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.
From VLSM.Lib Require Import Preamble ListExtras.
From VLSM.Core Require Import VLSM VLSMProjections Composition ProjectionTraces SubProjectionTraces.
From VLSM.Core Require Import Equivocation EquivocationProjections.
From VLSM.Core Require Import Equivocation.FixedSetEquivocation Equivocation.NoEquivocation.
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.
From VLSM.Core Require Import Equivocators.FullReplayTraces.
From VLSM.Core Require Import Equivocators.FixedEquivocation.
From VLSM.Core Require Import Equivocators.SimulatingFree.

Core: VLSM Equivocators Simulating Fixed Set Equivocation Composition

In this module, we show that the composition of equivocators with no message-equivocation and fixed-set state-equivocation can simulate the fixed-set message-equivocation composition of regular components.

The proof is based on generalized_equivocators_finite_valid_trace_init_to_rev, but also reuses seeded_equivocators_finite_valid_trace_init_to_rev to lift a trace of the free subcomposition of equivocating components generating a message (given by the fixed message-equivocation constraint) to a trace of the subcomposition of equivocating equivocators with no message-equivocation, which is then used to satisfy the replayable_message_property.

Section sec_fixed_equivocating.

Context
  {message : Type}
  {index : Type}
  `{FinSet index Ci}
  `{finite.Finite index}
  (IM : index -> VLSM message)
  (Free := free_composite_vlsm IM)
  `{forall i : index, HasBeenSentCapability (IM i)}
  (equivocating : Ci)
  (X : VLSM message := strong_fixed_equivocation_vlsm_composition IM equivocating)
  (XE : VLSM message := equivocators_fixed_equivocations_vlsm IM (elements equivocating))
  (FreeE := free_composite_vlsm (equivocator_IM IM))
  (SubFreeE := free_composite_vlsm (sub_IM (equivocator_IM IM) (elements equivocating)))
  (no_initial_messages_in_IM : no_initial_messages_in_IM_prop IM)
  .

Lemma no_initial_messages_in_sub_IM
  : forall i m, ~ initial_message_prop (sub_IM IM (elements equivocating) i) m.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
∀ (i : sub_index (elements equivocating)) (m : message),
  ¬ initial_message_prop m
Proof.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
∀ (i : sub_index (elements equivocating)) (m : message),
  ¬ initial_message_prop m
  intros [i Hi] m Hinit.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
i: index
Hi: bool_decide
  (sub_index_prop (elements equivocating) i)
m: message
Hinit: initial_message_prop m
False
  by apply (no_initial_messages_in_IM i m).
Qed.

Messages sent_by_non_equivocating components in the projection of a valid state for the fixed-set state-equivocation composition of equivocators with no message-equivocation are valid for that composition of equivocators.

Lemma fixed_equivocating_messages_sent_by_non_equivocating_are_valid
  eqv_state_s
  (Heqv_state_s : valid_state_prop XE eqv_state_s)
  (s := equivocators_total_state_project IM eqv_state_s)
  (Hs : valid_state_prop X s)
  m
  (Hm : sent_by_non_equivocating IM equivocating s m)
  : valid_message_prop XE m.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
s:= equivocators_total_state_project IM eqv_state_s: state (free_composite_vlsm IM)
Hs: valid_state_prop X s
m: message
Hm: sent_by_non_equivocating IM equivocating s m
valid_message_prop XE m
Proof.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
s:= equivocators_total_state_project IM eqv_state_s: state (free_composite_vlsm IM)
Hs: valid_state_prop X s
m: message
Hm: sent_by_non_equivocating IM equivocating s m
valid_message_prop XE m
  pose proof (HinclE :=
    equivocators_fixed_equivocations_vlsm_incl_PreFree IM (elements equivocating)).message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
s:= equivocators_total_state_project IM eqv_state_s: state (free_composite_vlsm IM)
Hs: valid_state_prop X s
m: message
Hm: sent_by_non_equivocating IM equivocating s m
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
valid_message_prop XE m
  apply sent_by_non_equivocating_are_sent in Hm.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
s:= equivocators_total_state_project IM eqv_state_s: state (free_composite_vlsm IM)
Hs: valid_state_prop X s
m: message
Hm: composite_has_been_sent IM s m
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
valid_message_prop XE m
  eapply VLSM_incl_valid_state in Hs; [| by apply constrained_preloaded_incl].message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
s:= equivocators_total_state_project IM eqv_state_s: state (free_composite_vlsm IM)
m: message
Hm: composite_has_been_sent IM s m
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hs: valid_state_prop
  {|
    vlsm_type := free_composite_vlsm IM;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm IM)
  |} s
valid_message_prop XE m
  eapply sent_valid; [done |].message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
s:= equivocators_total_state_project IM eqv_state_s: state (free_composite_vlsm IM)
m: message
Hm: composite_has_been_sent IM s m
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hs: valid_state_prop
  {|
    vlsm_type := free_composite_vlsm IM;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm IM)
  |} s
has_been_sent XE eqv_state_s m
  revert Hm; apply (VLSM_incl_valid_state HinclE) in Heqv_state_s.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop
  {|
    vlsm_type :=
      equivocators_fixed_equivocations_vlsm
        IM (elements equivocating);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM IM))
  |} eqv_state_s
s:= equivocators_total_state_project IM eqv_state_s: state (free_composite_vlsm IM)
m: message
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hs: valid_state_prop
  {|
    vlsm_type := free_composite_vlsm IM;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm IM)
  |} s
composite_has_been_sent IM s m
→ has_been_sent XE eqv_state_s m
  by specialize
    (VLSM_projection_has_been_sent_reflect
      (preloaded_equivocators_no_equivocations_vlsm_X_vlsm_projection IM)
      _ Heqv_state_s m).
Qed.

This is the constraint counterpart of the weak_embedding_valid_preservation property (for the equivocators_fixed_equivocations_constraint).

Lemma fixed_equivocating_non_equivocating_constraint_lifting
  eqv_state_s
  (Heqv_state_s : valid_state_prop XE eqv_state_s)
  l s om
  (Hv :
    input_valid
      (seeded_equivocators_no_equivocation_vlsm IM (elements equivocating)
        (sent_by_non_equivocating IM equivocating (equivocators_total_state_project IM eqv_state_s)))
        l (s, om))
  (Hlift_s : valid_state_prop XE (lift_equivocators_sub_state_to IM (elements equivocating) eqv_state_s s))
  : equivocators_fixed_equivocations_constraint IM (elements equivocating)
        (lift_equivocators_sub_label_to IM (elements equivocating) eqv_state_s l)
        (lift_equivocators_sub_state_to IM (elements equivocating) eqv_state_s s, om).message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
om: option message
Hv: input_valid
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) l (
  s, om)
Hlift_s: valid_state_prop XE
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
equivocators_fixed_equivocations_constraint IM
  (elements equivocating)
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s l)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s, om)
Proof.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
om: option message
Hv: input_valid
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) l (
  s, om)
Hlift_s: valid_state_prop XE
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
equivocators_fixed_equivocations_constraint IM
  (elements equivocating)
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s l)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s, om)
  specialize (equivocators_fixed_equivocations_vlsm_incl_PreFree IM (elements equivocating))
    as HinclE.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
om: option message
Hv: input_valid
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) l (
  s, om)
Hlift_s: valid_state_prop XE
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
equivocators_fixed_equivocations_constraint IM
  (elements equivocating)
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s l)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s, om)
  destruct Hv as [Hs [Hom [_ Hc]]].message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
om: option message
Hs: valid_state_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) s
Hom: option_valid_message_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_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)))
  (sent_by_non_equivocating IM equivocating
     (equivocators_total_state_project IM
        eqv_state_s)) l (
  s, om)
Hlift_s: valid_state_prop XE
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
equivocators_fixed_equivocations_constraint IM
  (elements equivocating)
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s l)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s, om)
  apply valid_state_has_fixed_equivocation in Hlift_s.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
om: option message
Hs: valid_state_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) s
Hom: option_valid_message_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_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)))
  (sent_by_non_equivocating IM equivocating
     (equivocators_total_state_project IM
        eqv_state_s)) l (
  s, om)
Hlift_s: state_has_fixed_equivocation IM
  (elements equivocating)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
equivocators_fixed_equivocations_constraint IM
  (elements equivocating)
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s l)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s, om)
  split.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
om: option message
Hs: valid_state_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) s
Hom: option_valid_message_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_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)))
  (sent_by_non_equivocating IM equivocating
     (equivocators_total_state_project IM
        eqv_state_s)) l (
  s, om)
Hlift_s: state_has_fixed_equivocation IM
  (elements equivocating)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
equivocators_no_equivocations_constraint IM
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s l)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s, om)
message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
om: option message
Hs: valid_state_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) s
Hom: option_valid_message_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_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)))
  (sent_by_non_equivocating IM equivocating
     (equivocators_total_state_project IM
        eqv_state_s)) l (
  s, om)
Hlift_s: state_has_fixed_equivocation IM
  (elements equivocating)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
state_has_fixed_equivocation IM
  (elements equivocating)
  (composite_transition 
     (equivocator_IM IM)
     (lift_equivocators_sub_label_to IM
        (elements equivocating) eqv_state_s l)
     (lift_equivocators_sub_state_to IM
        (elements equivocating) eqv_state_s s, om)).1
  -message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
om: option message
Hs: valid_state_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) s
Hom: option_valid_message_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_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)))
  (sent_by_non_equivocating IM equivocating
     (equivocators_total_state_project IM
        eqv_state_s)) l (
  s, om)
Hlift_s: state_has_fixed_equivocation IM
  (elements equivocating)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
equivocators_no_equivocations_constraint IM
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s l)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s, om) split; [| done].message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
om: option message
Hs: valid_state_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) s
Hom: option_valid_message_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_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)))
  (sent_by_non_equivocating IM equivocating
     (equivocators_total_state_project IM
        eqv_state_s)) l (
  s, om)
Hlift_s: state_has_fixed_equivocation IM
  (elements equivocating)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
composite_no_equivocations (equivocator_IM IM)
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s l)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s, om)
    destruct om as [m |]; [| done].message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
m: message
Hs: valid_state_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) s
Hom: option_valid_message_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) 
  (Some m)
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM)
     (elements equivocating))
  (free_constraint
     (sub_IM (equivocator_IM IM)
        (elements equivocating)))
  (sent_by_non_equivocating IM equivocating
     (equivocators_total_state_project IM
        eqv_state_s)) l (
  s, Some m)
Hlift_s: state_has_fixed_equivocation IM
  (elements equivocating)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
composite_no_equivocations (equivocator_IM IM)
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s l)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s, Some m)
    left.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
m: message
Hs: valid_state_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) s
Hom: option_valid_message_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) 
  (Some m)
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM)
     (elements equivocating))
  (free_constraint
     (sub_IM (equivocator_IM IM)
        (elements equivocating)))
  (sent_by_non_equivocating IM equivocating
     (equivocators_total_state_project IM
        eqv_state_s)) l (
  s, Some m)
Hlift_s: state_has_fixed_equivocation IM
  (elements equivocating)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
composite_has_been_sent (equivocator_IM IM)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s, Some m).1
  m
    apply proj1 in Hc as [Hsent | Hseed].message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
m: message
Hs: valid_state_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) s
Hom: option_valid_message_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) 
  (Some m)
Hsent: composite_has_been_sent
  (sub_IM (equivocator_IM IM)
     (elements equivocating)) (
  s, Some m).1 m
Hlift_s: state_has_fixed_equivocation IM
  (elements equivocating)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
composite_has_been_sent (equivocator_IM IM)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s, Some m).1
  m
message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
m: message
Hs: valid_state_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) s
Hom: option_valid_message_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) 
  (Some m)
Hseed: sent_by_non_equivocating IM equivocating
  (equivocators_total_state_project IM
     eqv_state_s) m
Hlift_s: state_has_fixed_equivocation IM
  (elements equivocating)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
composite_has_been_sent 
  (equivocator_IM IM)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s, 
   Some m).1 m
    +message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
m: message
Hs: valid_state_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) s
Hom: option_valid_message_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) 
  (Some m)
Hsent: composite_has_been_sent
  (sub_IM (equivocator_IM IM)
     (elements equivocating)) (
  s, Some m).1 m
Hlift_s: state_has_fixed_equivocation IM
  (elements equivocating)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
composite_has_been_sent (equivocator_IM IM)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s, Some m).1
  m revert Hsent.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
m: message
Hs: valid_state_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) s
Hom: option_valid_message_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) 
  (Some m)
Hlift_s: state_has_fixed_equivocation IM
  (elements equivocating)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
composite_has_been_sent
  (sub_IM (equivocator_IM IM) (elements equivocating))
  (s, Some m).1 m
→ composite_has_been_sent (equivocator_IM IM)
    (lift_equivocators_sub_state_to IM
       (elements equivocating) eqv_state_s s, Some m).1
    m simpl.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
m: message
Hs: valid_state_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) s
Hom: option_valid_message_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) 
  (Some m)
Hlift_s: state_has_fixed_equivocation IM
  (elements equivocating)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
composite_has_been_sent
  (sub_IM (equivocator_IM IM) (elements equivocating))
  s m
→ composite_has_been_sent (equivocator_IM IM)
    (lift_equivocators_sub_state_to IM
       (elements equivocating) eqv_state_s s) m
      apply (VLSM_incl_valid_state HinclE) in Heqv_state_s.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop
  {|
    vlsm_type :=
      equivocators_fixed_equivocations_vlsm
        IM (elements equivocating);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM IM))
  |} eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
m: message
Hs: valid_state_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) s
Hom: option_valid_message_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) 
  (Some m)
Hlift_s: state_has_fixed_equivocation IM
  (elements equivocating)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
composite_has_been_sent
  (sub_IM (equivocator_IM IM) (elements equivocating))
  s m
→ composite_has_been_sent (equivocator_IM IM)
    (lift_equivocators_sub_state_to IM
       (elements equivocating) eqv_state_s s) m
      apply (VLSM_incl_valid_state (seeded_no_equivocation_incl_preloaded IM (elements equivocating) _)) in Hs.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop
  {|
    vlsm_type :=
      equivocators_fixed_equivocations_vlsm
        IM (elements equivocating);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM IM))
  |} eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
m: message
Hs: valid_state_prop
  {|
    vlsm_type :=
      seeded_equivocators_no_equivocation_vlsm IM
        (elements equivocating)
        (sent_by_non_equivocating IM equivocating
           (equivocators_total_state_project IM
              eqv_state_s));
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (sub_IM (equivocator_IM IM)
              (elements equivocating)))
  |} s
Hom: option_valid_message_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) 
  (Some m)
Hlift_s: state_has_fixed_equivocation IM
  (elements equivocating)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
composite_has_been_sent
  (sub_IM (equivocator_IM IM) (elements equivocating))
  s m
→ composite_has_been_sent (equivocator_IM IM)
    (lift_equivocators_sub_state_to IM
       (elements equivocating) eqv_state_s s) m
      by specialize
        (VLSM_weak_embedding_has_been_sent
          (PreFreeSubE_PreFreeE_weak_embedding IM _ _ Heqv_state_s)
          _ Hs m).
    +message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
m: message
Hs: valid_state_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) s
Hom: option_valid_message_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) 
  (Some m)
Hseed: sent_by_non_equivocating IM equivocating
  (equivocators_total_state_project IM
     eqv_state_s) m
Hlift_s: state_has_fixed_equivocation IM
  (elements equivocating)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
composite_has_been_sent (equivocator_IM IM)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s, Some m).1
  m destruct Hseed as [i [Hi Hsent]].message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
m: message
Hs: valid_state_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) s
Hom: option_valid_message_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) 
  (Some m)
i: index
Hi: i ∉ elements equivocating
Hsent: has_been_sent (IM i)
  (equivocators_total_state_project IM
     eqv_state_s i) m
Hlift_s: state_has_fixed_equivocation IM
  (elements equivocating)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
composite_has_been_sent (equivocator_IM IM)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s, Some m).1
  m
      simpl.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
m: message
Hs: valid_state_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) s
Hom: option_valid_message_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) 
  (Some m)
i: index
Hi: i ∉ elements equivocating
Hsent: has_been_sent (IM i)
  (equivocators_total_state_project IM
     eqv_state_s i) m
Hlift_s: state_has_fixed_equivocation IM
  (elements equivocating)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
composite_has_been_sent (equivocator_IM IM)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s) m
      exists i.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
m: message
Hs: valid_state_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) s
Hom: option_valid_message_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) 
  (Some m)
i: index
Hi: i ∉ elements equivocating
Hsent: has_been_sent (IM i)
  (equivocators_total_state_project IM
     eqv_state_s i) m
Hlift_s: state_has_fixed_equivocation IM
  (elements equivocating)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
has_been_sent (equivocator_IM IM i)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s i) m
      unfold lift_equivocators_sub_state_to.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
m: message
Hs: valid_state_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) s
Hom: option_valid_message_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) 
  (Some m)
i: index
Hi: i ∉ elements equivocating
Hsent: has_been_sent (IM i)
  (equivocators_total_state_project IM
     eqv_state_s i) m
Hlift_s: state_has_fixed_equivocation IM
  (elements equivocating)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
has_been_sent (equivocator_IM IM i)
  match
    decide (sub_index_prop (elements equivocating) i)
  with
  | left e =>
      equivocator_state_append (eqv_state_s i)
        (s (dexist i e))
  | right _ => eqv_state_s i
  end m
      case_decide; [done |].message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
m: message
Hs: valid_state_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) s
Hom: option_valid_message_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) 
  (Some m)
i: index
Hi: i ∉ elements equivocating
Hsent: has_been_sent (IM i)
  (equivocators_total_state_project IM
     eqv_state_s i) m
Hlift_s: state_has_fixed_equivocation IM
  (elements equivocating)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
H9: ¬ sub_index_prop (elements equivocating) i
has_been_sent (equivocator_IM IM i) (eqv_state_s i) m
      unfold equivocator_IM.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
m: message
Hs: valid_state_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) s
Hom: option_valid_message_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) 
  (Some m)
i: index
Hi: i ∉ elements equivocating
Hsent: has_been_sent (IM i)
  (equivocators_total_state_project IM
     eqv_state_s i) m
Hlift_s: state_has_fixed_equivocation IM
  (elements equivocating)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
H9: ¬ sub_index_prop (elements equivocating) i
has_been_sent (equivocator_vlsm (IM i))
  (eqv_state_s i) m
      change (equivocators_total_state_project _ _ _)
        with (equivocator_state_zero (eqv_state_s i)) in Hsent.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
m: message
Hs: valid_state_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) s
Hom: option_valid_message_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) 
  (Some m)
i: index
Hi: i ∉ elements equivocating
Hsent: has_been_sent (IM i)
  (equivocator_state_zero (eqv_state_s i)) m
Hlift_s: state_has_fixed_equivocation IM
  (elements equivocating)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
H9: ¬ sub_index_prop (elements equivocating) i
has_been_sent (equivocator_vlsm (IM i))
  (eqv_state_s i) m
      revert Hsent.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
m: message
Hs: valid_state_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) s
Hom: option_valid_message_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) 
  (Some m)
i: index
Hi: i ∉ elements equivocating
Hlift_s: state_has_fixed_equivocation IM
  (elements equivocating)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
H9: ¬ sub_index_prop (elements equivocating) i
has_been_sent (IM i)
  (equivocator_state_zero (eqv_state_s i)) m
→ has_been_sent (equivocator_vlsm (IM i))
    (eqv_state_s i) m
      apply
        (VLSM_projection_has_been_sent_reflect
          (preloaded_equivocator_zero_projection (IM i))).message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
m: message
Hs: valid_state_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) s
Hom: option_valid_message_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) 
  (Some m)
i: index
Hi: i ∉ elements equivocating
Hlift_s: state_has_fixed_equivocation IM
  (elements equivocating)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
H9: ¬ sub_index_prop (elements equivocating) i
constrained_state_prop (equivocator_vlsm (IM i))
  (eqv_state_s i)
      apply (VLSM_incl_valid_state HinclE) in Heqv_state_s.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop
  {|
    vlsm_type :=
      equivocators_fixed_equivocations_vlsm
        IM (elements equivocating);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM IM))
  |} eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
m: message
Hs: valid_state_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) s
Hom: option_valid_message_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) 
  (Some m)
i: index
Hi: i ∉ elements equivocating
Hlift_s: state_has_fixed_equivocation IM
  (elements equivocating)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
H9: ¬ sub_index_prop (elements equivocating) i
constrained_state_prop (equivocator_vlsm (IM i))
  (eqv_state_s i)
      revert Heqv_state_s.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
m: message
Hs: valid_state_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) s
Hom: option_valid_message_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) 
  (Some m)
i: index
Hi: i ∉ elements equivocating
Hlift_s: state_has_fixed_equivocation IM
  (elements equivocating)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
H9: ¬ sub_index_prop (elements equivocating) i
valid_state_prop
  {|
    vlsm_type :=
      equivocators_fixed_equivocations_vlsm IM
        (elements equivocating);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm (equivocator_IM IM))
  |} eqv_state_s
→ constrained_state_prop (equivocator_vlsm (IM i))
    (eqv_state_s i)
      by apply (VLSM_projection_valid_state (preloaded_component_projection (equivocator_IM IM) i)).
  -message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
om: option message
Hs: valid_state_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) s
Hom: option_valid_message_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_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)))
  (sent_by_non_equivocating IM equivocating
     (equivocators_total_state_project IM
        eqv_state_s)) l (
  s, om)
Hlift_s: state_has_fixed_equivocation IM
  (elements equivocating)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
state_has_fixed_equivocation IM
  (elements equivocating)
  (composite_transition (equivocator_IM IM)
     (lift_equivocators_sub_label_to IM
        (elements equivocating) eqv_state_s l)
     (lift_equivocators_sub_state_to IM
        (elements equivocating) eqv_state_s s, om)).1 destruct (composite_transition _ _ _) as (s', om') eqn: Ht'.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
om: option message
Hs: valid_state_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) s
Hom: option_valid_message_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_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)))
  (sent_by_non_equivocating IM equivocating
     (equivocators_total_state_project IM
        eqv_state_s)) l (
  s, om)
Hlift_s: state_has_fixed_equivocation IM
  (elements equivocating)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
s': composite_state (equivocator_IM IM)
om': option message
Ht': composite_transition (equivocator_IM IM)
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s l)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s, om) =
(s', om')
state_has_fixed_equivocation IM
  (elements equivocating) (s', om').1
    apply
      (equivocating_transition_preserves_fixed_equivocation
        IM (elements equivocating) _ _ _ _ _ Ht' Hlift_s).message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
om: option message
Hs: valid_state_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) s
Hom: option_valid_message_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_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)))
  (sent_by_non_equivocating IM equivocating
     (equivocators_total_state_project IM
        eqv_state_s)) l (
  s, om)
Hlift_s: state_has_fixed_equivocation IM
  (elements equivocating)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
s': composite_state (equivocator_IM IM)
om': option message
Ht': composite_transition (equivocator_IM IM)
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s l)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s, om) =
(s', om')
projT1
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s l)
∈ elements equivocating
    destruct l as (sub_i, li).message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
sub_i: sub_index (elements equivocating)
li: label
  (sub_IM (equivocator_IM IM)
     (elements equivocating) sub_i)
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
om: option message
Hs: valid_state_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) s
Hom: option_valid_message_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_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)))
  (sent_by_non_equivocating IM equivocating
     (equivocators_total_state_project IM
        eqv_state_s)) (existT sub_i li) (
  s, om)
Hlift_s: state_has_fixed_equivocation IM
  (elements equivocating)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
s': composite_state (equivocator_IM IM)
om': option message
Ht': composite_transition (equivocator_IM IM)
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s
     (existT sub_i li))
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s, om) =
(s', om')
projT1
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s
     (existT sub_i li)) ∈ elements equivocating
    destruct_dec_sig sub_i i Hi Hsub_i.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state XE
Heqv_state_s: valid_state_prop XE eqv_state_s
sub_i: sub_index (elements equivocating)
li: label
  (sub_IM (equivocator_IM IM)
     (elements equivocating) sub_i)
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
om: option message
Hs: valid_state_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) s
Hom: option_valid_message_prop
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_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)))
  (sent_by_non_equivocating IM equivocating
     (equivocators_total_state_project IM
        eqv_state_s)) (existT sub_i li) (
  s, om)
Hlift_s: state_has_fixed_equivocation IM
  (elements equivocating)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
s': composite_state (equivocator_IM IM)
om': option message
Ht': composite_transition (equivocator_IM IM)
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s
     (existT sub_i li))
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s, om) =
(s', om')
i: index
Hi: sub_index_prop (elements equivocating) i
Hsub_i: sub_i = dexist i Hi
projT1
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s
     (existT sub_i li)) ∈ elements equivocating
    by subst.
Qed.

A message that can be generated from a state s of the free composition of equivocating equivocators preloaded with all messages has the composite_has_been_sent property for the state obtained upon "appending" state s to valid state for the composition of all equivocators.

This result plays an important role in satisfying the no-message equivocation constraint.

Lemma fixed_equivocation_replay_has_message
  eqv_state_s
  (Heqv_state_s : constrained_state_prop FreeE eqv_state_s)
  im s
  (Him :
    can_produce
    (preloaded_with_all_messages_vlsm
      (free_composite_vlsm (equivocator_IM (sub_IM IM (elements equivocating)))))
    s im)
  : composite_has_been_sent (equivocator_IM IM)
      (lift_equivocators_sub_state_to IM (elements equivocating) eqv_state_s s) im.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state
  (preloaded_with_all_messages_vlsm
     FreeE)
Heqv_state_s: constrained_state_prop FreeE
  eqv_state_s
im: message
s: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
Him: can_produce
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
  s im
composite_has_been_sent (equivocator_IM IM)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s) im
Proof.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state
  (preloaded_with_all_messages_vlsm
     FreeE)
Heqv_state_s: constrained_state_prop FreeE
  eqv_state_s
im: message
s: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
Him: can_produce
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
  s im
composite_has_been_sent (equivocator_IM IM)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s) im
  apply non_empty_valid_trace_from_can_produce in Him
     as (im_eis & im_etr & item & Him_etr & Hlast & Heqs & Him).message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state
  (preloaded_with_all_messages_vlsm
     FreeE)
Heqv_state_s: constrained_state_prop FreeE
  eqv_state_s
im: message
s, im_eis: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
im_etr: list transition_item
item: transition_item
Him_etr: finite_valid_trace
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  im_eis im_etr
Hlast: last_error im_etr = Some item
Heqs: destination item = s
Him: output item = Some im
composite_has_been_sent (equivocator_IM IM)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s) im
  specialize
    (PreFreeSubE_PreFreeE_weak_embedding IM (elements equivocating) _ Heqv_state_s)
    as Hproj.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state
  (preloaded_with_all_messages_vlsm
     FreeE)
Heqv_state_s: constrained_state_prop FreeE
  eqv_state_s
im: message
s, im_eis: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
im_etr: list transition_item
item: transition_item
Him_etr: finite_valid_trace
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  im_eis im_etr
Hlast: last_error im_etr = Some item
Heqs: destination item = s
Him: output item = Some im
Hproj: VLSM_weak_embedding
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s)
composite_has_been_sent (equivocator_IM IM)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s) im
  apply valid_trace_add_default_last in Him_etr.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state
  (preloaded_with_all_messages_vlsm
     FreeE)
Heqv_state_s: constrained_state_prop FreeE
  eqv_state_s
im: message
s, im_eis: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
im_etr: list transition_item
item: transition_item
Him_etr: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  im_eis (finite_trace_last im_eis im_etr)
  im_etr
Hlast: last_error im_etr = Some item
Heqs: destination item = s
Him: output item = Some im
Hproj: VLSM_weak_embedding
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s)
composite_has_been_sent (equivocator_IM IM)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s) im
  apply valid_trace_last_pstate in Him_etr as Him_etr_lst.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state
  (preloaded_with_all_messages_vlsm
     FreeE)
Heqv_state_s: constrained_state_prop FreeE
  eqv_state_s
im: message
s, im_eis: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
im_etr: list transition_item
item: transition_item
Him_etr: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  im_eis (finite_trace_last im_eis im_etr)
  im_etr
Hlast: last_error im_etr = Some item
Heqs: destination item = s
Him: output item = Some im
Hproj: VLSM_weak_embedding
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s)
Him_etr_lst: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  (finite_trace_last im_eis im_etr)
composite_has_been_sent (equivocator_IM IM)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s) im
  specialize
    (VLSM_weak_embedding_has_been_sent Hproj _ Him_etr_lst im) as Hsent.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state
  (preloaded_with_all_messages_vlsm
     FreeE)
Heqv_state_s: constrained_state_prop FreeE
  eqv_state_s
im: message
s, im_eis: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
im_etr: list transition_item
item: transition_item
Him_etr: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  im_eis (finite_trace_last im_eis im_etr)
  im_etr
Hlast: last_error im_etr = Some item
Heqs: destination item = s
Him: output item = Some im
Hproj: VLSM_weak_embedding
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s)
Him_etr_lst: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  (finite_trace_last im_eis im_etr)
Hsent: has_been_sent
  (free_composite_vlsm
     (sub_IM (equivocator_IM IM)
        (elements equivocating)))
  (finite_trace_last im_eis im_etr) im
→ has_been_sent
    (free_composite_vlsm (equivocator_IM IM))
    (lift_equivocators_sub_state_to IM
       (elements equivocating) eqv_state_s
       (finite_trace_last im_eis im_etr)) im
composite_has_been_sent (equivocator_IM IM)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s) im
  unfold has_been_sent in Hsent.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state
  (preloaded_with_all_messages_vlsm
     FreeE)
Heqv_state_s: constrained_state_prop FreeE
  eqv_state_s
im: message
s, im_eis: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
im_etr: list transition_item
item: transition_item
Him_etr: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  im_eis (finite_trace_last im_eis im_etr)
  im_etr
Hlast: last_error im_etr = Some item
Heqs: destination item = s
Him: output item = Some im
Hproj: VLSM_weak_embedding
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s)
Him_etr_lst: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  (finite_trace_last im_eis im_etr)
Hsent: (let (has_been_sent, _, _) :=
   free_composite_HasBeenSentCapability
     (sub_IM (equivocator_IM IM)
        (elements equivocating)) in
 has_been_sent)
  (finite_trace_last im_eis im_etr) im
→ (let (has_been_sent, _, _) :=
     free_composite_HasBeenSentCapability
       (equivocator_IM IM) in
   has_been_sent)
    (lift_equivocators_sub_state_to IM
       (elements equivocating) eqv_state_s
       (finite_trace_last im_eis im_etr)) im
composite_has_been_sent (equivocator_IM IM)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s) im simpl in Hsent.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state
  (preloaded_with_all_messages_vlsm
     FreeE)
Heqv_state_s: constrained_state_prop FreeE
  eqv_state_s
im: message
s, im_eis: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
im_etr: list transition_item
item: transition_item
Him_etr: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  im_eis (finite_trace_last im_eis im_etr)
  im_etr
Hlast: last_error im_etr = Some item
Heqs: destination item = s
Him: output item = Some im
Hproj: VLSM_weak_embedding
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s)
Him_etr_lst: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  (finite_trace_last im_eis im_etr)
Hsent: composite_has_been_sent
  (sub_IM (equivocator_IM IM)
     (elements equivocating))
  (finite_trace_last im_eis im_etr) im
→ composite_has_been_sent 
    (equivocator_IM IM)
    (lift_equivocators_sub_state_to IM
       (elements equivocating) eqv_state_s
       (finite_trace_last im_eis im_etr)) im
composite_has_been_sent (equivocator_IM IM)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s) im
  replace s with (finite_trace_last im_eis im_etr).message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state
  (preloaded_with_all_messages_vlsm
     FreeE)
Heqv_state_s: constrained_state_prop FreeE
  eqv_state_s
im: message
s, im_eis: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
im_etr: list transition_item
item: transition_item
Him_etr: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  im_eis (finite_trace_last im_eis im_etr)
  im_etr
Hlast: last_error im_etr = Some item
Heqs: destination item = s
Him: output item = Some im
Hproj: VLSM_weak_embedding
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s)
Him_etr_lst: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  (finite_trace_last im_eis im_etr)
Hsent: composite_has_been_sent
  (sub_IM (equivocator_IM IM)
     (elements equivocating))
  (finite_trace_last im_eis im_etr) im
→ composite_has_been_sent 
    (equivocator_IM IM)
    (lift_equivocators_sub_state_to IM
       (elements equivocating) eqv_state_s
       (finite_trace_last im_eis im_etr)) im
composite_has_been_sent (equivocator_IM IM)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s
     (finite_trace_last im_eis im_etr)) im
message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state
  (preloaded_with_all_messages_vlsm
     FreeE)
Heqv_state_s: constrained_state_prop FreeE
  eqv_state_s
im: message
s, im_eis: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
im_etr: list transition_item
item: transition_item
Him_etr: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  im_eis (finite_trace_last im_eis im_etr)
  im_etr
Hlast: last_error im_etr = Some item
Heqs: destination item = s
Him: output item = Some im
Hproj: VLSM_weak_embedding
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s)
Him_etr_lst: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  (finite_trace_last im_eis im_etr)
Hsent: composite_has_been_sent
  (sub_IM (equivocator_IM IM)
     (elements equivocating))
  (finite_trace_last im_eis im_etr) im
→ composite_has_been_sent 
    (equivocator_IM IM)
    (lift_equivocators_sub_state_to IM
       (elements equivocating) eqv_state_s
       (finite_trace_last im_eis im_etr)) im
finite_trace_last im_eis im_etr = s
  -message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state
  (preloaded_with_all_messages_vlsm
     FreeE)
Heqv_state_s: constrained_state_prop FreeE
  eqv_state_s
im: message
s, im_eis: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
im_etr: list transition_item
item: transition_item
Him_etr: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  im_eis (finite_trace_last im_eis im_etr)
  im_etr
Hlast: last_error im_etr = Some item
Heqs: destination item = s
Him: output item = Some im
Hproj: VLSM_weak_embedding
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s)
Him_etr_lst: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  (finite_trace_last im_eis im_etr)
Hsent: composite_has_been_sent
  (sub_IM (equivocator_IM IM)
     (elements equivocating))
  (finite_trace_last im_eis im_etr) im
→ composite_has_been_sent 
    (equivocator_IM IM)
    (lift_equivocators_sub_state_to IM
       (elements equivocating) eqv_state_s
       (finite_trace_last im_eis im_etr)) im
composite_has_been_sent (equivocator_IM IM)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s
     (finite_trace_last im_eis im_etr)) im apply Hsent.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state
  (preloaded_with_all_messages_vlsm
     FreeE)
Heqv_state_s: constrained_state_prop FreeE
  eqv_state_s
im: message
s, im_eis: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
im_etr: list transition_item
item: transition_item
Him_etr: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  im_eis (finite_trace_last im_eis im_etr)
  im_etr
Hlast: last_error im_etr = Some item
Heqs: destination item = s
Him: output item = Some im
Hproj: VLSM_weak_embedding
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s)
Him_etr_lst: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  (finite_trace_last im_eis im_etr)
Hsent: composite_has_been_sent
  (sub_IM (equivocator_IM IM)
     (elements equivocating))
  (finite_trace_last im_eis im_etr) im
→ composite_has_been_sent 
    (equivocator_IM IM)
    (lift_equivocators_sub_state_to IM
       (elements equivocating) eqv_state_s
       (finite_trace_last im_eis im_etr)) im
composite_has_been_sent
  (sub_IM (equivocator_IM IM) (elements equivocating))
  (finite_trace_last im_eis im_etr) im
    specialize (has_been_sent_examine_one_trace _ _ _ Him_etr im)
      as Hrew.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state
  (preloaded_with_all_messages_vlsm
     FreeE)
Heqv_state_s: constrained_state_prop FreeE
  eqv_state_s
im: message
s, im_eis: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
im_etr: list transition_item
item: transition_item
Him_etr: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  im_eis (finite_trace_last im_eis im_etr)
  im_etr
Hlast: last_error im_etr = Some item
Heqs: destination item = s
Him: output item = Some im
Hproj: VLSM_weak_embedding
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s)
Him_etr_lst: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  (finite_trace_last im_eis im_etr)
Hsent: composite_has_been_sent
  (sub_IM (equivocator_IM IM)
     (elements equivocating))
  (finite_trace_last im_eis im_etr) im
→ composite_has_been_sent 
    (equivocator_IM IM)
    (lift_equivocators_sub_state_to IM
       (elements equivocating) eqv_state_s
       (finite_trace_last im_eis im_etr)) im
Hrew: has_been_sent
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
  (finite_trace_last im_eis im_etr) im
↔ trace_has_message (field_selector output) im
    im_etr
composite_has_been_sent
  (sub_IM (equivocator_IM IM) (elements equivocating))
  (finite_trace_last im_eis im_etr) im
    unfold has_been_sent in Hrew; cbn; apply Hrew.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state
  (preloaded_with_all_messages_vlsm
     FreeE)
Heqv_state_s: constrained_state_prop FreeE
  eqv_state_s
im: message
s, im_eis: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
im_etr: list transition_item
item: transition_item
Him_etr: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  im_eis (finite_trace_last im_eis im_etr)
  im_etr
Hlast: last_error im_etr = Some item
Heqs: destination item = s
Him: output item = Some im
Hproj: VLSM_weak_embedding
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s)
Him_etr_lst: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  (finite_trace_last im_eis im_etr)
Hsent: composite_has_been_sent
  (sub_IM (equivocator_IM IM)
     (elements equivocating))
  (finite_trace_last im_eis im_etr) im
→ composite_has_been_sent 
    (equivocator_IM IM)
    (lift_equivocators_sub_state_to IM
       (elements equivocating) eqv_state_s
       (finite_trace_last im_eis im_etr)) im
Hrew: (let (has_been_sent, _, _) :=
   free_composite_HasBeenSentCapability
     (equivocator_IM
        (sub_IM IM (elements equivocating))) in
 has_been_sent)
  (finite_trace_last im_eis im_etr) im
↔ trace_has_message (field_selector output) im
    im_etr
trace_has_message (field_selector output) im im_etr
    apply Exists_exists.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state
  (preloaded_with_all_messages_vlsm
     FreeE)
Heqv_state_s: constrained_state_prop FreeE
  eqv_state_s
im: message
s, im_eis: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
im_etr: list transition_item
item: transition_item
Him_etr: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  im_eis (finite_trace_last im_eis im_etr)
  im_etr
Hlast: last_error im_etr = Some item
Heqs: destination item = s
Him: output item = Some im
Hproj: VLSM_weak_embedding
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s)
Him_etr_lst: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  (finite_trace_last im_eis im_etr)
Hsent: composite_has_been_sent
  (sub_IM (equivocator_IM IM)
     (elements equivocating))
  (finite_trace_last im_eis im_etr) im
→ composite_has_been_sent 
    (equivocator_IM IM)
    (lift_equivocators_sub_state_to IM
       (elements equivocating) eqv_state_s
       (finite_trace_last im_eis im_etr)) im
Hrew: (let (has_been_sent, _, _) :=
   free_composite_HasBeenSentCapability
     (equivocator_IM
        (sub_IM IM (elements equivocating))) in
 has_been_sent)
  (finite_trace_last im_eis im_etr) im
↔ trace_has_message (field_selector output) im
    im_etr
∃ x : transition_item,
  x ∈ im_etr ∧ field_selector output im x exists item.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state
  (preloaded_with_all_messages_vlsm
     FreeE)
Heqv_state_s: constrained_state_prop FreeE
  eqv_state_s
im: message
s, im_eis: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
im_etr: list transition_item
item: transition_item
Him_etr: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  im_eis (finite_trace_last im_eis im_etr)
  im_etr
Hlast: last_error im_etr = Some item
Heqs: destination item = s
Him: output item = Some im
Hproj: VLSM_weak_embedding
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s)
Him_etr_lst: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  (finite_trace_last im_eis im_etr)
Hsent: composite_has_been_sent
  (sub_IM (equivocator_IM IM)
     (elements equivocating))
  (finite_trace_last im_eis im_etr) im
→ composite_has_been_sent 
    (equivocator_IM IM)
    (lift_equivocators_sub_state_to IM
       (elements equivocating) eqv_state_s
       (finite_trace_last im_eis im_etr)) im
Hrew: (let (has_been_sent, _, _) :=
   free_composite_HasBeenSentCapability
     (equivocator_IM
        (sub_IM IM (elements equivocating))) in
 has_been_sent)
  (finite_trace_last im_eis im_etr) im
↔ trace_has_message (field_selector output) im
    im_etr
item ∈ im_etr ∧ field_selector output im item split; [| done].message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state
  (preloaded_with_all_messages_vlsm
     FreeE)
Heqv_state_s: constrained_state_prop FreeE
  eqv_state_s
im: message
s, im_eis: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
im_etr: list transition_item
item: transition_item
Him_etr: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  im_eis (finite_trace_last im_eis im_etr)
  im_etr
Hlast: last_error im_etr = Some item
Heqs: destination item = s
Him: output item = Some im
Hproj: VLSM_weak_embedding
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s)
Him_etr_lst: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  (finite_trace_last im_eis im_etr)
Hsent: composite_has_been_sent
  (sub_IM (equivocator_IM IM)
     (elements equivocating))
  (finite_trace_last im_eis im_etr) im
→ composite_has_been_sent 
    (equivocator_IM IM)
    (lift_equivocators_sub_state_to IM
       (elements equivocating) eqv_state_s
       (finite_trace_last im_eis im_etr)) im
Hrew: (let (has_been_sent, _, _) :=
   free_composite_HasBeenSentCapability
     (equivocator_IM
        (sub_IM IM (elements equivocating))) in
 has_been_sent)
  (finite_trace_last im_eis im_etr) im
↔ trace_has_message (field_selector output) im
    im_etr
item ∈ im_etr
    apply elem_of_list_lookup.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state
  (preloaded_with_all_messages_vlsm
     FreeE)
Heqv_state_s: constrained_state_prop FreeE
  eqv_state_s
im: message
s, im_eis: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
im_etr: list transition_item
item: transition_item
Him_etr: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  im_eis (finite_trace_last im_eis im_etr)
  im_etr
Hlast: last_error im_etr = Some item
Heqs: destination item = s
Him: output item = Some im
Hproj: VLSM_weak_embedding
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s)
Him_etr_lst: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  (finite_trace_last im_eis im_etr)
Hsent: composite_has_been_sent
  (sub_IM (equivocator_IM IM)
     (elements equivocating))
  (finite_trace_last im_eis im_etr) im
→ composite_has_been_sent 
    (equivocator_IM IM)
    (lift_equivocators_sub_state_to IM
       (elements equivocating) eqv_state_s
       (finite_trace_last im_eis im_etr)) im
Hrew: (let (has_been_sent, _, _) :=
   free_composite_HasBeenSentCapability
     (equivocator_IM
        (sub_IM IM (elements equivocating))) in
 has_been_sent)
  (finite_trace_last im_eis im_etr) im
↔ trace_has_message (field_selector output) im
    im_etr
∃ i : nat, im_etr !! i = Some item
    rewrite StdppExtras.last_last_error in Hlast.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state
  (preloaded_with_all_messages_vlsm
     FreeE)
Heqv_state_s: constrained_state_prop FreeE
  eqv_state_s
im: message
s, im_eis: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
im_etr: list transition_item
item: transition_item
Him_etr: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  im_eis (finite_trace_last im_eis im_etr)
  im_etr
Hlast: last im_etr = Some item
Heqs: destination item = s
Him: output item = Some im
Hproj: VLSM_weak_embedding
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s)
Him_etr_lst: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  (finite_trace_last im_eis im_etr)
Hsent: composite_has_been_sent
  (sub_IM (equivocator_IM IM)
     (elements equivocating))
  (finite_trace_last im_eis im_etr) im
→ composite_has_been_sent 
    (equivocator_IM IM)
    (lift_equivocators_sub_state_to IM
       (elements equivocating) eqv_state_s
       (finite_trace_last im_eis im_etr)) im
Hrew: (let (has_been_sent, _, _) :=
   free_composite_HasBeenSentCapability
     (equivocator_IM
        (sub_IM IM (elements equivocating))) in
 has_been_sent)
  (finite_trace_last im_eis im_etr) im
↔ trace_has_message (field_selector output) im
    im_etr
∃ i : nat, im_etr !! i = Some item
    replace (Some _) with (last im_etr).message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state
  (preloaded_with_all_messages_vlsm
     FreeE)
Heqv_state_s: constrained_state_prop FreeE
  eqv_state_s
im: message
s, im_eis: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
im_etr: list transition_item
item: transition_item
Him_etr: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  im_eis (finite_trace_last im_eis im_etr)
  im_etr
Hlast: last im_etr = Some item
Heqs: destination item = s
Him: output item = Some im
Hproj: VLSM_weak_embedding
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s)
Him_etr_lst: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  (finite_trace_last im_eis im_etr)
Hsent: composite_has_been_sent
  (sub_IM (equivocator_IM IM)
     (elements equivocating))
  (finite_trace_last im_eis im_etr) im
→ composite_has_been_sent 
    (equivocator_IM IM)
    (lift_equivocators_sub_state_to IM
       (elements equivocating) eqv_state_s
       (finite_trace_last im_eis im_etr)) im
Hrew: (let (has_been_sent, _, _) :=
   free_composite_HasBeenSentCapability
     (equivocator_IM
        (sub_IM IM (elements equivocating))) in
 has_been_sent)
  (finite_trace_last im_eis im_etr) im
↔ trace_has_message (field_selector output) im
    im_etr
∃ i : nat, im_etr !! i = last im_etr
    exists (pred (length im_etr)).message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state
  (preloaded_with_all_messages_vlsm
     FreeE)
Heqv_state_s: constrained_state_prop FreeE
  eqv_state_s
im: message
s, im_eis: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
im_etr: list transition_item
item: transition_item
Him_etr: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  im_eis (finite_trace_last im_eis im_etr)
  im_etr
Hlast: last im_etr = Some item
Heqs: destination item = s
Him: output item = Some im
Hproj: VLSM_weak_embedding
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s)
Him_etr_lst: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  (finite_trace_last im_eis im_etr)
Hsent: composite_has_been_sent
  (sub_IM (equivocator_IM IM)
     (elements equivocating))
  (finite_trace_last im_eis im_etr) im
→ composite_has_been_sent 
    (equivocator_IM IM)
    (lift_equivocators_sub_state_to IM
       (elements equivocating) eqv_state_s
       (finite_trace_last im_eis im_etr)) im
Hrew: (let (has_been_sent, _, _) :=
   free_composite_HasBeenSentCapability
     (equivocator_IM
        (sub_IM IM (elements equivocating))) in
 has_been_sent)
  (finite_trace_last im_eis im_etr) im
↔ trace_has_message (field_selector output) im
    im_etr
im_etr !! Init.Nat.pred (length im_etr) = last im_etr
    by rewrite list.last_lookup.
  -message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state
  (preloaded_with_all_messages_vlsm
     FreeE)
Heqv_state_s: constrained_state_prop FreeE
  eqv_state_s
im: message
s, im_eis: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
im_etr: list transition_item
item: transition_item
Him_etr: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  im_eis (finite_trace_last im_eis im_etr)
  im_etr
Hlast: last_error im_etr = Some item
Heqs: destination item = s
Him: output item = Some im
Hproj: VLSM_weak_embedding
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s)
Him_etr_lst: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  (finite_trace_last im_eis im_etr)
Hsent: composite_has_been_sent
  (sub_IM (equivocator_IM IM)
     (elements equivocating))
  (finite_trace_last im_eis im_etr) im
→ composite_has_been_sent 
    (equivocator_IM IM)
    (lift_equivocators_sub_state_to IM
       (elements equivocating) eqv_state_s
       (finite_trace_last im_eis im_etr)) im
finite_trace_last im_eis im_etr = s apply last_error_destination_last.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
eqv_state_s: state
  (preloaded_with_all_messages_vlsm
     FreeE)
Heqv_state_s: constrained_state_prop FreeE
  eqv_state_s
im: message
s, im_eis: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
im_etr: list transition_item
item: transition_item
Him_etr: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  im_eis (finite_trace_last im_eis im_etr)
  im_etr
Hlast: last_error im_etr = Some item
Heqs: destination item = s
Him: output item = Some im
Hproj: VLSM_weak_embedding
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (sub_IM (equivocator_IM IM)
           (elements equivocating))))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s)
Him_etr_lst: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
  (finite_trace_last im_eis im_etr)
Hsent: composite_has_been_sent
  (sub_IM (equivocator_IM IM)
     (elements equivocating))
  (finite_trace_last im_eis im_etr) im
→ composite_has_been_sent 
    (equivocator_IM IM)
    (lift_equivocators_sub_state_to IM
       (elements equivocating) eqv_state_s
       (finite_trace_last im_eis im_etr)) im
option_map destination (last_error im_etr) = Some s
    by rewrite Hlast; simpl; f_equal.
Qed.

Messages satisfying the strong_fixed_equivocation_constraint have the replayable_message_property for the equivocators_fixed_equivocations_constraint.

Lemma fixed_equivocation_has_replayable_message_prop
  : replayable_message_prop IM (fun _ : message => False)
    (strong_fixed_equivocation_constraint IM equivocating)
    (equivocators_fixed_equivocations_constraint IM (elements equivocating)).message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
replayable_message_prop IM (λ _ : message, False)
  (strong_fixed_equivocation_constraint IM
     equivocating)
  (equivocators_fixed_equivocations_constraint IM
     (elements equivocating))
Proof.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
replayable_message_prop IM (λ _ : message, False)
  (strong_fixed_equivocation_constraint IM
     equivocating)
  (equivocators_fixed_equivocations_constraint IM
     (elements equivocating))
  specialize (vlsm_is_preloaded_with_False XE) as HeqXE.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
replayable_message_prop IM (λ _ : message, False)
  (strong_fixed_equivocation_constraint IM
     equivocating)
  (equivocators_fixed_equivocations_constraint IM
     (elements equivocating))
  specialize (vlsm_is_preloaded_with_False X) as HeqX.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
replayable_message_prop IM (λ _ : message, False)
  (strong_fixed_equivocation_constraint IM
     equivocating)
  (equivocators_fixed_equivocations_constraint IM
     (elements equivocating))
  specialize (equivocators_fixed_equivocations_vlsm_incl_PreFree IM (elements equivocating)) as HinclE.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
replayable_message_prop IM (λ _ : message, False)
  (strong_fixed_equivocation_constraint IM
     equivocating)
  (equivocators_fixed_equivocations_constraint IM
     (elements equivocating))
  intro; intros.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
iom: option message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr iom
l: composite_label IM
HcX: strong_fixed_equivocation_constraint IM
  equivocating l (s, iom)
∃ (eqv_msg_tr : list transition_item) (lst_msg_tr : 
                                       state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (equivocators_fixed_equivocations_constraint
                                                 IM
                                                 (elements
                                                 equivocating)))
                                            (λ _ : message,
                                               False))),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM (elements equivocating)))
       (λ _ : message, False)) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr iom
  destruct iom as [im |]; swap 1 2; [| destruct HcX as [Hsent | Hemitted]].message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr None
l: composite_label IM
HcX: strong_fixed_equivocation_constraint IM
  equivocating l (s, None)
∃ (eqv_msg_tr : list transition_item) (lst_msg_tr : 
                                       state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (equivocators_fixed_equivocations_constraint
                                                 IM
                                                 (elements
                                                 equivocating)))
                                            (λ _ : message,
                                               False))),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM (elements equivocating)))
       (λ _ : message, False)) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr None
message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
Hsent: sent_by_non_equivocating IM equivocating s im
∃ (eqv_msg_tr : list transition_item) 
  (lst_msg_tr : state
                  (preloaded_vlsm
                     (composite_vlsm
                      (equivocator_IM IM)
                      (equivocators_fixed_equivocations_constraint
                      IM (elements equivocating)))
                     (λ _ : message, False))),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm 
          (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM (elements equivocating)))
       (λ _ : message, False)) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from 
          (equivocator_IM IM) initial_message_prop
          lst_msg_tr (Some im)
message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
Hemitted: can_emit
  (equivocators_composition_for_sent IM
     equivocating s) im
∃ (eqv_msg_tr : list transition_item) 
  (lst_msg_tr : state
                  (preloaded_vlsm
                     (composite_vlsm
                      (equivocator_IM IM)
                      (equivocators_fixed_equivocations_constraint
                      IM (elements equivocating)))
                     (λ _ : message, False))),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm 
          (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM (elements equivocating)))
       (λ _ : message, False)) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from 
          (equivocator_IM IM) initial_message_prop
          lst_msg_tr (Some im)
  -message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr None
l: composite_label IM
HcX: strong_fixed_equivocation_constraint IM
  equivocating l (s, None)
∃ (eqv_msg_tr : list transition_item) (lst_msg_tr : 
                                       state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (equivocators_fixed_equivocations_constraint
                                                 IM
                                                 (elements
                                                 equivocating)))
                                            (λ _ : message,
                                               False))),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM (elements equivocating)))
       (λ _ : message, False)) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr None exists [], eqv_state_s.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr None
l: composite_label IM
HcX: strong_fixed_equivocation_constraint IM
  equivocating l (s, None)
finite_valid_trace_from_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_state_s eqv_state_s
  []
∧ equivocators_total_trace_project IM [] = []
  ∧ equivocators_total_state_project IM eqv_state_s =
    s
    ∧ sent_except_from (equivocator_IM IM)
        initial_message_prop eqv_state_s None
    by split; [constructor | eauto].
  -message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
Hsent: sent_by_non_equivocating IM equivocating s im
∃ (eqv_msg_tr : list transition_item) (lst_msg_tr : 
                                       state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (equivocators_fixed_equivocations_constraint
                                                 IM
                                                 (elements
                                                 equivocating)))
                                            (λ _ : message,
                                               False))),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM (elements equivocating)))
       (λ _ : message, False)) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr (Some im) exists [].message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
Hsent: sent_by_non_equivocating IM equivocating s im
∃ lst_msg_tr : state
                 (preloaded_vlsm
                    (composite_vlsm
                       (equivocator_IM IM)
                       (equivocators_fixed_equivocations_constraint
                          IM (elements equivocating)))
                    (λ _ : message, False)),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM (elements equivocating)))
       (λ _ : message, False)) eqv_state_s lst_msg_tr
    []
  ∧ equivocators_total_trace_project IM [] = []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr (Some im) exists eqv_state_s.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
Hsent: sent_by_non_equivocating IM equivocating s im
finite_valid_trace_from_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_state_s eqv_state_s
  []
∧ equivocators_total_trace_project IM [] = []
  ∧ equivocators_total_state_project IM eqv_state_s =
    s
    ∧ sent_except_from (equivocator_IM IM)
        initial_message_prop eqv_state_s (Some im)
    split; [by constructor |].message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
Hsent: sent_by_non_equivocating IM equivocating s im
equivocators_total_trace_project IM [] = []
∧ equivocators_total_state_project IM eqv_state_s = s
  ∧ sent_except_from (equivocator_IM IM)
      initial_message_prop eqv_state_s (Some im)
    split; [done |].message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
Hsent: sent_by_non_equivocating IM equivocating s im
equivocators_total_state_project IM eqv_state_s = s
∧ sent_except_from (equivocator_IM IM)
    initial_message_prop eqv_state_s (Some im)
    split; [done |].message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
Hsent: sent_by_non_equivocating IM equivocating s im
sent_except_from (equivocator_IM IM)
  initial_message_prop eqv_state_s (Some im)
    apply sent_by_non_equivocating_are_sent in Hsent.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
Hsent: composite_has_been_sent IM s im
sent_except_from (equivocator_IM IM)
  initial_message_prop eqv_state_s (Some im)
    apply (VLSM_eq_valid_state HeqXE) in Hstate_valid.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  {|
    vlsm_type := XE; vlsm_machine := XE
  |} eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
Hsent: composite_has_been_sent IM s im
sent_except_from (equivocator_IM IM)
  initial_message_prop eqv_state_s (Some im)
    apply (VLSM_incl_valid_state HinclE) in Hstate_valid.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  {|
    vlsm_type :=
      equivocators_fixed_equivocations_vlsm
        IM (elements equivocating);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM IM))
  |} eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
Hsent: composite_has_been_sent IM s im
sent_except_from (equivocator_IM IM)
  initial_message_prop eqv_state_s (Some im)
    left.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  {|
    vlsm_type :=
      equivocators_fixed_equivocations_vlsm
        IM (elements equivocating);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM IM))
  |} eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
Hsent: composite_has_been_sent IM s im
composite_has_been_sent (equivocator_IM IM)
  eqv_state_s im
    revert Hsent.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  {|
    vlsm_type :=
      equivocators_fixed_equivocations_vlsm
        IM (elements equivocating);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM IM))
  |} eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
composite_has_been_sent IM s im
→ composite_has_been_sent (equivocator_IM IM)
    eqv_state_s im subst.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si
  (equivocators_total_state_project IM
     eqv_state_s) tr
Hstate_valid: valid_state_prop
  {|
    vlsm_type :=
      equivocators_fixed_equivocations_vlsm
        IM (elements equivocating);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM IM))
  |} eqv_state_s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
composite_has_been_sent IM
  (equivocators_total_state_project IM eqv_state_s) im
→ composite_has_been_sent (equivocator_IM IM)
    eqv_state_s im
    by specialize
      (VLSM_projection_has_been_sent_reflect
        (preloaded_equivocators_no_equivocations_vlsm_X_vlsm_projection IM)
        eqv_state_s Hstate_valid im) as Hsent.
  -message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
Hemitted: can_emit
  (equivocators_composition_for_sent IM
     equivocating s) im
∃ (eqv_msg_tr : list transition_item) (lst_msg_tr : 
                                       state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (equivocators_fixed_equivocations_constraint
                                                 IM
                                                 (elements
                                                 equivocating)))
                                            (λ _ : message,
                                               False))),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM (elements equivocating)))
       (λ _ : message, False)) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr (Some im) (*
      If <<im>> can be emitted by the free composition of equivocating components
      seeded with the messages [sent_by_non_equivocating] in <<s>>, then we can
      use Lemma [seeded_equivocators_finite_valid_trace_init_to_rev] to
      simulate that trace in the equivocator-composition of equivocating
      components with no-messages equivocation.
    *)
    apply can_emit_has_trace in Hemitted
        as [im_is [im_tr [im_item [Him_tr Him_item]]]].message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
Him_tr: finite_valid_trace
  (equivocators_composition_for_sent IM
     equivocating s) im_is
  (im_tr ++ [im_item])
Him_item: output im_item = Some im
∃ (eqv_msg_tr : list transition_item) (lst_msg_tr : 
                                       state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (equivocators_fixed_equivocations_constraint
                                                 IM
                                                 (elements
                                                 equivocating)))
                                            (λ _ : message,
                                               False))),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM (elements equivocating)))
       (λ _ : message, False)) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr (Some im)
    apply valid_trace_add_default_last in Him_tr.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
Him_tr: finite_valid_trace_init_to
  (equivocators_composition_for_sent IM
     equivocating s) im_is
  (finite_trace_last im_is
     (im_tr ++ [im_item]))
  (im_tr ++ [im_item])
Him_item: output im_item = Some im
∃ (eqv_msg_tr : list transition_item) (lst_msg_tr : 
                                       state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (equivocators_fixed_equivocations_constraint
                                                 IM
                                                 (elements
                                                 equivocating)))
                                            (λ _ : message,
                                               False))),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM (elements equivocating)))
       (λ _ : message, False)) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr (Some im)
    rewrite finite_trace_last_is_last in Him_tr.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
Him_tr: finite_valid_trace_init_to
  (equivocators_composition_for_sent IM
     equivocating s) im_is
  (destination im_item) 
  (im_tr ++ [im_item])
Him_item: output im_item = Some im
∃ (eqv_msg_tr : list transition_item) (lst_msg_tr : 
                                       state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (equivocators_fixed_equivocations_constraint
                                                 IM
                                                 (elements
                                                 equivocating)))
                                            (λ _ : message,
                                               False))),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM (elements equivocating)))
       (λ _ : message, False)) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr (Some im)
    apply
      (seeded_equivocators_finite_valid_trace_init_to_rev
        (sub_IM IM (elements equivocating)) _ no_initial_messages_in_sub_IM)
      in Him_tr
      as (im_eis & Him_eis & im_es & Him_es & im_etr & Him_etr_pr & Him_etr & Him_output).message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_es im_etr
Him_output: finite_trace_last_output
  (im_tr ++ [im_item]) =
finite_trace_last_output im_etr
Him_item: output im_item = Some im
∃ (eqv_msg_tr : list transition_item) (lst_msg_tr : 
                                       state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (equivocators_fixed_equivocations_constraint
                                                 IM
                                                 (elements
                                                 equivocating)))
                                            (λ _ : message,
                                               False))),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM (elements equivocating)))
       (λ _ : message, False)) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr (Some im)
    rewrite finite_trace_last_output_is_last in Him_output.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_es im_etr
Him_output: output im_item =
finite_trace_last_output im_etr
Him_item: output im_item = Some im
∃ (eqv_msg_tr : list transition_item) (lst_msg_tr : 
                                       state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (equivocators_fixed_equivocations_constraint
                                                 IM
                                                 (elements
                                                 equivocating)))
                                            (λ _ : message,
                                               False))),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM (elements equivocating)))
       (λ _ : message, False)) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr (Some im)
    rewrite Him_item in Him_output.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_es im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
∃ (eqv_msg_tr : list transition_item) (lst_msg_tr : 
                                       state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (equivocators_fixed_equivocations_constraint
                                                 IM
                                                 (elements
                                                 equivocating)))
                                            (λ _ : message,
                                               False))),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM (elements equivocating)))
       (λ _ : message, False)) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr (Some im)
    (*
      We will use Lemma
      [sub_preloaded_replayed_trace_from_valid_equivocating] to replay
      the trace obtained above on top of the given state, thus ensuring that
      state-equivocation is only introduced for the equivocating components.
      We will used Lemmas [fixed_equivocating_messages_sent_by_non_equivocating_are_valid]
      and [fixed_equivocating_non_equivocating_constraint_lifting] to satisfy
      the hypotheses of the replay lemma.
    *)
    pose proof (Hreplay :=
      (sub_preloaded_replayed_trace_from_valid_equivocating
        IM (sent_by_non_equivocating IM equivocating s)
        (elements equivocating)
        (equivocators_fixed_equivocations_constraint IM (elements equivocating))
        (fun m => False))).message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_es im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: (∀ (i : index) (ns : state (IM i)) 
   (s : state
          (preloaded_vlsm
             (composite_vlsm
                (equivocator_IM IM)
                (equivocators_fixed_equivocations_constraint
                   IM
                   (elements equivocating)))
             (λ _ : message, False))),
   i ∈ elements equivocating
   → valid_state_prop
       (preloaded_vlsm
          (composite_vlsm
             (equivocator_IM IM)
             (equivocators_fixed_equivocations_constraint
                IM (elements equivocating)))
          (λ _ : message, False)) s
     → equivocators_fixed_equivocations_constraint
         IM (elements equivocating)
         (existT i (Spawn ns)) (
         s, None))
→ (∀ m : message,
     sent_by_non_equivocating IM
       equivocating s m
     → valid_message_prop
         (preloaded_vlsm
            (composite_vlsm
               (equivocator_IM IM)
               (equivocators_fixed_equivocations_constraint
                  IM (elements equivocating)))
            (λ _ : message, False)) m)
  → ∀ full_replay_state : 
       composite_state 
         (equivocator_IM IM),
      valid_state_prop
        (preloaded_vlsm
           (composite_vlsm
              (equivocator_IM IM)
              (equivocators_fixed_equivocations_constraint
                 IM (elements equivocating)))
           (λ _ : message, False))
        full_replay_state
      → (∀ (l : label
                  (seeded_equivocators_no_equivocation_vlsm
                     IM
                     (elements equivocating)
                     (sent_by_non_equivocating
                      IM equivocating s))) 
           (s0 : state
                   (seeded_equivocators_no_equivocation_vlsm
                      IM
                      (elements equivocating)
                      (sent_by_non_equivocating
                      IM equivocating s))) 
           (om : option message),
           input_valid
             (seeded_equivocators_no_equivocation_vlsm
                IM (elements equivocating)
                (sent_by_non_equivocating IM
                   equivocating s)) l
             (s0, om)
           → valid_state_prop
               (preloaded_vlsm
                  (composite_vlsm
                     (equivocator_IM IM)
                     (equivocators_fixed_equivocations_constraint
                      IM
                      (elements equivocating)))
                  (λ _ : message, False))
               (lift_equivocators_sub_state_to
                  IM (elements equivocating)
                  full_replay_state s0)
             → equivocators_fixed_equivocations_constraint
                 IM (elements equivocating)
                 (lift_equivocators_sub_label_to
                    IM
                    (elements equivocating)
                    full_replay_state l)
                 (lift_equivocators_sub_state_to
                    IM
                    (elements equivocating)
                    full_replay_state s0, om))
        → ∀ (is : composite_state
                    (sub_IM
                      (equivocator_IM IM)
                      (elements equivocating))) 
            (tr : list
                    (composite_transition_item
                      (sub_IM
                      (equivocator_IM IM)
                      (elements equivocating)))),
            finite_valid_trace
              (seeded_equivocators_no_equivocation_vlsm
                 IM (elements equivocating)
                 (sent_by_non_equivocating
                    IM equivocating s)) is
              tr
            → finite_valid_trace_from
                (preloaded_vlsm
                   (composite_vlsm
                      (equivocator_IM IM)
                      (equivocators_fixed_equivocations_constraint
                      IM
                      (elements equivocating)))
                   (λ _ : message, False))
                full_replay_state
                (replayed_trace_from IM
                   (elements equivocating)
                   full_replay_state is tr)
∃ (eqv_msg_tr : list transition_item) (lst_msg_tr : 
                                       state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (equivocators_fixed_equivocations_constraint
                                                 IM
                                                 (elements
                                                 equivocating)))
                                            (λ _ : message,
                                               False))),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM (elements equivocating)))
       (λ _ : message, False)) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr (Some im)
    spec Hreplay.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_es im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: (∀ (i : index) (ns : state (IM i)) 
   (s : state
          (preloaded_vlsm
             (composite_vlsm
                (equivocator_IM IM)
                (equivocators_fixed_equivocations_constraint
                   IM
                   (elements equivocating)))
             (λ _ : message, False))),
   i ∈ elements equivocating
   → valid_state_prop
       (preloaded_vlsm
          (composite_vlsm
             (equivocator_IM IM)
             (equivocators_fixed_equivocations_constraint
                IM (elements equivocating)))
          (λ _ : message, False)) s
     → equivocators_fixed_equivocations_constraint
         IM (elements equivocating)
         (existT i (Spawn ns)) (
         s, None))
→ (∀ m : message,
     sent_by_non_equivocating IM
       equivocating s m
     → valid_message_prop
         (preloaded_vlsm
            (composite_vlsm
               (equivocator_IM IM)
               (equivocators_fixed_equivocations_constraint
                  IM (elements equivocating)))
            (λ _ : message, False)) m)
  → ∀ full_replay_state : 
       composite_state 
         (equivocator_IM IM),
      valid_state_prop
        (preloaded_vlsm
           (composite_vlsm
              (equivocator_IM IM)
              (equivocators_fixed_equivocations_constraint
                 IM (elements equivocating)))
           (λ _ : message, False))
        full_replay_state
      → (∀ (l : label
                  (seeded_equivocators_no_equivocation_vlsm
                     IM
                     (elements equivocating)
                     (sent_by_non_equivocating
                      IM equivocating s))) 
           (s0 : state
                   (seeded_equivocators_no_equivocation_vlsm
                      IM
                      (elements equivocating)
                      (sent_by_non_equivocating
                      IM equivocating s))) 
           (om : option message),
           input_valid
             (seeded_equivocators_no_equivocation_vlsm
                IM (elements equivocating)
                (sent_by_non_equivocating IM
                   equivocating s)) l
             (s0, om)
           → valid_state_prop
               (preloaded_vlsm
                  (composite_vlsm
                     (equivocator_IM IM)
                     (equivocators_fixed_equivocations_constraint
                      IM
                      (elements equivocating)))
                  (λ _ : message, False))
               (lift_equivocators_sub_state_to
                  IM (elements equivocating)
                  full_replay_state s0)
             → equivocators_fixed_equivocations_constraint
                 IM (elements equivocating)
                 (lift_equivocators_sub_label_to
                    IM
                    (elements equivocating)
                    full_replay_state l)
                 (lift_equivocators_sub_state_to
                    IM
                    (elements equivocating)
                    full_replay_state s0, om))
        → ∀ (is : composite_state
                    (sub_IM
                      (equivocator_IM IM)
                      (elements equivocating))) 
            (tr : list
                    (composite_transition_item
                      (sub_IM
                      (equivocator_IM IM)
                      (elements equivocating)))),
            finite_valid_trace
              (seeded_equivocators_no_equivocation_vlsm
                 IM (elements equivocating)
                 (sent_by_non_equivocating
                    IM equivocating s)) is
              tr
            → finite_valid_trace_from
                (preloaded_vlsm
                   (composite_vlsm
                      (equivocator_IM IM)
                      (equivocators_fixed_equivocations_constraint
                      IM
                      (elements equivocating)))
                   (λ _ : message, False))
                full_replay_state
                (replayed_trace_from IM
                   (elements equivocating)
                   full_replay_state is tr)
∀ (i : index) (ns : state (IM i)) (s : state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (equivocators_fixed_equivocations_constraint
                                                 IM
                                                 (elements
                                                 equivocating)))
                                            (λ _ : message,
                                               False))),
  i ∈ elements equivocating
  → valid_state_prop
      (preloaded_vlsm
         (composite_vlsm (equivocator_IM IM)
            (equivocators_fixed_equivocations_constraint
               IM (elements equivocating)))
         (λ _ : message, False)) s
    → equivocators_fixed_equivocations_constraint IM
        (elements equivocating) (existT i (Spawn ns))
        (s, None)
message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_es im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: (∀ m : message,
   sent_by_non_equivocating IM equivocating
     s m
   → valid_message_prop
       (preloaded_vlsm
          (composite_vlsm
             (equivocator_IM IM)
             (equivocators_fixed_equivocations_constraint
                IM (elements equivocating)))
          (λ _ : message, False)) m)
→ ∀ full_replay_state : 
     composite_state (equivocator_IM IM),
    valid_state_prop
      (preloaded_vlsm
         (composite_vlsm 
            (equivocator_IM IM)
            (equivocators_fixed_equivocations_constraint
               IM (elements equivocating)))
         (λ _ : message, False))
      full_replay_state
    → (∀ (l : label
                (seeded_equivocators_no_equivocation_vlsm
                   IM
                   (elements equivocating)
                   (sent_by_non_equivocating
                      IM equivocating s))) 
         (s0 : state
                 (seeded_equivocators_no_equivocation_vlsm
                    IM
                    (elements equivocating)
                    (sent_by_non_equivocating
                      IM equivocating s))) 
         (om : option message),
         input_valid
           (seeded_equivocators_no_equivocation_vlsm
              IM (elements equivocating)
              (sent_by_non_equivocating IM
                 equivocating s)) l (
           s0, om)
         → valid_state_prop
             (preloaded_vlsm
                (composite_vlsm
                   (equivocator_IM IM)
                   (equivocators_fixed_equivocations_constraint
                      IM
                      (elements equivocating)))
                (λ _ : message, False))
             (lift_equivocators_sub_state_to
                IM (elements equivocating)
                full_replay_state s0)
           → equivocators_fixed_equivocations_constraint
               IM (elements equivocating)
               (lift_equivocators_sub_label_to
                  IM (elements equivocating)
                  full_replay_state l)
               (lift_equivocators_sub_state_to
                  IM (elements equivocating)
                  full_replay_state s0, om))
      → ∀ (is : composite_state
                  (sub_IM
                     (equivocator_IM IM)
                     (elements equivocating))) 
          (tr : list
                  (composite_transition_item
                     (sub_IM
                      (equivocator_IM IM)
                      (elements equivocating)))),
          finite_valid_trace
            (seeded_equivocators_no_equivocation_vlsm
               IM (elements equivocating)
               (sent_by_non_equivocating IM
                  equivocating s)) is tr
          → finite_valid_trace_from
              (preloaded_vlsm
                 (composite_vlsm
                    (equivocator_IM IM)
                    (equivocators_fixed_equivocations_constraint
                      IM
                      (elements equivocating)))
                 (λ _ : message, False))
              full_replay_state
              (replayed_trace_from IM
                 (elements equivocating)
                 full_replay_state is tr)
∃ (eqv_msg_tr : list transition_item) 
  (lst_msg_tr : state
                  (preloaded_vlsm
                     (composite_vlsm
                      (equivocator_IM IM)
                      (equivocators_fixed_equivocations_constraint
                      IM (elements equivocating)))
                     (λ _ : message, False))),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm 
          (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM (elements equivocating)))
       (λ _ : message, False)) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from 
          (equivocator_IM IM) initial_message_prop
          lst_msg_tr (Some im)
    {message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_es im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: (∀ (i : index) (ns : state (IM i)) 
   (s : state
          (preloaded_vlsm
             (composite_vlsm
                (equivocator_IM IM)
                (equivocators_fixed_equivocations_constraint
                   IM
                   (elements equivocating)))
             (λ _ : message, False))),
   i ∈ elements equivocating
   → valid_state_prop
       (preloaded_vlsm
          (composite_vlsm
             (equivocator_IM IM)
             (equivocators_fixed_equivocations_constraint
                IM (elements equivocating)))
          (λ _ : message, False)) s
     → equivocators_fixed_equivocations_constraint
         IM (elements equivocating)
         (existT i (Spawn ns)) (
         s, None))
→ (∀ m : message,
     sent_by_non_equivocating IM
       equivocating s m
     → valid_message_prop
         (preloaded_vlsm
            (composite_vlsm
               (equivocator_IM IM)
               (equivocators_fixed_equivocations_constraint
                  IM (elements equivocating)))
            (λ _ : message, False)) m)
  → ∀ full_replay_state : 
       composite_state 
         (equivocator_IM IM),
      valid_state_prop
        (preloaded_vlsm
           (composite_vlsm
              (equivocator_IM IM)
              (equivocators_fixed_equivocations_constraint
                 IM (elements equivocating)))
           (λ _ : message, False))
        full_replay_state
      → (∀ (l : label
                  (seeded_equivocators_no_equivocation_vlsm
                     IM
                     (elements equivocating)
                     (sent_by_non_equivocating
                      IM equivocating s))) 
           (s0 : state
                   (seeded_equivocators_no_equivocation_vlsm
                      IM
                      (elements equivocating)
                      (sent_by_non_equivocating
                      IM equivocating s))) 
           (om : option message),
           input_valid
             (seeded_equivocators_no_equivocation_vlsm
                IM (elements equivocating)
                (sent_by_non_equivocating IM
                   equivocating s)) l
             (s0, om)
           → valid_state_prop
               (preloaded_vlsm
                  (composite_vlsm
                     (equivocator_IM IM)
                     (equivocators_fixed_equivocations_constraint
                      IM
                      (elements equivocating)))
                  (λ _ : message, False))
               (lift_equivocators_sub_state_to
                  IM (elements equivocating)
                  full_replay_state s0)
             → equivocators_fixed_equivocations_constraint
                 IM (elements equivocating)
                 (lift_equivocators_sub_label_to
                    IM
                    (elements equivocating)
                    full_replay_state l)
                 (lift_equivocators_sub_state_to
                    IM
                    (elements equivocating)
                    full_replay_state s0, om))
        → ∀ (is : composite_state
                    (sub_IM
                      (equivocator_IM IM)
                      (elements equivocating))) 
            (tr : list
                    (composite_transition_item
                      (sub_IM
                      (equivocator_IM IM)
                      (elements equivocating)))),
            finite_valid_trace
              (seeded_equivocators_no_equivocation_vlsm
                 IM (elements equivocating)
                 (sent_by_non_equivocating
                    IM equivocating s)) is
              tr
            → finite_valid_trace_from
                (preloaded_vlsm
                   (composite_vlsm
                      (equivocator_IM IM)
                      (equivocators_fixed_equivocations_constraint
                      IM
                      (elements equivocating)))
                   (λ _ : message, False))
                full_replay_state
                (replayed_trace_from IM
                   (elements equivocating)
                   full_replay_state is tr)
∀ (i : index) (ns : state (IM i)) (s : state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (equivocators_fixed_equivocations_constraint
                                                 IM
                                                 (elements
                                                 equivocating)))
                                            (λ _ : message,
                                               False))),
  i ∈ elements equivocating
  → valid_state_prop
      (preloaded_vlsm
         (composite_vlsm (equivocator_IM IM)
            (equivocators_fixed_equivocations_constraint
               IM (elements equivocating)))
         (λ _ : message, False)) s
    → equivocators_fixed_equivocations_constraint IM
        (elements equivocating) (existT i (Spawn ns))
        (s, None) clear -HeqXE.message, index, Ci: Type
H5: Elements index Ci
EqDecision1: 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
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
∀ (i : index) (ns : state (IM i)) (s : state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (equivocators_fixed_equivocations_constraint
                                                 IM
                                                 (elements
                                                 equivocating)))
                                            (λ _ : message,
                                               False))),
  i ∈ elements equivocating
  → valid_state_prop
      (preloaded_vlsm
         (composite_vlsm (equivocator_IM IM)
            (equivocators_fixed_equivocations_constraint
               IM (elements equivocating)))
         (λ _ : message, False)) s
    → equivocators_fixed_equivocations_constraint IM
        (elements equivocating) (existT i (Spawn ns))
        (s, None) intros i ns s Hi Hs.message, index, Ci: Type
H5: Elements index Ci
EqDecision1: 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
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
i: index
ns: state (IM i)
s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hi: i ∈ elements equivocating
Hs: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) s
equivocators_fixed_equivocations_constraint IM
  (elements equivocating) (existT i (Spawn ns))
  (s, None)
      split; [by split |].message, index, Ci: Type
H5: Elements index Ci
EqDecision1: 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
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
i: index
ns: state (IM i)
s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hi: i ∈ elements equivocating
Hs: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) s
state_has_fixed_equivocation IM
  (elements equivocating)
  (composite_transition (equivocator_IM IM)
     (existT i (Spawn ns)) (s, None)).1
      apply (VLSM_eq_valid_state HeqXE) in Hs.message, index, Ci: Type
H5: Elements index Ci
EqDecision1: 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
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
i: index
ns: state (IM i)
s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hi: i ∈ elements equivocating
Hs: valid_state_prop
  {| vlsm_type := XE; vlsm_machine := XE |} s
state_has_fixed_equivocation IM
  (elements equivocating)
  (composite_transition (equivocator_IM IM)
     (existT i (Spawn ns)) (s, None)).1
      apply valid_state_has_fixed_equivocation in Hs.message, index, Ci: Type
H5: Elements index Ci
EqDecision1: 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
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
i: index
ns: state (IM i)
s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hi: i ∈ elements equivocating
Hs: state_has_fixed_equivocation IM
  (elements equivocating) s
state_has_fixed_equivocation IM
  (elements equivocating)
  (composite_transition (equivocator_IM IM)
     (existT i (Spawn ns)) (s, None)).1
      destruct (composite_transition _ _ _) as (s', om') eqn: Ht.message, index, Ci: Type
H5: Elements index Ci
EqDecision1: 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
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
i: index
ns: state (IM i)
s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hi: i ∈ elements equivocating
Hs: state_has_fixed_equivocation IM
  (elements equivocating) s
s': composite_state (equivocator_IM IM)
om': option message
Ht: composite_transition (equivocator_IM IM)
  (existT i (Spawn ns)) (
  s, None) = (s', om')
state_has_fixed_equivocation IM
  (elements equivocating) (s', om').1
      by apply
        (equivocating_transition_preserves_fixed_equivocation
          IM (elements equivocating) _ _ _ _ _ Ht Hs).
    }message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_es im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: (∀ m : message,
   sent_by_non_equivocating IM equivocating
     s m
   → valid_message_prop
       (preloaded_vlsm
          (composite_vlsm
             (equivocator_IM IM)
             (equivocators_fixed_equivocations_constraint
                IM (elements equivocating)))
          (λ _ : message, False)) m)
→ ∀ full_replay_state : 
     composite_state (equivocator_IM IM),
    valid_state_prop
      (preloaded_vlsm
         (composite_vlsm 
            (equivocator_IM IM)
            (equivocators_fixed_equivocations_constraint
               IM (elements equivocating)))
         (λ _ : message, False))
      full_replay_state
    → (∀ (l : label
                (seeded_equivocators_no_equivocation_vlsm
                   IM
                   (elements equivocating)
                   (sent_by_non_equivocating
                      IM equivocating s))) 
         (s0 : state
                 (seeded_equivocators_no_equivocation_vlsm
                    IM
                    (elements equivocating)
                    (sent_by_non_equivocating
                      IM equivocating s))) 
         (om : option message),
         input_valid
           (seeded_equivocators_no_equivocation_vlsm
              IM (elements equivocating)
              (sent_by_non_equivocating IM
                 equivocating s)) l (
           s0, om)
         → valid_state_prop
             (preloaded_vlsm
                (composite_vlsm
                   (equivocator_IM IM)
                   (equivocators_fixed_equivocations_constraint
                      IM
                      (elements equivocating)))
                (λ _ : message, False))
             (lift_equivocators_sub_state_to
                IM (elements equivocating)
                full_replay_state s0)
           → equivocators_fixed_equivocations_constraint
               IM (elements equivocating)
               (lift_equivocators_sub_label_to
                  IM (elements equivocating)
                  full_replay_state l)
               (lift_equivocators_sub_state_to
                  IM (elements equivocating)
                  full_replay_state s0, om))
      → ∀ (is : composite_state
                  (sub_IM
                     (equivocator_IM IM)
                     (elements equivocating))) 
          (tr : list
                  (composite_transition_item
                     (sub_IM
                      (equivocator_IM IM)
                      (elements equivocating)))),
          finite_valid_trace
            (seeded_equivocators_no_equivocation_vlsm
               IM (elements equivocating)
               (sent_by_non_equivocating IM
                  equivocating s)) is tr
          → finite_valid_trace_from
              (preloaded_vlsm
                 (composite_vlsm
                    (equivocator_IM IM)
                    (equivocators_fixed_equivocations_constraint
                      IM
                      (elements equivocating)))
                 (λ _ : message, False))
              full_replay_state
              (replayed_trace_from IM
                 (elements equivocating)
                 full_replay_state is tr)
∃ (eqv_msg_tr : list transition_item) (lst_msg_tr : 
                                       state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (equivocators_fixed_equivocations_constraint
                                                 IM
                                                 (elements
                                                 equivocating)))
                                            (λ _ : message,
                                               False))),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM (elements equivocating)))
       (λ _ : message, False)) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr (Some im)
    spec Hreplay.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_es im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: (∀ m : message,
   sent_by_non_equivocating IM equivocating
     s m
   → valid_message_prop
       (preloaded_vlsm
          (composite_vlsm
             (equivocator_IM IM)
             (equivocators_fixed_equivocations_constraint
                IM (elements equivocating)))
          (λ _ : message, False)) m)
→ ∀ full_replay_state : 
     composite_state (equivocator_IM IM),
    valid_state_prop
      (preloaded_vlsm
         (composite_vlsm 
            (equivocator_IM IM)
            (equivocators_fixed_equivocations_constraint
               IM (elements equivocating)))
         (λ _ : message, False))
      full_replay_state
    → (∀ (l : label
                (seeded_equivocators_no_equivocation_vlsm
                   IM
                   (elements equivocating)
                   (sent_by_non_equivocating
                      IM equivocating s))) 
         (s0 : state
                 (seeded_equivocators_no_equivocation_vlsm
                    IM
                    (elements equivocating)
                    (sent_by_non_equivocating
                      IM equivocating s))) 
         (om : option message),
         input_valid
           (seeded_equivocators_no_equivocation_vlsm
              IM (elements equivocating)
              (sent_by_non_equivocating IM
                 equivocating s)) l (
           s0, om)
         → valid_state_prop
             (preloaded_vlsm
                (composite_vlsm
                   (equivocator_IM IM)
                   (equivocators_fixed_equivocations_constraint
                      IM
                      (elements equivocating)))
                (λ _ : message, False))
             (lift_equivocators_sub_state_to
                IM (elements equivocating)
                full_replay_state s0)
           → equivocators_fixed_equivocations_constraint
               IM (elements equivocating)
               (lift_equivocators_sub_label_to
                  IM (elements equivocating)
                  full_replay_state l)
               (lift_equivocators_sub_state_to
                  IM (elements equivocating)
                  full_replay_state s0, om))
      → ∀ (is : composite_state
                  (sub_IM
                     (equivocator_IM IM)
                     (elements equivocating))) 
          (tr : list
                  (composite_transition_item
                     (sub_IM
                      (equivocator_IM IM)
                      (elements equivocating)))),
          finite_valid_trace
            (seeded_equivocators_no_equivocation_vlsm
               IM (elements equivocating)
               (sent_by_non_equivocating IM
                  equivocating s)) is tr
          → finite_valid_trace_from
              (preloaded_vlsm
                 (composite_vlsm
                    (equivocator_IM IM)
                    (equivocators_fixed_equivocations_constraint
                      IM
                      (elements equivocating)))
                 (λ _ : message, False))
              full_replay_state
              (replayed_trace_from IM
                 (elements equivocating)
                 full_replay_state is tr)
∀ m : message,
  sent_by_non_equivocating IM equivocating s m
  → valid_message_prop
      (preloaded_vlsm
         (composite_vlsm (equivocator_IM IM)
            (equivocators_fixed_equivocations_constraint
               IM (elements equivocating)))
         (λ _ : message, False)) m
message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_es im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: ∀ full_replay_state : composite_state
                      (equivocator_IM IM),
  valid_state_prop
    (preloaded_vlsm
       (composite_vlsm 
          (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM (elements equivocating)))
       (λ _ : message, False))
    full_replay_state
  → (∀ (l : label
              (seeded_equivocators_no_equivocation_vlsm
                 IM (elements equivocating)
                 (sent_by_non_equivocating
                    IM equivocating s))) 
       (s0 : state
               (seeded_equivocators_no_equivocation_vlsm
                  IM (elements equivocating)
                  (sent_by_non_equivocating
                     IM equivocating s))) 
       (om : option message),
       input_valid
         (seeded_equivocators_no_equivocation_vlsm
            IM (elements equivocating)
            (sent_by_non_equivocating IM
               equivocating s)) l (
         s0, om)
       → valid_state_prop
           (preloaded_vlsm
              (composite_vlsm
                 (equivocator_IM IM)
                 (equivocators_fixed_equivocations_constraint
                    IM
                    (elements equivocating)))
              (λ _ : message, False))
           (lift_equivocators_sub_state_to
              IM (elements equivocating)
              full_replay_state s0)
         → equivocators_fixed_equivocations_constraint
             IM (elements equivocating)
             (lift_equivocators_sub_label_to
                IM (elements equivocating)
                full_replay_state l)
             (lift_equivocators_sub_state_to
                IM (elements equivocating)
                full_replay_state s0, om))
    → ∀ (is : composite_state
                (sub_IM 
                   (equivocator_IM IM)
                   (elements equivocating))) 
        (tr : list
                (composite_transition_item
                   (sub_IM
                      (equivocator_IM IM)
                      (elements equivocating)))),
        finite_valid_trace
          (seeded_equivocators_no_equivocation_vlsm
             IM (elements equivocating)
             (sent_by_non_equivocating IM
                equivocating s)) is tr
        → finite_valid_trace_from
            (preloaded_vlsm
               (composite_vlsm
                  (equivocator_IM IM)
                  (equivocators_fixed_equivocations_constraint
                     IM
                     (elements equivocating)))
               (λ _ : message, False))
            full_replay_state
            (replayed_trace_from IM
               (elements equivocating)
               full_replay_state is tr)
∃ (eqv_msg_tr : list transition_item) 
  (lst_msg_tr : state
                  (preloaded_vlsm
                     (composite_vlsm
                      (equivocator_IM IM)
                      (equivocators_fixed_equivocations_constraint
                      IM (elements equivocating)))
                     (λ _ : message, False))),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm 
          (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM (elements equivocating)))
       (λ _ : message, False)) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from 
          (equivocator_IM IM) initial_message_prop
          lst_msg_tr (Some im)
    {message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_es im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: (∀ m : message,
   sent_by_non_equivocating IM equivocating
     s m
   → valid_message_prop
       (preloaded_vlsm
          (composite_vlsm
             (equivocator_IM IM)
             (equivocators_fixed_equivocations_constraint
                IM (elements equivocating)))
          (λ _ : message, False)) m)
→ ∀ full_replay_state : 
     composite_state (equivocator_IM IM),
    valid_state_prop
      (preloaded_vlsm
         (composite_vlsm 
            (equivocator_IM IM)
            (equivocators_fixed_equivocations_constraint
               IM (elements equivocating)))
         (λ _ : message, False))
      full_replay_state
    → (∀ (l : label
                (seeded_equivocators_no_equivocation_vlsm
                   IM
                   (elements equivocating)
                   (sent_by_non_equivocating
                      IM equivocating s))) 
         (s0 : state
                 (seeded_equivocators_no_equivocation_vlsm
                    IM
                    (elements equivocating)
                    (sent_by_non_equivocating
                      IM equivocating s))) 
         (om : option message),
         input_valid
           (seeded_equivocators_no_equivocation_vlsm
              IM (elements equivocating)
              (sent_by_non_equivocating IM
                 equivocating s)) l (
           s0, om)
         → valid_state_prop
             (preloaded_vlsm
                (composite_vlsm
                   (equivocator_IM IM)
                   (equivocators_fixed_equivocations_constraint
                      IM
                      (elements equivocating)))
                (λ _ : message, False))
             (lift_equivocators_sub_state_to
                IM (elements equivocating)
                full_replay_state s0)
           → equivocators_fixed_equivocations_constraint
               IM (elements equivocating)
               (lift_equivocators_sub_label_to
                  IM (elements equivocating)
                  full_replay_state l)
               (lift_equivocators_sub_state_to
                  IM (elements equivocating)
                  full_replay_state s0, om))
      → ∀ (is : composite_state
                  (sub_IM
                     (equivocator_IM IM)
                     (elements equivocating))) 
          (tr : list
                  (composite_transition_item
                     (sub_IM
                      (equivocator_IM IM)
                      (elements equivocating)))),
          finite_valid_trace
            (seeded_equivocators_no_equivocation_vlsm
               IM (elements equivocating)
               (sent_by_non_equivocating IM
                  equivocating s)) is tr
          → finite_valid_trace_from
              (preloaded_vlsm
                 (composite_vlsm
                    (equivocator_IM IM)
                    (equivocators_fixed_equivocations_constraint
                      IM
                      (elements equivocating)))
                 (λ _ : message, False))
              full_replay_state
              (replayed_trace_from IM
                 (elements equivocating)
                 full_replay_state is tr)
∀ m : message,
  sent_by_non_equivocating IM equivocating s m
  → valid_message_prop
      (preloaded_vlsm
         (composite_vlsm (equivocator_IM IM)
            (equivocators_fixed_equivocations_constraint
               IM (elements equivocating)))
         (λ _ : message, False)) m
      subst s.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si
  (equivocators_total_state_project IM
     eqv_state_s) tr
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_item: transition_item
im_tr: list transition_item
im_is: state
  (equivocators_composition_for_sent IM
     equivocating
     (equivocators_total_state_project IM
        eqv_state_s))
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) im_eis im_es
  im_etr
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_output: Some im = finite_trace_last_output im_etr
Hreplay: (∀ m : message,
   sent_by_non_equivocating IM equivocating
     (equivocators_total_state_project IM
        eqv_state_s) m
   → valid_message_prop
       (preloaded_vlsm
          (composite_vlsm
             (equivocator_IM IM)
             (equivocators_fixed_equivocations_constraint
                IM (elements equivocating)))
          (λ _ : message, False)) m)
→ ∀ full_replay_state : 
     composite_state (equivocator_IM IM),
    valid_state_prop
      (preloaded_vlsm
         (composite_vlsm 
            (equivocator_IM IM)
            (equivocators_fixed_equivocations_constraint
               IM (elements equivocating)))
         (λ _ : message, False))
      full_replay_state
    → (∀ (l : label
                (seeded_equivocators_no_equivocation_vlsm
                   IM
                   (elements equivocating)
                   (sent_by_non_equivocating
                      IM equivocating
                      (equivocators_total_state_project
                      IM eqv_state_s)))) 
         (s : state
                (seeded_equivocators_no_equivocation_vlsm
                   IM
                   (elements equivocating)
                   (sent_by_non_equivocating
                      IM equivocating
                      (equivocators_total_state_project
                      IM eqv_state_s)))) 
         (om : option message),
         input_valid
           (seeded_equivocators_no_equivocation_vlsm
              IM (elements equivocating)
              (sent_by_non_equivocating IM
                 equivocating
                 (equivocators_total_state_project
                    IM eqv_state_s))) l
           (s, om)
         → valid_state_prop
             (preloaded_vlsm
                (composite_vlsm
                   (equivocator_IM IM)
                   (equivocators_fixed_equivocations_constraint
                      IM
                      (elements equivocating)))
                (λ _ : message, False))
             (lift_equivocators_sub_state_to
                IM (elements equivocating)
                full_replay_state s)
           → equivocators_fixed_equivocations_constraint
               IM (elements equivocating)
               (lift_equivocators_sub_label_to
                  IM (elements equivocating)
                  full_replay_state l)
               (lift_equivocators_sub_state_to
                  IM (elements equivocating)
                  full_replay_state s, om))
      → ∀ (is : composite_state
                  (sub_IM
                     (equivocator_IM IM)
                     (elements equivocating))) 
          (tr : list
                  (composite_transition_item
                     (sub_IM
                      (equivocator_IM IM)
                      (elements equivocating)))),
          finite_valid_trace
            (seeded_equivocators_no_equivocation_vlsm
               IM (elements equivocating)
               (sent_by_non_equivocating IM
                  equivocating
                  (equivocators_total_state_project
                     IM eqv_state_s))) is tr
          → finite_valid_trace_from
              (preloaded_vlsm
                 (composite_vlsm
                    (equivocator_IM IM)
                    (equivocators_fixed_equivocations_constraint
                      IM
                      (elements equivocating)))
                 (λ _ : message, False))
              full_replay_state
              (replayed_trace_from IM
                 (elements equivocating)
                 full_replay_state is tr)
Him_item: output im_item = Some im
∀ m : message,
  sent_by_non_equivocating IM equivocating
    (equivocators_total_state_project IM eqv_state_s)
    m
  → valid_message_prop
      (preloaded_vlsm
         (composite_vlsm (equivocator_IM IM)
            (equivocators_fixed_equivocations_constraint
               IM (elements equivocating)))
         (λ _ : message, False)) m
      apply valid_trace_last_pstate in HtrX.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
HtrX: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False))
  (equivocators_total_state_project IM
     eqv_state_s)
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_item: transition_item
im_tr: list transition_item
im_is: state
  (equivocators_composition_for_sent IM
     equivocating
     (equivocators_total_state_project IM
        eqv_state_s))
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) im_eis im_es
  im_etr
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_output: Some im = finite_trace_last_output im_etr
Hreplay: (∀ m : message,
   sent_by_non_equivocating IM equivocating
     (equivocators_total_state_project IM
        eqv_state_s) m
   → valid_message_prop
       (preloaded_vlsm
          (composite_vlsm
             (equivocator_IM IM)
             (equivocators_fixed_equivocations_constraint
                IM (elements equivocating)))
          (λ _ : message, False)) m)
→ ∀ full_replay_state : 
     composite_state (equivocator_IM IM),
    valid_state_prop
      (preloaded_vlsm
         (composite_vlsm 
            (equivocator_IM IM)
            (equivocators_fixed_equivocations_constraint
               IM (elements equivocating)))
         (λ _ : message, False))
      full_replay_state
    → (∀ (l : label
                (seeded_equivocators_no_equivocation_vlsm
                   IM
                   (elements equivocating)
                   (sent_by_non_equivocating
                      IM equivocating
                      (equivocators_total_state_project
                      IM eqv_state_s)))) 
         (s : state
                (seeded_equivocators_no_equivocation_vlsm
                   IM
                   (elements equivocating)
                   (sent_by_non_equivocating
                      IM equivocating
                      (equivocators_total_state_project
                      IM eqv_state_s)))) 
         (om : option message),
         input_valid
           (seeded_equivocators_no_equivocation_vlsm
              IM (elements equivocating)
              (sent_by_non_equivocating IM
                 equivocating
                 (equivocators_total_state_project
                    IM eqv_state_s))) l
           (s, om)
         → valid_state_prop
             (preloaded_vlsm
                (composite_vlsm
                   (equivocator_IM IM)
                   (equivocators_fixed_equivocations_constraint
                      IM
                      (elements equivocating)))
                (λ _ : message, False))
             (lift_equivocators_sub_state_to
                IM (elements equivocating)
                full_replay_state s)
           → equivocators_fixed_equivocations_constraint
               IM (elements equivocating)
               (lift_equivocators_sub_label_to
                  IM (elements equivocating)
                  full_replay_state l)
               (lift_equivocators_sub_state_to
                  IM (elements equivocating)
                  full_replay_state s, om))
      → ∀ (is : composite_state
                  (sub_IM
                     (equivocator_IM IM)
                     (elements equivocating))) 
          (tr : list
                  (composite_transition_item
                     (sub_IM
                      (equivocator_IM IM)
                      (elements equivocating)))),
          finite_valid_trace
            (seeded_equivocators_no_equivocation_vlsm
               IM (elements equivocating)
               (sent_by_non_equivocating IM
                  equivocating
                  (equivocators_total_state_project
                     IM eqv_state_s))) is tr
          → finite_valid_trace_from
              (preloaded_vlsm
                 (composite_vlsm
                    (equivocator_IM IM)
                    (equivocators_fixed_equivocations_constraint
                      IM
                      (elements equivocating)))
                 (λ _ : message, False))
              full_replay_state
              (replayed_trace_from IM
                 (elements equivocating)
                 full_replay_state is tr)
Him_item: output im_item = Some im
∀ m : message,
  sent_by_non_equivocating IM equivocating
    (equivocators_total_state_project IM eqv_state_s)
    m
  → valid_message_prop
      (preloaded_vlsm
         (composite_vlsm (equivocator_IM IM)
            (equivocators_fixed_equivocations_constraint
               IM (elements equivocating)))
         (λ _ : message, False)) m
      apply (VLSM_eq_valid_state HeqXE) in Hstate_valid.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
HtrX: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False))
  (equivocators_total_state_project IM
     eqv_state_s)
Hstate_valid: valid_state_prop
  {|
    vlsm_type := XE; vlsm_machine := XE
  |} eqv_state_s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_item: transition_item
im_tr: list transition_item
im_is: state
  (equivocators_composition_for_sent IM
     equivocating
     (equivocators_total_state_project IM
        eqv_state_s))
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) im_eis im_es
  im_etr
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_output: Some im = finite_trace_last_output im_etr
Hreplay: (∀ m : message,
   sent_by_non_equivocating IM equivocating
     (equivocators_total_state_project IM
        eqv_state_s) m
   → valid_message_prop
       (preloaded_vlsm
          (composite_vlsm
             (equivocator_IM IM)
             (equivocators_fixed_equivocations_constraint
                IM (elements equivocating)))
          (λ _ : message, False)) m)
→ ∀ full_replay_state : 
     composite_state (equivocator_IM IM),
    valid_state_prop
      (preloaded_vlsm
         (composite_vlsm 
            (equivocator_IM IM)
            (equivocators_fixed_equivocations_constraint
               IM (elements equivocating)))
         (λ _ : message, False))
      full_replay_state
    → (∀ (l : label
                (seeded_equivocators_no_equivocation_vlsm
                   IM
                   (elements equivocating)
                   (sent_by_non_equivocating
                      IM equivocating
                      (equivocators_total_state_project
                      IM eqv_state_s)))) 
         (s : state
                (seeded_equivocators_no_equivocation_vlsm
                   IM
                   (elements equivocating)
                   (sent_by_non_equivocating
                      IM equivocating
                      (equivocators_total_state_project
                      IM eqv_state_s)))) 
         (om : option message),
         input_valid
           (seeded_equivocators_no_equivocation_vlsm
              IM (elements equivocating)
              (sent_by_non_equivocating IM
                 equivocating
                 (equivocators_total_state_project
                    IM eqv_state_s))) l
           (s, om)
         → valid_state_prop
             (preloaded_vlsm
                (composite_vlsm
                   (equivocator_IM IM)
                   (equivocators_fixed_equivocations_constraint
                      IM
                      (elements equivocating)))
                (λ _ : message, False))
             (lift_equivocators_sub_state_to
                IM (elements equivocating)
                full_replay_state s)
           → equivocators_fixed_equivocations_constraint
               IM (elements equivocating)
               (lift_equivocators_sub_label_to
                  IM (elements equivocating)
                  full_replay_state l)
               (lift_equivocators_sub_state_to
                  IM (elements equivocating)
                  full_replay_state s, om))
      → ∀ (is : composite_state
                  (sub_IM
                     (equivocator_IM IM)
                     (elements equivocating))) 
          (tr : list
                  (composite_transition_item
                     (sub_IM
                      (equivocator_IM IM)
                      (elements equivocating)))),
          finite_valid_trace
            (seeded_equivocators_no_equivocation_vlsm
               IM (elements equivocating)
               (sent_by_non_equivocating IM
                  equivocating
                  (equivocators_total_state_project
                     IM eqv_state_s))) is tr
          → finite_valid_trace_from
              (preloaded_vlsm
                 (composite_vlsm
                    (equivocator_IM IM)
                    (equivocators_fixed_equivocations_constraint
                      IM
                      (elements equivocating)))
                 (λ _ : message, False))
              full_replay_state
              (replayed_trace_from IM
                 (elements equivocating)
                 full_replay_state is tr)
Him_item: output im_item = Some im
∀ m : message,
  sent_by_non_equivocating IM equivocating
    (equivocators_total_state_project IM eqv_state_s)
    m
  → valid_message_prop
      (preloaded_vlsm
         (composite_vlsm (equivocator_IM IM)
            (equivocators_fixed_equivocations_constraint
               IM (elements equivocating)))
         (λ _ : message, False)) m
      apply (VLSM_eq_valid_state HeqX) in HtrX.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
HtrX: valid_state_prop
  {| vlsm_type := X; vlsm_machine := X |}
  (equivocators_total_state_project IM
     eqv_state_s)
Hstate_valid: valid_state_prop
  {|
    vlsm_type := XE; vlsm_machine := XE
  |} eqv_state_s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_item: transition_item
im_tr: list transition_item
im_is: state
  (equivocators_composition_for_sent IM
     equivocating
     (equivocators_total_state_project IM
        eqv_state_s))
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) im_eis im_es
  im_etr
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_output: Some im = finite_trace_last_output im_etr
Hreplay: (∀ m : message,
   sent_by_non_equivocating IM equivocating
     (equivocators_total_state_project IM
        eqv_state_s) m
   → valid_message_prop
       (preloaded_vlsm
          (composite_vlsm
             (equivocator_IM IM)
             (equivocators_fixed_equivocations_constraint
                IM (elements equivocating)))
          (λ _ : message, False)) m)
→ ∀ full_replay_state : 
     composite_state (equivocator_IM IM),
    valid_state_prop
      (preloaded_vlsm
         (composite_vlsm 
            (equivocator_IM IM)
            (equivocators_fixed_equivocations_constraint
               IM (elements equivocating)))
         (λ _ : message, False))
      full_replay_state
    → (∀ (l : label
                (seeded_equivocators_no_equivocation_vlsm
                   IM
                   (elements equivocating)
                   (sent_by_non_equivocating
                      IM equivocating
                      (equivocators_total_state_project
                      IM eqv_state_s)))) 
         (s : state
                (seeded_equivocators_no_equivocation_vlsm
                   IM
                   (elements equivocating)
                   (sent_by_non_equivocating
                      IM equivocating
                      (equivocators_total_state_project
                      IM eqv_state_s)))) 
         (om : option message),
         input_valid
           (seeded_equivocators_no_equivocation_vlsm
              IM (elements equivocating)
              (sent_by_non_equivocating IM
                 equivocating
                 (equivocators_total_state_project
                    IM eqv_state_s))) l
           (s, om)
         → valid_state_prop
             (preloaded_vlsm
                (composite_vlsm
                   (equivocator_IM IM)
                   (equivocators_fixed_equivocations_constraint
                      IM
                      (elements equivocating)))
                (λ _ : message, False))
             (lift_equivocators_sub_state_to
                IM (elements equivocating)
                full_replay_state s)
           → equivocators_fixed_equivocations_constraint
               IM (elements equivocating)
               (lift_equivocators_sub_label_to
                  IM (elements equivocating)
                  full_replay_state l)
               (lift_equivocators_sub_state_to
                  IM (elements equivocating)
                  full_replay_state s, om))
      → ∀ (is : composite_state
                  (sub_IM
                     (equivocator_IM IM)
                     (elements equivocating))) 
          (tr : list
                  (composite_transition_item
                     (sub_IM
                      (equivocator_IM IM)
                      (elements equivocating)))),
          finite_valid_trace
            (seeded_equivocators_no_equivocation_vlsm
               IM (elements equivocating)
               (sent_by_non_equivocating IM
                  equivocating
                  (equivocators_total_state_project
                     IM eqv_state_s))) is tr
          → finite_valid_trace_from
              (preloaded_vlsm
                 (composite_vlsm
                    (equivocator_IM IM)
                    (equivocators_fixed_equivocations_constraint
                      IM
                      (elements equivocating)))
                 (λ _ : message, False))
              full_replay_state
              (replayed_trace_from IM
                 (elements equivocating)
                 full_replay_state is tr)
Him_item: output im_item = Some im
∀ m : message,
  sent_by_non_equivocating IM equivocating
    (equivocators_total_state_project IM eqv_state_s)
    m
  → valid_message_prop
      (preloaded_vlsm
         (composite_vlsm (equivocator_IM IM)
            (equivocators_fixed_equivocations_constraint
               IM (elements equivocating)))
         (λ _ : message, False)) m
      intros m Hsent.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
HtrX: valid_state_prop
  {| vlsm_type := X; vlsm_machine := X |}
  (equivocators_total_state_project IM
     eqv_state_s)
Hstate_valid: valid_state_prop
  {|
    vlsm_type := XE; vlsm_machine := XE
  |} eqv_state_s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_item: transition_item
im_tr: list transition_item
im_is: state
  (equivocators_composition_for_sent IM
     equivocating
     (equivocators_total_state_project IM
        eqv_state_s))
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) im_eis im_es
  im_etr
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_output: Some im = finite_trace_last_output im_etr
Hreplay: (∀ m : message,
   sent_by_non_equivocating IM equivocating
     (equivocators_total_state_project IM
        eqv_state_s) m
   → valid_message_prop
       (preloaded_vlsm
          (composite_vlsm
             (equivocator_IM IM)
             (equivocators_fixed_equivocations_constraint
                IM (elements equivocating)))
          (λ _ : message, False)) m)
→ ∀ full_replay_state : 
     composite_state (equivocator_IM IM),
    valid_state_prop
      (preloaded_vlsm
         (composite_vlsm 
            (equivocator_IM IM)
            (equivocators_fixed_equivocations_constraint
               IM (elements equivocating)))
         (λ _ : message, False))
      full_replay_state
    → (∀ (l : label
                (seeded_equivocators_no_equivocation_vlsm
                   IM
                   (elements equivocating)
                   (sent_by_non_equivocating
                      IM equivocating
                      (equivocators_total_state_project
                      IM eqv_state_s)))) 
         (s : state
                (seeded_equivocators_no_equivocation_vlsm
                   IM
                   (elements equivocating)
                   (sent_by_non_equivocating
                      IM equivocating
                      (equivocators_total_state_project
                      IM eqv_state_s)))) 
         (om : option message),
         input_valid
           (seeded_equivocators_no_equivocation_vlsm
              IM (elements equivocating)
              (sent_by_non_equivocating IM
                 equivocating
                 (equivocators_total_state_project
                    IM eqv_state_s))) l
           (s, om)
         → valid_state_prop
             (preloaded_vlsm
                (composite_vlsm
                   (equivocator_IM IM)
                   (equivocators_fixed_equivocations_constraint
                      IM
                      (elements equivocating)))
                (λ _ : message, False))
             (lift_equivocators_sub_state_to
                IM (elements equivocating)
                full_replay_state s)
           → equivocators_fixed_equivocations_constraint
               IM (elements equivocating)
               (lift_equivocators_sub_label_to
                  IM (elements equivocating)
                  full_replay_state l)
               (lift_equivocators_sub_state_to
                  IM (elements equivocating)
                  full_replay_state s, om))
      → ∀ (is : composite_state
                  (sub_IM
                     (equivocator_IM IM)
                     (elements equivocating))) 
          (tr : list
                  (composite_transition_item
                     (sub_IM
                      (equivocator_IM IM)
                      (elements equivocating)))),
          finite_valid_trace
            (seeded_equivocators_no_equivocation_vlsm
               IM (elements equivocating)
               (sent_by_non_equivocating IM
                  equivocating
                  (equivocators_total_state_project
                     IM eqv_state_s))) is tr
          → finite_valid_trace_from
              (preloaded_vlsm
                 (composite_vlsm
                    (equivocator_IM IM)
                    (equivocators_fixed_equivocations_constraint
                      IM
                      (elements equivocating)))
                 (λ _ : message, False))
              full_replay_state
              (replayed_trace_from IM
                 (elements equivocating)
                 full_replay_state is tr)
Him_item: output im_item = Some im
m: message
Hsent: sent_by_non_equivocating IM equivocating
  (equivocators_total_state_project IM
     eqv_state_s) m
valid_message_prop
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) m
      apply (VLSM_incl_valid_message (proj1 HeqXE)); [by left |].message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
HtrX: valid_state_prop
  {| vlsm_type := X; vlsm_machine := X |}
  (equivocators_total_state_project IM
     eqv_state_s)
Hstate_valid: valid_state_prop
  {|
    vlsm_type := XE; vlsm_machine := XE
  |} eqv_state_s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_item: transition_item
im_tr: list transition_item
im_is: state
  (equivocators_composition_for_sent IM
     equivocating
     (equivocators_total_state_project IM
        eqv_state_s))
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) im_eis im_es
  im_etr
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_output: Some im = finite_trace_last_output im_etr
Hreplay: (∀ m : message,
   sent_by_non_equivocating IM equivocating
     (equivocators_total_state_project IM
        eqv_state_s) m
   → valid_message_prop
       (preloaded_vlsm
          (composite_vlsm
             (equivocator_IM IM)
             (equivocators_fixed_equivocations_constraint
                IM (elements equivocating)))
          (λ _ : message, False)) m)
→ ∀ full_replay_state : 
     composite_state (equivocator_IM IM),
    valid_state_prop
      (preloaded_vlsm
         (composite_vlsm 
            (equivocator_IM IM)
            (equivocators_fixed_equivocations_constraint
               IM (elements equivocating)))
         (λ _ : message, False))
      full_replay_state
    → (∀ (l : label
                (seeded_equivocators_no_equivocation_vlsm
                   IM
                   (elements equivocating)
                   (sent_by_non_equivocating
                      IM equivocating
                      (equivocators_total_state_project
                      IM eqv_state_s)))) 
         (s : state
                (seeded_equivocators_no_equivocation_vlsm
                   IM
                   (elements equivocating)
                   (sent_by_non_equivocating
                      IM equivocating
                      (equivocators_total_state_project
                      IM eqv_state_s)))) 
         (om : option message),
         input_valid
           (seeded_equivocators_no_equivocation_vlsm
              IM (elements equivocating)
              (sent_by_non_equivocating IM
                 equivocating
                 (equivocators_total_state_project
                    IM eqv_state_s))) l
           (s, om)
         → valid_state_prop
             (preloaded_vlsm
                (composite_vlsm
                   (equivocator_IM IM)
                   (equivocators_fixed_equivocations_constraint
                      IM
                      (elements equivocating)))
                (λ _ : message, False))
             (lift_equivocators_sub_state_to
                IM (elements equivocating)
                full_replay_state s)
           → equivocators_fixed_equivocations_constraint
               IM (elements equivocating)
               (lift_equivocators_sub_label_to
                  IM (elements equivocating)
                  full_replay_state l)
               (lift_equivocators_sub_state_to
                  IM (elements equivocating)
                  full_replay_state s, om))
      → ∀ (is : composite_state
                  (sub_IM
                     (equivocator_IM IM)
                     (elements equivocating))) 
          (tr : list
                  (composite_transition_item
                     (sub_IM
                      (equivocator_IM IM)
                      (elements equivocating)))),
          finite_valid_trace
            (seeded_equivocators_no_equivocation_vlsm
               IM (elements equivocating)
               (sent_by_non_equivocating IM
                  equivocating
                  (equivocators_total_state_project
                     IM eqv_state_s))) is tr
          → finite_valid_trace_from
              (preloaded_vlsm
                 (composite_vlsm
                    (equivocator_IM IM)
                    (equivocators_fixed_equivocations_constraint
                      IM
                      (elements equivocating)))
                 (λ _ : message, False))
              full_replay_state
              (replayed_trace_from IM
                 (elements equivocating)
                 full_replay_state is tr)
Him_item: output im_item = Some im
m: message
Hsent: sent_by_non_equivocating IM equivocating
  (equivocators_total_state_project IM
     eqv_state_s) m
valid_message_prop
  {|
    vlsm_type :=
      {| vlsm_type := XE; vlsm_machine := XE |};
    vlsm_machine :=
      {| vlsm_type := XE; vlsm_machine := XE |}
  |} m
      by apply (fixed_equivocating_messages_sent_by_non_equivocating_are_valid eqv_state_s).
    }message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_es im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: ∀ full_replay_state : composite_state
                      (equivocator_IM IM),
  valid_state_prop
    (preloaded_vlsm
       (composite_vlsm 
          (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM (elements equivocating)))
       (λ _ : message, False))
    full_replay_state
  → (∀ (l : label
              (seeded_equivocators_no_equivocation_vlsm
                 IM (elements equivocating)
                 (sent_by_non_equivocating
                    IM equivocating s))) 
       (s0 : state
               (seeded_equivocators_no_equivocation_vlsm
                  IM (elements equivocating)
                  (sent_by_non_equivocating
                     IM equivocating s))) 
       (om : option message),
       input_valid
         (seeded_equivocators_no_equivocation_vlsm
            IM (elements equivocating)
            (sent_by_non_equivocating IM
               equivocating s)) l (
         s0, om)
       → valid_state_prop
           (preloaded_vlsm
              (composite_vlsm
                 (equivocator_IM IM)
                 (equivocators_fixed_equivocations_constraint
                    IM
                    (elements equivocating)))
              (λ _ : message, False))
           (lift_equivocators_sub_state_to
              IM (elements equivocating)
              full_replay_state s0)
         → equivocators_fixed_equivocations_constraint
             IM (elements equivocating)
             (lift_equivocators_sub_label_to
                IM (elements equivocating)
                full_replay_state l)
             (lift_equivocators_sub_state_to
                IM (elements equivocating)
                full_replay_state s0, om))
    → ∀ (is : composite_state
                (sub_IM 
                   (equivocator_IM IM)
                   (elements equivocating))) 
        (tr : list
                (composite_transition_item
                   (sub_IM
                      (equivocator_IM IM)
                      (elements equivocating)))),
        finite_valid_trace
          (seeded_equivocators_no_equivocation_vlsm
             IM (elements equivocating)
             (sent_by_non_equivocating IM
                equivocating s)) is tr
        → finite_valid_trace_from
            (preloaded_vlsm
               (composite_vlsm
                  (equivocator_IM IM)
                  (equivocators_fixed_equivocations_constraint
                     IM
                     (elements equivocating)))
               (λ _ : message, False))
            full_replay_state
            (replayed_trace_from IM
               (elements equivocating)
               full_replay_state is tr)
∃ (eqv_msg_tr : list transition_item) (lst_msg_tr : 
                                       state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (equivocators_fixed_equivocations_constraint
                                                 IM
                                                 (elements
                                                 equivocating)))
                                            (λ _ : message,
                                               False))),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM (elements equivocating)))
       (λ _ : message, False)) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr (Some im)
    specialize (Hreplay eqv_state_s).message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_es im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_state_s
→ (∀ (l : label
            (seeded_equivocators_no_equivocation_vlsm
               IM (elements equivocating)
               (sent_by_non_equivocating IM
                  equivocating s))) 
     (s0 : state
             (seeded_equivocators_no_equivocation_vlsm
                IM (elements equivocating)
                (sent_by_non_equivocating IM
                   equivocating s))) 
     (om : option message),
     input_valid
       (seeded_equivocators_no_equivocation_vlsm
          IM (elements equivocating)
          (sent_by_non_equivocating IM
             equivocating s)) l (
       s0, om)
     → valid_state_prop
         (preloaded_vlsm
            (composite_vlsm
               (equivocator_IM IM)
               (equivocators_fixed_equivocations_constraint
                  IM (elements equivocating)))
            (λ _ : message, False))
         (lift_equivocators_sub_state_to IM
            (elements equivocating)
            eqv_state_s s0)
       → equivocators_fixed_equivocations_constraint
           IM (elements equivocating)
           (lift_equivocators_sub_label_to
              IM (elements equivocating)
              eqv_state_s l)
           (lift_equivocators_sub_state_to
              IM (elements equivocating)
              eqv_state_s s0, om))
  → ∀ (is : composite_state
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) 
      (tr : list
              (composite_transition_item
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))),
      finite_valid_trace
        (seeded_equivocators_no_equivocation_vlsm
           IM (elements equivocating)
           (sent_by_non_equivocating IM
              equivocating s)) is tr
      → finite_valid_trace_from
          (preloaded_vlsm
             (composite_vlsm
                (equivocator_IM IM)
                (equivocators_fixed_equivocations_constraint
                   IM
                   (elements equivocating)))
             (λ _ : message, False))
          eqv_state_s
          (replayed_trace_from IM
             (elements equivocating)
             eqv_state_s is tr)
∃ (eqv_msg_tr : list transition_item) (lst_msg_tr : 
                                       state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (equivocators_fixed_equivocations_constraint
                                                 IM
                                                 (elements
                                                 equivocating)))
                                            (λ _ : message,
                                               False))),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM (elements equivocating)))
       (λ _ : message, False)) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr (Some im)
    spec Hreplay.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_es im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_state_s
→ (∀ (l : label
            (seeded_equivocators_no_equivocation_vlsm
               IM (elements equivocating)
               (sent_by_non_equivocating IM
                  equivocating s))) 
     (s0 : state
             (seeded_equivocators_no_equivocation_vlsm
                IM (elements equivocating)
                (sent_by_non_equivocating IM
                   equivocating s))) 
     (om : option message),
     input_valid
       (seeded_equivocators_no_equivocation_vlsm
          IM (elements equivocating)
          (sent_by_non_equivocating IM
             equivocating s)) l (
       s0, om)
     → valid_state_prop
         (preloaded_vlsm
            (composite_vlsm
               (equivocator_IM IM)
               (equivocators_fixed_equivocations_constraint
                  IM (elements equivocating)))
            (λ _ : message, False))
         (lift_equivocators_sub_state_to IM
            (elements equivocating)
            eqv_state_s s0)
       → equivocators_fixed_equivocations_constraint
           IM (elements equivocating)
           (lift_equivocators_sub_label_to
              IM (elements equivocating)
              eqv_state_s l)
           (lift_equivocators_sub_state_to
              IM (elements equivocating)
              eqv_state_s s0, om))
  → ∀ (is : composite_state
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) 
      (tr : list
              (composite_transition_item
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))),
      finite_valid_trace
        (seeded_equivocators_no_equivocation_vlsm
           IM (elements equivocating)
           (sent_by_non_equivocating IM
              equivocating s)) is tr
      → finite_valid_trace_from
          (preloaded_vlsm
             (composite_vlsm
                (equivocator_IM IM)
                (equivocators_fixed_equivocations_constraint
                   IM
                   (elements equivocating)))
             (λ _ : message, False))
          eqv_state_s
          (replayed_trace_from IM
             (elements equivocating)
             eqv_state_s is tr)
valid_state_prop
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_state_s
message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_es im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: (∀ (l : label
          (seeded_equivocators_no_equivocation_vlsm
             IM (elements equivocating)
             (sent_by_non_equivocating IM
                equivocating s))) 
   (s0 : state
           (seeded_equivocators_no_equivocation_vlsm
              IM (elements equivocating)
              (sent_by_non_equivocating IM
                 equivocating s))) 
   (om : option message),
   input_valid
     (seeded_equivocators_no_equivocation_vlsm
        IM (elements equivocating)
        (sent_by_non_equivocating IM
           equivocating s)) l (
     s0, om)
   → valid_state_prop
       (preloaded_vlsm
          (composite_vlsm
             (equivocator_IM IM)
             (equivocators_fixed_equivocations_constraint
                IM (elements equivocating)))
          (λ _ : message, False))
       (lift_equivocators_sub_state_to IM
          (elements equivocating)
          eqv_state_s s0)
     → equivocators_fixed_equivocations_constraint
         IM (elements equivocating)
         (lift_equivocators_sub_label_to IM
            (elements equivocating)
            eqv_state_s l)
         (lift_equivocators_sub_state_to IM
            (elements equivocating)
            eqv_state_s s0, om))
→ ∀ (is : composite_state
            (sub_IM (equivocator_IM IM)
               (elements equivocating))) 
    (tr : list
            (composite_transition_item
               (sub_IM 
                  (equivocator_IM IM)
                  (elements equivocating)))),
    finite_valid_trace
      (seeded_equivocators_no_equivocation_vlsm
         IM (elements equivocating)
         (sent_by_non_equivocating IM
            equivocating s)) is tr
    → finite_valid_trace_from
        (preloaded_vlsm
           (composite_vlsm
              (equivocator_IM IM)
              (equivocators_fixed_equivocations_constraint
                 IM (elements equivocating)))
           (λ _ : message, False))
        eqv_state_s
        (replayed_trace_from IM
           (elements equivocating)
           eqv_state_s is tr)
∃ (eqv_msg_tr : list transition_item) 
  (lst_msg_tr : state
                  (preloaded_vlsm
                     (composite_vlsm
                      (equivocator_IM IM)
                      (equivocators_fixed_equivocations_constraint
                      IM (elements equivocating)))
                     (λ _ : message, False))),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm 
          (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM (elements equivocating)))
       (λ _ : message, False)) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from 
          (equivocator_IM IM) initial_message_prop
          lst_msg_tr (Some im)
    {message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_es im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_state_s
→ (∀ (l : label
            (seeded_equivocators_no_equivocation_vlsm
               IM (elements equivocating)
               (sent_by_non_equivocating IM
                  equivocating s))) 
     (s0 : state
             (seeded_equivocators_no_equivocation_vlsm
                IM (elements equivocating)
                (sent_by_non_equivocating IM
                   equivocating s))) 
     (om : option message),
     input_valid
       (seeded_equivocators_no_equivocation_vlsm
          IM (elements equivocating)
          (sent_by_non_equivocating IM
             equivocating s)) l (
       s0, om)
     → valid_state_prop
         (preloaded_vlsm
            (composite_vlsm
               (equivocator_IM IM)
               (equivocators_fixed_equivocations_constraint
                  IM (elements equivocating)))
            (λ _ : message, False))
         (lift_equivocators_sub_state_to IM
            (elements equivocating)
            eqv_state_s s0)
       → equivocators_fixed_equivocations_constraint
           IM (elements equivocating)
           (lift_equivocators_sub_label_to
              IM (elements equivocating)
              eqv_state_s l)
           (lift_equivocators_sub_state_to
              IM (elements equivocating)
              eqv_state_s s0, om))
  → ∀ (is : composite_state
              (sub_IM (equivocator_IM IM)
                 (elements equivocating))) 
      (tr : list
              (composite_transition_item
                 (sub_IM 
                    (equivocator_IM IM)
                    (elements equivocating)))),
      finite_valid_trace
        (seeded_equivocators_no_equivocation_vlsm
           IM (elements equivocating)
           (sent_by_non_equivocating IM
              equivocating s)) is tr
      → finite_valid_trace_from
          (preloaded_vlsm
             (composite_vlsm
                (equivocator_IM IM)
                (equivocators_fixed_equivocations_constraint
                   IM
                   (elements equivocating)))
             (λ _ : message, False))
          eqv_state_s
          (replayed_trace_from IM
             (elements equivocating)
             eqv_state_s is tr)
valid_state_prop
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_state_s by apply (VLSM_eq_valid_state HeqXE), (VLSM_eq_valid_state HeqXE). }message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_es im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: (∀ (l : label
          (seeded_equivocators_no_equivocation_vlsm
             IM (elements equivocating)
             (sent_by_non_equivocating IM
                equivocating s))) 
   (s0 : state
           (seeded_equivocators_no_equivocation_vlsm
              IM (elements equivocating)
              (sent_by_non_equivocating IM
                 equivocating s))) 
   (om : option message),
   input_valid
     (seeded_equivocators_no_equivocation_vlsm
        IM (elements equivocating)
        (sent_by_non_equivocating IM
           equivocating s)) l (
     s0, om)
   → valid_state_prop
       (preloaded_vlsm
          (composite_vlsm
             (equivocator_IM IM)
             (equivocators_fixed_equivocations_constraint
                IM (elements equivocating)))
          (λ _ : message, False))
       (lift_equivocators_sub_state_to IM
          (elements equivocating)
          eqv_state_s s0)
     → equivocators_fixed_equivocations_constraint
         IM (elements equivocating)
         (lift_equivocators_sub_label_to IM
            (elements equivocating)
            eqv_state_s l)
         (lift_equivocators_sub_state_to IM
            (elements equivocating)
            eqv_state_s s0, om))
→ ∀ (is : composite_state
            (sub_IM (equivocator_IM IM)
               (elements equivocating))) 
    (tr : list
            (composite_transition_item
               (sub_IM 
                  (equivocator_IM IM)
                  (elements equivocating)))),
    finite_valid_trace
      (seeded_equivocators_no_equivocation_vlsm
         IM (elements equivocating)
         (sent_by_non_equivocating IM
            equivocating s)) is tr
    → finite_valid_trace_from
        (preloaded_vlsm
           (composite_vlsm
              (equivocator_IM IM)
              (equivocators_fixed_equivocations_constraint
                 IM (elements equivocating)))
           (λ _ : message, False))
        eqv_state_s
        (replayed_trace_from IM
           (elements equivocating)
           eqv_state_s is tr)
∃ (eqv_msg_tr : list transition_item) (lst_msg_tr : 
                                       state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (equivocators_fixed_equivocations_constraint
                                                 IM
                                                 (elements
                                                 equivocating)))
                                            (λ _ : message,
                                               False))),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM (elements equivocating)))
       (λ _ : message, False)) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr (Some im)
    spec Hreplay.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_es im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: (∀ (l : label
          (seeded_equivocators_no_equivocation_vlsm
             IM (elements equivocating)
             (sent_by_non_equivocating IM
                equivocating s))) 
   (s0 : state
           (seeded_equivocators_no_equivocation_vlsm
              IM (elements equivocating)
              (sent_by_non_equivocating IM
                 equivocating s))) 
   (om : option message),
   input_valid
     (seeded_equivocators_no_equivocation_vlsm
        IM (elements equivocating)
        (sent_by_non_equivocating IM
           equivocating s)) l (
     s0, om)
   → valid_state_prop
       (preloaded_vlsm
          (composite_vlsm
             (equivocator_IM IM)
             (equivocators_fixed_equivocations_constraint
                IM (elements equivocating)))
          (λ _ : message, False))
       (lift_equivocators_sub_state_to IM
          (elements equivocating)
          eqv_state_s s0)
     → equivocators_fixed_equivocations_constraint
         IM (elements equivocating)
         (lift_equivocators_sub_label_to IM
            (elements equivocating)
            eqv_state_s l)
         (lift_equivocators_sub_state_to IM
            (elements equivocating)
            eqv_state_s s0, om))
→ ∀ (is : composite_state
            (sub_IM (equivocator_IM IM)
               (elements equivocating))) 
    (tr : list
            (composite_transition_item
               (sub_IM 
                  (equivocator_IM IM)
                  (elements equivocating)))),
    finite_valid_trace
      (seeded_equivocators_no_equivocation_vlsm
         IM (elements equivocating)
         (sent_by_non_equivocating IM
            equivocating s)) is tr
    → finite_valid_trace_from
        (preloaded_vlsm
           (composite_vlsm
              (equivocator_IM IM)
              (equivocators_fixed_equivocations_constraint
                 IM (elements equivocating)))
           (λ _ : message, False))
        eqv_state_s
        (replayed_trace_from IM
           (elements equivocating)
           eqv_state_s is tr)
∀ (l : label
         (seeded_equivocators_no_equivocation_vlsm IM
            (elements equivocating)
            (sent_by_non_equivocating IM equivocating
               s))) (s0 : state
                            (seeded_equivocators_no_equivocation_vlsm
                               IM
                               (elements equivocating)
                               (sent_by_non_equivocating
                                  IM equivocating s))) 
  (om : option message),
  input_valid
    (seeded_equivocators_no_equivocation_vlsm IM
       (elements equivocating)
       (sent_by_non_equivocating IM equivocating s)) l
    (s0, om)
  → valid_state_prop
      (preloaded_vlsm
         (composite_vlsm (equivocator_IM IM)
            (equivocators_fixed_equivocations_constraint
               IM (elements equivocating)))
         (λ _ : message, False))
      (lift_equivocators_sub_state_to IM
         (elements equivocating) eqv_state_s s0)
    → equivocators_fixed_equivocations_constraint IM
        (elements equivocating)
        (lift_equivocators_sub_label_to IM
           (elements equivocating) eqv_state_s l)
        (lift_equivocators_sub_state_to IM
           (elements equivocating) eqv_state_s s0, om)
message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_es im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: ∀ (is : composite_state
          (sub_IM (equivocator_IM IM)
             (elements equivocating))) 
  (tr : list
          (composite_transition_item
             (sub_IM (equivocator_IM IM)
                (elements equivocating)))),
  finite_valid_trace
    (seeded_equivocators_no_equivocation_vlsm
       IM (elements equivocating)
       (sent_by_non_equivocating IM
          equivocating s)) is tr
  → finite_valid_trace_from
      (preloaded_vlsm
         (composite_vlsm 
            (equivocator_IM IM)
            (equivocators_fixed_equivocations_constraint
               IM (elements equivocating)))
         (λ _ : message, False)) eqv_state_s
      (replayed_trace_from IM
         (elements equivocating) eqv_state_s
         is tr)
∃ (eqv_msg_tr : list transition_item) 
  (lst_msg_tr : state
                  (preloaded_vlsm
                     (composite_vlsm
                      (equivocator_IM IM)
                      (equivocators_fixed_equivocations_constraint
                      IM (elements equivocating)))
                     (λ _ : message, False))),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm 
          (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM (elements equivocating)))
       (λ _ : message, False)) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from 
          (equivocator_IM IM) initial_message_prop
          lst_msg_tr (Some im)
    {message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_es im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: (∀ (l : label
          (seeded_equivocators_no_equivocation_vlsm
             IM (elements equivocating)
             (sent_by_non_equivocating IM
                equivocating s))) 
   (s0 : state
           (seeded_equivocators_no_equivocation_vlsm
              IM (elements equivocating)
              (sent_by_non_equivocating IM
                 equivocating s))) 
   (om : option message),
   input_valid
     (seeded_equivocators_no_equivocation_vlsm
        IM (elements equivocating)
        (sent_by_non_equivocating IM
           equivocating s)) l (
     s0, om)
   → valid_state_prop
       (preloaded_vlsm
          (composite_vlsm
             (equivocator_IM IM)
             (equivocators_fixed_equivocations_constraint
                IM (elements equivocating)))
          (λ _ : message, False))
       (lift_equivocators_sub_state_to IM
          (elements equivocating)
          eqv_state_s s0)
     → equivocators_fixed_equivocations_constraint
         IM (elements equivocating)
         (lift_equivocators_sub_label_to IM
            (elements equivocating)
            eqv_state_s l)
         (lift_equivocators_sub_state_to IM
            (elements equivocating)
            eqv_state_s s0, om))
→ ∀ (is : composite_state
            (sub_IM (equivocator_IM IM)
               (elements equivocating))) 
    (tr : list
            (composite_transition_item
               (sub_IM 
                  (equivocator_IM IM)
                  (elements equivocating)))),
    finite_valid_trace
      (seeded_equivocators_no_equivocation_vlsm
         IM (elements equivocating)
         (sent_by_non_equivocating IM
            equivocating s)) is tr
    → finite_valid_trace_from
        (preloaded_vlsm
           (composite_vlsm
              (equivocator_IM IM)
              (equivocators_fixed_equivocations_constraint
                 IM (elements equivocating)))
           (λ _ : message, False))
        eqv_state_s
        (replayed_trace_from IM
           (elements equivocating)
           eqv_state_s is tr)
∀ (l : label
         (seeded_equivocators_no_equivocation_vlsm IM
            (elements equivocating)
            (sent_by_non_equivocating IM equivocating
               s))) (s0 : state
                            (seeded_equivocators_no_equivocation_vlsm
                               IM
                               (elements equivocating)
                               (sent_by_non_equivocating
                                  IM equivocating s))) 
  (om : option message),
  input_valid
    (seeded_equivocators_no_equivocation_vlsm IM
       (elements equivocating)
       (sent_by_non_equivocating IM equivocating s)) l
    (s0, om)
  → valid_state_prop
      (preloaded_vlsm
         (composite_vlsm (equivocator_IM IM)
            (equivocators_fixed_equivocations_constraint
               IM (elements equivocating)))
         (λ _ : message, False))
      (lift_equivocators_sub_state_to IM
         (elements equivocating) eqv_state_s s0)
    → equivocators_fixed_equivocations_constraint IM
        (elements equivocating)
        (lift_equivocators_sub_label_to IM
           (elements equivocating) eqv_state_s l)
        (lift_equivocators_sub_state_to IM
           (elements equivocating) eqv_state_s s0, om) subst s.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si
  (equivocators_total_state_project IM
     eqv_state_s) tr
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_item: transition_item
im_tr: list transition_item
im_is: state
  (equivocators_composition_for_sent IM
     equivocating
     (equivocators_total_state_project IM
        eqv_state_s))
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) im_eis im_es
  im_etr
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_output: Some im = finite_trace_last_output im_etr
Hreplay: (∀ (l : label
          (seeded_equivocators_no_equivocation_vlsm
             IM (elements equivocating)
             (sent_by_non_equivocating IM
                equivocating
                (equivocators_total_state_project
                   IM eqv_state_s)))) 
   (s : state
          (seeded_equivocators_no_equivocation_vlsm
             IM (elements equivocating)
             (sent_by_non_equivocating IM
                equivocating
                (equivocators_total_state_project
                   IM eqv_state_s)))) 
   (om : option message),
   input_valid
     (seeded_equivocators_no_equivocation_vlsm
        IM (elements equivocating)
        (sent_by_non_equivocating IM
           equivocating
           (equivocators_total_state_project
              IM eqv_state_s))) l (
     s, om)
   → valid_state_prop
       (preloaded_vlsm
          (composite_vlsm
             (equivocator_IM IM)
             (equivocators_fixed_equivocations_constraint
                IM (elements equivocating)))
          (λ _ : message, False))
       (lift_equivocators_sub_state_to IM
          (elements equivocating)
          eqv_state_s s)
     → equivocators_fixed_equivocations_constraint
         IM (elements equivocating)
         (lift_equivocators_sub_label_to IM
            (elements equivocating)
            eqv_state_s l)
         (lift_equivocators_sub_state_to IM
            (elements equivocating)
            eqv_state_s s, om))
→ ∀ (is : composite_state
            (sub_IM (equivocator_IM IM)
               (elements equivocating))) 
    (tr : list
            (composite_transition_item
               (sub_IM 
                  (equivocator_IM IM)
                  (elements equivocating)))),
    finite_valid_trace
      (seeded_equivocators_no_equivocation_vlsm
         IM (elements equivocating)
         (sent_by_non_equivocating IM
            equivocating
            (equivocators_total_state_project
               IM eqv_state_s))) is tr
    → finite_valid_trace_from
        (preloaded_vlsm
           (composite_vlsm
              (equivocator_IM IM)
              (equivocators_fixed_equivocations_constraint
                 IM (elements equivocating)))
           (λ _ : message, False))
        eqv_state_s
        (replayed_trace_from IM
           (elements equivocating)
           eqv_state_s is tr)
Him_item: output im_item = Some im
∀ (l : label
         (seeded_equivocators_no_equivocation_vlsm IM
            (elements equivocating)
            (sent_by_non_equivocating IM equivocating
               (equivocators_total_state_project IM
                  eqv_state_s)))) (s : state
                                         (seeded_equivocators_no_equivocation_vlsm
                                            IM
                                            (elements
                                               equivocating)
                                            (sent_by_non_equivocating
                                               IM
                                               equivocating
                                               (equivocators_total_state_project
                                                 IM
                                                 eqv_state_s)))) 
  (om : option message),
  input_valid
    (seeded_equivocators_no_equivocation_vlsm IM
       (elements equivocating)
       (sent_by_non_equivocating IM equivocating
          (equivocators_total_state_project IM
             eqv_state_s))) l (s, om)
  → valid_state_prop
      (preloaded_vlsm
         (composite_vlsm (equivocator_IM IM)
            (equivocators_fixed_equivocations_constraint
               IM (elements equivocating)))
         (λ _ : message, False))
      (lift_equivocators_sub_state_to IM
         (elements equivocating) eqv_state_s s)
    → equivocators_fixed_equivocations_constraint IM
        (elements equivocating)
        (lift_equivocators_sub_label_to IM
           (elements equivocating) eqv_state_s l)
        (lift_equivocators_sub_state_to IM
           (elements equivocating) eqv_state_s s, om)
      clear -HeqXE Hstate_valid.message, index, Ci: Type
H5: Elements index Ci
EqDecision1: 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
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
∀ (l : label
         (seeded_equivocators_no_equivocation_vlsm IM
            (elements equivocating)
            (sent_by_non_equivocating IM equivocating
               (equivocators_total_state_project IM
                  eqv_state_s)))) (s : state
                                         (seeded_equivocators_no_equivocation_vlsm
                                            IM
                                            (elements
                                               equivocating)
                                            (sent_by_non_equivocating
                                               IM
                                               equivocating
                                               (equivocators_total_state_project
                                                 IM
                                                 eqv_state_s)))) 
  (om : option message),
  input_valid
    (seeded_equivocators_no_equivocation_vlsm IM
       (elements equivocating)
       (sent_by_non_equivocating IM equivocating
          (equivocators_total_state_project IM
             eqv_state_s))) l (s, om)
  → valid_state_prop
      (preloaded_vlsm
         (composite_vlsm (equivocator_IM IM)
            (equivocators_fixed_equivocations_constraint
               IM (elements equivocating)))
         (λ _ : message, False))
      (lift_equivocators_sub_state_to IM
         (elements equivocating) eqv_state_s s)
    → equivocators_fixed_equivocations_constraint IM
        (elements equivocating)
        (lift_equivocators_sub_label_to IM
           (elements equivocating) eqv_state_s l)
        (lift_equivocators_sub_state_to IM
           (elements equivocating) eqv_state_s s, om)
      intros l s om Hv Hlift_s.message, index, Ci: Type
H5: Elements index Ci
EqDecision1: 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
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
om: option message
Hv: input_valid
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) l (
  s, om)
Hlift_s: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
equivocators_fixed_equivocations_constraint IM
  (elements equivocating)
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s l)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s, om)
      apply (VLSM_eq_valid_state HeqXE) in Hstate_valid.message, index, Ci: Type
H5: Elements index Ci
EqDecision1: 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
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  {|
    vlsm_type := XE; vlsm_machine := XE
  |} eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
om: option message
Hv: input_valid
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) l (
  s, om)
Hlift_s: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
equivocators_fixed_equivocations_constraint IM
  (elements equivocating)
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s l)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s, om)
      apply (VLSM_eq_valid_state HeqXE) in Hlift_s.message, index, Ci: Type
H5: Elements index Ci
EqDecision1: 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
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  {|
    vlsm_type := XE; vlsm_machine := XE
  |} eqv_state_s
l: label
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
s: state
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s)))
om: option message
Hv: input_valid
  (seeded_equivocators_no_equivocation_vlsm IM
     (elements equivocating)
     (sent_by_non_equivocating IM equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) l (
  s, om)
Hlift_s: valid_state_prop
  {| vlsm_type := XE; vlsm_machine := XE |}
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s)
equivocators_fixed_equivocations_constraint IM
  (elements equivocating)
  (lift_equivocators_sub_label_to IM
     (elements equivocating) eqv_state_s l)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s s, om)
      by apply fixed_equivocating_non_equivocating_constraint_lifting.
    }message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_es im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: ∀ (is : composite_state
          (sub_IM (equivocator_IM IM)
             (elements equivocating))) 
  (tr : list
          (composite_transition_item
             (sub_IM (equivocator_IM IM)
                (elements equivocating)))),
  finite_valid_trace
    (seeded_equivocators_no_equivocation_vlsm
       IM (elements equivocating)
       (sent_by_non_equivocating IM
          equivocating s)) is tr
  → finite_valid_trace_from
      (preloaded_vlsm
         (composite_vlsm 
            (equivocator_IM IM)
            (equivocators_fixed_equivocations_constraint
               IM (elements equivocating)))
         (λ _ : message, False)) eqv_state_s
      (replayed_trace_from IM
         (elements equivocating) eqv_state_s
         is tr)
∃ (eqv_msg_tr : list transition_item) (lst_msg_tr : 
                                       state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (equivocators_fixed_equivocations_constraint
                                                 IM
                                                 (elements
                                                 equivocating)))
                                            (λ _ : message,
                                               False))),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM (elements equivocating)))
       (λ _ : message, False)) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr (Some im)
    apply valid_trace_get_last in Him_etr as Him_etr_lst.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_es im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: ∀ (is : composite_state
          (sub_IM (equivocator_IM IM)
             (elements equivocating))) 
  (tr : list
          (composite_transition_item
             (sub_IM (equivocator_IM IM)
                (elements equivocating)))),
  finite_valid_trace
    (seeded_equivocators_no_equivocation_vlsm
       IM (elements equivocating)
       (sent_by_non_equivocating IM
          equivocating s)) is tr
  → finite_valid_trace_from
      (preloaded_vlsm
         (composite_vlsm 
            (equivocator_IM IM)
            (equivocators_fixed_equivocations_constraint
               IM (elements equivocating)))
         (λ _ : message, False)) eqv_state_s
      (replayed_trace_from IM
         (elements equivocating) eqv_state_s
         is tr)
Him_etr_lst: finite_trace_last im_eis im_etr = im_es
∃ (eqv_msg_tr : list transition_item) (lst_msg_tr : 
                                       state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (equivocators_fixed_equivocations_constraint
                                                 IM
                                                 (elements
                                                 equivocating)))
                                            (λ _ : message,
                                               False))),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM (elements equivocating)))
       (λ _ : message, False)) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr (Some im)
    apply valid_trace_forget_last in Him_etr.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: ∀ (is : composite_state
          (sub_IM (equivocator_IM IM)
             (elements equivocating))) 
  (tr : list
          (composite_transition_item
             (sub_IM (equivocator_IM IM)
                (elements equivocating)))),
  finite_valid_trace
    (seeded_equivocators_no_equivocation_vlsm
       IM (elements equivocating)
       (sent_by_non_equivocating IM
          equivocating s)) is tr
  → finite_valid_trace_from
      (preloaded_vlsm
         (composite_vlsm 
            (equivocator_IM IM)
            (equivocators_fixed_equivocations_constraint
               IM (elements equivocating)))
         (λ _ : message, False)) eqv_state_s
      (replayed_trace_from IM
         (elements equivocating) eqv_state_s
         is tr)
Him_etr_lst: finite_trace_last im_eis im_etr = im_es
∃ (eqv_msg_tr : list transition_item) (lst_msg_tr : 
                                       state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (equivocators_fixed_equivocations_constraint
                                                 IM
                                                 (elements
                                                 equivocating)))
                                            (λ _ : message,
                                               False))),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM (elements equivocating)))
       (λ _ : message, False)) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr (Some im)
    specialize (Hreplay _ _ Him_etr).message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: finite_valid_trace_from
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_state_s
  (replayed_trace_from IM
     (elements equivocating) eqv_state_s
     im_eis im_etr)
Him_etr_lst: finite_trace_last im_eis im_etr = im_es
∃ (eqv_msg_tr : list transition_item) (lst_msg_tr : 
                                       state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (equivocators_fixed_equivocations_constraint
                                                 IM
                                                 (elements
                                                 equivocating)))
                                            (λ _ : message,
                                               False))),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM (elements equivocating)))
       (λ _ : message, False)) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr (Some im)
    apply valid_trace_add_default_last in Hreplay.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: finite_valid_trace_from_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_state_s
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM
        (elements equivocating) eqv_state_s
        im_eis im_etr))
  (replayed_trace_from IM
     (elements equivocating) eqv_state_s
     im_eis im_etr)
Him_etr_lst: finite_trace_last im_eis im_etr = im_es
∃ (eqv_msg_tr : list transition_item) (lst_msg_tr : 
                                       state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (equivocators_fixed_equivocations_constraint
                                                 IM
                                                 (elements
                                                 equivocating)))
                                            (λ _ : message,
                                               False))),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM (elements equivocating)))
       (λ _ : message, False)) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr (Some im)
    eexists _, _; split; [done |].message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: finite_valid_trace_from_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_state_s
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM
        (elements equivocating) eqv_state_s
        im_eis im_etr))
  (replayed_trace_from IM
     (elements equivocating) eqv_state_s
     im_eis im_etr)
Him_etr_lst: finite_trace_last im_eis im_etr = im_es
equivocators_total_trace_project IM
  (replayed_trace_from IM (elements equivocating)
     eqv_state_s im_eis im_etr) = []
∧ equivocators_total_state_project IM
    (finite_trace_last eqv_state_s
       (replayed_trace_from IM (elements equivocating)
          eqv_state_s im_eis im_etr)) = s
  ∧ sent_except_from (equivocator_IM IM)
      initial_message_prop
      (finite_trace_last eqv_state_s
         (replayed_trace_from IM
            (elements equivocating) eqv_state_s im_eis
            im_etr)) (Some im)
    (*
      Having verified the validity part of the conclusion, now we only
      need to show two projection properties, and the no message-equivocation
      constraint for which we employ Lemma [fixed_equivocation_replay_has_message].
    *)
    repeat split.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: finite_valid_trace_from_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_state_s
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM
        (elements equivocating) eqv_state_s
        im_eis im_etr))
  (replayed_trace_from IM
     (elements equivocating) eqv_state_s
     im_eis im_etr)
Him_etr_lst: finite_trace_last im_eis im_etr = im_es
equivocators_total_trace_project IM
  (replayed_trace_from IM (elements equivocating)
     eqv_state_s im_eis im_etr) = []
message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: finite_valid_trace_from_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_state_s
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM
        (elements equivocating) eqv_state_s
        im_eis im_etr))
  (replayed_trace_from IM
     (elements equivocating) eqv_state_s
     im_eis im_etr)
Him_etr_lst: finite_trace_last im_eis im_etr = im_es
equivocators_total_state_project IM
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM 
        (elements equivocating) eqv_state_s im_eis
        im_etr)) = s
message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: finite_valid_trace_from_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_state_s
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM
        (elements equivocating) eqv_state_s
        im_eis im_etr))
  (replayed_trace_from IM
     (elements equivocating) eqv_state_s
     im_eis im_etr)
Him_etr_lst: finite_trace_last im_eis im_etr = im_es
sent_except_from (equivocator_IM IM)
  initial_message_prop
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM 
        (elements equivocating) eqv_state_s im_eis
        im_etr)) (Some im)
    +message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: finite_valid_trace_from_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_state_s
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM
        (elements equivocating) eqv_state_s
        im_eis im_etr))
  (replayed_trace_from IM
     (elements equivocating) eqv_state_s
     im_eis im_etr)
Him_etr_lst: finite_trace_last im_eis im_etr = im_es
equivocators_total_trace_project IM
  (replayed_trace_from IM (elements equivocating)
     eqv_state_s im_eis im_etr) = [] by apply
      (equivocators_total_trace_project_replayed_trace_from
        IM (elements equivocating) eqv_state_s im_eis im_etr).
    +message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: finite_valid_trace_from_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_state_s
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM
        (elements equivocating) eqv_state_s
        im_eis im_etr))
  (replayed_trace_from IM
     (elements equivocating) eqv_state_s
     im_eis im_etr)
Him_etr_lst: finite_trace_last im_eis im_etr = im_es
equivocators_total_state_project IM
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM (elements equivocating)
        eqv_state_s im_eis im_etr)) = s subst s.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si
  (equivocators_total_state_project IM
     eqv_state_s) tr
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_item: transition_item
im_tr: list transition_item
im_is: state
  (equivocators_composition_for_sent IM
     equivocating
     (equivocators_total_state_project IM
        eqv_state_s))
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating
        (equivocators_total_state_project IM
           eqv_state_s))) im_eis im_etr
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: finite_valid_trace_from_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_state_s
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM
        (elements equivocating) eqv_state_s
        im_eis im_etr))
  (replayed_trace_from IM
     (elements equivocating) eqv_state_s
     im_eis im_etr)
Him_etr_lst: finite_trace_last im_eis im_etr = im_es
equivocators_total_state_project IM
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM (elements equivocating)
        eqv_state_s im_eis im_etr)) =
equivocators_total_state_project IM eqv_state_s
      by apply
        (equivocators_total_state_project_replayed_trace_from
          IM (elements equivocating) eqv_state_s im_eis im_etr).
    +message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: finite_valid_trace_from_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_state_s
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM
        (elements equivocating) eqv_state_s
        im_eis im_etr))
  (replayed_trace_from IM
     (elements equivocating) eqv_state_s
     im_eis im_etr)
Him_etr_lst: finite_trace_last im_eis im_etr = im_es
sent_except_from (equivocator_IM IM)
  initial_message_prop
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM (elements equivocating)
        eqv_state_s im_eis im_etr)) (Some im) apply (VLSM_eq_valid_state HeqXE) in Hstate_valid.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  {|
    vlsm_type := XE; vlsm_machine := XE
  |} eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: finite_valid_trace_from_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_state_s
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM
        (elements equivocating) eqv_state_s
        im_eis im_etr))
  (replayed_trace_from IM
     (elements equivocating) eqv_state_s
     im_eis im_etr)
Him_etr_lst: finite_trace_last im_eis im_etr = im_es
sent_except_from (equivocator_IM IM)
  initial_message_prop
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM (elements equivocating)
        eqv_state_s im_eis im_etr)) (Some im)
      apply (VLSM_incl_valid_state HinclE) in Hstate_valid as Hstate_pre.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  {|
    vlsm_type := XE; vlsm_machine := XE
  |} eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: finite_valid_trace_from_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_state_s
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM
        (elements equivocating) eqv_state_s
        im_eis im_etr))
  (replayed_trace_from IM
     (elements equivocating) eqv_state_s
     im_eis im_etr)
Him_etr_lst: finite_trace_last im_eis im_etr = im_es
Hstate_pre: valid_state_prop
  {|
    vlsm_type :=
      equivocators_fixed_equivocations_vlsm
        IM (elements equivocating);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM IM))
  |} eqv_state_s
sent_except_from (equivocator_IM IM)
  initial_message_prop
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM (elements equivocating)
        eqv_state_s im_eis im_etr)) (Some im)
      specialize (NoEquivocation.seeded_no_equivocation_incl_preloaded
        (equivocator_IM (sub_IM IM (elements equivocating)))
        (free_constraint _)
          (sent_by_non_equivocating IM equivocating s))
        as Hsub_incl.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  {|
    vlsm_type := XE; vlsm_machine := XE
  |} eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s)) im_eis im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: finite_valid_trace_from_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_state_s
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM
        (elements equivocating) eqv_state_s
        im_eis im_etr))
  (replayed_trace_from IM
     (elements equivocating) eqv_state_s
     im_eis im_etr)
Him_etr_lst: finite_trace_last im_eis im_etr = im_es
Hstate_pre: valid_state_prop
  {|
    vlsm_type :=
      equivocators_fixed_equivocations_vlsm
        IM (elements equivocating);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM IM))
  |} eqv_state_s
Hsub_incl: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
sent_except_from (equivocator_IM IM)
  initial_message_prop
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM (elements equivocating)
        eqv_state_s im_eis im_etr)) (Some im)
      apply (VLSM_incl_finite_valid_trace Hsub_incl) in Him_etr.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  {|
    vlsm_type := XE; vlsm_machine := XE
  |} eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))
        (free_constraint
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating))))
        (sent_by_non_equivocating IM
           equivocating s);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating))))
  |} im_eis im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: finite_valid_trace_from_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_state_s
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM
        (elements equivocating) eqv_state_s
        im_eis im_etr))
  (replayed_trace_from IM
     (elements equivocating) eqv_state_s
     im_eis im_etr)
Him_etr_lst: finite_trace_last im_eis im_etr = im_es
Hstate_pre: valid_state_prop
  {|
    vlsm_type :=
      equivocators_fixed_equivocations_vlsm
        IM (elements equivocating);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM IM))
  |} eqv_state_s
Hsub_incl: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
sent_except_from (equivocator_IM IM)
  initial_message_prop
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM (elements equivocating)
        eqv_state_s im_eis im_etr)) (Some im)
      left.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  {|
    vlsm_type := XE; vlsm_machine := XE
  |} eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))
        (free_constraint
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating))))
        (sent_by_non_equivocating IM
           equivocating s);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating))))
  |} im_eis im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: finite_valid_trace_from_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_state_s
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM
        (elements equivocating) eqv_state_s
        im_eis im_etr))
  (replayed_trace_from IM
     (elements equivocating) eqv_state_s
     im_eis im_etr)
Him_etr_lst: finite_trace_last im_eis im_etr = im_es
Hstate_pre: valid_state_prop
  {|
    vlsm_type :=
      equivocators_fixed_equivocations_vlsm
        IM (elements equivocating);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM IM))
  |} eqv_state_s
Hsub_incl: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
composite_has_been_sent (equivocator_IM IM)
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM (elements equivocating)
        eqv_state_s im_eis im_etr)) im
      specialize (replayed_trace_from_finite_trace_last IM (elements equivocating) eqv_state_s
        im_eis im_etr (proj2 Him_etr)).message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  {|
    vlsm_type := XE; vlsm_machine := XE
  |} eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))
        (free_constraint
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating))))
        (sent_by_non_equivocating IM
           equivocating s);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating))))
  |} im_eis im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: finite_valid_trace_from_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_state_s
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM
        (elements equivocating) eqv_state_s
        im_eis im_etr))
  (replayed_trace_from IM
     (elements equivocating) eqv_state_s
     im_eis im_etr)
Him_etr_lst: finite_trace_last im_eis im_etr = im_es
Hstate_pre: valid_state_prop
  {|
    vlsm_type :=
      equivocators_fixed_equivocations_vlsm
        IM (elements equivocating);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM IM))
  |} eqv_state_s
Hsub_incl: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
finite_trace_last eqv_state_s
  (replayed_trace_from IM (elements equivocating)
     eqv_state_s im_eis im_etr) =
lift_equivocators_sub_state_to IM
  (elements equivocating) eqv_state_s
  (finite_trace_last im_eis im_etr)
→ composite_has_been_sent (equivocator_IM IM)
    (finite_trace_last eqv_state_s
       (replayed_trace_from IM (elements equivocating)
          eqv_state_s im_eis im_etr)) im
      simpl.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  {|
    vlsm_type := XE; vlsm_machine := XE
  |} eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))
        (free_constraint
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating))))
        (sent_by_non_equivocating IM
           equivocating s);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating))))
  |} im_eis im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: finite_valid_trace_from_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_state_s
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM
        (elements equivocating) eqv_state_s
        im_eis im_etr))
  (replayed_trace_from IM
     (elements equivocating) eqv_state_s
     im_eis im_etr)
Him_etr_lst: finite_trace_last im_eis im_etr = im_es
Hstate_pre: valid_state_prop
  {|
    vlsm_type :=
      equivocators_fixed_equivocations_vlsm
        IM (elements equivocating);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM IM))
  |} eqv_state_s
Hsub_incl: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
finite_trace_last eqv_state_s
  (replayed_trace_from IM (elements equivocating)
     eqv_state_s im_eis im_etr) =
lift_equivocators_sub_state_to IM
  (elements equivocating) eqv_state_s
  (finite_trace_last im_eis im_etr)
→ composite_has_been_sent (equivocator_IM IM)
    (finite_trace_last eqv_state_s
       (replayed_trace_from IM (elements equivocating)
          eqv_state_s im_eis im_etr)) im intro Hrew.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  {|
    vlsm_type := XE; vlsm_machine := XE
  |} eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))
        (free_constraint
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating))))
        (sent_by_non_equivocating IM
           equivocating s);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating))))
  |} im_eis im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: finite_valid_trace_from_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_state_s
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM
        (elements equivocating) eqv_state_s
        im_eis im_etr))
  (replayed_trace_from IM
     (elements equivocating) eqv_state_s
     im_eis im_etr)
Him_etr_lst: finite_trace_last im_eis im_etr = im_es
Hstate_pre: valid_state_prop
  {|
    vlsm_type :=
      equivocators_fixed_equivocations_vlsm
        IM (elements equivocating);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM IM))
  |} eqv_state_s
Hsub_incl: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
Hrew: finite_trace_last eqv_state_s
  (replayed_trace_from IM
     (elements equivocating) eqv_state_s im_eis
     im_etr) =
lift_equivocators_sub_state_to IM
  (elements equivocating) eqv_state_s
  (finite_trace_last im_eis im_etr)
composite_has_been_sent (equivocator_IM IM)
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM (elements equivocating)
        eqv_state_s im_eis im_etr)) im rewrite Hrew.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  {|
    vlsm_type := XE; vlsm_machine := XE
  |} eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))
        (free_constraint
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating))))
        (sent_by_non_equivocating IM
           equivocating s);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating))))
  |} im_eis im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: finite_valid_trace_from_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_state_s
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM
        (elements equivocating) eqv_state_s
        im_eis im_etr))
  (replayed_trace_from IM
     (elements equivocating) eqv_state_s
     im_eis im_etr)
Him_etr_lst: finite_trace_last im_eis im_etr = im_es
Hstate_pre: valid_state_prop
  {|
    vlsm_type :=
      equivocators_fixed_equivocations_vlsm
        IM (elements equivocating);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM IM))
  |} eqv_state_s
Hsub_incl: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
Hrew: finite_trace_last eqv_state_s
  (replayed_trace_from IM
     (elements equivocating) eqv_state_s im_eis
     im_etr) =
lift_equivocators_sub_state_to IM
  (elements equivocating) eqv_state_s
  (finite_trace_last im_eis im_etr)
composite_has_been_sent (equivocator_IM IM)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s
     (finite_trace_last im_eis im_etr)) im clear Hrew.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  {|
    vlsm_type := XE; vlsm_machine := XE
  |} eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))
        (free_constraint
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating))))
        (sent_by_non_equivocating IM
           equivocating s);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating))))
  |} im_eis im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: finite_valid_trace_from_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_state_s
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM
        (elements equivocating) eqv_state_s
        im_eis im_etr))
  (replayed_trace_from IM
     (elements equivocating) eqv_state_s
     im_eis im_etr)
Him_etr_lst: finite_trace_last im_eis im_etr = im_es
Hstate_pre: valid_state_prop
  {|
    vlsm_type :=
      equivocators_fixed_equivocations_vlsm
        IM (elements equivocating);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM IM))
  |} eqv_state_s
Hsub_incl: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
composite_has_been_sent (equivocator_IM IM)
  (lift_equivocators_sub_state_to IM
     (elements equivocating) eqv_state_s
     (finite_trace_last im_eis im_etr)) im
      apply fixed_equivocation_replay_has_message; [done |].message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     (elements equivocating))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint
           IM equivocating))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  {|
    vlsm_type := XE; vlsm_machine := XE
  |} eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_msg_is
  eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
im_is: state
  (equivocators_composition_for_sent IM
     equivocating s)
im_tr: list transition_item
im_item: transition_item
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_eis: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_eis =
im_is
im_es: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_es: equivocators_total_state_project
  (sub_IM IM (elements equivocating)) im_es =
destination im_item
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr_pr: equivocators_total_trace_project
  (sub_IM IM (elements equivocating))
  im_etr = im_tr ++ [im_item]
Him_etr: finite_valid_trace
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))
        (free_constraint
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating))))
        (sent_by_non_equivocating IM
           equivocating s);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating))))
  |} im_eis im_etr
Him_output: Some im = finite_trace_last_output im_etr
Him_item: output im_item = Some im
Hreplay: finite_valid_trace_from_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) eqv_state_s
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM
        (elements equivocating) eqv_state_s
        im_eis im_etr))
  (replayed_trace_from IM
     (elements equivocating) eqv_state_s
     im_eis im_etr)
Him_etr_lst: finite_trace_last im_eis im_etr = im_es
Hstate_pre: valid_state_prop
  {|
    vlsm_type :=
      equivocators_fixed_equivocations_vlsm
        IM (elements equivocating);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM IM))
  |} eqv_state_s
Hsub_incl: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM
        (sub_IM IM (elements equivocating)))
     (free_constraint
        (equivocator_IM
           (sub_IM IM
              (elements equivocating))))
     (sent_by_non_equivocating IM
        equivocating s))
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))))
can_produce
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
  (finite_trace_last im_eis im_etr) im
      clear -Him_etr Him_output.message, index, Ci: Type
H5: Elements index Ci
EqDecision1: EqDecision index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
im: message
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
im_etr: list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
Him_etr: finite_valid_trace
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))
        (free_constraint
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating))))
        (sent_by_non_equivocating IM
           equivocating s);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating))))
  |} im_eis im_etr
Him_output: Some im = finite_trace_last_output im_etr
can_produce
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
  (finite_trace_last im_eis im_etr) im
      destruct_list_last im_etr im_ert' item Heqim_etr; [by inversion Him_output |].message, index, Ci: Type
H5: Elements index Ci
EqDecision1: EqDecision index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
im: message
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
im_etr, im_ert': list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
item: composite_transition_item
  (equivocator_IM
     (sub_IM IM (elements equivocating)))
Him_etr: finite_valid_trace
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))
        (free_constraint
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating))))
        (sent_by_non_equivocating IM
           equivocating s);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating))))
  |} im_eis (im_ert' ++ [item])
Him_output: Some im =
finite_trace_last_output
  (im_ert' ++ [item])
Heqim_etr: im_etr = im_ert' ++ [item]
can_produce
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
  (finite_trace_last im_eis (im_ert' ++ [item])) im
      apply proj1 in Him_etr.message, index, Ci: Type
H5: Elements index Ci
EqDecision1: EqDecision index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
im: message
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
im_etr, im_ert': list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
item: composite_transition_item
  (equivocator_IM
     (sub_IM IM (elements equivocating)))
Him_etr: finite_valid_trace_from
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))
        (free_constraint
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating))))
        (sent_by_non_equivocating IM
           equivocating s);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating))))
  |} im_eis (im_ert' ++ [item])
Him_output: Some im =
finite_trace_last_output
  (im_ert' ++ [item])
Heqim_etr: im_etr = im_ert' ++ [item]
can_produce
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
  (finite_trace_last im_eis (im_ert' ++ [item])) im
      replace (im_ert' ++ [item]) with (im_ert' ++ [item] ++ []) in Him_etr
        by (rewrite app_assoc; apply app_nil_r).message, index, Ci: Type
H5: Elements index Ci
EqDecision1: EqDecision index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
im: message
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
im_etr, im_ert': list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
item: composite_transition_item
  (equivocator_IM
     (sub_IM IM (elements equivocating)))
Him_etr: finite_valid_trace_from
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))
        (free_constraint
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating))))
        (sent_by_non_equivocating IM
           equivocating s);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating))))
  |} im_eis (im_ert' ++ [item] ++ [])
Him_output: Some im =
finite_trace_last_output
  (im_ert' ++ [item])
Heqim_etr: im_etr = im_ert' ++ [item]
can_produce
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
  (finite_trace_last im_eis (im_ert' ++ [item])) im
      specialize (input_constrained_transition_to
        (free_composite_vlsm (equivocator_IM (sub_IM IM (elements equivocating))))
        _ _ _ _ _ Him_etr eq_refl)
        as Ht.message, index, Ci: Type
H5: Elements index Ci
EqDecision1: EqDecision index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
im: message
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
im_etr, im_ert': list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
item: composite_transition_item
  (equivocator_IM
     (sub_IM IM (elements equivocating)))
Him_etr: finite_valid_trace_from
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))
        (free_constraint
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating))))
        (sent_by_non_equivocating IM
           equivocating s);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating))))
  |} im_eis (im_ert' ++ [item] ++ [])
Him_output: Some im =
finite_trace_last_output
  (im_ert' ++ [item])
Heqim_etr: im_etr = im_ert' ++ [item]
Ht: let lst1 := finite_trace_last im_eis im_ert' in
input_valid_transition
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
  (l item) (lst1, input item)
  (destination item, output item)
can_produce
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
  (finite_trace_last im_eis (im_ert' ++ [item])) im
      rewrite finite_trace_last_is_last.message, index, Ci: Type
H5: Elements index Ci
EqDecision1: EqDecision index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
im: message
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
im_etr, im_ert': list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
item: composite_transition_item
  (equivocator_IM
     (sub_IM IM (elements equivocating)))
Him_etr: finite_valid_trace_from
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))
        (free_constraint
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating))))
        (sent_by_non_equivocating IM
           equivocating s);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating))))
  |} im_eis (im_ert' ++ [item] ++ [])
Him_output: Some im =
finite_trace_last_output
  (im_ert' ++ [item])
Heqim_etr: im_etr = im_ert' ++ [item]
Ht: let lst1 := finite_trace_last im_eis im_ert' in
input_valid_transition
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
  (l item) (lst1, input item)
  (destination item, output item)
can_produce
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
  (destination item) im
      rewrite finite_trace_last_output_is_last in Him_output.message, index, Ci: Type
H5: Elements index Ci
EqDecision1: EqDecision index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
im: message
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
im_etr, im_ert': list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
item: composite_transition_item
  (equivocator_IM
     (sub_IM IM (elements equivocating)))
Him_etr: finite_valid_trace_from
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))
        (free_constraint
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating))))
        (sent_by_non_equivocating IM
           equivocating s);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating))))
  |} im_eis (im_ert' ++ [item] ++ [])
Him_output: Some im = output item
Heqim_etr: im_etr = im_ert' ++ [item]
Ht: let lst1 := finite_trace_last im_eis im_ert' in
input_valid_transition
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
  (l item) (lst1, input item)
  (destination item, output item)
can_produce
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
  (destination item) im
      replace (output _) with (Some im) in Ht.message, index, Ci: Type
H5: Elements index Ci
EqDecision1: EqDecision index
IM: index → VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (strong_fixed_equivocation_constraint IM
           equivocating)) 
     (λ _ : message, False))
im: message
im_eis: state
  (free_composite_vlsm
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
im_etr, im_ert': list
  (composite_transition_item
     (equivocator_IM
        (sub_IM IM (elements equivocating))))
item: composite_transition_item
  (equivocator_IM
     (sub_IM IM (elements equivocating)))
Him_etr: finite_valid_trace_from
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (equivocator_IM
           (sub_IM IM
              (elements equivocating)))
        (free_constraint
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating))))
        (sent_by_non_equivocating IM
           equivocating s);
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM
              (sub_IM IM
                 (elements equivocating))))
  |} im_eis (im_ert' ++ [item] ++ [])
Him_output: Some im = output item
Heqim_etr: im_etr = im_ert' ++ [item]
Ht: let lst1 := finite_trace_last im_eis im_ert' in
input_valid_transition
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
  (l item) (lst1, input item)
  (destination item, Some im)
can_produce
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM
           (sub_IM IM (elements equivocating)))))
  (destination item) im
      by eexists _, _.
Qed.

The main result

The main result, showing that fixed-set message-equivocation traces can be simulated by fixed-set state-equivocation traces is obtained by instantiating Lemma generalized_equivocators_finite_valid_trace_init_to_rev, discharging the most complex assumption of the lemma (about replayable_message_prop) with Lemma fixed_equivocation_has_replayable_message_prop.

Lemma fixed_equivocators_finite_valid_trace_init_to_rev
  isX sX trX
  (HtrX : finite_valid_trace_init_to X isX sX trX)
  : exists is s tr,
      equivocators_total_state_project IM is = isX /\
      equivocators_total_state_project IM s = sX /\
      equivocators_total_trace_project IM tr = trX /\
      finite_valid_trace_init_to XE is s tr /\
      finite_trace_last_output trX = finite_trace_last_output tr.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to X isX sX trX
∃ (is
   s : state (free_composite_vlsm (equivocator_IM IM))) 
  (tr : list
          (composite_transition_item
             (equivocator_IM IM))),
  equivocators_total_state_project IM is = isX
  ∧ equivocators_total_state_project IM s = sX
    ∧ equivocators_total_trace_project IM tr = trX
      ∧ finite_valid_trace_init_to XE is s tr
        ∧ finite_trace_last_output trX =
          finite_trace_last_output tr
Proof.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to X isX sX trX
∃ (is
   s : state (free_composite_vlsm (equivocator_IM IM))) 
  (tr : list
          (composite_transition_item
             (equivocator_IM IM))),
  equivocators_total_state_project IM is = isX
  ∧ equivocators_total_state_project IM s = sX
    ∧ equivocators_total_trace_project IM tr = trX
      ∧ finite_valid_trace_init_to XE is s tr
        ∧ finite_trace_last_output trX =
          finite_trace_last_output tr
  (*
    Since the base result works with preloaded vlsms, some massaging of the
    hypothesis and conclusion is done to fit the applied lemma.
  *)
  assert (no_initial_messages_in_XE :
    forall m, ~ initial_message_prop (preloaded_vlsm XE (fun _ => False)) m).message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to X isX sX trX
∀ m : message, ¬ initial_message_prop m
message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to X isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
∃ (is
   s : state (free_composite_vlsm (equivocator_IM IM))) 
  (tr : list
          (composite_transition_item
             (equivocator_IM IM))),
  equivocators_total_state_project IM is = isX
  ∧ equivocators_total_state_project IM s = sX
    ∧ equivocators_total_trace_project IM tr = trX
      ∧ finite_valid_trace_init_to XE is s tr
        ∧ finite_trace_last_output trX =
          finite_trace_last_output tr
  {message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to X isX sX trX
∀ m : message, ¬ initial_message_prop m intros m [[i [[mi Hmi] Him]] | Hseeded]; [| done].message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to X isX sX trX
m: message
i: index
mi: message
Hmi: initial_message_prop mi
Him: `(mi ↾ Hmi) = m
False
    by elim (no_initial_messages_in_IM i mi).
  }message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to X isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
∃ (is
   s : state (free_composite_vlsm (equivocator_IM IM))) 
  (tr : list
          (composite_transition_item
             (equivocator_IM IM))),
  equivocators_total_state_project IM is = isX
  ∧ equivocators_total_state_project IM s = sX
    ∧ equivocators_total_trace_project IM tr = trX
      ∧ finite_valid_trace_init_to XE is s tr
        ∧ finite_trace_last_output trX =
          finite_trace_last_output tr
  specialize (vlsm_is_preloaded_with_False X) as HeqX.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to X isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
∃ (is
   s : state (free_composite_vlsm (equivocator_IM IM))) 
  (tr : list
          (composite_transition_item
             (equivocator_IM IM))),
  equivocators_total_state_project IM is = isX
  ∧ equivocators_total_state_project IM s = sX
    ∧ equivocators_total_trace_project IM tr = trX
      ∧ finite_valid_trace_init_to XE is s tr
        ∧ finite_trace_last_output trX =
          finite_trace_last_output tr
  specialize (vlsm_is_preloaded_with_False XE) as HeqXE.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to X isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
∃ (is
   s : state (free_composite_vlsm (equivocator_IM IM))) 
  (tr : list
          (composite_transition_item
             (equivocator_IM IM))),
  equivocators_total_state_project IM is = isX
  ∧ equivocators_total_state_project IM s = sX
    ∧ equivocators_total_trace_project IM tr = trX
      ∧ finite_valid_trace_init_to XE is s tr
        ∧ finite_trace_last_output trX =
          finite_trace_last_output tr
  apply (VLSM_eq_finite_valid_trace_init_to HeqX) in HtrX.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
∃ (is
   s : state (free_composite_vlsm (equivocator_IM IM))) 
  (tr : list
          (composite_transition_item
             (equivocator_IM IM))),
  equivocators_total_state_project IM is = isX
  ∧ equivocators_total_state_project IM s = sX
    ∧ equivocators_total_trace_project IM tr = trX
      ∧ finite_valid_trace_init_to XE is s tr
        ∧ finite_trace_last_output trX =
          finite_trace_last_output tr
  apply
    (generalized_equivocators_finite_valid_trace_init_to_rev
      IM _ _ (equivocators_fixed_equivocations_constraint IM (elements equivocating)))
    in HtrX
    as (is & His & s & Hs & tr & Htr & Hptr & Houtput).message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
is: state (free_composite_vlsm (equivocator_IM IM))
His: equivocators_total_state_project IM is = isX
s: state (free_composite_vlsm (equivocator_IM IM))
Hs: equivocators_total_state_project IM s = sX
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: equivocators_total_trace_project IM tr = trX
Hptr: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) is s tr
Houtput: finite_trace_last_output trX =
finite_trace_last_output tr
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
∃ (is
   s : state (free_composite_vlsm (equivocator_IM IM))) 
  (tr : list
          (composite_transition_item
             (equivocator_IM IM))),
  equivocators_total_state_project IM is = isX
  ∧ equivocators_total_state_project IM s = sX
    ∧ equivocators_total_trace_project IM tr = trX
      ∧ finite_valid_trace_init_to XE is s tr
        ∧ finite_trace_last_output trX =
          finite_trace_last_output tr
message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
zero_descriptor_constraint_lifting_prop IM
  (λ _ : message, False)
  (equivocators_fixed_equivocations_constraint IM
     (elements equivocating))
message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
replayable_message_prop IM 
  (λ _ : message, False)
  (strong_fixed_equivocation_constraint IM
     equivocating)
  (equivocators_fixed_equivocations_constraint IM
     (elements equivocating))
  -message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
is: state (free_composite_vlsm (equivocator_IM IM))
His: equivocators_total_state_project IM is = isX
s: state (free_composite_vlsm (equivocator_IM IM))
Hs: equivocators_total_state_project IM s = sX
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: equivocators_total_trace_project IM tr = trX
Hptr: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) is s tr
Houtput: finite_trace_last_output trX =
finite_trace_last_output tr
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
∃ (is
   s : state (free_composite_vlsm (equivocator_IM IM))) 
  (tr : list
          (composite_transition_item
             (equivocator_IM IM))),
  equivocators_total_state_project IM is = isX
  ∧ equivocators_total_state_project IM s = sX
    ∧ equivocators_total_trace_project IM tr = trX
      ∧ finite_valid_trace_init_to XE is s tr
        ∧ finite_trace_last_output trX =
          finite_trace_last_output tr exists is, s, tr; split_and!; try itauto.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
is: state (free_composite_vlsm (equivocator_IM IM))
His: equivocators_total_state_project IM is = isX
s: state (free_composite_vlsm (equivocator_IM IM))
Hs: equivocators_total_state_project IM s = sX
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: equivocators_total_trace_project IM tr = trX
Hptr: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) is s tr
Houtput: finite_trace_last_output trX =
finite_trace_last_output tr
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
finite_valid_trace_init_to XE is s tr
    by apply (VLSM_eq_finite_valid_trace_init_to HeqXE).
  -message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
zero_descriptor_constraint_lifting_prop IM
  (λ _ : message, False)
  (equivocators_fixed_equivocations_constraint IM
     (elements equivocating)) intro; intros.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
es: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hes: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False)) es
om: option message
Hom: sent_except_from (equivocator_IM IM)
  initial_message_prop es om
eqv: index
li: label (IM eqv)
equivocators_fixed_equivocations_constraint IM
  (elements equivocating)
  (existT eqv (ContinueWith 0 li)) (es, om)
    apply (VLSM_eq_valid_state HeqXE) in Hes.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
es: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hes: valid_state_prop
  {| vlsm_type := XE; vlsm_machine := XE |} es
om: option message
Hom: sent_except_from (equivocator_IM IM)
  initial_message_prop es om
eqv: index
li: label (IM eqv)
equivocators_fixed_equivocations_constraint IM
  (elements equivocating)
  (existT eqv (ContinueWith 0 li)) (es, om)
    split.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
es: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hes: valid_state_prop
  {| vlsm_type := XE; vlsm_machine := XE |} es
om: option message
Hom: sent_except_from (equivocator_IM IM)
  initial_message_prop es om
eqv: index
li: label (IM eqv)
equivocators_no_equivocations_constraint IM
  (existT eqv (ContinueWith 0 li)) (es, om)
message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
es: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hes: valid_state_prop
  {| vlsm_type := XE; vlsm_machine := XE |} es
om: option message
Hom: sent_except_from (equivocator_IM IM)
  initial_message_prop es om
eqv: index
li: label (IM eqv)
state_has_fixed_equivocation IM
  (elements equivocating)
  (composite_transition 
     (equivocator_IM IM)
     (existT eqv (ContinueWith 0 li)) (
     es, om)).1
    +message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
es: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hes: valid_state_prop
  {| vlsm_type := XE; vlsm_machine := XE |} es
om: option message
Hom: sent_except_from (equivocator_IM IM)
  initial_message_prop es om
eqv: index
li: label (IM eqv)
equivocators_no_equivocations_constraint IM
  (existT eqv (ContinueWith 0 li)) (es, om) split; [| done].message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
es: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hes: valid_state_prop
  {| vlsm_type := XE; vlsm_machine := XE |} es
om: option message
Hom: sent_except_from (equivocator_IM IM)
  initial_message_prop es om
eqv: index
li: label (IM eqv)
composite_no_equivocations (equivocator_IM IM)
  (existT eqv (ContinueWith 0 li)) (es, om)  destruct om as [im |]; [| done].message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
es: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hes: valid_state_prop
  {| vlsm_type := XE; vlsm_machine := XE |} es
im: message
Hom: sent_except_from (equivocator_IM IM)
  initial_message_prop es 
  (Some im)
eqv: index
li: label (IM eqv)
composite_no_equivocations (equivocator_IM IM)
  (existT eqv (ContinueWith 0 li)) (es, Some im)
      destruct Hom as [Hom | Hinitial]; [by left | exfalso].message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
es: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hes: valid_state_prop
  {| vlsm_type := XE; vlsm_machine := XE |} es
im: message
Hinitial: initial_message_prop im
eqv: index
li: label (IM eqv)
False
      by apply no_initial_messages_in_XE in Hinitial.
    +message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
es: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hes: valid_state_prop
  {| vlsm_type := XE; vlsm_machine := XE |} es
om: option message
Hom: sent_except_from (equivocator_IM IM)
  initial_message_prop es om
eqv: index
li: label (IM eqv)
state_has_fixed_equivocation IM
  (elements equivocating)
  (composite_transition (equivocator_IM IM)
     (existT eqv (ContinueWith 0 li)) (es, om)).1 apply valid_state_has_fixed_equivocation in Hes.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
es: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hes: state_has_fixed_equivocation IM
  (elements equivocating) es
om: option message
Hom: sent_except_from (equivocator_IM IM)
  initial_message_prop es om
eqv: index
li: label (IM eqv)
state_has_fixed_equivocation IM
  (elements equivocating)
  (composite_transition (equivocator_IM IM)
     (existT eqv (ContinueWith 0 li)) (es, om)).1
      destruct (composite_transition _ _ _) as (es', om') eqn: Het.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
es: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hes: state_has_fixed_equivocation IM
  (elements equivocating) es
om: option message
Hom: sent_except_from (equivocator_IM IM)
  initial_message_prop es om
eqv: index
li: label (IM eqv)
es': composite_state (equivocator_IM IM)
om': option message
Het: composite_transition (equivocator_IM IM)
  (existT eqv (ContinueWith 0 li)) (
  es, om) = (es', om')
state_has_fixed_equivocation IM
  (elements equivocating) (es', om').1
      simpl.message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
es: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM (elements equivocating)))
     (λ _ : message, False))
Hes: state_has_fixed_equivocation IM
  (elements equivocating) es
om: option message
Hom: sent_except_from (equivocator_IM IM)
  initial_message_prop es om
eqv: index
li: label (IM eqv)
es': composite_state (equivocator_IM IM)
om': option message
Het: composite_transition (equivocator_IM IM)
  (existT eqv (ContinueWith 0 li)) (
  es, om) = (es', om')
state_has_fixed_equivocation IM
  (elements equivocating) es'
      by apply
        (zero_descriptor_transition_preserves_fixed_equivocation
          IM (elements equivocating) _ _ _ _ _ Het Hes li).
  -message, index, Ci: Type
H: ElemOf index Ci
H0: Empty Ci
H1: Singleton index Ci
H2: Union Ci
H3: Intersection Ci
H4: Difference Ci
H5: Elements index Ci
EqDecision0: EqDecision index
H6: FinSet index Ci
EqDecision1: EqDecision index
H7: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H8: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: Ci
X:= strong_fixed_equivocation_vlsm_composition IM
  equivocating: VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM
  (elements equivocating): VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM)
     (elements equivocating)): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
replayable_message_prop IM (λ _ : message, False)
  (strong_fixed_equivocation_constraint IM
     equivocating)
  (equivocators_fixed_equivocations_constraint IM
     (elements equivocating)) by apply fixed_equivocation_has_replayable_message_prop.
Qed.

End sec_fixed_equivocating.

No-equivocation simulation as a particular case of fixed set simulation

In this section we show that traces of the composition with no message equivocation can be simulated by the composition of equivocators with no message equivocation in which no component is allowed to state-equivocate.

Section sec_no_equivocation.

Context
  {message : Type}
  `{finite.Finite index}
  (IM : index -> VLSM message)
  (Free := free_composite_vlsm IM)
  `{forall i : index, HasBeenSentCapability (IM i)}
  (X : VLSM message := composite_vlsm IM (composite_no_equivocations IM))
  (XE : VLSM message := equivocators_fixed_equivocations_vlsm IM [])
  (FreeE := free_composite_vlsm (equivocator_IM IM))
  .

Lemma no_equivocating_equivocators_finite_valid_trace_init_to_rev
  (no_initial_messages_in_IM : no_initial_messages_in_IM_prop IM)
  isX sX trX
  (HtrX : finite_valid_trace_init_to X isX sX trX)
  : exists is, equivocators_total_state_project IM is = isX /\
    exists s, equivocators_total_state_project IM s = sX /\
    exists tr, equivocators_total_trace_project IM tr = trX /\
    finite_valid_trace_init_to XE is s tr /\
    finite_trace_last_output trX = finite_trace_last_output tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to X isX sX trX
∃ is : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM is = isX
  ∧ (∃ s : state
             (free_composite_vlsm (equivocator_IM IM)),
       equivocators_total_state_project IM s = sX
       ∧ (∃ tr : list
                   (composite_transition_item
                      (equivocator_IM IM)),
            equivocators_total_trace_project IM tr =
            trX
            ∧ finite_valid_trace_init_to XE is s tr
              ∧ finite_trace_last_output trX =
                finite_trace_last_output tr))
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to X isX sX trX
∃ is : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM is = isX
  ∧ (∃ s : state
             (free_composite_vlsm (equivocator_IM IM)),
       equivocators_total_state_project IM s = sX
       ∧ (∃ tr : list
                   (composite_transition_item
                      (equivocator_IM IM)),
            equivocators_total_trace_project IM tr =
            trX
            ∧ finite_valid_trace_init_to XE is s tr
              ∧ finite_trace_last_output trX =
                finite_trace_last_output tr))
  assert (no_initial_messages_in_XE :
    forall m, ~ initial_message_prop (preloaded_vlsm XE (fun _ => False)) m).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to X isX sX trX
∀ m : message, ¬ initial_message_prop m
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to X isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
∃ is : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM is = isX
  ∧ (∃ s : state
             (free_composite_vlsm (equivocator_IM IM)),
       equivocators_total_state_project IM s = sX
       ∧ (∃ tr : list
                   (composite_transition_item
                      (equivocator_IM IM)),
            equivocators_total_trace_project IM tr =
            trX
            ∧ finite_valid_trace_init_to XE is s tr
              ∧ finite_trace_last_output trX =
                finite_trace_last_output tr))
  {message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to X isX sX trX
∀ m : message, ¬ initial_message_prop m
    intros m [[i [[mi Hmi] Him]] | Hseeded]; [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to X isX sX trX
m: message
i: index
mi: message
Hmi: initial_message_prop mi
Him: `(mi ↾ Hmi) = m
False
    by elim (no_initial_messages_in_IM i mi).
  }message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to X isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
∃ is : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM is = isX
  ∧ (∃ s : state
             (free_composite_vlsm (equivocator_IM IM)),
       equivocators_total_state_project IM s = sX
       ∧ (∃ tr : list
                   (composite_transition_item
                      (equivocator_IM IM)),
            equivocators_total_trace_project IM tr =
            trX
            ∧ finite_valid_trace_init_to XE is s tr
              ∧ finite_trace_last_output trX =
                finite_trace_last_output tr))
  specialize (vlsm_is_preloaded_with_False X) as HeqX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to X isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
∃ is : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM is = isX
  ∧ (∃ s : state
             (free_composite_vlsm (equivocator_IM IM)),
       equivocators_total_state_project IM s = sX
       ∧ (∃ tr : list
                   (composite_transition_item
                      (equivocator_IM IM)),
            equivocators_total_trace_project IM tr =
            trX
            ∧ finite_valid_trace_init_to XE is s tr
              ∧ finite_trace_last_output trX =
                finite_trace_last_output tr))
  specialize (vlsm_is_preloaded_with_False XE) as HeqXE.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to X isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
∃ is : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM is = isX
  ∧ (∃ s : state
             (free_composite_vlsm (equivocator_IM IM)),
       equivocators_total_state_project IM s = sX
       ∧ (∃ tr : list
                   (composite_transition_item
                      (equivocator_IM IM)),
            equivocators_total_trace_project IM tr =
            trX
            ∧ finite_valid_trace_init_to XE is s tr
              ∧ finite_trace_last_output trX =
                finite_trace_last_output tr))
  apply (VLSM_eq_finite_valid_trace_init_to HeqX) in HtrX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
∃ is : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM is = isX
  ∧ (∃ s : state
             (free_composite_vlsm (equivocator_IM IM)),
       equivocators_total_state_project IM s = sX
       ∧ (∃ tr : list
                   (composite_transition_item
                      (equivocator_IM IM)),
            equivocators_total_trace_project IM tr =
            trX
            ∧ finite_valid_trace_init_to XE is s tr
              ∧ finite_trace_last_output trX =
                finite_trace_last_output tr))
  apply
    (generalized_equivocators_finite_valid_trace_init_to_rev
      IM _ _ (equivocators_fixed_equivocations_constraint IM []))
    in HtrX
    as [is [His [s [Hs [tr [Htr [Hptr Houtput]]]]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
is: state (free_composite_vlsm (equivocator_IM IM))
His: equivocators_total_state_project IM is = isX
s: state (free_composite_vlsm (equivocator_IM IM))
Hs: equivocators_total_state_project IM s = sX
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: equivocators_total_trace_project IM tr = trX
Hptr: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False)) is s
  tr
Houtput: finite_trace_last_output trX =
finite_trace_last_output tr
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
∃ is : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM is = isX
  ∧ (∃ s : state
             (free_composite_vlsm (equivocator_IM IM)),
       equivocators_total_state_project IM s = sX
       ∧ (∃ tr : list
                   (composite_transition_item
                      (equivocator_IM IM)),
            equivocators_total_trace_project IM tr =
            trX
            ∧ finite_valid_trace_init_to XE is s tr
              ∧ finite_trace_last_output trX =
                finite_trace_last_output tr))
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
zero_descriptor_constraint_lifting_prop IM
  (λ _ : message, False)
  (equivocators_fixed_equivocations_constraint IM [])
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
replayable_message_prop IM 
  (λ _ : message, False)
  (composite_no_equivocations IM)
  (equivocators_fixed_equivocations_constraint IM [])
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
is: state (free_composite_vlsm (equivocator_IM IM))
His: equivocators_total_state_project IM is = isX
s: state (free_composite_vlsm (equivocator_IM IM))
Hs: equivocators_total_state_project IM s = sX
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: equivocators_total_trace_project IM tr = trX
Hptr: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False)) is s
  tr
Houtput: finite_trace_last_output trX =
finite_trace_last_output tr
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
∃ is : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM is = isX
  ∧ (∃ s : state
             (free_composite_vlsm (equivocator_IM IM)),
       equivocators_total_state_project IM s = sX
       ∧ (∃ tr : list
                   (composite_transition_item
                      (equivocator_IM IM)),
            equivocators_total_trace_project IM tr =
            trX
            ∧ finite_valid_trace_init_to XE is s tr
              ∧ finite_trace_last_output trX =
                finite_trace_last_output tr)) exists is.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
is: state (free_composite_vlsm (equivocator_IM IM))
His: equivocators_total_state_project IM is = isX
s: state (free_composite_vlsm (equivocator_IM IM))
Hs: equivocators_total_state_project IM s = sX
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: equivocators_total_trace_project IM tr = trX
Hptr: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False)) is s
  tr
Houtput: finite_trace_last_output trX =
finite_trace_last_output tr
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
equivocators_total_state_project IM is = isX
∧ (∃ s : state
           (free_composite_vlsm (equivocator_IM IM)),
     equivocators_total_state_project IM s = sX
     ∧ (∃ tr : list
                 (composite_transition_item
                    (equivocator_IM IM)),
          equivocators_total_trace_project IM tr = trX
          ∧ finite_valid_trace_init_to XE is s tr
            ∧ finite_trace_last_output trX =
              finite_trace_last_output tr)) split; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
is: state (free_composite_vlsm (equivocator_IM IM))
His: equivocators_total_state_project IM is = isX
s: state (free_composite_vlsm (equivocator_IM IM))
Hs: equivocators_total_state_project IM s = sX
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: equivocators_total_trace_project IM tr = trX
Hptr: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False)) is s
  tr
Houtput: finite_trace_last_output trX =
finite_trace_last_output tr
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
∃ s : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM s = sX
  ∧ (∃ tr : list
              (composite_transition_item
                 (equivocator_IM IM)),
       equivocators_total_trace_project IM tr = trX
       ∧ finite_valid_trace_init_to XE is s tr
         ∧ finite_trace_last_output trX =
           finite_trace_last_output tr)
    exists s.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
is: state (free_composite_vlsm (equivocator_IM IM))
His: equivocators_total_state_project IM is = isX
s: state (free_composite_vlsm (equivocator_IM IM))
Hs: equivocators_total_state_project IM s = sX
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: equivocators_total_trace_project IM tr = trX
Hptr: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False)) is s
  tr
Houtput: finite_trace_last_output trX =
finite_trace_last_output tr
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
equivocators_total_state_project IM s = sX
∧ (∃ tr : list
            (composite_transition_item
               (equivocator_IM IM)),
     equivocators_total_trace_project IM tr = trX
     ∧ finite_valid_trace_init_to XE is s tr
       ∧ finite_trace_last_output trX =
         finite_trace_last_output tr) split; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
is: state (free_composite_vlsm (equivocator_IM IM))
His: equivocators_total_state_project IM is = isX
s: state (free_composite_vlsm (equivocator_IM IM))
Hs: equivocators_total_state_project IM s = sX
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: equivocators_total_trace_project IM tr = trX
Hptr: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False)) is s
  tr
Houtput: finite_trace_last_output trX =
finite_trace_last_output tr
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
∃ tr : list
         (composite_transition_item
            (equivocator_IM IM)),
  equivocators_total_trace_project IM tr = trX
  ∧ finite_valid_trace_init_to XE is s tr
    ∧ finite_trace_last_output trX =
      finite_trace_last_output tr
    exists tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
is: state (free_composite_vlsm (equivocator_IM IM))
His: equivocators_total_state_project IM is = isX
s: state (free_composite_vlsm (equivocator_IM IM))
Hs: equivocators_total_state_project IM s = sX
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: equivocators_total_trace_project IM tr = trX
Hptr: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False)) is s
  tr
Houtput: finite_trace_last_output trX =
finite_trace_last_output tr
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
equivocators_total_trace_project IM tr = trX
∧ finite_valid_trace_init_to XE is s tr
  ∧ finite_trace_last_output trX =
    finite_trace_last_output tr split; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
is: state (free_composite_vlsm (equivocator_IM IM))
His: equivocators_total_state_project IM is = isX
s: state (free_composite_vlsm (equivocator_IM IM))
Hs: equivocators_total_state_project IM s = sX
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: equivocators_total_trace_project IM tr = trX
Hptr: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False)) is s
  tr
Houtput: finite_trace_last_output trX =
finite_trace_last_output tr
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
finite_valid_trace_init_to XE is s tr
∧ finite_trace_last_output trX =
  finite_trace_last_output tr
    split; [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
is: state (free_composite_vlsm (equivocator_IM IM))
His: equivocators_total_state_project IM is = isX
s: state (free_composite_vlsm (equivocator_IM IM))
Hs: equivocators_total_state_project IM s = sX
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: equivocators_total_trace_project IM tr = trX
Hptr: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False)) is s
  tr
Houtput: finite_trace_last_output trX =
finite_trace_last_output tr
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
finite_valid_trace_init_to XE is s tr
    by apply (VLSM_eq_finite_valid_trace_init_to HeqXE).
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
zero_descriptor_constraint_lifting_prop IM
  (λ _ : message, False)
  (equivocators_fixed_equivocations_constraint IM []) intro; intros.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
es: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
Hes: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False)) es
om: option message
Hom: sent_except_from (equivocator_IM IM)
  initial_message_prop es om
eqv: index
li: label (IM eqv)
equivocators_fixed_equivocations_constraint IM []
  (existT eqv (ContinueWith 0 li)) (es, om)
    apply (VLSM_eq_valid_state HeqXE) in Hes.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
es: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
Hes: valid_state_prop
  {| vlsm_type := XE; vlsm_machine := XE |} es
om: option message
Hom: sent_except_from (equivocator_IM IM)
  initial_message_prop es om
eqv: index
li: label (IM eqv)
equivocators_fixed_equivocations_constraint IM []
  (existT eqv (ContinueWith 0 li)) (es, om)
    split.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
es: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
Hes: valid_state_prop
  {| vlsm_type := XE; vlsm_machine := XE |} es
om: option message
Hom: sent_except_from (equivocator_IM IM)
  initial_message_prop es om
eqv: index
li: label (IM eqv)
equivocators_no_equivocations_constraint IM
  (existT eqv (ContinueWith 0 li)) (es, om)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
es: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
Hes: valid_state_prop
  {| vlsm_type := XE; vlsm_machine := XE |} es
om: option message
Hom: sent_except_from (equivocator_IM IM)
  initial_message_prop es om
eqv: index
li: label (IM eqv)
state_has_fixed_equivocation IM []
  (composite_transition 
     (equivocator_IM IM)
     (existT eqv (ContinueWith 0 li)) (
     es, om)).1
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
es: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
Hes: valid_state_prop
  {| vlsm_type := XE; vlsm_machine := XE |} es
om: option message
Hom: sent_except_from (equivocator_IM IM)
  initial_message_prop es om
eqv: index
li: label (IM eqv)
equivocators_no_equivocations_constraint IM
  (existT eqv (ContinueWith 0 li)) (es, om) split; [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
es: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
Hes: valid_state_prop
  {| vlsm_type := XE; vlsm_machine := XE |} es
om: option message
Hom: sent_except_from (equivocator_IM IM)
  initial_message_prop es om
eqv: index
li: label (IM eqv)
composite_no_equivocations (equivocator_IM IM)
  (existT eqv (ContinueWith 0 li)) (es, om) destruct om as [im |]; [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
es: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
Hes: valid_state_prop
  {| vlsm_type := XE; vlsm_machine := XE |} es
im: message
Hom: sent_except_from (equivocator_IM IM)
  initial_message_prop es 
  (Some im)
eqv: index
li: label (IM eqv)
composite_no_equivocations (equivocator_IM IM)
  (existT eqv (ContinueWith 0 li)) (es, Some im)
      destruct Hom as [Hom | Hinitial]; [by left | exfalso].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
es: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
Hes: valid_state_prop
  {| vlsm_type := XE; vlsm_machine := XE |} es
im: message
Hinitial: initial_message_prop im
eqv: index
li: label (IM eqv)
False
      by apply no_initial_messages_in_XE in Hinitial.
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
es: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
Hes: valid_state_prop
  {| vlsm_type := XE; vlsm_machine := XE |} es
om: option message
Hom: sent_except_from (equivocator_IM IM)
  initial_message_prop es om
eqv: index
li: label (IM eqv)
state_has_fixed_equivocation IM []
  (composite_transition (equivocator_IM IM)
     (existT eqv (ContinueWith 0 li)) (es, om)).1 apply valid_state_has_fixed_equivocation in Hes.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
es: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
Hes: state_has_fixed_equivocation IM [] es
om: option message
Hom: sent_except_from (equivocator_IM IM)
  initial_message_prop es om
eqv: index
li: label (IM eqv)
state_has_fixed_equivocation IM []
  (composite_transition (equivocator_IM IM)
     (existT eqv (ContinueWith 0 li)) (es, om)).1
      destruct (composite_transition _ _ _) as (es', om') eqn: Het.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
es: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
Hes: state_has_fixed_equivocation IM [] es
om: option message
Hom: sent_except_from (equivocator_IM IM)
  initial_message_prop es om
eqv: index
li: label (IM eqv)
es': composite_state (equivocator_IM IM)
om': option message
Het: composite_transition (equivocator_IM IM)
  (existT eqv (ContinueWith 0 li)) (
  es, om) = (es', om')
state_has_fixed_equivocation IM [] (es', om').1
      simpl.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
es: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
Hes: state_has_fixed_equivocation IM [] es
om: option message
Hom: sent_except_from (equivocator_IM IM)
  initial_message_prop es om
eqv: index
li: label (IM eqv)
es': composite_state (equivocator_IM IM)
om': option message
Het: composite_transition (equivocator_IM IM)
  (existT eqv (ContinueWith 0 li)) (
  es, om) = (es', om')
state_has_fixed_equivocation IM [] es'
      by apply
        (zero_descriptor_transition_preserves_fixed_equivocation
          IM [] _ _ _ _ _ Het Hes li).
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state X
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := X;
    vlsm_machine :=
      preloaded_vlsm X (λ _ : message, False)
  |} isX sX trX
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
replayable_message_prop IM (λ _ : message, False)
  (composite_no_equivocations IM)
  (equivocators_fixed_equivocations_constraint IM []) clear isX sX trX HtrX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
replayable_message_prop IM (λ _ : message, False)
  (composite_no_equivocations IM)
  (equivocators_fixed_equivocations_constraint IM []) intro; intros.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (composite_no_equivocations IM))
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (composite_no_equivocations IM))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM []))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM []))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
  eqv_msg_is eqv_msg_s eqv_msg_tr
iom: option message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr iom
l: composite_label IM
HcX: composite_no_equivocations IM l (s, iom)
∃ (eqv_msg_tr : list transition_item) (lst_msg_tr : 
                                       state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (equivocators_fixed_equivocations_constraint
                                                 IM []))
                                            (λ _ : message,
                                               False))),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM [])) (λ _ : message, False))
    eqv_state_s lst_msg_tr eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr iom
    specialize (equivocators_fixed_equivocations_vlsm_incl_PreFree IM []) as HinclE.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (composite_no_equivocations IM))
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (composite_no_equivocations IM))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM []))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM []))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
  eqv_msg_is eqv_msg_s eqv_msg_tr
iom: option message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr iom
l: composite_label IM
HcX: composite_no_equivocations IM l (s, iom)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     [])
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
∃ (eqv_msg_tr : list transition_item) (lst_msg_tr : 
                                       state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (equivocators_fixed_equivocations_constraint
                                                 IM []))
                                            (λ _ : message,
                                               False))),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM [])) (λ _ : message, False))
    eqv_state_s lst_msg_tr eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr iom
    destruct iom as [im |]; [| by exists [], eqv_state_s; split; constructor].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (composite_no_equivocations IM))
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (composite_no_equivocations IM))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM []))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM []))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
  eqv_msg_is eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
HcX: composite_no_equivocations IM l (s, Some im)
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     [])
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
∃ (eqv_msg_tr : list transition_item) (lst_msg_tr : 
                                       state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (equivocators_fixed_equivocations_constraint
                                                 IM []))
                                            (λ _ : message,
                                               False))),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM [])) (λ _ : message, False))
    eqv_state_s lst_msg_tr eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr (Some im)
    destruct HcX as [Hsent | Hemitted]; [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (composite_no_equivocations IM))
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (composite_no_equivocations IM))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM []))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM []))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
  eqv_msg_is eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
Hsent: composite_has_been_sent IM (s, Some im).1 im
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     [])
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
∃ (eqv_msg_tr : list transition_item) (lst_msg_tr : 
                                       state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (equivocators_fixed_equivocations_constraint
                                                 IM []))
                                            (λ _ : message,
                                               False))),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM [])) (λ _ : message, False))
    eqv_state_s lst_msg_tr eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr (Some im)
    exists [].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (composite_no_equivocations IM))
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (composite_no_equivocations IM))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM []))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM []))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
  eqv_msg_is eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
Hsent: composite_has_been_sent IM (s, Some im).1 im
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     [])
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
∃ lst_msg_tr : state
                 (preloaded_vlsm
                    (composite_vlsm
                       (equivocator_IM IM)
                       (equivocators_fixed_equivocations_constraint
                          IM []))
                    (λ _ : message, False)),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM [])) (λ _ : message, False))
    eqv_state_s lst_msg_tr []
  ∧ equivocators_total_trace_project IM [] = []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr (Some im) exists eqv_state_s.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (composite_no_equivocations IM))
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (composite_no_equivocations IM))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM []))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM []))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
  eqv_msg_is eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
Hsent: composite_has_been_sent IM (s, Some im).1 im
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     [])
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
finite_valid_trace_from_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False)) eqv_state_s
  eqv_state_s []
∧ equivocators_total_trace_project IM [] = []
  ∧ equivocators_total_state_project IM eqv_state_s =
    s
    ∧ sent_except_from (equivocator_IM IM)
        initial_message_prop eqv_state_s (Some im)
    split; [by constructor |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (composite_no_equivocations IM))
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (composite_no_equivocations IM))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM []))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM []))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
  eqv_msg_is eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
Hsent: composite_has_been_sent IM (s, Some im).1 im
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     [])
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
equivocators_total_trace_project IM [] = []
∧ equivocators_total_state_project IM eqv_state_s = s
  ∧ sent_except_from (equivocator_IM IM)
      initial_message_prop eqv_state_s (Some im)
    split; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (composite_no_equivocations IM))
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (composite_no_equivocations IM))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM []))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM []))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
  eqv_msg_is eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
Hsent: composite_has_been_sent IM (s, Some im).1 im
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     [])
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
equivocators_total_state_project IM eqv_state_s = s
∧ sent_except_from (equivocator_IM IM)
    initial_message_prop eqv_state_s (Some im)
    split; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (composite_no_equivocations IM))
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (composite_no_equivocations IM))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM []))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM []))
     (λ _ : message, False))
  eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
  eqv_msg_is eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
Hsent: composite_has_been_sent IM (s, Some im).1 im
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     [])
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
sent_except_from (equivocator_IM IM)
  initial_message_prop eqv_state_s (Some im)
    apply (VLSM_eq_valid_state HeqXE) in Hstate_valid.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (composite_no_equivocations IM))
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (composite_no_equivocations IM))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM []))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  {|
    vlsm_type := XE; vlsm_machine := XE
  |} eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
  eqv_msg_is eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
Hsent: composite_has_been_sent IM (s, Some im).1 im
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     [])
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
sent_except_from (equivocator_IM IM)
  initial_message_prop eqv_state_s (Some im)
    apply (VLSM_incl_valid_state HinclE) in Hstate_valid.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (composite_no_equivocations IM))
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (composite_no_equivocations IM))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM []))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  {|
    vlsm_type :=
      equivocators_fixed_equivocations_vlsm
        IM [];
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM IM))
  |} eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
  eqv_msg_is eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
Hsent: composite_has_been_sent IM (s, Some im).1 im
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     [])
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
sent_except_from (equivocator_IM IM)
  initial_message_prop eqv_state_s (Some im)
    left.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (composite_no_equivocations IM))
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (composite_no_equivocations IM))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM []))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  {|
    vlsm_type :=
      equivocators_fixed_equivocations_vlsm
        IM [];
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM IM))
  |} eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
  eqv_msg_is eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
Hsent: composite_has_been_sent IM (s, Some im).1 im
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     [])
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
composite_has_been_sent (equivocator_IM IM)
  eqv_state_s im
    revert Hsent.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM
        (composite_no_equivocations IM))
     (λ _ : message, False))
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (composite_no_equivocations IM))
     (λ _ : message, False)) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM []))
     (λ _ : message, False))
Hstate_valid: valid_state_prop
  {|
    vlsm_type :=
      equivocators_fixed_equivocations_vlsm
        IM [];
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM IM))
  |} eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
  eqv_msg_is eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     [])
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
composite_has_been_sent IM (s, Some im).1 im
→ composite_has_been_sent (equivocator_IM IM)
    eqv_state_s im subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
X:= composite_vlsm IM (composite_no_equivocations IM): VLSM message
XE:= equivocators_fixed_equivocations_vlsm IM []: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
no_initial_messages_in_XE: ∀ m : message,
  ¬ initial_message_prop m
HeqX: VLSM_eq X
  (preloaded_vlsm X (λ _ : message, False))
HeqXE: VLSM_eq XE
  (preloaded_vlsm XE (λ _ : message, False))
si: state
  (preloaded_vlsm
     (composite_vlsm IM
        (composite_no_equivocations IM))
     (λ _ : message, False))
tr: list transition_item
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM []))
     (λ _ : message, False))
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM
        (composite_no_equivocations IM))
     (λ _ : message, False)) si
  (equivocators_total_state_project IM
     eqv_state_s) tr
Hstate_valid: valid_state_prop
  {|
    vlsm_type :=
      equivocators_fixed_equivocations_vlsm
        IM [];
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM IM))
  |} eqv_state_s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM [])) (λ _ : message, False))
  eqv_msg_is eqv_msg_s eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
HinclE: VLSM_incl
  (equivocators_fixed_equivocations_vlsm IM
     [])
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
composite_has_been_sent IM
  (equivocators_total_state_project IM eqv_state_s,
   Some im).1 im
→ composite_has_been_sent (equivocator_IM IM)
    eqv_state_s im
    by specialize
      (VLSM_projection_has_been_sent_reflect
        (preloaded_equivocators_no_equivocations_vlsm_X_vlsm_projection IM)
        eqv_state_s Hstate_valid im) as Hsent.
Qed.

End sec_no_equivocation.