hs: valid_state_prop XE s
eqv_descriptors: equivocator_descriptors IM
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors s
proper_fixed_equivocator_descriptors eqv_descriptors s
  apply not_equivocating_equivocator_descriptors_proper in Heqv_descriptors as Hproper.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
Hs: valid_state_prop XE s
eqv_descriptors: equivocator_descriptors IM
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors s
Hproper: proper_equivocator_descriptors IM
  eqv_descriptors s
proper_fixed_equivocator_descriptors eqv_descriptors s
  split; [done |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
Hs: valid_state_prop XE s
eqv_descriptors: equivocator_descriptors IM
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors s
Hproper: proper_equivocator_descriptors IM
  eqv_descriptors s
∀ i : index,
  i ∉ equivocating → eqv_descriptors i = Existing 0
  intros i Hi; rewrite <- elem_of_elements in Hi.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
Hs: valid_state_prop XE s
eqv_descriptors: equivocator_descriptors IM
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors s
Hproper: proper_equivocator_descriptors IM
  eqv_descriptors s
i: index
Hi: i ∉ elements equivocating
eqv_descriptors i = Existing 0
  pose proof (Hzero := not_equivocating_index_has_singleton_state
    IM (elements equivocating) _ Hs _ Hi); unfold is_singleton_state in Hzero.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
Hs: valid_state_prop XE s
eqv_descriptors: equivocator_descriptors IM
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors s
Hproper: proper_equivocator_descriptors IM
  eqv_descriptors s
i: index
Hi: i ∉ elements equivocating
Hzero: equivocator_state_n (s i) = 1
eqv_descriptors i = Existing 0
  specialize (Heqv_descriptors i); destruct (eqv_descriptors i); [done |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
Hs: valid_state_prop XE s
eqv_descriptors: equivocator_descriptors IM
i: index
n: nat
Heqv_descriptors: existing_descriptor (IM i)
  (Existing n) (s i)
Hproper: proper_equivocator_descriptors IM
  eqv_descriptors s
Hi: i ∉ elements equivocating
Hzero: equivocator_state_n (s i) = 1
Existing n = Existing 0
  destruct Heqv_descriptors as [s_i_n Heqv_descriptors].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
Hs: valid_state_prop XE s
eqv_descriptors: equivocator_descriptors IM
i: index
n: nat
s_i_n: state (IM i)
Heqv_descriptors: equivocator_state_project (s i) n =
Some s_i_n
Hproper: proper_equivocator_descriptors IM
  eqv_descriptors s
Hi: i ∉ elements equivocating
Hzero: equivocator_state_n (s i) = 1
Existing n = Existing 0
  apply equivocator_state_project_Some_rev in Heqv_descriptors.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
Hs: valid_state_prop XE s
eqv_descriptors: equivocator_descriptors IM
i: index
n: nat
s_i_n: state (IM i)
Heqv_descriptors: n < equivocator_state_n (s i)
Hproper: proper_equivocator_descriptors IM
  eqv_descriptors s
Hi: i ∉ elements equivocating
Hzero: equivocator_state_n (s i) = 1
Existing n = Existing 0
  by f_equal; lia.
Qed.

Projections of (valid) traces of the composition of equivocators preserve proper_fixed_equivocator_descriptors.

Lemma equivocators_trace_project_preserves_proper_fixed_equivocator_descriptors
  (is : composite_state (equivocator_IM IM))
  (tr : list (composite_transition_item (equivocator_IM IM)))
  (Htr : finite_constrained_trace_from FreeE is tr)
  (s := finite_trace_last is tr)
  (descriptors : equivocator_descriptors IM)
  (idescriptors : equivocator_descriptors IM)
  (trX : list (composite_transition_item IM))
  (HtrX : equivocators_trace_project IM descriptors tr = Some (trX, idescriptors))
  : proper_fixed_equivocator_descriptors descriptors s ->
      proper_fixed_equivocator_descriptors idescriptors is.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from FreeE is tr
s:= finite_trace_last is tr: state (composite_type (equivocator_IM IM))
descriptors, idescriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
HtrX: equivocators_trace_project IM descriptors tr =
Some (trX, idescriptors)
proper_fixed_equivocator_descriptors descriptors s
→ proper_fixed_equivocator_descriptors idescriptors is
Proof.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from FreeE is tr
s:= finite_trace_last is tr: state (composite_type (equivocator_IM IM))
descriptors, idescriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
HtrX: equivocators_trace_project IM descriptors tr =
Some (trX, idescriptors)
proper_fixed_equivocator_descriptors descriptors s
→ proper_fixed_equivocator_descriptors idescriptors is
  intros [Hproper Hfixed].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from FreeE is tr
s:= finite_trace_last is tr: state (composite_type (equivocator_IM IM))
descriptors, idescriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
HtrX: equivocators_trace_project IM descriptors tr =
Some (trX, idescriptors)
Hproper: proper_equivocator_descriptors IM
  descriptors s
Hfixed: ∀ i : index,
  i ∉ equivocating
  → descriptors i = Existing 0
proper_fixed_equivocator_descriptors idescriptors is
  specialize
    (preloaded_equivocators_valid_trace_project_proper_initial IM
      _ _ Htr _ _ _ HtrX Hproper)
    as Hiproper.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from FreeE is tr
s:= finite_trace_last is tr: state (composite_type (equivocator_IM IM))
descriptors, idescriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
HtrX: equivocators_trace_project IM descriptors tr =
Some (trX, idescriptors)
Hproper: proper_equivocator_descriptors IM
  descriptors s
Hfixed: ∀ i : index,
  i ∉ equivocating
  → descriptors i = Existing 0
Hiproper: proper_equivocator_descriptors IM
  idescriptors is
proper_fixed_equivocator_descriptors idescriptors is
  split; [done |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from FreeE is tr
s:= finite_trace_last is tr: state (composite_type (equivocator_IM IM))
descriptors, idescriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
HtrX: equivocators_trace_project IM descriptors tr =
Some (trX, idescriptors)
Hproper: proper_equivocator_descriptors IM
  descriptors s
Hfixed: ∀ i : index,
  i ∉ equivocating
  → descriptors i = Existing 0
Hiproper: proper_equivocator_descriptors IM
  idescriptors is
∀ i : index,
  i ∉ equivocating → idescriptors i = Existing 0
  intros i Hi.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from FreeE is tr
s:= finite_trace_last is tr: state (composite_type (equivocator_IM IM))
descriptors, idescriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
HtrX: equivocators_trace_project IM descriptors tr =
Some (trX, idescriptors)
Hproper: proper_equivocator_descriptors IM
  descriptors s
Hfixed: ∀ i : index,
  i ∉ equivocating
  → descriptors i = Existing 0
Hiproper: proper_equivocator_descriptors IM
  idescriptors is
i: index
Hi: i ∉ equivocating
idescriptors i = Existing 0 specialize (Hfixed i Hi).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from FreeE is tr
s:= finite_trace_last is tr: state (composite_type (equivocator_IM IM))
descriptors, idescriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
HtrX: equivocators_trace_project IM descriptors tr =
Some (trX, idescriptors)
Hproper: proper_equivocator_descriptors IM
  descriptors s
i: index
Hfixed: descriptors i = Existing 0
Hiproper: proper_equivocator_descriptors IM
  idescriptors is
Hi: i ∉ equivocating
idescriptors i = Existing 0
  revert Hfixed.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from FreeE is tr
s:= finite_trace_last is tr: state (composite_type (equivocator_IM IM))
descriptors, idescriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
HtrX: equivocators_trace_project IM descriptors tr =
Some (trX, idescriptors)
Hproper: proper_equivocator_descriptors IM
  descriptors s
i: index
Hiproper: proper_equivocator_descriptors IM
  idescriptors is
Hi: i ∉ equivocating
descriptors i = Existing 0
→ idescriptors i = Existing 0 clear Hi.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from FreeE is tr
s:= finite_trace_last is tr: state (composite_type (equivocator_IM IM))
descriptors, idescriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
HtrX: equivocators_trace_project IM descriptors tr =
Some (trX, idescriptors)
Hproper: proper_equivocator_descriptors IM
  descriptors s
i: index
Hiproper: proper_equivocator_descriptors IM
  idescriptors is
descriptors i = Existing 0
→ idescriptors i = Existing 0 revert i.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from FreeE is tr
s:= finite_trace_last is tr: state (composite_type (equivocator_IM IM))
descriptors, idescriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
HtrX: equivocators_trace_project IM descriptors tr =
Some (trX, idescriptors)
Hproper: proper_equivocator_descriptors IM
  descriptors s
Hiproper: proper_equivocator_descriptors IM
  idescriptors is
∀ i : index,
  descriptors i = Existing 0
  → idescriptors i = Existing 0
  by eapply equivocators_trace_project_preserves_zero_descriptors.
Qed.

Lemma fixed_equivocators_initial_state_project
  (es : state XE)
  (Hes : initial_state_prop XE es)
  (eqv_descriptors : equivocator_descriptors IM)
  (Heqv : proper_equivocator_descriptors IM eqv_descriptors es)
  : initial_state_prop X (equivocators_state_project IM eqv_descriptors es).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
es: state XE
Hes: initial_state_prop es
eqv_descriptors: equivocator_descriptors IM
Heqv: proper_equivocator_descriptors IM
  eqv_descriptors es
initial_state_prop
  (equivocators_state_project IM eqv_descriptors es)
Proof.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
es: state XE
Hes: initial_state_prop es
eqv_descriptors: equivocator_descriptors IM
Heqv: proper_equivocator_descriptors IM
  eqv_descriptors es
initial_state_prop
  (equivocators_state_project IM eqv_descriptors es)
  intro eqv.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
es: state XE
Hes: initial_state_prop es
eqv_descriptors: equivocator_descriptors IM
Heqv: proper_equivocator_descriptors IM
  eqv_descriptors es
eqv: index
initial_state_prop
  (equivocators_state_project IM eqv_descriptors es
     eqv) specialize (Hes eqv).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
es: state XE
eqv: index
Hes: initial_state_prop (es eqv)
eqv_descriptors: equivocator_descriptors IM
Heqv: proper_equivocator_descriptors IM
  eqv_descriptors es
initial_state_prop
  (equivocators_state_project IM eqv_descriptors es
     eqv)
  unfold equivocator_IM in Hes.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
es: state XE
eqv: index
Hes: initial_state_prop (es eqv)
eqv_descriptors: equivocator_descriptors IM
Heqv: proper_equivocator_descriptors IM
  eqv_descriptors es
initial_state_prop
  (equivocators_state_project IM eqv_descriptors es
     eqv)
  unfold equivocators_state_project.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
es: state XE
eqv: index
Hes: initial_state_prop (es eqv)
eqv_descriptors: equivocator_descriptors IM
Heqv: proper_equivocator_descriptors IM
  eqv_descriptors es
initial_state_prop
  (equivocator_state_descriptor_project (es eqv)
     (eqv_descriptors eqv))
  specialize (Heqv eqv).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
es: state XE
eqv: index
Hes: initial_state_prop (es eqv)
eqv_descriptors: equivocator_descriptors IM
Heqv: proper_descriptor (IM eqv)
  (eqv_descriptors eqv) 
  (es eqv)
initial_state_prop
  (equivocator_state_descriptor_project (es eqv)
     (eqv_descriptors eqv))
  destruct (eqv_descriptors eqv) as [sn | i]; [done |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
es: state XE
eqv: index
Hes: initial_state_prop (es eqv)
eqv_descriptors: equivocator_descriptors IM
i: nat
Heqv: proper_descriptor (IM eqv) 
  (Existing i) (es eqv)
initial_state_prop
  (equivocator_state_descriptor_project (es eqv)
     (Existing i))
  destruct Heqv as [esi Heqsi].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
es: state XE
eqv: index
Hes: initial_state_prop (es eqv)
eqv_descriptors: equivocator_descriptors IM
i: nat
esi: state (IM eqv)
Heqsi: equivocator_state_project (es eqv) i =
Some esi
initial_state_prop
  (equivocator_state_descriptor_project (es eqv)
     (Existing i))
  simpl.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
es: state XE
eqv: index
Hes: initial_state_prop (es eqv)
eqv_descriptors: equivocator_descriptors IM
i: nat
esi: state (IM eqv)
Heqsi: equivocator_state_project (es eqv) i =
Some esi
initial_state_prop
  (default (equivocator_state_zero (es eqv))
     (equivocator_state_project (es eqv) i)) rewrite Heqsi.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
es: state XE
eqv: index
Hes: initial_state_prop (es eqv)
eqv_descriptors: equivocator_descriptors IM
i: nat
esi: state (IM eqv)
Heqsi: equivocator_state_project (es eqv) i =
Some esi
initial_state_prop
  (default (equivocator_state_zero (es eqv))
     (Some esi))
  by eapply equivocator_vlsm_initial_state_preservation_rev.
Qed.

This is a property of the fixed_equivocation_constraint which also trivially holds for the free constraint. This property is sufficient for proving the _equivocators_valid_trace_project lemma, which lets that lemma be used for both the composition of equivocators with fixed state-equivocation and the free composition.

It basically says that if a message has_been_sent for a state of the composition of equivocators with no-message equivocations and fixed state-equivocations, then any of its projections should be allowed to receive it.

Definition constraint_has_been_sent_prop
  (constraint : composite_label IM -> composite_state IM * option message -> Prop)
  : Prop :=
  forall
    (s : composite_state (equivocator_IM IM))
    (Hs : valid_state_prop XE s)
    (descriptors : equivocator_descriptors IM)
    (Hdescriptors : proper_fixed_equivocator_descriptors descriptors s)
    (sX := equivocators_state_project IM descriptors s)
    (m : message)
    (Hm : has_been_sent FreeE s m)
    l,
  constraint l (sX, Some m).

Generic proof that the projection of a trace of the composition of equivocators with no message equivocation and fixed state equivocation is valid w.r.t. the composition of the regular components constrained by any constraint satisfying several properties, including the constraint_has_been_sent_property.

The proof proceeds by well founded induction on the length of the trace, performing an analysis on the final transition item of the trace.

It uses the fact that the trace has no message equivocation to extract a subtrace producing the message being received at the last transition and proves that it's a valid message for the destination machine by using the induction hypothesis (that is why well-founded induction is used rather than a simpler induction principle).

The constraint satisfaction for the projection of the final transition is for now assumes as hypothesis Hconstraint_hbs.

Lemma _equivocators_valid_trace_project
  (final_descriptors : equivocator_descriptors IM)
  (is : composite_state (equivocator_IM IM))
  (tr : list (composite_transition_item (equivocator_IM IM)))
  (final_state := finite_trace_last is tr)
  (Hproper : proper_fixed_equivocator_descriptors final_descriptors final_state)
  (Htr : finite_valid_trace XE is tr)
  (constraint : composite_label IM -> composite_state IM * option message -> Prop)
  (HconstraintNone : forall l s, constraint l (s, None))
  (Hconstraint_hbs :  constraint_has_been_sent_prop constraint)
  (X' := composite_vlsm IM constraint)
  : exists
    (trX : list (composite_transition_item IM))
    (initial_descriptors : equivocator_descriptors IM)
    (isX := equivocators_state_project IM initial_descriptors is)
    (final_stateX := finite_trace_last isX trX),
    proper_fixed_equivocator_descriptors initial_descriptors is /\
    equivocators_trace_project IM final_descriptors tr = Some (trX, initial_descriptors) /\
    equivocators_state_project IM final_descriptors final_state = final_stateX /\
    finite_valid_trace X' isX trX.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
final_descriptors: equivocator_descriptors IM
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr: finite_valid_trace XE is tr
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
Proof.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
final_descriptors: equivocator_descriptors IM
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr: finite_valid_trace XE is tr
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
  remember (length tr) as len_tr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
final_descriptors: equivocator_descriptors IM
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr: finite_valid_trace XE is tr
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
len_tr: nat
Heqlen_tr: len_tr = length tr
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
  revert tr Htr Heqlen_tr final_state final_descriptors Hproper.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
len_tr: nat
∀ tr : list
         (composite_transition_item
            (equivocator_IM IM)),
  finite_valid_trace XE is tr
  → len_tr = length tr
    → ∀ (final_state := finite_trace_last is tr) (final_descriptors : 
                                                 equivocator_descriptors
                                                 IM),
        proper_fixed_equivocator_descriptors
          final_descriptors final_state
        → ∃ (trX : list (composite_transition_item IM)) 
            (initial_descriptors : equivocator_descriptors
                                     IM),
            let isX :=
              equivocators_state_project IM
                initial_descriptors is in
            let final_stateX :=
              finite_trace_last isX trX in
            proper_fixed_equivocator_descriptors
              initial_descriptors is
            ∧ equivocators_trace_project IM
                final_descriptors tr =
              Some (trX, initial_descriptors)
              ∧ equivocators_state_project IM
                  final_descriptors final_state =
                final_stateX
                ∧ finite_valid_trace X' isX trX
  induction len_tr as [len_tr IHlen] using (well_founded_induction Wf_nat.lt_wf); intros.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
len_tr: nat
IHlen: ∀ y : nat,
  y < len_tr
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ (final_state :=
             finite_trace_last is tr) 
            (final_descriptors : 
             equivocator_descriptors IM),
            proper_fixed_equivocator_descriptors
              final_descriptors final_state
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                let isX :=
                  equivocators_state_project
                    IM initial_descriptors is
                  in
                let final_stateX :=
                  finite_trace_last isX trX in
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      final_state =
                    final_stateX
                    ∧ finite_valid_trace X'
                      isX trX
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
Heqlen_tr: len_tr = length tr
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
  subst len_tr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
IHlen: ∀ y : nat,
  y < length tr
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ (final_state :=
             finite_trace_last is tr) 
            (final_descriptors : 
             equivocator_descriptors IM),
            proper_fixed_equivocator_descriptors
              final_descriptors final_state
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                let isX :=
                  equivocators_state_project
                    IM initial_descriptors is
                  in
                let final_stateX :=
                  finite_trace_last isX trX in
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      final_state =
                    final_stateX
                    ∧ finite_valid_trace X'
                      isX trX
Htr: finite_valid_trace XE is tr
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
  destruct_list_last tr tr' lst Htr_lst.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
IHlen: ∀ y : nat,
  y < length []
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ (final_state :=
             finite_trace_last is tr) 
            (final_descriptors : 
             equivocator_descriptors IM),
            proper_fixed_equivocator_descriptors
              final_descriptors final_state
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                let isX :=
                  equivocators_state_project
                    IM initial_descriptors is
                  in
                let final_stateX :=
                  finite_trace_last isX trX in
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      final_state =
                    final_stateX
                    ∧ finite_valid_trace X'
                      isX trX
Htr: finite_valid_trace XE is []
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr_lst: tr = []
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors [] =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
IHlen: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ (final_state :=
             finite_trace_last is tr) 
            (final_descriptors : 
             equivocator_descriptors IM),
            proper_fixed_equivocator_descriptors
              final_descriptors final_state
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                let isX :=
                  equivocators_state_project
                    IM initial_descriptors is
                  in
                let final_stateX :=
                  finite_trace_last isX trX in
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      final_state =
                    final_stateX
                    ∧ finite_valid_trace X'
                      isX trX
Htr: finite_valid_trace XE is (tr' ++ [lst])
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr_lst: tr = tr' ++ [lst]
∃ (trX : list (composite_transition_item IM)) 
  (initial_descriptors : 
   equivocator_descriptors IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors
      (tr' ++ [lst]) = 
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
  -message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
IHlen: ∀ y : nat,
  y < length []
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ (final_state :=
             finite_trace_last is tr) 
            (final_descriptors : 
             equivocator_descriptors IM),
            proper_fixed_equivocator_descriptors
              final_descriptors final_state
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                let isX :=
                  equivocators_state_project
                    IM initial_descriptors is
                  in
                let final_stateX :=
                  finite_trace_last isX trX in
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      final_state =
                    final_stateX
                    ∧ finite_valid_trace X'
                      isX trX
Htr: finite_valid_trace XE is []
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr_lst: tr = []
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors [] =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX clear IHlen.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is []
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr_lst: tr = []
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors [] =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX subst.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
Htr: finite_valid_trace XE is []
final_state:= finite_trace_last is []: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors [] =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX exists [], final_descriptors.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
Htr: finite_valid_trace XE is []
final_state:= finite_trace_last is []: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
let isX :=
  equivocators_state_project IM final_descriptors is
  in
let final_stateX := finite_trace_last isX [] in
proper_fixed_equivocator_descriptors final_descriptors
  is
∧ equivocators_trace_project IM final_descriptors [] =
  Some ([], final_descriptors)
  ∧ equivocators_state_project IM final_descriptors
      final_state = final_stateX
    ∧ finite_valid_trace X' isX []
    split; [done |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
Htr: finite_valid_trace XE is []
final_state:= finite_trace_last is []: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
equivocators_trace_project IM final_descriptors [] =
Some ([], final_descriptors)
∧ equivocators_state_project IM final_descriptors
    final_state =
  finite_trace_last
    (equivocators_state_project IM final_descriptors
       is) []
  ∧ finite_valid_trace X'
      (equivocators_state_project IM final_descriptors
         is) [] split; [done |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
Htr: finite_valid_trace XE is []
final_state:= finite_trace_last is []: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
equivocators_state_project IM final_descriptors
  final_state =
finite_trace_last
  (equivocators_state_project IM final_descriptors is)
  []
∧ finite_valid_trace X'
    (equivocators_state_project IM final_descriptors
       is) [] split; [done |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
Htr: finite_valid_trace XE is []
final_state:= finite_trace_last is []: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
finite_valid_trace X'
  (equivocators_state_project IM final_descriptors is)
  []
    remember (equivocators_state_project IM final_descriptors is) as isx.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
Htr: finite_valid_trace XE is []
final_state:= finite_trace_last is []: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
isx: state (free_composite_vlsm IM)
Heqisx: isx =
equivocators_state_project IM
  final_descriptors is
finite_valid_trace X' isx []
    cut (initial_state_prop X' isx).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
Htr: finite_valid_trace XE is []
final_state:= finite_trace_last is []: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
isx: state (free_composite_vlsm IM)
Heqisx: isx =
equivocators_state_project IM
  final_descriptors is
initial_state_prop isx → finite_valid_trace X' isx []
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
Htr: finite_valid_trace XE is []
final_state:= finite_trace_last is []: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
isx: state (free_composite_vlsm IM)
Heqisx: isx =
equivocators_state_project IM
  final_descriptors is
initial_state_prop isx
    {message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
Htr: finite_valid_trace XE is []
final_state:= finite_trace_last is []: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
isx: state (free_composite_vlsm IM)
Heqisx: isx =
equivocators_state_project IM
  final_descriptors is
initial_state_prop isx → finite_valid_trace X' isx [] intro His.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
Htr: finite_valid_trace XE is []
final_state:= finite_trace_last is []: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
isx: state (free_composite_vlsm IM)
Heqisx: isx =
equivocators_state_project IM
  final_descriptors is
His: initial_state_prop isx
finite_valid_trace X' isx [] split; [| done].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
Htr: finite_valid_trace XE is []
final_state:= finite_trace_last is []: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
isx: state (free_composite_vlsm IM)
Heqisx: isx =
equivocators_state_project IM
  final_descriptors is
His: initial_state_prop isx
finite_valid_trace_from X' isx [] constructor.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
Htr: finite_valid_trace XE is []
final_state:= finite_trace_last is []: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
isx: state (free_composite_vlsm IM)
Heqisx: isx =
equivocators_state_project IM
  final_descriptors is
His: initial_state_prop isx
valid_state_prop X' isx
      apply valid_state_prop_iff.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
Htr: finite_valid_trace XE is []
final_state:= finite_trace_last is []: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
isx: state (free_composite_vlsm IM)
Heqisx: isx =
equivocators_state_project IM
  final_descriptors is
His: initial_state_prop isx
(∃ is : initial_state X', isx = `is)
∨ (∃ (l : label X') (som : state X' * option message) 
     (om' : option message),
     input_valid_transition X' l som (isx, om')) left.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
Htr: finite_valid_trace XE is []
final_state:= finite_trace_last is []: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
isx: state (free_composite_vlsm IM)
Heqisx: isx =
equivocators_state_project IM
  final_descriptors is
His: initial_state_prop isx
∃ is : initial_state X', isx = `is
      by exists (exist _ _ His).
    }message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
Htr: finite_valid_trace XE is []
final_state:= finite_trace_last is []: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
isx: state (free_composite_vlsm IM)
Heqisx: isx =
equivocators_state_project IM
  final_descriptors is
initial_state_prop isx
    by subst; apply fixed_equivocators_initial_state_project; [apply Htr | apply Hproper].
  -message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
IHlen: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ (final_state :=
             finite_trace_last is tr) 
            (final_descriptors : 
             equivocator_descriptors IM),
            proper_fixed_equivocator_descriptors
              final_descriptors final_state
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                let isX :=
                  equivocators_state_project
                    IM initial_descriptors is
                  in
                let final_stateX :=
                  finite_trace_last isX trX in
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      final_state =
                    final_stateX
                    ∧ finite_valid_trace X'
                      isX trX
Htr: finite_valid_trace XE is (tr' ++ [lst])
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr_lst: tr = tr' ++ [lst]
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors
      (tr' ++ [lst]) = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX specialize (IHlen (length tr')) as IH'.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
IHlen: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ (final_state :=
             finite_trace_last is tr) 
            (final_descriptors : 
             equivocator_descriptors IM),
            proper_fixed_equivocator_descriptors
              final_descriptors final_state
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                let isX :=
                  equivocators_state_project
                    IM initial_descriptors is
                  in
                let final_stateX :=
                  finite_trace_last isX trX in
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      final_state =
                    final_stateX
                    ∧ finite_valid_trace X'
                      isX trX
Htr: finite_valid_trace XE is (tr' ++ [lst])
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr_lst: tr = tr' ++ [lst]
IH': length tr' < length (tr' ++ [lst])
→ ∀ tr : list
           (composite_transition_item
              (equivocator_IM IM)),
    finite_valid_trace XE is tr
    → length tr' = length tr
      → ∀ (final_state :=
           finite_trace_last is tr) 
          (final_descriptors : 
           equivocator_descriptors IM),
          proper_fixed_equivocator_descriptors
            final_descriptors final_state
          → ∃ (trX : list
                      (composite_transition_item
                      IM)) 
              (initial_descriptors : 
               equivocator_descriptors IM),
              let isX :=
                equivocators_state_project IM
                  initial_descriptors is in
              let final_stateX :=
                finite_trace_last isX trX in
              proper_fixed_equivocator_descriptors
                initial_descriptors is
              ∧ equivocators_trace_project IM
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                ∧ equivocators_state_project IM
                    final_descriptors
                    final_state = final_stateX
                  ∧ finite_valid_trace X' isX
                      trX
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors
      (tr' ++ [lst]) = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
    spec IH'; [by rewrite app_length; cbn; lia |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
IHlen: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ (final_state :=
             finite_trace_last is tr) 
            (final_descriptors : 
             equivocator_descriptors IM),
            proper_fixed_equivocator_descriptors
              final_descriptors final_state
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                let isX :=
                  equivocators_state_project
                    IM initial_descriptors is
                  in
                let final_stateX :=
                  finite_trace_last isX trX in
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      final_state =
                    final_stateX
                    ∧ finite_valid_trace X'
                      isX trX
Htr: finite_valid_trace XE is (tr' ++ [lst])
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr_lst: tr = tr' ++ [lst]
IH': ∀ tr : list
         (composite_transition_item
            (equivocator_IM IM)),
  finite_valid_trace XE is tr
  → length tr' = length tr
    → ∀ (final_state := finite_trace_last is tr) 
        (final_descriptors : 
         equivocator_descriptors IM),
        proper_fixed_equivocator_descriptors
          final_descriptors final_state
        → ∃ (trX : list
                     (composite_transition_item
                      IM)) 
            (initial_descriptors : 
             equivocator_descriptors IM),
            let isX :=
              equivocators_state_project IM
                initial_descriptors is in
            let final_stateX :=
              finite_trace_last isX trX in
            proper_fixed_equivocator_descriptors
              initial_descriptors is
            ∧ equivocators_trace_project IM
                final_descriptors tr =
              Some (trX, initial_descriptors)
              ∧ equivocators_state_project IM
                  final_descriptors final_state =
                final_stateX
                ∧ finite_valid_trace X' isX trX
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors
      (tr' ++ [lst]) = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
    destruct Htr as [Htr Hinit].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
IHlen: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ (final_state :=
             finite_trace_last is tr) 
            (final_descriptors : 
             equivocator_descriptors IM),
            proper_fixed_equivocator_descriptors
              final_descriptors final_state
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                let isX :=
                  equivocators_state_project
                    IM initial_descriptors is
                  in
                let final_stateX :=
                  finite_trace_last isX trX in
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      final_state =
                    final_stateX
                    ∧ finite_valid_trace X'
                      isX trX
Htr: finite_valid_trace_from XE is (tr' ++ [lst])
Hinit: initial_state_prop is
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr_lst: tr = tr' ++ [lst]
IH': ∀ tr : list
         (composite_transition_item
            (equivocator_IM IM)),
  finite_valid_trace XE is tr
  → length tr' = length tr
    → ∀ (final_state := finite_trace_last is tr) 
        (final_descriptors : 
         equivocator_descriptors IM),
        proper_fixed_equivocator_descriptors
          final_descriptors final_state
        → ∃ (trX : list
                     (composite_transition_item
                      IM)) 
            (initial_descriptors : 
             equivocator_descriptors IM),
            let isX :=
              equivocators_state_project IM
                initial_descriptors is in
            let final_stateX :=
              finite_trace_last isX trX in
            proper_fixed_equivocator_descriptors
              initial_descriptors is
            ∧ equivocators_trace_project IM
                final_descriptors tr =
              Some (trX, initial_descriptors)
              ∧ equivocators_state_project IM
                  final_descriptors final_state =
                final_stateX
                ∧ finite_valid_trace X' isX trX
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors
      (tr' ++ [lst]) = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
    apply finite_valid_trace_from_app_iff in Htr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
IHlen: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ (final_state :=
             finite_trace_last is tr) 
            (final_descriptors : 
             equivocator_descriptors IM),
            proper_fixed_equivocator_descriptors
              final_descriptors final_state
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                let isX :=
                  equivocators_state_project
                    IM initial_descriptors is
                  in
                let final_stateX :=
                  finite_trace_last isX trX in
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      final_state =
                    final_stateX
                    ∧ finite_valid_trace X'
                      isX trX
Htr: finite_valid_trace_from XE is tr'
∧ finite_valid_trace_from XE
    (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr_lst: tr = tr' ++ [lst]
IH': ∀ tr : list
         (composite_transition_item
            (equivocator_IM IM)),
  finite_valid_trace XE is tr
  → length tr' = length tr
    → ∀ (final_state := finite_trace_last is tr) 
        (final_descriptors : 
         equivocator_descriptors IM),
        proper_fixed_equivocator_descriptors
          final_descriptors final_state
        → ∃ (trX : list
                     (composite_transition_item
                      IM)) 
            (initial_descriptors : 
             equivocator_descriptors IM),
            let isX :=
              equivocators_state_project IM
                initial_descriptors is in
            let final_stateX :=
              finite_trace_last isX trX in
            proper_fixed_equivocator_descriptors
              initial_descriptors is
            ∧ equivocators_trace_project IM
                final_descriptors tr =
              Some (trX, initial_descriptors)
              ∧ equivocators_state_project IM
                  final_descriptors final_state =
                final_stateX
                ∧ finite_valid_trace X' isX trX
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors
      (tr' ++ [lst]) = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
    destruct Htr as [Htr Hlst].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
IHlen: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ (final_state :=
             finite_trace_last is tr) 
            (final_descriptors : 
             equivocator_descriptors IM),
            proper_fixed_equivocator_descriptors
              final_descriptors final_state
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                let isX :=
                  equivocators_state_project
                    IM initial_descriptors is
                  in
                let final_stateX :=
                  finite_trace_last isX trX in
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      final_state =
                    final_stateX
                    ∧ finite_valid_trace X'
                      isX trX
Htr: finite_valid_trace_from XE is tr'
Hlst: finite_valid_trace_from XE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr_lst: tr = tr' ++ [lst]
IH': ∀ tr : list
         (composite_transition_item
            (equivocator_IM IM)),
  finite_valid_trace XE is tr
  → length tr' = length tr
    → ∀ (final_state := finite_trace_last is tr) 
        (final_descriptors : 
         equivocator_descriptors IM),
        proper_fixed_equivocator_descriptors
          final_descriptors final_state
        → ∃ (trX : list
                     (composite_transition_item
                      IM)) 
            (initial_descriptors : 
             equivocator_descriptors IM),
            let isX :=
              equivocators_state_project IM
                initial_descriptors is in
            let final_stateX :=
              finite_trace_last isX trX in
            proper_fixed_equivocator_descriptors
              initial_descriptors is
            ∧ equivocators_trace_project IM
                final_descriptors tr =
              Some (trX, initial_descriptors)
              ∧ equivocators_state_project IM
                  final_descriptors final_state =
                final_stateX
                ∧ finite_valid_trace X' isX trX
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors
      (tr' ++ [lst]) = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
    specialize (IH' tr' (conj Htr Hinit) eq_refl).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
IHlen: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ (final_state :=
             finite_trace_last is tr) 
            (final_descriptors : 
             equivocator_descriptors IM),
            proper_fixed_equivocator_descriptors
              final_descriptors final_state
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                let isX :=
                  equivocators_state_project
                    IM initial_descriptors is
                  in
                let final_stateX :=
                  finite_trace_last isX trX in
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      final_state =
                    final_stateX
                    ∧ finite_valid_trace X'
                      isX trX
Htr: finite_valid_trace_from XE is tr'
Hlst: finite_valid_trace_from XE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr_lst: tr = tr' ++ [lst]
IH': ∀ (final_state := finite_trace_last is tr') 
  (final_descriptors : 
   equivocator_descriptors IM),
  proper_fixed_equivocator_descriptors
    final_descriptors final_state
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      let isX :=
        equivocators_state_project IM
          initial_descriptors is in
      let final_stateX :=
        finite_trace_last isX trX in
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
          final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ equivocators_state_project IM
            final_descriptors final_state =
          final_stateX
          ∧ finite_valid_trace X' isX trX
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors
      (tr' ++ [lst]) = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
    specialize (equivocators_transition_item_project_proper_characterization IM final_descriptors lst)
      as Hproperx.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
IHlen: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ (final_state :=
             finite_trace_last is tr) 
            (final_descriptors : 
             equivocator_descriptors IM),
            proper_fixed_equivocator_descriptors
              final_descriptors final_state
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                let isX :=
                  equivocators_state_project
                    IM initial_descriptors is
                  in
                let final_stateX :=
                  finite_trace_last isX trX in
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      final_state =
                    final_stateX
                    ∧ finite_valid_trace X'
                      isX trX
Htr: finite_valid_trace_from XE is tr'
Hlst: finite_valid_trace_from XE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr_lst: tr = tr' ++ [lst]
IH': ∀ (final_state := finite_trace_last is tr') 
  (final_descriptors : 
   equivocator_descriptors IM),
  proper_fixed_equivocator_descriptors
    final_descriptors final_state
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      let isX :=
        equivocators_state_project IM
          initial_descriptors is in
      let final_stateX :=
        finite_trace_last isX trX in
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
          final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ equivocators_state_project IM
            final_descriptors final_state =
          final_stateX
          ∧ finite_valid_trace X' isX trX
Hproperx: proper_equivocator_descriptors IM
  final_descriptors (destination lst)
→ ∃ (oitem : option
               (composite_transition_item
                  IM)) 
    (eqv_descriptors' : 
     equivocator_descriptors IM),
    equivocators_transition_item_project IM
      final_descriptors lst =
    Some (oitem, eqv_descriptors')
    ∧ match oitem with
      | Some itemx =>
          (∃ Hex : existing_equivocator_label
                     (IM (projT1 (l lst)))
                     (projT2 (l lst)),
             existT (projT1 (l lst))
               (existing_equivocator_label_extract
                  (IM (projT1 (l lst)))
                  (projT2 (l lst)) Hex) =
             l itemx)
          ∧ input lst = input itemx
            ∧ output lst = output itemx
              ∧ equivocators_state_project
                  IM final_descriptors
                  (destination lst) =
                destination itemx
                ∧ eqv_descriptors'
                    (projT1 (l lst)) =
                  equivocator_label_descriptor
                    (projT2 (l lst))
      | None => True
      end
      ∧ (∀ s : composite_state
                 (equivocator_IM IM),
           composite_valid
             (equivocator_IM IM) 
             (l lst) (s, input lst)
           → composite_transition
               (equivocator_IM IM) 
               (l lst) (
               s, input lst) =
             (destination lst, output lst)
             → proper_equivocator_descriptors
                 IM eqv_descriptors' s
               ∧ eqv_descriptors' =
                 equivocator_descriptors_update
                   IM final_descriptors
                   (projT1 (l lst))
                   (eqv_descriptors'
                      (projT1 (l lst)))
                 ∧ s =
                   state_update
                     (equivocator_IM IM)
                     (destination lst)
                     (projT1 (l lst))
                     (s (projT1 (l lst)))
                   ∧ previous_state_descriptor_prop
                      (IM (projT1 (l lst)))
                      (final_descriptors
                      (projT1 (l lst)))
                      (s (projT1 (l lst)))
                      (eqv_descriptors'
                      (projT1 (l lst)))
                     ∧ 
                     match oitem with
                     | Some itemx =>
                      ∀ sx : 
                      composite_state IM,
                      sx =
                      equivocators_state_project
                      IM eqv_descriptors' s
                      → 
                      composite_valid IM
                      (l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition
                      IM 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
                     | None =>
                      equivocators_state_project
                      IM final_descriptors
                      (destination lst) =
                      equivocators_state_project
                      IM eqv_descriptors' s
                     end)
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors
      (tr' ++ [lst]) = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
    specialize
      (equivocators_transition_item_project_preserves_zero_descriptors IM final_descriptors lst)
      as Hzero.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
IHlen: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ (final_state :=
             finite_trace_last is tr) 
            (final_descriptors : 
             equivocator_descriptors IM),
            proper_fixed_equivocator_descriptors
              final_descriptors final_state
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                let isX :=
                  equivocators_state_project
                    IM initial_descriptors is
                  in
                let final_stateX :=
                  finite_trace_last isX trX in
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      final_state =
                    final_stateX
                    ∧ finite_valid_trace X'
                      isX trX
Htr: finite_valid_trace_from XE is tr'
Hlst: finite_valid_trace_from XE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr_lst: tr = tr' ++ [lst]
IH': ∀ (final_state := finite_trace_last is tr') 
  (final_descriptors : 
   equivocator_descriptors IM),
  proper_fixed_equivocator_descriptors
    final_descriptors final_state
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      let isX :=
        equivocators_state_project IM
          initial_descriptors is in
      let final_stateX :=
        finite_trace_last isX trX in
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
          final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ equivocators_state_project IM
            final_descriptors final_state =
          final_stateX
          ∧ finite_valid_trace X' isX trX
Hproperx: proper_equivocator_descriptors IM
  final_descriptors (destination lst)
→ ∃ (oitem : option
               (composite_transition_item
                  IM)) 
    (eqv_descriptors' : 
     equivocator_descriptors IM),
    equivocators_transition_item_project IM
      final_descriptors lst =
    Some (oitem, eqv_descriptors')
    ∧ match oitem with
      | Some itemx =>
          (∃ Hex : existing_equivocator_label
                     (IM (projT1 (l lst)))
                     (projT2 (l lst)),
             existT (projT1 (l lst))
               (existing_equivocator_label_extract
                  (IM (projT1 (l lst)))
                  (projT2 (l lst)) Hex) =
             l itemx)
          ∧ input lst = input itemx
            ∧ output lst = output itemx
              ∧ equivocators_state_project
                  IM final_descriptors
                  (destination lst) =
                destination itemx
                ∧ eqv_descriptors'
                    (projT1 (l lst)) =
                  equivocator_label_descriptor
                    (projT2 (l lst))
      | None => True
      end
      ∧ (∀ s : composite_state
                 (equivocator_IM IM),
           composite_valid
             (equivocator_IM IM) 
             (l lst) (s, input lst)
           → composite_transition
               (equivocator_IM IM) 
               (l lst) (
               s, input lst) =
             (destination lst, output lst)
             → proper_equivocator_descriptors
                 IM eqv_descriptors' s
               ∧ eqv_descriptors' =
                 equivocator_descriptors_update
                   IM final_descriptors
                   (projT1 (l lst))
                   (eqv_descriptors'
                      (projT1 (l lst)))
                 ∧ s =
                   state_update
                     (equivocator_IM IM)
                     (destination lst)
                     (projT1 (l lst))
                     (s (projT1 (l lst)))
                   ∧ previous_state_descriptor_prop
                      (IM (projT1 (l lst)))
                      (final_descriptors
                      (projT1 (l lst)))
                      (s (projT1 (l lst)))
                      (eqv_descriptors'
                      (projT1 (l lst)))
                     ∧ 
                     match oitem with
                     | Some itemx =>
                      ∀ sx : 
                      composite_state IM,
                      sx =
                      equivocators_state_project
                      IM eqv_descriptors' s
                      → 
                      composite_valid IM
                      (l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition
                      IM 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
                     | None =>
                      equivocators_state_project
                      IM final_descriptors
                      (destination lst) =
                      equivocators_state_project
                      IM eqv_descriptors' s
                     end)
Hzero: ∀ (oitem : option
             (composite_transition_item IM)) 
  (idescriptors : equivocator_descriptors IM) 
  (s : composite_state (equivocator_IM IM)),
  composite_transition 
    (equivocator_IM IM) 
    (l lst) (s, input lst) =
  (destination lst, output lst)
  → composite_valid (equivocator_IM IM)
      (l lst) (s, input lst)
    → equivocators_transition_item_project IM
        final_descriptors lst =
      Some (oitem, idescriptors)
      → ∀ i : index,
          final_descriptors i = Existing 0
          → idescriptors i = Existing 0
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors
      (tr' ++ [lst]) = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
    unfold final_state in Hproper.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
IHlen: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ (final_state :=
             finite_trace_last is tr) 
            (final_descriptors : 
             equivocator_descriptors IM),
            proper_fixed_equivocator_descriptors
              final_descriptors final_state
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                let isX :=
                  equivocators_state_project
                    IM initial_descriptors is
                  in
                let final_stateX :=
                  finite_trace_last isX trX in
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      final_state =
                    final_stateX
                    ∧ finite_valid_trace X'
                      isX trX
Htr: finite_valid_trace_from XE is tr'
Hlst: finite_valid_trace_from XE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is tr)
Htr_lst: tr = tr' ++ [lst]
IH': ∀ (final_state := finite_trace_last is tr') 
  (final_descriptors : 
   equivocator_descriptors IM),
  proper_fixed_equivocator_descriptors
    final_descriptors final_state
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      let isX :=
        equivocators_state_project IM
          initial_descriptors is in
      let final_stateX :=
        finite_trace_last isX trX in
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
          final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ equivocators_state_project IM
            final_descriptors final_state =
          final_stateX
          ∧ finite_valid_trace X' isX trX
Hproperx: proper_equivocator_descriptors IM
  final_descriptors (destination lst)
→ ∃ (oitem : option
               (composite_transition_item
                  IM)) 
    (eqv_descriptors' : 
     equivocator_descriptors IM),
    equivocators_transition_item_project IM
      final_descriptors lst =
    Some (oitem, eqv_descriptors')
    ∧ match oitem with
      | Some itemx =>
          (∃ Hex : existing_equivocator_label
                     (IM (projT1 (l lst)))
                     (projT2 (l lst)),
             existT (projT1 (l lst))
               (existing_equivocator_label_extract
                  (IM (projT1 (l lst)))
                  (projT2 (l lst)) Hex) =
             l itemx)
          ∧ input lst = input itemx
            ∧ output lst = output itemx
              ∧ equivocators_state_project
                  IM final_descriptors
                  (destination lst) =
                destination itemx
                ∧ eqv_descriptors'
                    (projT1 (l lst)) =
                  equivocator_label_descriptor
                    (projT2 (l lst))
      | None => True
      end
      ∧ (∀ s : composite_state
                 (equivocator_IM IM),
           composite_valid
             (equivocator_IM IM) 
             (l lst) (s, input lst)
           → composite_transition
               (equivocator_IM IM) 
               (l lst) (
               s, input lst) =
             (destination lst, output lst)
             → proper_equivocator_descriptors
                 IM eqv_descriptors' s
               ∧ eqv_descriptors' =
                 equivocator_descriptors_update
                   IM final_descriptors
                   (projT1 (l lst))
                   (eqv_descriptors'
                      (projT1 (l lst)))
                 ∧ s =
                   state_update
                     (equivocator_IM IM)
                     (destination lst)
                     (projT1 (l lst))
                     (s (projT1 (l lst)))
                   ∧ previous_state_descriptor_prop
                      (IM (projT1 (l lst)))
                      (final_descriptors
                      (projT1 (l lst)))
                      (s (projT1 (l lst)))
                      (eqv_descriptors'
                      (projT1 (l lst)))
                     ∧ 
                     match oitem with
                     | Some itemx =>
                      ∀ sx : 
                      composite_state IM,
                      sx =
                      equivocators_state_project
                      IM eqv_descriptors' s
                      → 
                      composite_valid IM
                      (l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition
                      IM 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
                     | None =>
                      equivocators_state_project
                      IM final_descriptors
                      (destination lst) =
                      equivocators_state_project
                      IM eqv_descriptors' s
                     end)
Hzero: ∀ (oitem : option
             (composite_transition_item IM)) 
  (idescriptors : equivocator_descriptors IM) 
  (s : composite_state (equivocator_IM IM)),
  composite_transition 
    (equivocator_IM IM) 
    (l lst) (s, input lst) =
  (destination lst, output lst)
  → composite_valid (equivocator_IM IM)
      (l lst) (s, input lst)
    → equivocators_transition_item_project IM
        final_descriptors lst =
      Some (oitem, idescriptors)
      → ∀ i : index,
          final_descriptors i = Existing 0
          → idescriptors i = Existing 0
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors
      (tr' ++ [lst]) = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
    rewrite Htr_lst, finite_trace_last_is_last in Hproper.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
IHlen: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ (final_state :=
             finite_trace_last is tr) 
            (final_descriptors : 
             equivocator_descriptors IM),
            proper_fixed_equivocator_descriptors
              final_descriptors final_state
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                let isX :=
                  equivocators_state_project
                    IM initial_descriptors is
                  in
                let final_stateX :=
                  finite_trace_last isX trX in
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      final_state =
                    final_stateX
                    ∧ finite_valid_trace X'
                      isX trX
Htr: finite_valid_trace_from XE is tr'
Hlst: finite_valid_trace_from XE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors (destination lst)
Htr_lst: tr = tr' ++ [lst]
IH': ∀ (final_state := finite_trace_last is tr') 
  (final_descriptors : 
   equivocator_descriptors IM),
  proper_fixed_equivocator_descriptors
    final_descriptors final_state
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      let isX :=
        equivocators_state_project IM
          initial_descriptors is in
      let final_stateX :=
        finite_trace_last isX trX in
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
          final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ equivocators_state_project IM
            final_descriptors final_state =
          final_stateX
          ∧ finite_valid_trace X' isX trX
Hproperx: proper_equivocator_descriptors IM
  final_descriptors (destination lst)
→ ∃ (oitem : option
               (composite_transition_item
                  IM)) 
    (eqv_descriptors' : 
     equivocator_descriptors IM),
    equivocators_transition_item_project IM
      final_descriptors lst =
    Some (oitem, eqv_descriptors')
    ∧ match oitem with
      | Some itemx =>
          (∃ Hex : existing_equivocator_label
                     (IM (projT1 (l lst)))
                     (projT2 (l lst)),
             existT (projT1 (l lst))
               (existing_equivocator_label_extract
                  (IM (projT1 (l lst)))
                  (projT2 (l lst)) Hex) =
             l itemx)
          ∧ input lst = input itemx
            ∧ output lst = output itemx
              ∧ equivocators_state_project
                  IM final_descriptors
                  (destination lst) =
                destination itemx
                ∧ eqv_descriptors'
                    (projT1 (l lst)) =
                  equivocator_label_descriptor
                    (projT2 (l lst))
      | None => True
      end
      ∧ (∀ s : composite_state
                 (equivocator_IM IM),
           composite_valid
             (equivocator_IM IM) 
             (l lst) (s, input lst)
           → composite_transition
               (equivocator_IM IM) 
               (l lst) (
               s, input lst) =
             (destination lst, output lst)
             → proper_equivocator_descriptors
                 IM eqv_descriptors' s
               ∧ eqv_descriptors' =
                 equivocator_descriptors_update
                   IM final_descriptors
                   (projT1 (l lst))
                   (eqv_descriptors'
                      (projT1 (l lst)))
                 ∧ s =
                   state_update
                     (equivocator_IM IM)
                     (destination lst)
                     (projT1 (l lst))
                     (s (projT1 (l lst)))
                   ∧ previous_state_descriptor_prop
                      (IM (projT1 (l lst)))
                      (final_descriptors
                      (projT1 (l lst)))
                      (s (projT1 (l lst)))
                      (eqv_descriptors'
                      (projT1 (l lst)))
                     ∧ 
                     match oitem with
                     | Some itemx =>
                      ∀ sx : 
                      composite_state IM,
                      sx =
                      equivocators_state_project
                      IM eqv_descriptors' s
                      → 
                      composite_valid IM
                      (l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition
                      IM 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
                     | None =>
                      equivocators_state_project
                      IM final_descriptors
                      (destination lst) =
                      equivocators_state_project
                      IM eqv_descriptors' s
                     end)
Hzero: ∀ (oitem : option
             (composite_transition_item IM)) 
  (idescriptors : equivocator_descriptors IM) 
  (s : composite_state (equivocator_IM IM)),
  composite_transition 
    (equivocator_IM IM) 
    (l lst) (s, input lst) =
  (destination lst, output lst)
  → composite_valid (equivocator_IM IM)
      (l lst) (s, input lst)
    → equivocators_transition_item_project IM
        final_descriptors lst =
      Some (oitem, idescriptors)
      → ∀ i : index,
          final_descriptors i = Existing 0
          → idescriptors i = Existing 0
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors
      (tr' ++ [lst]) = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
    specialize (Hproperx (proj1 Hproper)).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
IHlen: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ (final_state :=
             finite_trace_last is tr) 
            (final_descriptors : 
             equivocator_descriptors IM),
            proper_fixed_equivocator_descriptors
              final_descriptors final_state
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                let isX :=
                  equivocators_state_project
                    IM initial_descriptors is
                  in
                let final_stateX :=
                  finite_trace_last isX trX in
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      final_state =
                    final_stateX
                    ∧ finite_valid_trace X'
                      isX trX
Htr: finite_valid_trace_from XE is tr'
Hlst: finite_valid_trace_from XE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors (destination lst)
Htr_lst: tr = tr' ++ [lst]
IH': ∀ (final_state := finite_trace_last is tr') 
  (final_descriptors : 
   equivocator_descriptors IM),
  proper_fixed_equivocator_descriptors
    final_descriptors final_state
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      let isX :=
        equivocators_state_project IM
          initial_descriptors is in
      let final_stateX :=
        finite_trace_last isX trX in
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
          final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ equivocators_state_project IM
            final_descriptors final_state =
          final_stateX
          ∧ finite_valid_trace X' isX trX
Hproperx: ∃ (oitem : option
             (composite_transition_item IM)) 
  (eqv_descriptors' : equivocator_descriptors
                      IM),
  equivocators_transition_item_project IM
    final_descriptors lst =
  Some (oitem, eqv_descriptors')
  ∧ match oitem with
    | Some itemx =>
        (∃ Hex : existing_equivocator_label
                   (IM (projT1 (l lst)))
                   (projT2 (l lst)),
           existT (projT1 (l lst))
             (existing_equivocator_label_extract
                (IM (projT1 (l lst)))
                (projT2 (l lst)) Hex) =
           l itemx)
        ∧ input lst = input itemx
          ∧ output lst = output itemx
            ∧ equivocators_state_project IM
                final_descriptors
                (destination lst) =
              destination itemx
              ∧ eqv_descriptors'
                  (projT1 (l lst)) =
                equivocator_label_descriptor
                  (projT2 (l lst))
    | None => True
    end
    ∧ (∀ s : composite_state
               (equivocator_IM IM),
         composite_valid
           (equivocator_IM IM) 
           (l lst) (s, input lst)
         → composite_transition
             (equivocator_IM IM) 
             (l lst) (s, input lst) =
           (destination lst, output lst)
           → proper_equivocator_descriptors
               IM eqv_descriptors' s
             ∧ eqv_descriptors' =
               equivocator_descriptors_update
                 IM final_descriptors
                 (projT1 (l lst))
                 (eqv_descriptors'
                    (projT1 (l lst)))
               ∧ s =
                 state_update
                   (equivocator_IM IM)
                   (destination lst)
                   (projT1 (l lst))
                   (s (projT1 (l lst)))
                 ∧ previous_state_descriptor_prop
                     (IM (projT1 (l lst)))
                     (final_descriptors
                      (projT1 (l lst)))
                     (s (projT1 (l lst)))
                     (eqv_descriptors'
                      (projT1 (l lst)))
                   ∧ match oitem with
                     | Some itemx =>
                      ∀ sx : 
                      composite_state IM,
                      sx =
                      equivocators_state_project
                      IM eqv_descriptors' s
                      → 
                      composite_valid IM
                      (l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition
                      IM 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
                     | None =>
                      equivocators_state_project
                      IM final_descriptors
                      (destination lst) =
                      equivocators_state_project
                      IM eqv_descriptors' s
                     end)
Hzero: ∀ (oitem : option
             (composite_transition_item IM)) 
  (idescriptors : equivocator_descriptors IM) 
  (s : composite_state (equivocator_IM IM)),
  composite_transition 
    (equivocator_IM IM) 
    (l lst) (s, input lst) =
  (destination lst, output lst)
  → composite_valid (equivocator_IM IM)
      (l lst) (s, input lst)
    → equivocators_transition_item_project IM
        final_descriptors lst =
      Some (oitem, idescriptors)
      → ∀ i : index,
          final_descriptors i = Existing 0
          → idescriptors i = Existing 0
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors
      (tr' ++ [lst]) = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
    destruct Hproperx as [oitem [final_descriptors' [Hprojectx [Hitemx Hproperx]]]].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
IHlen: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ (final_state :=
             finite_trace_last is tr) 
            (final_descriptors : 
             equivocator_descriptors IM),
            proper_fixed_equivocator_descriptors
              final_descriptors final_state
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                let isX :=
                  equivocators_state_project
                    IM initial_descriptors is
                  in
                let final_stateX :=
                  finite_trace_last isX trX in
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      final_state =
                    final_stateX
                    ∧ finite_valid_trace X'
                      isX trX
Htr: finite_valid_trace_from XE is tr'
Hlst: finite_valid_trace_from XE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors (destination lst)
Htr_lst: tr = tr' ++ [lst]
IH': ∀ (final_state := finite_trace_last is tr') 
  (final_descriptors : 
   equivocator_descriptors IM),
  proper_fixed_equivocator_descriptors
    final_descriptors final_state
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      let isX :=
        equivocators_state_project IM
          initial_descriptors is in
      let final_stateX :=
        finite_trace_last isX trX in
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
          final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ equivocators_state_project IM
            final_descriptors final_state =
          final_stateX
          ∧ finite_valid_trace X' isX trX
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors lst =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l lst)))
               (projT2 (l lst)),
       existT (projT1 (l lst))
         (existing_equivocator_label_extract
            (IM (projT1 (l lst)))
            (projT2 (l lst)) Hex) = 
       l itemx)
    ∧ input lst = input itemx
      ∧ output lst = output itemx
        ∧ equivocators_state_project IM
            final_descriptors
            (destination lst) =
          destination itemx
          ∧ final_descriptors'
              (projT1 (l lst)) =
            equivocator_label_descriptor
              (projT2 (l lst))
| None => True
end
Hproperx: ∀ s : composite_state (equivocator_IM IM),
  composite_valid (equivocator_IM IM)
    (l lst) (s, input lst)
  → composite_transition
      (equivocator_IM IM) 
      (l lst) (s, input lst) =
    (destination lst, output lst)
    → proper_equivocator_descriptors IM
        final_descriptors' s
      ∧ final_descriptors' =
        equivocator_descriptors_update IM
          final_descriptors
          (projT1 (l lst))
          (final_descriptors'
             (projT1 (l lst)))
        ∧ s =
          state_update 
            (equivocator_IM IM)
            (destination lst)
            (projT1 (l lst))
            (s (projT1 (l lst)))
          ∧ previous_state_descriptor_prop
              (IM (projT1 (l lst)))
              (final_descriptors
                 (projT1 (l lst)))
              (s (projT1 (l lst)))
              (final_descriptors'
                 (projT1 (l lst)))
            ∧ match oitem with
              | Some itemx =>
                  ∀ sx : composite_state IM,
                    sx =
                    equivocators_state_project
                      IM final_descriptors'
                      s
                    → composite_valid IM
                      (l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition
                      IM 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
              | None =>
                  equivocators_state_project
                    IM final_descriptors
                    (destination lst) =
                  equivocators_state_project
                    IM final_descriptors' s
              end
Hzero: ∀ (oitem : option
             (composite_transition_item IM)) 
  (idescriptors : equivocator_descriptors IM) 
  (s : composite_state (equivocator_IM IM)),
  composite_transition 
    (equivocator_IM IM) 
    (l lst) (s, input lst) =
  (destination lst, output lst)
  → composite_valid (equivocator_IM IM)
      (l lst) (s, input lst)
    → equivocators_transition_item_project IM
        final_descriptors lst =
      Some (oitem, idescriptors)
      → ∀ i : index,
          final_descriptors i = Existing 0
          → idescriptors i = Existing 0
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors
      (tr' ++ [lst]) = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
    specialize (Hproperx (finite_trace_last is tr')).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
IHlen: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ (final_state :=
             finite_trace_last is tr) 
            (final_descriptors : 
             equivocator_descriptors IM),
            proper_fixed_equivocator_descriptors
              final_descriptors final_state
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                let isX :=
                  equivocators_state_project
                    IM initial_descriptors is
                  in
                let final_stateX :=
                  finite_trace_last isX trX in
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      final_state =
                    final_stateX
                    ∧ finite_valid_trace X'
                      isX trX
Htr: finite_valid_trace_from XE is tr'
Hlst: finite_valid_trace_from XE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors (destination lst)
Htr_lst: tr = tr' ++ [lst]
IH': ∀ (final_state := finite_trace_last is tr') 
  (final_descriptors : 
   equivocator_descriptors IM),
  proper_fixed_equivocator_descriptors
    final_descriptors final_state
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      let isX :=
        equivocators_state_project IM
          initial_descriptors is in
      let final_stateX :=
        finite_trace_last isX trX in
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
          final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ equivocators_state_project IM
            final_descriptors final_state =
          final_stateX
          ∧ finite_valid_trace X' isX trX
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors lst =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l lst)))
               (projT2 (l lst)),
       existT (projT1 (l lst))
         (existing_equivocator_label_extract
            (IM (projT1 (l lst)))
            (projT2 (l lst)) Hex) = 
       l itemx)
    ∧ input lst = input itemx
      ∧ output lst = output itemx
        ∧ equivocators_state_project IM
            final_descriptors
            (destination lst) =
          destination itemx
          ∧ final_descriptors'
              (projT1 (l lst)) =
            equivocator_label_descriptor
              (projT2 (l lst))
| None => True
end
Hproperx: composite_valid (equivocator_IM IM) 
  (l lst)
  (finite_trace_last is tr', input lst)
→ composite_transition 
    (equivocator_IM IM) 
    (l lst)
    (finite_trace_last is tr', input lst) =
  (destination lst, output lst)
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr')
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 (l lst))
        (final_descriptors'
           (projT1 (l lst)))
      ∧ finite_trace_last is tr' =
        state_update (equivocator_IM IM)
          (destination lst)
          (projT1 (l lst))
          (finite_trace_last is tr'
             (projT1 (l lst)))
        ∧ previous_state_descriptor_prop
            (IM (projT1 (l lst)))
            (final_descriptors
               (projT1 (l lst)))
            (finite_trace_last is tr'
               (projT1 (l lst)))
            (final_descriptors'
               (projT1 (l lst)))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM,
                  sx =
                  equivocators_state_project
                    IM final_descriptors'
                    (finite_trace_last is
                      tr')
                  → composite_valid IM
                      (l itemx)
                      (
                      sx, input itemx)
                    ∧ composite_transition
                      IM 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM final_descriptors
                  (destination lst) =
                equivocators_state_project
                  IM final_descriptors'
                  (finite_trace_last is tr')
            end
Hzero: ∀ (oitem : option
             (composite_transition_item IM)) 
  (idescriptors : equivocator_descriptors IM) 
  (s : composite_state (equivocator_IM IM)),
  composite_transition 
    (equivocator_IM IM) 
    (l lst) (s, input lst) =
  (destination lst, output lst)
  → composite_valid (equivocator_IM IM)
      (l lst) (s, input lst)
    → equivocators_transition_item_project IM
        final_descriptors lst =
      Some (oitem, idescriptors)
      → ∀ i : index,
          final_descriptors i = Existing 0
          → idescriptors i = Existing 0
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors
      (tr' ++ [lst]) = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
    unfold equivocators_trace_project.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
IHlen: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ (final_state :=
             finite_trace_last is tr) 
            (final_descriptors : 
             equivocator_descriptors IM),
            proper_fixed_equivocator_descriptors
              final_descriptors final_state
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                let isX :=
                  equivocators_state_project
                    IM initial_descriptors is
                  in
                let final_stateX :=
                  finite_trace_last isX trX in
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      final_state =
                    final_stateX
                    ∧ finite_valid_trace X'
                      isX trX
Htr: finite_valid_trace_from XE is tr'
Hlst: finite_valid_trace_from XE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors (destination lst)
Htr_lst: tr = tr' ++ [lst]
IH': ∀ (final_state := finite_trace_last is tr') 
  (final_descriptors : 
   equivocator_descriptors IM),
  proper_fixed_equivocator_descriptors
    final_descriptors final_state
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      let isX :=
        equivocators_state_project IM
          initial_descriptors is in
      let final_stateX :=
        finite_trace_last isX trX in
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
          final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ equivocators_state_project IM
            final_descriptors final_state =
          final_stateX
          ∧ finite_valid_trace X' isX trX
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors lst =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l lst)))
               (projT2 (l lst)),
       existT (projT1 (l lst))
         (existing_equivocator_label_extract
            (IM (projT1 (l lst)))
            (projT2 (l lst)) Hex) = 
       l itemx)
    ∧ input lst = input itemx
      ∧ output lst = output itemx
        ∧ equivocators_state_project IM
            final_descriptors
            (destination lst) =
          destination itemx
          ∧ final_descriptors'
              (projT1 (l lst)) =
            equivocator_label_descriptor
              (projT2 (l lst))
| None => True
end
Hproperx: composite_valid (equivocator_IM IM) 
  (l lst)
  (finite_trace_last is tr', input lst)
→ composite_transition 
    (equivocator_IM IM) 
    (l lst)
    (finite_trace_last is tr', input lst) =
  (destination lst, output lst)
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr')
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 (l lst))
        (final_descriptors'
           (projT1 (l lst)))
      ∧ finite_trace_last is tr' =
        state_update (equivocator_IM IM)
          (destination lst)
          (projT1 (l lst))
          (finite_trace_last is tr'
             (projT1 (l lst)))
        ∧ previous_state_descriptor_prop
            (IM (projT1 (l lst)))
            (final_descriptors
               (projT1 (l lst)))
            (finite_trace_last is tr'
               (projT1 (l lst)))
            (final_descriptors'
               (projT1 (l lst)))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM,
                  sx =
                  equivocators_state_project
                    IM final_descriptors'
                    (finite_trace_last is
                      tr')
                  → composite_valid IM
                      (l itemx)
                      (
                      sx, input itemx)
                    ∧ composite_transition
                      IM 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM final_descriptors
                  (destination lst) =
                equivocators_state_project
                  IM final_descriptors'
                  (finite_trace_last is tr')
            end
Hzero: ∀ (oitem : option
             (composite_transition_item IM)) 
  (idescriptors : equivocator_descriptors IM) 
  (s : composite_state (equivocator_IM IM)),
  composite_transition 
    (equivocator_IM IM) 
    (l lst) (s, input lst) =
  (destination lst, output lst)
  → composite_valid (equivocator_IM IM)
      (l lst) (s, input lst)
    → equivocators_transition_item_project IM
        final_descriptors lst =
      Some (oitem, idescriptors)
      → ∀ i : index,
          final_descriptors i = Existing 0
          → idescriptors i = Existing 0
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM)
      (Some ([], final_descriptors)) (tr' ++ [lst]) =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    rewrite fold_right_app.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
IHlen: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ (final_state :=
             finite_trace_last is tr) 
            (final_descriptors : 
             equivocator_descriptors IM),
            proper_fixed_equivocator_descriptors
              final_descriptors final_state
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                let isX :=
                  equivocators_state_project
                    IM initial_descriptors is
                  in
                let final_stateX :=
                  finite_trace_last isX trX in
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      final_state =
                    final_stateX
                    ∧ finite_valid_trace X'
                      isX trX
Htr: finite_valid_trace_from XE is tr'
Hlst: finite_valid_trace_from XE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors (destination lst)
Htr_lst: tr = tr' ++ [lst]
IH': ∀ (final_state := finite_trace_last is tr') 
  (final_descriptors : 
   equivocator_descriptors IM),
  proper_fixed_equivocator_descriptors
    final_descriptors final_state
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      let isX :=
        equivocators_state_project IM
          initial_descriptors is in
      let final_stateX :=
        finite_trace_last isX trX in
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
          final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ equivocators_state_project IM
            final_descriptors final_state =
          final_stateX
          ∧ finite_valid_trace X' isX trX
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors lst =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l lst)))
               (projT2 (l lst)),
       existT (projT1 (l lst))
         (existing_equivocator_label_extract
            (IM (projT1 (l lst)))
            (projT2 (l lst)) Hex) = 
       l itemx)
    ∧ input lst = input itemx
      ∧ output lst = output itemx
        ∧ equivocators_state_project IM
            final_descriptors
            (destination lst) =
          destination itemx
          ∧ final_descriptors'
              (projT1 (l lst)) =
            equivocator_label_descriptor
              (projT2 (l lst))
| None => True
end
Hproperx: composite_valid (equivocator_IM IM) 
  (l lst)
  (finite_trace_last is tr', input lst)
→ composite_transition 
    (equivocator_IM IM) 
    (l lst)
    (finite_trace_last is tr', input lst) =
  (destination lst, output lst)
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr')
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 (l lst))
        (final_descriptors'
           (projT1 (l lst)))
      ∧ finite_trace_last is tr' =
        state_update (equivocator_IM IM)
          (destination lst)
          (projT1 (l lst))
          (finite_trace_last is tr'
             (projT1 (l lst)))
        ∧ previous_state_descriptor_prop
            (IM (projT1 (l lst)))
            (final_descriptors
               (projT1 (l lst)))
            (finite_trace_last is tr'
               (projT1 (l lst)))
            (final_descriptors'
               (projT1 (l lst)))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM,
                  sx =
                  equivocators_state_project
                    IM final_descriptors'
                    (finite_trace_last is
                      tr')
                  → composite_valid IM
                      (l itemx)
                      (
                      sx, input itemx)
                    ∧ composite_transition
                      IM 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM final_descriptors
                  (destination lst) =
                equivocators_state_project
                  IM final_descriptors'
                  (finite_trace_last is tr')
            end
Hzero: ∀ (oitem : option
             (composite_transition_item IM)) 
  (idescriptors : equivocator_descriptors IM) 
  (s : composite_state (equivocator_IM IM)),
  composite_transition 
    (equivocator_IM IM) 
    (l lst) (s, input lst) =
  (destination lst, output lst)
  → composite_valid (equivocator_IM IM)
      (l lst) (s, input lst)
    → equivocators_transition_item_project IM
        final_descriptors lst =
      Some (oitem, idescriptors)
      → ∀ i : index,
          final_descriptors i = Existing 0
          → idescriptors i = Existing 0
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM)
      (foldr (equivocators_trace_project_folder IM)
         (Some ([], final_descriptors)) [lst]) tr' =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    match goal with
    |- context [fold_right _ ?fld _] => remember fld as foldx
    end.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
IHlen: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ (final_state :=
             finite_trace_last is tr) 
            (final_descriptors : 
             equivocator_descriptors IM),
            proper_fixed_equivocator_descriptors
              final_descriptors final_state
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                let isX :=
                  equivocators_state_project
                    IM initial_descriptors is
                  in
                let final_stateX :=
                  finite_trace_last isX trX in
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      final_state =
                    final_stateX
                    ∧ finite_valid_trace X'
                      isX trX
Htr: finite_valid_trace_from XE is tr'
Hlst: finite_valid_trace_from XE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors (destination lst)
Htr_lst: tr = tr' ++ [lst]
IH': ∀ (final_state := finite_trace_last is tr') 
  (final_descriptors : 
   equivocator_descriptors IM),
  proper_fixed_equivocator_descriptors
    final_descriptors final_state
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      let isX :=
        equivocators_state_project IM
          initial_descriptors is in
      let final_stateX :=
        finite_trace_last isX trX in
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
          final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ equivocators_state_project IM
            final_descriptors final_state =
          final_stateX
          ∧ finite_valid_trace X' isX trX
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors lst =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l lst)))
               (projT2 (l lst)),
       existT (projT1 (l lst))
         (existing_equivocator_label_extract
            (IM (projT1 (l lst)))
            (projT2 (l lst)) Hex) = 
       l itemx)
    ∧ input lst = input itemx
      ∧ output lst = output itemx
        ∧ equivocators_state_project IM
            final_descriptors
            (destination lst) =
          destination itemx
          ∧ final_descriptors'
              (projT1 (l lst)) =
            equivocator_label_descriptor
              (projT2 (l lst))
| None => True
end
Hproperx: composite_valid (equivocator_IM IM) 
  (l lst)
  (finite_trace_last is tr', input lst)
→ composite_transition 
    (equivocator_IM IM) 
    (l lst)
    (finite_trace_last is tr', input lst) =
  (destination lst, output lst)
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr')
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 (l lst))
        (final_descriptors'
           (projT1 (l lst)))
      ∧ finite_trace_last is tr' =
        state_update (equivocator_IM IM)
          (destination lst)
          (projT1 (l lst))
          (finite_trace_last is tr'
             (projT1 (l lst)))
        ∧ previous_state_descriptor_prop
            (IM (projT1 (l lst)))
            (final_descriptors
               (projT1 (l lst)))
            (finite_trace_last is tr'
               (projT1 (l lst)))
            (final_descriptors'
               (projT1 (l lst)))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM,
                  sx =
                  equivocators_state_project
                    IM final_descriptors'
                    (finite_trace_last is
                      tr')
                  → composite_valid IM
                      (l itemx)
                      (
                      sx, input itemx)
                    ∧ composite_transition
                      IM 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM final_descriptors
                  (destination lst) =
                equivocators_state_project
                  IM final_descriptors'
                  (finite_trace_last is tr')
            end
Hzero: ∀ (oitem : option
             (composite_transition_item IM)) 
  (idescriptors : equivocator_descriptors IM) 
  (s : composite_state (equivocator_IM IM)),
  composite_transition 
    (equivocator_IM IM) 
    (l lst) (s, input lst) =
  (destination lst, output lst)
  → composite_valid (equivocator_IM IM)
      (l lst) (s, input lst)
    → equivocators_transition_item_project IM
        final_descriptors lst =
      Some (oitem, idescriptors)
      → ∀ i : index,
          final_descriptors i = Existing 0
          → idescriptors i = Existing 0
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
foldr
  (equivocators_trace_project_folder IM)
  (Some ([], final_descriptors)) [lst]
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    simpl in Heqfoldx.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
IHlen: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ (final_state :=
             finite_trace_last is tr) 
            (final_descriptors : 
             equivocator_descriptors IM),
            proper_fixed_equivocator_descriptors
              final_descriptors final_state
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                let isX :=
                  equivocators_state_project
                    IM initial_descriptors is
                  in
                let final_stateX :=
                  finite_trace_last isX trX in
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      final_state =
                    final_stateX
                    ∧ finite_valid_trace X'
                      isX trX
Htr: finite_valid_trace_from XE is tr'
Hlst: finite_valid_trace_from XE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors (destination lst)
Htr_lst: tr = tr' ++ [lst]
IH': ∀ (final_state := finite_trace_last is tr') 
  (final_descriptors : 
   equivocator_descriptors IM),
  proper_fixed_equivocator_descriptors
    final_descriptors final_state
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      let isX :=
        equivocators_state_project IM
          initial_descriptors is in
      let final_stateX :=
        finite_trace_last isX trX in
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
          final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ equivocators_state_project IM
            final_descriptors final_state =
          final_stateX
          ∧ finite_valid_trace X' isX trX
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors lst =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l lst)))
               (projT2 (l lst)),
       existT (projT1 (l lst))
         (existing_equivocator_label_extract
            (IM (projT1 (l lst)))
            (projT2 (l lst)) Hex) = 
       l itemx)
    ∧ input lst = input itemx
      ∧ output lst = output itemx
        ∧ equivocators_state_project IM
            final_descriptors
            (destination lst) =
          destination itemx
          ∧ final_descriptors'
              (projT1 (l lst)) =
            equivocator_label_descriptor
              (projT2 (l lst))
| None => True
end
Hproperx: composite_valid (equivocator_IM IM) 
  (l lst)
  (finite_trace_last is tr', input lst)
→ composite_transition 
    (equivocator_IM IM) 
    (l lst)
    (finite_trace_last is tr', input lst) =
  (destination lst, output lst)
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr')
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 (l lst))
        (final_descriptors'
           (projT1 (l lst)))
      ∧ finite_trace_last is tr' =
        state_update (equivocator_IM IM)
          (destination lst)
          (projT1 (l lst))
          (finite_trace_last is tr'
             (projT1 (l lst)))
        ∧ previous_state_descriptor_prop
            (IM (projT1 (l lst)))
            (final_descriptors
               (projT1 (l lst)))
            (finite_trace_last is tr'
               (projT1 (l lst)))
            (final_descriptors'
               (projT1 (l lst)))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM,
                  sx =
                  equivocators_state_project
                    IM final_descriptors'
                    (finite_trace_last is
                      tr')
                  → composite_valid IM
                      (l itemx)
                      (
                      sx, input itemx)
                    ∧ composite_transition
                      IM 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM final_descriptors
                  (destination lst) =
                equivocators_state_project
                  IM final_descriptors'
                  (finite_trace_last is tr')
            end
Hzero: ∀ (oitem : option
             (composite_transition_item IM)) 
  (idescriptors : equivocator_descriptors IM) 
  (s : composite_state (equivocator_IM IM)),
  composite_transition 
    (equivocator_IM IM) 
    (l lst) (s, input lst) =
  (destination lst, output lst)
  → composite_valid (equivocator_IM IM)
      (l lst) (s, input lst)
    → equivocators_transition_item_project IM
        final_descriptors lst =
      Some (oitem, idescriptors)
      → ∀ i : index,
          final_descriptors i = Existing 0
          → idescriptors i = Existing 0
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match
  equivocators_transition_item_project IM
    final_descriptors lst
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) =>
    Some ([], odescriptor)
| None => None
end
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    rewrite Hprojectx in Heqfoldx.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
IHlen: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ (final_state :=
             finite_trace_last is tr) 
            (final_descriptors : 
             equivocator_descriptors IM),
            proper_fixed_equivocator_descriptors
              final_descriptors final_state
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                let isX :=
                  equivocators_state_project
                    IM initial_descriptors is
                  in
                let final_stateX :=
                  finite_trace_last isX trX in
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      final_state =
                    final_stateX
                    ∧ finite_valid_trace X'
                      isX trX
Htr: finite_valid_trace_from XE is tr'
Hlst: finite_valid_trace_from XE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors (destination lst)
Htr_lst: tr = tr' ++ [lst]
IH': ∀ (final_state := finite_trace_last is tr') 
  (final_descriptors : 
   equivocator_descriptors IM),
  proper_fixed_equivocator_descriptors
    final_descriptors final_state
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      let isX :=
        equivocators_state_project IM
          initial_descriptors is in
      let final_stateX :=
        finite_trace_last isX trX in
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
          final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ equivocators_state_project IM
            final_descriptors final_state =
          final_stateX
          ∧ finite_valid_trace X' isX trX
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors lst =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l lst)))
               (projT2 (l lst)),
       existT (projT1 (l lst))
         (existing_equivocator_label_extract
            (IM (projT1 (l lst)))
            (projT2 (l lst)) Hex) = 
       l itemx)
    ∧ input lst = input itemx
      ∧ output lst = output itemx
        ∧ equivocators_state_project IM
            final_descriptors
            (destination lst) =
          destination itemx
          ∧ final_descriptors'
              (projT1 (l lst)) =
            equivocator_label_descriptor
              (projT2 (l lst))
| None => True
end
Hproperx: composite_valid (equivocator_IM IM) 
  (l lst)
  (finite_trace_last is tr', input lst)
→ composite_transition 
    (equivocator_IM IM) 
    (l lst)
    (finite_trace_last is tr', input lst) =
  (destination lst, output lst)
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr')
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 (l lst))
        (final_descriptors'
           (projT1 (l lst)))
      ∧ finite_trace_last is tr' =
        state_update (equivocator_IM IM)
          (destination lst)
          (projT1 (l lst))
          (finite_trace_last is tr'
             (projT1 (l lst)))
        ∧ previous_state_descriptor_prop
            (IM (projT1 (l lst)))
            (final_descriptors
               (projT1 (l lst)))
            (finite_trace_last is tr'
               (projT1 (l lst)))
            (final_descriptors'
               (projT1 (l lst)))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM,
                  sx =
                  equivocators_state_project
                    IM final_descriptors'
                    (finite_trace_last is
                      tr')
                  → composite_valid IM
                      (l itemx)
                      (
                      sx, input itemx)
                    ∧ composite_transition
                      IM 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM final_descriptors
                  (destination lst) =
                equivocators_state_project
                  IM final_descriptors'
                  (finite_trace_last is tr')
            end
Hzero: ∀ (oitem : option
             (composite_transition_item IM)) 
  (idescriptors : equivocator_descriptors IM) 
  (s : composite_state (equivocator_IM IM)),
  composite_transition 
    (equivocator_IM IM) 
    (l lst) (s, input lst) =
  (destination lst, output lst)
  → composite_valid (equivocator_IM IM)
      (l lst) (s, input lst)
    → equivocators_transition_item_project IM
        final_descriptors lst =
      Some (oitem, idescriptors)
      → ∀ i : index,
          final_descriptors i = Existing 0
          → idescriptors i = Existing 0
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    apply first_transition_valid in Hlst.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
IHlen: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ (final_state :=
             finite_trace_last is tr) 
            (final_descriptors : 
             equivocator_descriptors IM),
            proper_fixed_equivocator_descriptors
              final_descriptors final_state
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                let isX :=
                  equivocators_state_project
                    IM initial_descriptors is
                  in
                let final_stateX :=
                  finite_trace_last isX trX in
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      final_state =
                    final_stateX
                    ∧ finite_valid_trace X'
                      isX trX
Htr: finite_valid_trace_from XE is tr'
Hlst: input_valid_transition XE 
  (l lst) (finite_trace_last is tr', input lst)
  (destination lst, output lst)
Hinit: initial_state_prop is
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors (destination lst)
Htr_lst: tr = tr' ++ [lst]
IH': ∀ (final_state := finite_trace_last is tr') 
  (final_descriptors : 
   equivocator_descriptors IM),
  proper_fixed_equivocator_descriptors
    final_descriptors final_state
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      let isX :=
        equivocators_state_project IM
          initial_descriptors is in
      let final_stateX :=
        finite_trace_last isX trX in
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
          final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ equivocators_state_project IM
            final_descriptors final_state =
          final_stateX
          ∧ finite_valid_trace X' isX trX
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors lst =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l lst)))
               (projT2 (l lst)),
       existT (projT1 (l lst))
         (existing_equivocator_label_extract
            (IM (projT1 (l lst)))
            (projT2 (l lst)) Hex) = 
       l itemx)
    ∧ input lst = input itemx
      ∧ output lst = output itemx
        ∧ equivocators_state_project IM
            final_descriptors
            (destination lst) =
          destination itemx
          ∧ final_descriptors'
              (projT1 (l lst)) =
            equivocator_label_descriptor
              (projT2 (l lst))
| None => True
end
Hproperx: composite_valid (equivocator_IM IM) 
  (l lst)
  (finite_trace_last is tr', input lst)
→ composite_transition 
    (equivocator_IM IM) 
    (l lst)
    (finite_trace_last is tr', input lst) =
  (destination lst, output lst)
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr')
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 (l lst))
        (final_descriptors'
           (projT1 (l lst)))
      ∧ finite_trace_last is tr' =
        state_update (equivocator_IM IM)
          (destination lst)
          (projT1 (l lst))
          (finite_trace_last is tr'
             (projT1 (l lst)))
        ∧ previous_state_descriptor_prop
            (IM (projT1 (l lst)))
            (final_descriptors
               (projT1 (l lst)))
            (finite_trace_last is tr'
               (projT1 (l lst)))
            (final_descriptors'
               (projT1 (l lst)))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM,
                  sx =
                  equivocators_state_project
                    IM final_descriptors'
                    (finite_trace_last is
                      tr')
                  → composite_valid IM
                      (l itemx)
                      (
                      sx, input itemx)
                    ∧ composite_transition
                      IM 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM final_descriptors
                  (destination lst) =
                equivocators_state_project
                  IM final_descriptors'
                  (finite_trace_last is tr')
            end
Hzero: ∀ (oitem : option
             (composite_transition_item IM)) 
  (idescriptors : equivocator_descriptors IM) 
  (s : composite_state (equivocator_IM IM)),
  composite_transition 
    (equivocator_IM IM) 
    (l lst) (s, input lst) =
  (destination lst, output lst)
  → composite_valid (equivocator_IM IM)
      (l lst) (s, input lst)
    → equivocators_transition_item_project IM
        final_descriptors lst =
      Some (oitem, idescriptors)
      → ∀ i : index,
          final_descriptors i = Existing 0
          → idescriptors i = Existing 0
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX destruct lst as (l, iom, s, oom).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
iom: option message
s: state (composite_type (equivocator_IM IM))
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ (final_state :=
             finite_trace_last is tr) 
            (final_descriptors : 
             equivocator_descriptors IM),
            proper_fixed_equivocator_descriptors
              final_descriptors final_state
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                let isX :=
                  equivocators_state_project
                    IM initial_descriptors is
                  in
                let final_stateX :=
                  finite_trace_last isX trX in
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      final_state =
                    final_stateX
                    ∧ finite_valid_trace X'
                      isX trX
Htr: finite_valid_trace_from XE is tr'
Hlst: input_valid_transition XE
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  (finite_trace_last is tr',
   input
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |},
   output
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
Hinit: initial_state_prop is
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
IH': ∀ (final_state := finite_trace_last is tr') 
  (final_descriptors : 
   equivocator_descriptors IM),
  proper_fixed_equivocator_descriptors
    final_descriptors final_state
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      let isX :=
        equivocators_state_project IM
          initial_descriptors is in
      let final_stateX :=
        finite_trace_last isX trX in
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
          final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ equivocators_state_project IM
            final_descriptors final_state =
          final_stateX
          ∧ finite_valid_trace X' isX trX
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM
            final_descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hproperx: composite_valid (equivocator_IM IM)
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  (finite_trace_last is tr',
   input
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
→ composite_transition 
    (equivocator_IM IM)
    (VLSM.l
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
    (finite_trace_last is tr',
     input
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |}) =
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |},
   output
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr')
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        final_descriptors
        (projT1
           (VLSM.l
              {|
                l := l;
                input := iom;
                destination := s;
                output := oom
              |}))
        (final_descriptors'
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})))
      ∧ finite_trace_last is tr' =
        state_update (equivocator_IM IM)
          (destination
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |})
          (projT1
             (VLSM.l
                {|
                  l := l;
                  input := iom;
                  destination := s;
                  output := oom
                |}))
          (finite_trace_last is tr'
             (projT1
                (VLSM.l
                   {|
                     l := l;
                     input := iom;
                     destination := s;
                     output := oom
                   |})))
        ∧ previous_state_descriptor_prop
            (IM
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (final_descriptors
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (finite_trace_last is tr'
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (final_descriptors'
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM,
                  sx =
                  equivocators_state_project
                    IM final_descriptors'
                    (finite_trace_last is
                      tr')
                  → composite_valid IM
                      (VLSM.l itemx)
                      (
                      sx, input itemx)
                    ∧ composite_transition
                      IM 
                      (VLSM.l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM final_descriptors
                  (destination
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |}) =
                equivocators_state_project
                  IM final_descriptors'
                  (finite_trace_last is tr')
            end
Hzero: ∀ (oitem : option
             (composite_transition_item IM)) 
  (idescriptors : equivocator_descriptors IM) 
  (s0 : composite_state (equivocator_IM IM)),
  composite_transition 
    (equivocator_IM IM)
    (VLSM.l
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
    (s0,
     input
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |}) =
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |},
   output
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  → composite_valid (equivocator_IM IM)
      (VLSM.l
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |})
      (s0,
       input
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |})
    → equivocators_transition_item_project IM
        final_descriptors
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = Some (oitem, idescriptors)
      → ∀ i : index,
          final_descriptors i = Existing 0
          → idescriptors i = Existing 0
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX simpl in *.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hlst: input_valid_transition XE l
  (finite_trace_last is tr', iom) (
  s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
IH': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
          final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ equivocators_state_project IM
            final_descriptors
            (finite_trace_last is tr') =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
          ∧ finite_valid_trace X'
              (equivocators_state_project IM
                 initial_descriptors is) trX
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
Hproperx: (let (i, li) := l in
 equivocator_valid (IM i) li
   (finite_trace_last is tr' i, iom))
→ (let (i, li) := l in
   let (si', om') :=
     equivocator_transition 
       (IM i) li
       (finite_trace_last is tr' i, iom) in
   (state_update (equivocator_IM IM)
      (finite_trace_last is tr') i si', om')) =
  (s, oom)
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr')
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 l)
        (final_descriptors' (projT1 l))
      ∧ finite_trace_last is tr' =
        state_update (equivocator_IM IM) s
          (projT1 l)
          (finite_trace_last is tr'
             (projT1 l))
        ∧ previous_state_descriptor_prop
            (IM (projT1 l))
            (final_descriptors (projT1 l))
            (finite_trace_last is tr'
               (projT1 l))
            (final_descriptors' (projT1 l))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM,
                  sx =
                  equivocators_state_project
                    IM final_descriptors'
                    (finite_trace_last is
                      tr')
                  → (let 
                      (i, li) :=
                      VLSM.l itemx in
                     valid li
                      (sx i, input itemx))
                    ∧ (let 
                      (i, li) :=
                      VLSM.l itemx in
                      let 
                      (si', om') :=
                      transition li
                      (sx i, input itemx) in
                      (
                      state_update IM sx i
                      si', om')) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM final_descriptors s =
                equivocators_state_project
                  IM final_descriptors'
                  (finite_trace_last is tr')
            end
Hzero: ∀ (oitem : option
             (composite_transition_item IM)) 
  (idescriptors : equivocator_descriptors IM) 
  (s0 : composite_state (equivocator_IM IM)),
  (let (i, li) := l in
   let (si', om') :=
     equivocator_transition 
       (IM i) li (s0 i, iom) in
   (state_update (equivocator_IM IM) s0 i si',
    om')) = (s, oom)
  → (let (i, li) := l in
     equivocator_valid (IM i) li (s0 i, iom))
    → equivocators_transition_item_project IM
        final_descriptors
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = Some (oitem, idescriptors)
      → ∀ i : index,
          final_descriptors i = Existing 0
          → idescriptors i = Existing 0
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    destruct Hlst as [[Hs [Hiom [Hv Hc]]] Ht].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
IH': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
          final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ equivocators_state_project IM
            final_descriptors
            (finite_trace_last is tr') =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
          ∧ finite_valid_trace X'
              (equivocators_state_project IM
                 initial_descriptors is) trX
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
Hproperx: (let (i, li) := l in
 equivocator_valid (IM i) li
   (finite_trace_last is tr' i, iom))
→ (let (i, li) := l in
   let (si', om') :=
     equivocator_transition 
       (IM i) li
       (finite_trace_last is tr' i, iom) in
   (state_update (equivocator_IM IM)
      (finite_trace_last is tr') i si', om')) =
  (s, oom)
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr')
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 l)
        (final_descriptors' (projT1 l))
      ∧ finite_trace_last is tr' =
        state_update (equivocator_IM IM) s
          (projT1 l)
          (finite_trace_last is tr'
             (projT1 l))
        ∧ previous_state_descriptor_prop
            (IM (projT1 l))
            (final_descriptors (projT1 l))
            (finite_trace_last is tr'
               (projT1 l))
            (final_descriptors' (projT1 l))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM,
                  sx =
                  equivocators_state_project
                    IM final_descriptors'
                    (finite_trace_last is
                      tr')
                  → (let 
                      (i, li) :=
                      VLSM.l itemx in
                     valid li
                      (sx i, input itemx))
                    ∧ (let 
                      (i, li) :=
                      VLSM.l itemx in
                      let 
                      (si', om') :=
                      transition li
                      (sx i, input itemx) in
                      (
                      state_update IM sx i
                      si', om')) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM final_descriptors s =
                equivocators_state_project
                  IM final_descriptors'
                  (finite_trace_last is tr')
            end
Hzero: ∀ (oitem : option
             (composite_transition_item IM)) 
  (idescriptors : equivocator_descriptors IM) 
  (s0 : composite_state (equivocator_IM IM)),
  (let (i, li) := l in
   let (si', om') :=
     equivocator_transition 
       (IM i) li (s0 i, iom) in
   (state_update (equivocator_IM IM) s0 i si',
    om')) = (s, oom)
  → (let (i, li) := l in
     equivocator_valid (IM i) li (s0 i, iom))
    → equivocators_transition_item_project IM
        final_descriptors
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = Some (oitem, idescriptors)
      → ∀ i : index,
          final_descriptors i = Existing 0
          → idescriptors i = Existing 0
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    specialize (Hzero oitem final_descriptors' _ Ht Hv Hprojectx).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
IH': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
          final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ equivocators_state_project IM
            final_descriptors
            (finite_trace_last is tr') =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
          ∧ finite_valid_trace X'
              (equivocators_state_project IM
                 initial_descriptors is) trX
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
Hproperx: (let (i, li) := l in
 equivocator_valid (IM i) li
   (finite_trace_last is tr' i, iom))
→ (let (i, li) := l in
   let (si', om') :=
     equivocator_transition 
       (IM i) li
       (finite_trace_last is tr' i, iom) in
   (state_update (equivocator_IM IM)
      (finite_trace_last is tr') i si', om')) =
  (s, oom)
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr')
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 l)
        (final_descriptors' (projT1 l))
      ∧ finite_trace_last is tr' =
        state_update (equivocator_IM IM) s
          (projT1 l)
          (finite_trace_last is tr'
             (projT1 l))
        ∧ previous_state_descriptor_prop
            (IM (projT1 l))
            (final_descriptors (projT1 l))
            (finite_trace_last is tr'
               (projT1 l))
            (final_descriptors' (projT1 l))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM,
                  sx =
                  equivocators_state_project
                    IM final_descriptors'
                    (finite_trace_last is
                      tr')
                  → (let 
                      (i, li) :=
                      VLSM.l itemx in
                     valid li
                      (sx i, input itemx))
                    ∧ (let 
                      (i, li) :=
                      VLSM.l itemx in
                      let 
                      (si', om') :=
                      transition li
                      (sx i, input itemx) in
                      (
                      state_update IM sx i
                      si', om')) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM final_descriptors s =
                equivocators_state_project
                  IM final_descriptors'
                  (finite_trace_last is tr')
            end
Hzero: ∀ i : index,
  final_descriptors i = Existing 0
  → final_descriptors' i = Existing 0
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    specialize (Hproperx Hv Ht).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
IH': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
          final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ equivocators_state_project IM
            final_descriptors
            (finite_trace_last is tr') =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
          ∧ finite_valid_trace X'
              (equivocators_state_project IM
                 initial_descriptors is) trX
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
Hproperx: proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr')
∧ final_descriptors' =
  equivocator_descriptors_update IM
    final_descriptors (projT1 l)
    (final_descriptors' (projT1 l))
  ∧ finite_trace_last is tr' =
    state_update (equivocator_IM IM) s
      (projT1 l)
      (finite_trace_last is tr' (projT1 l))
    ∧ previous_state_descriptor_prop
        (IM (projT1 l))
        (final_descriptors (projT1 l))
        (finite_trace_last is tr'
           (projT1 l))
        (final_descriptors' (projT1 l))
      ∧ match oitem with
        | Some itemx =>
            ∀ sx : composite_state IM,
              sx =
              equivocators_state_project IM
                final_descriptors'
                (finite_trace_last is tr')
              → (let (i, li) :=
                   VLSM.l itemx in
                 valid li
                   (sx i, input itemx))
                ∧ (let 
                     (i, li) :=
                     VLSM.l itemx in
                   let 
                     (si', om') :=
                     transition li
                      (sx i, input itemx) in
                   (state_update IM sx i
                      si', om')) =
                  (destination itemx,
                   output itemx)
        | None =>
            equivocators_state_project IM
              final_descriptors s =
            equivocators_state_project IM
              final_descriptors'
              (finite_trace_last is tr')
        end
Hzero: ∀ i : index,
  final_descriptors i = Existing 0
  → final_descriptors' i = Existing 0
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    destruct Hproperx as [_Hproper' [_ [_ [_ Hx]]]].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
IH': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
          final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ equivocators_state_project IM
            final_descriptors
            (finite_trace_last is tr') =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
          ∧ finite_valid_trace X'
              (equivocators_state_project IM
                 initial_descriptors is) trX
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
_Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr')
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
Hzero: ∀ i : index,
  final_descriptors i = Existing 0
  → final_descriptors' i = Existing 0
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    assert (Hproper' :
      proper_fixed_equivocator_descriptors final_descriptors' (finite_trace_last is tr')).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
IH': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
          final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ equivocators_state_project IM
            final_descriptors
            (finite_trace_last is tr') =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
          ∧ finite_valid_trace X'
              (equivocators_state_project IM
                 initial_descriptors is) trX
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
_Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr')
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
Hzero: ∀ i : index,
  final_descriptors i = Existing 0
  → final_descriptors' i = Existing 0
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
proper_fixed_equivocator_descriptors
  final_descriptors' (finite_trace_last is tr')
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
IH': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
          final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ equivocators_state_project IM
            final_descriptors
            (finite_trace_last is tr') =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
          ∧ finite_valid_trace X'
              (equivocators_state_project IM
                 initial_descriptors is) trX
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
_Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr')
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
Hzero: ∀ i : index,
  final_descriptors i = Existing 0
  → final_descriptors' i = Existing 0
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
∃ (trX : list (composite_transition_item IM)) 
  (initial_descriptors : 
   equivocator_descriptors IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    {message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
IH': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
          final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ equivocators_state_project IM
            final_descriptors
            (finite_trace_last is tr') =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
          ∧ finite_valid_trace X'
              (equivocators_state_project IM
                 initial_descriptors is) trX
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
_Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr')
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
Hzero: ∀ i : index,
  final_descriptors i = Existing 0
  → final_descriptors' i = Existing 0
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
proper_fixed_equivocator_descriptors
  final_descriptors' (finite_trace_last is tr') split; [done |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
IH': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
          final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ equivocators_state_project IM
            final_descriptors
            (finite_trace_last is tr') =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
          ∧ finite_valid_trace X'
              (equivocators_state_project IM
                 initial_descriptors is) trX
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
_Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr')
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
Hzero: ∀ i : index,
  final_descriptors i = Existing 0
  → final_descriptors' i = Existing 0
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
∀ i : index,
  i ∉ equivocating → final_descriptors' i = Existing 0
      intros i Hi.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
IH': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
          final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ equivocators_state_project IM
            final_descriptors
            (finite_trace_last is tr') =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
          ∧ finite_valid_trace X'
              (equivocators_state_project IM
                 initial_descriptors is) trX
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
_Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr')
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
Hzero: ∀ i : index,
  final_descriptors i = Existing 0
  → final_descriptors' i = Existing 0
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
i: index
Hi: i ∉ equivocating
final_descriptors' i = Existing 0 apply Hzero.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
IH': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
          final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ equivocators_state_project IM
            final_descriptors
            (finite_trace_last is tr') =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
          ∧ finite_valid_trace X'
              (equivocators_state_project IM
                 initial_descriptors is) trX
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
_Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr')
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
Hzero: ∀ i : index,
  final_descriptors i = Existing 0
  → final_descriptors' i = Existing 0
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
i: index
Hi: i ∉ equivocating
final_descriptors i = Existing 0 clear Hzero.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
IH': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
          final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ equivocators_state_project IM
            final_descriptors
            (finite_trace_last is tr') =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
          ∧ finite_valid_trace X'
              (equivocators_state_project IM
                 initial_descriptors is) trX
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
_Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr')
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
i: index
Hi: i ∉ equivocating
final_descriptors i = Existing 0 destruct Hproper as [_ Hzero].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hzero: ∀ i : index,
  i ∉ equivocating
  → final_descriptors i = Existing 0
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
IH': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
          final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ equivocators_state_project IM
            final_descriptors
            (finite_trace_last is tr') =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
          ∧ finite_valid_trace X'
              (equivocators_state_project IM
                 initial_descriptors is) trX
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
_Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr')
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
i: index
Hi: i ∉ equivocating
final_descriptors i = Existing 0
      by apply Hzero.
    }message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
IH': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
          final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ equivocators_state_project IM
            final_descriptors
            (finite_trace_last is tr') =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
          ∧ finite_valid_trace X'
              (equivocators_state_project IM
                 initial_descriptors is) trX
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
_Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr')
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
Hzero: ∀ i : index,
  final_descriptors i = Existing 0
  → final_descriptors' i = Existing 0
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    clear _Hproper' Hzero.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
IH': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
          final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ equivocators_state_project IM
            final_descriptors
            (finite_trace_last is tr') =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
          ∧ finite_valid_trace X'
              (equivocators_state_project IM
                 initial_descriptors is) trX
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    specialize (IH' _ Hproper')
      as (trX' & initial_descriptors & _ & Htr_project & Hstate_project & HtrX').message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    assert
      (Htr' : finite_valid_trace FreeE is tr').message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
finite_valid_trace FreeE is tr'
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
∃ (trX : list (composite_transition_item IM)) 
  (initial_descriptors : 
   equivocator_descriptors IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    {message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
finite_valid_trace FreeE is tr'
      split; [| done].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
finite_valid_trace_from FreeE is tr'
      revert Htr; apply VLSM_incl_finite_valid_trace_from.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
VLSM_incl_part
  (constrained_vlsm_machine
     (free_composite_vlsm (equivocator_IM IM))
     (equivocators_fixed_equivocations_constraint IM
        (elements equivocating)))
  (free_composite_vlsm_machine (equivocator_IM IM))
      by apply equivocators_fixed_equivocations_vlsm_incl_free.
    }message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    assert
      (Htr'pre : finite_constrained_trace FreeE is tr').message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
finite_constrained_trace FreeE is tr'
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
∃ (trX : list (composite_transition_item IM)) 
  (initial_descriptors : 
   equivocator_descriptors IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    {message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
finite_constrained_trace FreeE is tr'
      split; [| done].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr'
      specialize (vlsm_incl_preloaded_with_all_messages_vlsm FreeE) as Hincl.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
Hincl: VLSM_incl FreeE
  (preloaded_with_all_messages_vlsm FreeE)
finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr'
      by apply (VLSM_incl_finite_valid_trace_from Hincl), Htr'.
    }message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    specialize (equivocators_trace_project_preserves_proper_fixed_equivocator_descriptors
      _ _ (proj1 Htr'pre) _ _ _ Htr_project Hproper')
      as Hproper_initial.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    destruct oitem as [item |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
item: composite_transition_item IM
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hitemx: (∃ Hex : existing_equivocator_label
           (IM (projT1 l)) 
           (projT2 l),
   existT (projT1 l)
     (existing_equivocator_label_extract
        (IM (projT1 l)) 
        (projT2 l) Hex) = 
   VLSM.l item)
∧ iom = input item
  ∧ oom = output item
    ∧ equivocators_state_project IM
        final_descriptors s =
      destination item
      ∧ final_descriptors' (projT1 l) =
        equivocator_label_descriptor
          (projT2 l)
Hx: ∀ sx : composite_state IM,
  sx =
  equivocators_state_project IM
    final_descriptors' 
    (finite_trace_last is tr')
  → (let (i, li) := VLSM.l item in
     valid li (sx i, input item))
    ∧ (let (i, li) := VLSM.l item in
       let (si', om') :=
         transition li (sx i, input item) in
       (state_update IM sx i si', om')) =
      (destination item, output item)
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx = Some ([item], final_descriptors')
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (None, final_descriptors')
Hitemx: True
Hx: equivocators_state_project IM final_descriptors s =
equivocators_state_project IM final_descriptors'
  (finite_trace_last is tr')
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx = Some ([], final_descriptors')
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
∃ (trX : list (composite_transition_item IM)) 
  (initial_descriptors : 
   equivocator_descriptors IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    +message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
item: composite_transition_item IM
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hitemx: (∃ Hex : existing_equivocator_label
           (IM (projT1 l)) 
           (projT2 l),
   existT (projT1 l)
     (existing_equivocator_label_extract
        (IM (projT1 l)) 
        (projT2 l) Hex) = 
   VLSM.l item)
∧ iom = input item
  ∧ oom = output item
    ∧ equivocators_state_project IM
        final_descriptors s =
      destination item
      ∧ final_descriptors' (projT1 l) =
        equivocator_label_descriptor
          (projT2 l)
Hx: ∀ sx : composite_state IM,
  sx =
  equivocators_state_project IM
    final_descriptors' 
    (finite_trace_last is tr')
  → (let (i, li) := VLSM.l item in
     valid li (sx i, input item))
    ∧ (let (i, li) := VLSM.l item in
       let (si', om') :=
         transition li (sx i, input item) in
       (state_update IM sx i si', om')) =
      (destination item, output item)
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx = Some ([item], final_descriptors')
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX  simpl in Hitemx.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
item: composite_transition_item IM
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hitemx: (∃ Hex : existing_equivocator_label
           (IM (projT1 l)) 
           (projT2 l),
   existT (projT1 l)
     (existing_equivocator_label_extract
        (IM (projT1 l)) 
        (projT2 l) Hex) = 
   VLSM.l item)
∧ iom = input item
  ∧ oom = output item
    ∧ equivocators_state_project IM
        final_descriptors s =
      destination item
      ∧ final_descriptors' (projT1 l) =
        equivocator_label_descriptor
          (projT2 l)
Hx: ∀ sx : composite_state IM,
  sx =
  equivocators_state_project IM
    final_descriptors' 
    (finite_trace_last is tr')
  → (let (i, li) := VLSM.l item in
     valid li (sx i, input item))
    ∧ (let (i, li) := VLSM.l item in
       let (si', om') :=
         transition li (sx i, input item) in
       (state_update IM sx i si', om')) =
      (destination item, output item)
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx = Some ([item], final_descriptors')
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX destruct Hitemx as [Hl [Hinput [Houtput [Hdestination _]]]].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
item: composite_transition_item IM
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hx: ∀ sx : composite_state IM,
  sx =
  equivocators_state_project IM
    final_descriptors' 
    (finite_trace_last is tr')
  → (let (i, li) := VLSM.l item in
     valid li (sx i, input item))
    ∧ (let (i, li) := VLSM.l item in
       let (si', om') :=
         transition li (sx i, input item) in
       (state_update IM sx i si', om')) =
      (destination item, output item)
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx = Some ([item], final_descriptors')
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
      specialize (Hx _ eq_refl).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
item: composite_transition_item IM
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hx: (let (i, li) := VLSM.l item in
 valid li
   (equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr') i, 
    input item))
∧ (let (i, li) := VLSM.l item in
   let (si', om') :=
     transition li
       (equivocators_state_project IM
          final_descriptors'
          (finite_trace_last is tr') i,
        input item) in
   (state_update IM
      (equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr')) i si', om')) =
  (destination item, output item)
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx = Some ([item], final_descriptors')
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
      destruct Hx as [Hvx Htx].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
item: composite_transition_item IM
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
valid li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
   input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   transition li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i, 
      input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx = Some ([item], final_descriptors')
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
      exists (trX' ++ [item]), initial_descriptors.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
item: composite_transition_item IM
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
valid li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
   input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   transition li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i, 
      input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx = Some ([item], final_descriptors')
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
proper_fixed_equivocator_descriptors
  initial_descriptors is
∧ foldr (equivocators_trace_project_folder IM) foldx
    tr' = Some (trX' ++ [item], initial_descriptors)
  ∧ equivocators_state_project IM final_descriptors
      final_state =
    finite_trace_last
      (equivocators_state_project IM
         initial_descriptors is) (trX' ++ [item])
    ∧ finite_valid_trace X'
        (equivocators_state_project IM
           initial_descriptors is) (trX' ++ [item]) subst foldx.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
item: composite_transition_item IM
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
valid li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
   input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   transition li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i, 
      input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
proper_fixed_equivocator_descriptors
  initial_descriptors is
∧ foldr (equivocators_trace_project_folder IM)
    (Some ([item], final_descriptors')) tr' =
  Some (trX' ++ [item], initial_descriptors)
  ∧ equivocators_state_project IM final_descriptors
      final_state =
    finite_trace_last
      (equivocators_state_project IM
         initial_descriptors is) (trX' ++ [item])
    ∧ finite_valid_trace X'
        (equivocators_state_project IM
           initial_descriptors is) (trX' ++ [item])
      rewrite equivocators_trace_project_folder_additive
        with (trX := trX') (eqv_descriptors := initial_descriptors)
      ; [| done].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
item: composite_transition_item IM
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
valid li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
   input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   transition li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i, 
      input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
proper_fixed_equivocator_descriptors
  initial_descriptors is
∧ Some (trX' ++ [item], initial_descriptors) =
  Some (trX' ++ [item], initial_descriptors)
  ∧ equivocators_state_project IM final_descriptors
      final_state =
    finite_trace_last
      (equivocators_state_project IM
         initial_descriptors is) (trX' ++ [item])
    ∧ finite_valid_trace X'
        (equivocators_state_project IM
           initial_descriptors is) (trX' ++ [item])
      split; [done |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
item: composite_transition_item IM
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
valid li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
   input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   transition li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i, 
      input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Some (trX' ++ [item], initial_descriptors) =
Some (trX' ++ [item], initial_descriptors)
∧ equivocators_state_project IM final_descriptors
    final_state =
  finite_trace_last
    (equivocators_state_project IM initial_descriptors
       is) (trX' ++ [item])
  ∧ finite_valid_trace X'
      (equivocators_state_project IM
         initial_descriptors is) (trX' ++ [item])
      split; [done |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
item: composite_transition_item IM
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
valid li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
   input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   transition li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i, 
      input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
equivocators_state_project IM final_descriptors
  final_state =
finite_trace_last
  (equivocators_state_project IM initial_descriptors
     is) (trX' ++ [item])
∧ finite_valid_trace X'
    (equivocators_state_project IM initial_descriptors
       is) (trX' ++ [item])
      rewrite finite_trace_last_is_last.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
item: composite_transition_item IM
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
valid li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
   input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   transition li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i, 
      input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
equivocators_state_project IM final_descriptors
  final_state = destination item
∧ finite_valid_trace X'
    (equivocators_state_project IM initial_descriptors
       is) (trX' ++ [item])
      unfold final_state.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
item: composite_transition_item IM
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
valid li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
   input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   transition li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i, 
      input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
equivocators_state_project IM final_descriptors
  (finite_trace_last is tr) = destination item
∧ finite_valid_trace X'
    (equivocators_state_project IM initial_descriptors
       is) (trX' ++ [item]) subst tr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
item: composite_transition_item IM
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
valid li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
   input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   transition li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i, 
      input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
equivocators_state_project IM final_descriptors
  (finite_trace_last is
     (tr' ++
      [{|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |}])) = destination item
∧ finite_valid_trace X'
    (equivocators_state_project IM initial_descriptors
       is) (trX' ++ [item])
      rewrite finite_trace_last_is_last.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
item: composite_transition_item IM
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
valid li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
   input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   transition li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i, 
      input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
equivocators_state_project IM final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) = destination item
∧ finite_valid_trace X'
    (equivocators_state_project IM initial_descriptors
       is) (trX' ++ [item])
      split; [done |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
item: composite_transition_item IM
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
valid li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
   input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   transition li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i, 
      input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
finite_valid_trace X'
  (equivocators_state_project IM initial_descriptors
     is) (trX' ++ [item])
      destruct HtrX' as [HtrX' His].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
item: composite_transition_item IM
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
valid li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
   input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   transition li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i, 
      input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
finite_valid_trace X'
  (equivocators_state_project IM initial_descriptors
     is) (trX' ++ [item])
      split; [| done].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
item: composite_transition_item IM
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
valid li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
   input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   transition li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i, 
      input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
finite_valid_trace_from X'
  (equivocators_state_project IM initial_descriptors
     is) (trX' ++ [item])
      apply finite_valid_trace_from_app_iff.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
item: composite_transition_item IM
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
valid li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
   input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   transition li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i, 
      input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
finite_valid_trace_from X'
  (equivocators_state_project IM initial_descriptors
     is) trX'
∧ finite_valid_trace_from X'
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') [item]
      split; [done |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
item: composite_transition_item IM
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
valid li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
   input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   transition li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i, 
      input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
finite_valid_trace_from X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX') [item]
      change [item] with ([] ++ [item]).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
item: composite_transition_item IM
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
valid li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
   input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   transition li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i, 
      input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
finite_valid_trace_from X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX') ([] ++ [item])
      match goal with
      |- finite_valid_trace_from _ ?l _ => remember l as lst
      end.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
item: composite_transition_item IM
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
valid li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
   input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   transition li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i, 
      input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
lst: state X'
Heqlst: lst =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
finite_valid_trace_from X' lst ([] ++ [item])
      destruct item.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
l0: label (composite_type IM)
input: option message
destination: state (composite_type IM)
output: option message
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) =
  VLSM.l
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |}
Hinput: iom =
VLSM.input
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Houtput: oom =
VLSM.output
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) :=
  VLSM.l
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |} in
valid li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i,
   VLSM.input
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |})
Htx: (let (i, li) :=
   VLSM.l
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |} in
 let (si', om') :=
   transition li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
      VLSM.input
        {|
          l := l0;
          input := input;
          destination := destination;
          output := output
        |}) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(VLSM.destination
   {|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |},
 VLSM.output
   {|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |})
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
lst: state X'
Heqlst: lst =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
finite_valid_trace_from X' lst
  ([] ++
   [{|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |}])
      assert (Hplst : valid_state_prop X' lst)
        by (apply finite_valid_trace_last_pstate in HtrX'; subst; done).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
l0: label (composite_type IM)
input: option message
destination: state (composite_type IM)
output: option message
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) =
  VLSM.l
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |}
Hinput: iom =
VLSM.input
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Houtput: oom =
VLSM.output
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) :=
  VLSM.l
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |} in
valid li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i,
   VLSM.input
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |})
Htx: (let (i, li) :=
   VLSM.l
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |} in
 let (si', om') :=
   transition li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
      VLSM.input
        {|
          l := l0;
          input := input;
          destination := destination;
          output := output
        |}) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(VLSM.destination
   {|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |},
 VLSM.output
   {|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |})
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
lst: state X'
Heqlst: lst =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
Hplst: valid_state_prop X' lst
finite_valid_trace_from X' lst
  ([] ++
   [{|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |}])
      apply (extend_right_finite_trace_from X'); [by constructor |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
l0: label (composite_type IM)
input: option message
destination: state (composite_type IM)
output: option message
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) =
  VLSM.l
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |}
Hinput: iom =
VLSM.input
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Houtput: oom =
VLSM.output
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) :=
  VLSM.l
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |} in
valid li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i,
   VLSM.input
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |})
Htx: (let (i, li) :=
   VLSM.l
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |} in
 let (si', om') :=
   transition li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
      VLSM.input
        {|
          l := l0;
          input := input;
          destination := destination;
          output := output
        |}) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(VLSM.destination
   {|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |},
 VLSM.output
   {|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |})
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
lst: state X'
Heqlst: lst =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
Hplst: valid_state_prop X' lst
input_valid_transition X' l0
  (finite_trace_last lst [], input)
  (destination, output)
      simpl in Hl.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
l0: label (composite_type IM)
input: option message
destination: state (composite_type IM)
output: option message
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hinput: iom =
VLSM.input
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Houtput: oom =
VLSM.output
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) :=
  VLSM.l
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |} in
valid li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i,
   VLSM.input
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |})
Htx: (let (i, li) :=
   VLSM.l
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |} in
 let (si', om') :=
   transition li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
      VLSM.input
        {|
          l := l0;
          input := input;
          destination := destination;
          output := output
        |}) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(VLSM.destination
   {|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |},
 VLSM.output
   {|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |})
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
lst: state X'
Heqlst: lst =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
Hplst: valid_state_prop X' lst
input_valid_transition X' l0
  (finite_trace_last lst [], input)
  (destination, output) subst.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: label (composite_type IM)
input: option message
destination: state (composite_type IM)
output: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
          VLSM.input
            {|
              l := l0;
              input := input;
              destination := destination;
              output := output
            |};
        destination := s;
        output :=
          VLSM.output
            {|
              l := l0;
              input := input;
              destination := destination;
              output := output
            |}
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |}) =
(s,
 VLSM.output
   {|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |})
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |})
Hv: valid l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := input;
            destination := destination;
            output := output
          |}
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
          l := l0;
          input := input;
          destination := destination;
          output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
          l := l0;
          input := input;
          destination := destination;
          output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) :=
  VLSM.l
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |} in
valid li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i,
   VLSM.input
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |})
Htx: (let (i, li) :=
   VLSM.l
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |} in
 let (si', om') :=
   transition li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
      VLSM.input
        {|
          l := l0;
          input := input;
          destination := destination;
          output := output
        |}) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(VLSM.destination
   {|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |},
 VLSM.output
   {|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |})
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [], input)
  (destination, output)
      simpl in Hc.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: label (composite_type IM)
input: option message
destination: state (composite_type IM)
output: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
          VLSM.input
            {|
              l := l0;
              input := input;
              destination := destination;
              output := output
            |};
        destination := s;
        output :=
          VLSM.output
            {|
              l := l0;
              input := input;
              destination := destination;
              output := output
            |}
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |}) =
(s,
 VLSM.output
   {|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |})
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', input)
Hv: valid l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := input;
            destination := destination;
            output := output
          |}
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
          l := l0;
          input := input;
          destination := destination;
          output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
          l := l0;
          input := input;
          destination := destination;
          output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) :=
  VLSM.l
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |} in
valid li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i,
   VLSM.input
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |})
Htx: (let (i, li) :=
   VLSM.l
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |} in
 let (si', om') :=
   transition li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
      VLSM.input
        {|
          l := l0;
          input := input;
          destination := destination;
          output := output
        |}) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(VLSM.destination
   {|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |},
 VLSM.output
   {|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |})
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [], input)
  (destination, output)
      destruct Hc as [[Hno_equiv _] _].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: label (composite_type IM)
input: option message
destination: state (composite_type IM)
output: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
          VLSM.input
            {|
              l := l0;
              input := input;
              destination := destination;
              output := output
            |};
        destination := s;
        output :=
          VLSM.output
            {|
              l := l0;
              input := input;
              destination := destination;
              output := output
            |}
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |}) =
(s,
 VLSM.output
   {|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |})
Hno_equiv: composite_no_equivocations
  (equivocator_IM IM) l
  (finite_trace_last is tr', input)
Hv: valid l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := input;
            destination := destination;
            output := output
          |}
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
          l := l0;
          input := input;
          destination := destination;
          output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
          l := l0;
          input := input;
          destination := destination;
          output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) :=
  VLSM.l
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |} in
valid li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i,
   VLSM.input
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |})
Htx: (let (i, li) :=
   VLSM.l
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |} in
 let (si', om') :=
   transition li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
      VLSM.input
        {|
          l := l0;
          input := input;
          destination := destination;
          output := output
        |}) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(VLSM.destination
   {|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |},
 VLSM.output
   {|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |})
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [], input)
  (destination, output)
      simpl in Htx, Hvx, Hstate_project.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: label (composite_type IM)
input: option message
destination: state (composite_type IM)
output: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
          VLSM.input
            {|
              l := l0;
              input := input;
              destination := destination;
              output := output
            |};
        destination := s;
        output :=
          VLSM.output
            {|
              l := l0;
              input := input;
              destination := destination;
              output := output
            |}
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |}) =
(s,
 VLSM.output
   {|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |})
Hno_equiv: composite_no_equivocations
  (equivocator_IM IM) l
  (finite_trace_last is tr', input)
Hv: valid l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := input;
            destination := destination;
            output := output
          |}
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
          l := l0;
          input := input;
          destination := destination;
          output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
          l := l0;
          input := input;
          destination := destination;
          output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) := l0 in
valid li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i, input) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [], input)
  (destination, output)
      rewrite Hstate_project in Hvx, Htx.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: label (composite_type IM)
input: option message
destination: state (composite_type IM)
output: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
          VLSM.input
            {|
              l := l0;
              input := input;
              destination := destination;
              output := output
            |};
        destination := s;
        output :=
          VLSM.output
            {|
              l := l0;
              input := input;
              destination := destination;
              output := output
            |}
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |}) =
(s,
 VLSM.output
   {|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |})
Hno_equiv: composite_no_equivocations
  (equivocator_IM IM) l
  (finite_trace_last is tr', input)
Hv: valid l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := input;
            destination := destination;
            output := output
          |}
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
          l := l0;
          input := input;
          destination := destination;
          output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
          l := l0;
          input := input;
          destination := destination;
          output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i, input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i, input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [], input)
  (destination, output)
      destruct input as [input |]; [| by repeat split;
        [| apply option_valid_message_None | | apply HconstraintNone |]].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: label (composite_type IM)
input: message
destination: state (composite_type IM)
output: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
          VLSM.input
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |};
        destination := s;
        output :=
          VLSM.output
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |}
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}) =
(s,
 VLSM.output
   {|
     l := l0;
     input := Some input;
     destination := destination;
     output := output
   |})
Hno_equiv: composite_no_equivocations
  (equivocator_IM IM) l
  (finite_trace_last is tr', Some input)
Hv: valid l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
   Some input) (destination, output)
      simpl in Hno_equiv.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: label (composite_type IM)
input: message
destination: state (composite_type IM)
output: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
          VLSM.input
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |};
        destination := s;
        output :=
          VLSM.output
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |}
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}) =
(s,
 VLSM.output
   {|
     l := l0;
     input := Some input;
     destination := destination;
     output := output
   |})
Hno_equiv: composite_no_equivocations
  (equivocator_IM IM) l
  (finite_trace_last is tr', Some input)
Hv: valid l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
   Some input) (destination, output)
      apply or_comm in Hno_equiv.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: label (composite_type IM)
input: message
destination: state (composite_type IM)
output: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
          VLSM.input
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |};
        destination := s;
        output :=
          VLSM.output
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |}
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}) =
(s,
 VLSM.output
   {|
     l := l0;
     input := Some input;
     destination := destination;
     output := output
   |})
Hno_equiv: False
∨ composite_has_been_sent
    (equivocator_IM IM)
    (finite_trace_last is tr', Some input).1
    input
Hv: valid l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
   Some input) (destination, output)
      destruct Hno_equiv as [Hinit_input | Hno_equiv]
      ; [done |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: label (composite_type IM)
input: message
destination: state (composite_type IM)
output: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
          VLSM.input
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |};
        destination := s;
        output :=
          VLSM.output
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |}
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}) =
(s,
 VLSM.output
   {|
     l := l0;
     input := Some input;
     destination := destination;
     output := output
   |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr', Some input).1
  input
Hv: valid l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
   Some input) (destination, output)
      assert
        (Hs_free : valid_state_prop FreeE (finite_trace_last is tr')).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: label (composite_type IM)
input: message
destination: state (composite_type IM)
output: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
          VLSM.input
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |};
        destination := s;
        output :=
          VLSM.output
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |}
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}) =
(s,
 VLSM.output
   {|
     l := l0;
     input := Some input;
     destination := destination;
     output := output
   |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr', Some input).1
  input
Hv: valid l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
valid_state_prop FreeE (finite_trace_last is tr')
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: label (composite_type IM)
input: message
destination: state (composite_type IM)
output: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
          VLSM.input
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |};
        destination := s;
        output :=
          VLSM.output
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |}
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}) =
(s,
 VLSM.output
   {|
     l := l0;
     input := Some input;
     destination := destination;
     output := output
   |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr', Some input).1
  input
Hv: valid l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
   Some input) (destination, output)
      {message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: label (composite_type IM)
input: message
destination: state (composite_type IM)
output: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
          VLSM.input
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |};
        destination := s;
        output :=
          VLSM.output
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |}
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}) =
(s,
 VLSM.output
   {|
     l := l0;
     input := Some input;
     destination := destination;
     output := output
   |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr', Some input).1
  input
Hv: valid l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
valid_state_prop FreeE (finite_trace_last is tr')
        destruct Hs as [_om Hs].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: label (composite_type IM)
input: message
destination: state (composite_type IM)
output: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
          VLSM.input
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |};
        destination := s;
        output :=
          VLSM.output
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |}
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
_om: option message
Hs: valid_state_message_prop XE
  (finite_trace_last is tr') _om
Ht: transition l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}) =
(s,
 VLSM.output
   {|
     l := l0;
     input := Some input;
     destination := destination;
     output := output
   |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr', Some input).1
  input
Hv: valid l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
valid_state_prop FreeE (finite_trace_last is tr')
        exists _om.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: label (composite_type IM)
input: message
destination: state (composite_type IM)
output: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
          VLSM.input
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |};
        destination := s;
        output :=
          VLSM.output
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |}
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
_om: option message
Hs: valid_state_message_prop XE
  (finite_trace_last is tr') _om
Ht: transition l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}) =
(s,
 VLSM.output
   {|
     l := l0;
     input := Some input;
     destination := destination;
     output := output
   |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr', Some input).1
  input
Hv: valid l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
valid_state_message_prop FreeE
  (finite_trace_last is tr') _om
        eapply VLSM_incl_valid_state_message; [by apply free_composite_vlsm_spec | by do 2 red |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: label (composite_type IM)
input: message
destination: state (composite_type IM)
output: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
          VLSM.input
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |};
        destination := s;
        output :=
          VLSM.output
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |}
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
_om: option message
Hs: valid_state_message_prop XE
  (finite_trace_last is tr') _om
Ht: transition l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}) =
(s,
 VLSM.output
   {|
     l := l0;
     input := Some input;
     destination := destination;
     output := output
   |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr', Some input).1
  input
Hv: valid l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
valid_state_message_prop
  {|
    vlsm_type := composite_type (equivocator_IM IM);
    vlsm_machine :=
      {|
        vlsm_type :=
          free_composite_vlsm (equivocator_IM IM);
        vlsm_machine :=
          composite_vlsm (equivocator_IM IM)
            (free_constraint (equivocator_IM IM))
      |}
  |} (finite_trace_last is tr') _om
        by eapply (constraint_subsumption_valid_state_message_preservation (free_composite_vlsm _)).
      }message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: label (composite_type IM)
input: message
destination: state (composite_type IM)
output: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
          VLSM.input
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |};
        destination := s;
        output :=
          VLSM.output
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |}
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}) =
(s,
 VLSM.output
   {|
     l := l0;
     input := Some input;
     destination := destination;
     output := output
   |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr', Some input).1
  input
Hv: valid l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
   Some input) (destination, output)
      specialize
        (specialized_proper_sent_rev FreeE _ Hs_free _ Hno_equiv) as Hall.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: label (composite_type IM)
input: message
destination: state (composite_type IM)
output: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
          VLSM.input
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |};
        destination := s;
        output :=
          VLSM.output
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |}
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}) =
(s,
 VLSM.output
   {|
     l := l0;
     input := Some input;
     destination := destination;
     output := output
   |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr', Some input).1
  input
Hv: valid l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: specialized_selected_message_exists_in_all_traces
  FreeE (field_selector VLSM.output)
  (finite_trace_last is tr') input
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
   Some input) (destination, output)
      specialize (Hall is tr' (valid_trace_add_default_last Htr')).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: label (composite_type IM)
input: message
destination: state (composite_type IM)
output: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
          VLSM.input
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |};
        destination := s;
        output :=
          VLSM.output
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |}
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}) =
(s,
 VLSM.output
   {|
     l := l0;
     input := Some input;
     destination := destination;
     output := output
   |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr', Some input).1
  input
Hv: valid l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message (field_selector VLSM.output)
  input tr'
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
   Some input) (destination, output)
      destruct (equivocators_trace_project_output_reflecting_inv IM _ _ (proj1 Htr'pre) _ Hall)
        as [final_descriptors_m [initial_descriptors_m [trXm [_Hfinal_descriptors_m
            [Hproject_trXm Hex]]]]].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: label (composite_type IM)
input: message
destination: state (composite_type IM)
output: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
          VLSM.input
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |};
        destination := s;
        output :=
          VLSM.output
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |}
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}) =
(s,
 VLSM.output
   {|
     l := l0;
     input := Some input;
     destination := destination;
     output := output
   |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr', Some input).1
  input
Hv: valid l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message (field_selector VLSM.output)
  input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
   Some input) (destination, output)
      assert (Hfinal_descriptors_m :
        proper_fixed_equivocator_descriptors final_descriptors_m (finite_trace_last is tr')).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: label (composite_type IM)
input: message
destination: state (composite_type IM)
output: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
          VLSM.input
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |};
        destination := s;
        output :=
          VLSM.output
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |}
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}) =
(s,
 VLSM.output
   {|
     l := l0;
     input := Some input;
     destination := destination;
     output := output
   |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr', Some input).1
  input
Hv: valid l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message (field_selector VLSM.output)
  input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
proper_fixed_equivocator_descriptors
  final_descriptors_m (finite_trace_last is tr')
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: label (composite_type IM)
input: message
destination: state (composite_type IM)
output: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
          VLSM.input
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |};
        destination := s;
        output :=
          VLSM.output
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |}
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}) =
(s,
 VLSM.output
   {|
     l := l0;
     input := Some input;
     destination := destination;
     output := output
   |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr', Some input).1
  input
Hv: valid l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message (field_selector VLSM.output)
  input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
   Some input) (destination, output)
      {message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: label (composite_type IM)
input: message
destination: state (composite_type IM)
output: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
          VLSM.input
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |};
        destination := s;
        output :=
          VLSM.output
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |}
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}) =
(s,
 VLSM.output
   {|
     l := l0;
     input := Some input;
     destination := destination;
     output := output
   |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr', Some input).1
  input
Hv: valid l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message (field_selector VLSM.output)
  input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
proper_fixed_equivocator_descriptors
  final_descriptors_m (finite_trace_last is tr') apply not_equivocating_equivocator_descriptors_proper_fixed; [| done].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: label (composite_type IM)
input: message
destination: state (composite_type IM)
output: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
          VLSM.input
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |};
        destination := s;
        output :=
          VLSM.output
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |}
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}) =
(s,
 VLSM.output
   {|
     l := l0;
     input := Some input;
     destination := destination;
     output := output
   |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr', Some input).1
  input
Hv: valid l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message (field_selector VLSM.output)
  input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
valid_state_prop XE (finite_trace_last is tr')
        by apply finite_valid_trace_last_pstate.
      }message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: label (composite_type IM)
input: message
destination: state (composite_type IM)
output: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
          VLSM.input
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |};
        destination := s;
        output :=
          VLSM.output
            {|
              l := l0;
              input := Some input;
              destination := destination;
              output := output
            |}
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}) =
(s,
 VLSM.output
   {|
     l := l0;
     input := Some input;
     destination := destination;
     output := output
   |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr', Some input).1
  input
Hv: valid l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message (field_selector VLSM.output)
  input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
   Some input) (destination, output)
      specialize (IHlen (length tr')).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: label (composite_type IM)
input: message
destination: state (composite_type IM)
output: option message
IHlen: length tr' <
length
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}])
→ ∀ tr : list
           (composite_transition_item
              (equivocator_IM IM)),
    finite_valid_trace XE is tr
    → length tr' = length tr
      → ∀ final_descriptors : 
           equivocator_descriptors IM,
          proper_fixed_equivocator_descriptors
            final_descriptors
            (finite_trace_last is tr)
          → ∃ (trX : list
                      (composite_transition_item
                      IM)) 
              (initial_descriptors : 
               equivocator_descriptors IM),
              proper_fixed_equivocator_descriptors
                initial_descriptors is
              ∧ equivocators_trace_project IM
                  final_descriptors tr =
                Some
                  (trX, initial_descriptors)
                ∧ equivocators_state_project
                    IM final_descriptors
                    (finite_trace_last is tr) =
                  finite_trace_last
                    (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                  ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}) =
(s,
 VLSM.output
   {|
     l := l0;
     input := Some input;
     destination := destination;
     output := output
   |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr', Some input).1
  input
Hv: valid l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message (field_selector VLSM.output)
  input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
   Some input) (destination, output)
      spec IHlen.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: label (composite_type IM)
input: message
destination: state (composite_type IM)
output: option message
IHlen: length tr' <
length
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}])
→ ∀ tr : list
           (composite_transition_item
              (equivocator_IM IM)),
    finite_valid_trace XE is tr
    → length tr' = length tr
      → ∀ final_descriptors : 
           equivocator_descriptors IM,
          proper_fixed_equivocator_descriptors
            final_descriptors
            (finite_trace_last is tr)
          → ∃ (trX : list
                      (composite_transition_item
                      IM)) 
              (initial_descriptors : 
               equivocator_descriptors IM),
              proper_fixed_equivocator_descriptors
                initial_descriptors is
              ∧ equivocators_trace_project IM
                  final_descriptors tr =
                Some
                  (trX, initial_descriptors)
                ∧ equivocators_state_project
                    IM final_descriptors
                    (finite_trace_last is tr) =
                  finite_trace_last
                    (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                  ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}) =
(s,
 VLSM.output
   {|
     l := l0;
     input := Some input;
     destination := destination;
     output := output
   |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr', Some input).1
  input
Hv: valid l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message (field_selector VLSM.output)
  input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
length tr' <
length
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}])
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: label (composite_type IM)
input: message
destination: state (composite_type IM)
output: option message
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}) =
(s,
 VLSM.output
   {|
     l := l0;
     input := Some input;
     destination := destination;
     output := output
   |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr', Some input).1
  input
Hv: valid l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message (field_selector VLSM.output)
  input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
IHlen: ∀ tr : list
         (composite_transition_item
            (equivocator_IM IM)),
  finite_valid_trace XE is tr
  → length tr' = length tr
    → ∀ final_descriptors : 
         equivocator_descriptors IM,
        proper_fixed_equivocator_descriptors
          final_descriptors
          (finite_trace_last is tr)
        → ∃ (trX : list
                     (composite_transition_item
                      IM)) 
            (initial_descriptors : 
             equivocator_descriptors IM),
            proper_fixed_equivocator_descriptors
              initial_descriptors is
            ∧ equivocators_trace_project IM
                final_descriptors tr =
              Some (trX, initial_descriptors)
              ∧ equivocators_state_project IM
                  final_descriptors
                  (finite_trace_last is tr) =
                finite_trace_last
                  (equivocators_state_project
                     IM initial_descriptors is)
                  trX
                ∧ finite_valid_trace X'
                    (equivocators_state_project
                      IM initial_descriptors
                      is) trX
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
   Some input) (destination, output) {message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: label (composite_type IM)
input: message
destination: state (composite_type IM)
output: option message
IHlen: length tr' <
length
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}])
→ ∀ tr : list
           (composite_transition_item
              (equivocator_IM IM)),
    finite_valid_trace XE is tr
    → length tr' = length tr
      → ∀ final_descriptors : 
           equivocator_descriptors IM,
          proper_fixed_equivocator_descriptors
            final_descriptors
            (finite_trace_last is tr)
          → ∃ (trX : list
                      (composite_transition_item
                      IM)) 
              (initial_descriptors : 
               equivocator_descriptors IM),
              proper_fixed_equivocator_descriptors
                initial_descriptors is
              ∧ equivocators_trace_project IM
                  final_descriptors tr =
                Some
                  (trX, initial_descriptors)
                ∧ equivocators_state_project
                    IM final_descriptors
                    (finite_trace_last is tr) =
                  finite_trace_last
                    (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                  ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}) =
(s,
 VLSM.output
   {|
     l := l0;
     input := Some input;
     destination := destination;
     output := output
   |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr', Some input).1
  input
Hv: valid l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message (field_selector VLSM.output)
  input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
length tr' <
length
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}]) rewrite app_length.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: label (composite_type IM)
input: message
destination: state (composite_type IM)
output: option message
IHlen: length tr' <
length
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}])
→ ∀ tr : list
           (composite_transition_item
              (equivocator_IM IM)),
    finite_valid_trace XE is tr
    → length tr' = length tr
      → ∀ final_descriptors : 
           equivocator_descriptors IM,
          proper_fixed_equivocator_descriptors
            final_descriptors
            (finite_trace_last is tr)
          → ∃ (trX : list
                      (composite_transition_item
                      IM)) 
              (initial_descriptors : 
               equivocator_descriptors IM),
              proper_fixed_equivocator_descriptors
                initial_descriptors is
              ∧ equivocators_trace_project IM
                  final_descriptors tr =
                Some
                  (trX, initial_descriptors)
                ∧ equivocators_state_project
                    IM final_descriptors
                    (finite_trace_last is tr) =
                  finite_trace_last
                    (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                  ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}) =
(s,
 VLSM.output
   {|
     l := l0;
     input := Some input;
     destination := destination;
     output := output
   |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr', Some input).1
  input
Hv: valid l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message (field_selector VLSM.output)
  input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
length tr' <
length tr' +
length
  [{|
     l := l;
     input :=
       VLSM.input
         {|
           l := l0;
           input := Some input;
           destination := destination;
           output := output
         |};
     destination := s;
     output :=
       VLSM.output
         {|
           l := l0;
           input := Some input;
           destination := destination;
           output := output
         |}
   |}] simpl.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: label (composite_type IM)
input: message
destination: state (composite_type IM)
output: option message
IHlen: length tr' <
length
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}])
→ ∀ tr : list
           (composite_transition_item
              (equivocator_IM IM)),
    finite_valid_trace XE is tr
    → length tr' = length tr
      → ∀ final_descriptors : 
           equivocator_descriptors IM,
          proper_fixed_equivocator_descriptors
            final_descriptors
            (finite_trace_last is tr)
          → ∃ (trX : list
                      (composite_transition_item
                      IM)) 
              (initial_descriptors : 
               equivocator_descriptors IM),
              proper_fixed_equivocator_descriptors
                initial_descriptors is
              ∧ equivocators_trace_project IM
                  final_descriptors tr =
                Some
                  (trX, initial_descriptors)
                ∧ equivocators_state_project
                    IM final_descriptors
                    (finite_trace_last is tr) =
                  finite_trace_last
                    (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                  ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}) =
(s,
 VLSM.output
   {|
     l := l0;
     input := Some input;
     destination := destination;
     output := output
   |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr', Some input).1
  input
Hv: valid l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message (field_selector VLSM.output)
  input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
length tr' < length tr' + 1 lia. }message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: label (composite_type IM)
input: message
destination: state (composite_type IM)
output: option message
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}) =
(s,
 VLSM.output
   {|
     l := l0;
     input := Some input;
     destination := destination;
     output := output
   |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr', Some input).1
  input
Hv: valid l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message (field_selector VLSM.output)
  input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
IHlen: ∀ tr : list
         (composite_transition_item
            (equivocator_IM IM)),
  finite_valid_trace XE is tr
  → length tr' = length tr
    → ∀ final_descriptors : 
         equivocator_descriptors IM,
        proper_fixed_equivocator_descriptors
          final_descriptors
          (finite_trace_last is tr)
        → ∃ (trX : list
                     (composite_transition_item
                      IM)) 
            (initial_descriptors : 
             equivocator_descriptors IM),
            proper_fixed_equivocator_descriptors
              initial_descriptors is
            ∧ equivocators_trace_project IM
                final_descriptors tr =
              Some (trX, initial_descriptors)
              ∧ equivocators_state_project IM
                  final_descriptors
                  (finite_trace_last is tr) =
                finite_trace_last
                  (equivocators_state_project
                     IM initial_descriptors is)
                  trX
                ∧ finite_valid_trace X'
                    (equivocators_state_project
                      IM initial_descriptors
                      is) trX
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
   Some input) (destination, output)
      specialize (IHlen tr' (conj Htr Hinit) eq_refl final_descriptors_m Hfinal_descriptors_m)
        as (trXm' & initial_descriptors_m' & Hproper_initial_m &
            Hproject_trXm' & Hpr_fin_tr' & HtrXm).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: label (composite_type IM)
input: message
destination: state (composite_type IM)
output: option message
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}) =
(s,
 VLSM.output
   {|
     l := l0;
     input := Some input;
     destination := destination;
     output := output
   |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr', Some input).1
  input
Hv: valid l
  (finite_trace_last is tr',
   VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
          l := l0;
          input := Some input;
          destination := destination;
          output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message (field_selector VLSM.output)
  input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
trXm': list (composite_transition_item IM)
initial_descriptors_m': equivocator_descriptors IM
Hproper_initial_m: proper_fixed_equivocator_descriptors
  initial_descriptors_m' is
Hproject_trXm': equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm', initial_descriptors_m')
Hpr_fin_tr': equivocators_state_project IM
  final_descriptors_m
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors_m' is) trXm'
HtrXm: finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors_m' is) trXm'
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
   Some input) (destination, output)
      simpl in *.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: composite_label IM
input: message
destination: composite_state IM
output: option message
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li
     (finite_trace_last is tr' i, Some input) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr') i si', om')) =
(s, output)
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr') input
Hv: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr' i, Some input)
Hiom: option_valid_message_prop XE (Some input)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s = destination
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message (field_selector VLSM.output)
  input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
trXm': list (composite_transition_item IM)
initial_descriptors_m': equivocator_descriptors IM
Hproper_initial_m: proper_fixed_equivocator_descriptors
  initial_descriptors_m' is
Hproject_trXm': equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm', initial_descriptors_m')
Hpr_fin_tr': equivocators_state_project IM
  final_descriptors_m
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors_m' is) trXm'
HtrXm: finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors_m' is) trXm'
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
   Some input) (destination, output) rewrite Hproject_trXm in Hproject_trXm'.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: composite_label IM
input: message
destination: composite_state IM
output: option message
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li
     (finite_trace_last is tr' i, Some input) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr') i si', om')) =
(s, output)
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr') input
Hv: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr' i, Some input)
Hiom: option_valid_message_prop XE (Some input)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s = destination
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message (field_selector VLSM.output)
  input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
trXm': list (composite_transition_item IM)
initial_descriptors_m': equivocator_descriptors IM
Hproper_initial_m: proper_fixed_equivocator_descriptors
  initial_descriptors_m' is
Hproject_trXm': Some (trXm, initial_descriptors_m) =
Some (trXm', initial_descriptors_m')
Hpr_fin_tr': equivocators_state_project IM
  final_descriptors_m
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors_m' is) trXm'
HtrXm: finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors_m' is) trXm'
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
   Some input) (destination, output)
      inversion Hproject_trXm'.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: composite_label IM
input: message
destination: composite_state IM
output: option message
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li
     (finite_trace_last is tr' i, Some input) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr') i si', om')) =
(s, output)
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr') input
Hv: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr' i, Some input)
Hiom: option_valid_message_prop XE (Some input)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s = destination
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message (field_selector VLSM.output)
  input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
trXm': list (composite_transition_item IM)
initial_descriptors_m': equivocator_descriptors IM
Hproper_initial_m: proper_fixed_equivocator_descriptors
  initial_descriptors_m' is
Hproject_trXm': Some (trXm, initial_descriptors_m) =
Some (trXm', initial_descriptors_m')
Hpr_fin_tr': equivocators_state_project IM
  final_descriptors_m
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors_m' is) trXm'
HtrXm: finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors_m' is) trXm'
H11: trXm = trXm'
H12: initial_descriptors_m = initial_descriptors_m'
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
   Some input) (destination, output) subst trXm' initial_descriptors_m'.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: composite_label IM
input: message
destination: composite_state IM
output: option message
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li
     (finite_trace_last is tr' i, Some input) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr') i si', om')) =
(s, output)
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr') input
Hv: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr' i, Some input)
Hiom: option_valid_message_prop XE (Some input)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s = destination
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message (field_selector VLSM.output)
  input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm': Some (trXm, initial_descriptors_m) =
Some (trXm, initial_descriptors_m)
Hpr_fin_tr': equivocators_state_project IM
  final_descriptors_m
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
HtrXm: finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
Hproper_initial_m: proper_fixed_equivocator_descriptors
  initial_descriptors_m is
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
   Some input) (destination, output) clear Hproject_trXm'.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: composite_label IM
input: message
destination: composite_state IM
output: option message
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li
     (finite_trace_last is tr' i, Some input) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr') i si', om')) =
(s, output)
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr') input
Hv: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr' i, Some input)
Hiom: option_valid_message_prop XE (Some input)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s = destination
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message (field_selector VLSM.output)
  input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
Hpr_fin_tr': equivocators_state_project IM
  final_descriptors_m
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
HtrXm: finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
Hproper_initial_m: proper_fixed_equivocator_descriptors
  initial_descriptors_m is
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
   Some input) (destination, output)
      repeat split; [done | | done | | done].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: composite_label IM
input: message
destination: composite_state IM
output: option message
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li
     (finite_trace_last is tr' i, Some input) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr') i si', om')) =
(s, output)
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr') input
Hv: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr' i, Some input)
Hiom: option_valid_message_prop XE (Some input)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s = destination
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message (field_selector VLSM.output)
  input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
Hpr_fin_tr': equivocators_state_project IM
  final_descriptors_m
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
HtrXm: finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
Hproper_initial_m: proper_fixed_equivocator_descriptors
  initial_descriptors_m is
option_valid_message_prop X' (Some input)
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: composite_label IM
input: message
destination: composite_state IM
output: option message
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li
     (finite_trace_last is tr' i, Some input) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr') i si', om')) =
(s, output)
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr') input
Hv: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr' i, Some input)
Hiom: option_valid_message_prop XE (Some input)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s = destination
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message (field_selector VLSM.output)
  input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
Hpr_fin_tr': equivocators_state_project IM
  final_descriptors_m
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
HtrXm: finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
Hproper_initial_m: proper_fixed_equivocator_descriptors
  initial_descriptors_m is
constraint l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
   Some input)
      *message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: composite_label IM
input: message
destination: composite_state IM
output: option message
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li
     (finite_trace_last is tr' i, Some input) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr') i si', om')) =
(s, output)
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr') input
Hv: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr' i, Some input)
Hiom: option_valid_message_prop XE (Some input)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s = destination
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message (field_selector VLSM.output)
  input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
Hpr_fin_tr': equivocators_state_project IM
  final_descriptors_m
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
HtrXm: finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
Hproper_initial_m: proper_fixed_equivocator_descriptors
  initial_descriptors_m is
option_valid_message_prop X' (Some input) by eapply valid_trace_output_is_valid; [apply HtrXm |].
      *message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: composite_label IM
input: message
destination: composite_state IM
output: option message
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li
     (finite_trace_last is tr' i, Some input) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr') i si', om')) =
(s, output)
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr') input
Hv: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr' i, Some input)
Hiom: option_valid_message_prop XE (Some input)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s = destination
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message (field_selector VLSM.output)
  input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
Hpr_fin_tr': equivocators_state_project IM
  final_descriptors_m
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
HtrXm: finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
Hproper_initial_m: proper_fixed_equivocator_descriptors
  initial_descriptors_m is
constraint l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
   Some input) rewrite <- Hstate_project.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: composite_label IM
input: message
destination: composite_state IM
output: option message
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li
     (finite_trace_last is tr' i, Some input) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr') i si', om')) =
(s, output)
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr') input
Hv: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr' i, Some input)
Hiom: option_valid_message_prop XE (Some input)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s = destination
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message (field_selector VLSM.output)
  input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
Hpr_fin_tr': equivocators_state_project IM
  final_descriptors_m
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
HtrXm: finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
Hproper_initial_m: proper_fixed_equivocator_descriptors
  initial_descriptors_m is
constraint l0
  (finite_trace_last
     (equivocators_state_project IM final_descriptors'
        (finite_trace_last is tr')) [], Some input)
        apply Hconstraint_hbs; [done | apply Hproper' |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: composite_label IM
input: message
destination: composite_state IM
output: option message
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li
     (finite_trace_last is tr' i, Some input) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr') i si', om')) =
(s, output)
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr') input
Hv: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr' i, Some input)
Hiom: option_valid_message_prop XE (Some input)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s = destination
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message (field_selector VLSM.output)
  input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
Hpr_fin_tr': equivocators_state_project IM
  final_descriptors_m
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
HtrXm: finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
Hproper_initial_m: proper_fixed_equivocator_descriptors
  initial_descriptors_m is
has_been_sent FreeE (finite_trace_last is tr') input
        assert (Hlst'pre : constrained_state_prop FreeE (finite_trace_last is tr'))
            by (apply finite_valid_trace_last_pstate, Htr'pre).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: composite_label IM
input: message
destination: composite_state IM
output: option message
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li
     (finite_trace_last is tr' i, Some input) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr') i si', om')) =
(s, output)
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr') input
Hv: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr' i, Some input)
Hiom: option_valid_message_prop XE (Some input)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s = destination
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message (field_selector VLSM.output)
  input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
Hpr_fin_tr': equivocators_state_project IM
  final_descriptors_m
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
HtrXm: finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
Hproper_initial_m: proper_fixed_equivocator_descriptors
  initial_descriptors_m is
Hlst'pre: constrained_state_prop FreeE
  (finite_trace_last is tr')
has_been_sent FreeE (finite_trace_last is tr') input
        apply proper_sent; [done |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: composite_label IM
input: message
destination: composite_state IM
output: option message
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li
     (finite_trace_last is tr' i, Some input) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr') i si', om')) =
(s, output)
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr') input
Hv: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr' i, Some input)
Hiom: option_valid_message_prop XE (Some input)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s = destination
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message (field_selector VLSM.output)
  input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
Hpr_fin_tr': equivocators_state_project IM
  final_descriptors_m
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
HtrXm: finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
Hproper_initial_m: proper_fixed_equivocator_descriptors
  initial_descriptors_m is
Hlst'pre: constrained_state_prop FreeE
  (finite_trace_last is tr')
selected_message_exists_in_all_preloaded_traces FreeE
  (field_selector VLSM.output)
  (finite_trace_last is tr') input
        apply has_been_sent_consistency; [typeclasses eauto | done |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
l0: composite_label IM
input: message
destination: composite_state IM
output: option message
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li
     (finite_trace_last is tr' i, Some input) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr') i si', om')) =
(s, output)
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (finite_trace_last is tr') input
Hv: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr' i, Some input)
Hiom: option_valid_message_prop XE (Some input)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s = destination
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') i si',
  om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message (field_selector VLSM.output)
  input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
Hpr_fin_tr': equivocators_state_project IM
  final_descriptors_m
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
HtrXm: finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
Hproper_initial_m: proper_fixed_equivocator_descriptors
  initial_descriptors_m is
Hlst'pre: constrained_state_prop FreeE
  (finite_trace_last is tr')
selected_message_exists_in_some_preloaded_traces FreeE
  (field_selector VLSM.output)
  (finite_trace_last is tr') input
        by red in Htr'pre; exists is, tr', (valid_trace_add_default_last Htr'pre).
    +message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (None, final_descriptors')
Hitemx: True
Hx: equivocators_state_project IM final_descriptors s =
equivocators_state_project IM final_descriptors'
  (finite_trace_last is tr')
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx = Some ([], final_descriptors')
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX exists trX'.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (None, final_descriptors')
Hitemx: True
Hx: equivocators_state_project IM final_descriptors s =
equivocators_state_project IM final_descriptors'
  (finite_trace_last is tr')
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx = Some ([], final_descriptors')
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
∃ initial_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX', initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX'
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX' exists initial_descriptors.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (None, final_descriptors')
Hitemx: True
Hx: equivocators_state_project IM final_descriptors s =
equivocators_state_project IM final_descriptors'
  (finite_trace_last is tr')
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx = Some ([], final_descriptors')
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
proper_fixed_equivocator_descriptors
  initial_descriptors is
∧ foldr (equivocators_trace_project_folder IM) foldx
    tr' = Some (trX', initial_descriptors)
  ∧ equivocators_state_project IM final_descriptors
      final_state =
    finite_trace_last
      (equivocators_state_project IM
         initial_descriptors is) trX'
    ∧ finite_valid_trace X'
        (equivocators_state_project IM
           initial_descriptors is) trX' subst foldx.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (None, final_descriptors')
Hitemx: True
Hx: equivocators_state_project IM final_descriptors s =
equivocators_state_project IM final_descriptors'
  (finite_trace_last is tr')
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
proper_fixed_equivocator_descriptors
  initial_descriptors is
∧ foldr (equivocators_trace_project_folder IM)
    (Some ([], final_descriptors')) tr' =
  Some (trX', initial_descriptors)
  ∧ equivocators_state_project IM final_descriptors
      final_state =
    finite_trace_last
      (equivocators_state_project IM
         initial_descriptors is) trX'
    ∧ finite_valid_trace X'
        (equivocators_state_project IM
           initial_descriptors is) trX' split; [done |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (None, final_descriptors')
Hitemx: True
Hx: equivocators_state_project IM final_descriptors s =
equivocators_state_project IM final_descriptors'
  (finite_trace_last is tr')
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
foldr (equivocators_trace_project_folder IM)
  (Some ([], final_descriptors')) tr' =
Some (trX', initial_descriptors)
∧ equivocators_state_project IM final_descriptors
    final_state =
  finite_trace_last
    (equivocators_state_project IM initial_descriptors
       is) trX'
  ∧ finite_valid_trace X'
      (equivocators_state_project IM
         initial_descriptors is) trX'
      split; [by apply Htr_project |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (None, final_descriptors')
Hitemx: True
Hx: equivocators_state_project IM final_descriptors s =
equivocators_state_project IM final_descriptors'
  (finite_trace_last is tr')
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
equivocators_state_project IM final_descriptors
  final_state =
finite_trace_last
  (equivocators_state_project IM initial_descriptors
     is) trX'
∧ finite_valid_trace X'
    (equivocators_state_project IM initial_descriptors
       is) trX'
      split; [| done].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is tr: composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (None, final_descriptors')
Hitemx: True
Hx: equivocators_state_project IM final_descriptors s =
equivocators_state_project IM final_descriptors'
  (finite_trace_last is tr')
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
equivocators_state_project IM final_descriptors
  final_state =
finite_trace_last
  (equivocators_state_project IM initial_descriptors
     is) trX'
      subst tr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
IHlen: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM,
            proper_fixed_equivocator_descriptors
              final_descriptors
              (finite_trace_last is tr)
            → ∃ (trX : 
                 list
                   (composite_transition_item
                      IM)) 
                (initial_descriptors : 
                 equivocator_descriptors IM),
                proper_fixed_equivocator_descriptors
                  initial_descriptors is
                ∧ equivocators_trace_project
                    IM final_descriptors tr =
                  Some
                    (trX, initial_descriptors)
                  ∧ equivocators_state_project
                      IM final_descriptors
                      (finite_trace_last is tr) =
                    finite_trace_last
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
                    ∧ finite_valid_trace X'
                      (equivocators_state_project
                      IM initial_descriptors
                      is) trX
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): composite_state (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (None, final_descriptors')
Hitemx: True
Hx: equivocators_state_project IM final_descriptors s =
equivocators_state_project IM final_descriptors'
  (finite_trace_last is tr')
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_constrained_trace FreeE is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
equivocators_state_project IM final_descriptors
  final_state =
finite_trace_last
  (equivocators_state_project IM initial_descriptors
     is) trX' clear -Hstate_project Hx.message, index: Type
EqDecision2: EqDecision index
IM: index → VLSM message
is: composite_state (equivocator_IM IM)
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): composite_state (equivocator_IM IM)
final_descriptors, final_descriptors': equivocator_descriptors IM
Hx: equivocators_state_project IM final_descriptors s =
equivocators_state_project IM final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
equivocators_state_project IM final_descriptors
  final_state =
finite_trace_last
  (equivocators_state_project IM initial_descriptors
     is) trX'
      unfold final_state.message, index: Type
EqDecision2: EqDecision index
IM: index → VLSM message
is: composite_state (equivocator_IM IM)
tr': list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
iom: option message
s: composite_state (equivocator_IM IM)
oom: option message
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): composite_state (equivocator_IM IM)
final_descriptors, final_descriptors': equivocator_descriptors IM
Hx: equivocators_state_project IM final_descriptors s =
equivocators_state_project IM final_descriptors'
  (finite_trace_last is tr')
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
equivocators_state_project IM final_descriptors
  (finite_trace_last is
     (tr' ++
      [{|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |}])) =
finite_trace_last
  (equivocators_state_project IM initial_descriptors
     is) trX'
      by rewrite <- Hstate_project, <- Hx, finite_trace_last_is_last.
Qed.

Instantiating the lemma above with the free constraint.

Lemma free_equivocators_valid_trace_project
  (final_descriptors : equivocator_descriptors IM)
  (is : composite_state (equivocator_IM IM))
  (tr : list (composite_transition_item (equivocator_IM IM)))
  (final_state := @finite_trace_last _ XE is tr)
  (Hproper : proper_fixed_equivocator_descriptors final_descriptors final_state)
  (Htr : finite_valid_trace XE is tr)
  : exists
    (trX : list (composite_transition_item IM))
    (initial_descriptors : equivocator_descriptors IM)
    (isX := equivocators_state_project IM initial_descriptors is)
    (final_stateX := finite_trace_last isX trX),
    proper_fixed_equivocator_descriptors initial_descriptors is /\
    equivocators_trace_project IM final_descriptors tr = Some (trX, initial_descriptors) /\
    equivocators_state_project IM final_descriptors final_state = final_stateX /\
    finite_valid_trace (free_composite_vlsm IM) isX trX.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
final_descriptors: equivocator_descriptors IM
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state XE
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr: finite_valid_trace XE is tr
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace (free_composite_vlsm IM)
          isX trX
Proof.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
final_descriptors: equivocator_descriptors IM
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state XE
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr: finite_valid_trace XE is tr
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace (free_composite_vlsm IM)
          isX trX
  destruct (_equivocators_valid_trace_project final_descriptors is tr Hproper Htr
    (free_constraint IM)) as (trX & idesc & Hex); [done.. |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
final_descriptors: equivocator_descriptors IM
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state XE
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr: finite_valid_trace XE is tr
trX: list (composite_transition_item IM)
idesc: equivocator_descriptors IM
Hex: let isX :=
  equivocators_state_project IM idesc is in
let final_stateX := finite_trace_last isX trX in
proper_fixed_equivocator_descriptors idesc is
∧ equivocators_trace_project IM
    final_descriptors tr = 
  Some (trX, idesc)
  ∧ equivocators_state_project IM
      final_descriptors
      (finite_trace_last is tr) = final_stateX
    ∧ finite_valid_trace
        (composite_vlsm IM (free_constraint IM))
        isX trX
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace (free_composite_vlsm IM)
          isX trX
  exists trX, idesc.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
final_descriptors: equivocator_descriptors IM
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state XE
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr: finite_valid_trace XE is tr
trX: list (composite_transition_item IM)
idesc: equivocator_descriptors IM
Hex: let isX :=
  equivocators_state_project IM idesc is in
let final_stateX := finite_trace_last isX trX in
proper_fixed_equivocator_descriptors idesc is
∧ equivocators_trace_project IM
    final_descriptors tr = 
  Some (trX, idesc)
  ∧ equivocators_state_project IM
      final_descriptors
      (finite_trace_last is tr) = final_stateX
    ∧ finite_valid_trace
        (composite_vlsm IM (free_constraint IM))
        isX trX
let isX := equivocators_state_project IM idesc is in
let final_stateX := finite_trace_last isX trX in
proper_fixed_equivocator_descriptors idesc is
∧ equivocators_trace_project IM final_descriptors tr =
  Some (trX, idesc)
  ∧ equivocators_state_project IM final_descriptors
      final_state = final_stateX
    ∧ finite_valid_trace (free_composite_vlsm IM) isX
        trX
  split_and!; [by itauto.. |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
final_descriptors: equivocator_descriptors IM
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state XE
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr: finite_valid_trace XE is tr
trX: list (composite_transition_item IM)
idesc: equivocator_descriptors IM
Hex: let isX :=
  equivocators_state_project IM idesc is in
let final_stateX := finite_trace_last isX trX in
proper_fixed_equivocator_descriptors idesc is
∧ equivocators_trace_project IM
    final_descriptors tr = 
  Some (trX, idesc)
  ∧ equivocators_state_project IM
      final_descriptors
      (finite_trace_last is tr) = final_stateX
    ∧ finite_valid_trace
        (composite_vlsm IM (free_constraint IM))
        isX trX
finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM idesc is) trX
  apply (@VLSM_incl_finite_valid_trace _ _ (composite_vlsm IM (free_constraint IM))); [| by itauto].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
final_descriptors: equivocator_descriptors IM
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state XE
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr: finite_valid_trace XE is tr
trX: list (composite_transition_item IM)
idesc: equivocator_descriptors IM
Hex: let isX :=
  equivocators_state_project IM idesc is in
let final_stateX := finite_trace_last isX trX in
proper_fixed_equivocator_descriptors idesc is
∧ equivocators_trace_project IM
    final_descriptors tr = 
  Some (trX, idesc)
  ∧ equivocators_state_project IM
      final_descriptors
      (finite_trace_last is tr) = final_stateX
    ∧ finite_valid_trace
        (composite_vlsm IM (free_constraint IM))
        isX trX
VLSM_incl_part
  (composite_vlsm IM (free_constraint IM))
  (free_composite_vlsm_machine IM)
  by apply free_composite_vlsm_spec.
Qed.

A message sent by a non-state-equivocating machine can be directly observed in any projection of the final state.

Lemma not_equivocating_sent_message_has_been_directly_observed_in_projection
  (is : state XE)
  (tr : list (composite_transition_item (equivocator_IM IM)))
  (Htr : finite_valid_trace XE is tr)
  (lst := finite_trace_last is tr)
  (item : transition_item)
  (Hitem : item ∈ tr)
  (Hitem_not_equiv : projT1 (l item) ∉ equivocating)
  (m : message)
  (Hm : field_selector output m item)
  (descriptors : equivocator_descriptors IM)
  (Hdescriptors : proper_fixed_equivocator_descriptors descriptors lst)
  : has_been_directly_observed Free (equivocators_state_project IM descriptors lst) m.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
lst:= finite_trace_last is tr: state XE
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors lst
has_been_directly_observed Free
  (equivocators_state_project IM descriptors lst) m
Proof.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
lst:= finite_trace_last is tr: state XE
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors lst
has_been_directly_observed Free
  (equivocators_state_project IM descriptors lst) m
  destruct (free_equivocators_valid_trace_project descriptors is tr Hdescriptors Htr)
    as [trX [initial_descriptors [_ [Htr_project [Hfinal_state HtrX_Free]]]]].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
lst:= finite_trace_last is tr: state XE
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors lst
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
has_been_directly_observed Free
  (equivocators_state_project IM descriptors lst) m
  subst lst.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
has_been_directly_observed Free
  (equivocators_state_project IM descriptors
     (finite_trace_last is tr)) m
  rewrite Hfinal_state.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
has_been_directly_observed Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) m

  assert (Htr_Pre : finite_constrained_trace FreeE is tr).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
finite_constrained_trace FreeE is tr
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is tr
has_been_directly_observed Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) m
  {message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
finite_constrained_trace FreeE is tr
    apply VLSM_incl_finite_valid_trace; [| done].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
VLSM_incl_part
  (constrained_vlsm_machine
     (free_composite_vlsm (equivocator_IM IM))
     (equivocators_fixed_equivocations_constraint IM
        (elements equivocating)))
  (preloaded_vlsm_machine FreeE (λ _ : message, True))
    by apply constrained_preloaded_incl.
  }message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is tr
has_been_directly_observed Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) m

  assert (HtrX_Pre : finite_constrained_trace Free
    (equivocators_state_project IM initial_descriptors is) trX).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is tr
finite_constrained_trace Free
  (equivocators_state_project IM initial_descriptors
     is) trX
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is tr
HtrX_Pre: finite_constrained_trace Free
  (equivocators_state_project IM
     initial_descriptors is) trX
has_been_directly_observed Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) m
  {message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is tr
finite_constrained_trace Free
  (equivocators_state_project IM initial_descriptors
     is) trX
    revert HtrX_Free; apply VLSM_incl_finite_valid_trace.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is tr
VLSM_incl_part (free_composite_vlsm_machine IM)
  (preloaded_vlsm_machine Free (λ _ : message, True))
    by apply vlsm_incl_preloaded_with_all_messages_vlsm.
  }message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is tr
HtrX_Pre: finite_constrained_trace Free
  (equivocators_state_project IM
     initial_descriptors is) trX
has_been_directly_observed Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) m

  assert (Hlst_preX : constrained_state_prop Free
    (finite_trace_last (equivocators_state_project IM initial_descriptors is) trX)).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is tr
HtrX_Pre: finite_constrained_trace Free
  (equivocators_state_project IM
     initial_descriptors is) trX
constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is tr
HtrX_Pre: finite_constrained_trace Free
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
has_been_directly_observed Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) m
  {message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is tr
HtrX_Pre: finite_constrained_trace Free
  (equivocators_state_project IM
     initial_descriptors is) trX
constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
    apply (finite_valid_trace_last_pstate (preloaded_with_all_messages_vlsm Free)).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is tr
HtrX_Pre: finite_constrained_trace Free
  (equivocators_state_project IM
     initial_descriptors is) trX
finite_valid_trace_from
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM initial_descriptors
     is) trX
    by apply HtrX_Pre.
  }message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is tr
HtrX_Pre: finite_constrained_trace Free
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
has_been_directly_observed Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) m
  rewrite (has_been_directly_observed_sent_received_iff Free) by done.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is tr
HtrX_Pre: finite_constrained_trace Free
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
has_been_received Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) m
∨ has_been_sent Free
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX) m
  right.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is tr
HtrX_Pre: finite_constrained_trace Free
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
has_been_sent Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) m apply proper_sent; [done |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is tr
HtrX_Pre: finite_constrained_trace Free
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
selected_message_exists_in_all_preloaded_traces Free
  (field_selector output)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) m
  apply has_been_sent_consistency; [by typeclasses eauto | done |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is tr
HtrX_Pre: finite_constrained_trace Free
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
selected_message_exists_in_some_preloaded_traces Free
  (field_selector output)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) m
  red in HtrX_Pre.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is tr
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
selected_message_exists_in_some_preloaded_traces Free
  (field_selector output)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) m
  exists (equivocators_state_project IM initial_descriptors is), trX,
    (valid_trace_add_default_last HtrX_Pre).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is tr
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
trace_has_message (field_selector output) m trX

  apply elem_of_list_split in Hitem.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: ∃ l1 l2 : list transition_item,
  tr = l1 ++ item :: l2
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is tr
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
trace_has_message (field_selector output) m trX
  destruct Hitem as [pre [suf Hitem]].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
pre, suf: list transition_item
Hitem: tr = pre ++ item :: suf
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is tr
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
trace_has_message (field_selector output) m trX
  change (item :: suf) with ([item] ++ suf) in Hitem.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
pre, suf: list transition_item
Hitem: tr = pre ++ [item] ++ suf
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is tr
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
trace_has_message (field_selector output) m trX
  subst tr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
trace_has_message (field_selector output) m trX

  assert (Hsingleton_d_item :
    is_singleton_state (IM (projT1 (VLSM.l item))) (destination item (projT1 (VLSM.l item)))).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
is_singleton_state (IM (projT1 (l item)))
  (destination item (projT1 (l item)))
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
trace_has_message (field_selector output) m trX
  {message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
is_singleton_state (IM (projT1 (l item)))
  (destination item (projT1 (l item)))
    apply (not_equivocating_index_has_singleton_state IM (elements equivocating));
      [| by rewrite elem_of_elements].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
valid_state_prop
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating)) (destination item)
    apply proj1 in Htr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace_from XE is
  (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
valid_state_prop
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating)) (destination item)
    rewrite app_assoc in Htr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
valid_state_prop
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating)) (destination item)
    apply finite_valid_trace_from_app_iff in Htr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace_from XE is (pre ++ [item])
∧ finite_valid_trace_from XE
    (finite_trace_last is (pre ++ [item])) suf
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
valid_state_prop
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating)) (destination item)
    destruct Htr as [Htr _].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace_from XE is (pre ++ [item])
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
valid_state_prop
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating)) (destination item)
    apply finite_valid_trace_last_pstate in Htr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
item: transition_item
pre, suf: list transition_item
Htr: valid_state_prop XE
  (finite_trace_last is (pre ++ [item]))
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
valid_state_prop
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating)) (destination item)
    by rewrite finite_trace_last_is_last in Htr.
  }message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
trace_has_message (field_selector output) m trX

  apply equivocators_trace_project_app_iff in Htr_project.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  equivocators_trace_project IM
    descriptors ([item] ++ suf) =
  Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project IM
      eqv_descriptors' pre =
    Some (preX, initial_descriptors)
    ∧ trX = preX ++ sufX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
trace_has_message (field_selector output) m trX
  destruct Htr_project as [preX' [sufX' [eqv_descriptors' [Hpr_suf [Hpr_pre HpreX]]]]].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
preX', sufX': list (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM descriptors
  ([item] ++ suf) =
Some (sufX', eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HpreX: trX = preX' ++ sufX'
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
trace_has_message (field_selector output) m trX
  subst trX.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', sufX': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
eqv_descriptors': equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM descriptors
  ([item] ++ suf) =
Some (sufX', eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
Htr_Pre: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ sufX'))
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
trace_has_message (field_selector output) m
  (preX' ++ sufX') apply Exists_app.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', sufX': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
eqv_descriptors': equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM descriptors
  ([item] ++ suf) =
Some (sufX', eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
Htr_Pre: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ sufX'))
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Exists (field_selector output m) preX'
∨ Exists (field_selector output m) sufX' right.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', sufX': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
eqv_descriptors': equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM descriptors
  ([item] ++ suf) =
Some (sufX', eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
Htr_Pre: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ sufX'))
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Exists (field_selector output m) sufX'

  apply proj1 in Htr_Pre.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', sufX': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
eqv_descriptors': equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM descriptors
  ([item] ++ suf) =
Some (sufX', eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
Htr_Pre: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ sufX'))
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Exists (field_selector output m) sufX'
  apply finite_valid_trace_from_app_iff in Htr_Pre.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', sufX': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
eqv_descriptors': equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM descriptors
  ([item] ++ suf) =
Some (sufX', eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
Htr_Pre: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
∧ finite_valid_trace_from
    (preloaded_with_all_messages_vlsm FreeE)
    (finite_trace_last is pre)
    ([item] ++ suf)
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ sufX'))
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Exists (field_selector output m) sufX'
  destruct Htr_Pre as [Hpre'_free Hsuf'_free].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', sufX': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
eqv_descriptors': equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM descriptors
  ([item] ++ suf) =
Some (sufX', eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([item] ++ suf)
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ sufX'))
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Exists (field_selector output m) sufX'

  apply equivocators_trace_project_app_iff in Hpr_suf.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', sufX': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
eqv_descriptors': equivocator_descriptors IM
Hpr_suf: ∃ (preX
   sufX : list
            (composite_transition_item IM)) 
  (eqv_descriptors'0 : 
   equivocator_descriptors IM),
  equivocators_trace_project IM descriptors
    suf = Some (sufX, eqv_descriptors'0)
  ∧ equivocators_trace_project IM
      eqv_descriptors'0 [item] =
    Some (preX, eqv_descriptors')
    ∧ sufX' = preX ++ sufX
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([item] ++ suf)
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ sufX'))
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Exists (field_selector output m) sufX'
  destruct Hpr_suf as [pre_itemX [sufX'' [eqv_descriptors'' [Hpr_suf' [Hpr_pre_item HsufX']]]]].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', sufX': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
eqv_descriptors': equivocator_descriptors IM
pre_itemX, sufX'': list (composite_transition_item IM)
eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre_item: equivocators_trace_project IM
  eqv_descriptors'' [item] =
Some (pre_itemX, eqv_descriptors')
HsufX': sufX' = pre_itemX ++ sufX''
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([item] ++ suf)
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ sufX'))
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Exists (field_selector output m) sufX'
  subst sufX'.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre_item: equivocators_trace_project IM
  eqv_descriptors'' [item] =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([item] ++ suf)
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Exists (field_selector output m) (pre_itemX ++ sufX'') apply Exists_app.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre_item: equivocators_trace_project IM
  eqv_descriptors'' [item] =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([item] ++ suf)
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Exists (field_selector output m) pre_itemX
∨ Exists (field_selector output m) sufX'' left.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre_item: equivocators_trace_project IM
  eqv_descriptors'' [item] =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([item] ++ suf)
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Exists (field_selector output m) pre_itemX

  apply finite_valid_trace_from_app_iff in Hsuf'_free.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre_item: equivocators_trace_project IM
  eqv_descriptors'' [item] =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre) [item]
∧ finite_valid_trace_from
    (preloaded_with_all_messages_vlsm
       FreeE)
    (finite_trace_last
       (finite_trace_last is pre) [item])
    suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Exists (field_selector output m) pre_itemX
  destruct Hsuf'_free as [Hpre_item_free Hsuf'_free].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre_item: equivocators_trace_project IM
  eqv_descriptors'' [item] =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hpre_item_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last is pre) [item]
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last
     (finite_trace_last is pre) [item])
  suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Exists (field_selector output m) pre_itemX
  assert (Heqv_descriptors'' : forall i, i ∉ equivocating -> eqv_descriptors'' i = Existing 0).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre_item: equivocators_trace_project IM
  eqv_descriptors'' [item] =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hpre_item_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last is pre) [item]
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last
     (finite_trace_last is pre) [item])
  suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
∀ i : index,
  i ∉ equivocating → eqv_descriptors'' i = Existing 0
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre_item: equivocators_trace_project IM
  eqv_descriptors'' [item] =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hpre_item_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last is pre) [item]
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last
     (finite_trace_last is pre) [item])
  suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Heqv_descriptors'': ∀ i : index,
  i ∉ equivocating
  → eqv_descriptors'' i =
    Existing 0
Exists (field_selector output m) pre_itemX
  {message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre_item: equivocators_trace_project IM
  eqv_descriptors'' [item] =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hpre_item_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last is pre) [item]
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last
     (finite_trace_last is pre) [item])
  suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
∀ i : index,
  i ∉ equivocating → eqv_descriptors'' i = Existing 0
    by intros i Hi; eapply equivocators_trace_project_preserves_zero_descriptors, Hdescriptors.
  }message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre_item: equivocators_trace_project IM
  eqv_descriptors'' [item] =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hpre_item_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last is pre) [item]
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last
     (finite_trace_last is pre) [item])
  suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Heqv_descriptors'': ∀ i : index,
  i ∉ equivocating
  → eqv_descriptors'' i =
    Existing 0
Exists (field_selector output m) pre_itemX

  specialize (Heqv_descriptors'' _ Hitem_not_equiv).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre_item: equivocators_trace_project IM
  eqv_descriptors'' [item] =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hpre_item_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last is pre) [item]
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last
     (finite_trace_last is pre) [item])
  suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Heqv_descriptors'': eqv_descriptors''
  (projT1 (l item)) = Existing 0
Exists (field_selector output m) pre_itemX
  simpl in *.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ item :: suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: output item = Some m
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ item :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ item :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre_item: match
  equivocators_transition_item_project
    IM eqv_descriptors'' item
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) =>
    Some ([], odescriptor)
| None => None
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hpre_item_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last is pre) [item]
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last
     (finite_trace_last is pre) [item])
  suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Heqv_descriptors'': eqv_descriptors''
  (projT1 (l item)) = Existing 0
Exists (field_selector output m) pre_itemX
  destruct (equivocators_transition_item_project IM eqv_descriptors'' item)
    as [(o, odescriptor) |] eqn: Hpr
  ; [| by congruence].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ item :: suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: output item = Some m
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ item :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ item :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: equivocators_transition_item_project IM
  eqv_descriptors'' item = 
Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hpre_item_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last is pre) [item]
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last
     (finite_trace_last is pre) [item])
  suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Heqv_descriptors'': eqv_descriptors''
  (projT1 (l item)) = Existing 0
Exists (field_selector output m) pre_itemX
  destruct item.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
l: label (composite_type (equivocator_IM IM))
input: option message
destination: state
  (composite_type (equivocator_IM IM))
output: option message
pre, suf: list transition_item
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := l;
     input := input;
     destination := destination;
     output := output
   |} :: suf)
Hitem_not_equiv: projT1
  (VLSM.l
     {|
       l := l;
       input := input;
       destination := destination;
       output := output
     |}) ∉ equivocating
m: message
Hm: VLSM.output
  {|
    l := l;
    input := input;
    destination := destination;
    output := output
  |} = Some m
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := input;
        destination := destination;
        output := output
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := input;
        destination := destination;
        output := output
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: equivocators_transition_item_project IM
  eqv_descriptors''
  {|
    l := l;
    input := input;
    destination := destination;
    output := output
  |} = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hpre_item_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last is pre)
  [{|
     l := l;
     input := input;
     destination := destination;
     output := output
   |}]
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := l;
        input := input;
        destination := destination;
        output := output
      |}]) suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state
  (IM
     (projT1
        (VLSM.l
           {|
             l := l;
             input := input;
             destination :=
               destination;
             output := output
           |})))
  (VLSM.destination
     {|
       l := l;
       input := input;
       destination := destination;
       output := output
     |}
     (projT1
        (VLSM.l
           {|
             l := l;
             input := input;
             destination :=
               destination;
             output := output
           |})))
Heqv_descriptors'': eqv_descriptors''
  (projT1
     (VLSM.l
        {|
          l := l;
          input := input;
          destination :=
            destination;
          output := output
        |})) = Existing 0
Exists (field_selector VLSM.output m) pre_itemX simpl in *.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
l: composite_label (equivocator_IM IM)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
pre, suf: list transition_item
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := l;
     input := input;
     destination := destination;
     output := output
   |} :: suf)
Hitem_not_equiv: projT1 l ∉ equivocating
m: message
Hm: output = Some m
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := input;
        destination := destination;
        output := output
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := input;
        destination := destination;
        output := output
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: equivocators_transition_item_project IM
  eqv_descriptors''
  {|
    l := l;
    input := input;
    destination := destination;
    output := output
  |} = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hpre_item_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last is pre)
  [{|
     l := l;
     input := input;
     destination := destination;
     output := output
   |}]
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := l;
        input := input;
        destination := destination;
        output := output
      |}]) suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM (projT1 l))
  (destination (projT1 l))
Heqv_descriptors'': eqv_descriptors'' (projT1 l) =
Existing 0
Exists (field_selector VLSM.output m) pre_itemX
  apply first_transition_valid in Hpre_item_free.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
l: composite_label (equivocator_IM IM)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
pre, suf: list transition_item
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := l;
     input := input;
     destination := destination;
     output := output
   |} :: suf)
Hitem_not_equiv: projT1 l ∉ equivocating
m: message
Hm: output = Some m
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := input;
        destination := destination;
        output := output
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := input;
        destination := destination;
        output := output
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: equivocators_transition_item_project IM
  eqv_descriptors''
  {|
    l := l;
    input := input;
    destination := destination;
    output := output
  |} = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hpre_item_free: input_valid_transition
  (preloaded_with_all_messages_vlsm
     FreeE)
  (VLSM.l
     {|
       l := l;
       input := input;
       destination := destination;
       output := output
     |})
  (finite_trace_last is pre,
   VLSM.input
     {|
       l := l;
       input := input;
       destination := destination;
       output := output
     |})
  (VLSM.destination
     {|
       l := l;
       input := input;
       destination := destination;
       output := output
     |},
   VLSM.output
     {|
       l := l;
       input := input;
       destination := destination;
       output := output
     |})
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := l;
        input := input;
        destination := destination;
        output := output
      |}]) suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM (projT1 l))
  (destination (projT1 l))
Heqv_descriptors'': eqv_descriptors'' (projT1 l) =
Existing 0
Exists (field_selector VLSM.output m) pre_itemX simpl in Hpre_item_free.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
l: composite_label (equivocator_IM IM)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
pre, suf: list transition_item
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := l;
     input := input;
     destination := destination;
     output := output
   |} :: suf)
Hitem_not_equiv: projT1 l ∉ equivocating
m: message
Hm: output = Some m
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := input;
        destination := destination;
        output := output
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := input;
        destination := destination;
        output := output
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: equivocators_transition_item_project IM
  eqv_descriptors''
  {|
    l := l;
    input := input;
    destination := destination;
    output := output
  |} = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hpre_item_free: input_valid_transition
  (preloaded_with_all_messages_vlsm
     FreeE) l
  (finite_trace_last is pre, input)
  (destination, output)
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := l;
        input := input;
        destination := destination;
        output := output
      |}]) suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM (projT1 l))
  (destination (projT1 l))
Heqv_descriptors'': eqv_descriptors'' (projT1 l) =
Existing 0
Exists (field_selector VLSM.output m) pre_itemX
  destruct Hpre_item_free as [[_ [_ Hv]] Ht].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
l: composite_label (equivocator_IM IM)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
pre, suf: list transition_item
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := l;
     input := input;
     destination := destination;
     output := output
   |} :: suf)
Hitem_not_equiv: projT1 l ∉ equivocating
m: message
Hm: output = Some m
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := input;
        destination := destination;
        output := output
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := input;
        destination := destination;
        output := output
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: equivocators_transition_item_project IM
  eqv_descriptors''
  {|
    l := l;
    input := input;
    destination := destination;
    output := output
  |} = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hv: valid l (finite_trace_last is pre, input)
Ht: transition l (finite_trace_last is pre, input) =
(destination, output)
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := l;
        input := input;
        destination := destination;
        output := output
      |}]) suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM (projT1 l))
  (destination (projT1 l))
Heqv_descriptors'': eqv_descriptors'' (projT1 l) =
Existing 0
Exists (field_selector VLSM.output m) pre_itemX
  destruct l as [x l].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
x: index
l: label (equivocator_IM IM x)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
pre, suf: list transition_item
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x l;
     input := input;
     destination := destination;
     output := output
   |} :: suf)
Hitem_not_equiv: projT1 (existT x l) ∉ equivocating
m: message
Hm: output = Some m
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x l;
        input := input;
        destination := destination;
        output := output
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x l;
        input := input;
        destination := destination;
        output := output
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: equivocators_transition_item_project IM
  eqv_descriptors''
  {|
    l := existT x l;
    input := input;
    destination := destination;
    output := output
  |} = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hv: valid (existT x l)
  (finite_trace_last is pre, input)
Ht: transition (existT x l)
  (finite_trace_last is pre, input) =
(destination, output)
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := existT x l;
        input := input;
        destination := destination;
        output := output
      |}]) suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (existT x l)))
  (destination
     (projT1 (existT x l)))
Heqv_descriptors'': eqv_descriptors''
  (projT1 (existT x l)) =
Existing 0
Exists (field_selector VLSM.output m) pre_itemX
  cbn in *.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
x: index
l: EquivocatorLabel (IM x)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
pre, suf: list transition_item
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x l;
     input := input;
     destination := destination;
     output := output
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
m: message
Hm: output = Some m
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x l;
        input := input;
        destination := destination;
        output := output
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x l;
        input := input;
        destination := destination;
        output := output
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: equivocators_transition_item_project IM
  eqv_descriptors''
  {|
    l := existT x l;
    input := input;
    destination := destination;
    output := output
  |} = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hv: equivocator_valid (IM x) l
  (finite_trace_last is pre x, input)
Ht: (let (si', om') :=
   equivocator_transition 
     (IM x) l (finite_trace_last is pre x, input) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is pre) x si', om')) =
(destination, output)
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  destination suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x)
  (destination x)
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector VLSM.output m) pre_itemX
  destruct (equivocator_transition (IM x) l _) as [si' om'] eqn: Hti.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
x: index
l: EquivocatorLabel (IM x)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
pre, suf: list transition_item
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x l;
     input := input;
     destination := destination;
     output := output
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
m: message
Hm: output = Some m
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x l;
        input := input;
        destination := destination;
        output := output
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x l;
        input := input;
        destination := destination;
        output := output
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: equivocators_transition_item_project IM
  eqv_descriptors''
  {|
    l := existT x l;
    input := input;
    destination := destination;
    output := output
  |} = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hv: equivocator_valid (IM x) l
  (finite_trace_last is pre x, input)
si': equivocator_state (IM x)
om': option message
Hti: equivocator_transition 
  (IM x) l (finite_trace_last is pre x, input) =
(si', om')
Ht: (state_update (equivocator_IM IM)
   (finite_trace_last is pre) x si', om') =
(destination, output)
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  destination suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x)
  (destination x)
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector VLSM.output m) pre_itemX
  inversion Ht; subst; clear Ht.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
x: index
l: EquivocatorLabel (IM x)
input: option message
pre, suf: list transition_item
m: message
si': equivocator_state (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x l;
     input := input;
     destination :=
       state_update (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x l;
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x l;
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: equivocators_transition_item_project IM
  eqv_descriptors''
  {|
    l := existT x l;
    input := input;
    destination :=
      state_update (equivocator_IM IM)
        (finite_trace_last is pre) x si';
    output := Some m
  |} = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hv: equivocator_valid (IM x) l
  (finite_trace_last is pre x, input)
Hti: equivocator_transition 
  (IM x) l (finite_trace_last is pre x, input) =
(si', Some m)
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (state_update (equivocator_IM IM)
     (finite_trace_last is pre) x si')
  suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x)
  (state_update
     (equivocator_IM IM)
     (finite_trace_last is pre) x
     si' x)
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m) pre_itemX
  state_update_simpl.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
x: index
l: EquivocatorLabel (IM x)
input: option message
pre, suf: list transition_item
m: message
si': equivocator_state (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x l;
     input := input;
     destination :=
       state_update (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x l;
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x l;
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: equivocators_transition_item_project IM
  eqv_descriptors''
  {|
    l := existT x l;
    input := input;
    destination :=
      state_update (equivocator_IM IM)
        (finite_trace_last is pre) x si';
    output := Some m
  |} = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hv: equivocator_valid (IM x) l
  (finite_trace_last is pre x, input)
Hti: equivocator_transition 
  (IM x) l (finite_trace_last is pre x, input) =
(si', Some m)
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (state_update (equivocator_IM IM)
     (finite_trace_last is pre) x si')
  suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m) pre_itemX
  destruct (equivocator_transition_no_equivocation_zero_descriptor
    (IM x) _ _ _ _ _ Hv Hti Hsingleton_d_item) as [li Hsndv].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
x: index
l: EquivocatorLabel (IM x)
input: option message
pre, suf: list transition_item
m: message
si': equivocator_state (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x l;
     input := input;
     destination :=
       state_update (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x l;
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x l;
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: equivocators_transition_item_project IM
  eqv_descriptors''
  {|
    l := existT x l;
    input := input;
    destination :=
      state_update (equivocator_IM IM)
        (finite_trace_last is pre) x si';
    output := Some m
  |} = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hv: equivocator_valid (IM x) l
  (finite_trace_last is pre x, input)
Hti: equivocator_transition 
  (IM x) l (finite_trace_last is pre x, input) =
(si', Some m)
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (state_update (equivocator_IM IM)
     (finite_trace_last is pre) x si')
  suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
li: label (IM x)
Hsndv: l = ContinueWith 0 li
Exists (field_selector output m) pre_itemX
  unfold equivocators_transition_item_project in Hpr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
x: index
l: EquivocatorLabel (IM x)
input: option message
pre, suf: list transition_item
m: message
si': equivocator_state (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x l;
     input := input;
     destination :=
       state_update (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x l;
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x l;
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM
       (projT1
          (VLSM.l
             {|
               l := existT x l;
               input := input;
               destination :=
                 state_update
                   (equivocator_IM IM)
                   (finite_trace_last is pre) x
                   si';
               output := Some m
             |})))
    (composite_transition_item_projection
       (equivocator_IM IM)
       {|
         l := existT x l;
         input := input;
         destination :=
           state_update 
             (equivocator_IM IM)
             (finite_trace_last is pre) x si';
         output := Some m
       |})
    (eqv_descriptors''
       (projT1
          (VLSM.l
             {|
               l := existT x l;
               input := input;
               destination :=
                 state_update
                   (equivocator_IM IM)
                   (finite_trace_last is pre) x
                   si';
               output := Some m
             |})))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT
               (projT1
                  (VLSM.l
                     {|
                      l := existT x l;
                      input := input;
                      destination :=
                      state_update
                      (equivocator_IM IM)
                      (finite_trace_last is pre)
                      x si';
                      output := Some m
                     |})) 
               (VLSM.l item');
           input :=
             VLSM.input
               {|
                 l := existT x l;
                 input := input;
                 destination :=
                   state_update
                     (equivocator_IM IM)
                     (finite_trace_last is pre)
                     x si';
                 output := Some m
               |};
           destination :=
             equivocators_state_project IM
               eqv_descriptors''
               (destination
                  {|
                    l := existT x l;
                    input := input;
                    destination :=
                      state_update
                      (equivocator_IM IM)
                      (finite_trace_last is pre)
                      x si';
                    output := Some m
                  |});
           output :=
             output
               {|
                 l := existT x l;
                 input := input;
                 destination :=
                   state_update
                     (equivocator_IM IM)
                     (finite_trace_last is pre)
                     x si';
                 output := Some m
               |}
         |},
       equivocator_descriptors_update IM
         eqv_descriptors''
         (projT1
            (VLSM.l
               {|
                 l := existT x l;
                 input := input;
                 destination :=
                   state_update
                     (equivocator_IM IM)
                     (finite_trace_last is pre)
                     x si';
                 output := Some m
               |})) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         eqv_descriptors''
         (projT1
            (VLSM.l
               {|
                 l := existT x l;
                 input := input;
                 destination :=
                   state_update
                     (equivocator_IM IM)
                     (finite_trace_last is pre)
                     x si';
                 output := Some m
               |})) deqv')
| None => None
end = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hv: equivocator_valid (IM x) l
  (finite_trace_last is pre x, input)
Hti: equivocator_transition 
  (IM x) l (finite_trace_last is pre x, input) =
(si', Some m)
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (state_update (equivocator_IM IM)
     (finite_trace_last is pre) x si')
  suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
li: label (IM x)
Hsndv: l = ContinueWith 0 li
Exists (field_selector output m) pre_itemX
  simpl in Hpr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
x: index
l: EquivocatorLabel (IM x)
input: option message
pre, suf: list transition_item
m: message
si': equivocator_state (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x l;
     input := input;
     destination :=
       state_update (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x l;
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x l;
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM x)
    (composite_transition_item_projection
       (equivocator_IM IM)
       {|
         l := existT x l;
         input := input;
         destination :=
           state_update 
             (equivocator_IM IM)
             (finite_trace_last is pre) x si';
         output := Some m
       |}) (eqv_descriptors'' x)
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l := existT x (VLSM.l item');
           input := input;
           destination :=
             equivocators_state_project IM
               eqv_descriptors''
               (state_update 
                  (equivocator_IM IM)
                  (finite_trace_last is pre) x
                  si');
           output := Some m
         |},
       equivocator_descriptors_update IM
         eqv_descriptors'' x deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         eqv_descriptors'' x deqv')
| None => None
end = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hv: equivocator_valid (IM x) l
  (finite_trace_last is pre x, input)
Hti: equivocator_transition 
  (IM x) l (finite_trace_last is pre x, input) =
(si', Some m)
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (state_update (equivocator_IM IM)
     (finite_trace_last is pre) x si')
  suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
li: label (IM x)
Hsndv: l = ContinueWith 0 li
Exists (field_selector output m) pre_itemX
  subst l.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
x: index
input: option message
pre, suf: list transition_item
m: message
si': equivocator_state (IM x)
li: label (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM x)
    (composite_transition_item_projection
       (equivocator_IM IM)
       {|
         l := existT x (ContinueWith 0 li);
         input := input;
         destination :=
           state_update 
             (equivocator_IM IM)
             (finite_trace_last is pre) x si';
         output := Some m
       |}) (eqv_descriptors'' x)
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l := existT x (l item');
           input := input;
           destination :=
             equivocators_state_project IM
               eqv_descriptors''
               (state_update 
                  (equivocator_IM IM)
                  (finite_trace_last is pre) x
                  si');
           output := Some m
         |},
       equivocator_descriptors_update IM
         eqv_descriptors'' x deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         eqv_descriptors'' x deqv')
| None => None
end = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hti: equivocator_transition 
  (IM x) (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: equivocator_valid (IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (state_update (equivocator_IM IM)
     (finite_trace_last is pre) x si')
  suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m) pre_itemX
  unfold ProjectionTraces.composite_transition_item_projection in Hpr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
x: index
input: option message
pre, suf: list transition_item
m: message
si': equivocator_state (IM x)
li: label (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM x)
    (composite_transition_item_projection_from_eq
       (equivocator_IM IM)
       {|
         l := existT x (ContinueWith 0 li);
         input := input;
         destination :=
           state_update 
             (equivocator_IM IM)
             (finite_trace_last is pre) x si';
         output := Some m
       |}
       (projT1
          (l
             {|
               l := existT x (ContinueWith 0 li);
               input := input;
               destination :=
                 state_update
                   (equivocator_IM IM)
                   (finite_trace_last is pre) x
                   si';
               output := Some m
             |})) eq_refl) 
    (eqv_descriptors'' x)
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l := existT x (l item');
           input := input;
           destination :=
             equivocators_state_project IM
               eqv_descriptors''
               (state_update 
                  (equivocator_IM IM)
                  (finite_trace_last is pre) x
                  si');
           output := Some m
         |},
       equivocator_descriptors_update IM
         eqv_descriptors'' x deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         eqv_descriptors'' x deqv')
| None => None
end = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hti: equivocator_transition 
  (IM x) (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: equivocator_valid (IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (state_update (equivocator_IM IM)
     (finite_trace_last is pre) x si')
  suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m) pre_itemX
  unfold ProjectionTraces.composite_transition_item_projection_from_eq in Hpr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
x: index
input: option message
pre, suf: list transition_item
m: message
si': equivocator_state (IM x)
li: label (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM x)
    {|
      l :=
        eq_rect_r
          (λ n : index,
             label (equivocator_IM IM n))
          (projT2
             (l
                {|
                  l :=
                    existT x (ContinueWith 0 li);
                  input := input;
                  destination :=
                    state_update
                      (equivocator_IM IM)
                      (finite_trace_last is pre)
                      x si';
                  output := Some m
                |})) eq_refl;
      input :=
        VLSM.input
          {|
            l := existT x (ContinueWith 0 li);
            input := input;
            destination :=
              state_update 
                (equivocator_IM IM)
                (finite_trace_last is pre) x si';
            output := Some m
          |};
      destination :=
        destination
          {|
            l := existT x (ContinueWith 0 li);
            input := input;
            destination :=
              state_update 
                (equivocator_IM IM)
                (finite_trace_last is pre) x si';
            output := Some m
          |}
          (projT1
             (l
                {|
                  l :=
                    existT x (ContinueWith 0 li);
                  input := input;
                  destination :=
                    state_update
                      (equivocator_IM IM)
                      (finite_trace_last is pre)
                      x si';
                  output := Some m
                |}));
      output :=
        output
          {|
            l := existT x (ContinueWith 0 li);
            input := input;
            destination :=
              state_update 
                (equivocator_IM IM)
                (finite_trace_last is pre) x si';
            output := Some m
          |}
    |} (eqv_descriptors'' x)
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l := existT x (l item');
           input := input;
           destination :=
             equivocators_state_project IM
               eqv_descriptors''
               (state_update 
                  (equivocator_IM IM)
                  (finite_trace_last is pre) x
                  si');
           output := Some m
         |},
       equivocator_descriptors_update IM
         eqv_descriptors'' x deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         eqv_descriptors'' x deqv')
| None => None
end = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hti: equivocator_transition 
  (IM x) (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: equivocator_valid (IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (state_update (equivocator_IM IM)
     (finite_trace_last is pre) x si')
  suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m) pre_itemX
  simpl in Hpr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
x: index
input: option message
pre, suf: list transition_item
m: message
si': equivocator_state (IM x)
li: label (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM x)
    {|
      l :=
        eq_rect_r
          (λ n : index, EquivocatorLabel (IM n))
          (ContinueWith 0 li) eq_refl;
      input := input;
      destination :=
        state_update (equivocator_IM IM)
          (finite_trace_last is pre) x si' x;
      output := Some m
    |} (eqv_descriptors'' x)
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l := existT x (l item');
           input := input;
           destination :=
             equivocators_state_project IM
               eqv_descriptors''
               (state_update 
                  (equivocator_IM IM)
                  (finite_trace_last is pre) x
                  si');
           output := Some m
         |},
       equivocator_descriptors_update IM
         eqv_descriptors'' x deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         eqv_descriptors'' x deqv')
| None => None
end = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hti: equivocator_transition 
  (IM x) (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: equivocator_valid (IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (state_update (equivocator_IM IM)
     (finite_trace_last is pre) x si')
  suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m) pre_itemX
  unfold eq_rect_r in Hpr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
x: index
input: option message
pre, suf: list transition_item
m: message
si': equivocator_state (IM x)
li: label (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM x)
    {|
      l :=
        eq_rect x
          (λ y : index, EquivocatorLabel (IM y))
          (ContinueWith 0 li) x 
          (eq_sym eq_refl);
      input := input;
      destination :=
        state_update (equivocator_IM IM)
          (finite_trace_last is pre) x si' x;
      output := Some m
    |} (eqv_descriptors'' x)
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l := existT x (l item');
           input := input;
           destination :=
             equivocators_state_project IM
               eqv_descriptors''
               (state_update 
                  (equivocator_IM IM)
                  (finite_trace_last is pre) x
                  si');
           output := Some m
         |},
       equivocator_descriptors_update IM
         eqv_descriptors'' x deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         eqv_descriptors'' x deqv')
| None => None
end = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hti: equivocator_transition 
  (IM x) (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: equivocator_valid (IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (state_update (equivocator_IM IM)
     (finite_trace_last is pre) x si')
  suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m) pre_itemX simpl in Hpr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
x: index
input: option message
pre, suf: list transition_item
m: message
si': equivocator_state (IM x)
li: label (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM x)
    {|
      l := ContinueWith 0 li;
      input := input;
      destination :=
        state_update (equivocator_IM IM)
          (finite_trace_last is pre) x si' x;
      output := Some m
    |} (eqv_descriptors'' x)
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l := existT x (l item');
           input := input;
           destination :=
             equivocators_state_project IM
               eqv_descriptors''
               (state_update 
                  (equivocator_IM IM)
                  (finite_trace_last is pre) x
                  si');
           output := Some m
         |},
       equivocator_descriptors_update IM
         eqv_descriptors'' x deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         eqv_descriptors'' x deqv')
| None => None
end = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hti: equivocator_transition 
  (IM x) (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: equivocator_valid (IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (state_update (equivocator_IM IM)
     (finite_trace_last is pre) x si')
  suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m) pre_itemX
  rewrite Heqv_descriptors'' in Hpr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
x: index
input: option message
pre, suf: list transition_item
m: message
si': equivocator_state (IM x)
li: label (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM x)
    {|
      l := ContinueWith 0 li;
      input := input;
      destination :=
        state_update (equivocator_IM IM)
          (finite_trace_last is pre) x si' x;
      output := Some m
    |} (Existing 0)
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l := existT x (l item');
           input := input;
           destination :=
             equivocators_state_project IM
               eqv_descriptors''
               (state_update 
                  (equivocator_IM IM)
                  (finite_trace_last is pre) x
                  si');
           output := Some m
         |},
       equivocator_descriptors_update IM
         eqv_descriptors'' x deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         eqv_descriptors'' x deqv')
| None => None
end = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hti: equivocator_transition 
  (IM x) (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: equivocator_valid (IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (state_update (equivocator_IM IM)
     (finite_trace_last is pre) x si')
  suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m) pre_itemX
  unfold equivocator_vlsm_transition_item_project in Hpr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
x: index
input: option message
pre, suf: list transition_item
m: message
si': equivocator_state (IM x)
li: label (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: match
  match
    equivocator_state_project
      (state_update (equivocator_IM IM)
         (finite_trace_last is pre) x si' x) 0
  with
  | Some sj =>
      if decide (0 = 0)
      then
       Some
         (Some
            {|
              l := li;
              input := input;
              destination := sj;
              output := Some m
            |}, Existing 0)
      else Some (None, Existing 0)
  | None => None
  end
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l := existT x (l item');
           input := input;
           destination :=
             equivocators_state_project IM
               eqv_descriptors''
               (state_update 
                  (equivocator_IM IM)
                  (finite_trace_last is pre) x
                  si');
           output := Some m
         |},
       equivocator_descriptors_update IM
         eqv_descriptors'' x deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         eqv_descriptors'' x deqv')
| None => None
end = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hti: equivocator_transition 
  (IM x) (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: equivocator_valid (IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (state_update (equivocator_IM IM)
     (finite_trace_last is pre) x si')
  suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m) pre_itemX
  state_update_simpl.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
x: index
input: option message
pre, suf: list transition_item
m: message
si': equivocator_state (IM x)
li: label (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: match
  match equivocator_state_project si' 0 with
  | Some sj =>
      if decide (0 = 0)
      then
       Some
         (Some
            {|
              l := li;
              input := input;
              destination := sj;
              output := Some m
            |}, Existing 0)
      else Some (None, Existing 0)
  | None => None
  end
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l := existT x (l item');
           input := input;
           destination :=
             state_update IM
               (equivocators_state_project IM
                  eqv_descriptors''
                  (finite_trace_last is pre)) x
               match eqv_descriptors'' x with
               | NewMachine sn => sn
               | Existing i =>
                   match
                     equivocator_state_project
                      si' i
                   with
                   | Some si => si
                   | None =>
                      equivocator_state_zero si'
                   end
               end;
           output := Some m
         |},
       equivocator_descriptors_update IM
         eqv_descriptors'' x deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         eqv_descriptors'' x deqv')
| None => None
end = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hti: equivocator_transition 
  (IM x) (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: equivocator_valid (IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (state_update (equivocator_IM IM)
     (finite_trace_last is pre) x si')
  suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m) pre_itemX
  rewrite equivocator_state_project_zero in Hpr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
x: index
input: option message
pre, suf: list transition_item
m: message
si': equivocator_state (IM x)
li: label (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: match
  (if decide (0 = 0)
   then
    Some
      (Some
         {|
           l := li;
           input := input;
           destination :=
             equivocator_state_zero si';
           output := Some m
         |}, Existing 0)
   else Some (None, Existing 0))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l := existT x (l item');
           input := input;
           destination :=
             state_update IM
               (equivocators_state_project IM
                  eqv_descriptors''
                  (finite_trace_last is pre)) x
               match eqv_descriptors'' x with
               | NewMachine sn => sn
               | Existing i =>
                   match
                     equivocator_state_project
                      si' i
                   with
                   | Some si => si
                   | None =>
                      equivocator_state_zero si'
                   end
               end;
           output := Some m
         |},
       equivocator_descriptors_update IM
         eqv_descriptors'' x deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         eqv_descriptors'' x deqv')
| None => None
end = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hti: equivocator_transition 
  (IM x) (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: equivocator_valid (IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (state_update (equivocator_IM IM)
     (finite_trace_last is pre) x si')
  suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m) pre_itemX
  rewrite decide_True in Hpr by done.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
x: index
input: option message
pre, suf: list transition_item
m: message
si': equivocator_state (IM x)
li: label (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: Some
  (Some
     {|
       l :=
         existT x
           (l
              {|
                l := li;
                input := input;
                destination :=
                  equivocator_state_zero si';
                output := Some m
              |});
       input := input;
       destination :=
         state_update IM
           (equivocators_state_project IM
              eqv_descriptors''
              (finite_trace_last is pre)) x
           match eqv_descriptors'' x with
           | NewMachine sn => sn
           | Existing i =>
               match
                 equivocator_state_project si' i
               with
               | Some si => si
               | None =>
                   equivocator_state_zero si'
               end
           end;
       output := Some m
     |},
   equivocator_descriptors_update IM
     eqv_descriptors'' x 
     (Existing 0)) = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hti: equivocator_transition 
  (IM x) (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: equivocator_valid (IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (state_update (equivocator_IM IM)
     (finite_trace_last is pre) x si')
  suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m) pre_itemX
  inversion Hpr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
x: index
input: option message
pre, suf: list transition_item
m: message
si': equivocator_state (IM x)
li: label (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: Some
  (Some
     {|
       l :=
         existT x
           (l
              {|
                l := li;
                input := input;
                destination :=
                  equivocator_state_zero si';
                output := Some m
              |});
       input := input;
       destination :=
         state_update IM
           (equivocators_state_project IM
              eqv_descriptors''
              (finite_trace_last is pre)) x
           match eqv_descriptors'' x with
           | NewMachine sn => sn
           | Existing i =>
               match
                 equivocator_state_project si' i
               with
               | Some si => si
               | None =>
                   equivocator_state_zero si'
               end
           end;
       output := Some m
     |},
   equivocator_descriptors_update IM
     eqv_descriptors'' x 
     (Existing 0)) = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hti: equivocator_transition 
  (IM x) (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: equivocator_valid (IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (state_update (equivocator_IM IM)
     (finite_trace_last is pre) x si')
  suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
H11: Some
  {|
    l := existT x li;
    input := input;
    destination :=
      state_update IM
        (equivocators_state_project IM
           eqv_descriptors''
           (finite_trace_last is pre)) x
        match eqv_descriptors'' x with
        | NewMachine sn => sn
        | Existing i =>
            match
              equivocator_state_project si' i
            with
            | Some si => si
            | None => equivocator_state_zero si'
            end
        end;
    output := Some m
  |} = o
H12: equivocator_descriptors_update IM
  eqv_descriptors'' x (Existing 0) = odescriptor
Exists (field_selector output m) pre_itemX subst.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
x: index
input: option message
pre, suf: list transition_item
m: message
si': equivocator_state (IM x)
li: label (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr: Some
  (Some
     {|
       l :=
         existT x
           (l
              {|
                l := li;
                input := input;
                destination :=
                  equivocator_state_zero si';
                output := Some m
              |});
       input := input;
       destination :=
         state_update IM
           (equivocators_state_project IM
              eqv_descriptors''
              (finite_trace_last is pre)) x
           match eqv_descriptors'' x with
           | NewMachine sn => sn
           | Existing i =>
               match
                 equivocator_state_project si' i
               with
               | Some si => si
               | None =>
                   equivocator_state_zero si'
               end
           end;
       output := Some m
     |},
   equivocator_descriptors_update IM
     eqv_descriptors'' x 
     (Existing 0)) =
Some
  (Some
     {|
       l := existT x li;
       input := input;
       destination :=
         state_update IM
           (equivocators_state_project IM
              eqv_descriptors''
              (finite_trace_last is pre)) x
           match eqv_descriptors'' x with
           | NewMachine sn => sn
           | Existing i =>
               match
                 equivocator_state_project si' i
               with
               | Some si => si
               | None =>
                   equivocator_state_zero si'
               end
           end;
       output := Some m
     |},
   equivocator_descriptors_update IM
     eqv_descriptors'' x 
     (Existing 0))
Hpr_pre_item: Some
  ([{|
      l := existT x li;
      input := input;
      destination :=
        state_update IM
          (equivocators_state_project
             IM eqv_descriptors''
             (finite_trace_last is pre))
          x
          match
            eqv_descriptors'' x
          with
          | NewMachine sn => sn
          | Existing i =>
              match
                equivocator_state_project
                  si' i
              with
              | Some si => si
              | None =>
                  equivocator_state_zero
                    si'
              end
          end;
      output := Some m
    |}],
   equivocator_descriptors_update IM
     eqv_descriptors'' x 
     (Existing 0)) =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hti: equivocator_transition 
  (IM x) (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: equivocator_valid (IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (state_update (equivocator_IM IM)
     (finite_trace_last is pre) x si')
  suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m) pre_itemX clear Hpr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
x: index
input: option message
pre, suf: list transition_item
m: message
si': equivocator_state (IM x)
li: label (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre_item: Some
  ([{|
      l := existT x li;
      input := input;
      destination :=
        state_update IM
          (equivocators_state_project
             IM eqv_descriptors''
             (finite_trace_last is pre))
          x
          match
            eqv_descriptors'' x
          with
          | NewMachine sn => sn
          | Existing i =>
              match
                equivocator_state_project
                  si' i
              with
              | Some si => si
              | None =>
                  equivocator_state_zero
                    si'
              end
          end;
      output := Some m
    |}],
   equivocator_descriptors_update IM
     eqv_descriptors'' x 
     (Existing 0)) =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hti: equivocator_transition 
  (IM x) (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: equivocator_valid (IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (state_update (equivocator_IM IM)
     (finite_trace_last is pre) x si')
  suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m) pre_itemX
  inversion Hpr_pre_item.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
x: index
input: option message
pre, suf: list transition_item
m: message
si': equivocator_state (IM x)
li: label (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre_item: Some
  ([{|
      l := existT x li;
      input := input;
      destination :=
        state_update IM
          (equivocators_state_project
             IM eqv_descriptors''
             (finite_trace_last is pre))
          x
          match
            eqv_descriptors'' x
          with
          | NewMachine sn => sn
          | Existing i =>
              match
                equivocator_state_project
                  si' i
              with
              | Some si => si
              | None =>
                  equivocator_state_zero
                    si'
              end
          end;
      output := Some m
    |}],
   equivocator_descriptors_update IM
     eqv_descriptors'' x 
     (Existing 0)) =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hti: equivocator_transition 
  (IM x) (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: equivocator_valid (IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (state_update (equivocator_IM IM)
     (finite_trace_last is pre) x si')
  suf
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
H11: [{|
   l := existT x li;
   input := input;
   destination :=
     state_update IM
       (equivocators_state_project IM
          eqv_descriptors''
          (finite_trace_last is pre)) x
       match eqv_descriptors'' x with
       | NewMachine sn => sn
       | Existing i =>
           match
             equivocator_state_project si' i
           with
           | Some si => si
           | None => equivocator_state_zero si'
           end
       end;
   output := Some m
 |}] = pre_itemX
H12: equivocator_descriptors_update IM
  eqv_descriptors'' x (Existing 0) =
eqv_descriptors'
Exists (field_selector output m)
  [{|
     l := existT x li;
     input := input;
     destination :=
       state_update IM
         (equivocators_state_project IM
            eqv_descriptors''
            (finite_trace_last is pre)) x
         match eqv_descriptors'' x with
         | NewMachine sn => sn
         | Existing i =>
             match
               equivocator_state_project si' i
             with
             | Some si => si
             | None => equivocator_state_zero si'
             end
         end;
     output := Some m
   |}] subst.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
x: index
input: option message
pre, suf: list transition_item
m: message
si': equivocator_state (IM x)
li: label (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', sufX'': list (composite_transition_item IM)
eqv_descriptors'': equivocator_descriptors IM
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++
   [{|
      l := existT x li;
      input := input;
      destination :=
        state_update IM
          (equivocators_state_project
             IM eqv_descriptors''
             (finite_trace_last is pre))
          x
          match
            eqv_descriptors'' x
          with
          | NewMachine sn => sn
          | Existing i =>
              match
                equivocator_state_project
                  si' i
              with
              | Some si => si
              | None =>
                  equivocator_state_zero
                    si'
              end
          end;
      output := Some m
    |}] ++ sufX'')
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre: equivocators_trace_project IM
  (equivocator_descriptors_update IM
     eqv_descriptors'' x 
     (Existing 0)) pre =
Some (preX', initial_descriptors)
Hpr_pre_item: Some
  ([{|
      l := existT x li;
      input := input;
      destination :=
        state_update IM
          (equivocators_state_project
             IM eqv_descriptors''
             (finite_trace_last is pre))
          x
          match
            eqv_descriptors'' x
          with
          | NewMachine sn => sn
          | Existing i =>
              match
                equivocator_state_project
                  si' i
              with
              | Some si => si
              | None =>
                  equivocator_state_zero
                    si'
              end
          end;
      output := Some m
    |}],
   equivocator_descriptors_update IM
     eqv_descriptors'' x 
     (Existing 0)) =
Some
  ([{|
      l := existT x li;
      input := input;
      destination :=
        state_update IM
          (equivocators_state_project
             IM eqv_descriptors''
             (finite_trace_last is pre))
          x
          match
            eqv_descriptors'' x
          with
          | NewMachine sn => sn
          | Existing i =>
              match
                equivocator_state_project
                  si' i
              with
              | Some si => si
              | None =>
                  equivocator_state_zero
                    si'
              end
          end;
      output := Some m
    |}],
   equivocator_descriptors_update IM
     eqv_descriptors'' x 
     (Existing 0))
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++
   [{|
      l := existT x li;
      input := input;
      destination :=
        state_update IM
          (equivocators_state_project IM
             eqv_descriptors''
             (finite_trace_last is pre)) x
          match eqv_descriptors'' x with
          | NewMachine sn => sn
          | Existing i =>
              match
                equivocator_state_project
                  si' i
              with
              | Some si => si
              | None =>
                  equivocator_state_zero
                    si'
              end
          end;
      output := Some m
    |}] ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hti: equivocator_transition 
  (IM x) (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: equivocator_valid (IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (state_update (equivocator_IM IM)
     (finite_trace_last is pre) x si')
  suf
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++
      [{|
         l := existT x li;
         input := input;
         destination :=
           state_update IM
             (equivocators_state_project
                IM eqv_descriptors''
                (finite_trace_last is pre))
             x
             match
               eqv_descriptors'' x
             with
             | NewMachine sn => sn
             | Existing i =>
                 match
                   equivocator_state_project
                     si' i
                 with
                 | Some si => si
                 | None =>
                     equivocator_state_zero
                      si'
                 end
             end;
         output := Some m
       |}] ++ sufX''))
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++
   [{|
      l := existT x li;
      input := input;
      destination :=
        state_update IM
          (equivocators_state_project IM
             eqv_descriptors''
             (finite_trace_last is pre)) x
          match eqv_descriptors'' x with
          | NewMachine sn => sn
          | Existing i =>
              match
                equivocator_state_project
                  si' i
              with
              | Some si => si
              | None =>
                  equivocator_state_zero
                    si'
              end
          end;
      output := Some m
    |}] ++ sufX'')
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m)
  [{|
     l := existT x li;
     input := input;
     destination :=
       state_update IM
         (equivocators_state_project IM
            eqv_descriptors''
            (finite_trace_last is pre)) x
         match eqv_descriptors'' x with
         | NewMachine sn => sn
         | Existing i =>
             match
               equivocator_state_project si' i
             with
             | Some si => si
             | None => equivocator_state_zero si'
             end
         end;
     output := Some m
   |}] clear Hpr_pre_item.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
x: index
input: option message
pre, suf: list transition_item
m: message
si': equivocator_state (IM x)
li: label (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', sufX'': list (composite_transition_item IM)
eqv_descriptors'': equivocator_descriptors IM
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
          existT x (ContinueWith 0 li);
        input := input;
        destination :=
          state_update
            (equivocator_IM IM)
            (finite_trace_last is pre)
            x si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++
   [{|
      l := existT x li;
      input := input;
      destination :=
        state_update IM
          (equivocators_state_project
             IM eqv_descriptors''
             (finite_trace_last is pre))
          x
          match
            eqv_descriptors'' x
          with
          | NewMachine sn => sn
          | Existing i =>
              match
                equivocator_state_project
                  si' i
              with
              | Some si => si
              | None =>
                  equivocator_state_zero
                    si'
              end
          end;
      output := Some m
    |}] ++ sufX'')
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre: equivocators_trace_project IM
  (equivocator_descriptors_update IM
     eqv_descriptors'' x 
     (Existing 0)) pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++
   [{|
      l := existT x li;
      input := input;
      destination :=
        state_update IM
          (equivocators_state_project IM
             eqv_descriptors''
             (finite_trace_last is pre)) x
          match eqv_descriptors'' x with
          | NewMachine sn => sn
          | Existing i =>
              match
                equivocator_state_project
                  si' i
              with
              | Some si => si
              | None =>
                  equivocator_state_zero
                    si'
              end
          end;
      output := Some m
    |}] ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is pre
Hti: equivocator_transition 
  (IM x) (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: equivocator_valid (IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
Hsuf'_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (state_update (equivocator_IM IM)
     (finite_trace_last is pre) x si')
  suf
Hlst_preX: constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++
      [{|
         l := existT x li;
         input := input;
         destination :=
           state_update IM
             (equivocators_state_project
                IM eqv_descriptors''
                (finite_trace_last is pre))
             x
             match
               eqv_descriptors'' x
             with
             | NewMachine sn => sn
             | Existing i =>
                 match
                   equivocator_state_project
                     si' i
                 with
                 | Some si => si
                 | None =>
                     equivocator_state_zero
                      si'
                 end
             end;
         output := Some m
       |}] ++ sufX''))
HtrX_Pre: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++
   [{|
      l := existT x li;
      input := input;
      destination :=
        state_update IM
          (equivocators_state_project IM
             eqv_descriptors''
             (finite_trace_last is pre)) x
          match eqv_descriptors'' x with
          | NewMachine sn => sn
          | Existing i =>
              match
                equivocator_state_project
                  si' i
              with
              | Some si => si
              | None =>
                  equivocator_state_zero
                    si'
              end
          end;
      output := Some m
    |}] ++ sufX'')
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m)
  [{|
     l := existT x li;
     input := input;
     destination :=
       state_update IM
         (equivocators_state_project IM
            eqv_descriptors''
            (finite_trace_last is pre)) x
         match eqv_descriptors'' x with
         | NewMachine sn => sn
         | Existing i =>
             match
               equivocator_state_project si' i
             with
             | Some si => si
             | None => equivocator_state_zero si'
             end
         end;
     output := Some m
   |}]
  by constructor.
Qed.

Consider a valid_trace for the composition of equivocators with no message equivocation and fixed state equivocation.

Because any of its projections to the composition of original components contains all transitions from components not allowed to equivocate, then a final state of such a projection will be able to observe all messages sent or received by non-equivocating components in the initial trace.

Therefore if seeding the composition of equivocating components with these messages, the restriction of the initial trace to only the equivocating components will satisfy the trace_sub_item_input_is_seeded_or_sub_previously_sent property w.r.t. these messages, a sufficient condition for it being valid (finite_valid_trace_sub_projection).

Lemma equivocators_trace_sub_item_input_is_seeded_or_sub_previously_sent
  (is : state XE)
  (tr : list (transition_item XE))
  (s := finite_trace_last is tr)
  (Htr : finite_valid_trace XE is tr)
  (descriptors : equivocator_descriptors IM)
  (Hproper : proper_fixed_equivocator_descriptors descriptors s)
  (lst_trX := equivocators_state_project IM descriptors s)
  : trace_sub_item_input_is_seeded_or_sub_previously_sent
    (equivocator_IM IM) (elements equivocating)
    (composite_has_been_directly_observed IM lst_trX) tr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list transition_item
s:= finite_trace_last is tr: state XE
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
trace_sub_item_input_is_seeded_or_sub_previously_sent
  (equivocator_IM IM) (elements equivocating)
  (composite_has_been_directly_observed IM lst_trX) tr
Proof.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list transition_item
s:= finite_trace_last is tr: state XE
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
trace_sub_item_input_is_seeded_or_sub_previously_sent
  (equivocator_IM IM) (elements equivocating)
  (composite_has_been_directly_observed IM lst_trX) tr
  intros pre item suf m Heq Hm Hitem.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list transition_item
s:= finite_trace_last is tr: state XE
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
composite_has_been_directly_observed IM lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection (equivocator_IM IM)
           (elements equivocating) pre_item)
  destruct (free_equivocators_valid_trace_project descriptors is tr Hproper Htr)
    as [trX [initial_descriptors [Hinitial_descriptors [Htr_project [Hfinal_state HtrXFree]]]]].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list transition_item
s:= finite_trace_last is tr: state XE
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
composite_has_been_directly_observed IM lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection (equivocator_IM IM)
           (elements equivocating) pre_item)
  assert (HtrXPre : finite_constrained_trace (free_composite_vlsm IM)
    (equivocators_state_project IM initial_descriptors is) trX).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list transition_item
s:= finite_trace_last is tr: state XE
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
finite_constrained_trace (free_composite_vlsm IM)
  (equivocators_state_project IM initial_descriptors
     is) trX
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list transition_item
s:= finite_trace_last is tr: state XE
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
composite_has_been_directly_observed IM lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection 
           (equivocator_IM IM) 
           (elements equivocating) pre_item)
  {message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list transition_item
s:= finite_trace_last is tr: state XE
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
finite_constrained_trace (free_composite_vlsm IM)
  (equivocators_state_project IM initial_descriptors
     is) trX
    revert HtrXFree; apply VLSM_incl_finite_valid_trace.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list transition_item
s:= finite_trace_last is tr: state XE
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
VLSM_incl_part (free_composite_vlsm_machine IM)
  (preloaded_vlsm_machine (free_composite_vlsm IM)
     (λ _ : message, True))
    by apply vlsm_incl_preloaded_with_all_messages_vlsm.
  }message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list transition_item
s:= finite_trace_last is tr: state XE
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
composite_has_been_directly_observed IM lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection (equivocator_IM IM)
           (elements equivocating) pre_item)
  assert (Hlst_trX : constrained_state_prop Free lst_trX).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list transition_item
s:= finite_trace_last is tr: state XE
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
constrained_state_prop Free lst_trX
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list transition_item
s:= finite_trace_last is tr: state XE
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_trX: constrained_state_prop Free lst_trX
composite_has_been_directly_observed IM lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection 
           (equivocator_IM IM) 
           (elements equivocating) pre_item)
  {message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list transition_item
s:= finite_trace_last is tr: state XE
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
constrained_state_prop Free lst_trX
    subst lst_trX s; cbn in Hfinal_state |- *; rewrite Hfinal_state.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list transition_item
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
constrained_state_prop Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
    by apply finite_valid_trace_last_pstate, HtrXPre.
  }message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list transition_item
s:= finite_trace_last is tr: state XE
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_trX: constrained_state_prop Free lst_trX
composite_has_been_directly_observed IM lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection (equivocator_IM IM)
           (elements equivocating) pre_item)
  assert (Htr_free : finite_constrained_trace FreeE is tr).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list transition_item
s:= finite_trace_last is tr: state XE
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_trX: constrained_state_prop Free lst_trX
finite_constrained_trace FreeE is tr
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list transition_item
s:= finite_trace_last is tr: state XE
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is tr
composite_has_been_directly_observed IM lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection 
           (equivocator_IM IM) 
           (elements equivocating) pre_item)
  {message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list transition_item
s:= finite_trace_last is tr: state XE
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_trX: constrained_state_prop Free lst_trX
finite_constrained_trace FreeE is tr
    apply VLSM_incl_finite_valid_trace; [| done].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list transition_item
s:= finite_trace_last is tr: state XE
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_trX: constrained_state_prop Free lst_trX
VLSM_incl_part
  (constrained_vlsm_machine
     (free_composite_vlsm (equivocator_IM IM))
     (equivocators_fixed_equivocations_constraint IM
        (elements equivocating)))
  (preloaded_vlsm_machine FreeE (λ _ : message, True))
    by apply constrained_preloaded_incl.
  }message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
tr: list transition_item
s:= finite_trace_last is tr: state XE
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is tr
composite_has_been_directly_observed IM lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection (equivocator_IM IM)
           (elements equivocating) pre_item)
  subst tr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
s:= finite_trace_last is (pre ++ [item] ++ suf): state XE
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
composite_has_been_directly_observed IM lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection (equivocator_IM IM)
           (elements equivocating) pre_item)
  destruct Htr as [Htr His].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  (pre ++ [item] ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state XE
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
composite_has_been_directly_observed IM lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection (equivocator_IM IM)
           (elements equivocating) pre_item)
  apply finite_valid_trace_from_app_iff in Htr as Hsuf.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  (pre ++ [item] ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state XE
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Hsuf: finite_valid_trace_from XE is pre
∧ finite_valid_trace_from XE
    (finite_trace_last is pre) 
    ([item] ++ suf)
composite_has_been_directly_observed IM lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection (equivocator_IM IM)
           (elements equivocating) pre_item) destruct Hsuf as [_ Hsuf].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  (pre ++ [item] ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state XE
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
composite_has_been_directly_observed IM lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection (equivocator_IM IM)
           (elements equivocating) pre_item)
  rewrite app_assoc in Htr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state XE
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
composite_has_been_directly_observed IM lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection (equivocator_IM IM)
           (elements equivocating) pre_item)
  apply finite_valid_trace_from_app_iff in Htr as Hpre.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state XE
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
Hpre: finite_valid_trace_from XE is (pre ++ [item])
∧ finite_valid_trace_from XE
    (finite_trace_last is (pre ++ [item])) suf
composite_has_been_directly_observed IM lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection (equivocator_IM IM)
           (elements equivocating) pre_item) destruct Hpre as [Hpre _].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state XE
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
Hpre: finite_valid_trace_from XE is (pre ++ [item])
composite_has_been_directly_observed IM lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection (equivocator_IM IM)
           (elements equivocating) pre_item)
  apply finite_valid_trace_from_app_iff in Hpre.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state XE
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
Hpre: finite_valid_trace_from XE is pre
∧ finite_valid_trace_from XE
    (finite_trace_last is pre) [item]
composite_has_been_directly_observed IM lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection (equivocator_IM IM)
           (elements equivocating) pre_item) destruct Hpre as [Hpre Hivt].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state XE
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
composite_has_been_directly_observed IM lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection (equivocator_IM IM)
           (elements equivocating) pre_item)
  apply first_transition_valid in Hivt.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state XE
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE 
  (l item)
  (finite_trace_last is pre, input item)
  (destination item, output item)
composite_has_been_directly_observed IM lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection (equivocator_IM IM)
           (elements equivocating) pre_item)
  destruct (composite_has_been_directly_observed_dec IM lst_trX m) as [| Heqv]
  ; [by left | right].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state XE
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE 
  (l item)
  (finite_trace_last is pre, input item)
  (destination item, output item)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        (elements equivocating) pre_item
  assert (Hsuf_free : finite_constrained_trace_from FreeE
    (finite_trace_last is pre) ([item] ++ suf)).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state XE
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE 
  (l item)
  (finite_trace_last is pre, input item)
  (destination item, output item)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
finite_constrained_trace_from FreeE
  (finite_trace_last is pre) ([item] ++ suf)
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state XE
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE 
  (l item)
  (finite_trace_last is pre, input item)
  (destination item, output item)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([item] ++ suf)
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection 
        (equivocator_IM IM) 
        (elements equivocating) pre_item
  {message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state XE
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE 
  (l item)
  (finite_trace_last is pre, input item)
  (destination item, output item)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
finite_constrained_trace_from FreeE
  (finite_trace_last is pre) ([item] ++ suf)
    apply VLSM_incl_finite_valid_trace_from; [| done].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state XE
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE 
  (l item)
  (finite_trace_last is pre, input item)
  (destination item, output item)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
VLSM_incl_part
  (constrained_vlsm_machine
     (free_composite_vlsm (equivocator_IM IM))
     (equivocators_fixed_equivocations_constraint IM
        (elements equivocating)))
  (preloaded_vlsm_machine FreeE (λ _ : message, True))
    by apply constrained_preloaded_incl.
  }message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state XE
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE 
  (l item)
  (finite_trace_last is pre, input item)
  (destination item, output item)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([item] ++ suf)
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        (elements equivocating) pre_item
  assert (Hs_free : constrained_state_prop FreeE s).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state XE
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE 
  (l item)
  (finite_trace_last is pre, input item)
  (destination item, output item)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([item] ++ suf)
constrained_state_prop FreeE s
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state XE
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE 
  (l item)
  (finite_trace_last is pre, input item)
  (destination item, output item)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([item] ++ suf)
Hs_free: constrained_state_prop FreeE s
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection 
        (equivocator_IM IM) 
        (elements equivocating) pre_item
  {message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state XE
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE 
  (l item)
  (finite_trace_last is pre, input item)
  (destination item, output item)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([item] ++ suf)
constrained_state_prop FreeE s apply finite_valid_trace_last_pstate in Hsuf_free.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state XE
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE 
  (l item)
  (finite_trace_last is pre, input item)
  (destination item, output item)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
Hsuf_free: valid_state_prop
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     ([item] ++ suf))
constrained_state_prop FreeE s subst s.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
His: initial_state_prop is
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)): state (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
m: message
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE 
  (l item)
  (finite_trace_last is pre, input item)
  (destination item, output item)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
Hsuf_free: valid_state_prop
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     ([item] ++ suf))
constrained_state_prop FreeE
  (finite_trace_last is (pre ++ [item] ++ suf))
    by rewrite finite_trace_last_app.
  }message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state XE
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating) item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++ [item] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE 
  (l item)
  (finite_trace_last is pre, input item)
  (destination item, output item)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([item] ++ suf)
Hs_free: constrained_state_prop FreeE s
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        (elements equivocating) pre_item
  destruct item as (l, iom, s0, oom).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
iom: option message
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := iom;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := iom;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input
  {|
    l := l;
    input := iom;
    destination := s0;
    output := oom
  |} = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := iom;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := iom;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := iom;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := iom;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := iom;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s0;
       output := oom
     |})
  (finite_trace_last is pre,
   input
     {|
       l := l;
       input := iom;
       destination := s0;
       output := oom
     |})
  (destination
     {|
       l := l;
       input := iom;
       destination := s0;
       output := oom
     |},
   output
     {|
       l := l;
       input := iom;
       destination := s0;
       output := oom
     |})
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := iom;
      destination := s0;
      output := oom
    |}] ++ suf)
Hs_free: constrained_state_prop FreeE s
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        (elements equivocating) pre_item
  simpl in Hm.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
iom: option message
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := iom;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := iom;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: iom = Some m
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := iom;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := iom;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := iom;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := iom;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := iom;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s0;
       output := oom
     |})
  (finite_trace_last is pre,
   input
     {|
       l := l;
       input := iom;
       destination := s0;
       output := oom
     |})
  (destination
     {|
       l := l;
       input := iom;
       destination := s0;
       output := oom
     |},
   output
     {|
       l := l;
       input := iom;
       destination := s0;
       output := oom
     |})
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := iom;
      destination := s0;
      output := oom
    |}] ++ suf)
Hs_free: constrained_state_prop FreeE s
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        (elements equivocating) pre_item subst iom.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
Hivt: input_valid_transition XE
  (VLSM.l
     {|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |})
  (finite_trace_last is pre,
   input
     {|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |})
  (destination
     {|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |},
   output
     {|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |})
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        (elements equivocating) pre_item
  destruct Hivt as [[_ [_ [_ [[Hc _] Hfixed]]]] Ht].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
Hc: composite_no_equivocations 
  (equivocator_IM IM)
  (VLSM.l
     {|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |})
  (finite_trace_last is pre,
   input
     {|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |})
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating)
  (composite_transition 
     (equivocator_IM IM)
     (VLSM.l
        {|
          l := l;
          input := Some m;
          destination := s0;
          output := oom
        |})
     (finite_trace_last is pre,
      input
        {|
          l := l;
          input := Some m;
          destination := s0;
          output := oom
        |})).1
Ht: transition
  (VLSM.l
     {|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |})
  (finite_trace_last is pre,
   input
     {|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}) =
(destination
   {|
     l := l;
     input := Some m;
     destination := s0;
     output := oom
   |},
 output
   {|
     l := l;
     input := Some m;
     destination := s0;
     output := oom
   |})
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        (elements equivocating) pre_item
  cbn in Ht, Hfixed; rewrite Ht in Hfixed.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
Hc: composite_no_equivocations 
  (equivocator_IM IM)
  (VLSM.l
     {|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |})
  (finite_trace_last is pre,
   input
     {|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |})
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Ht: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li
     (finite_trace_last is pre i, Some m) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is pre) i si', om')) =
(s0, oom)
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        (elements equivocating) pre_item
  clear Ht.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
Hc: composite_no_equivocations 
  (equivocator_IM IM)
  (VLSM.l
     {|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |})
  (finite_trace_last is pre,
   input
     {|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |})
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        (elements equivocating) pre_item
  destruct Hc as [Hc | Hinit]; [| done].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
Hc: composite_has_been_sent 
  (equivocator_IM IM)
  (finite_trace_last is pre,
   input
     {|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}).1 m
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        (elements equivocating) pre_item
  assert (Hpre_free : finite_valid_trace FreeE is pre).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
Hc: composite_has_been_sent 
  (equivocator_IM IM)
  (finite_trace_last is pre,
   input
     {|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}).1 m
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
finite_valid_trace FreeE is pre
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
Hc: composite_has_been_sent 
  (equivocator_IM IM)
  (finite_trace_last is pre,
   input
     {|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}).1 m
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection 
        (equivocator_IM IM) 
        (elements equivocating) pre_item
  {message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
Hc: composite_has_been_sent 
  (equivocator_IM IM)
  (finite_trace_last is pre,
   input
     {|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}).1 m
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
finite_valid_trace FreeE is pre
    split; [| done].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
Hc: composite_has_been_sent 
  (equivocator_IM IM)
  (finite_trace_last is pre,
   input
     {|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}).1 m
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
finite_valid_trace_from FreeE is pre
    revert Hpre; apply VLSM_incl_finite_valid_trace_from.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
Hc: composite_has_been_sent 
  (equivocator_IM IM)
  (finite_trace_last is pre,
   input
     {|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}).1 m
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
VLSM_incl_part
  (constrained_vlsm_machine
     (free_composite_vlsm (equivocator_IM IM))
     (equivocators_fixed_equivocations_constraint IM
        (elements equivocating)))
  (free_composite_vlsm_machine (equivocator_IM IM))
    by apply equivocators_fixed_equivocations_vlsm_incl_free.
  }message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
Hc: composite_has_been_sent 
  (equivocator_IM IM)
  (finite_trace_last is pre,
   input
     {|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}).1 m
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        (elements equivocating) pre_item
  assert (Hpre_lst_free : valid_state_prop FreeE (finite_trace_last is pre))
    by (apply (finite_valid_trace_last_pstate FreeE), Hpre_free; done).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
Hc: composite_has_been_sent 
  (equivocator_IM IM)
  (finite_trace_last is pre,
   input
     {|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}).1 m
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        (elements equivocating) pre_item
  specialize (specialized_proper_sent_rev FreeE _ Hpre_lst_free m) as Hproper_sent.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
Hc: composite_has_been_sent 
  (equivocator_IM IM)
  (finite_trace_last is pre,
   input
     {|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}).1 m
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hproper_sent: has_been_sent FreeE
  (finite_trace_last is pre) m
→ specialized_selected_message_exists_in_all_traces
    FreeE (field_selector output)
    (finite_trace_last is pre) m
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        (elements equivocating) pre_item
  apply Hproper_sent in Hc.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
Hc: specialized_selected_message_exists_in_all_traces
  FreeE (field_selector output)
  (finite_trace_last is pre) m
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hproper_sent: has_been_sent FreeE
  (finite_trace_last is pre) m
→ specialized_selected_message_exists_in_all_traces
    FreeE (field_selector output)
    (finite_trace_last is pre) m
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        (elements equivocating) pre_item clear Hproper_sent.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
Hc: specialized_selected_message_exists_in_all_traces
  FreeE (field_selector output)
  (finite_trace_last is pre) m
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        (elements equivocating) pre_item
  specialize (Hc  is pre (valid_trace_add_default_last Hpre_free)).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
Hc: trace_has_message (field_selector output) m pre
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        (elements equivocating) pre_item
  apply Exists_exists in Hc.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
Hc: ∃ x : transition_item,
  x ∈ pre ∧ field_selector output m x
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        (elements equivocating) pre_item
  destruct Hc as [pre_item [Hpre_item Hpre_m]].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        (elements equivocating) pre_item
  exists pre_item.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
pre_item ∈ pre
∧ output pre_item = Some m
  ∧ from_sub_projection (equivocator_IM IM)
      (elements equivocating) pre_item split; [done |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
output pre_item = Some m
∧ from_sub_projection (equivocator_IM IM)
    (elements equivocating) pre_item split; [done |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
from_sub_projection (equivocator_IM IM)
  (elements equivocating) pre_item

  apply equivocators_trace_project_app_iff in Htr_project.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  equivocators_trace_project IM
    descriptors
    ([{|
        l := l;
        input := Some m;
        destination := s0;
        output := oom
      |}] ++ suf) =
  Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project IM
      eqv_descriptors' pre =
    Some (preX, initial_descriptors)
    ∧ trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
from_sub_projection (equivocator_IM IM)
  (elements equivocating) pre_item
  destruct Htr_project as [preX [sufX [final_descriptors' [Hsuf_project [Htr_project Heq]]]]].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
from_sub_projection (equivocator_IM IM)
  (elements equivocating) pre_item

  assert (Hfinal' :
    proper_fixed_equivocator_descriptors final_descriptors' (finite_trace_last is pre)).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
proper_fixed_equivocator_descriptors
  final_descriptors' (finite_trace_last is pre)
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
from_sub_projection (equivocator_IM IM)
  (elements equivocating) pre_item
  {message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
proper_fixed_equivocator_descriptors
  final_descriptors' (finite_trace_last is pre) split.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
proper_equivocator_descriptors IM final_descriptors'
  (finite_trace_last is pre)
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
∀ i : index,
  i ∉ equivocating → final_descriptors' i = Existing 0
    -message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
proper_equivocator_descriptors IM final_descriptors'
  (finite_trace_last is pre) apply proj1 in Hproper.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
proper_equivocator_descriptors IM final_descriptors'
  (finite_trace_last is pre) subst s.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
lst_trX:= equivocators_state_project IM descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)): state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
proper_equivocator_descriptors IM final_descriptors'
  (finite_trace_last is pre) rewrite finite_trace_last_app in Hproper.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  descriptors
  (finite_trace_last
     (finite_trace_last is pre)
     ([{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
lst_trX:= equivocators_state_project IM descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)): state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
proper_equivocator_descriptors IM final_descriptors'
  (finite_trace_last is pre)
      destruct (preloaded_equivocators_valid_trace_from_project IM _ _ _ Hproper Hsuf_free)
        as [_sufX [_final_descriptors' [_Hsuf_project [Hproper' _]]]].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  descriptors
  (finite_trace_last
     (finite_trace_last is pre)
     ([{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
lst_trX:= equivocators_state_project IM descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)): state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
_sufX: list (composite_transition_item IM)
_final_descriptors': equivocator_descriptors IM
_Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (_sufX, _final_descriptors')
Hproper': proper_equivocator_descriptors IM
  _final_descriptors'
  (finite_trace_last is pre)
proper_equivocator_descriptors IM final_descriptors'
  (finite_trace_last is pre)
      rewrite Hsuf_project in _Hsuf_project.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  descriptors
  (finite_trace_last
     (finite_trace_last is pre)
     ([{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
lst_trX:= equivocators_state_project IM descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)): state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
_sufX: list (composite_transition_item IM)
_final_descriptors': equivocator_descriptors IM
_Hsuf_project: Some (sufX, final_descriptors') =
Some (_sufX, _final_descriptors')
Hproper': proper_equivocator_descriptors IM
  _final_descriptors'
  (finite_trace_last is pre)
proper_equivocator_descriptors IM final_descriptors'
  (finite_trace_last is pre)
      by inversion _Hsuf_project; subst.
    -message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
∀ i : index,
  i ∉ equivocating → final_descriptors' i = Existing 0 specialize (equivocators_trace_project_preserves_zero_descriptors IM _ _ Hsuf_free descriptors)
        as Hpr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hpr: ∀ (idescriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project IM descriptors
    ([{|
        l := l;
        input := Some m;
        destination := s0;
        output := oom
      |}] ++ suf) = Some (trX, idescriptors)
  → ∀ i : index,
      descriptors i = Existing 0
      → idescriptors i = Existing 0
∀ i : index,
  i ∉ equivocating → final_descriptors' i = Existing 0
      specialize (Hpr _ _ Hsuf_project).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hpr: ∀ i : index,
  descriptors i = Existing 0
  → final_descriptors' i = Existing 0
∀ i : index,
  i ∉ equivocating → final_descriptors' i = Existing 0
      by intros i Hi; apply Hpr, (proj2 Hproper).
  }message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
from_sub_projection (equivocator_IM IM)
  (elements equivocating) pre_item
  assert
    (Hfutures : in_futures (preloaded_with_all_messages_vlsm FreeE)
      (destination pre_item) s0).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
in_futures (preloaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: in_futures
  (preloaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0
from_sub_projection (equivocator_IM IM)
  (elements equivocating) pre_item
  {message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
in_futures (preloaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0 apply elem_of_list_split in Hpre_item.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: ∃ l1 l2 : list transition_item,
  pre = l1 ++ pre_item :: l2
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
in_futures (preloaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0
    destruct Hpre_item as [pre_pre [pre_suf Hpre_item]].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
pre_pre, pre_suf: list transition_item
Hpre_item: pre = pre_pre ++ pre_item :: pre_suf
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
in_futures (preloaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0
    change (pre_item :: pre_suf) with ([pre_item] ++ pre_suf) in Hpre_item.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
pre_pre, pre_suf: list transition_item
Hpre_item: pre = pre_pre ++ [pre_item] ++ pre_suf
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
in_futures (preloaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0
    rewrite app_assoc in Hpre_item.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
pre_pre, pre_suf: list transition_item
Hpre_item: pre = (pre_pre ++ [pre_item]) ++ pre_suf
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
in_futures (preloaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0
    rewrite app_assoc in Htr_free.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
pre_pre, pre_suf: list transition_item
Hpre_item: pre = (pre_pre ++ [pre_item]) ++ pre_suf
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
in_futures (preloaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0
    destruct Htr_free as [Htr_free _].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
pre_pre, pre_suf: list transition_item
Hpre_item: pre = (pre_pre ++ [pre_item]) ++ pre_suf
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
in_futures (preloaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0
    apply finite_valid_trace_from_app_iff in Htr_free.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}])
∧ finite_valid_trace_from
    (preloaded_with_all_messages_vlsm FreeE)
    (finite_trace_last is
       (pre ++
        [{|
           l := l;
           input := Some m;
           destination := s0;
           output := oom
         |}])) suf
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
pre_pre, pre_suf: list transition_item
Hpre_item: pre = (pre_pre ++ [pre_item]) ++ pre_suf
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
in_futures (preloaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0
    destruct Htr_free as [Htr_s0 _].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_s0: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}])
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
pre_pre, pre_suf: list transition_item
Hpre_item: pre = (pre_pre ++ [pre_item]) ++ pre_suf
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
in_futures (preloaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0
    subst pre.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
pre_item: transition_item
pre_pre, pre_suf: list transition_item
Htr: finite_valid_trace_from XE is
  ((((pre_pre ++ [pre_item]) ++ pre_suf) ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (((pre_pre ++ [pre_item]) ++ pre_suf) ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors'
  ((pre_pre ++ [pre_item]) ++ pre_suf) =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (((pre_pre ++ [pre_item]) ++
       pre_suf) ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is
     ((pre_pre ++ [pre_item]) ++ pre_suf))
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hs_free: constrained_state_prop FreeE s
Hpre: finite_valid_trace_from XE is
  ((pre_pre ++ [pre_item]) ++ pre_suf)
Htr_s0: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is
  (((pre_pre ++ [pre_item]) ++ pre_suf) ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}])
Hlst_trX: constrained_state_prop Free lst_trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is
     ((pre_pre ++ [pre_item]) ++ pre_suf))
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is
     ((pre_pre ++ [pre_item]) ++ pre_suf))
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is
     ((pre_pre ++ [pre_item]) ++
      pre_suf))
Hpre_free: finite_valid_trace FreeE is
  ((pre_pre ++ [pre_item]) ++ pre_suf)
in_futures (preloaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0 clear -Htr_s0.message, index, Ci: Type
H5: Elements index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
is: state XE
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
m: message
pre_item: transition_item
pre_pre, pre_suf: list transition_item
Htr_s0: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is
  (((pre_pre ++ [pre_item]) ++ pre_suf) ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}])
in_futures (preloaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0
    rewrite <- app_assoc in Htr_s0.message, index, Ci: Type
H5: Elements index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
is: state XE
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
m: message
pre_item: transition_item
pre_pre, pre_suf: list transition_item
Htr_s0: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is
  ((pre_pre ++ [pre_item]) ++
   pre_suf ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}])
in_futures (preloaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0
    apply (finite_valid_trace_from_app_iff (preloaded_with_all_messages_vlsm FreeE)) in Htr_s0.message, index, Ci: Type
H5: Elements index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
is: state XE
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
m: message
pre_item: transition_item
pre_pre, pre_suf: list transition_item
Htr_s0: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is
  (pre_pre ++ [pre_item])
∧ finite_valid_trace_from
    (preloaded_with_all_messages_vlsm FreeE)
    (finite_trace_last is
       (pre_pre ++ [pre_item]))
    (pre_suf ++
     [{|
        l := l;
        input := Some m;
        destination := s0;
        output := oom
      |}])
in_futures (preloaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0
    destruct Htr_s0 as [_ Hfuture].message, index, Ci: Type
H5: Elements index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
is: state XE
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
m: message
pre_item: transition_item
pre_pre, pre_suf: list transition_item
Hfuture: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is
     (pre_pre ++ [pre_item]))
  (pre_suf ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}])
in_futures (preloaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0
    rewrite finite_trace_last_is_last in Hfuture.message, index, Ci: Type
H5: Elements index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
is: state XE
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
m: message
pre_item: transition_item
pre_pre, pre_suf: list transition_item
Hfuture: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (destination pre_item)
  (pre_suf ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}])
in_futures (preloaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0
    eexists.message, index, Ci: Type
H5: Elements index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
is: state XE
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
m: message
pre_item: transition_item
pre_pre, pre_suf: list transition_item
Hfuture: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (destination pre_item)
  (pre_suf ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}])
finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0 ?tr
    apply valid_trace_add_last; [by apply Hfuture |].message, index, Ci: Type
H5: Elements index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
is: state XE
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
m: message
pre_item: transition_item
pre_pre, pre_suf: list transition_item
Hfuture: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (destination pre_item)
  (pre_suf ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}])
finite_trace_last (destination pre_item)
  (pre_suf ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}]) = s0
    by rewrite finite_trace_last_is_last.
  }message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: in_futures
  (preloaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0
from_sub_projection (equivocator_IM IM)
  (elements equivocating) pre_item
  eapply (in_futures_reflects_fixed_equivocation) in Hfutures; [| done].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  (elements equivocating)
  (destination pre_item)
from_sub_projection (equivocator_IM IM)
  (elements equivocating) pre_item
  destruct (free_equivocators_valid_trace_project final_descriptors' is pre Hfinal' (conj Hpre His))
    as [_preX [_initial_descriptors [_ [_Htr_project [Hpre_final_state _]]]]].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  (elements equivocating)
  (destination pre_item)
_preX: list (composite_transition_item IM)
_initial_descriptors: equivocator_descriptors IM
_Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (_preX, _initial_descriptors)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     _initial_descriptors is) _preX
from_sub_projection (equivocator_IM IM)
  (elements equivocating) pre_item
  rewrite Htr_project in _Htr_project.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  (elements equivocating)
  (destination pre_item)
_preX: list (composite_transition_item IM)
_initial_descriptors: equivocator_descriptors IM
_Htr_project: Some (preX, initial_descriptors) =
Some (_preX, _initial_descriptors)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     _initial_descriptors is) _preX
from_sub_projection (equivocator_IM IM)
  (elements equivocating) pre_item inversion _Htr_project.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  (elements equivocating)
  (destination pre_item)
_preX: list (composite_transition_item IM)
_initial_descriptors: equivocator_descriptors IM
_Htr_project: Some (preX, initial_descriptors) =
Some (_preX, _initial_descriptors)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     _initial_descriptors is) _preX
H11: preX = _preX
H12: initial_descriptors = _initial_descriptors
from_sub_projection (equivocator_IM IM)
  (elements equivocating) pre_item subst _preX _initial_descriptors.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  (elements equivocating)
  (destination pre_item)
_Htr_project: Some (preX, initial_descriptors) =
Some (preX, initial_descriptors)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
from_sub_projection (equivocator_IM IM)
  (elements equivocating) pre_item
  clear _Htr_project.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  (elements equivocating)
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
from_sub_projection (equivocator_IM IM)
  (elements equivocating) pre_item
  unfold from_sub_projection, pre_VLSM_projection_in_projection,
    composite_label_sub_projection_option.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  (elements equivocating)
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
is_Some
  match
    decide
      (projT1 (VLSM.l pre_item)
       ∈ elements equivocating)
  with
  | left i_in =>
      Some
        (composite_label_sub_projection
           (equivocator_IM IM) (elements equivocating)
           (VLSM.l pre_item) i_in)
  | right _ => None
  end
  case_decide as Hl; [by eexists |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Heqv: ¬ composite_has_been_directly_observed IM
    lst_trX m
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  (elements equivocating)
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
Hl: projT1 (VLSM.l pre_item) ∉ elements equivocating
is_Some None
  contradict Heqv.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  (elements equivocating)
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
Hl: projT1 (VLSM.l pre_item) ∉ elements equivocating
composite_has_been_directly_observed IM lst_trX m
  apply composite_has_been_directly_observed_free_iff.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  (elements equivocating)
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
Hl: projT1 (VLSM.l pre_item) ∉ elements equivocating
has_been_directly_observed (free_composite_vlsm IM)
  lst_trX m
  eapply in_futures_preserving_oracle_from_stepwise; cycle 2
  ; [| by apply has_been_directly_observed_stepwise_props |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  (elements equivocating)
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
Hl: projT1 (VLSM.l pre_item) ∉ elements equivocating
has_been_directly_observed (free_composite_vlsm IM)
  ?s1 m
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  (elements equivocating)
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
Hl: projT1 (VLSM.l pre_item) ∉ elements equivocating
in_futures
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm IM)) 
  ?s1 lst_trX
  -message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  (elements equivocating)
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
Hl: projT1 (VLSM.l pre_item) ∉ elements equivocating
has_been_directly_observed (free_composite_vlsm IM)
  ?s1 m by eapply not_equivocating_sent_message_has_been_directly_observed_in_projection;
      only 3: rewrite <- elem_of_elements.
  -message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free lst_trX
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  (elements equivocating)
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
Hl: projT1 (VLSM.l pre_item) ∉ elements equivocating
in_futures
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM final_descriptors'
     (finite_trace_last is pre)) lst_trX subst lst_trX.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state XE
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free
  (equivocators_state_project IM
     descriptors s)
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE s
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  (elements equivocating)
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
Hl: projT1 (VLSM.l pre_item) ∉ elements equivocating
in_futures
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM final_descriptors'
     (finite_trace_last is pre))
  (equivocators_state_project IM descriptors s) subst s.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free
  (equivocators_state_project IM
     descriptors
     (finite_trace_last is
        (pre ++
         [{|
            l := l;
            input := Some m;
            destination := s0;
            output := oom
          |}] ++ suf)))
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  (elements equivocating)
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
Hl: projT1 (VLSM.l pre_item) ∉ elements equivocating
in_futures
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM final_descriptors'
     (finite_trace_last is pre))
  (equivocators_state_project IM descriptors
     (finite_trace_last is
        (pre ++
         [{|
            l := l;
            input := Some m;
            destination := s0;
            output := oom
          |}] ++ suf))) simpl.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free
  (equivocators_state_project IM
     descriptors
     (finite_trace_last is
        (pre ++
         [{|
            l := l;
            input := Some m;
            destination := s0;
            output := oom
          |}] ++ suf)))
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  (elements equivocating)
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
Hl: projT1 (VLSM.l pre_item) ∉ elements equivocating
in_futures
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM final_descriptors'
     (finite_trace_last is pre))
  (equivocators_state_project IM descriptors
     (finite_trace_last is
        (pre ++
         {|
           l := l;
           input := Some m;
           destination := s0;
           output := oom
         |} :: suf))) simpl in Hfinal_state.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := Some m;
        destination := s0;
        output := oom
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free
  (equivocators_state_project IM
     descriptors
     (finite_trace_last is
        (pre ++
         [{|
            l := l;
            input := Some m;
            destination := s0;
            output := oom
          |}] ++ suf)))
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  (elements equivocating)
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
Hl: projT1 (VLSM.l pre_item) ∉ elements equivocating
in_futures
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM final_descriptors'
     (finite_trace_last is pre))
  (equivocators_state_project IM descriptors
     (finite_trace_last is
        (pre ++
         {|
           l := l;
           input := Some m;
           destination := s0;
           output := oom
         |} :: suf)))
    rewrite Hfinal_state.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := Some m;
        destination := s0;
        output := oom
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free
  (equivocators_state_project IM
     descriptors
     (finite_trace_last is
        (pre ++
         [{|
            l := l;
            input := Some m;
            destination := s0;
            output := oom
          |}] ++ suf)))
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  (elements equivocating)
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
Hl: projT1 (VLSM.l pre_item) ∉ elements equivocating
in_futures
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM final_descriptors'
     (finite_trace_last is pre))
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) subst trX.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) 
  (preX ++ sufX)
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) 
  (preX ++ sufX)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := Some m;
        destination := s0;
        output := oom
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX ++ sufX)
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free
  (equivocators_state_project IM
     descriptors
     (finite_trace_last is
        (pre ++
         [{|
            l := l;
            input := Some m;
            destination := s0;
            output := oom
          |}] ++ suf)))
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  (elements equivocating)
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
Hl: projT1 (VLSM.l pre_item) ∉ elements equivocating
in_futures
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM final_descriptors'
     (finite_trace_last is pre))
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) (preX ++ sufX))
    rewrite finite_trace_last_app.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) 
  (preX ++ sufX)
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) 
  (preX ++ sufX)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := Some m;
        destination := s0;
        output := oom
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX ++ sufX)
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free
  (equivocators_state_project IM
     descriptors
     (finite_trace_last is
        (pre ++
         [{|
            l := l;
            input := Some m;
            destination := s0;
            output := oom
          |}] ++ suf)))
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  (elements equivocating)
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
Hl: projT1 (VLSM.l pre_item) ∉ elements equivocating
in_futures
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM final_descriptors'
     (finite_trace_last is pre))
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) preX) sufX)
    exists sufX.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
HtrXPre: finite_constrained_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) 
  (preX ++ sufX)
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) 
  (preX ++ sufX)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := Some m;
        destination := s0;
        output := oom
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX ++ sufX)
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free
  (equivocators_state_project IM
     descriptors
     (finite_trace_last is
        (pre ++
         [{|
            l := l;
            input := Some m;
            destination := s0;
            output := oom
          |}] ++ suf)))
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  (elements equivocating)
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
Hl: projT1 (VLSM.l pre_item) ∉ elements equivocating
finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM final_descriptors'
     (finite_trace_last is pre))
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) preX) sufX) sufX
    apply proj1 in HtrXPre.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
HtrXPre: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) 
  (preX ++ sufX)
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) 
  (preX ++ sufX)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := Some m;
        destination := s0;
        output := oom
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX ++ sufX)
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free
  (equivocators_state_project IM
     descriptors
     (finite_trace_last is
        (pre ++
         [{|
            l := l;
            input := Some m;
            destination := s0;
            output := oom
          |}] ++ suf)))
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  (elements equivocating)
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
Hl: projT1 (VLSM.l pre_item) ∉ elements equivocating
finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM final_descriptors'
     (finite_trace_last is pre))
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) preX) sufX) sufX
    apply finite_valid_trace_from_app_iff in HtrXPre.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: state XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label (composite_type (equivocator_IM IM))
s0: state (composite_type (equivocator_IM IM))
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: initial_state_prop is
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
HtrXPre: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) preX
∧ finite_valid_trace_from
    (preloaded_with_all_messages_vlsm
       (free_composite_vlsm IM))
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) preX) sufX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) 
  (preX ++ sufX)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := Some m;
        destination := s0;
        output := oom
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX ++ sufX)
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: constrained_state_prop Free
  (equivocators_state_project IM
     descriptors
     (finite_trace_last is
        (pre ++
         [{|
            l := l;
            input := Some m;
            destination := s0;
            output := oom
          |}] ++ suf)))
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) (
  s0, oom).1
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  (elements equivocating)
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
Hl: projT1 (VLSM.l pre_item) ∉ elements equivocating
finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM final_descriptors'
     (finite_trace_last is pre))
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) preX) sufX) sufX
    simpl in *.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s0: composite_state (equivocator_IM IM)
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: composite_initial_state_prop 
  (equivocator_IM IM) is
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := Some m;
        destination := s0;
        output := oom
      |} :: suf))
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project_folder IM
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
  (equivocators_trace_project IM
     descriptors suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
HtrXPre: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) preX
∧ finite_valid_trace_from
    (preloaded_with_all_messages_vlsm
       (free_composite_vlsm IM))
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) preX) sufX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) 
  (preX ++ sufX)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := Some m;
        destination := s0;
        output := oom
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX ++ sufX)
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ({|
     l := l;
     input := Some m;
     destination := s0;
     output := oom
   |} :: suf)
Hlst_trX: constrained_state_prop Free
  (equivocators_state_project IM
     descriptors
     (finite_trace_last is
        (pre ++
         {|
           l := l;
           input := Some m;
           destination := s0;
           output := oom
         |} :: suf)))
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   {|
     l := l;
     input := Some m;
     destination := s0;
     output := oom
   |} :: suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := Some m;
        destination := s0;
        output := oom
      |} :: suf))
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ({|
     l := l;
     input := Some m;
     destination := s0;
     output := oom
   |} :: suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: output pre_item = Some m
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) s0
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  (elements equivocating)
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
Hl: projT1 (VLSM.l pre_item) ∉ elements equivocating
finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM final_descriptors'
     (finite_trace_last is pre))
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) preX) sufX) sufX rewrite Hpre_final_state.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s0: composite_state (equivocator_IM IM)
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: composite_initial_state_prop 
  (equivocator_IM IM) is
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := Some m;
        destination := s0;
        output := oom
      |} :: suf))
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project_folder IM
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
  (equivocators_trace_project IM
     descriptors suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
HtrXPre: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) preX
∧ finite_valid_trace_from
    (preloaded_with_all_messages_vlsm
       (free_composite_vlsm IM))
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) preX) sufX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) 
  (preX ++ sufX)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := Some m;
        destination := s0;
        output := oom
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX ++ sufX)
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ({|
     l := l;
     input := Some m;
     destination := s0;
     output := oom
   |} :: suf)
Hlst_trX: constrained_state_prop Free
  (equivocators_state_project IM
     descriptors
     (finite_trace_last is
        (pre ++
         {|
           l := l;
           input := Some m;
           destination := s0;
           output := oom
         |} :: suf)))
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   {|
     l := l;
     input := Some m;
     destination := s0;
     output := oom
   |} :: suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := Some m;
        destination := s0;
        output := oom
      |} :: suf))
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ({|
     l := l;
     input := Some m;
     destination := s0;
     output := oom
   |} :: suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: output pre_item = Some m
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) s0
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  (elements equivocating)
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
Hl: projT1 (VLSM.l pre_item) ∉ elements equivocating
finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) preX)
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) preX) sufX) sufX
    apply valid_trace_add_default_last.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
l: composite_label (equivocator_IM IM)
s0: composite_state (equivocator_IM IM)
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
His: composite_initial_state_prop 
  (equivocator_IM IM) is
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := Some m;
        destination := s0;
        output := oom
      |} :: suf))
Hitem: from_sub_projection (equivocator_IM IM)
  (elements equivocating)
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project_folder IM
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
  (equivocators_trace_project IM
     descriptors suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
HtrXPre: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) preX
∧ finite_valid_trace_from
    (preloaded_with_all_messages_vlsm
       (free_composite_vlsm IM))
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) preX) sufX
HtrXFree: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) 
  (preX ++ sufX)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := Some m;
        destination := s0;
        output := oom
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX ++ sufX)
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ({|
     l := l;
     input := Some m;
     destination := s0;
     output := oom
   |} :: suf)
Hlst_trX: constrained_state_prop Free
  (equivocators_state_project IM
     descriptors
     (finite_trace_last is
        (pre ++
         {|
           l := l;
           input := Some m;
           destination := s0;
           output := oom
         |} :: suf)))
Htr_free: finite_constrained_trace FreeE is
  (pre ++
   {|
     l := l;
     input := Some m;
     destination := s0;
     output := oom
   |} :: suf)
Hpre: finite_valid_trace_from XE is pre
Hs_free: constrained_state_prop FreeE
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := Some m;
        destination := s0;
        output := oom
      |} :: suf))
Hsuf_free: finite_constrained_trace_from FreeE
  (finite_trace_last is pre)
  ({|
     l := l;
     input := Some m;
     destination := s0;
     output := oom
   |} :: suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: output pre_item = Some m
Hfixed: state_has_fixed_equivocation IM
  (elements equivocating) s0
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  (elements equivocating)
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
Hl: projT1 (VLSM.l pre_item) ∉ elements equivocating
finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) preX) sufX
    by apply HtrXPre.
Qed.

Because any of its projections to the composition of original components contains all transitions from components not allowed to equivocate, all messages sent by non-equivocating components in the original trace will also be sent in any projection.

Lemma equivocator_vlsm_trace_project_reflect_non_equivocating
  (is : composite_state (equivocator_IM IM))
  (tr : list (composite_transition_item (equivocator_IM IM)))
  (Htr : finite_valid_trace XE is tr)
  (m : message)
  (final_descriptors : equivocator_descriptors IM)
  (Hproper : proper_fixed_equivocator_descriptors final_descriptors (finite_trace_last is tr))
  (trX : list (composite_transition_item IM))
  (initial_descriptors : equivocator_descriptors IM)
  (Htr_project : equivocators_trace_project IM final_descriptors tr = Some (trX, initial_descriptors))
  (item : composite_transition_item (equivocator_IM IM))
  (Hitem : item ∈ tr)
  (Houtput : output item = Some m)
  (Hno_equiv_item : projT1 (l item) ∉ equivocating)
  : trace_has_message (field_selector output) m trX.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
trace_has_message (field_selector output) m trX
Proof.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
trace_has_message (field_selector output) m trX
  apply elem_of_list_split in Hitem.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
item: composite_transition_item (equivocator_IM IM)
Hitem: ∃ l1
   l2 : list
          (composite_transition_item
             (equivocator_IM IM)),
  tr = l1 ++ item :: l2
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
trace_has_message (field_selector output) m trX
  destruct Hitem as [pre [suf Heq_tr]].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heq_tr: tr = pre ++ item :: suf
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
trace_has_message (field_selector output) m trX
  change (item :: suf) with ([item] ++ suf) in Heq_tr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heq_tr: tr = pre ++ [item] ++ suf
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
trace_has_message (field_selector output) m trX
  subst tr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
trace_has_message (field_selector output) m trX
  apply equivocators_trace_project_app_iff in Htr_project.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  equivocators_trace_project IM
    final_descriptors ([item] ++ suf) =
  Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project IM
      eqv_descriptors' pre =
    Some (preX, initial_descriptors)
    ∧ trX = preX ++ sufX
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
trace_has_message (field_selector output) m trX
  destruct Htr_project as [preX [item_sufX [eqv_pre [Hpr_item_suf [Hpr_pre Heq_trX]]]]].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
preX, item_sufX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
Hpr_item_suf: equivocators_trace_project IM
  final_descriptors ([item] ++ suf) =
Some (item_sufX, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Heq_trX: trX = preX ++ item_sufX
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
trace_has_message (field_selector output) m trX
  subst trX.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX, item_sufX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
Hpr_item_suf: equivocators_trace_project IM
  final_descriptors ([item] ++ suf) =
Some (item_sufX, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
trace_has_message (field_selector output) m
  (preX ++ item_sufX)
  apply Exists_app.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX, item_sufX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
Hpr_item_suf: equivocators_trace_project IM
  final_descriptors ([item] ++ suf) =
Some (item_sufX, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Exists (field_selector output m) preX
∨ Exists (field_selector output m) item_sufX right.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX, item_sufX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
Hpr_item_suf: equivocators_trace_project IM
  final_descriptors ([item] ++ suf) =
Some (item_sufX, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Exists (field_selector output m) item_sufX
  apply equivocators_trace_project_app_iff in Hpr_item_suf.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX, item_sufX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
Hpr_item_suf: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  equivocators_trace_project IM
    final_descriptors suf =
  Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project IM
      eqv_descriptors' [item] =
    Some (preX, eqv_pre)
    ∧ item_sufX = preX ++ sufX
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Exists (field_selector output m) item_sufX
  destruct Hpr_item_suf as [itemXs [sufX [eqv_item [Hpr_suf [Hpr_item Heq_item_sufX]]]]].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX, item_sufX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = Some (itemXs, eqv_pre)
Heq_item_sufX: item_sufX = itemXs ++ sufX
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Exists (field_selector output m) item_sufX
  subst item_sufX.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Exists (field_selector output m) (itemXs ++ sufX)
  apply Exists_app.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Exists (field_selector output m) itemXs
∨ Exists (field_selector output m) sufX left.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Exists (field_selector output m) itemXs
  destruct Htr as [Htr _].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  (pre ++ [item] ++ suf)
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Exists (field_selector output m) itemXs
  rewrite app_assoc in Htr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Exists (field_selector output m) itemXs
  apply finite_valid_trace_from_app_iff in Htr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is (pre ++ [item])
∧ finite_valid_trace_from XE
    (finite_trace_last is (pre ++ [item])) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Exists (field_selector output m) itemXs
  destruct Htr as [Hpre Hsuf].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hpre: finite_valid_trace_from XE is (pre ++ [item])
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is (pre ++ [item])) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Exists (field_selector output m) itemXs
  apply finite_valid_trace_from_app_iff in Hpre.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hpre: finite_valid_trace_from XE is pre
∧ finite_valid_trace_from XE
    (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is (pre ++ [item])) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Exists (field_selector output m) itemXs
  destruct Hpre as [_ Hitem].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is (pre ++ [item])) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Exists (field_selector output m) itemXs
  rewrite app_assoc, finite_trace_last_app in Hproper.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is (pre ++ [item])) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last
     (finite_trace_last is (pre ++ [item]))
     suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Exists (field_selector output m) itemXs
  rewrite finite_trace_last_is_last in Hsuf, Hproper.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE 
  (destination item) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Exists (field_selector output m) itemXs
  assert (Hsufpre : finite_constrained_trace_from FreeE (destination item) suf).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE 
  (destination item) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
finite_constrained_trace_from FreeE (destination item)
  suf
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE 
  (destination item) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Hsufpre: finite_constrained_trace_from FreeE
  (destination item) suf
Exists (field_selector output m) itemXs
  {message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE 
  (destination item) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
finite_constrained_trace_from FreeE (destination item)
  suf
    eapply VLSM_incl_finite_valid_trace_from; [| done].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE 
  (destination item) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
VLSM_incl_part
  (constrained_vlsm_machine
     (free_composite_vlsm (equivocator_IM IM))
     (equivocators_fixed_equivocations_constraint IM
        (elements equivocating)))
  (preloaded_vlsm_machine FreeE (λ _ : message, True))
    by apply constrained_preloaded_incl.
  }message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE 
  (destination item) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Hsufpre: finite_constrained_trace_from FreeE
  (destination item) suf
Exists (field_selector output m) itemXs
  specialize
    (equivocators_trace_project_preserves_proper_fixed_equivocator_descriptors _ _ Hsufpre
      final_descriptors eqv_item _ Hpr_suf Hproper)
    as Hproper_item.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE 
  (destination item) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Hsufpre: finite_constrained_trace_from FreeE
  (destination item) suf
Hproper_item: proper_fixed_equivocator_descriptors
  eqv_item (destination item)
Exists (field_selector output m) itemXs
  destruct Hproper_item as [Hproper_item Hproper_fixed_item].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE 
  (destination item) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Hsufpre: finite_constrained_trace_from FreeE
  (destination item) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item (destination item)
Hproper_fixed_item: ∀ i : index,
  i ∉ equivocating
  → eqv_item i = Existing 0
Exists (field_selector output m) itemXs
  specialize (Hproper_fixed_item _ Hno_equiv_item).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE 
  (destination item) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Hsufpre: finite_constrained_trace_from FreeE
  (destination item) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item (destination item)
Hproper_fixed_item: eqv_item (projT1 (l item)) =
Existing 0
Exists (field_selector output m) itemXs
  specialize
    (no_equivocating_equivocators_transition_item_project IM eqv_item item
      Hproper_fixed_item)
    as Hex.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE 
  (destination item) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Hsufpre: finite_constrained_trace_from FreeE
  (destination item) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item (destination item)
Hproper_fixed_item: eqv_item (projT1 (l item)) =
Existing 0
Hex: is_singleton_state (IM (projT1 (l item)))
  (destination item (projT1 (l item)))
→ ∀ s : composite_state (equivocator_IM IM),
    composite_valid (equivocator_IM IM) 
      (l item) (s, input item)
    → composite_transition 
        (equivocator_IM IM) 
        (l item) (s, input item) =
      (destination item, output item)
      → ∃ Hex : existing_equivocator_label
                  (IM (projT1 (l item)))
                  (projT2 (l item)),
          let lx :=
            existT (projT1 (l item))
              (existing_equivocator_label_extract
                 (IM (projT1 (l item)))
                 (projT2 (l item)) Hex) in
          equivocators_transition_item_project
            IM eqv_item item =
          Some
            (Some
               {|
                 l := lx;
                 input := input item;
                 destination :=
                   equivocators_state_project IM
                     eqv_item 
                     (destination item);
                 output := output item
               |}, eqv_item)
Exists (field_selector output m) itemXs
  spec Hex.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE 
  (destination item) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Hsufpre: finite_constrained_trace_from FreeE
  (destination item) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item (destination item)
Hproper_fixed_item: eqv_item (projT1 (l item)) =
Existing 0
Hex: is_singleton_state (IM (projT1 (l item)))
  (destination item (projT1 (l item)))
→ ∀ s : composite_state (equivocator_IM IM),
    composite_valid (equivocator_IM IM) 
      (l item) (s, input item)
    → composite_transition 
        (equivocator_IM IM) 
        (l item) (s, input item) =
      (destination item, output item)
      → ∃ Hex : existing_equivocator_label
                  (IM (projT1 (l item)))
                  (projT2 (l item)),
          let lx :=
            existT (projT1 (l item))
              (existing_equivocator_label_extract
                 (IM (projT1 (l item)))
                 (projT2 (l item)) Hex) in
          equivocators_transition_item_project
            IM eqv_item item =
          Some
            (Some
               {|
                 l := lx;
                 input := input item;
                 destination :=
                   equivocators_state_project IM
                     eqv_item 
                     (destination item);
                 output := output item
               |}, eqv_item)
is_singleton_state (IM (projT1 (l item)))
  (destination item (projT1 (l item)))
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE 
  (destination item) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Hsufpre: finite_constrained_trace_from FreeE
  (destination item) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item (destination item)
Hproper_fixed_item: eqv_item (projT1 (l item)) =
Existing 0
Hex: ∀ s : composite_state (equivocator_IM IM),
  composite_valid (equivocator_IM IM) 
    (l item) (s, input item)
  → composite_transition 
      (equivocator_IM IM) 
      (l item) (s, input item) =
    (destination item, output item)
    → ∃ Hex : existing_equivocator_label
                (IM (projT1 (l item)))
                (projT2 (l item)),
        let lx :=
          existT (projT1 (l item))
            (existing_equivocator_label_extract
               (IM (projT1 (l item)))
               (projT2 (l item)) Hex) in
        equivocators_transition_item_project IM
          eqv_item item =
        Some
          (Some
             {|
               l := lx;
               input := input item;
               destination :=
                 equivocators_state_project IM
                   eqv_item 
                   (destination item);
               output := output item
             |}, eqv_item)
Exists (field_selector output m) itemXs
  {message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE 
  (destination item) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Hsufpre: finite_constrained_trace_from FreeE
  (destination item) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item (destination item)
Hproper_fixed_item: eqv_item (projT1 (l item)) =
Existing 0
Hex: is_singleton_state (IM (projT1 (l item)))
  (destination item (projT1 (l item)))
→ ∀ s : composite_state (equivocator_IM IM),
    composite_valid (equivocator_IM IM) 
      (l item) (s, input item)
    → composite_transition 
        (equivocator_IM IM) 
        (l item) (s, input item) =
      (destination item, output item)
      → ∃ Hex : existing_equivocator_label
                  (IM (projT1 (l item)))
                  (projT2 (l item)),
          let lx :=
            existT (projT1 (l item))
              (existing_equivocator_label_extract
                 (IM (projT1 (l item)))
                 (projT2 (l item)) Hex) in
          equivocators_transition_item_project
            IM eqv_item item =
          Some
            (Some
               {|
                 l := lx;
                 input := input item;
                 destination :=
                   equivocators_state_project IM
                     eqv_item 
                     (destination item);
                 output := output item
               |}, eqv_item)
is_singleton_state (IM (projT1 (l item)))
  (destination item (projT1 (l item)))
    apply (not_equivocating_index_has_singleton_state _ (elements equivocating));
      [| by rewrite elem_of_elements].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE 
  (destination item) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Hsufpre: finite_constrained_trace_from FreeE
  (destination item) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item (destination item)
Hproper_fixed_item: eqv_item (projT1 (l item)) =
Existing 0
Hex: is_singleton_state (IM (projT1 (l item)))
  (destination item (projT1 (l item)))
→ ∀ s : composite_state (equivocator_IM IM),
    composite_valid (equivocator_IM IM) 
      (l item) (s, input item)
    → composite_transition 
        (equivocator_IM IM) 
        (l item) (s, input item) =
      (destination item, output item)
      → ∃ Hex : existing_equivocator_label
                  (IM (projT1 (l item)))
                  (projT2 (l item)),
          let lx :=
            existT (projT1 (l item))
              (existing_equivocator_label_extract
                 (IM (projT1 (l item)))
                 (projT2 (l item)) Hex) in
          equivocators_transition_item_project
            IM eqv_item item =
          Some
            (Some
               {|
                 l := lx;
                 input := input item;
                 destination :=
                   equivocators_state_project IM
                     eqv_item 
                     (destination item);
                 output := output item
               |}, eqv_item)
valid_state_prop
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating)) (destination item)
    by apply finite_valid_trace_last_pstate in Hitem.
  }message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE 
  (destination item) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Hsufpre: finite_constrained_trace_from FreeE
  (destination item) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item (destination item)
Hproper_fixed_item: eqv_item (projT1 (l item)) =
Existing 0
Hex: ∀ s : composite_state (equivocator_IM IM),
  composite_valid (equivocator_IM IM) 
    (l item) (s, input item)
  → composite_transition 
      (equivocator_IM IM) 
      (l item) (s, input item) =
    (destination item, output item)
    → ∃ Hex : existing_equivocator_label
                (IM (projT1 (l item)))
                (projT2 (l item)),
        let lx :=
          existT (projT1 (l item))
            (existing_equivocator_label_extract
               (IM (projT1 (l item)))
               (projT2 (l item)) Hex) in
        equivocators_transition_item_project IM
          eqv_item item =
        Some
          (Some
             {|
               l := lx;
               input := input item;
               destination :=
                 equivocators_state_project IM
                   eqv_item 
                   (destination item);
               output := output item
             |}, eqv_item)
Exists (field_selector output m) itemXs
  destruct item as (l, iom, s, oom).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
l: label (composite_type (equivocator_IM IM))
iom: option message
s: state (composite_type (equivocator_IM IM))
oom: option message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre)
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}]
Hsuf: finite_valid_trace_from XE
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last
     (destination
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some m
Hno_equiv_item: projT1
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) ∉ equivocating
Hsufpre: finite_constrained_trace_from FreeE
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
Hproper_fixed_item: eqv_item
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |})) = Existing 0
Hex: ∀ s0 : composite_state (equivocator_IM IM),
  composite_valid (equivocator_IM IM)
    (VLSM.l
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
    (s0,
     input
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
  → composite_transition 
      (equivocator_IM IM)
      (VLSM.l
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |})
      (s0,
       input
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |}) =
    (destination
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |},
     output
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
    → ∃ Hex : existing_equivocator_label
                (IM
                   (projT1
                      (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
                (projT2
                   (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})),
        let lx :=
          existT
            (projT1
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |}))
            (existing_equivocator_label_extract
               (IM
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})) Hex) in
        equivocators_transition_item_project IM
          eqv_item
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} =
        Some
          (Some
             {|
               l := lx;
               input :=
                 input
                   {|
                     l := l;
                     input := iom;
                     destination := s;
                     output := oom
                   |};
               destination :=
                 equivocators_state_project IM
                   eqv_item
                   (destination
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |});
               output :=
                 output
                   {|
                     l := l;
                     input := iom;
                     destination := s;
                     output := oom
                   |}
             |}, eqv_item)
Exists (field_selector output m) itemXs apply first_transition_valid in Hitem.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
l: label (composite_type (equivocator_IM IM))
iom: option message
s: state (composite_type (equivocator_IM IM))
oom: option message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: input_valid_transition XE
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  (finite_trace_last is pre,
   input
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |},
   output
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
Hsuf: finite_valid_trace_from XE
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last
     (destination
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some m
Hno_equiv_item: projT1
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) ∉ equivocating
Hsufpre: finite_constrained_trace_from FreeE
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
Hproper_fixed_item: eqv_item
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |})) = Existing 0
Hex: ∀ s0 : composite_state (equivocator_IM IM),
  composite_valid (equivocator_IM IM)
    (VLSM.l
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
    (s0,
     input
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
  → composite_transition 
      (equivocator_IM IM)
      (VLSM.l
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |})
      (s0,
       input
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |}) =
    (destination
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |},
     output
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
    → ∃ Hex : existing_equivocator_label
                (IM
                   (projT1
                      (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
                (projT2
                   (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})),
        let lx :=
          existT
            (projT1
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |}))
            (existing_equivocator_label_extract
               (IM
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})) Hex) in
        equivocators_transition_item_project IM
          eqv_item
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} =
        Some
          (Some
             {|
               l := lx;
               input :=
                 input
                   {|
                     l := l;
                     input := iom;
                     destination := s;
                     output := oom
                   |};
               destination :=
                 equivocators_state_project IM
                   eqv_item
                   (destination
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |});
               output :=
                 output
                   {|
                     l := l;
                     input := iom;
                     destination := s;
                     output := oom
                   |}
             |}, eqv_item)
Exists (field_selector output m) itemXs simpl in Hitem.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
l: label (composite_type (equivocator_IM IM))
iom: option message
s: state (composite_type (equivocator_IM IM))
oom: option message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: input_valid_transition XE l
  (finite_trace_last is pre, iom) (
  s, oom)
Hsuf: finite_valid_trace_from XE
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last
     (destination
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some m
Hno_equiv_item: projT1
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) ∉ equivocating
Hsufpre: finite_constrained_trace_from FreeE
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
Hproper_fixed_item: eqv_item
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |})) = Existing 0
Hex: ∀ s0 : composite_state (equivocator_IM IM),
  composite_valid (equivocator_IM IM)
    (VLSM.l
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
    (s0,
     input
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
  → composite_transition 
      (equivocator_IM IM)
      (VLSM.l
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |})
      (s0,
       input
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |}) =
    (destination
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |},
     output
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
    → ∃ Hex : existing_equivocator_label
                (IM
                   (projT1
                      (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
                (projT2
                   (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})),
        let lx :=
          existT
            (projT1
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |}))
            (existing_equivocator_label_extract
               (IM
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})) Hex) in
        equivocators_transition_item_project IM
          eqv_item
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} =
        Some
          (Some
             {|
               l := lx;
               input :=
                 input
                   {|
                     l := l;
                     input := iom;
                     destination := s;
                     output := oom
                   |};
               destination :=
                 equivocators_state_project IM
                   eqv_item
                   (destination
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |});
               output :=
                 output
                   {|
                     l := l;
                     input := iom;
                     destination := s;
                     output := oom
                   |}
             |}, eqv_item)
Exists (field_selector output m) itemXs
  destruct Hitem as [[Hs [Hiom [Hv Hc]]] Ht].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
l: label (composite_type (equivocator_IM IM))
iom: option message
s: state (composite_type (equivocator_IM IM))
oom: option message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hs: valid_state_prop XE (finite_trace_last is pre)
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is pre, iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is pre, iom)
Ht: transition l (finite_trace_last is pre, iom) =
(s, oom)
Hsuf: finite_valid_trace_from XE
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last
     (destination
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some m
Hno_equiv_item: projT1
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) ∉ equivocating
Hsufpre: finite_constrained_trace_from FreeE
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
Hproper_fixed_item: eqv_item
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |})) = Existing 0
Hex: ∀ s0 : composite_state (equivocator_IM IM),
  composite_valid (equivocator_IM IM)
    (VLSM.l
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
    (s0,
     input
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
  → composite_transition 
      (equivocator_IM IM)
      (VLSM.l
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |})
      (s0,
       input
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |}) =
    (destination
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |},
     output
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
    → ∃ Hex : existing_equivocator_label
                (IM
                   (projT1
                      (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
                (projT2
                   (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})),
        let lx :=
          existT
            (projT1
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |}))
            (existing_equivocator_label_extract
               (IM
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})) Hex) in
        equivocators_transition_item_project IM
          eqv_item
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} =
        Some
          (Some
             {|
               l := lx;
               input :=
                 input
                   {|
                     l := l;
                     input := iom;
                     destination := s;
                     output := oom
                   |};
               destination :=
                 equivocators_state_project IM
                   eqv_item
                   (destination
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |});
               output :=
                 output
                   {|
                     l := l;
                     input := iom;
                     destination := s;
                     output := oom
                   |}
             |}, eqv_item)
Exists (field_selector output m) itemXs
  simpl in Hex.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
l: label (composite_type (equivocator_IM IM))
iom: option message
s: state (composite_type (equivocator_IM IM))
oom: option message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hs: valid_state_prop XE (finite_trace_last is pre)
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is pre, iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is pre, iom)
Ht: transition l (finite_trace_last is pre, iom) =
(s, oom)
Hsuf: finite_valid_trace_from XE
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last
     (destination
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some m
Hno_equiv_item: projT1
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) ∉ equivocating
Hsufpre: finite_constrained_trace_from FreeE
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
Hproper_fixed_item: eqv_item
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |})) = Existing 0
Hex: ∀ s0 : composite_state (equivocator_IM IM),
  (let (i, li) := l in
   equivocator_valid (IM i) li (s0 i, iom))
  → (let (i, li) := l in
     let (si', om') :=
       equivocator_transition 
         (IM i) li (s0 i, iom) in
     (state_update (equivocator_IM IM) s0 i si',
      om')) = (s, oom)
    → ∃ Hex : existing_equivocator_label
                (IM (projT1 l)) 
                (projT2 l),
        equivocators_transition_item_project IM
          eqv_item
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} =
        Some
          (Some
             {|
               l :=
                 existT 
                   (projT1 l)
                   (existing_equivocator_label_extract
                      (IM (projT1 l)) 
                      (projT2 l) Hex);
               input := iom;
               destination :=
                 equivocators_state_project IM
                   eqv_item s;
               output := oom
             |}, eqv_item)
Exists (field_selector output m) itemXs
  specialize (Hex _ Hv Ht) as [Hex' Hex].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
l: label (composite_type (equivocator_IM IM))
iom: option message
s: state (composite_type (equivocator_IM IM))
oom: option message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hs: valid_state_prop XE (finite_trace_last is pre)
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is pre, iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is pre, iom)
Ht: transition l (finite_trace_last is pre, iom) =
(s, oom)
Hsuf: finite_valid_trace_from XE
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last
     (destination
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some m
Hno_equiv_item: projT1
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) ∉ equivocating
Hsufpre: finite_constrained_trace_from FreeE
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
Hproper_fixed_item: eqv_item
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |})) = Existing 0
Hex': existing_equivocator_label 
  (IM (projT1 l)) (projT2 l)
Hex: equivocators_transition_item_project IM eqv_item
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some
  (Some
     {|
       l :=
         existT (projT1 l)
           (existing_equivocator_label_extract
              (IM (projT1 l)) 
              (projT2 l) Hex');
       input := iom;
       destination :=
         equivocators_state_project IM eqv_item
           s;
       output := oom
     |}, eqv_item)
Exists (field_selector output m) itemXs simpl in Hex.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
l: label (composite_type (equivocator_IM IM))
iom: option message
s: state (composite_type (equivocator_IM IM))
oom: option message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hs: valid_state_prop XE (finite_trace_last is pre)
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is pre, iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is pre, iom)
Ht: transition l (finite_trace_last is pre, iom) =
(s, oom)
Hsuf: finite_valid_trace_from XE
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last
     (destination
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some m
Hno_equiv_item: projT1
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) ∉ equivocating
Hsufpre: finite_constrained_trace_from FreeE
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
Hproper_fixed_item: eqv_item
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |})) = Existing 0
Hex': existing_equivocator_label 
  (IM (projT1 l)) (projT2 l)
Hex: equivocators_transition_item_project IM eqv_item
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some
  (Some
     {|
       l :=
         existT (projT1 l)
           (existing_equivocator_label_extract
              (IM (projT1 l)) 
              (projT2 l) Hex');
       input := iom;
       destination :=
         equivocators_state_project IM eqv_item
           s;
       output := oom
     |}, eqv_item)
Exists (field_selector output m) itemXs
  simpl in Hpr_item.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
l: label (composite_type (equivocator_IM IM))
iom: option message
s: state (composite_type (equivocator_IM IM))
oom: option message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hs: valid_state_prop XE (finite_trace_last is pre)
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is pre, iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is pre, iom)
Ht: transition l (finite_trace_last is pre, iom) =
(s, oom)
Hsuf: finite_valid_trace_from XE
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last
     (destination
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: match
  equivocators_transition_item_project IM
    eqv_item
    {|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) =>
    Some ([], odescriptor)
| None => None
end = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some m
Hno_equiv_item: projT1
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) ∉ equivocating
Hsufpre: finite_constrained_trace_from FreeE
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
Hproper_fixed_item: eqv_item
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |})) = Existing 0
Hex': existing_equivocator_label 
  (IM (projT1 l)) (projT2 l)
Hex: equivocators_transition_item_project IM eqv_item
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some
  (Some
     {|
       l :=
         existT (projT1 l)
           (existing_equivocator_label_extract
              (IM (projT1 l)) 
              (projT2 l) Hex');
       input := iom;
       destination :=
         equivocators_state_project IM eqv_item
           s;
       output := oom
     |}, eqv_item)
Exists (field_selector output m) itemXs rewrite Hex in Hpr_item.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
l: label (composite_type (equivocator_IM IM))
iom: option message
s: state (composite_type (equivocator_IM IM))
oom: option message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hs: valid_state_prop XE (finite_trace_last is pre)
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is pre, iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is pre, iom)
Ht: transition l (finite_trace_last is pre, iom) =
(s, oom)
Hsuf: finite_valid_trace_from XE
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last
     (destination
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hex': existing_equivocator_label 
  (IM (projT1 l)) (projT2 l)
Hpr_item: Some
  ([{|
      l :=
        existT (projT1 l)
          (existing_equivocator_label_extract
             (IM (projT1 l)) 
             (projT2 l) Hex');
      input := iom;
      destination :=
        equivocators_state_project IM
          eqv_item s;
      output := oom
    |}], eqv_item) = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some m
Hno_equiv_item: projT1
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) ∉ equivocating
Hsufpre: finite_constrained_trace_from FreeE
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
Hproper_fixed_item: eqv_item
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |})) = Existing 0
Hex: equivocators_transition_item_project IM eqv_item
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some
  (Some
     {|
       l :=
         existT (projT1 l)
           (existing_equivocator_label_extract
              (IM (projT1 l)) 
              (projT2 l) Hex');
       input := iom;
       destination :=
         equivocators_state_project IM eqv_item
           s;
       output := oom
     |}, eqv_item)
Exists (field_selector output m) itemXs
  inversion_clear Hpr_item.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
l: label (composite_type (equivocator_IM IM))
iom: option message
s: state (composite_type (equivocator_IM IM))
oom: option message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hs: valid_state_prop XE (finite_trace_last is pre)
Hiom: option_valid_message_prop XE iom
Hv: valid l (finite_trace_last is pre, iom)
Hc: equivocators_fixed_equivocations_constraint IM
  (elements equivocating) l
  (finite_trace_last is pre, iom)
Ht: transition l (finite_trace_last is pre, iom) =
(s, oom)
Hsuf: finite_valid_trace_from XE
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last
     (destination
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hex': existing_equivocator_label 
  (IM (projT1 l)) (projT2 l)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some m
Hno_equiv_item: projT1
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) ∉ equivocating
Hsufpre: finite_constrained_trace_from FreeE
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
Hproper_fixed_item: eqv_item
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |})) = Existing 0
Hex: equivocators_transition_item_project IM eqv_item
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some
  (Some
     {|
       l :=
         existT (projT1 l)
           (existing_equivocator_label_extract
              (IM (projT1 l)) 
              (projT2 l) Hex');
       input := iom;
       destination :=
         equivocators_state_project IM eqv_item
           s;
       output := oom
     |}, eqv_item)
Exists (field_selector output m)
  [{|
     l :=
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) (projT2 l) Hex');
     input := iom;
     destination :=
       equivocators_state_project IM eqv_pre s;
     output := oom
   |}]
  by constructor.
Qed.

As a consequence of the equivocator_vlsm_trace_project_reflect_non_equivocating lemma, if a message emmitted by a trace cannot be directly observed in a projection of the trace's final state, then it must be that it was emitted by one of the components allowed to equivocate.

Lemma projection_has_not_been_directly_observed_is_equivocating
  (is : composite_state (equivocator_IM IM))
  (tr : list (composite_transition_item (equivocator_IM IM)))
  (Htr : finite_valid_trace XE is tr)
  (s := @finite_trace_last _ (composite_type (equivocator_IM IM)) is tr)
  (descriptors : equivocator_descriptors IM)
  (Hproper : proper_fixed_equivocator_descriptors descriptors s)
  (sX := equivocators_state_project IM descriptors s)
  (m : message)
  (Hno : ~ composite_has_been_directly_observed IM sX m)
  : forall item : composite_transition_item (equivocator_IM IM),
      item ∈ tr -> output item = Some m -> projT1 (l item) ∈ equivocating.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
s:= finite_trace_last is tr: state (composite_type (equivocator_IM IM))
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
sX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
m: message
Hno: ¬ composite_has_been_directly_observed IM sX m
∀ item : composite_transition_item (equivocator_IM IM),
  item ∈ tr
  → output item = Some m
    → projT1 (l item) ∈ equivocating
Proof.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
s:= finite_trace_last is tr: state (composite_type (equivocator_IM IM))
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
sX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
m: message
Hno: ¬ composite_has_been_directly_observed IM sX m
∀ item : composite_transition_item (equivocator_IM IM),
  item ∈ tr
  → output item = Some m
    → projT1 (l item) ∈ equivocating
  destruct (free_equivocators_valid_trace_project descriptors is tr Hproper Htr)
    as [trX [initial_descriptors [_ [Htr_project [Hlast_state HtrX]]]]].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
s:= finite_trace_last is tr: state (composite_type (equivocator_IM IM))
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
sX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
m: message
Hno: ¬ composite_has_been_directly_observed IM sX m
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
∀ item : composite_transition_item (equivocator_IM IM),
  item ∈ tr
  → output item = Some m
    → projT1 (l item) ∈ equivocating
  intros item Hitem Houtput.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
s:= finite_trace_last is tr: state (composite_type (equivocator_IM IM))
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
sX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
m: message
Hno: ¬ composite_has_been_directly_observed IM sX m
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
projT1 (l item) ∈ equivocating
  destruct (decide (projT1 (l item) ∈ elements equivocating));
    [by apply elem_of_elements |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
s:= finite_trace_last is tr: state (composite_type (equivocator_IM IM))
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
sX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
m: message
Hno: ¬ composite_has_been_directly_observed IM sX m
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ elements equivocating
projT1 (l item) ∈ equivocating
  elim Hno.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
s:= finite_trace_last is tr: state (composite_type (equivocator_IM IM))
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
sX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
m: message
Hno: ¬ composite_has_been_directly_observed IM sX m
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ elements equivocating
composite_has_been_directly_observed IM sX m clear Hno.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
s:= finite_trace_last is tr: state (composite_type (equivocator_IM IM))
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
sX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
m: message
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ elements equivocating
composite_has_been_directly_observed IM sX m
  apply composite_has_been_directly_observed_sent_received_iff.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
s:= finite_trace_last is tr: state (composite_type (equivocator_IM IM))
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
sX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
m: message
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ elements equivocating
composite_has_been_sent IM sX m
∨ composite_has_been_received IM sX m
  left.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
s:= finite_trace_last is tr: state (composite_type (equivocator_IM IM))
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
sX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
m: message
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ elements equivocating
composite_has_been_sent IM sX m
  assert (HtrX_free : finite_constrained_trace Free
    (equivocators_state_project IM initial_descriptors is) trX).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
s:= finite_trace_last is tr: state (composite_type (equivocator_IM IM))
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
sX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
m: message
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ elements equivocating
finite_constrained_trace Free
  (equivocators_state_project IM initial_descriptors
     is) trX
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
s:= finite_trace_last is tr: state (composite_type (equivocator_IM IM))
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
sX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
m: message
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ elements equivocating
HtrX_free: finite_constrained_trace Free
  (equivocators_state_project IM
     initial_descriptors is) trX
composite_has_been_sent IM sX m
  {message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
s:= finite_trace_last is tr: state (composite_type (equivocator_IM IM))
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
sX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
m: message
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ elements equivocating
finite_constrained_trace Free
  (equivocators_state_project IM initial_descriptors
     is) trX
    revert HtrX; apply VLSM_incl_finite_valid_trace.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
s:= finite_trace_last is tr: state (composite_type (equivocator_IM IM))
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
sX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
m: message
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ elements equivocating
VLSM_incl_part (free_composite_vlsm_machine IM)
  (preloaded_vlsm_machine Free (λ _ : message, True))
    by apply vlsm_incl_preloaded_with_all_messages_vlsm.
  }message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
s:= finite_trace_last is tr: state (composite_type (equivocator_IM IM))
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
sX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
m: message
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ elements equivocating
HtrX_free: finite_constrained_trace Free
  (equivocators_state_project IM
     initial_descriptors is) trX
composite_has_been_sent IM sX m
  specialize (finite_valid_trace_last_pstate _ _ _ (proj1 HtrX_free)) as Hfree_lst.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
s:= finite_trace_last is tr: state (composite_type (equivocator_IM IM))
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
sX:= equivocators_state_project IM descriptors s: state (free_composite_vlsm IM)
m: message
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ elements equivocating
HtrX_free: finite_constrained_trace Free
  (equivocators_state_project IM
     initial_descriptors is) trX
Hfree_lst: valid_state_prop
  (preloaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
composite_has_been_sent IM sX m
  subst sX s.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
m: message
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ elements equivocating
HtrX_free: finite_constrained_trace Free
  (equivocators_state_project IM
     initial_descriptors is) trX
Hfree_lst: valid_state_prop
  (preloaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
composite_has_been_sent IM
  (equivocators_state_project IM descriptors
     (finite_trace_last is tr)) m
  simpl in *.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
m: message
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ elements equivocating
HtrX_free: finite_constrained_trace Free
  (equivocators_state_project IM
     initial_descriptors is) trX
Hfree_lst: valid_state_prop
  (preloaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
composite_has_been_sent IM
  (equivocators_state_project IM descriptors
     (finite_trace_last is tr)) m
  rewrite Hlast_state.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
m: message
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ elements equivocating
HtrX_free: finite_constrained_trace Free
  (equivocators_state_project IM
     initial_descriptors is) trX
Hfree_lst: valid_state_prop
  (preloaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
composite_has_been_sent IM
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) m
  apply (composite_proper_sent IM); [done |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
m: message
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ elements equivocating
HtrX_free: finite_constrained_trace Free
  (equivocators_state_project IM
     initial_descriptors is) trX
Hfree_lst: valid_state_prop
  (preloaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
selected_message_exists_in_all_preloaded_traces
  (free_composite_vlsm IM) (field_selector output)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) m
  apply has_been_sent_consistency; [by typeclasses eauto | done |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
m: message
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ elements equivocating
HtrX_free: finite_constrained_trace Free
  (equivocators_state_project IM
     initial_descriptors is) trX
Hfree_lst: valid_state_prop
  (preloaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
selected_message_exists_in_some_preloaded_traces
  (free_composite_vlsm IM) (field_selector output)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) m
  red in HtrX_free.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
m: message
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ elements equivocating
HtrX_free: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hfree_lst: valid_state_prop
  (preloaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
selected_message_exists_in_some_preloaded_traces
  (free_composite_vlsm IM) (field_selector output)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) m
  exists (equivocators_state_project IM initial_descriptors is), trX,
    (valid_trace_add_default_last HtrX_free).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
m: message
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ elements equivocating
HtrX_free: finite_valid_trace
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hfree_lst: valid_state_prop
  (preloaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
trace_has_message (field_selector output) m trX
  clear HtrX HtrX_free Hfree_lst.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
m: message
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ elements equivocating
trace_has_message (field_selector output) m trX
  by eapply equivocator_vlsm_trace_project_reflect_non_equivocating;
    [.. | contradict n; apply elem_of_elements].
Qed.

The next two lemmas are rather technical, but their basic meaning is that if we take the composition of a subset of equivocators, it doesn't matter the way we do it (either first obtain the indexed subset of components and then transform that into equivocators and take their composition, or we first start with the indexed full set of equivocators and then select a subset of them and take their composition).

Lemma preloaded_equivocators_composition_sub_projection_commute
  (seed1 : message -> Prop)
  (seed2 : message -> Prop)
  (Hseed12 : forall m, seed1 m -> seed2 m)
  : VLSM_incl
    (composite_no_equivocation_vlsm_with_preloaded
      (sub_IM (equivocator_IM IM) (elements equivocating))
      (free_constraint _)
      seed1)
    (composite_no_equivocation_vlsm_with_preloaded
      (equivocator_IM (sub_IM IM (elements equivocating)))
      (free_constraint _)
      seed2).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating))) seed1)
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM (elements equivocating)))) seed2)
Proof.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating))) seed1)
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM (elements equivocating)))) seed2)
  apply basic_VLSM_incl.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
strong_incl_initial_state_preservation
  (preloaded_vlsm_machine
     (composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1))
     seed1)
  (preloaded_vlsm_machine
     (composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM (elements equivocating))))
           seed2)) seed2)
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
weak_incl_initial_message_preservation
  (preloaded_vlsm_machine
     (composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1))
     seed1)
  (preloaded_vlsm_machine
     (composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM (elements equivocating))))
           seed2)) seed2)
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
weak_incl_valid_preservation
  (preloaded_vlsm_machine
     (composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1))
     seed1)
  (preloaded_vlsm_machine
     (composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM (elements equivocating))))
           seed2)) seed2)
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
weak_incl_transition_preservation
  (preloaded_vlsm_machine
     (composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1))
     seed1)
  (preloaded_vlsm_machine
     (composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM (elements equivocating))))
           seed2)) seed2)
  -message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
strong_incl_initial_state_preservation
  (preloaded_vlsm_machine
     (composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1))
     seed1)
  (preloaded_vlsm_machine
     (composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM (elements equivocating))))
           seed2)) seed2) by intro; intros.
  -message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
weak_incl_initial_message_preservation
  (preloaded_vlsm_machine
     (composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1))
     seed1)
  (preloaded_vlsm_machine
     (composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM (elements equivocating))))
           seed2)) seed2) intro; intros.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
m: message
Hv: input_valid
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |} l (s, Some m)
HsY: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} (id s)
HmX: initial_message_prop m
valid_message_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM (elements equivocating))))
              seed2)) seed2
  |} m
    apply initial_message_is_valid; cbn.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
m: message
Hv: input_valid
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |} l (s, Some m)
HsY: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} (id s)
HmX: initial_message_prop m
composite_initial_message_prop
  (equivocator_IM (sub_IM IM (elements equivocating)))
  m ∨ seed2 m
    by destruct HmX; auto.
  -message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
weak_incl_valid_preservation
  (preloaded_vlsm_machine
     (composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1))
     seed1)
  (preloaded_vlsm_machine
     (composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM (elements equivocating))))
           seed2)) seed2) intros l s om [Hs [_ [Hv Hc]]].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
om: option message
Hs: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |} s
Hv: valid l (s, om)
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM)
     (elements equivocating))
  (free_constraint
     (sub_IM (equivocator_IM IM)
        (elements equivocating))) seed1 l (
  s, om)
valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM (elements equivocating))))
              seed2)) seed2
  |} (id s)
→ option_valid_message_prop
    {|
      vlsm_type :=
        composite_vlsm
          (sub_IM (equivocator_IM IM)
             (elements equivocating))
          (no_equivocations_additional_constraint_with_preloaded
             (sub_IM (equivocator_IM IM)
                (elements equivocating))
             (free_constraint
                (sub_IM (equivocator_IM IM)
                   (elements equivocating))) seed1);
      vlsm_machine :=
        preloaded_vlsm_machine
          (composite_vlsm
             (equivocator_IM
                (sub_IM IM (elements equivocating)))
             (no_equivocations_additional_constraint_with_preloaded
                (equivocator_IM
                   (sub_IM IM (elements equivocating)))
                (free_constraint
                   (equivocator_IM
                      (sub_IM IM
                         (elements equivocating))))
                seed2)) seed2
    |} om → valid (id l) (id s, om)
    split; [done |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
om: option message
Hs: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |} s
Hv: valid l (s, om)
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM)
     (elements equivocating))
  (free_constraint
     (sub_IM (equivocator_IM IM)
        (elements equivocating))) seed1 l (
  s, om)
HsY: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} (id s)
HomY: option_valid_message_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating)))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} om
no_equivocations_additional_constraint_with_preloaded
  (equivocator_IM (sub_IM IM (elements equivocating)))
  (free_constraint
     (equivocator_IM
        (sub_IM IM (elements equivocating)))) seed2
  (id l) (id s, om)
    destruct Hc as [Hc _].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
om: option message
Hs: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |} s
Hv: valid l (s, om)
Hc: composite_no_equivocations_except_from
  (sub_IM (equivocator_IM IM)
     (elements equivocating)) seed1 l (
  s, om)
HsY: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} (id s)
HomY: option_valid_message_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating)))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} om
no_equivocations_additional_constraint_with_preloaded
  (equivocator_IM (sub_IM IM (elements equivocating)))
  (free_constraint
     (equivocator_IM
        (sub_IM IM (elements equivocating)))) seed2
  (id l) (id s, om)
    split; [| done].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
om: option message
Hs: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |} s
Hv: valid l (s, om)
Hc: composite_no_equivocations_except_from
  (sub_IM (equivocator_IM IM)
     (elements equivocating)) seed1 l (
  s, om)
HsY: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} (id s)
HomY: option_valid_message_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating)))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} om
composite_no_equivocations_except_from
  (equivocator_IM (sub_IM IM (elements equivocating)))
  seed2 (id l) (id s, om)
    destruct om; [| done].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
m: message
Hs: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |} s
Hv: valid l (s, Some m)
Hc: composite_no_equivocations_except_from
  (sub_IM (equivocator_IM IM)
     (elements equivocating)) seed1 l (
  s, Some m)
HsY: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} (id s)
HomY: option_valid_message_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating)))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} (Some m)
composite_no_equivocations_except_from
  (equivocator_IM (sub_IM IM (elements equivocating)))
  seed2 (id l) (id s, Some m)
    simpl in Hc.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
m: message
Hs: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |} s
Hv: valid l (s, Some m)
Hc: composite_no_equivocations_except_from
  (sub_IM (equivocator_IM IM)
     (elements equivocating)) seed1 l (
  s, Some m)
HsY: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} (id s)
HomY: option_valid_message_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating)))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} (Some m)
composite_no_equivocations_except_from
  (equivocator_IM (sub_IM IM (elements equivocating)))
  seed2 (id l) (id s, Some m) simpl.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
m: message
Hs: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |} s
Hv: valid l (s, Some m)
Hc: composite_no_equivocations_except_from
  (sub_IM (equivocator_IM IM)
     (elements equivocating)) seed1 l (
  s, Some m)
HsY: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} (id s)
HomY: option_valid_message_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating)))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} (Some m)
composite_no_equivocations_except_from
  (equivocator_IM (sub_IM IM (elements equivocating)))
  seed2 l (s, Some m)
    destruct Hc as [Hc | Hc]
    ; [| by right; apply Hseed12].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
m: message
Hs: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |} s
Hv: valid l (s, Some m)
Hc: composite_has_been_sent
  (sub_IM (equivocator_IM IM)
     (elements equivocating)) (
  s, Some m).1 m
HsY: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} (id s)
HomY: option_valid_message_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating)))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} (Some m)
composite_no_equivocations_except_from
  (equivocator_IM (sub_IM IM (elements equivocating)))
  seed2 l (s, Some m)
    left.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
m: message
Hs: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |} s
Hv: valid l (s, Some m)
Hc: composite_has_been_sent
  (sub_IM (equivocator_IM IM)
     (elements equivocating)) (
  s, Some m).1 m
HsY: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} (id s)
HomY: option_valid_message_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating)))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} (Some m)
composite_has_been_sent
  (equivocator_IM (sub_IM IM (elements equivocating)))
  (s, Some m).1 m
    destruct Hc as [subi Hibs].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
m: message
Hs: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |} s
Hv: valid l (s, Some m)
subi: sub_index (elements equivocating)
Hibs: has_been_sent
  (sub_IM (equivocator_IM IM)
     (elements equivocating) subi)
  ((s, Some m).1 subi) m
HsY: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} (id s)
HomY: option_valid_message_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating)))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} (Some m)
composite_has_been_sent
  (equivocator_IM (sub_IM IM (elements equivocating)))
  (s, Some m).1 m
    exists subi.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
m: message
Hs: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |} s
Hv: valid l (s, Some m)
subi: sub_index (elements equivocating)
Hibs: has_been_sent
  (sub_IM (equivocator_IM IM)
     (elements equivocating) subi)
  ((s, Some m).1 subi) m
HsY: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} (id s)
HomY: option_valid_message_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating)))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} (Some m)
has_been_sent
  (equivocator_IM (sub_IM IM (elements equivocating))
     subi) ((s, Some m).1 subi) m revert Hibs.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
m: message
Hs: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |} s
Hv: valid l (s, Some m)
subi: sub_index (elements equivocating)
HsY: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} (id s)
HomY: option_valid_message_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating)))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} (Some m)
has_been_sent
  (sub_IM (equivocator_IM IM) (elements equivocating)
     subi) ((s, Some m).1 subi) m
→ has_been_sent
    (equivocator_IM
       (sub_IM IM (elements equivocating)) subi)
    ((s, Some m).1 subi) m
    apply has_been_sent_irrelevance.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
m: message
Hs: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |} s
Hv: valid l (s, Some m)
subi: sub_index (elements equivocating)
HsY: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} (id s)
HomY: option_valid_message_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating)))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} (Some m)
constrained_state_prop
  (equivocator_IM (sub_IM IM (elements equivocating))
     subi) ((s, Some m).1 subi)
    simpl.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
m: message
Hs: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |} s
Hv: valid l (s, Some m)
subi: sub_index (elements equivocating)
HsY: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} (id s)
HomY: option_valid_message_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating)))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} (Some m)
constrained_state_prop
  (equivocator_IM (sub_IM IM (elements equivocating))
     subi) (s subi)
    apply (preloaded_valid_state_projection (equivocator_IM (sub_IM IM (elements equivocating))) subi).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
m: message
Hs: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |} s
Hv: valid l (s, Some m)
subi: sub_index (elements equivocating)
HsY: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} (id s)
HomY: option_valid_message_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating)))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} (Some m)
constrained_state_prop
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating)))) s
    revert Hs.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
m: message
Hv: valid l (s, Some m)
subi: sub_index (elements equivocating)
HsY: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} (id s)
HomY: option_valid_message_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating)))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} (Some m)
valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM (equivocator_IM IM)
                    (elements equivocating))) seed1))
        seed1
  |} s
→ constrained_state_prop
    (free_composite_vlsm
       (equivocator_IM
          (sub_IM IM (elements equivocating)))) s
    apply VLSM_incl_valid_state.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed1)) seed1
  |}
m: message
Hv: valid l (s, Some m)
subi: sub_index (elements equivocating)
HsY: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} (id s)
HomY: option_valid_message_prop
  {|
    vlsm_type :=
      composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating)))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed2)) seed2
  |} (Some m)
VLSM_incl_part
  (preloaded_vlsm_machine
     (composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1))
     seed1)
  (preloaded_vlsm_machine
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating))))
     (λ _ : message, True))
    by apply constrained_preloaded_vlsm_incl_preloaded_with_all_messages.
  -message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
weak_incl_transition_preservation
  (preloaded_vlsm_machine
     (composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) seed1))
     seed1)
  (preloaded_vlsm_machine
     (composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM (elements equivocating))))
           seed2)) seed2) by destruct 1.
Qed.

Lemma preloaded_equivocators_composition_sub_projection_commute_inv
  (seed1 : message -> Prop)
  (seed2 : message -> Prop)
  (Hseed12 : forall m, seed1 m -> seed2 m)
  : VLSM_incl
    (composite_no_equivocation_vlsm_with_preloaded
      (equivocator_IM (sub_IM IM (elements equivocating)))
      (free_constraint _)
      seed1)
    (composite_no_equivocation_vlsm_with_preloaded
      (sub_IM (equivocator_IM IM) (elements equivocating))
      (free_constraint _)
      seed2).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM (elements equivocating)))) seed1)
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating))) seed2)
Proof.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM (elements equivocating)))) seed1)
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating))) seed2)
  apply basic_VLSM_incl; intros ? *; [done | | | by destruct 1].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
m: message
input_valid
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM (elements equivocating))))
              seed1)) seed1
  |} l (s, Some m)
→ valid_state_prop
    {|
      vlsm_type :=
        composite_vlsm
          (equivocator_IM
             (sub_IM IM (elements equivocating)))
          (no_equivocations_additional_constraint_with_preloaded
             (equivocator_IM
                (sub_IM IM (elements equivocating)))
             (free_constraint
                (equivocator_IM
                   (sub_IM IM (elements equivocating))))
             seed1);
      vlsm_machine :=
        preloaded_vlsm_machine
          (composite_vlsm
             (sub_IM (equivocator_IM IM)
                (elements equivocating))
             (no_equivocations_additional_constraint_with_preloaded
                (sub_IM (equivocator_IM IM)
                   (elements equivocating))
                (free_constraint
                   (sub_IM (equivocator_IM IM)
                      (elements equivocating))) seed2))
          seed2
    |} (id s)
  → initial_message_prop m
    → valid_message_prop
        {|
          vlsm_type :=
            composite_vlsm
              (equivocator_IM
                 (sub_IM IM (elements equivocating)))
              (no_equivocations_additional_constraint_with_preloaded
                 (equivocator_IM
                    (sub_IM IM (elements equivocating)))
                 (free_constraint
                    (equivocator_IM
                       (sub_IM IM
                          (elements equivocating))))
                 seed1);
          vlsm_machine :=
            preloaded_vlsm_machine
              (composite_vlsm
                 (sub_IM (equivocator_IM IM)
                    (elements equivocating))
                 (no_equivocations_additional_constraint_with_preloaded
                    (sub_IM (equivocator_IM IM)
                       (elements equivocating))
                    (free_constraint
                       (sub_IM (equivocator_IM IM)
                          (elements equivocating)))
                    seed2)) seed2
        |} m
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
om: option message
input_valid
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM (elements equivocating))))
              seed1)) seed1
  |} l (s, om)
→ valid_state_prop
    {|
      vlsm_type :=
        composite_vlsm
          (equivocator_IM
             (sub_IM IM (elements equivocating)))
          (no_equivocations_additional_constraint_with_preloaded
             (equivocator_IM
                (sub_IM IM (elements equivocating)))
             (free_constraint
                (equivocator_IM
                   (sub_IM IM (elements equivocating))))
             seed1);
      vlsm_machine :=
        preloaded_vlsm_machine
          (composite_vlsm
             (sub_IM (equivocator_IM IM)
                (elements equivocating))
             (no_equivocations_additional_constraint_with_preloaded
                (sub_IM 
                   (equivocator_IM IM)
                   (elements equivocating))
                (free_constraint
                   (sub_IM 
                      (equivocator_IM IM)
                      (elements equivocating))) seed2))
          seed2
    |} (id s)
  → option_valid_message_prop
      {|
        vlsm_type :=
          composite_vlsm
            (equivocator_IM
               (sub_IM IM (elements equivocating)))
            (no_equivocations_additional_constraint_with_preloaded
               (equivocator_IM
                  (sub_IM IM (elements equivocating)))
               (free_constraint
                  (equivocator_IM
                     (sub_IM IM
                      (elements equivocating)))) seed1);
        vlsm_machine :=
          preloaded_vlsm_machine
            (composite_vlsm
               (sub_IM 
                  (equivocator_IM IM)
                  (elements equivocating))
               (no_equivocations_additional_constraint_with_preloaded
                  (sub_IM 
                     (equivocator_IM IM)
                     (elements equivocating))
                  (free_constraint
                     (sub_IM 
                      (equivocator_IM IM)
                      (elements equivocating))) seed2))
            seed2
      |} om → valid (id l) (id s, om)
  -message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
m: message
input_valid
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM (elements equivocating))))
              seed1)) seed1
  |} l (s, Some m)
→ valid_state_prop
    {|
      vlsm_type :=
        composite_vlsm
          (equivocator_IM
             (sub_IM IM (elements equivocating)))
          (no_equivocations_additional_constraint_with_preloaded
             (equivocator_IM
                (sub_IM IM (elements equivocating)))
             (free_constraint
                (equivocator_IM
                   (sub_IM IM (elements equivocating))))
             seed1);
      vlsm_machine :=
        preloaded_vlsm_machine
          (composite_vlsm
             (sub_IM (equivocator_IM IM)
                (elements equivocating))
             (no_equivocations_additional_constraint_with_preloaded
                (sub_IM (equivocator_IM IM)
                   (elements equivocating))
                (free_constraint
                   (sub_IM (equivocator_IM IM)
                      (elements equivocating))) seed2))
          seed2
    |} (id s)
  → initial_message_prop m
    → valid_message_prop
        {|
          vlsm_type :=
            composite_vlsm
              (equivocator_IM
                 (sub_IM IM (elements equivocating)))
              (no_equivocations_additional_constraint_with_preloaded
                 (equivocator_IM
                    (sub_IM IM (elements equivocating)))
                 (free_constraint
                    (equivocator_IM
                       (sub_IM IM
                          (elements equivocating))))
                 seed1);
          vlsm_machine :=
            preloaded_vlsm_machine
              (composite_vlsm
                 (sub_IM (equivocator_IM IM)
                    (elements equivocating))
                 (no_equivocations_additional_constraint_with_preloaded
                    (sub_IM (equivocator_IM IM)
                       (elements equivocating))
                    (free_constraint
                       (sub_IM (equivocator_IM IM)
                          (elements equivocating)))
                    seed2)) seed2
        |} m intros.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
m: message
Hv: input_valid
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |} l (s, Some m)
HsY: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed2)) seed2
  |} (id s)
HmX: initial_message_prop m
valid_message_prop
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM (equivocator_IM IM)
                    (elements equivocating))) seed2))
        seed2
  |} m
    apply initial_message_is_valid; cbn.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
m: message
Hv: input_valid
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |} l (s, Some m)
HsY: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (no_equivocations_additional_constraint_with_preloaded
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))
              (free_constraint
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))
              seed2)) seed2
  |} (id s)
HmX: initial_message_prop m
composite_initial_message_prop
  (sub_IM (equivocator_IM IM) (elements equivocating))
  m ∨ seed2 m
    by destruct HmX; auto.
  -message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
om: option message
input_valid
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM (elements equivocating))))
              seed1)) seed1
  |} l (s, om)
→ valid_state_prop
    {|
      vlsm_type :=
        composite_vlsm
          (equivocator_IM
             (sub_IM IM (elements equivocating)))
          (no_equivocations_additional_constraint_with_preloaded
             (equivocator_IM
                (sub_IM IM (elements equivocating)))
             (free_constraint
                (equivocator_IM
                   (sub_IM IM (elements equivocating))))
             seed1);
      vlsm_machine :=
        preloaded_vlsm_machine
          (composite_vlsm
             (sub_IM (equivocator_IM IM)
                (elements equivocating))
             (no_equivocations_additional_constraint_with_preloaded
                (sub_IM (equivocator_IM IM)
                   (elements equivocating))
                (free_constraint
                   (sub_IM (equivocator_IM IM)
                      (elements equivocating))) seed2))
          seed2
    |} (id s)
  → option_valid_message_prop
      {|
        vlsm_type :=
          composite_vlsm
            (equivocator_IM
               (sub_IM IM (elements equivocating)))
            (no_equivocations_additional_constraint_with_preloaded
               (equivocator_IM
                  (sub_IM IM (elements equivocating)))
               (free_constraint
                  (equivocator_IM
                     (sub_IM IM
                        (elements equivocating))))
               seed1);
        vlsm_machine :=
          preloaded_vlsm_machine
            (composite_vlsm
               (sub_IM (equivocator_IM IM)
                  (elements equivocating))
               (no_equivocations_additional_constraint_with_preloaded
                  (sub_IM (equivocator_IM IM)
                     (elements equivocating))
                  (free_constraint
                     (sub_IM (equivocator_IM IM)
                        (elements equivocating)))
                  seed2)) seed2
      |} om → valid (id l) (id s, om) intros (Hs & _ & Hv & Hc & _) _ _.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
om: option message
Hs: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |} s
Hv: valid l (s, om)
Hc: composite_no_equivocations_except_from
  (equivocator_IM
     (sub_IM IM (elements equivocating))) seed1 l
  (s, om)
valid (id l) (id s, om)
    split; [done |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
om: option message
Hs: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |} s
Hv: valid l (s, om)
Hc: composite_no_equivocations_except_from
  (equivocator_IM
     (sub_IM IM (elements equivocating))) seed1 l
  (s, om)
no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) (elements equivocating))
  (free_constraint
     (sub_IM (equivocator_IM IM)
        (elements equivocating))) seed2 (id l)
  (id s, om)
    split; [| done].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
om: option message
Hs: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |} s
Hv: valid l (s, om)
Hc: composite_no_equivocations_except_from
  (equivocator_IM
     (sub_IM IM (elements equivocating))) seed1 l
  (s, om)
composite_no_equivocations_except_from
  (sub_IM (equivocator_IM IM) (elements equivocating))
  seed2 (id l) (id s, om)
    destruct om; [| done].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
m: message
Hs: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |} s
Hv: valid l (s, Some m)
Hc: composite_no_equivocations_except_from
  (equivocator_IM
     (sub_IM IM (elements equivocating))) seed1 l
  (s, Some m)
composite_no_equivocations_except_from
  (sub_IM (equivocator_IM IM) (elements equivocating))
  seed2 (id l) (id s, Some m)
    simpl in Hc.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
m: message
Hs: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |} s
Hv: valid l (s, Some m)
Hc: composite_no_equivocations_except_from
  (equivocator_IM
     (sub_IM IM (elements equivocating))) seed1 l
  (s, Some m)
composite_no_equivocations_except_from
  (sub_IM (equivocator_IM IM) (elements equivocating))
  seed2 (id l) (id s, Some m) simpl.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
m: message
Hs: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |} s
Hv: valid l (s, Some m)
Hc: composite_no_equivocations_except_from
  (equivocator_IM
     (sub_IM IM (elements equivocating))) seed1 l
  (s, Some m)
composite_no_equivocations_except_from
  (sub_IM (equivocator_IM IM) (elements equivocating))
  seed2 l (s, Some m)
    destruct Hc as [Hc | Hc]
    ; [| by right; apply Hseed12].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
m: message
Hs: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |} s
Hv: valid l (s, Some m)
Hc: composite_has_been_sent
  (equivocator_IM
     (sub_IM IM (elements equivocating)))
  (s, Some m).1 m
composite_no_equivocations_except_from
  (sub_IM (equivocator_IM IM) (elements equivocating))
  seed2 l (s, Some m)
    left.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
m: message
Hs: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |} s
Hv: valid l (s, Some m)
Hc: composite_has_been_sent
  (equivocator_IM
     (sub_IM IM (elements equivocating)))
  (s, Some m).1 m
composite_has_been_sent
  (sub_IM (equivocator_IM IM) (elements equivocating))
  (s, Some m).1 m
    destruct Hc as [subi Hibs].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
m: message
Hs: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |} s
Hv: valid l (s, Some m)
subi: sub_index (elements equivocating)
Hibs: has_been_sent
  (equivocator_IM
     (sub_IM IM (elements equivocating)) subi)
  ((s, Some m).1 subi) m
composite_has_been_sent
  (sub_IM (equivocator_IM IM) (elements equivocating))
  (s, Some m).1 m
    exists subi.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
m: message
Hs: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |} s
Hv: valid l (s, Some m)
subi: sub_index (elements equivocating)
Hibs: has_been_sent
  (equivocator_IM
     (sub_IM IM (elements equivocating)) subi)
  ((s, Some m).1 subi) m
has_been_sent
  (sub_IM (equivocator_IM IM) (elements equivocating)
     subi) ((s, Some m).1 subi) m revert Hibs.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
m: message
Hs: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |} s
Hv: valid l (s, Some m)
subi: sub_index (elements equivocating)
has_been_sent
  (equivocator_IM (sub_IM IM (elements equivocating))
     subi) ((s, Some m).1 subi) m
→ has_been_sent
    (sub_IM (equivocator_IM IM)
       (elements equivocating) subi)
    ((s, Some m).1 subi) m
    apply has_been_sent_irrelevance.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
m: message
Hs: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |} s
Hv: valid l (s, Some m)
subi: sub_index (elements equivocating)
constrained_state_prop
  (sub_IM (equivocator_IM IM) (elements equivocating)
     subi) ((s, Some m).1 subi)
    simpl.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
m: message
Hs: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |} s
Hv: valid l (s, Some m)
subi: sub_index (elements equivocating)
constrained_state_prop
  (sub_IM (equivocator_IM IM) (elements equivocating)
     subi) (s subi)
    apply (preloaded_valid_state_projection (equivocator_IM (sub_IM IM (elements equivocating))) subi).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
m: message
Hs: valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |} s
Hv: valid l (s, Some m)
subi: sub_index (elements equivocating)
constrained_state_prop
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating)))) s
    revert Hs.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
m: message
Hv: valid l (s, Some m)
subi: sub_index (elements equivocating)
valid_state_prop
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM (elements equivocating))))
              seed1)) seed1
  |} s
→ constrained_state_prop
    (free_composite_vlsm
       (equivocator_IM
          (sub_IM IM (elements equivocating)))) s
    apply VLSM_incl_valid_state.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
s: state
  {|
    vlsm_type :=
      composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating))))
           seed1);
    vlsm_machine :=
      preloaded_vlsm_machine
        (composite_vlsm
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM
                 (sub_IM IM
                    (elements equivocating)))
              (free_constraint
                 (equivocator_IM
                    (sub_IM IM
                      (elements equivocating))))
              seed1)) seed1
  |}
m: message
Hv: valid l (s, Some m)
subi: sub_index (elements equivocating)
VLSM_incl_part
  (preloaded_vlsm_machine
     (composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM
              (sub_IM IM (elements equivocating)))
           (free_constraint
              (equivocator_IM
                 (sub_IM IM (elements equivocating))))
           seed1)) seed1)
  (preloaded_vlsm_machine
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating))))
     (λ _ : message, True))
    by apply constrained_preloaded_vlsm_incl_preloaded_with_all_messages.
Qed.

The intermediary results above allow us to prove that the fixed_equivocation_constraint has the constraint_has_been_sent_property.

The core of this result is proving that given a valid_state s of the composition of equivocators with no message equivocation and fixed state equivocation, a message which has_been_sent for that state but not has_been_directly_observed for a projection of that state sx, can nevertheless be generated by the composition of the components allowed to equivocate, preloaded with the messages directly observed in the state sx.

To prove that, we consider a trace witness for the message having been sent, we use projection_has_not_been_directly_observed_is_equivocating to derive that it must have been sent by one of the machines allowed to equivocate, from this we derive that it can be sent by the restriction of the composition of equivocators to just the equivocating components, preloaded with the messages directly observed in the projection, then we use the seeded_equivocators_valid_trace_project result to reach our conclusion.

Lemma fixed_equivocation_constraint_has_constraint_has_been_sent_prop
  : constraint_has_been_sent_prop
    (fixed_equivocation_constraint IM equivocating).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
constraint_has_been_sent_prop
  (fixed_equivocation_constraint IM equivocating)
Proof.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
constraint_has_been_sent_prop
  (fixed_equivocation_constraint IM equivocating)
  unfold constraint_has_been_sent_prop.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
∀ s : composite_state (equivocator_IM IM),
  valid_state_prop XE s
  → ∀ descriptors : equivocator_descriptors IM,
      proper_fixed_equivocator_descriptors descriptors
        s
      → ∀ m : message,
          has_been_sent FreeE s m
          → ∀ l : composite_label IM,
              fixed_equivocation_constraint IM
                equivocating l
                (equivocators_state_project IM
                   descriptors s, Some m) intros.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
Hs: valid_state_prop XE s
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
l: composite_label IM
fixed_equivocation_constraint IM equivocating l
  (equivocators_state_project IM descriptors s, Some m)
  remember (equivocators_state_project _ _ _) as sX.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
Hs: valid_state_prop XE s
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
l: composite_label IM
sX: state (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
fixed_equivocation_constraint IM equivocating l
  (sX, Some m)
  destruct (composite_has_been_directly_observed_dec IM sX m)
  ; [by left | right].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
Hs: valid_state_prop XE s
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
l: composite_label IM
sX: state (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_directly_observed IM sX m
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  clear l.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
Hs: valid_state_prop XE s
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
sX: state (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_directly_observed IM sX m
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  unfold no_additional_equivocations in n.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
Hs: valid_state_prop XE s
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
sX: state (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_directly_observed IM sX m
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  (* Phase I: exhibiting a [valid_trace] ending in tr s and sending m *)
  apply valid_state_has_trace in Hs.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
Hs: ∃ (is : state XE) (tr : list transition_item),
  finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
sX: state (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_directly_observed IM sX m
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  destruct Hs as [is [tr Htr]].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
sX: state (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_directly_observed IM sX m
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  assert (Htr'pre : finite_constrained_trace_init_to FreeE is s tr).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
sX: state (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_directly_observed IM sX m
finite_constrained_trace_init_to FreeE is s tr
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
sX: state (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_directly_observed IM sX m
Htr'pre: finite_constrained_trace_init_to FreeE is s
  tr
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  {message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
sX: state (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_directly_observed IM sX m
finite_constrained_trace_init_to FreeE is s tr
    apply VLSM_incl_finite_valid_trace_init_to; [| done].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
sX: state (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_directly_observed IM sX m
VLSM_incl_part
  (constrained_vlsm_machine
     (free_composite_vlsm (equivocator_IM IM))
     (equivocators_fixed_equivocations_constraint IM
        (elements equivocating)))
  (preloaded_vlsm_machine FreeE (λ _ : message, True))
    by apply constrained_preloaded_incl.
  }message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
sX: state (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_directly_observed IM sX m
Htr'pre: finite_constrained_trace_init_to FreeE is s
  tr
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  assert (Hplst : constrained_state_prop FreeE s)
    by (red in Htr'pre; apply valid_trace_last_pstate in Htr'pre; done).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
sX: state (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_directly_observed IM sX m
Htr'pre: finite_constrained_trace_init_to FreeE is s
  tr
Hplst: constrained_state_prop FreeE s
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  apply proper_sent in Hm; [| done].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: selected_message_exists_in_all_preloaded_traces
  FreeE (field_selector output) s m
sX: state (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_directly_observed IM sX m
Htr'pre: finite_constrained_trace_init_to FreeE is s
  tr
Hplst: constrained_state_prop FreeE s
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m clear Hplst.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: selected_message_exists_in_all_preloaded_traces
  FreeE (field_selector output) s m
sX: state (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_directly_observed IM sX m
Htr'pre: finite_constrained_trace_init_to FreeE is s
  tr
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  specialize (Hm is tr Htr'pre).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: trace_has_message (field_selector output) m tr
sX: state (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_directly_observed IM sX m
Htr'pre: finite_constrained_trace_init_to FreeE is s
  tr
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  (*
    Phase II (a): The restriction of tr to the equivocators allowed to
    state-equivocate is valid for the corresponding composition
    preloaded with the messages directly observed in the projection sX of s.
  *)
  specialize
    (finite_valid_trace_sub_projection (equivocator_IM IM) (elements equivocating)
      (equivocators_fixed_equivocations_constraint IM (elements equivocating))
      (composite_has_been_directly_observed IM sX))
    as Hproject.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: trace_has_message (field_selector output) m tr
sX: state (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_directly_observed IM sX m
Htr'pre: finite_constrained_trace_init_to FreeE is s
  tr
Hproject: ∀ (s : composite_state (equivocator_IM IM)) 
  (tr : list
          (composite_transition_item
             (equivocator_IM IM))),
  trace_sub_item_input_is_seeded_or_sub_previously_sent
    (equivocator_IM IM)
    (elements equivocating)
    (composite_has_been_directly_observed
       IM sX) tr
  → finite_valid_trace
      (composite_vlsm (equivocator_IM IM)
         (equivocators_fixed_equivocations_constraint
            IM (elements equivocating))) s
      tr
    → finite_valid_trace
        (composite_no_equivocation_vlsm_with_preloaded
           (sub_IM (equivocator_IM IM)
              (elements equivocating))
           (free_constraint
              (sub_IM (equivocator_IM IM)
                 (elements equivocating)))
           (composite_has_been_directly_observed
              IM sX))
        (composite_state_sub_projection
           (equivocator_IM IM)
           (elements equivocating) s)
        (finite_trace_sub_projection
           (equivocator_IM IM)
           (elements equivocating) tr)
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  specialize (Hproject is tr).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: trace_has_message (field_selector output) m tr
sX: state (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_directly_observed IM sX m
Htr'pre: finite_constrained_trace_init_to FreeE is s
  tr
Hproject: trace_sub_item_input_is_seeded_or_sub_previously_sent
  (equivocator_IM IM)
  (elements equivocating)
  (composite_has_been_directly_observed IM
     sX) tr
→ finite_valid_trace
    (composite_vlsm (equivocator_IM IM)
       (equivocators_fixed_equivocations_constraint
          IM (elements equivocating))) is
    tr
  → finite_valid_trace
      (composite_no_equivocation_vlsm_with_preloaded
         (sub_IM (equivocator_IM IM)
            (elements equivocating))
         (free_constraint
            (sub_IM (equivocator_IM IM)
               (elements equivocating)))
         (composite_has_been_directly_observed
            IM sX))
      (composite_state_sub_projection
         (equivocator_IM IM)
         (elements equivocating) is)
      (finite_trace_sub_projection
         (equivocator_IM IM)
         (elements equivocating) tr)
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  spec Hproject.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: trace_has_message (field_selector output) m tr
sX: state (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_directly_observed IM sX m
Htr'pre: finite_constrained_trace_init_to FreeE is s
  tr
Hproject: trace_sub_item_input_is_seeded_or_sub_previously_sent
  (equivocator_IM IM)
  (elements equivocating)
  (composite_has_been_directly_observed IM
     sX) tr
→ finite_valid_trace
    (composite_vlsm (equivocator_IM IM)
       (equivocators_fixed_equivocations_constraint
          IM (elements equivocating))) is
    tr
  → finite_valid_trace
      (composite_no_equivocation_vlsm_with_preloaded
         (sub_IM (equivocator_IM IM)
            (elements equivocating))
         (free_constraint
            (sub_IM (equivocator_IM IM)
               (elements equivocating)))
         (composite_has_been_directly_observed
            IM sX))
      (composite_state_sub_projection
         (equivocator_IM IM)
         (elements equivocating) is)
      (finite_trace_sub_projection
         (equivocator_IM IM)
         (elements equivocating) tr)
trace_sub_item_input_is_seeded_or_sub_previously_sent
  (equivocator_IM IM) (elements equivocating)
  (composite_has_been_directly_observed IM sX) tr
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: trace_has_message (field_selector output) m tr
sX: state (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_directly_observed IM sX m
Htr'pre: finite_constrained_trace_init_to FreeE is s
  tr
Hproject: finite_valid_trace
  (composite_vlsm (equivocator_IM IM)
     (equivocators_fixed_equivocations_constraint
        IM (elements equivocating))) is tr
→ finite_valid_trace
    (composite_no_equivocation_vlsm_with_preloaded
       (sub_IM (equivocator_IM IM)
          (elements equivocating))
       (free_constraint
          (sub_IM (equivocator_IM IM)
             (elements equivocating)))
       (composite_has_been_directly_observed
          IM sX))
    (composite_state_sub_projection
       (equivocator_IM IM)
       (elements equivocating) is)
    (finite_trace_sub_projection
       (equivocator_IM IM)
       (elements equivocating) tr)
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  {message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: trace_has_message (field_selector output) m tr
sX: state (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_directly_observed IM sX m
Htr'pre: finite_constrained_trace_init_to FreeE is s
  tr
Hproject: trace_sub_item_input_is_seeded_or_sub_previously_sent
  (equivocator_IM IM)
  (elements equivocating)
  (composite_has_been_directly_observed IM
     sX) tr
→ finite_valid_trace
    (composite_vlsm (equivocator_IM IM)
       (equivocators_fixed_equivocations_constraint
          IM (elements equivocating))) is
    tr
  → finite_valid_trace
      (composite_no_equivocation_vlsm_with_preloaded
         (sub_IM (equivocator_IM IM)
            (elements equivocating))
         (free_constraint
            (sub_IM (equivocator_IM IM)
               (elements equivocating)))
         (composite_has_been_directly_observed
            IM sX))
      (composite_state_sub_projection
         (equivocator_IM IM)
         (elements equivocating) is)
      (finite_trace_sub_projection
         (equivocator_IM IM)
         (elements equivocating) tr)
trace_sub_item_input_is_seeded_or_sub_previously_sent
  (equivocator_IM IM) (elements equivocating)
  (composite_has_been_directly_observed IM sX) tr subst sX.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: trace_has_message (field_selector output) m tr
n: ¬ composite_has_been_directly_observed IM
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_constrained_trace_init_to FreeE is s
  tr
Hproject: trace_sub_item_input_is_seeded_or_sub_previously_sent
  (equivocator_IM IM)
  (elements equivocating)
  (composite_has_been_directly_observed IM
     (equivocators_state_project IM
        descriptors s)) tr
→ finite_valid_trace
    (composite_vlsm (equivocator_IM IM)
       (equivocators_fixed_equivocations_constraint
          IM (elements equivocating))) is
    tr
  → finite_valid_trace
      (composite_no_equivocation_vlsm_with_preloaded
         (sub_IM (equivocator_IM IM)
            (elements equivocating))
         (free_constraint
            (sub_IM (equivocator_IM IM)
               (elements equivocating)))
         (composite_has_been_directly_observed
            IM
            (equivocators_state_project IM
               descriptors s)))
      (composite_state_sub_projection
         (equivocator_IM IM)
         (elements equivocating) is)
      (finite_trace_sub_projection
         (equivocator_IM IM)
         (elements equivocating) tr)
trace_sub_item_input_is_seeded_or_sub_previously_sent
  (equivocator_IM IM) (elements equivocating)
  (composite_has_been_directly_observed IM
     (equivocators_state_project IM descriptors s)) tr
    rewrite <- (valid_trace_get_last Htr) in Hdescriptors |- *.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
m: message
Hm: trace_has_message (field_selector output) m tr
n: ¬ composite_has_been_directly_observed IM
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_constrained_trace_init_to FreeE is s
  tr
Hproject: trace_sub_item_input_is_seeded_or_sub_previously_sent
  (equivocator_IM IM)
  (elements equivocating)
  (composite_has_been_directly_observed IM
     (equivocators_state_project IM
        descriptors s)) tr
→ finite_valid_trace
    (composite_vlsm (equivocator_IM IM)
       (equivocators_fixed_equivocations_constraint
          IM (elements equivocating))) is
    tr
  → finite_valid_trace
      (composite_no_equivocation_vlsm_with_preloaded
         (sub_IM (equivocator_IM IM)
            (elements equivocating))
         (free_constraint
            (sub_IM (equivocator_IM IM)
               (elements equivocating)))
         (composite_has_been_directly_observed
            IM
            (equivocators_state_project IM
               descriptors s)))
      (composite_state_sub_projection
         (equivocator_IM IM)
         (elements equivocating) is)
      (finite_trace_sub_projection
         (equivocator_IM IM)
         (elements equivocating) tr)
trace_sub_item_input_is_seeded_or_sub_previously_sent
  (equivocator_IM IM) (elements equivocating)
  (composite_has_been_directly_observed IM
     (equivocators_state_project IM descriptors
        (finite_trace_last is tr))) tr
    apply
      (equivocators_trace_sub_item_input_is_seeded_or_sub_previously_sent
        _ _ (valid_trace_forget_last Htr) descriptors Hdescriptors).
  }message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: trace_has_message (field_selector output) m tr
sX: state (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_directly_observed IM sX m
Htr'pre: finite_constrained_trace_init_to FreeE is s
  tr
Hproject: finite_valid_trace
  (composite_vlsm (equivocator_IM IM)
     (equivocators_fixed_equivocations_constraint
        IM (elements equivocating))) is tr
→ finite_valid_trace
    (composite_no_equivocation_vlsm_with_preloaded
       (sub_IM (equivocator_IM IM)
          (elements equivocating))
       (free_constraint
          (sub_IM (equivocator_IM IM)
             (elements equivocating)))
       (composite_has_been_directly_observed
          IM sX))
    (composite_state_sub_projection
       (equivocator_IM IM)
       (elements equivocating) is)
    (finite_trace_sub_projection
       (equivocator_IM IM)
       (elements equivocating) tr)
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  specialize (Hproject (valid_trace_forget_last Htr)).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: trace_has_message (field_selector output) m tr
sX: state (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_directly_observed IM sX m
Htr'pre: finite_constrained_trace_init_to FreeE is s
  tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m

  rewrite HeqsX in n.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: trace_has_message (field_selector output) m tr
sX: state (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_directly_observed IM
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_constrained_trace_init_to FreeE is s
  tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  clear HeqsX.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: trace_has_message (field_selector output) m tr
sX: state (free_composite_vlsm IM)
n: ¬ composite_has_been_directly_observed IM
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_constrained_trace_init_to FreeE is s
  tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  (* Phase III (a): Obtain a projection trXm of tr outputing m using a final_descriptor_m *)
  red in Htr'pre.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: trace_has_message (field_selector output) m tr
sX: state (free_composite_vlsm IM)
n: ¬ composite_has_been_directly_observed IM
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  apply (equivocators_trace_project_output_reflecting_iff _ _ _
    (proj1 (valid_trace_forget_last Htr'pre))) in Hm
    as (final_descriptors_m & initial_descriptors_m & trXm & Hfinal_descriptors_m &
        Hproject_trXm & Hex).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector output m) trXm
sX: state (free_composite_vlsm IM)
n: ¬ composite_has_been_directly_observed IM
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m

  (* Identify the item outputing m in trXm an its corresponding item in tr. *)

  apply Exists_exists in Hex.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
Hex: ∃ x : transition_item,
  x ∈ trXm ∧ field_selector output m x
sX: state (free_composite_vlsm IM)
n: ¬ composite_has_been_directly_observed IM
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  destruct Hex as [output_itemX [Hin Houtput_select]].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
output_itemX: transition_item
Hin: output_itemX ∈ trXm
Houtput_select: field_selector output m output_itemX
sX: state (free_composite_vlsm IM)
n: ¬ composite_has_been_directly_observed IM
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  apply elem_of_list_split in Hin.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
output_itemX: transition_item
Hin: ∃ l1 l2 : list transition_item,
  trXm = l1 ++ output_itemX :: l2
Houtput_select: field_selector output m output_itemX
sX: state (free_composite_vlsm IM)
n: ¬ composite_has_been_directly_observed IM
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  destruct Hin as [preX [sufX Heq_trXm]].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
output_itemX: transition_item
preX, sufX: list transition_item
Heq_trXm: trXm = preX ++ output_itemX :: sufX
Houtput_select: field_selector output m output_itemX
sX: state (free_composite_vlsm IM)
n: ¬ composite_has_been_directly_observed IM
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  change (output_itemX :: sufX) with ([output_itemX] ++ sufX) in Heq_trXm.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
output_itemX: transition_item
preX, sufX: list transition_item
Heq_trXm: trXm = preX ++ [output_itemX] ++ sufX
Houtput_select: field_selector output m output_itemX
sX: state (free_composite_vlsm IM)
n: ¬ composite_has_been_directly_observed IM
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  assert (Hpr_item := Hproject_trXm).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
output_itemX: transition_item
preX, sufX: list transition_item
Heq_trXm: trXm = preX ++ [output_itemX] ++ sufX
Houtput_select: field_selector output m output_itemX
sX: state (free_composite_vlsm IM)
n: ¬ composite_has_been_directly_observed IM
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
Hpr_item: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  rewrite Heq_trXm in Hpr_item.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
output_itemX: transition_item
preX, sufX: list transition_item
Heq_trXm: trXm = preX ++ [output_itemX] ++ sufX
Houtput_select: field_selector output m output_itemX
sX: state (free_composite_vlsm IM)
n: ¬ composite_has_been_directly_observed IM
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
Hpr_item: equivocators_trace_project IM
  final_descriptors_m tr =
Some
  (preX ++ [output_itemX] ++ sufX,
   initial_descriptors_m)
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  apply equivocators_trace_project_app_inv_item in Hpr_item.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
output_itemX: transition_item
preX, sufX: list transition_item
Heq_trXm: trXm = preX ++ [output_itemX] ++ sufX
Houtput_select: field_selector output m output_itemX
sX: state (free_composite_vlsm IM)
n: ¬ composite_has_been_directly_observed IM
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
Hpr_item: ∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) 
  (item : composite_transition_item
            (equivocator_IM IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors
                      IM),
  equivocators_trace_project IM
    final_descriptors_m suf =
  Some (sufX, item_descriptors)
  ∧ equivocators_transition_item_project IM
      item_descriptors item =
    Some
      (Some output_itemX, pre_descriptors)
    ∧ equivocators_trace_project IM
        pre_descriptors pre =
      Some (preX, initial_descriptors_m)
      ∧ tr = pre ++ [item] ++ suf
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  destruct Hpr_item as [pre [suf [item [item_descriptors [pre_descriptors
    [_ [Hpr_item [_ Heqtr]]]]]]]].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
output_itemX: transition_item
preX, sufX: list transition_item
Heq_trXm: trXm = preX ++ [output_itemX] ++ sufX
Houtput_select: field_selector output m output_itemX
sX: state (free_composite_vlsm IM)
n: ¬ composite_has_been_directly_observed IM
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_item: equivocators_transition_item_project IM
  item_descriptors item =
Some (Some output_itemX, pre_descriptors)
Heqtr: tr = pre ++ [item] ++ suf
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  unfold equivocators_transition_item_project in Hpr_item.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
output_itemX: transition_item
preX, sufX: list transition_item
Heq_trXm: trXm = preX ++ [output_itemX] ++ sufX
Houtput_select: field_selector output m output_itemX
sX: state (free_composite_vlsm IM)
n: ¬ composite_has_been_directly_observed IM
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_item: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l item)))
    (composite_transition_item_projection
       (equivocator_IM IM) item)
    (item_descriptors (projT1 (l item)))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project IM
               item_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         item_descriptors 
         (projT1 (l item)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         item_descriptors 
         (projT1 (l item)) deqv')
| None => None
end =
Some (Some output_itemX, pre_descriptors)
Heqtr: tr = pre ++ [item] ++ suf
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  destruct
    (equivocator_vlsm_transition_item_project (IM (projT1 (l item)))
      (composite_transition_item_projection (equivocator_IM IM) item)
      (item_descriptors (projT1 (l item))))
    as [(o, deqv') |]
    ; [| by congruence].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
output_itemX: transition_item
preX, sufX: list transition_item
Heq_trXm: trXm = preX ++ [output_itemX] ++ sufX
Houtput_select: field_selector output m output_itemX
sX: state (free_composite_vlsm IM)
n: ¬ composite_has_been_directly_observed IM
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
o: option transition_item
deqv': MachineDescriptor (IM (projT1 (l item)))
Hpr_item: match o with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project IM
               item_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         item_descriptors 
         (projT1 (l item)) deqv')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         item_descriptors 
         (projT1 (l item)) deqv')
end =
Some (Some output_itemX, pre_descriptors)
Heqtr: tr = pre ++ [item] ++ suf
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  destruct o as [item' |]; [| by congruence].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
output_itemX: transition_item
preX, sufX: list transition_item
Heq_trXm: trXm = preX ++ [output_itemX] ++ sufX
Houtput_select: field_selector output m output_itemX
sX: state (free_composite_vlsm IM)
n: ¬ composite_has_been_directly_observed IM
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
item': transition_item
deqv': MachineDescriptor (IM (projT1 (l item)))
Hpr_item: Some
  (Some
     {|
       l :=
         existT (projT1 (l item)) (l item');
       input := input item;
       destination :=
         equivocators_state_project IM
           item_descriptors
           (destination item);
       output := output item
     |},
   equivocator_descriptors_update IM
     item_descriptors (projT1 (l item))
     deqv') =
Some (Some output_itemX, pre_descriptors)
Heqtr: tr = pre ++ [item] ++ suf
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  inversion Hpr_item.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
output_itemX: transition_item
preX, sufX: list transition_item
Heq_trXm: trXm = preX ++ [output_itemX] ++ sufX
Houtput_select: field_selector output m output_itemX
sX: state (free_composite_vlsm IM)
n: ¬ composite_has_been_directly_observed IM
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
item': transition_item
deqv': MachineDescriptor (IM (projT1 (l item)))
Hpr_item: Some
  (Some
     {|
       l :=
         existT (projT1 (l item)) (l item');
       input := input item;
       destination :=
         equivocators_state_project IM
           item_descriptors
           (destination item);
       output := output item
     |},
   equivocator_descriptors_update IM
     item_descriptors (projT1 (l item))
     deqv') =
Some (Some output_itemX, pre_descriptors)
Heqtr: tr = pre ++ [item] ++ suf
H11: {|
  l := existT (projT1 (l item)) (l item');
  input := input item;
  destination :=
    equivocators_state_project IM
      item_descriptors 
      (destination item);
  output := output item
|} = output_itemX
H12: equivocator_descriptors_update IM
  item_descriptors (projT1 (l item)) deqv' =
pre_descriptors
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m subst output_itemX pre_descriptors.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: field_selector output m
  {|
    l :=
      existT (projT1 (l item))
        (l item');
    input := input item;
    destination :=
      equivocators_state_project IM
        item_descriptors
        (destination item);
    output := output item
  |}
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: state (free_composite_vlsm IM)
n: ¬ composite_has_been_directly_observed IM
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
deqv': MachineDescriptor (IM (projT1 (l item)))
Hpr_item: Some
  (Some
     {|
       l :=
         existT (projT1 (l item)) (l item');
       input := input item;
       destination :=
         equivocators_state_project IM
           item_descriptors
           (destination item);
       output := output item
     |},
   equivocator_descriptors_update IM
     item_descriptors (projT1 (l item))
     deqv') =
Some
  (Some
     {|
       l :=
         existT (projT1 (l item)) (l item');
       input := input item;
       destination :=
         equivocators_state_project IM
           item_descriptors
           (destination item);
       output := output item
     |},
   equivocator_descriptors_update IM
     item_descriptors (projT1 (l item))
     deqv')
Heqtr: tr = pre ++ [item] ++ suf
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m clear Hpr_item deqv'.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: field_selector output m
  {|
    l :=
      existT (projT1 (l item))
        (l item');
    input := input item;
    destination :=
      equivocators_state_project IM
        item_descriptors
        (destination item);
    output := output item
  |}
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: state (free_composite_vlsm IM)
n: ¬ composite_has_been_directly_observed IM
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  simpl in Houtput_select.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: state (free_composite_vlsm IM)
n: ¬ composite_has_been_directly_observed IM
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m

  (* show that that item must be specifying a transition for an equivocating component *)

  rewrite <- (valid_trace_get_last Htr) in Hdescriptors, n.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: state (free_composite_vlsm IM)
n: ¬ composite_has_been_directly_observed IM
    (equivocators_state_project IM descriptors
       (finite_trace_last is tr)) m
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  specialize
    (projection_has_not_been_directly_observed_is_equivocating _ _ (valid_trace_forget_last Htr)
      _ Hdescriptors
      _ n item)
    as Hitem_equivocating.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors (finite_trace_last is tr)
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: state (free_composite_vlsm IM)
n: ¬ composite_has_been_directly_observed IM
    (equivocators_state_project IM descriptors
       (finite_trace_last is tr)) m
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: item ∈ tr
→ output item = Some m
  → projT1 (l item)
    ∈ equivocating
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  clear Hdescriptors n.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: state (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: item ∈ tr
→ output item = Some m
  → projT1 (l item)
    ∈ equivocating
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  spec Hitem_equivocating; [by rewrite Heqtr, !elem_of_app, elem_of_cons; auto |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: state (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: output item = Some m
→ projT1 (l item) ∈ equivocating
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  specialize (Hitem_equivocating Houtput_select).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: state (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
can_emit
  (equivocators_composition_for_directly_observed IM
     equivocating sX) m
  (*
    Phase III (b):
    Consider a projection trX' obtained using the final_descriptor_m as above,
    but first restricting the components to just the equivocators allowed to equivocate.
    We will show that we can use [seeded_equivocators_valid_trace_project]
    and leverage the result from Phase II (a)
    to derive that the resulting projection is valid.
  *)
  unfold equivocators_composition_for_directly_observed,
    preloaded_free_equivocating_vlsm_composition.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: state (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
can_emit
  (preloaded_vlsm
     (free_equivocating_vlsm_composition IM
        equivocating)
     (composite_has_been_directly_observed IM sX)) m
  specialize
    (seeded_equivocators_valid_trace_project IM (elements equivocating)
      (composite_has_been_directly_observed IM sX)
      (composite_state_sub_projection (equivocator_IM IM) (elements equivocating) is)
      (finite_trace_sub_projection (equivocator_IM IM) (elements equivocating) tr)
      Hproject
      (fun i => final_descriptors_m (proj1_sig i)))
    as Hsub_project.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: state (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
Hsub_project: proper_equivocator_descriptors
  (λ i : sub_index
           (elements equivocating),
     IM (`i))
  (λ i : sub_index
           (elements equivocating),
     final_descriptors_m (`i))
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM)
        (elements equivocating) is)
     (finite_trace_sub_projection
        (equivocator_IM IM)
        (elements equivocating) tr))
→ ∃ (trX : list
             (composite_transition_item
                (λ i : 
                    sub_index
                      (elements
                      equivocating),
                   IM (`i)))) 
    (initial_descriptors : 
     equivocator_descriptors
       (λ i : sub_index
                (elements equivocating),
          IM (`i))),
    let isX :=
      equivocators_state_project
        (λ i : sub_index
                 (elements equivocating),
           IM (`i)) initial_descriptors
        (composite_state_sub_projection
           (equivocator_IM IM)
           (elements equivocating) is)
      in
    let final_stateX :=
      finite_trace_last isX trX in
    proper_equivocator_descriptors
      (λ i : sub_index
               (elements equivocating),
         IM (`i)) initial_descriptors
      (composite_state_sub_projection
         (equivocator_IM IM)
         (elements equivocating) is)
    ∧ equivocators_trace_project
        (λ i : sub_index
                 (elements equivocating),
           IM (`i))
        (λ i : sub_index
                 (elements equivocating),
           final_descriptors_m (`i))
        (finite_trace_sub_projection
           (equivocator_IM IM)
           (elements equivocating) tr) =
      Some (trX, initial_descriptors)
      ∧ equivocators_state_project
          (λ i : sub_index
                   (elements
                      equivocating),
             IM (`i))
          (λ i : sub_index
                   (elements
                      equivocating),
             final_descriptors_m (`i))
          (finite_trace_last
             (composite_state_sub_projection
                (equivocator_IM IM)
                (elements equivocating)
                is)
             (finite_trace_sub_projection
                (equivocator_IM IM)
                (elements equivocating)
                tr)) = final_stateX
        ∧ finite_valid_trace
            (preloaded_vlsm
               (free_composite_vlsm
                  (sub_IM IM
                     (elements
                      equivocating)))
               (composite_has_been_directly_observed
                  IM sX)) isX trX
can_emit
  (preloaded_vlsm
     (free_equivocating_vlsm_composition IM
        equivocating)
     (composite_has_been_directly_observed IM sX)) m
  simpl in Hsub_project.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: state (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
Hsub_project: proper_equivocator_descriptors
  (λ i : sub_index
           (elements equivocating),
     IM (`i))
  (λ i : sub_index
           (elements equivocating),
     final_descriptors_m (`i))
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM)
        (elements equivocating) is)
     (finite_trace_sub_projection
        (equivocator_IM IM)
        (elements equivocating) tr))
→ ∃ (trX : list
             (composite_transition_item
                (λ i : 
                    sub_index
                      (elements
                      equivocating),
                   IM (`i)))) 
    (initial_descriptors : 
     equivocator_descriptors
       (λ i : sub_index
                (elements equivocating),
          IM (`i))),
    proper_equivocator_descriptors
      (λ i : sub_index
               (elements equivocating),
         IM (`i)) initial_descriptors
      (composite_state_sub_projection
         (equivocator_IM IM)
         (elements equivocating) is)
    ∧ equivocators_trace_project
        (λ i : sub_index
                 (elements equivocating),
           IM (`i))
        (λ i : sub_index
                 (elements equivocating),
           final_descriptors_m (`i))
        (finite_trace_sub_projection
           (equivocator_IM IM)
           (elements equivocating) tr) =
      Some (trX, initial_descriptors)
      ∧ equivocators_state_project
          (λ i : sub_index
                   (elements
                      equivocating),
             IM (`i))
          (λ i : sub_index
                   (elements
                      equivocating),
             final_descriptors_m (`i))
          (finite_trace_last
             (composite_state_sub_projection
                (equivocator_IM IM)
                (elements equivocating)
                is)
             (finite_trace_sub_projection
                (equivocator_IM IM)
                (elements equivocating)
                tr)) =
        finite_trace_last
          (equivocators_state_project
             (λ i : sub_index
                      (elements
                      equivocating),
                IM (`i))
             initial_descriptors
             (composite_state_sub_projection
                (equivocator_IM IM)
                (elements equivocating)
                is)) trX
        ∧ finite_valid_trace
            (preloaded_vlsm
               (free_composite_vlsm
                  (sub_IM IM
                     (elements
                      equivocating)))
               (composite_has_been_directly_observed
                  IM sX))
            (equivocators_state_project
               (λ i : sub_index
                      (elements
                      equivocating),
                  IM (`i))
               initial_descriptors
               (composite_state_sub_projection
                  (equivocator_IM IM)
                  (elements
                     equivocating) is))
            trX
can_emit
  (preloaded_vlsm
     (free_equivocating_vlsm_composition IM
        equivocating)
     (composite_has_been_directly_observed IM sX)) m
  spec Hsub_project.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: state (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
Hsub_project: proper_equivocator_descriptors
  (λ i : sub_index
           (elements equivocating),
     IM (`i))
  (λ i : sub_index
           (elements equivocating),
     final_descriptors_m (`i))
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM)
        (elements equivocating) is)
     (finite_trace_sub_projection
        (equivocator_IM IM)
        (elements equivocating) tr))
→ ∃ (trX : list
             (composite_transition_item
                (λ i : 
                    sub_index
                      (elements
                      equivocating),
                   IM (`i)))) 
    (initial_descriptors : 
     equivocator_descriptors
       (λ i : sub_index
                (elements equivocating),
          IM (`i))),
    proper_equivocator_descriptors
      (λ i : sub_index
               (elements equivocating),
         IM (`i)) initial_descriptors
      (composite_state_sub_projection
         (equivocator_IM IM)
         (elements equivocating) is)
    ∧ equivocators_trace_project
        (λ i : sub_index
                 (elements equivocating),
           IM (`i))
        (λ i : sub_index
                 (elements equivocating),
           final_descriptors_m (`i))
        (finite_trace_sub_projection
           (equivocator_IM IM)
           (elements equivocating) tr) =
      Some (trX, initial_descriptors)
      ∧ equivocators_state_project
          (λ i : sub_index
                   (elements
                      equivocating),
             IM (`i))
          (λ i : sub_index
                   (elements
                      equivocating),
             final_descriptors_m (`i))
          (finite_trace_last
             (composite_state_sub_projection
                (equivocator_IM IM)
                (elements equivocating)
                is)
             (finite_trace_sub_projection
                (equivocator_IM IM)
                (elements equivocating)
                tr)) =
        finite_trace_last
          (equivocators_state_project
             (λ i : sub_index
                      (elements
                      equivocating),
                IM (`i))
             initial_descriptors
             (composite_state_sub_projection
                (equivocator_IM IM)
                (elements equivocating)
                is)) trX
        ∧ finite_valid_trace
            (preloaded_vlsm
               (free_composite_vlsm
                  (sub_IM IM
                     (elements
                      equivocating)))
               (composite_has_been_directly_observed
                  IM sX))
            (equivocators_state_project
               (λ i : sub_index
                      (elements
                      equivocating),
                  IM (`i))
               initial_descriptors
               (composite_state_sub_projection
                  (equivocator_IM IM)
                  (elements
                     equivocating) is))
            trX
proper_equivocator_descriptors
  (λ i : sub_index (elements equivocating), IM (`i))
  (λ i : sub_index (elements equivocating),
     final_descriptors_m (`i))
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM) (elements equivocating) is)
     (finite_trace_sub_projection (equivocator_IM IM)
        (elements equivocating) tr))
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: state (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
Hsub_project: ∃ (trX : list
           (composite_transition_item
              (λ i : sub_index
                      (elements
                      equivocating),
                 IM (`i)))) 
  (initial_descriptors : 
   equivocator_descriptors
     (λ i : sub_index
              (elements equivocating),
        IM (`i))),
  proper_equivocator_descriptors
    (λ i : sub_index
             (elements equivocating),
       IM (`i)) initial_descriptors
    (composite_state_sub_projection
       (equivocator_IM IM)
       (elements equivocating) is)
  ∧ equivocators_trace_project
      (λ i : sub_index
               (elements equivocating),
         IM (`i))
      (λ i : sub_index
               (elements equivocating),
         final_descriptors_m (`i))
      (finite_trace_sub_projection
         (equivocator_IM IM)
         (elements equivocating) tr) =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project
        (λ i : sub_index
                 (elements equivocating),
           IM (`i))
        (λ i : sub_index
                 (elements equivocating),
           final_descriptors_m (`i))
        (finite_trace_last
           (composite_state_sub_projection
              (equivocator_IM IM)
              (elements equivocating)
              is)
           (finite_trace_sub_projection
              (equivocator_IM IM)
              (elements equivocating)
              tr)) =
      finite_trace_last
        (equivocators_state_project
           (λ i : sub_index
                    (elements
                      equivocating),
              IM (`i))
           initial_descriptors
           (composite_state_sub_projection
              (equivocator_IM IM)
              (elements equivocating)
              is)) trX
      ∧ finite_valid_trace
          (preloaded_vlsm
             (free_composite_vlsm
                (sub_IM IM
                   (elements
                      equivocating)))
             (composite_has_been_directly_observed
                IM sX))
          (equivocators_state_project
             (λ i : sub_index
                      (elements
                      equivocating),
                IM (`i))
             initial_descriptors
             (composite_state_sub_projection
                (equivocator_IM IM)
                (elements equivocating)
                is)) trX
can_emit
  (preloaded_vlsm
     (free_equivocating_vlsm_composition IM
        equivocating)
     (composite_has_been_directly_observed IM sX)) m
  {message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: state (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
Hsub_project: proper_equivocator_descriptors
  (λ i : sub_index
           (elements equivocating),
     IM (`i))
  (λ i : sub_index
           (elements equivocating),
     final_descriptors_m (`i))
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM)
        (elements equivocating) is)
     (finite_trace_sub_projection
        (equivocator_IM IM)
        (elements equivocating) tr))
→ ∃ (trX : list
             (composite_transition_item
                (λ i : 
                    sub_index
                      (elements
                      equivocating),
                   IM (`i)))) 
    (initial_descriptors : 
     equivocator_descriptors
       (λ i : sub_index
                (elements equivocating),
          IM (`i))),
    proper_equivocator_descriptors
      (λ i : sub_index
               (elements equivocating),
         IM (`i)) initial_descriptors
      (composite_state_sub_projection
         (equivocator_IM IM)
         (elements equivocating) is)
    ∧ equivocators_trace_project
        (λ i : sub_index
                 (elements equivocating),
           IM (`i))
        (λ i : sub_index
                 (elements equivocating),
           final_descriptors_m (`i))
        (finite_trace_sub_projection
           (equivocator_IM IM)
           (elements equivocating) tr) =
      Some (trX, initial_descriptors)
      ∧ equivocators_state_project
          (λ i : sub_index
                   (elements
                      equivocating),
             IM (`i))
          (λ i : sub_index
                   (elements
                      equivocating),
             final_descriptors_m (`i))
          (finite_trace_last
             (composite_state_sub_projection
                (equivocator_IM IM)
                (elements equivocating)
                is)
             (finite_trace_sub_projection
                (equivocator_IM IM)
                (elements equivocating)
                tr)) =
        finite_trace_last
          (equivocators_state_project
             (λ i : sub_index
                      (elements
                      equivocating),
                IM (`i))
             initial_descriptors
             (composite_state_sub_projection
                (equivocator_IM IM)
                (elements equivocating)
                is)) trX
        ∧ finite_valid_trace
            (preloaded_vlsm
               (free_composite_vlsm
                  (sub_IM IM
                     (elements
                      equivocating)))
               (composite_has_been_directly_observed
                  IM sX))
            (equivocators_state_project
               (λ i : sub_index
                      (elements
                      equivocating),
                  IM (`i))
               initial_descriptors
               (composite_state_sub_projection
                  (equivocator_IM IM)
                  (elements
                     equivocating) is))
            trX
proper_equivocator_descriptors
  (λ i : sub_index (elements equivocating), IM (`i))
  (λ i : sub_index (elements equivocating),
     final_descriptors_m (`i))
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM) (elements equivocating) is)
     (finite_trace_sub_projection (equivocator_IM IM)
        (elements equivocating) tr))
    specialize (finite_trace_sub_projection_last_state (equivocator_IM IM) (elements equivocating)
      (equivocators_fixed_equivocations_constraint IM (elements equivocating)) _ _
        (proj1 (valid_trace_forget_last Htr)))
      as Heq_lst.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: state (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
Hsub_project: proper_equivocator_descriptors
  (λ i : sub_index
           (elements equivocating),
     IM (`i))
  (λ i : sub_index
           (elements equivocating),
     final_descriptors_m (`i))
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM)
        (elements equivocating) is)
     (finite_trace_sub_projection
        (equivocator_IM IM)
        (elements equivocating) tr))
→ ∃ (trX : list
             (composite_transition_item
                (λ i : 
                    sub_index
                      (elements
                      equivocating),
                   IM (`i)))) 
    (initial_descriptors : 
     equivocator_descriptors
       (λ i : sub_index
                (elements equivocating),
          IM (`i))),
    proper_equivocator_descriptors
      (λ i : sub_index
               (elements equivocating),
         IM (`i)) initial_descriptors
      (composite_state_sub_projection
         (equivocator_IM IM)
         (elements equivocating) is)
    ∧ equivocators_trace_project
        (λ i : sub_index
                 (elements equivocating),
           IM (`i))
        (λ i : sub_index
                 (elements equivocating),
           final_descriptors_m (`i))
        (finite_trace_sub_projection
           (equivocator_IM IM)
           (elements equivocating) tr) =
      Some (trX, initial_descriptors)
      ∧ equivocators_state_project
          (λ i : sub_index
                   (elements
                      equivocating),
             IM (`i))
          (λ i : sub_index
                   (elements
                      equivocating),
             final_descriptors_m (`i))
          (finite_trace_last
             (composite_state_sub_projection
                (equivocator_IM IM)
                (elements equivocating)
                is)
             (finite_trace_sub_projection
                (equivocator_IM IM)
                (elements equivocating)
                tr)) =
        finite_trace_last
          (equivocators_state_project
             (λ i : sub_index
                      (elements
                      equivocating),
                IM (`i))
             initial_descriptors
             (composite_state_sub_projection
                (equivocator_IM IM)
                (elements equivocating)
                is)) trX
        ∧ finite_valid_trace
            (preloaded_vlsm
               (free_composite_vlsm
                  (sub_IM IM
                     (elements
                      equivocating)))
               (composite_has_been_directly_observed
                  IM sX))
            (equivocators_state_project
               (λ i : sub_index
                      (elements
                      equivocating),
                  IM (`i))
               initial_descriptors
               (composite_state_sub_projection
                  (equivocator_IM IM)
                  (elements
                     equivocating) is))
            trX
Heq_lst: let lstx := finite_trace_last is tr in
let lstj :=
  finite_trace_last
    (composite_state_sub_projection
       (equivocator_IM IM)
       (elements equivocating) is)
    (finite_trace_sub_projection
       (equivocator_IM IM)
       (elements equivocating) tr) in
lstj =
composite_state_sub_projection
  (equivocator_IM IM)
  (elements equivocating) lstx
proper_equivocator_descriptors
  (λ i : sub_index (elements equivocating), IM (`i))
  (λ i : sub_index (elements equivocating),
     final_descriptors_m (`i))
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM) (elements equivocating) is)
     (finite_trace_sub_projection (equivocator_IM IM)
        (elements equivocating) tr))
    simpl in Heq_lst.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: state (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
Hsub_project: proper_equivocator_descriptors
  (λ i : sub_index
           (elements equivocating),
     IM (`i))
  (λ i : sub_index
           (elements equivocating),
     final_descriptors_m (`i))
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM)
        (elements equivocating) is)
     (finite_trace_sub_projection
        (equivocator_IM IM)
        (elements equivocating) tr))
→ ∃ (trX : list
             (composite_transition_item
                (λ i : 
                    sub_index
                      (elements
                      equivocating),
                   IM (`i)))) 
    (initial_descriptors : 
     equivocator_descriptors
       (λ i : sub_index
                (elements equivocating),
          IM (`i))),
    proper_equivocator_descriptors
      (λ i : sub_index
               (elements equivocating),
         IM (`i)) initial_descriptors
      (composite_state_sub_projection
         (equivocator_IM IM)
         (elements equivocating) is)
    ∧ equivocators_trace_project
        (λ i : sub_index
                 (elements equivocating),
           IM (`i))
        (λ i : sub_index
                 (elements equivocating),
           final_descriptors_m (`i))
        (finite_trace_sub_projection
           (equivocator_IM IM)
           (elements equivocating) tr) =
      Some (trX, initial_descriptors)
      ∧ equivocators_state_project
          (λ i : sub_index
                   (elements
                      equivocating),
             IM (`i))
          (λ i : sub_index
                   (elements
                      equivocating),
             final_descriptors_m (`i))
          (finite_trace_last
             (composite_state_sub_projection
                (equivocator_IM IM)
                (elements equivocating)
                is)
             (finite_trace_sub_projection
                (equivocator_IM IM)
                (elements equivocating)
                tr)) =
        finite_trace_last
          (equivocators_state_project
             (λ i : sub_index
                      (elements
                      equivocating),
                IM (`i))
             initial_descriptors
             (composite_state_sub_projection
                (equivocator_IM IM)
                (elements equivocating)
                is)) trX
        ∧ finite_valid_trace
            (preloaded_vlsm
               (free_composite_vlsm
                  (sub_IM IM
                     (elements
                      equivocating)))
               (composite_has_been_directly_observed
                  IM sX))
            (equivocators_state_project
               (λ i : sub_index
                      (elements
                      equivocating),
                  IM (`i))
               initial_descriptors
               (composite_state_sub_projection
                  (equivocator_IM IM)
                  (elements
                     equivocating) is))
            trX
Heq_lst: finite_trace_last
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr) =
composite_state_sub_projection
  (equivocator_IM IM)
  (elements equivocating)
  (finite_trace_last is tr)
proper_equivocator_descriptors
  (λ i : sub_index (elements equivocating), IM (`i))
  (λ i : sub_index (elements equivocating),
     final_descriptors_m (`i))
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM) (elements equivocating) is)
     (finite_trace_sub_projection (equivocator_IM IM)
        (elements equivocating) tr))
    rewrite Heq_lst.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: state (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
Hsub_project: proper_equivocator_descriptors
  (λ i : sub_index
           (elements equivocating),
     IM (`i))
  (λ i : sub_index
           (elements equivocating),
     final_descriptors_m (`i))
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM)
        (elements equivocating) is)
     (finite_trace_sub_projection
        (equivocator_IM IM)
        (elements equivocating) tr))
→ ∃ (trX : list
             (composite_transition_item
                (λ i : 
                    sub_index
                      (elements
                      equivocating),
                   IM (`i)))) 
    (initial_descriptors : 
     equivocator_descriptors
       (λ i : sub_index
                (elements equivocating),
          IM (`i))),
    proper_equivocator_descriptors
      (λ i : sub_index
               (elements equivocating),
         IM (`i)) initial_descriptors
      (composite_state_sub_projection
         (equivocator_IM IM)
         (elements equivocating) is)
    ∧ equivocators_trace_project
        (λ i : sub_index
                 (elements equivocating),
           IM (`i))
        (λ i : sub_index
                 (elements equivocating),
           final_descriptors_m (`i))
        (finite_trace_sub_projection
           (equivocator_IM IM)
           (elements equivocating) tr) =
      Some (trX, initial_descriptors)
      ∧ equivocators_state_project
          (λ i : sub_index
                   (elements
                      equivocating),
             IM (`i))
          (λ i : sub_index
                   (elements
                      equivocating),
             final_descriptors_m (`i))
          (finite_trace_last
             (composite_state_sub_projection
                (equivocator_IM IM)
                (elements equivocating)
                is)
             (finite_trace_sub_projection
                (equivocator_IM IM)
                (elements equivocating)
                tr)) =
        finite_trace_last
          (equivocators_state_project
             (λ i : sub_index
                      (elements
                      equivocating),
                IM (`i))
             initial_descriptors
             (composite_state_sub_projection
                (equivocator_IM IM)
                (elements equivocating)
                is)) trX
        ∧ finite_valid_trace
            (preloaded_vlsm
               (free_composite_vlsm
                  (sub_IM IM
                     (elements
                      equivocating)))
               (composite_has_been_directly_observed
                  IM sX))
            (equivocators_state_project
               (λ i : sub_index
                      (elements
                      equivocating),
                  IM (`i))
               initial_descriptors
               (composite_state_sub_projection
                  (equivocator_IM IM)
                  (elements
                     equivocating) is))
            trX
Heq_lst: finite_trace_last
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr) =
composite_state_sub_projection
  (equivocator_IM IM)
  (elements equivocating)
  (finite_trace_last is tr)
proper_equivocator_descriptors
  (λ i : sub_index (elements equivocating), IM (`i))
  (λ i : sub_index (elements equivocating),
     final_descriptors_m (`i))
  (composite_state_sub_projection (equivocator_IM IM)
     (elements equivocating) (finite_trace_last is tr))
    intros e.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: state (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
Hsub_project: proper_equivocator_descriptors
  (λ i : sub_index
           (elements equivocating),
     IM (`i))
  (λ i : sub_index
           (elements equivocating),
     final_descriptors_m (`i))
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM)
        (elements equivocating) is)
     (finite_trace_sub_projection
        (equivocator_IM IM)
        (elements equivocating) tr))
→ ∃ (trX : list
             (composite_transition_item
                (λ i : 
                    sub_index
                      (elements
                      equivocating),
                   IM (`i)))) 
    (initial_descriptors : 
     equivocator_descriptors
       (λ i : sub_index
                (elements equivocating),
          IM (`i))),
    proper_equivocator_descriptors
      (λ i : sub_index
               (elements equivocating),
         IM (`i)) initial_descriptors
      (composite_state_sub_projection
         (equivocator_IM IM)
         (elements equivocating) is)
    ∧ equivocators_trace_project
        (λ i : sub_index
                 (elements equivocating),
           IM (`i))
        (λ i : sub_index
                 (elements equivocating),
           final_descriptors_m (`i))
        (finite_trace_sub_projection
           (equivocator_IM IM)
           (elements equivocating) tr) =
      Some (trX, initial_descriptors)
      ∧ equivocators_state_project
          (λ i : sub_index
                   (elements
                      equivocating),
             IM (`i))
          (λ i : sub_index
                   (elements
                      equivocating),
             final_descriptors_m (`i))
          (finite_trace_last
             (composite_state_sub_projection
                (equivocator_IM IM)
                (elements equivocating)
                is)
             (finite_trace_sub_projection
                (equivocator_IM IM)
                (elements equivocating)
                tr)) =
        finite_trace_last
          (equivocators_state_project
             (λ i : sub_index
                      (elements
                      equivocating),
                IM (`i))
             initial_descriptors
             (composite_state_sub_projection
                (equivocator_IM IM)
                (elements equivocating)
                is)) trX
        ∧ finite_valid_trace
            (preloaded_vlsm
               (free_composite_vlsm
                  (sub_IM IM
                     (elements
                      equivocating)))
               (composite_has_been_directly_observed
                  IM sX))
            (equivocators_state_project
               (λ i : sub_index
                      (elements
                      equivocating),
                  IM (`i))
               initial_descriptors
               (composite_state_sub_projection
                  (equivocator_IM IM)
                  (elements
                     equivocating) is))
            trX
Heq_lst: finite_trace_last
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr) =
composite_state_sub_projection
  (equivocator_IM IM)
  (elements equivocating)
  (finite_trace_last is tr)
e: sub_index (elements equivocating)
proper_descriptor (IM (`e)) (final_descriptors_m (`e))
  (composite_state_sub_projection (equivocator_IM IM)
     (elements equivocating) (finite_trace_last is tr)
     e)
    destruct e.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: state (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
Hsub_project: proper_equivocator_descriptors
  (λ i : sub_index
           (elements equivocating),
     IM (`i))
  (λ i : sub_index
           (elements equivocating),
     final_descriptors_m (`i))
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM)
        (elements equivocating) is)
     (finite_trace_sub_projection
        (equivocator_IM IM)
        (elements equivocating) tr))
→ ∃ (trX : list
             (composite_transition_item
                (λ i : 
                    sub_index
                      (elements
                      equivocating),
                   IM (`i)))) 
    (initial_descriptors : 
     equivocator_descriptors
       (λ i : sub_index
                (elements equivocating),
          IM (`i))),
    proper_equivocator_descriptors
      (λ i : sub_index
               (elements equivocating),
         IM (`i)) initial_descriptors
      (composite_state_sub_projection
         (equivocator_IM IM)
         (elements equivocating) is)
    ∧ equivocators_trace_project
        (λ i : sub_index
                 (elements equivocating),
           IM (`i))
        (λ i : sub_index
                 (elements equivocating),
           final_descriptors_m (`i))
        (finite_trace_sub_projection
           (equivocator_IM IM)
           (elements equivocating) tr) =
      Some (trX, initial_descriptors)
      ∧ equivocators_state_project
          (λ i : sub_index
                   (elements
                      equivocating),
             IM (`i))
          (λ i : sub_index
                   (elements
                      equivocating),
             final_descriptors_m (`i))
          (finite_trace_last
             (composite_state_sub_projection
                (equivocator_IM IM)
                (elements equivocating)
                is)
             (finite_trace_sub_projection
                (equivocator_IM IM)
                (elements equivocating)
                tr)) =
        finite_trace_last
          (equivocators_state_project
             (λ i : sub_index
                      (elements
                      equivocating),
                IM (`i))
             initial_descriptors
             (composite_state_sub_projection
                (equivocator_IM IM)
                (elements equivocating)
                is)) trX
        ∧ finite_valid_trace
            (preloaded_vlsm
               (free_composite_vlsm
                  (sub_IM IM
                     (elements
                      equivocating)))
               (composite_has_been_directly_observed
                  IM sX))
            (equivocators_state_project
               (λ i : sub_index
                      (elements
                      equivocating),
                  IM (`i))
               initial_descriptors
               (composite_state_sub_projection
                  (equivocator_IM IM)
                  (elements
                     equivocating) is))
            trX
Heq_lst: finite_trace_last
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr) =
composite_state_sub_projection
  (equivocator_IM IM)
  (elements equivocating)
  (finite_trace_last is tr)
x: index
i: bool_decide
  (sub_index_prop (elements equivocating) x)
proper_descriptor (IM (`(x ↾ i)))
  (final_descriptors_m (`(x ↾ i)))
  (composite_state_sub_projection (equivocator_IM IM)
     (elements equivocating) (finite_trace_last is tr)
     (x ↾ i))
    apply not_equivocating_equivocator_descriptors_proper in Hfinal_descriptors_m.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: proper_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: state (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
Hsub_project: proper_equivocator_descriptors
  (λ i : sub_index
           (elements equivocating),
     IM (`i))
  (λ i : sub_index
           (elements equivocating),
     final_descriptors_m (`i))
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM)
        (elements equivocating) is)
     (finite_trace_sub_projection
        (equivocator_IM IM)
        (elements equivocating) tr))
→ ∃ (trX : list
             (composite_transition_item
                (λ i : 
                    sub_index
                      (elements
                      equivocating),
                   IM (`i)))) 
    (initial_descriptors : 
     equivocator_descriptors
       (λ i : sub_index
                (elements equivocating),
          IM (`i))),
    proper_equivocator_descriptors
      (λ i : sub_index
               (elements equivocating),
         IM (`i)) initial_descriptors
      (composite_state_sub_projection
         (equivocator_IM IM)
         (elements equivocating) is)
    ∧ equivocators_trace_project
        (λ i : sub_index
                 (elements equivocating),
           IM (`i))
        (λ i : sub_index
                 (elements equivocating),
           final_descriptors_m (`i))
        (finite_trace_sub_projection
           (equivocator_IM IM)
           (elements equivocating) tr) =
      Some (trX, initial_descriptors)
      ∧ equivocators_state_project
          (λ i : sub_index
                   (elements
                      equivocating),
             IM (`i))
          (λ i : sub_index
                   (elements
                      equivocating),
             final_descriptors_m (`i))
          (finite_trace_last
             (composite_state_sub_projection
                (equivocator_IM IM)
                (elements equivocating)
                is)
             (finite_trace_sub_projection
                (equivocator_IM IM)
                (elements equivocating)
                tr)) =
        finite_trace_last
          (equivocators_state_project
             (λ i : sub_index
                      (elements
                      equivocating),
                IM (`i))
             initial_descriptors
             (composite_state_sub_projection
                (equivocator_IM IM)
                (elements equivocating)
                is)) trX
        ∧ finite_valid_trace
            (preloaded_vlsm
               (free_composite_vlsm
                  (sub_IM IM
                     (elements
                      equivocating)))
               (composite_has_been_directly_observed
                  IM sX))
            (equivocators_state_project
               (λ i : sub_index
                      (elements
                      equivocating),
                  IM (`i))
               initial_descriptors
               (composite_state_sub_projection
                  (equivocator_IM IM)
                  (elements
                     equivocating) is))
            trX
Heq_lst: finite_trace_last
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr) =
composite_state_sub_projection
  (equivocator_IM IM)
  (elements equivocating)
  (finite_trace_last is tr)
x: index
i: bool_decide
  (sub_index_prop (elements equivocating) x)
proper_descriptor (IM (`(x ↾ i)))
  (final_descriptors_m (`(x ↾ i)))
  (composite_state_sub_projection (equivocator_IM IM)
     (elements equivocating) (finite_trace_last is tr)
     (x ↾ i))
    by apply Hfinal_descriptors_m.
  }message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: state (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
Hsub_project: ∃ (trX : list
           (composite_transition_item
              (λ i : sub_index
                      (elements
                      equivocating),
                 IM (`i)))) 
  (initial_descriptors : 
   equivocator_descriptors
     (λ i : sub_index
              (elements equivocating),
        IM (`i))),
  proper_equivocator_descriptors
    (λ i : sub_index
             (elements equivocating),
       IM (`i)) initial_descriptors
    (composite_state_sub_projection
       (equivocator_IM IM)
       (elements equivocating) is)
  ∧ equivocators_trace_project
      (λ i : sub_index
               (elements equivocating),
         IM (`i))
      (λ i : sub_index
               (elements equivocating),
         final_descriptors_m (`i))
      (finite_trace_sub_projection
         (equivocator_IM IM)
         (elements equivocating) tr) =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project
        (λ i : sub_index
                 (elements equivocating),
           IM (`i))
        (λ i : sub_index
                 (elements equivocating),
           final_descriptors_m (`i))
        (finite_trace_last
           (composite_state_sub_projection
              (equivocator_IM IM)
              (elements equivocating)
              is)
           (finite_trace_sub_projection
              (equivocator_IM IM)
              (elements equivocating)
              tr)) =
      finite_trace_last
        (equivocators_state_project
           (λ i : sub_index
                    (elements
                      equivocating),
              IM (`i))
           initial_descriptors
           (composite_state_sub_projection
              (equivocator_IM IM)
              (elements equivocating)
              is)) trX
      ∧ finite_valid_trace
          (preloaded_vlsm
             (free_composite_vlsm
                (sub_IM IM
                   (elements
                      equivocating)))
             (composite_has_been_directly_observed
                IM sX))
          (equivocators_state_project
             (λ i : sub_index
                      (elements
                      equivocating),
                IM (`i))
             initial_descriptors
             (composite_state_sub_projection
                (equivocator_IM IM)
                (elements equivocating)
                is)) trX
can_emit
  (preloaded_vlsm
     (free_equivocating_vlsm_composition IM
        equivocating)
     (composite_has_been_directly_observed IM sX)) m
  destruct Hsub_project
    as [trX' [initial_descriptors' [_ [Hpr_tr' [_ HtrX]]]]].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: state (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM)
        (elements equivocating))
     (free_constraint
        (sub_IM (equivocator_IM IM)
           (elements equivocating)))
     (composite_has_been_directly_observed
        IM sX))
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
trX': list
  (composite_transition_item
     (λ i : sub_index (elements equivocating),
        IM (`i)))
initial_descriptors': equivocator_descriptors
  (λ i : sub_index
           (elements
              equivocating),
     IM (`i))
Hpr_tr': equivocators_trace_project
  (λ i : sub_index (elements equivocating),
     IM (`i))
  (λ i : sub_index (elements equivocating),
     final_descriptors_m (`i))
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr) =
Some (trX', initial_descriptors')
HtrX: finite_valid_trace
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (elements equivocating)))
     (composite_has_been_directly_observed IM
        sX))
  (equivocators_state_project
     (λ i : sub_index (elements equivocating),
        IM (`i)) initial_descriptors'
     (composite_state_sub_projection
        (equivocator_IM IM)
        (elements equivocating) is)) trX'
can_emit
  (preloaded_vlsm
     (free_equivocating_vlsm_composition IM
        equivocating)
     (composite_has_been_directly_observed IM sX)) m
  clear Htr'pre Hproject.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: state (free_composite_vlsm IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
trX': list
  (composite_transition_item
     (λ i : sub_index (elements equivocating),
        IM (`i)))
initial_descriptors': equivocator_descriptors
  (λ i : sub_index
           (elements
              equivocating),
     IM (`i))
Hpr_tr': equivocators_trace_project
  (λ i : sub_index (elements equivocating),
     IM (`i))
  (λ i : sub_index (elements equivocating),
     final_descriptors_m (`i))
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr) =
Some (trX', initial_descriptors')
HtrX: finite_valid_trace
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (elements equivocating)))
     (composite_has_been_directly_observed IM
        sX))
  (equivocators_state_project
     (λ i : sub_index (elements equivocating),
        IM (`i)) initial_descriptors'
     (composite_state_sub_projection
        (equivocator_IM IM)
        (elements equivocating) is)) trX'
can_emit
  (preloaded_vlsm
     (free_equivocating_vlsm_composition IM
        equivocating)
     (composite_has_been_directly_observed IM sX)) m

  (*
    State that by restricting trXm to the subset of equivocating components
    we obtain the same trX' trace.
  *)

  destruct
    (equivocators_trace_project_finite_trace_sub_projection_commute IM (elements equivocating)
      final_descriptors_m initial_descriptors_m initial_descriptors' tr trXm trX'
      Hproject_trXm Hpr_tr')
    as [_ HeqtrX].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: state (free_composite_vlsm IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
trX': list
  (composite_transition_item
     (λ i : sub_index (elements equivocating),
        IM (`i)))
initial_descriptors': equivocator_descriptors
  (λ i : sub_index
           (elements
              equivocating),
     IM (`i))
Hpr_tr': equivocators_trace_project
  (λ i : sub_index (elements equivocating),
     IM (`i))
  (λ i : sub_index (elements equivocating),
     final_descriptors_m (`i))
  (finite_trace_sub_projection
     (equivocator_IM IM)
     (elements equivocating) tr) =
Some (trX', initial_descriptors')
HtrX: finite_valid_trace
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (elements equivocating)))
     (composite_has_been_directly_observed IM
        sX))
  (equivocators_state_project
     (λ i : sub_index (elements equivocating),
        IM (`i)) initial_descriptors'
     (composite_state_sub_projection
        (equivocator_IM IM)
        (elements equivocating) is)) trX'
HeqtrX: finite_trace_sub_projection IM
  (elements equivocating) trXm = trX'
can_emit
  (preloaded_vlsm
     (free_equivocating_vlsm_composition IM
        equivocating)
     (composite_has_been_directly_observed IM sX)) m
  clear Hproject_trXm Hpr_tr'.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: state (free_composite_vlsm IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
trX': list
  (composite_transition_item
     (λ i : sub_index (elements equivocating),
        IM (`i)))
initial_descriptors': equivocator_descriptors
  (λ i : sub_index
           (elements
              equivocating),
     IM (`i))
HtrX: finite_valid_trace
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (elements equivocating)))
     (composite_has_been_directly_observed IM
        sX))
  (equivocators_state_project
     (λ i : sub_index (elements equivocating),
        IM (`i)) initial_descriptors'
     (composite_state_sub_projection
        (equivocator_IM IM)
        (elements equivocating) is)) trX'
HeqtrX: finite_trace_sub_projection IM
  (elements equivocating) trXm = trX'
can_emit
  (preloaded_vlsm
     (free_equivocating_vlsm_composition IM
        equivocating)
     (composite_has_been_directly_observed IM sX)) m

  (* reduce the goal to showing that the message appears in trX'. *)

  remember
    (equivocators_state_project (sub_IM IM (elements equivocating)) initial_descriptors'
      (composite_state_sub_projection (equivocator_IM IM) (elements equivocating) is))
    as isX.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: state (free_composite_vlsm IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
trX': list
  (composite_transition_item
     (λ i : sub_index (elements equivocating),
        IM (`i)))
initial_descriptors': equivocator_descriptors
  (λ i : sub_index
           (elements
              equivocating),
     IM (`i))
HtrX: finite_valid_trace
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (elements equivocating)))
     (composite_has_been_directly_observed IM
        sX))
  (equivocators_state_project
     (λ i : sub_index (elements equivocating),
        IM (`i)) initial_descriptors'
     (composite_state_sub_projection
        (equivocator_IM IM)
        (elements equivocating) is)) trX'
HeqtrX: finite_trace_sub_projection IM
  (elements equivocating) trXm = trX'
isX: state
  (free_composite_vlsm
     (sub_IM IM (elements equivocating)))
HeqisX: isX =
equivocators_state_project
  (sub_IM IM (elements equivocating))
  initial_descriptors'
  (composite_state_sub_projection
     (equivocator_IM IM)
     (elements equivocating) is)
can_emit
  (preloaded_vlsm
     (free_equivocating_vlsm_composition IM
        equivocating)
     (composite_has_been_directly_observed IM sX)) m clear HeqisX.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: state (free_composite_vlsm IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
trX': list
  (composite_transition_item
     (λ i : sub_index (elements equivocating),
        IM (`i)))
initial_descriptors': equivocator_descriptors
  (λ i : sub_index
           (elements
              equivocating),
     IM (`i))
HtrX: finite_valid_trace
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (elements equivocating)))
     (composite_has_been_directly_observed IM
        sX))
  (equivocators_state_project
     (λ i : sub_index (elements equivocating),
        IM (`i)) initial_descriptors'
     (composite_state_sub_projection
        (equivocator_IM IM)
        (elements equivocating) is)) trX'
HeqtrX: finite_trace_sub_projection IM
  (elements equivocating) trXm = trX'
isX: state
  (free_composite_vlsm
     (sub_IM IM (elements equivocating)))
can_emit
  (preloaded_vlsm
     (free_equivocating_vlsm_composition IM
        equivocating)
     (composite_has_been_directly_observed IM sX)) m
  eapply can_emit_from_valid_trace; [done |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: state (free_composite_vlsm IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
trX': list
  (composite_transition_item
     (λ i : sub_index (elements equivocating),
        IM (`i)))
initial_descriptors': equivocator_descriptors
  (λ i : sub_index
           (elements
              equivocating),
     IM (`i))
HtrX: finite_valid_trace
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (elements equivocating)))
     (composite_has_been_directly_observed IM
        sX))
  (equivocators_state_project
     (λ i : sub_index (elements equivocating),
        IM (`i)) initial_descriptors'
     (composite_state_sub_projection
        (equivocator_IM IM)
        (elements equivocating) is)) trX'
HeqtrX: finite_trace_sub_projection IM
  (elements equivocating) trXm = trX'
isX: state
  (free_composite_vlsm
     (sub_IM IM (elements equivocating)))
trace_has_message (field_selector output) m trX'

  subst.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_init_to XE is s
  (pre ++ [item] ++ suf)
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is
     (pre ++ [item] ++ suf))
preX, sufX: list transition_item
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
sX: state (free_composite_vlsm IM)
Hitem_equivocating: projT1 (l item) ∈ equivocating
initial_descriptors': equivocator_descriptors
  (λ i : sub_index
           (elements
              equivocating),
     IM (`i))
HtrX: finite_valid_trace
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (elements equivocating)))
     (composite_has_been_directly_observed IM
        sX))
  (equivocators_state_project
     (λ i : sub_index (elements equivocating),
        IM (`i)) initial_descriptors'
     (composite_state_sub_projection
        (equivocator_IM IM)
        (elements equivocating) is))
  (finite_trace_sub_projection IM
     (elements equivocating)
     (preX ++
      [{|
         l :=
           existT (projT1 (l item)) (l item');
         input := input item;
         destination :=
           equivocators_state_project IM
             item_descriptors
             (destination item);
         output := output item
       |}] ++ sufX))
isX: state
  (free_composite_vlsm
     (sub_IM IM (elements equivocating)))
trace_has_message (field_selector output) m
  (finite_trace_sub_projection IM
     (elements equivocating)
     (preX ++
      [{|
         l := existT (projT1 (l item)) (l item');
         input := input item;
         destination :=
           equivocators_state_project IM
             item_descriptors (destination item);
         output := output item
       |}] ++ sufX))
  rewrite! (finite_trace_sub_projection_app IM).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_init_to XE is s
  (pre ++ [item] ++ suf)
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is
     (pre ++ [item] ++ suf))
preX, sufX: list transition_item
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
sX: state (free_composite_vlsm IM)
Hitem_equivocating: projT1 (l item) ∈ equivocating
initial_descriptors': equivocator_descriptors
  (λ i : sub_index
           (elements
              equivocating),
     IM (`i))
HtrX: finite_valid_trace
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (elements equivocating)))
     (composite_has_been_directly_observed IM
        sX))
  (equivocators_state_project
     (λ i : sub_index (elements equivocating),
        IM (`i)) initial_descriptors'
     (composite_state_sub_projection
        (equivocator_IM IM)
        (elements equivocating) is))
  (finite_trace_sub_projection IM
     (elements equivocating)
     (preX ++
      [{|
         l :=
           existT (projT1 (l item)) (l item');
         input := input item;
         destination :=
           equivocators_state_project IM
             item_descriptors
             (destination item);
         output := output item
       |}] ++ sufX))
isX: state
  (free_composite_vlsm
     (sub_IM IM (elements equivocating)))
trace_has_message (field_selector output) m
  (finite_trace_sub_projection IM
     (elements equivocating) preX ++
   finite_trace_sub_projection IM
     (elements equivocating)
     [{|
        l := existT (projT1 (l item)) (l item');
        input := input item;
        destination :=
          equivocators_state_project IM
            item_descriptors (destination item);
        output := output item
      |}] ++
   finite_trace_sub_projection IM
     (elements equivocating) sufX)
  unfold trace_has_message.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_init_to XE is s
  (pre ++ [item] ++ suf)
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is
     (pre ++ [item] ++ suf))
preX, sufX: list transition_item
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
sX: state (free_composite_vlsm IM)
Hitem_equivocating: projT1 (l item) ∈ equivocating
initial_descriptors': equivocator_descriptors
  (λ i : sub_index
           (elements
              equivocating),
     IM (`i))
HtrX: finite_valid_trace
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (elements equivocating)))
     (composite_has_been_directly_observed IM
        sX))
  (equivocators_state_project
     (λ i : sub_index (elements equivocating),
        IM (`i)) initial_descriptors'
     (composite_state_sub_projection
        (equivocator_IM IM)
        (elements equivocating) is))
  (finite_trace_sub_projection IM
     (elements equivocating)
     (preX ++
      [{|
         l :=
           existT (projT1 (l item)) (l item');
         input := input item;
         destination :=
           equivocators_state_project IM
             item_descriptors
             (destination item);
         output := output item
       |}] ++ sufX))
isX: state
  (free_composite_vlsm
     (sub_IM IM (elements equivocating)))
Exists (field_selector output m)
  (finite_trace_sub_projection IM
     (elements equivocating) preX ++
   finite_trace_sub_projection IM
     (elements equivocating)
     [{|
        l := existT (projT1 (l item)) (l item');
        input := input item;
        destination :=
          equivocators_state_project IM
            item_descriptors (destination item);
        output := output item
      |}] ++
   finite_trace_sub_projection IM
     (elements equivocating) sufX)
  rewrite! Exists_app.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_init_to XE is s
  (pre ++ [item] ++ suf)
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is
     (pre ++ [item] ++ suf))
preX, sufX: list transition_item
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
sX: state (free_composite_vlsm IM)
Hitem_equivocating: projT1 (l item) ∈ equivocating
initial_descriptors': equivocator_descriptors
  (λ i : sub_index
           (elements
              equivocating),
     IM (`i))
HtrX: finite_valid_trace
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (elements equivocating)))
     (composite_has_been_directly_observed IM
        sX))
  (equivocators_state_project
     (λ i : sub_index (elements equivocating),
        IM (`i)) initial_descriptors'
     (composite_state_sub_projection
        (equivocator_IM IM)
        (elements equivocating) is))
  (finite_trace_sub_projection IM
     (elements equivocating)
     (preX ++
      [{|
         l :=
           existT (projT1 (l item)) (l item');
         input := input item;
         destination :=
           equivocators_state_project IM
             item_descriptors
             (destination item);
         output := output item
       |}] ++ sufX))
isX: state
  (free_composite_vlsm
     (sub_IM IM (elements equivocating)))
Exists (field_selector output m)
  (finite_trace_sub_projection IM
     (elements equivocating) preX)
∨ Exists (field_selector output m)
    (finite_trace_sub_projection IM
       (elements equivocating)
       [{|
          l := existT (projT1 (l item)) (l item');
          input := input item;
          destination :=
            equivocators_state_project IM
              item_descriptors (destination item);
          output := output item
        |}])
  ∨ Exists (field_selector output m)
      (finite_trace_sub_projection IM
         (elements equivocating) sufX) right.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_init_to XE is s
  (pre ++ [item] ++ suf)
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is
     (pre ++ [item] ++ suf))
preX, sufX: list transition_item
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
sX: state (free_composite_vlsm IM)
Hitem_equivocating: projT1 (l item) ∈ equivocating
initial_descriptors': equivocator_descriptors
  (λ i : sub_index
           (elements
              equivocating),
     IM (`i))
HtrX: finite_valid_trace
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (elements equivocating)))
     (composite_has_been_directly_observed IM
        sX))
  (equivocators_state_project
     (λ i : sub_index (elements equivocating),
        IM (`i)) initial_descriptors'
     (composite_state_sub_projection
        (equivocator_IM IM)
        (elements equivocating) is))
  (finite_trace_sub_projection IM
     (elements equivocating)
     (preX ++
      [{|
         l :=
           existT (projT1 (l item)) (l item');
         input := input item;
         destination :=
           equivocators_state_project IM
             item_descriptors
             (destination item);
         output := output item
       |}] ++ sufX))
isX: state
  (free_composite_vlsm
     (sub_IM IM (elements equivocating)))
Exists (field_selector output m)
  (finite_trace_sub_projection IM
     (elements equivocating)
     [{|
        l := existT (projT1 (l item)) (l item');
        input := input item;
        destination :=
          equivocators_state_project IM
            item_descriptors (destination item);
        output := output item
      |}])
∨ Exists (field_selector output m)
    (finite_trace_sub_projection IM
       (elements equivocating) sufX) left.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_init_to XE is s
  (pre ++ [item] ++ suf)
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is
     (pre ++ [item] ++ suf))
preX, sufX: list transition_item
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
sX: state (free_composite_vlsm IM)
Hitem_equivocating: projT1 (l item) ∈ equivocating
initial_descriptors': equivocator_descriptors
  (λ i : sub_index
           (elements
              equivocating),
     IM (`i))
HtrX: finite_valid_trace
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (elements equivocating)))
     (composite_has_been_directly_observed IM
        sX))
  (equivocators_state_project
     (λ i : sub_index (elements equivocating),
        IM (`i)) initial_descriptors'
     (composite_state_sub_projection
        (equivocator_IM IM)
        (elements equivocating) is))
  (finite_trace_sub_projection IM
     (elements equivocating)
     (preX ++
      [{|
         l :=
           existT (projT1 (l item)) (l item');
         input := input item;
         destination :=
           equivocators_state_project IM
             item_descriptors
             (destination item);
         output := output item
       |}] ++ sufX))
isX: state
  (free_composite_vlsm
     (sub_IM IM (elements equivocating)))
Exists (field_selector output m)
  (finite_trace_sub_projection IM
     (elements equivocating)
     [{|
        l := existT (projT1 (l item)) (l item');
        input := input item;
        destination :=
          equivocators_state_project IM
            item_descriptors (destination item);
        output := output item
      |}]) simpl.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_init_to XE is s
  (pre ++ [item] ++ suf)
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is
     (pre ++ [item] ++ suf))
preX, sufX: list transition_item
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
sX: state (free_composite_vlsm IM)
Hitem_equivocating: projT1 (l item) ∈ equivocating
initial_descriptors': equivocator_descriptors
  (λ i : sub_index
           (elements
              equivocating),
     IM (`i))
HtrX: finite_valid_trace
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (elements equivocating)))
     (composite_has_been_directly_observed IM
        sX))
  (equivocators_state_project
     (λ i : sub_index (elements equivocating),
        IM (`i)) initial_descriptors'
     (composite_state_sub_projection
        (equivocator_IM IM)
        (elements equivocating) is))
  (finite_trace_sub_projection IM
     (elements equivocating)
     (preX ++
      [{|
         l :=
           existT (projT1 (l item)) (l item');
         input := input item;
         destination :=
           equivocators_state_project IM
             item_descriptors
             (destination item);
         output := output item
       |}] ++ sufX))
isX: state
  (free_composite_vlsm
     (sub_IM IM (elements equivocating)))
Exists (field_selector output m)
  match
    pre_VLSM_projection_transition_item_project
      (composite_type IM)
      (composite_type
         (sub_IM IM (elements equivocating)))
      (composite_label_sub_projection_option IM
         (elements equivocating))
      (composite_state_sub_projection IM
         (elements equivocating))
      {|
        l := existT (projT1 (l item)) (l item');
        input := input item;
        destination :=
          equivocators_state_project IM
            item_descriptors (destination item);
        output := output item
      |}
  with
  | Some y => [y]
  | None => []
  end

  unfold pre_VLSM_projection_transition_item_project,
    composite_label_sub_projection_option.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_init_to XE is s
  (pre ++ [item] ++ suf)
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is
     (pre ++ [item] ++ suf))
preX, sufX: list transition_item
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
sX: state (free_composite_vlsm IM)
Hitem_equivocating: projT1 (l item) ∈ equivocating
initial_descriptors': equivocator_descriptors
  (λ i : sub_index
           (elements
              equivocating),
     IM (`i))
HtrX: finite_valid_trace
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (elements equivocating)))
     (composite_has_been_directly_observed IM
        sX))
  (equivocators_state_project
     (λ i : sub_index (elements equivocating),
        IM (`i)) initial_descriptors'
     (composite_state_sub_projection
        (equivocator_IM IM)
        (elements equivocating) is))
  (finite_trace_sub_projection IM
     (elements equivocating)
     (preX ++
      [{|
         l :=
           existT (projT1 (l item)) (l item');
         input := input item;
         destination :=
           equivocators_state_project IM
             item_descriptors
             (destination item);
         output := output item
       |}] ++ sufX))
isX: state
  (free_composite_vlsm
     (sub_IM IM (elements equivocating)))
Exists (field_selector output m)
  match
    match
      match
        decide
          (projT1
             (l
                {|
                  l :=
                    existT (projT1 (l item)) (l item');
                  input := input item;
                  destination :=
                    equivocators_state_project IM
                      item_descriptors
                      (destination item);
                  output := output item
                |}) ∈ elements equivocating)
      with
      | left i_in =>
          Some
            (composite_label_sub_projection IM
               (elements equivocating)
               (l
                  {|
                    l :=
                      existT (projT1 (l item))
                        (l item');
                    input := input item;
                    destination :=
                      equivocators_state_project IM
                        item_descriptors
                        (destination item);
                    output := output item
                  |}) i_in)
      | right _ => None
      end
    with
    | Some lY =>
        Some
          {|
            l := lY;
            input :=
              input
                {|
                  l :=
                    existT (projT1 (l item)) (l item');
                  input := input item;
                  destination :=
                    equivocators_state_project IM
                      item_descriptors
                      (destination item);
                  output := output item
                |};
            destination :=
              composite_state_sub_projection IM
                (elements equivocating)
                (destination
                   {|
                     l :=
                       existT (projT1 (l item))
                         (l item');
                     input := input item;
                     destination :=
                       equivocators_state_project IM
                         item_descriptors
                         (destination item);
                     output := output item
                   |});
            output :=
              output
                {|
                  l :=
                    existT (projT1 (l item)) (l item');
                  input := input item;
                  destination :=
                    equivocators_state_project IM
                      item_descriptors
                      (destination item);
                  output := output item
                |}
          |}
    | None => None
    end
  with
  | Some y => [y]
  | None => []
  end
  apply elem_of_elements in Hitem_equivocating.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: composite_state (equivocator_IM IM)
is: state XE
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_init_to XE is s
  (pre ++ [item] ++ suf)
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is
     (pre ++ [item] ++ suf))
preX, sufX: list transition_item
item_descriptors: equivocator_descriptors IM
item': transition_item
Houtput_select: output item = Some m
sX: state (free_composite_vlsm IM)
Hitem_equivocating: projT1 (l item)
∈ elements equivocating
initial_descriptors': equivocator_descriptors
  (λ i : sub_index
           (elements
              equivocating),
     IM (`i))
HtrX: finite_valid_trace
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (elements equivocating)))
     (composite_has_been_directly_observed IM
        sX))
  (equivocators_state_project
     (λ i : sub_index (elements equivocating),
        IM (`i)) initial_descriptors'
     (composite_state_sub_projection
        (equivocator_IM IM)
        (elements equivocating) is))
  (finite_trace_sub_projection IM
     (elements equivocating)
     (preX ++
      [{|
         l :=
           existT (projT1 (l item)) (l item');
         input := input item;
         destination :=
           equivocators_state_project IM
             item_descriptors
             (destination item);
         output := output item
       |}] ++ sufX))
isX: state
  (free_composite_vlsm
     (sub_IM IM (elements equivocating)))
Exists (field_selector output m)
  match
    match
      match
        decide
          (projT1
             (l
                {|
                  l :=
                    existT (projT1 (l item)) (l item');
                  input := input item;
                  destination :=
                    equivocators_state_project IM
                      item_descriptors
                      (destination item);
                  output := output item
                |}) ∈ elements equivocating)
      with
      | left i_in =>
          Some
            (composite_label_sub_projection IM
               (elements equivocating)
               (l
                  {|
                    l :=
                      existT (projT1 (l item))
                        (l item');
                    input := input item;
                    destination :=
                      equivocators_state_project IM
                        item_descriptors
                        (destination item);
                    output := output item
                  |}) i_in)
      | right _ => None
      end
    with
    | Some lY =>
        Some
          {|
            l := lY;
            input :=
              input
                {|
                  l :=
                    existT (projT1 (l item)) (l item');
                  input := input item;
                  destination :=
                    equivocators_state_project IM
                      item_descriptors
                      (destination item);
                  output := output item
                |};
            destination :=
              composite_state_sub_projection IM
                (elements equivocating)
                (destination
                   {|
                     l :=
                       existT (projT1 (l item))
                         (l item');
                     input := input item;
                     destination :=
                       equivocators_state_project IM
                         item_descriptors
                         (destination item);
                     output := output item
                   |});
            output :=
              output
                {|
                  l :=
                    existT (projT1 (l item)) (l item');
                  input := input item;
                  destination :=
                    equivocators_state_project IM
                      item_descriptors
                      (destination item);
                  output := output item
                |}
          |}
    | None => None
    end
  with
  | Some y => [y]
  | None => []
  end
  by case_decide; [constructor |].
Qed.

Main result of this section, stating that traces which are valid for the equivocator-based definition of fixed equivocation project to traces which are valid for the simple-components definition of fixed equivocation.

Theorem fixed_equivocators_valid_trace_project
  (final_descriptors : equivocator_descriptors IM)
  (is : composite_state (equivocator_IM IM))
  (tr : list (composite_transition_item (equivocator_IM IM)))
  (final_state := finite_trace_last is tr)
  (Hproper : proper_fixed_equivocator_descriptors final_descriptors final_state)
  (Htr : finite_valid_trace XE is tr)
  : exists
    (trX : list (composite_transition_item IM))
    (initial_descriptors : equivocator_descriptors IM)
    (isX := equivocators_state_project IM initial_descriptors is)
    (final_stateX := finite_trace_last isX trX),
    proper_fixed_equivocator_descriptors initial_descriptors is /\
    equivocators_trace_project IM final_descriptors tr = Some (trX, initial_descriptors) /\
    equivocators_state_project IM final_descriptors final_state = final_stateX /\
    finite_valid_trace X isX trX.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
final_descriptors: equivocator_descriptors IM
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr: finite_valid_trace XE is tr
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X isX trX
Proof.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
final_descriptors: equivocator_descriptors IM
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr: finite_valid_trace XE is tr
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X isX trX
  apply _equivocators_valid_trace_project; [done | done | done |]; intros.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
final_descriptors: equivocator_descriptors IM
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr: finite_valid_trace XE is tr
constraint_has_been_sent_prop
  (fixed_equivocation_constraint IM equivocating)
  by apply fixed_equivocation_constraint_has_constraint_has_been_sent_prop.
Qed.

Lemma fixed_equivocators_vlsm_partial_projection
  (final_descriptors : equivocator_descriptors IM)
  : VLSM_partial_projection XE X (equivocators_partial_trace_project IM final_descriptors).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
final_descriptors: equivocator_descriptors IM
VLSM_partial_projection XE X
  (equivocators_partial_trace_project IM
     final_descriptors)
Proof.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
final_descriptors: equivocator_descriptors IM
VLSM_partial_projection XE X
  (equivocators_partial_trace_project IM
     final_descriptors)
  split; [split |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
final_descriptors: equivocator_descriptors IM
∀ (sX : state XE) (trX : list transition_item) (sY : 
                                                state
                                                 X) 
  (trY : list transition_item),
  equivocators_partial_trace_project IM
    final_descriptors (sX, trX) = Some (sY, trY)
  → ∀ (s'X : state XE) (preX : list transition_item),
      finite_trace_last s'X preX = sX
      → finite_valid_trace_from XE s'X (preX ++ trX)
        → ∃ (s'Y : state X) (preY : list
                                      transition_item),
            equivocators_partial_trace_project IM
              final_descriptors (s'X, preX ++ trX) =
            Some (s'Y, preY ++ trY)
            ∧ finite_trace_last s'Y preY = sY
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
final_descriptors: equivocator_descriptors IM
∀ (sX : state XE) (trX : list transition_item) 
  (sY : state X) (trY : list transition_item),
  equivocators_partial_trace_project IM
    final_descriptors (
    sX, trX) = Some (sY, trY)
  → finite_valid_trace XE sX trX
    → finite_valid_trace X sY trY
  -message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
final_descriptors: equivocator_descriptors IM
∀ (sX : state XE) (trX : list transition_item) (sY : 
                                                state
                                                 X) 
  (trY : list transition_item),
  equivocators_partial_trace_project IM
    final_descriptors (sX, trX) = Some (sY, trY)
  → ∀ (s'X : state XE) (preX : list transition_item),
      finite_trace_last s'X preX = sX
      → finite_valid_trace_from XE s'X (preX ++ trX)
        → ∃ (s'Y : state X) (preY : list
                                      transition_item),
            equivocators_partial_trace_project IM
              final_descriptors (s'X, preX ++ trX) =
            Some (s'Y, preY ++ trY)
            ∧ finite_trace_last s'Y preY = sY intros s tr sX trX Hpr_tr s_pre pre Hs_lst Hpre_tr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
final_descriptors: equivocator_descriptors IM
s: state XE
tr: list transition_item
sX: state X
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
s_pre: state XE
pre: list transition_item
Hs_lst: finite_trace_last s_pre pre = s
Hpre_tr: finite_valid_trace_from XE s_pre (pre ++ tr)
∃ (s'Y : state X) (preY : list transition_item),
  equivocators_partial_trace_project IM
    final_descriptors (s_pre, pre ++ tr) =
  Some (s'Y, preY ++ trX)
  ∧ finite_trace_last s'Y preY = sX
    assert (HPreFree_pre_tr : finite_constrained_trace_from FreeE s_pre (pre ++ tr)).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
final_descriptors: equivocator_descriptors IM
s: state XE
tr: list transition_item
sX: state X
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
s_pre: state XE
pre: list transition_item
Hs_lst: finite_trace_last s_pre pre = s
Hpre_tr: finite_valid_trace_from XE s_pre (pre ++ tr)
finite_constrained_trace_from FreeE s_pre (pre ++ tr)
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
final_descriptors: equivocator_descriptors IM
s: state XE
tr: list transition_item
sX: state X
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
s_pre: state XE
pre: list transition_item
Hs_lst: finite_trace_last s_pre pre = s
Hpre_tr: finite_valid_trace_from XE s_pre (pre ++ tr)
HPreFree_pre_tr: finite_constrained_trace_from FreeE
  s_pre (pre ++ tr)
∃ (s'Y : state X) (preY : list transition_item),
  equivocators_partial_trace_project IM
    final_descriptors (
    s_pre, pre ++ tr) = 
  Some (s'Y, preY ++ trX)
  ∧ finite_trace_last s'Y preY = sX
    {message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
final_descriptors: equivocator_descriptors IM
s: state XE
tr: list transition_item
sX: state X
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
s_pre: state XE
pre: list transition_item
Hs_lst: finite_trace_last s_pre pre = s
Hpre_tr: finite_valid_trace_from XE s_pre (pre ++ tr)
finite_constrained_trace_from FreeE s_pre (pre ++ tr)
      revert Hpre_tr; apply VLSM_incl_finite_valid_trace_from.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
final_descriptors: equivocator_descriptors IM
s: state XE
tr: list transition_item
sX: state X
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
s_pre: state XE
pre: list transition_item
Hs_lst: finite_trace_last s_pre pre = s
VLSM_incl_part
  (constrained_vlsm_machine
     (free_composite_vlsm (equivocator_IM IM))
     (equivocators_fixed_equivocations_constraint IM
        (elements equivocating)))
  (preloaded_vlsm_machine FreeE (λ _ : message, True))
      by apply equivocators_fixed_equivocations_vlsm_incl_PreFree.
    }message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
final_descriptors: equivocator_descriptors IM
s: state XE
tr: list transition_item
sX: state X
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
s_pre: state XE
pre: list transition_item
Hs_lst: finite_trace_last s_pre pre = s
Hpre_tr: finite_valid_trace_from XE s_pre (pre ++ tr)
HPreFree_pre_tr: finite_constrained_trace_from FreeE
  s_pre (pre ++ tr)
∃ (s'Y : state X) (preY : list transition_item),
  equivocators_partial_trace_project IM
    final_descriptors (s_pre, pre ++ tr) =
  Some (s'Y, preY ++ trX)
  ∧ finite_trace_last s'Y preY = sX
    clear Hpre_tr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
final_descriptors: equivocator_descriptors IM
s: state XE
tr: list transition_item
sX: state X
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
s_pre: state XE
pre: list transition_item
Hs_lst: finite_trace_last s_pre pre = s
HPreFree_pre_tr: finite_constrained_trace_from FreeE
  s_pre (pre ++ tr)
∃ (s'Y : state X) (preY : list transition_item),
  equivocators_partial_trace_project IM
    final_descriptors (s_pre, pre ++ tr) =
  Some (s'Y, preY ++ trX)
  ∧ finite_trace_last s'Y preY = sX  revert s tr sX trX Hpr_tr s_pre pre Hs_lst HPreFree_pre_tr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
final_descriptors: equivocator_descriptors IM
∀ (s : state XE) (tr : list transition_item) (sX : 
                                              state X) 
  (trX : list transition_item),
  equivocators_partial_trace_project IM
    final_descriptors (s, tr) = Some (sX, trX)
  → ∀ (s_pre : state XE) (pre : list transition_item),
      finite_trace_last s_pre pre = s
      → finite_constrained_trace_from FreeE s_pre
          (pre ++ tr)
        → ∃ (s'Y : state X) (preY : list
                                      transition_item),
            equivocators_partial_trace_project IM
              final_descriptors (s_pre, pre ++ tr) =
            Some (s'Y, preY ++ trX)
            ∧ finite_trace_last s'Y preY = sX
    apply equivocators_partial_trace_project_extends_left.
  -message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
final_descriptors: equivocator_descriptors IM
∀ (sX : state XE) (trX : list transition_item) (sY : 
                                                state
                                                 X) 
  (trY : list transition_item),
  equivocators_partial_trace_project IM
    final_descriptors (sX, trX) = Some (sY, trY)
  → finite_valid_trace XE sX trX
    → finite_valid_trace X sY trY intros s tr sX trX Hpr_tr Htr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
final_descriptors: equivocator_descriptors IM
s: state XE
tr: list transition_item
sX: state X
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
Htr: finite_valid_trace XE s tr
finite_valid_trace X sX trX
    destruct (destruct_equivocators_partial_trace_project IM Hpr_tr)
      as [Hnot_equiv [initial_descriptors [Htr_project Hs_project]]].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
final_descriptors: equivocator_descriptors IM
s: state XE
tr: list transition_item
sX: state X
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
Htr: finite_valid_trace XE s tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
finite_valid_trace X sX trX

    specialize (finite_valid_trace_last_pstate XE _ _ (proj1 Htr)) as Hlst.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
final_descriptors: equivocator_descriptors IM
s: state XE
tr: list transition_item
sX: state X
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
Htr: finite_valid_trace XE s tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hlst: valid_state_prop XE (finite_trace_last s tr)
finite_valid_trace X sX trX
    apply not_equivocating_equivocator_descriptors_proper_fixed in Hnot_equiv as Hproper
    ; [| done].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
final_descriptors: equivocator_descriptors IM
s: state XE
tr: list transition_item
sX: state X
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
Htr: finite_valid_trace XE s tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hlst: valid_state_prop XE (finite_trace_last s tr)
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors (finite_trace_last s tr)
finite_valid_trace X sX trX
    destruct (fixed_equivocators_valid_trace_project  _ _ _ Hproper Htr)
      as [_trX [_initial_descriptors [_ [_Htr_project [_ HtrX]]]]].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
final_descriptors: equivocator_descriptors IM
s: state XE
tr: list transition_item
sX: state X
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
Htr: finite_valid_trace XE s tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hlst: valid_state_prop XE (finite_trace_last s tr)
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors (finite_trace_last s tr)
_trX: list (composite_transition_item IM)
_initial_descriptors: equivocator_descriptors IM
_Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (_trX, _initial_descriptors)
HtrX: finite_valid_trace X
  (equivocators_state_project IM
     _initial_descriptors s) _trX
finite_valid_trace X sX trX
    rewrite Htr_project in _Htr_project.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
final_descriptors: equivocator_descriptors IM
s: state XE
tr: list transition_item
sX: state X
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
Htr: finite_valid_trace XE s tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hlst: valid_state_prop XE (finite_trace_last s tr)
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors (finite_trace_last s tr)
_trX: list (composite_transition_item IM)
_initial_descriptors: equivocator_descriptors IM
_Htr_project: Some (trX, initial_descriptors) =
Some (_trX, _initial_descriptors)
HtrX: finite_valid_trace X
  (equivocators_state_project IM
     _initial_descriptors s) _trX
finite_valid_trace X sX trX
    by inversion _Htr_project; subst.
Qed.

Lemma fixed_equivocators_vlsm_projection
  : VLSM_projection XE X (equivocators_total_label_project IM) (equivocators_total_state_project IM).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
VLSM_projection XE X
  (equivocators_total_label_project IM)
  (equivocators_total_state_project IM)
Proof.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
VLSM_projection XE X
  (equivocators_total_label_project IM)
  (equivocators_total_state_project IM)
  constructor; [constructor |]; intros sX trX HtrX.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
sX: state XE
trX: list transition_item
HtrX: finite_valid_trace_from XE sX trX
equivocators_total_state_project IM
  (finite_trace_last sX trX) =
finite_trace_last
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_finite_trace_project XE X
     (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX)
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
sX: state XE
trX: list transition_item
HtrX: finite_valid_trace XE sX trX
finite_valid_trace X
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_finite_trace_project XE X
     (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX)
  -message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
sX: state XE
trX: list transition_item
HtrX: finite_valid_trace_from XE sX trX
equivocators_total_state_project IM
  (finite_trace_last sX trX) =
finite_trace_last
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_finite_trace_project XE X
     (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX) apply PreFreeE_Free_vlsm_projection_type.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
sX: state XE
trX: list transition_item
HtrX: finite_valid_trace_from XE sX trX
finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM))) sX trX
    apply VLSM_incl_finite_valid_trace_from; [| done].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
sX: state XE
trX: list transition_item
HtrX: finite_valid_trace_from XE sX trX
VLSM_incl_part
  (constrained_vlsm_machine
     (free_composite_vlsm (equivocator_IM IM))
     (equivocators_fixed_equivocations_constraint IM
        (elements equivocating)))
  (preloaded_vlsm_machine
     (free_composite_vlsm (equivocator_IM IM))
     (λ _ : message, True))
    by apply equivocators_fixed_equivocations_vlsm_incl_PreFree.
  -message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
sX: state XE
trX: list transition_item
HtrX: finite_valid_trace XE sX trX
finite_valid_trace X
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_finite_trace_project XE X
     (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX) assert (Hpre_tr : finite_constrained_trace FreeE sX trX).message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
sX: state XE
trX: list transition_item
HtrX: finite_valid_trace XE sX trX
finite_constrained_trace FreeE sX trX
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
sX: state XE
trX: list transition_item
HtrX: finite_valid_trace XE sX trX
Hpre_tr: finite_constrained_trace FreeE sX trX
finite_valid_trace X
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_finite_trace_project XE X
     (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX)
    {message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
sX: state XE
trX: list transition_item
HtrX: finite_valid_trace XE sX trX
finite_constrained_trace FreeE sX trX
      apply VLSM_incl_finite_valid_trace; [| done].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
sX: state XE
trX: list transition_item
HtrX: finite_valid_trace XE sX trX
VLSM_incl_part
  (constrained_vlsm_machine
     (free_composite_vlsm (equivocator_IM IM))
     (equivocators_fixed_equivocations_constraint IM
        (elements equivocating)))
  (preloaded_vlsm_machine FreeE (λ _ : message, True))
      by apply equivocators_fixed_equivocations_vlsm_incl_PreFree.
    }message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
sX: state XE
trX: list transition_item
HtrX: finite_valid_trace XE sX trX
Hpre_tr: finite_constrained_trace FreeE sX trX
finite_valid_trace X
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_finite_trace_project XE X
     (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX)
    specialize (VLSM_partial_projection_finite_valid_trace
      (fixed_equivocators_vlsm_partial_projection (zero_descriptor IM))
      sX trX (equivocators_state_project IM (zero_descriptor IM) sX)
      (equivocators_total_trace_project IM trX)) as Hsim.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
sX: state XE
trX: list transition_item
HtrX: finite_valid_trace XE sX trX
Hpre_tr: finite_constrained_trace FreeE sX trX
Hsim: equivocators_partial_trace_project IM
  (zero_descriptor IM) (
  sX, trX) =
Some
  (equivocators_state_project IM
     (zero_descriptor IM) sX,
   equivocators_total_trace_project IM trX)
→ finite_valid_trace XE sX trX
  → finite_valid_trace X
      (equivocators_state_project IM
         (zero_descriptor IM) sX)
      (equivocators_total_trace_project IM trX)
finite_valid_trace X
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_finite_trace_project XE X
     (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX)
    spec Hsim.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
sX: state XE
trX: list transition_item
HtrX: finite_valid_trace XE sX trX
Hpre_tr: finite_constrained_trace FreeE sX trX
Hsim: equivocators_partial_trace_project IM
  (zero_descriptor IM) (
  sX, trX) =
Some
  (equivocators_state_project IM
     (zero_descriptor IM) sX,
   equivocators_total_trace_project IM trX)
→ finite_valid_trace XE sX trX
  → finite_valid_trace X
      (equivocators_state_project IM
         (zero_descriptor IM) sX)
      (equivocators_total_trace_project IM trX)
equivocators_partial_trace_project IM
  (zero_descriptor IM) (sX, trX) =
Some
  (equivocators_state_project IM (zero_descriptor IM)
     sX, equivocators_total_trace_project IM trX)
message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
sX: state XE
trX: list transition_item
HtrX: finite_valid_trace XE sX trX
Hpre_tr: finite_constrained_trace FreeE sX trX
Hsim: finite_valid_trace XE sX trX
→ finite_valid_trace X
    (equivocators_state_project IM
       (zero_descriptor IM) sX)
    (equivocators_total_trace_project IM trX)
finite_valid_trace X
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_finite_trace_project XE X
     (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX)
    {message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
sX: state XE
trX: list transition_item
HtrX: finite_valid_trace XE sX trX
Hpre_tr: finite_constrained_trace FreeE sX trX
Hsim: equivocators_partial_trace_project IM
  (zero_descriptor IM) (
  sX, trX) =
Some
  (equivocators_state_project IM
     (zero_descriptor IM) sX,
   equivocators_total_trace_project IM trX)
→ finite_valid_trace XE sX trX
  → finite_valid_trace X
      (equivocators_state_project IM
         (zero_descriptor IM) sX)
      (equivocators_total_trace_project IM trX)
equivocators_partial_trace_project IM
  (zero_descriptor IM) (sX, trX) =
Some
  (equivocators_state_project IM (zero_descriptor IM)
     sX, equivocators_total_trace_project IM trX) simpl.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
sX: state XE
trX: list transition_item
HtrX: finite_valid_trace XE sX trX
Hpre_tr: finite_constrained_trace FreeE sX trX
Hsim: equivocators_partial_trace_project IM
  (zero_descriptor IM) (
  sX, trX) =
Some
  (equivocators_state_project IM
     (zero_descriptor IM) sX,
   equivocators_total_trace_project IM trX)
→ finite_valid_trace XE sX trX
  → finite_valid_trace X
      (equivocators_state_project IM
         (zero_descriptor IM) sX)
      (equivocators_total_trace_project IM trX)
(if
  decide
    (not_equivocating_equivocator_descriptors IM
       (zero_descriptor IM) (finite_trace_last sX trX))
 then
  match
    equivocators_trace_project IM (zero_descriptor IM)
      trX
  with
  | Some (trX, initial_descriptors) =>
      Some
        (equivocators_state_project IM
           initial_descriptors sX, trX)
  | None => None
  end
 else None) =
Some
  (equivocators_state_project IM (zero_descriptor IM)
     sX, equivocators_total_trace_project IM trX) rewrite decide_True by apply zero_descriptor_not_equivocating.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
sX: state XE
trX: list transition_item
HtrX: finite_valid_trace XE sX trX
Hpre_tr: finite_constrained_trace FreeE sX trX
Hsim: equivocators_partial_trace_project IM
  (zero_descriptor IM) (
  sX, trX) =
Some
  (equivocators_state_project IM
     (zero_descriptor IM) sX,
   equivocators_total_trace_project IM trX)
→ finite_valid_trace XE sX trX
  → finite_valid_trace X
      (equivocators_state_project IM
         (zero_descriptor IM) sX)
      (equivocators_total_trace_project IM trX)
match
  equivocators_trace_project IM (zero_descriptor IM)
    trX
with
| Some (trX, initial_descriptors) =>
    Some
      (equivocators_state_project IM
         initial_descriptors sX, trX)
| None => None
end =
Some
  (equivocators_state_project IM (zero_descriptor IM)
     sX, equivocators_total_trace_project IM trX)
      by rewrite (equivocators_total_trace_project_characterization IM (proj1 Hpre_tr)).
    }message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
sX: state XE
trX: list transition_item
HtrX: finite_valid_trace XE sX trX
Hpre_tr: finite_constrained_trace FreeE sX trX
Hsim: finite_valid_trace XE sX trX
→ finite_valid_trace X
    (equivocators_state_project IM
       (zero_descriptor IM) sX)
    (equivocators_total_trace_project IM trX)
finite_valid_trace X
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_finite_trace_project XE X
     (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX)
    apply Hsim in HtrX.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
sX: state XE
trX: list transition_item
HtrX: finite_valid_trace X
  (equivocators_state_project IM
     (zero_descriptor IM) sX)
  (equivocators_total_trace_project IM trX)
Hpre_tr: finite_constrained_trace FreeE sX trX
Hsim: finite_valid_trace XE sX trX
→ finite_valid_trace X
    (equivocators_state_project IM
       (zero_descriptor IM) sX)
    (equivocators_total_trace_project IM trX)
finite_valid_trace X
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_finite_trace_project XE X
     (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX)
    remember (pre_VLSM_projection_finite_trace_project _ _ _ _ _) as tr.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
sX: state XE
trX: list transition_item
HtrX: finite_valid_trace X
  (equivocators_state_project IM
     (zero_descriptor IM) sX)
  (equivocators_total_trace_project IM trX)
Hpre_tr: finite_constrained_trace FreeE sX trX
Hsim: finite_valid_trace XE sX trX
→ finite_valid_trace X
    (equivocators_state_project IM
       (zero_descriptor IM) sX)
    (equivocators_total_trace_project IM trX)
tr: list transition_item
Heqtr: tr =
pre_VLSM_projection_finite_trace_project XE X
  (equivocators_total_label_project IM)
  (equivocators_total_state_project IM) trX
finite_valid_trace X
  (equivocators_total_state_project IM sX) tr
    replace tr with (equivocators_total_trace_project IM trX); [done |].message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
sX: state XE
trX: list transition_item
HtrX: finite_valid_trace X
  (equivocators_state_project IM
     (zero_descriptor IM) sX)
  (equivocators_total_trace_project IM trX)
Hpre_tr: finite_constrained_trace FreeE sX trX
Hsim: finite_valid_trace XE sX trX
→ finite_valid_trace X
    (equivocators_state_project IM
       (zero_descriptor IM) sX)
    (equivocators_total_trace_project IM trX)
tr: list transition_item
Heqtr: tr =
pre_VLSM_projection_finite_trace_project XE X
  (equivocators_total_label_project IM)
  (equivocators_total_state_project IM) trX
equivocators_total_trace_project IM trX = tr
    subst.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
sX: state XE
trX: list transition_item
HtrX: finite_valid_trace X
  (equivocators_state_project IM
     (zero_descriptor IM) sX)
  (equivocators_total_trace_project IM trX)
Hpre_tr: finite_constrained_trace FreeE sX trX
Hsim: finite_valid_trace XE sX trX
→ finite_valid_trace X
    (equivocators_state_project IM
       (zero_descriptor IM) sX)
    (equivocators_total_trace_project IM trX)
equivocators_total_trace_project IM trX =
pre_VLSM_projection_finite_trace_project XE X
  (equivocators_total_label_project IM)
  (equivocators_total_state_project IM) trX symmetry.message: Type
EqDecision0: EqDecision message
index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision1: EqDecision index
H6: FinSet index Ci
EqDecision2: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
H9: ∀ i : index, HasBeenReceivedCapability (IM i)
equivocating: Ci
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
X:= fixed_equivocation_vlsm_composition IM equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hdec_init: ∀ i : index,
  decidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
sX: state XE
trX: list transition_item
HtrX: finite_valid_trace X
  (equivocators_state_project IM
     (zero_descriptor IM) sX)
  (equivocators_total_trace_project IM trX)
Hpre_tr: finite_constrained_trace FreeE sX trX
Hsim: finite_valid_trace XE sX trX
→ finite_valid_trace X
    (equivocators_state_project IM
       (zero_descriptor IM) sX)
    (equivocators_total_trace_project IM trX)
pre_VLSM_projection_finite_trace_project XE X
  (equivocators_total_label_project IM)
  (equivocators_total_state_project IM) trX =
equivocators_total_trace_project IM trX
    by apply (equivocators_total_VLSM_projection_finite_trace_project IM (proj1 Hpre_tr)).
Qed.

End sec_from_equivocators_to_components.

Section sec_all_equivocating.

Fixed Equivocation for all Equivocators

In this section we show that if the set of components allowed to equivocate in the fixed set equivocation is the entire set of components, then the composition under the fixed-set equivocation constraint is the same as the composition with only the no-message-equivocation constraint.

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

Lemma strong_constraint_subsumption_fixed_all
  : strong_constraint_subsumption (free_composite_vlsm (equivocator_IM IM))
    (equivocators_no_equivocations_constraint IM)
    (equivocators_fixed_equivocations_constraint IM (enum index)).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
XE:= equivocators_fixed_equivocations_vlsm IM
  (enum index): VLSM message
NE:= equivocators_no_equivocations_vlsm IM: VLSM message
strong_constraint_subsumption
  (free_composite_vlsm (equivocator_IM IM))
  (equivocators_no_equivocations_constraint IM)
  (equivocators_fixed_equivocations_constraint IM
     (enum index))
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
XE:= equivocators_fixed_equivocations_vlsm IM
  (enum index): VLSM message
NE:= equivocators_no_equivocations_vlsm IM: VLSM message
strong_constraint_subsumption
  (free_composite_vlsm (equivocator_IM IM))
  (equivocators_no_equivocations_constraint IM)
  (equivocators_fixed_equivocations_constraint IM
     (enum index))
  by split; [| intro; intros; apply elem_of_enum].
Qed.

Lemma strong_constraint_subsumption_all_fixed
  : strong_constraint_subsumption (free_composite_vlsm (equivocator_IM IM))
    (equivocators_fixed_equivocations_constraint IM (enum index))
    (equivocators_no_equivocations_constraint IM).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
XE:= equivocators_fixed_equivocations_vlsm IM
  (enum index): VLSM message
NE:= equivocators_no_equivocations_vlsm IM: VLSM message
strong_constraint_subsumption
  (free_composite_vlsm (equivocator_IM IM))
  (equivocators_fixed_equivocations_constraint IM
     (enum index))
  (equivocators_no_equivocations_constraint IM)
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
XE:= equivocators_fixed_equivocations_vlsm IM
  (enum index): VLSM message
NE:= equivocators_no_equivocations_vlsm IM: VLSM message
strong_constraint_subsumption
  (free_composite_vlsm (equivocator_IM IM))
  (equivocators_fixed_equivocations_constraint IM
     (enum index))
  (equivocators_no_equivocations_constraint IM)
  by intros l [s om] Hv; apply Hv.
Qed.

Lemma equivocators_fixed_equivocations_all_eq : VLSM_eq XE NE.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
XE:= equivocators_fixed_equivocations_vlsm IM
  (enum index): VLSM message
NE:= equivocators_no_equivocations_vlsm IM: VLSM message
VLSM_eq XE NE
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
XE:= equivocators_fixed_equivocations_vlsm IM
  (enum index): VLSM message
NE:= equivocators_no_equivocations_vlsm IM: VLSM message
VLSM_eq XE NE
  split.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
XE:= equivocators_fixed_equivocations_vlsm IM
  (enum index): VLSM message
NE:= equivocators_no_equivocations_vlsm IM: VLSM message
VLSM_incl {| vlsm_type := XE; vlsm_machine := XE |}
  {| vlsm_type := XE; vlsm_machine := NE |}
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
XE:= equivocators_fixed_equivocations_vlsm IM
  (enum index): VLSM message
NE:= equivocators_no_equivocations_vlsm IM: VLSM message
VLSM_incl {| vlsm_type := XE; vlsm_machine := NE |}
  {| vlsm_type := XE; vlsm_machine := XE |}
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
XE:= equivocators_fixed_equivocations_vlsm IM
  (enum index): VLSM message
NE:= equivocators_no_equivocations_vlsm IM: VLSM message
VLSM_incl {| vlsm_type := XE; vlsm_machine := XE |}
  {| vlsm_type := XE; vlsm_machine := NE |} apply (constraint_subsumption_incl (free_composite_vlsm _)).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
XE:= equivocators_fixed_equivocations_vlsm IM
  (enum index): VLSM message
NE:= equivocators_no_equivocations_vlsm IM: VLSM message
input_valid_constraint_subsumption
  (free_composite_vlsm (equivocator_IM IM))
  (equivocators_fixed_equivocations_constraint IM
     (enum index))
  (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)
XE:= equivocators_fixed_equivocations_vlsm IM
  (enum index): VLSM message
NE:= equivocators_no_equivocations_vlsm IM: VLSM message
preloaded_constraint_subsumption
  (free_composite_vlsm (equivocator_IM IM))
  (equivocators_fixed_equivocations_constraint IM
     (enum index))
  (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)
XE:= equivocators_fixed_equivocations_vlsm IM
  (enum index): VLSM message
NE:= equivocators_no_equivocations_vlsm IM: VLSM message
strong_constraint_subsumption
  (free_composite_vlsm (equivocator_IM IM))
  (equivocators_fixed_equivocations_constraint IM
     (enum index))
  (equivocators_no_equivocations_constraint IM)
    by apply strong_constraint_subsumption_all_fixed.
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
XE:= equivocators_fixed_equivocations_vlsm IM
  (enum index): VLSM message
NE:= equivocators_no_equivocations_vlsm IM: VLSM message
VLSM_incl {| vlsm_type := XE; vlsm_machine := NE |}
  {| vlsm_type := XE; vlsm_machine := XE |} apply
      (constraint_subsumption_incl (free_composite_vlsm (equivocator_IM IM))
        (equivocators_no_equivocations_constraint IM)
        (equivocators_fixed_equivocations_constraint IM (enum index))).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
XE:= equivocators_fixed_equivocations_vlsm IM
  (enum index): VLSM message
NE:= equivocators_no_equivocations_vlsm IM: VLSM message
input_valid_constraint_subsumption
  (free_composite_vlsm (equivocator_IM IM))
  (equivocators_no_equivocations_constraint IM)
  (equivocators_fixed_equivocations_constraint IM
     (enum index))
    apply preloaded_constraint_subsumption_stronger.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
XE:= equivocators_fixed_equivocations_vlsm IM
  (enum index): VLSM message
NE:= equivocators_no_equivocations_vlsm IM: VLSM message
preloaded_constraint_subsumption
  (free_composite_vlsm (equivocator_IM IM))
  (equivocators_no_equivocations_constraint IM)
  (equivocators_fixed_equivocations_constraint IM
     (enum index))
    apply strong_constraint_subsumption_strongest.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
XE:= equivocators_fixed_equivocations_vlsm IM
  (enum index): VLSM message
NE:= equivocators_no_equivocations_vlsm IM: VLSM message
strong_constraint_subsumption
  (free_composite_vlsm (equivocator_IM IM))
  (equivocators_no_equivocations_constraint IM)
  (equivocators_fixed_equivocations_constraint IM
     (enum index))
    by apply strong_constraint_subsumption_fixed_all.
Qed.

End sec_all_equivocating.