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 StdppListSet.
From VLSM.Lib Require Import ListSetExtras NatExtras.[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 VLSM Equivocation.
From VLSM.Core Require Import Equivocators.Equivocators Equivocators.EquivocatorsProjections.

Core: VLSM Message Properties

Section sec_equivocator_vlsm_message_properties.

Lifting properties about sent messages to the equivocators

In this section we first prove some general properties about sent messages being preserved and reflected by the equivocator_vlsm, and then we show that the HasBeenSentCapability and the ComputableSentMessages can be lifted (each separately) from the original machine to the equivocator_vlsm.

Context
  {message : Type}
  (X : VLSM message)
  .

If a projection of an equivocator_vlsm trace outputs a message, then the original trace must do so too.

Lemma equivocator_vlsm_trace_project_output_reflecting
  (tr : list (transition_item (equivocator_vlsm X)))
  (trX : list (transition_item X))
  (j i : MachineDescriptor X)
  (HtrX : equivocator_vlsm_trace_project _ tr j = Some (trX, i))
  (m : message)
  (Hjbs : Exists (field_selector output m) trX)
  : Exists (field_selector output m) tr.message: Type
X: VLSM message
tr, trX: list transition_item
j, i: MachineDescriptor X
HtrX: equivocator_vlsm_trace_project X tr j =
Some (trX, i)
m: message
Hjbs: Exists (field_selector output m) trX
Exists (field_selector output m) tr
Proof.message: Type
X: VLSM message
tr, trX: list transition_item
j, i: MachineDescriptor X
HtrX: equivocator_vlsm_trace_project X tr j =
Some (trX, i)
m: message
Hjbs: Exists (field_selector output m) trX
Exists (field_selector output m) tr
  revert trX i HtrX Hjbs.message: Type
X: VLSM message
tr: list transition_item
j: MachineDescriptor X
m: message
∀ (trX : list transition_item) (i : MachineDescriptor
                                      X),
  equivocator_vlsm_trace_project X tr j =
  Some (trX, i)
  → Exists (field_selector output m) trX
    → Exists (field_selector output m) tr
  induction tr; intros.message: Type
X: VLSM message
j: MachineDescriptor X
m: message
trX: list transition_item
i: MachineDescriptor X
HtrX: equivocator_vlsm_trace_project X [] j =
Some (trX, i)
Hjbs: Exists (field_selector output m) trX
Exists (field_selector output m) []
message: Type
X: VLSM message
a: transition_item
tr: list transition_item
j: MachineDescriptor X
m: message
IHtr: ∀ (trX : list transition_item) 
  (i : MachineDescriptor X),
  equivocator_vlsm_trace_project X tr j =
  Some (trX, i)
  → Exists (field_selector output m) trX
    → Exists (field_selector output m) tr
trX: list transition_item
i: MachineDescriptor X
HtrX: equivocator_vlsm_trace_project X (a :: tr) j =
Some (trX, i)
Hjbs: Exists (field_selector output m) trX
Exists (field_selector output m) (a :: tr)
  -message: Type
X: VLSM message
j: MachineDescriptor X
m: message
trX: list transition_item
i: MachineDescriptor X
HtrX: equivocator_vlsm_trace_project X [] j =
Some (trX, i)
Hjbs: Exists (field_selector output m) trX
Exists (field_selector output m) [] by inversion HtrX; subst; inversion Hjbs.
  -message: Type
X: VLSM message
a: transition_item
tr: list transition_item
j: MachineDescriptor X
m: message
IHtr: ∀ (trX : list transition_item) 
  (i : MachineDescriptor X),
  equivocator_vlsm_trace_project X tr j =
  Some (trX, i)
  → Exists (field_selector output m) trX
    → Exists (field_selector output m) tr
trX: list transition_item
i: MachineDescriptor X
HtrX: equivocator_vlsm_trace_project X (a :: tr) j =
Some (trX, i)
Hjbs: Exists (field_selector output m) trX
Exists (field_selector output m) (a :: tr) simpl in HtrX.message: Type
X: VLSM message
a: transition_item
tr: list transition_item
j: MachineDescriptor X
m: message
IHtr: ∀ (trX : list transition_item) 
  (i : MachineDescriptor X),
  equivocator_vlsm_trace_project X tr j =
  Some (trX, i)
  → Exists (field_selector output m) trX
    → Exists (field_selector output m) tr
trX: list transition_item
i: MachineDescriptor X
HtrX: match
  equivocator_vlsm_trace_project X tr j
with
| Some (r, idescriptor) =>
    match
      equivocator_vlsm_transition_item_project
        X a idescriptor
    with
    | Some (Some item', odescriptor) =>
        Some (item' :: r, odescriptor)
    | Some (None, odescriptor) =>
        Some (r, odescriptor)
    | None => None
    end
| None => None
end = Some (trX, i)
Hjbs: Exists (field_selector output m) trX
Exists (field_selector output m) (a :: tr)
    destruct (equivocator_vlsm_trace_project _ tr j) as [(trX', i') |]
      eqn: Htr; [| by congruence].message: Type
X: VLSM message
a: transition_item
tr: list transition_item
j: MachineDescriptor X
m: message
trX': list transition_item
i': MachineDescriptor X
Htr: equivocator_vlsm_trace_project X tr j =
Some (trX', i')
IHtr: ∀ (trX : list transition_item) 
  (i : MachineDescriptor X),
  Some (trX', i') = Some (trX, i)
  → Exists (field_selector output m) trX
    → Exists (field_selector output m) tr
trX: list transition_item
i: MachineDescriptor X
HtrX: match
  equivocator_vlsm_transition_item_project X a
    i'
with
| Some (Some item', odescriptor) =>
    Some (item' :: trX', odescriptor)
| Some (None, odescriptor) =>
    Some (trX', odescriptor)
| None => None
end = Some (trX, i)
Hjbs: Exists (field_selector output m) trX
Exists (field_selector output m) (a :: tr)
    specialize (IHtr trX').message: Type
X: VLSM message
a: transition_item
tr: list transition_item
j: MachineDescriptor X
m: message
trX': list transition_item
i': MachineDescriptor X
Htr: equivocator_vlsm_trace_project X tr j =
Some (trX', i')
IHtr: ∀ i : MachineDescriptor X,
  Some (trX', i') = Some (trX', i)
  → Exists (field_selector output m) trX'
    → Exists (field_selector output m) tr
trX: list transition_item
i: MachineDescriptor X
HtrX: match
  equivocator_vlsm_transition_item_project X a
    i'
with
| Some (Some item', odescriptor) =>
    Some (item' :: trX', odescriptor)
| Some (None, odescriptor) =>
    Some (trX', odescriptor)
| None => None
end = Some (trX, i)
Hjbs: Exists (field_selector output m) trX
Exists (field_selector output m) (a :: tr)
    destruct (equivocator_vlsm_transition_item_project _ a i') as [[[item' |] i''] |]
      eqn: Hitem'
    ; inversion HtrX; subst; clear HtrX.message: Type
X: VLSM message
a: transition_item
tr: list transition_item
j: MachineDescriptor X
m: message
trX': list transition_item
i': MachineDescriptor X
Htr: equivocator_vlsm_trace_project X tr j =
Some (trX', i')
IHtr: ∀ i : MachineDescriptor X,
  Some (trX', i') = Some (trX', i)
  → Exists (field_selector output m) trX'
    → Exists (field_selector output m) tr
i: MachineDescriptor X
item': transition_item
Hitem': equivocator_vlsm_transition_item_project X a
  i' = Some (Some item', i)
Hjbs: Exists (field_selector output m)
  (item' :: trX')
Exists (field_selector output m) (a :: tr)
message: Type
X: VLSM message
a: transition_item
tr: list transition_item
j: MachineDescriptor X
m: message
i': MachineDescriptor X
trX: list transition_item
IHtr: ∀ i : MachineDescriptor X,
  Some (trX, i') = Some (trX, i)
  → Exists (field_selector output m) trX
    → Exists (field_selector output m) tr
Htr: equivocator_vlsm_trace_project X tr j =
Some (trX, i')
i: MachineDescriptor X
Hitem': equivocator_vlsm_transition_item_project X a
  i' = Some (None, i)
Hjbs: Exists (field_selector output m) trX
Exists (field_selector output m) (a :: tr)
    +message: Type
X: VLSM message
a: transition_item
tr: list transition_item
j: MachineDescriptor X
m: message
trX': list transition_item
i': MachineDescriptor X
Htr: equivocator_vlsm_trace_project X tr j =
Some (trX', i')
IHtr: ∀ i : MachineDescriptor X,
  Some (trX', i') = Some (trX', i)
  → Exists (field_selector output m) trX'
    → Exists (field_selector output m) tr
i: MachineDescriptor X
item': transition_item
Hitem': equivocator_vlsm_transition_item_project X a
  i' = Some (Some item', i)
Hjbs: Exists (field_selector output m)
  (item' :: trX')
Exists (field_selector output m) (a :: tr) apply equivocator_transition_item_project_inv_messages in Hitem'.message: Type
X: VLSM message
a: transition_item
tr: list transition_item
j: MachineDescriptor X
m: message
trX': list transition_item
i': MachineDescriptor X
Htr: equivocator_vlsm_trace_project X tr j =
Some (trX', i')
IHtr: ∀ i : MachineDescriptor X,
  Some (trX', i') = Some (trX', i)
  → Exists (field_selector output m) trX'
    → Exists (field_selector output m) tr
i: MachineDescriptor X
item': transition_item
Hitem': ∃ i : nat,
  i' = Existing i
  ∧ proper_descriptor X i' (destination a)
    ∧ input a = input item'
      ∧ output a = output item'
Hjbs: Exists (field_selector output m)
  (item' :: trX')
Exists (field_selector output m) (a :: tr)
      destruct Hitem' as [_ [_ [_ [_ Ha]]]].message: Type
X: VLSM message
a: transition_item
tr: list transition_item
j: MachineDescriptor X
m: message
trX': list transition_item
i': MachineDescriptor X
Htr: equivocator_vlsm_trace_project X tr j =
Some (trX', i')
IHtr: ∀ i : MachineDescriptor X,
  Some (trX', i') = Some (trX', i)
  → Exists (field_selector output m) trX'
    → Exists (field_selector output m) tr
i: MachineDescriptor X
item': transition_item
Ha: output a = output item'
Hjbs: Exists (field_selector output m)
  (item' :: trX')
Exists (field_selector output m) (a :: tr)
      inversion Hjbs; subst.message: Type
X: VLSM message
a: transition_item
tr: list transition_item
j: MachineDescriptor X
m: message
trX': list transition_item
i': MachineDescriptor X
Htr: equivocator_vlsm_trace_project X tr j =
Some (trX', i')
IHtr: ∀ i : MachineDescriptor X,
  Some (trX', i') = Some (trX', i)
  → Exists (field_selector output m) trX'
    → Exists (field_selector output m) tr
i: MachineDescriptor X
item': transition_item
Ha: output a = output item'
Hjbs: Exists (field_selector output m)
  (item' :: trX')
H0: field_selector output m item'
Exists (field_selector output m) (a :: tr)
message: Type
X: VLSM message
a: transition_item
tr: list transition_item
j: MachineDescriptor X
m: message
trX': list transition_item
i': MachineDescriptor X
Htr: equivocator_vlsm_trace_project X tr j =
Some (trX', i')
IHtr: ∀ i : MachineDescriptor X,
  Some (trX', i') = Some (trX', i)
  → Exists (field_selector output m) trX'
    → Exists (field_selector output m) tr
i: MachineDescriptor X
item': transition_item
Ha: output a = output item'
Hjbs: Exists (field_selector output m)
  (item' :: trX')
H0: Exists (field_selector output m) trX'
Exists (field_selector output m) (a :: tr)
      *message: Type
X: VLSM message
a: transition_item
tr: list transition_item
j: MachineDescriptor X
m: message
trX': list transition_item
i': MachineDescriptor X
Htr: equivocator_vlsm_trace_project X tr j =
Some (trX', i')
IHtr: ∀ i : MachineDescriptor X,
  Some (trX', i') = Some (trX', i)
  → Exists (field_selector output m) trX'
    → Exists (field_selector output m) tr
i: MachineDescriptor X
item': transition_item
Ha: output a = output item'
Hjbs: Exists (field_selector output m)
  (item' :: trX')
H0: field_selector output m item'
Exists (field_selector output m) (a :: tr) by left; cbn in *; rewrite Ha.
      *message: Type
X: VLSM message
a: transition_item
tr: list transition_item
j: MachineDescriptor X
m: message
trX': list transition_item
i': MachineDescriptor X
Htr: equivocator_vlsm_trace_project X tr j =
Some (trX', i')
IHtr: ∀ i : MachineDescriptor X,
  Some (trX', i') = Some (trX', i)
  → Exists (field_selector output m) trX'
    → Exists (field_selector output m) tr
i: MachineDescriptor X
item': transition_item
Ha: output a = output item'
Hjbs: Exists (field_selector output m)
  (item' :: trX')
H0: Exists (field_selector output m) trX'
Exists (field_selector output m) (a :: tr) by right; eapply IHtr.
    +message: Type
X: VLSM message
a: transition_item
tr: list transition_item
j: MachineDescriptor X
m: message
i': MachineDescriptor X
trX: list transition_item
IHtr: ∀ i : MachineDescriptor X,
  Some (trX, i') = Some (trX, i)
  → Exists (field_selector output m) trX
    → Exists (field_selector output m) tr
Htr: equivocator_vlsm_trace_project X tr j =
Some (trX, i')
i: MachineDescriptor X
Hitem': equivocator_vlsm_transition_item_project X a
  i' = Some (None, i)
Hjbs: Exists (field_selector output m) trX
Exists (field_selector output m) (a :: tr) by right; eapply IHtr.
Qed.

Lemma preloaded_equivocator_vlsm_trace_project_valid_item_new_machine
  (bs : state (equivocator_vlsm X))
  (btr : list (transition_item (equivocator_vlsm X)))
  (Hbtr : finite_constrained_trace_from (equivocator_vlsm X) bs btr)
  (bitem : transition_item (equivocator_vlsm X))
  (Hitem : bitem ∈ btr)
  (sn : state X)
  (Hnew : l bitem = Spawn sn)
  : input bitem = None /\ output bitem = None /\
    exists
      (d : MachineDescriptor X),
      equivocator_vlsm_transition_item_project _ bitem d = Some (None, NewMachine sn).message: Type
X: VLSM message
bs: state (equivocator_vlsm X)
btr: list transition_item
Hbtr: finite_constrained_trace_from
  (equivocator_vlsm X) bs btr
bitem: transition_item
Hitem: bitem ∈ btr
sn: state X
Hnew: l bitem = Spawn sn
input bitem = None
∧ output bitem = None
  ∧ (∃ d : MachineDescriptor X,
       equivocator_vlsm_transition_item_project X
         bitem d = Some (None, NewMachine sn))
Proof.message: Type
X: VLSM message
bs: state (equivocator_vlsm X)
btr: list transition_item
Hbtr: finite_constrained_trace_from
  (equivocator_vlsm X) bs btr
bitem: transition_item
Hitem: bitem ∈ btr
sn: state X
Hnew: l bitem = Spawn sn
input bitem = None
∧ output bitem = None
  ∧ (∃ d : MachineDescriptor X,
       equivocator_vlsm_transition_item_project X
         bitem d = Some (None, NewMachine sn))
  apply elem_of_list_split in Hitem.message: Type
X: VLSM message
bs: state (equivocator_vlsm X)
btr: list transition_item
Hbtr: finite_constrained_trace_from
  (equivocator_vlsm X) bs btr
bitem: transition_item
Hitem: ∃ l1 l2 : list transition_item,
  btr = l1 ++ bitem :: l2
sn: state X
Hnew: l bitem = Spawn sn
input bitem = None
∧ output bitem = None
  ∧ (∃ d : MachineDescriptor X,
       equivocator_vlsm_transition_item_project X
         bitem d = Some (None, NewMachine sn))
  destruct Hitem as [bprefix [bsuffix Heq]].message: Type
X: VLSM message
bs: state (equivocator_vlsm X)
btr: list transition_item
Hbtr: finite_constrained_trace_from
  (equivocator_vlsm X) bs btr
bitem: transition_item
bprefix, bsuffix: list transition_item
Heq: btr = bprefix ++ bitem :: bsuffix
sn: state X
Hnew: l bitem = Spawn sn
input bitem = None
∧ output bitem = None
  ∧ (∃ d : MachineDescriptor X,
       equivocator_vlsm_transition_item_project X
         bitem d = Some (None, NewMachine sn))
  subst btr.message: Type
X: VLSM message
bs: state (equivocator_vlsm X)
bitem: transition_item
bprefix, bsuffix: list transition_item
Hbtr: finite_constrained_trace_from
  (equivocator_vlsm X) bs
  (bprefix ++ bitem :: bsuffix)
sn: state X
Hnew: l bitem = Spawn sn
input bitem = None
∧ output bitem = None
  ∧ (∃ d : MachineDescriptor X,
       equivocator_vlsm_transition_item_project X
         bitem d = Some (None, NewMachine sn))
  apply (finite_valid_trace_from_app_iff (preloaded_with_all_messages_vlsm (equivocator_vlsm X)))
    in Hbtr.message: Type
X: VLSM message
bs: state (equivocator_vlsm X)
bitem: transition_item
bprefix, bsuffix: list transition_item
Hbtr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs bprefix
∧ finite_valid_trace_from
    (preloaded_with_all_messages_vlsm
       (equivocator_vlsm X))
    (finite_trace_last bs bprefix)
    (bitem :: bsuffix)
sn: state X
Hnew: l bitem = Spawn sn
input bitem = None
∧ output bitem = None
  ∧ (∃ d : MachineDescriptor X,
       equivocator_vlsm_transition_item_project X
         bitem d = Some (None, NewMachine sn))
  destruct Hbtr as [Hbprefix Hbsuffix].message: Type
X: VLSM message
bs: state (equivocator_vlsm X)
bitem: transition_item
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs bprefix
Hbsuffix: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
  (finite_trace_last bs bprefix)
  (bitem :: bsuffix)
sn: state X
Hnew: l bitem = Spawn sn
input bitem = None
∧ output bitem = None
  ∧ (∃ d : MachineDescriptor X,
       equivocator_vlsm_transition_item_project X
         bitem d = Some (None, NewMachine sn))
  remember (finite_trace_last _ _) as lst.message: Type
X: VLSM message
bs: state (equivocator_vlsm X)
bitem: transition_item
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs bprefix
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
Hbsuffix: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst
  (bitem :: bsuffix)
sn: state X
Hnew: l bitem = Spawn sn
input bitem = None
∧ output bitem = None
  ∧ (∃ d : MachineDescriptor X,
       equivocator_vlsm_transition_item_project X
         bitem d = Some (None, NewMachine sn))
  inversion Hbsuffix.message: Type
X: VLSM message
bs: state (equivocator_vlsm X)
bitem: transition_item
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs bprefix
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
Hbsuffix: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst
  (bitem :: bsuffix)
sn: state X
Hnew: VLSM.l bitem = Spawn sn
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
tl: list transition_item
s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) l (
  lst, iom) (s, oom)
H: s' = lst
H1: {|
  l := l;
  input := iom;
  destination := s;
  output := oom
|} = bitem
H2: tl = bsuffix
input
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = None
∧ output
    {|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |} = None
  ∧ (∃ d : MachineDescriptor X,
       equivocator_vlsm_transition_item_project X
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |} d = Some (None, NewMachine sn)) subst s' tl bitem.message: Type
X: VLSM message
bs: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs bprefix
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
sn: state X
Hnew: VLSM.l
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Spawn sn
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) l (
  lst, iom) (s, oom)
input
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = None
∧ output
    {|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |} = None
  ∧ (∃ d : MachineDescriptor X,
       equivocator_vlsm_transition_item_project X
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |} d = Some (None, NewMachine sn))
  destruct Ht as [[_ [_ Hv]] Ht].message: Type
X: VLSM message
bs: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs bprefix
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
sn: state X
Hnew: VLSM.l
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Spawn sn
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Hv: valid l (lst, iom)
Ht: transition l (lst, iom) = (s, oom)
input
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = None
∧ output
    {|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |} = None
  ∧ (∃ d : MachineDescriptor X,
       equivocator_vlsm_transition_item_project X
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |} d = Some (None, NewMachine sn))
  simpl.message: Type
X: VLSM message
bs: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs bprefix
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
sn: state X
Hnew: VLSM.l
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Spawn sn
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Hv: valid l (lst, iom)
Ht: transition l (lst, iom) = (s, oom)
iom = None
∧ oom = None
  ∧ (∃ d : MachineDescriptor X,
       equivocator_vlsm_transition_item_project X
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |} d = Some (None, NewMachine sn))
  specialize
    (equivocator_valid_transition_project_inv5_new_machine
      _ l s lst iom oom Ht)
    as Hpitem.message: Type
X: VLSM message
bs: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs bprefix
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
sn: state X
Hnew: VLSM.l
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Spawn sn
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Hv: valid l (lst, iom)
Ht: transition l (lst, iom) = (s, oom)
Hpitem: ∀ (item :=
   {|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}) (sn : state X),
  l = Spawn sn
  → ∃ (i : nat) (si : option (state X)),
      equivocator_state_project s i = si
      ∧ equivocator_vlsm_transition_item_project
          X item (Existing i) =
        Some (None, NewMachine sn)
iom = None
∧ oom = None
  ∧ (∃ d : MachineDescriptor X,
       equivocator_vlsm_transition_item_project X
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |} d = Some (None, NewMachine sn))
  unfold VLSM.l in *.message: Type
X: VLSM message
bs: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs bprefix
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
sn: state X
Hnew: l = Spawn sn
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Hv: valid l (lst, iom)
Ht: transition l (lst, iom) = (s, oom)
Hpitem: ∀ sn : state X,
  l = Spawn sn
  → ∃ (i : nat) (si : option (state X)),
      equivocator_state_project s i = si
      ∧ equivocator_vlsm_transition_item_project
          X
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} (Existing i) =
        Some (None, NewMachine sn)
iom = None
∧ oom = None
  ∧ (∃ d : MachineDescriptor X,
       equivocator_vlsm_transition_item_project X
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |} d = Some (None, NewMachine sn))
  subst l.message: Type
X: VLSM message
bs: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs bprefix
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
sn: state X
Hbsuffix: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst
  ({|
     l := Spawn sn;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Hpitem: ∀ sn0 : state X,
  Spawn sn = Spawn sn0
  → ∃ (i : nat) (si : option (state X)),
      equivocator_state_project s i = si
      ∧ equivocator_vlsm_transition_item_project
          X
          {|
            l := Spawn sn;
            input := iom;
            destination := s;
            output := oom
          |} (Existing i) =
        Some (None, NewMachine sn0)
Ht: transition (Spawn sn) (lst, iom) = (s, oom)
Hv: valid (Spawn sn) (lst, iom)
iom = None
∧ oom = None
  ∧ (∃ d : MachineDescriptor X,
       equivocator_vlsm_transition_item_project X
         {|
           l := Spawn sn;
           input := iom;
           destination := s;
           output := oom
         |} d = Some (None, NewMachine sn))
  specialize (Hpitem _ eq_refl) as [i [lst_i [Hi Hpitem]]].message: Type
X: VLSM message
bs: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs bprefix
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
sn: state X
Hbsuffix: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst
  ({|
     l := Spawn sn;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Ht: transition (Spawn sn) (lst, iom) = (s, oom)
Hv: valid (Spawn sn) (lst, iom)
i: nat
lst_i: option (state X)
Hi: equivocator_state_project s i = lst_i
Hpitem: equivocator_vlsm_transition_item_project X
  {|
    l := Spawn sn;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (None, NewMachine sn)
iom = None
∧ oom = None
  ∧ (∃ d : MachineDescriptor X,
       equivocator_vlsm_transition_item_project X
         {|
           l := Spawn sn;
           input := iom;
           destination := s;
           output := oom
         |} d = Some (None, NewMachine sn))
  by inversion Ht; destruct Hv as [Hsndl Hiom]; subst; eauto.
Qed.

For any transition_item in a valid trace segment of an equivocator_vlsm there exists a projection of that trace containing the projection of the item.

Lemma preloaded_equivocator_vlsm_trace_project_valid_item
  (bs bf : state (equivocator_vlsm X))
  (btr : list (transition_item (equivocator_vlsm X)))
  (Hbtr : finite_constrained_trace_from_to (equivocator_vlsm X) bs bf btr)
  (bitem : transition_item (equivocator_vlsm X))
  (Hitem : bitem ∈ btr)
  (idl : nat)
  (Hlbitem : equivocator_label_descriptor (l bitem) = Existing idl)
  : exists (item : transition_item X),
      (exists (d : MachineDescriptor X),
        equivocator_vlsm_transition_item_project _ bitem d = Some (Some item, Existing idl))
      /\ exists (tr : list (transition_item X)),
        item ∈ tr /\
        exists (dfinal dfirst : MachineDescriptor X),
          proper_descriptor X dfirst bs /\
          existing_descriptor X dfinal (finite_trace_last bs btr) /\
          equivocator_vlsm_trace_project _ btr dfinal = Some (tr, dfirst).message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
btr: list transition_item
Hbtr: finite_constrained_trace_from_to
  (equivocator_vlsm X) bs bf btr
bitem: transition_item
Hitem: bitem ∈ btr
idl: nat
Hlbitem: equivocator_label_descriptor (l bitem) =
Existing idl
∃ item : transition_item,
  (∃ d : MachineDescriptor X,
     equivocator_vlsm_transition_item_project X bitem
       d = Some (Some item, Existing idl))
  ∧ (∃ tr : list transition_item,
       item ∈ tr
       ∧ (∃ dfinal dfirst : MachineDescriptor X,
            proper_descriptor X dfirst bs
            ∧ existing_descriptor X dfinal
                (finite_trace_last bs btr)
              ∧ equivocator_vlsm_trace_project X btr
                  dfinal = Some (tr, dfirst)))
Proof.message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
btr: list transition_item
Hbtr: finite_constrained_trace_from_to
  (equivocator_vlsm X) bs bf btr
bitem: transition_item
Hitem: bitem ∈ btr
idl: nat
Hlbitem: equivocator_label_descriptor (l bitem) =
Existing idl
∃ item : transition_item,
  (∃ d : MachineDescriptor X,
     equivocator_vlsm_transition_item_project X bitem
       d = Some (Some item, Existing idl))
  ∧ (∃ tr : list transition_item,
       item ∈ tr
       ∧ (∃ dfinal dfirst : MachineDescriptor X,
            proper_descriptor X dfirst bs
            ∧ existing_descriptor X dfinal
                (finite_trace_last bs btr)
              ∧ equivocator_vlsm_trace_project X btr
                  dfinal = Some (tr, dfirst)))
  specialize (preloaded_equivocator_vlsm_valid_trace_project_inv2 X bs bf btr) as Hinv2.message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
btr: list transition_item
Hbtr: finite_constrained_trace_from_to
  (equivocator_vlsm X) bs bf btr
bitem: transition_item
Hitem: bitem ∈ btr
idl: nat
Hlbitem: equivocator_label_descriptor (l bitem) =
Existing idl
Hinv2: btr ≠ []
→ finite_constrained_trace_from_to
    (equivocator_vlsm X) bs bf btr
  → ∀ (j : nat) (di : MachineDescriptor X) 
      (trX : list transition_item),
      equivocator_vlsm_trace_project X btr
        (Existing j) = 
      Some (trX, di)
      → ∃ fsj : state X,
          equivocator_state_project bf j =
          Some fsj
          ∧ match di with
            | NewMachine sn =>
                finite_constrained_trace_init_to
                  X sn fsj trX
            | Existing i =>
                ∃ isi : state X,
                  equivocator_state_project bs
                    i = 
                  Some isi
                  ∧ finite_constrained_trace_from_to
                      X isi fsj trX
                    ∧ (initial_state_prop bs
                      → 
                      initial_state_prop isi)
            end
∃ item : transition_item,
  (∃ d : MachineDescriptor X,
     equivocator_vlsm_transition_item_project X bitem
       d = Some (Some item, Existing idl))
  ∧ (∃ tr : list transition_item,
       item ∈ tr
       ∧ (∃ dfinal dfirst : MachineDescriptor X,
            proper_descriptor X dfirst bs
            ∧ existing_descriptor X dfinal
                (finite_trace_last bs btr)
              ∧ equivocator_vlsm_trace_project X btr
                  dfinal = Some (tr, dfirst)))
  spec Hinv2; [by intro contra; subst; inversion Hitem |].message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
btr: list transition_item
Hbtr: finite_constrained_trace_from_to
  (equivocator_vlsm X) bs bf btr
bitem: transition_item
Hitem: bitem ∈ btr
idl: nat
Hlbitem: equivocator_label_descriptor (l bitem) =
Existing idl
Hinv2: finite_constrained_trace_from_to
  (equivocator_vlsm X) bs bf btr
→ ∀ (j : nat) (di : MachineDescriptor X) 
    (trX : list transition_item),
    equivocator_vlsm_trace_project X btr
      (Existing j) = Some (trX, di)
    → ∃ fsj : state X,
        equivocator_state_project bf j =
        Some fsj
        ∧ match di with
          | NewMachine sn =>
              finite_constrained_trace_init_to
                X sn fsj trX
          | Existing i =>
              ∃ isi : state X,
                equivocator_state_project bs i =
                Some isi
                ∧ finite_constrained_trace_from_to
                    X isi fsj trX
                  ∧ (initial_state_prop bs
                     → 
                     initial_state_prop isi)
          end
∃ item : transition_item,
  (∃ d : MachineDescriptor X,
     equivocator_vlsm_transition_item_project X bitem
       d = Some (Some item, Existing idl))
  ∧ (∃ tr : list transition_item,
       item ∈ tr
       ∧ (∃ dfinal dfirst : MachineDescriptor X,
            proper_descriptor X dfirst bs
            ∧ existing_descriptor X dfinal
                (finite_trace_last bs btr)
              ∧ equivocator_vlsm_trace_project X btr
                  dfinal = Some (tr, dfirst)))
  specialize (Hinv2 Hbtr).message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
btr: list transition_item
Hbtr: finite_constrained_trace_from_to
  (equivocator_vlsm X) bs bf btr
bitem: transition_item
Hitem: bitem ∈ btr
idl: nat
Hlbitem: equivocator_label_descriptor (l bitem) =
Existing idl
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X btr
    (Existing j) = Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
∃ item : transition_item,
  (∃ d : MachineDescriptor X,
     equivocator_vlsm_transition_item_project X bitem
       d = Some (Some item, Existing idl))
  ∧ (∃ tr : list transition_item,
       item ∈ tr
       ∧ (∃ dfinal dfirst : MachineDescriptor X,
            proper_descriptor X dfirst bs
            ∧ existing_descriptor X dfinal
                (finite_trace_last bs btr)
              ∧ equivocator_vlsm_trace_project X btr
                  dfinal = Some (tr, dfirst)))
  apply elem_of_list_split in Hitem.message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
btr: list transition_item
Hbtr: finite_constrained_trace_from_to
  (equivocator_vlsm X) bs bf btr
bitem: transition_item
Hitem: ∃ l1 l2 : list transition_item,
  btr = l1 ++ bitem :: l2
idl: nat
Hlbitem: equivocator_label_descriptor (l bitem) =
Existing idl
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X btr
    (Existing j) = Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
∃ item : transition_item,
  (∃ d : MachineDescriptor X,
     equivocator_vlsm_transition_item_project X bitem
       d = Some (Some item, Existing idl))
  ∧ (∃ tr : list transition_item,
       item ∈ tr
       ∧ (∃ dfinal dfirst : MachineDescriptor X,
            proper_descriptor X dfirst bs
            ∧ existing_descriptor X dfinal
                (finite_trace_last bs btr)
              ∧ equivocator_vlsm_trace_project X btr
                  dfinal = Some (tr, dfirst)))
  destruct Hitem as [bprefix [bsuffix Heq]].message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
btr: list transition_item
Hbtr: finite_constrained_trace_from_to
  (equivocator_vlsm X) bs bf btr
bitem: transition_item
bprefix, bsuffix: list transition_item
Heq: btr = bprefix ++ bitem :: bsuffix
idl: nat
Hlbitem: equivocator_label_descriptor (l bitem) =
Existing idl
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X btr
    (Existing j) = Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
∃ item : transition_item,
  (∃ d : MachineDescriptor X,
     equivocator_vlsm_transition_item_project X bitem
       d = Some (Some item, Existing idl))
  ∧ (∃ tr : list transition_item,
       item ∈ tr
       ∧ (∃ dfinal dfirst : MachineDescriptor X,
            proper_descriptor X dfirst bs
            ∧ existing_descriptor X dfinal
                (finite_trace_last bs btr)
              ∧ equivocator_vlsm_trace_project X btr
                  dfinal = Some (tr, dfirst)))
  subst btr.message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bitem: transition_item
bprefix, bsuffix: list transition_item
Hbtr: finite_constrained_trace_from_to
  (equivocator_vlsm X) bs bf
  (bprefix ++ bitem :: bsuffix)
idl: nat
Hlbitem: equivocator_label_descriptor (l bitem) =
Existing idl
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X
    (bprefix ++ bitem :: bsuffix) 
    (Existing j) = Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
∃ item : transition_item,
  (∃ d : MachineDescriptor X,
     equivocator_vlsm_transition_item_project X bitem
       d = Some (Some item, Existing idl))
  ∧ (∃ tr : list transition_item,
       item ∈ tr
       ∧ (∃ dfinal dfirst : MachineDescriptor X,
            proper_descriptor X dfirst bs
            ∧ existing_descriptor X dfinal
                (finite_trace_last bs
                   (bprefix ++ bitem :: bsuffix))
              ∧ equivocator_vlsm_trace_project X
                  (bprefix ++ bitem :: bsuffix) dfinal =
                Some (tr, dfirst)))
  apply (finite_valid_trace_from_to_app_split (preloaded_with_all_messages_vlsm (equivocator_vlsm X)))
    in Hbtr.message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bitem: transition_item
bprefix, bsuffix: list transition_item
Hbtr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs
  (finite_trace_last bs bprefix) bprefix
∧ finite_valid_trace_from_to
    (preloaded_with_all_messages_vlsm
       (equivocator_vlsm X))
    (finite_trace_last bs bprefix) bf
    (bitem :: bsuffix)
idl: nat
Hlbitem: equivocator_label_descriptor (l bitem) =
Existing idl
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X
    (bprefix ++ bitem :: bsuffix) 
    (Existing j) = Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
∃ item : transition_item,
  (∃ d : MachineDescriptor X,
     equivocator_vlsm_transition_item_project X bitem
       d = Some (Some item, Existing idl))
  ∧ (∃ tr : list transition_item,
       item ∈ tr
       ∧ (∃ dfinal dfirst : MachineDescriptor X,
            proper_descriptor X dfirst bs
            ∧ existing_descriptor X dfinal
                (finite_trace_last bs
                   (bprefix ++ bitem :: bsuffix))
              ∧ equivocator_vlsm_trace_project X
                  (bprefix ++ bitem :: bsuffix) dfinal =
                Some (tr, dfirst)))
  destruct Hbtr as [Hbprefix Hbsuffix].message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bitem: transition_item
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs
  (finite_trace_last bs bprefix) bprefix
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
  (finite_trace_last bs bprefix) bf
  (bitem :: bsuffix)
idl: nat
Hlbitem: equivocator_label_descriptor (l bitem) =
Existing idl
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X
    (bprefix ++ bitem :: bsuffix) 
    (Existing j) = Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
∃ item : transition_item,
  (∃ d : MachineDescriptor X,
     equivocator_vlsm_transition_item_project X bitem
       d = Some (Some item, Existing idl))
  ∧ (∃ tr : list transition_item,
       item ∈ tr
       ∧ (∃ dfinal dfirst : MachineDescriptor X,
            proper_descriptor X dfirst bs
            ∧ existing_descriptor X dfinal
                (finite_trace_last bs
                   (bprefix ++ bitem :: bsuffix))
              ∧ equivocator_vlsm_trace_project X
                  (bprefix ++ bitem :: bsuffix) dfinal =
                Some (tr, dfirst)))
  remember (finite_trace_last _ _) as lst.message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bitem: transition_item
bprefix, bsuffix: list transition_item
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs lst bprefix
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst bf
  (bitem :: bsuffix)
idl: nat
Hlbitem: equivocator_label_descriptor (l bitem) =
Existing idl
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X
    (bprefix ++ bitem :: bsuffix) 
    (Existing j) = Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
∃ item : transition_item,
  (∃ d : MachineDescriptor X,
     equivocator_vlsm_transition_item_project X bitem
       d = Some (Some item, Existing idl))
  ∧ (∃ tr : list transition_item,
       item ∈ tr
       ∧ (∃ dfinal dfirst : MachineDescriptor X,
            proper_descriptor X dfirst bs
            ∧ existing_descriptor X dfinal
                (finite_trace_last bs
                   (bprefix ++ bitem :: bsuffix))
              ∧ equivocator_vlsm_trace_project X
                  (bprefix ++ bitem :: bsuffix) dfinal =
                Some (tr, dfirst)))
  inversion Hbsuffix; subst s' f bitem tl.message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs lst bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X
    (bprefix ++
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |} :: bsuffix) (Existing j) =
  Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
Hlbitem: equivocator_label_descriptor
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) = Existing idl
Htl: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bf bsuffix
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) l (
  lst, iom) (s, oom)
∃ item : transition_item,
  (∃ d : MachineDescriptor X,
     equivocator_vlsm_transition_item_project X
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |} d = Some (Some item, Existing idl))
  ∧ (∃ tr : list transition_item,
       item ∈ tr
       ∧ (∃ dfinal dfirst : MachineDescriptor X,
            proper_descriptor X dfirst bs
            ∧ existing_descriptor X dfinal
                (finite_trace_last bs
                   (bprefix ++
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |} :: bsuffix))
              ∧ equivocator_vlsm_trace_project X
                  (bprefix ++
                   {|
                     l := l;
                     input := iom;
                     destination := s;
                     output := oom
                   |} :: bsuffix) dfinal =
                Some (tr, dfirst)))
  destruct Ht as [[_ [_ Hv]] Ht].message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs lst bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X
    (bprefix ++
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |} :: bsuffix) (Existing j) =
  Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
Hlbitem: equivocator_label_descriptor
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) = Existing idl
Htl: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bf bsuffix
Hv: valid l (lst, iom)
Ht: transition l (lst, iom) = (s, oom)
∃ item : transition_item,
  (∃ d : MachineDescriptor X,
     equivocator_vlsm_transition_item_project X
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |} d = Some (Some item, Existing idl))
  ∧ (∃ tr : list transition_item,
       item ∈ tr
       ∧ (∃ dfinal dfirst : MachineDescriptor X,
            proper_descriptor X dfirst bs
            ∧ existing_descriptor X dfinal
                (finite_trace_last bs
                   (bprefix ++
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |} :: bsuffix))
              ∧ equivocator_vlsm_trace_project X
                  (bprefix ++
                   {|
                     l := l;
                     input := iom;
                     destination := s;
                     output := oom
                   |} :: bsuffix) dfinal =
                Some (tr, dfirst)))
  specialize
    (equivocator_valid_transition_project_inv5 _ l s lst iom oom Hv Ht) as Hpitem.message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs lst bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X
    (bprefix ++
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |} :: bsuffix) (Existing j) =
  Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
Hlbitem: equivocator_label_descriptor
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) = Existing idl
Htl: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bf bsuffix
Hv: valid l (lst, iom)
Ht: transition l (lst, iom) = (s, oom)
Hpitem: ∀ (item :=
   {|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}) (_i : nat),
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X item (Existing i) =
           Some (Some itemx, Existing _i))
∃ item : transition_item,
  (∃ d : MachineDescriptor X,
     equivocator_vlsm_transition_item_project X
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |} d = Some (Some item, Existing idl))
  ∧ (∃ tr : list transition_item,
       item ∈ tr
       ∧ (∃ dfinal dfirst : MachineDescriptor X,
            proper_descriptor X dfirst bs
            ∧ existing_descriptor X dfinal
                (finite_trace_last bs
                   (bprefix ++
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |} :: bsuffix))
              ∧ equivocator_vlsm_trace_project X
                  (bprefix ++
                   {|
                     l := l;
                     input := iom;
                     destination := s;
                     output := oom
                   |} :: bsuffix) dfinal =
                Some (tr, dfirst)))
  unfold VLSM.l in *.message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs lst bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X
    (bprefix ++
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |} :: bsuffix) (Existing j) =
  Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bf bsuffix
Hv: valid l (lst, iom)
Ht: transition l (lst, iom) = (s, oom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
∃ item : transition_item,
  (∃ d : MachineDescriptor X,
     equivocator_vlsm_transition_item_project X
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |} d = Some (Some item, Existing idl))
  ∧ (∃ tr : list transition_item,
       item ∈ tr
       ∧ (∃ dfinal dfirst : MachineDescriptor X,
            proper_descriptor X dfirst bs
            ∧ existing_descriptor X dfinal
                (finite_trace_last bs
                   (bprefix ++
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |} :: bsuffix))
              ∧ equivocator_vlsm_trace_project X
                  (bprefix ++
                   {|
                     l := l;
                     input := iom;
                     destination := s;
                     output := oom
                   |} :: bsuffix) dfinal =
                Some (tr, dfirst)))
  destruct (Hpitem _ Hlbitem) as [i [si [Hi [itemx Hitemx]]]].message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs lst bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X
    (bprefix ++
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |} :: bsuffix) (Existing j) =
  Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bf bsuffix
Hv: valid l (lst, iom)
Ht: transition l (lst, iom) = (s, oom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
∃ item : transition_item,
  (∃ d : MachineDescriptor X,
     equivocator_vlsm_transition_item_project X
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |} d = Some (Some item, Existing idl))
  ∧ (∃ tr : list transition_item,
       item ∈ tr
       ∧ (∃ dfinal dfirst : MachineDescriptor X,
            proper_descriptor X dfirst bs
            ∧ existing_descriptor X dfinal
                (finite_trace_last bs
                   (bprefix ++
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |} :: bsuffix))
              ∧ equivocator_vlsm_trace_project X
                  (bprefix ++
                   {|
                     l := l;
                     input := iom;
                     destination := s;
                     output := oom
                   |} :: bsuffix) dfinal =
                Some (tr, dfirst)))

  apply valid_trace_forget_last in Htl.message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs lst bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X
    (bprefix ++
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |} :: bsuffix) (Existing j) =
  Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Hv: valid l (lst, iom)
Ht: transition l (lst, iom) = (s, oom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
∃ item : transition_item,
  (∃ d : MachineDescriptor X,
     equivocator_vlsm_transition_item_project X
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |} d = Some (Some item, Existing idl))
  ∧ (∃ tr : list transition_item,
       item ∈ tr
       ∧ (∃ dfinal dfirst : MachineDescriptor X,
            proper_descriptor X dfirst bs
            ∧ existing_descriptor X dfinal
                (finite_trace_last bs
                   (bprefix ++
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |} :: bsuffix))
              ∧ equivocator_vlsm_trace_project X
                  (bprefix ++
                   {|
                     l := l;
                     input := iom;
                     destination := s;
                     output := oom
                   |} :: bsuffix) dfinal =
                Some (tr, dfirst)))
  destruct
    (preloaded_equivocator_vlsm_trace_project_valid_inv _ _ _ Htl _ _ Hi)
    as [suffix Hsuffix].message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs lst bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X
    (bprefix ++
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |} :: bsuffix) (Existing j) =
  Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Hv: valid l (lst, iom)
Ht: transition l (lst, iom) = (s, oom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
∃ item : transition_item,
  (∃ d : MachineDescriptor X,
     equivocator_vlsm_transition_item_project X
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |} d = Some (Some item, Existing idl))
  ∧ (∃ tr : list transition_item,
       item ∈ tr
       ∧ (∃ dfinal dfirst : MachineDescriptor X,
            proper_descriptor X dfirst bs
            ∧ existing_descriptor X dfinal
                (finite_trace_last bs
                   (bprefix ++
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |} :: bsuffix))
              ∧ equivocator_vlsm_trace_project X
                  (bprefix ++
                   {|
                     l := l;
                     input := iom;
                     destination := s;
                     output := oom
                   |} :: bsuffix) dfinal =
                Some (tr, dfirst)))
  exists itemx.message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs lst bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X
    (bprefix ++
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |} :: bsuffix) (Existing j) =
  Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Hv: valid l (lst, iom)
Ht: transition l (lst, iom) = (s, oom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
(∃ d : MachineDescriptor X,
   equivocator_vlsm_transition_item_project X
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |} d = Some (Some itemx, Existing idl))
∧ (∃ tr : list transition_item,
     itemx ∈ tr
     ∧ (∃ dfinal dfirst : MachineDescriptor X,
          proper_descriptor X dfirst bs
          ∧ existing_descriptor X dfinal
              (finite_trace_last bs
                 (bprefix ++
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |} :: bsuffix))
            ∧ equivocator_vlsm_trace_project X
                (bprefix ++
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |} :: bsuffix) dfinal =
              Some (tr, dfirst))) split; [eexists; apply Hitemx |].message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs lst bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X
    (bprefix ++
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |} :: bsuffix) (Existing j) =
  Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Hv: valid l (lst, iom)
Ht: transition l (lst, iom) = (s, oom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
∃ tr : list transition_item,
  itemx ∈ tr
  ∧ (∃ dfinal dfirst : MachineDescriptor X,
       proper_descriptor X dfirst bs
       ∧ existing_descriptor X dfinal
           (finite_trace_last bs
              (bprefix ++
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |} :: bsuffix))
         ∧ equivocator_vlsm_trace_project X
             (bprefix ++
              {|
                l := l;
                input := iom;
                destination := s;
                output := oom
              |} :: bsuffix) dfinal =
           Some (tr, dfirst))
  remember (Existing i) as dsuffix.message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs lst bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X
    (bprefix ++
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |} :: bsuffix) (Existing j) =
  Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Hv: valid l (lst, iom)
Ht: transition l (lst, iom) = (s, oom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
dsuffix: MachineDescriptor X
Heqdsuffix: dsuffix = Existing i
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} dsuffix =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  dsuffix = Some (suffix, dsuffix)
∃ tr : list transition_item,
  itemx ∈ tr
  ∧ (∃ dfinal dfirst : MachineDescriptor X,
       proper_descriptor X dfirst bs
       ∧ existing_descriptor X dfinal
           (finite_trace_last bs
              (bprefix ++
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |} :: bsuffix))
         ∧ equivocator_vlsm_trace_project X
             (bprefix ++
              {|
                l := l;
                input := iom;
                destination := s;
                output := oom
              |} :: bsuffix) dfinal =
           Some (tr, dfirst))
  remember {| input := iom |} as bitem.message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs lst bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
bitem: transition_item
Heqbitem: bitem =
{|
  l := l;
  input := iom;
  destination := s;
  output := oom
|}
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst bf
  (bitem :: bsuffix)
idl: nat
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X
    (bprefix ++ bitem :: bsuffix) 
    (Existing j) = Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Hv: valid l (lst, iom)
Ht: transition l (lst, iom) = (s, oom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X bitem (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
dsuffix: MachineDescriptor X
Heqdsuffix: dsuffix = Existing i
Hitemx: equivocator_vlsm_transition_item_project X
  bitem dsuffix =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  dsuffix = Some (suffix, dsuffix)
∃ tr : list transition_item,
  itemx ∈ tr
  ∧ (∃ dfinal dfirst : MachineDescriptor X,
       proper_descriptor X dfirst bs
       ∧ existing_descriptor X dfinal
           (finite_trace_last bs
              (bprefix ++ bitem :: bsuffix))
         ∧ equivocator_vlsm_trace_project X
             (bprefix ++ bitem :: bsuffix) dfinal =
           Some (tr, dfirst))
  specialize (equivocator_vlsm_trace_project_app_inv
    _ [bitem] bsuffix (Existing i) (Existing idl) dsuffix [itemx] suffix)
    as Hsuffix'.message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs lst bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
bitem: transition_item
Heqbitem: bitem =
{|
  l := l;
  input := iom;
  destination := s;
  output := oom
|}
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst bf
  (bitem :: bsuffix)
idl: nat
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X
    (bprefix ++ bitem :: bsuffix) 
    (Existing j) = Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Hv: valid l (lst, iom)
Ht: transition l (lst, iom) = (s, oom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X bitem (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
dsuffix: MachineDescriptor X
Heqdsuffix: dsuffix = Existing i
Hitemx: equivocator_vlsm_transition_item_project X
  bitem dsuffix =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  dsuffix = Some (suffix, dsuffix)
Hsuffix': equivocator_vlsm_trace_project X [bitem]
  dsuffix = Some ([itemx], Existing idl)
→ equivocator_vlsm_trace_project X bsuffix
    (Existing i) = Some (suffix, dsuffix)
  → equivocator_vlsm_trace_project X
      ([bitem] ++ bsuffix) 
      (Existing i) =
    Some ([itemx] ++ suffix, Existing idl)
∃ tr : list transition_item,
  itemx ∈ tr
  ∧ (∃ dfinal dfirst : MachineDescriptor X,
       proper_descriptor X dfirst bs
       ∧ existing_descriptor X dfinal
           (finite_trace_last bs
              (bprefix ++ bitem :: bsuffix))
         ∧ equivocator_vlsm_trace_project X
             (bprefix ++ bitem :: bsuffix) dfinal =
           Some (tr, dfirst))
  spec Hsuffix'; [by cbn; rewrite Hitemx |].message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs lst bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
bitem: transition_item
Heqbitem: bitem =
{|
  l := l;
  input := iom;
  destination := s;
  output := oom
|}
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst bf
  (bitem :: bsuffix)
idl: nat
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X
    (bprefix ++ bitem :: bsuffix) 
    (Existing j) = Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Hv: valid l (lst, iom)
Ht: transition l (lst, iom) = (s, oom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X bitem (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
dsuffix: MachineDescriptor X
Heqdsuffix: dsuffix = Existing i
Hitemx: equivocator_vlsm_transition_item_project X
  bitem dsuffix =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  dsuffix = Some (suffix, dsuffix)
Hsuffix': equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, dsuffix)
→ equivocator_vlsm_trace_project X
    ([bitem] ++ bsuffix) 
    (Existing i) =
  Some ([itemx] ++ suffix, Existing idl)
∃ tr : list transition_item,
  itemx ∈ tr
  ∧ (∃ dfinal dfirst : MachineDescriptor X,
       proper_descriptor X dfirst bs
       ∧ existing_descriptor X dfinal
           (finite_trace_last bs
              (bprefix ++ bitem :: bsuffix))
         ∧ equivocator_vlsm_trace_project X
             (bprefix ++ bitem :: bsuffix) dfinal =
           Some (tr, dfirst))
  subst dsuffix.message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs lst bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
bitem: transition_item
Heqbitem: bitem =
{|
  l := l;
  input := iom;
  destination := s;
  output := oom
|}
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst bf
  (bitem :: bsuffix)
idl: nat
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X
    (bprefix ++ bitem :: bsuffix) 
    (Existing j) = Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Hv: valid l (lst, iom)
Ht: transition l (lst, iom) = (s, oom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X bitem (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  bitem (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
→ equivocator_vlsm_trace_project X
    ([bitem] ++ bsuffix) 
    (Existing i) =
  Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
∃ tr : list transition_item,
  itemx ∈ tr
  ∧ (∃ dfinal dfirst : MachineDescriptor X,
       proper_descriptor X dfirst bs
       ∧ existing_descriptor X dfinal
           (finite_trace_last bs
              (bprefix ++ bitem :: bsuffix))
         ∧ equivocator_vlsm_trace_project X
             (bprefix ++ bitem :: bsuffix) dfinal =
           Some (tr, dfirst))
  specialize (Hsuffix' Hsuffix).message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs lst bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
bitem: transition_item
Heqbitem: bitem =
{|
  l := l;
  input := iom;
  destination := s;
  output := oom
|}
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst bf
  (bitem :: bsuffix)
idl: nat
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X
    (bprefix ++ bitem :: bsuffix) 
    (Existing j) = Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Hv: valid l (lst, iom)
Ht: transition l (lst, iom) = (s, oom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X bitem (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  bitem (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X
  ([bitem] ++ bsuffix) 
  (Existing i) =
Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
∃ tr : list transition_item,
  itemx ∈ tr
  ∧ (∃ dfinal dfirst : MachineDescriptor X,
       proper_descriptor X dfirst bs
       ∧ existing_descriptor X dfinal
           (finite_trace_last bs
              (bprefix ++ bitem :: bsuffix))
         ∧ equivocator_vlsm_trace_project X
             (bprefix ++ bitem :: bsuffix) dfinal =
           Some (tr, dfirst))
  subst bitem.message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs lst bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X
    (bprefix ++
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |} :: bsuffix) (Existing j) =
  Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Hv: valid l (lst, iom)
Ht: transition l (lst, iom) = (s, oom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X
  ([{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}] ++ bsuffix) (Existing i) =
Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
∃ tr : list transition_item,
  itemx ∈ tr
  ∧ (∃ dfinal dfirst : MachineDescriptor X,
       proper_descriptor X dfirst bs
       ∧ existing_descriptor X dfinal
           (finite_trace_last bs
              (bprefix ++
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |} :: bsuffix))
         ∧ equivocator_vlsm_trace_project X
             (bprefix ++
              {|
                l := l;
                input := iom;
                destination := s;
                output := oom
              |} :: bsuffix) dfinal =
           Some (tr, dfirst))
  destruct
    (equivocator_valid_transition_project_inv2 _ l lst s iom oom Hv Ht _ _ _ Hitemx)
    as [_i [_Hdsuffix [s_i [_Hi Hd']]]].message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs lst bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X
    (bprefix ++
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |} :: bsuffix) (Existing j) =
  Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Hv: valid l (lst, iom)
Ht: transition l (lst, iom) = (s, oom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X
  ([{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}] ++ bsuffix) (Existing i) =
Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
_i: nat
_Hdsuffix: Existing i = Existing _i
s_i: state X
_Hi: equivocator_state_project s _i = Some s_i
Hd': ∃ i' : nat,
  Existing idl = Existing i'
  ∧ (∃ s'x : state X,
       equivocator_state_project lst i' =
       Some s'x
       ∧ (∃ Hex : existing_equivocator_label X l,
            let lx :=
              existing_equivocator_label_extract
                X l Hex in
            itemx =
            {|
              l := lx;
              input := iom;
              destination := s_i;
              output := oom
            |}
            ∧ valid lx (s'x, iom)
              ∧ transition lx (s'x, iom) =
                (s_i, oom)))
∃ tr : list transition_item,
  itemx ∈ tr
  ∧ (∃ dfinal dfirst : MachineDescriptor X,
       proper_descriptor X dfirst bs
       ∧ existing_descriptor X dfinal
           (finite_trace_last bs
              (bprefix ++
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |} :: bsuffix))
         ∧ equivocator_vlsm_trace_project X
             (bprefix ++
              {|
                l := l;
                input := iom;
                destination := s;
                output := oom
              |} :: bsuffix) dfinal =
           Some (tr, dfirst))
  inversion _Hdsuffix.message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs lst bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X
    (bprefix ++
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |} :: bsuffix) (Existing j) =
  Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Hv: valid l (lst, iom)
Ht: transition l (lst, iom) = (s, oom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X
  ([{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}] ++ bsuffix) (Existing i) =
Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
_i: nat
_Hdsuffix: Existing i = Existing _i
s_i: state X
_Hi: equivocator_state_project s _i = Some s_i
Hd': ∃ i' : nat,
  Existing idl = Existing i'
  ∧ (∃ s'x : state X,
       equivocator_state_project lst i' =
       Some s'x
       ∧ (∃ Hex : existing_equivocator_label X l,
            let lx :=
              existing_equivocator_label_extract
                X l Hex in
            itemx =
            {|
              l := lx;
              input := iom;
              destination := s_i;
              output := oom
            |}
            ∧ valid lx (s'x, iom)
              ∧ transition lx (s'x, iom) =
                (s_i, oom)))
H0: i = _i
∃ tr : list transition_item,
  itemx ∈ tr
  ∧ (∃ dfinal dfirst : MachineDescriptor X,
       proper_descriptor X dfirst bs
       ∧ existing_descriptor X dfinal
           (finite_trace_last bs
              (bprefix ++
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |} :: bsuffix))
         ∧ equivocator_vlsm_trace_project X
             (bprefix ++
              {|
                l := l;
                input := iom;
                destination := s;
                output := oom
              |} :: bsuffix) dfinal =
           Some (tr, dfirst)) subst _i.message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs lst bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X
    (bprefix ++
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |} :: bsuffix) (Existing j) =
  Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Hv: valid l (lst, iom)
Ht: transition l (lst, iom) = (s, oom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X
  ([{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}] ++ bsuffix) (Existing i) =
Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
_Hdsuffix: Existing i = Existing i
s_i: state X
_Hi: equivocator_state_project s i = Some s_i
Hd': ∃ i' : nat,
  Existing idl = Existing i'
  ∧ (∃ s'x : state X,
       equivocator_state_project lst i' =
       Some s'x
       ∧ (∃ Hex : existing_equivocator_label X l,
            let lx :=
              existing_equivocator_label_extract
                X l Hex in
            itemx =
            {|
              l := lx;
              input := iom;
              destination := s_i;
              output := oom
            |}
            ∧ valid lx (s'x, iom)
              ∧ transition lx (s'x, iom) =
                (s_i, oom)))
∃ tr : list transition_item,
  itemx ∈ tr
  ∧ (∃ dfinal dfirst : MachineDescriptor X,
       proper_descriptor X dfirst bs
       ∧ existing_descriptor X dfinal
           (finite_trace_last bs
              (bprefix ++
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |} :: bsuffix))
         ∧ equivocator_vlsm_trace_project X
             (bprefix ++
              {|
                l := l;
                input := iom;
                destination := s;
                output := oom
              |} :: bsuffix) dfinal =
           Some (tr, dfirst)) clear _Hdsuffix.message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs lst bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X
    (bprefix ++
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |} :: bsuffix) (Existing j) =
  Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Hv: valid l (lst, iom)
Ht: transition l (lst, iom) = (s, oom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X
  ([{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}] ++ bsuffix) (Existing i) =
Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
s_i: state X
_Hi: equivocator_state_project s i = Some s_i
Hd': ∃ i' : nat,
  Existing idl = Existing i'
  ∧ (∃ s'x : state X,
       equivocator_state_project lst i' =
       Some s'x
       ∧ (∃ Hex : existing_equivocator_label X l,
            let lx :=
              existing_equivocator_label_extract
                X l Hex in
            itemx =
            {|
              l := lx;
              input := iom;
              destination := s_i;
              output := oom
            |}
            ∧ valid lx (s'x, iom)
              ∧ transition lx (s'x, iom) =
                (s_i, oom)))
∃ tr : list transition_item,
  itemx ∈ tr
  ∧ (∃ dfinal dfirst : MachineDescriptor X,
       proper_descriptor X dfirst bs
       ∧ existing_descriptor X dfinal
           (finite_trace_last bs
              (bprefix ++
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |} :: bsuffix))
         ∧ equivocator_vlsm_trace_project X
             (bprefix ++
              {|
                l := l;
                input := iom;
                destination := s;
                output := oom
              |} :: bsuffix) dfinal =
           Some (tr, dfirst))
  destruct Hd' as [_i' [_Heq [lst_i' [id [Hex [_Hitemx [Hvs' Hts']]]]]]].message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs lst bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X
    (bprefix ++
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |} :: bsuffix) (Existing j) =
  Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Hv: valid l (lst, iom)
Ht: transition l (lst, iom) = (s, oom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X
  ([{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}] ++ bsuffix) (Existing i) =
Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
s_i: state X
_Hi: equivocator_state_project s i = Some s_i
_i': nat
_Heq: Existing idl = Existing _i'
lst_i': state X
id: equivocator_state_project lst _i' = Some lst_i'
Hex: existing_equivocator_label X l
_Hitemx: itemx =
{|
  l :=
    existing_equivocator_label_extract X l
      Hex;
  input := iom;
  destination := s_i;
  output := oom
|}
Hvs': valid
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom)
Hts': transition
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom) = (s_i, oom)
∃ tr : list transition_item,
  itemx ∈ tr
  ∧ (∃ dfinal dfirst : MachineDescriptor X,
       proper_descriptor X dfirst bs
       ∧ existing_descriptor X dfinal
           (finite_trace_last bs
              (bprefix ++
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |} :: bsuffix))
         ∧ equivocator_vlsm_trace_project X
             (bprefix ++
              {|
                l := l;
                input := iom;
                destination := s;
                output := oom
              |} :: bsuffix) dfinal =
           Some (tr, dfirst))
  inversion _Heq.message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs lst bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X
    (bprefix ++
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |} :: bsuffix) (Existing j) =
  Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Hv: valid l (lst, iom)
Ht: transition l (lst, iom) = (s, oom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X
  ([{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}] ++ bsuffix) (Existing i) =
Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
s_i: state X
_Hi: equivocator_state_project s i = Some s_i
_i': nat
_Heq: Existing idl = Existing _i'
lst_i': state X
id: equivocator_state_project lst _i' = Some lst_i'
Hex: existing_equivocator_label X l
_Hitemx: itemx =
{|
  l :=
    existing_equivocator_label_extract X l
      Hex;
  input := iom;
  destination := s_i;
  output := oom
|}
Hvs': valid
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom)
Hts': transition
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom) = (s_i, oom)
H0: idl = _i'
∃ tr : list transition_item,
  itemx ∈ tr
  ∧ (∃ dfinal dfirst : MachineDescriptor X,
       proper_descriptor X dfirst bs
       ∧ existing_descriptor X dfinal
           (finite_trace_last bs
              (bprefix ++
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |} :: bsuffix))
         ∧ equivocator_vlsm_trace_project X
             (bprefix ++
              {|
                l := l;
                input := iom;
                destination := s;
                output := oom
              |} :: bsuffix) dfinal =
           Some (tr, dfirst)) subst _i'.message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs lst bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X
    (bprefix ++
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |} :: bsuffix) (Existing j) =
  Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Hv: valid l (lst, iom)
Ht: transition l (lst, iom) = (s, oom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X
  ([{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}] ++ bsuffix) (Existing i) =
Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
s_i: state X
_Hi: equivocator_state_project s i = Some s_i
_Heq: Existing idl = Existing idl
lst_i': state X
id: equivocator_state_project lst idl = Some lst_i'
Hex: existing_equivocator_label X l
_Hitemx: itemx =
{|
  l :=
    existing_equivocator_label_extract X l
      Hex;
  input := iom;
  destination := s_i;
  output := oom
|}
Hvs': valid
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom)
Hts': transition
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom) = (s_i, oom)
∃ tr : list transition_item,
  itemx ∈ tr
  ∧ (∃ dfinal dfirst : MachineDescriptor X,
       proper_descriptor X dfirst bs
       ∧ existing_descriptor X dfinal
           (finite_trace_last bs
              (bprefix ++
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |} :: bsuffix))
         ∧ equivocator_vlsm_trace_project X
             (bprefix ++
              {|
                l := l;
                input := iom;
                destination := s;
                output := oom
              |} :: bsuffix) dfinal =
           Some (tr, dfirst)) clear _Heq.message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
lst: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Heqlst: lst = finite_trace_last bs bprefix
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs lst bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) lst bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X
    (bprefix ++
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |} :: bsuffix) (Existing j) =
  Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Hv: valid l (lst, iom)
Ht: transition l (lst, iom) = (s, oom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X
  ([{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}] ++ bsuffix) (Existing i) =
Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
s_i: state X
_Hi: equivocator_state_project s i = Some s_i
lst_i': state X
id: equivocator_state_project lst idl = Some lst_i'
Hex: existing_equivocator_label X l
_Hitemx: itemx =
{|
  l :=
    existing_equivocator_label_extract X l
      Hex;
  input := iom;
  destination := s_i;
  output := oom
|}
Hvs': valid
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom)
Hts': transition
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom) = (s_i, oom)
∃ tr : list transition_item,
  itemx ∈ tr
  ∧ (∃ dfinal dfirst : MachineDescriptor X,
       proper_descriptor X dfirst bs
       ∧ existing_descriptor X dfinal
           (finite_trace_last bs
              (bprefix ++
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |} :: bsuffix))
         ∧ equivocator_vlsm_trace_project X
             (bprefix ++
              {|
                l := l;
                input := iom;
                destination := s;
                output := oom
              |} :: bsuffix) dfinal =
           Some (tr, dfirst))
  subst lst.message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs
  (finite_trace_last bs bprefix) bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
  (finite_trace_last bs bprefix) bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X
    (bprefix ++
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |} :: bsuffix) (Existing j) =
  Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Ht: transition l (finite_trace_last bs bprefix, iom) =
(s, oom)
Hv: valid l (finite_trace_last bs bprefix, iom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X
  ([{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}] ++ bsuffix) (Existing i) =
Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
s_i: state X
_Hi: equivocator_state_project s i = Some s_i
lst_i': state X
id: equivocator_state_project
  (finite_trace_last bs bprefix) idl =
Some lst_i'
Hex: existing_equivocator_label X l
_Hitemx: itemx =
{|
  l :=
    existing_equivocator_label_extract X l
      Hex;
  input := iom;
  destination := s_i;
  output := oom
|}
Hvs': valid
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom)
Hts': transition
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom) = (s_i, oom)
∃ tr : list transition_item,
  itemx ∈ tr
  ∧ (∃ dfinal dfirst : MachineDescriptor X,
       proper_descriptor X dfirst bs
       ∧ existing_descriptor X dfinal
           (finite_trace_last bs
              (bprefix ++
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |} :: bsuffix))
         ∧ equivocator_vlsm_trace_project X
             (bprefix ++
              {|
                l := l;
                input := iom;
                destination := s;
                output := oom
              |} :: bsuffix) dfinal =
           Some (tr, dfirst))
  destruct
    (preloaded_equivocator_vlsm_trace_project_valid _ _ _ _ Hbprefix idl _ id)
    as [prefix [dfirst [Hprefix _]]].message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs
  (finite_trace_last bs bprefix) bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
  (finite_trace_last bs bprefix) bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X
    (bprefix ++
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |} :: bsuffix) (Existing j) =
  Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Ht: transition l (finite_trace_last bs bprefix, iom) =
(s, oom)
Hv: valid l (finite_trace_last bs bprefix, iom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X
  ([{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}] ++ bsuffix) (Existing i) =
Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
s_i: state X
_Hi: equivocator_state_project s i = Some s_i
lst_i': state X
id: equivocator_state_project
  (finite_trace_last bs bprefix) idl =
Some lst_i'
Hex: existing_equivocator_label X l
_Hitemx: itemx =
{|
  l :=
    existing_equivocator_label_extract X l
      Hex;
  input := iom;
  destination := s_i;
  output := oom
|}
Hvs': valid
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom)
Hts': transition
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom) = (s_i, oom)
prefix: list transition_item
dfirst: MachineDescriptor X
Hprefix: equivocator_vlsm_trace_project X bprefix
  (Existing idl) = Some (prefix, dfirst)
∃ tr : list transition_item,
  itemx ∈ tr
  ∧ (∃ dfinal dfirst : MachineDescriptor X,
       proper_descriptor X dfirst bs
       ∧ existing_descriptor X dfinal
           (finite_trace_last bs
              (bprefix ++
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |} :: bsuffix))
         ∧ equivocator_vlsm_trace_project X
             (bprefix ++
              {|
                l := l;
                input := iom;
                destination := s;
                output := oom
              |} :: bsuffix) dfinal =
           Some (tr, dfirst))
  specialize
    (equivocator_vlsm_trace_project_app_inv _ _ _ _ _ _ _ _ Hprefix Hsuffix')
    as Htr.message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs
  (finite_trace_last bs bprefix) bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
  (finite_trace_last bs bprefix) bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X
    (bprefix ++
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |} :: bsuffix) (Existing j) =
  Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Ht: transition l (finite_trace_last bs bprefix, iom) =
(s, oom)
Hv: valid l (finite_trace_last bs bprefix, iom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X
  ([{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}] ++ bsuffix) (Existing i) =
Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
s_i: state X
_Hi: equivocator_state_project s i = Some s_i
lst_i': state X
id: equivocator_state_project
  (finite_trace_last bs bprefix) idl =
Some lst_i'
Hex: existing_equivocator_label X l
_Hitemx: itemx =
{|
  l :=
    existing_equivocator_label_extract X l
      Hex;
  input := iom;
  destination := s_i;
  output := oom
|}
Hvs': valid
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom)
Hts': transition
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom) = (s_i, oom)
prefix: list transition_item
dfirst: MachineDescriptor X
Hprefix: equivocator_vlsm_trace_project X bprefix
  (Existing idl) = Some (prefix, dfirst)
Htr: equivocator_vlsm_trace_project X
  (bprefix ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}] ++ bsuffix) (Existing i) =
Some (prefix ++ [itemx] ++ suffix, dfirst)
∃ tr : list transition_item,
  itemx ∈ tr
  ∧ (∃ dfinal dfirst : MachineDescriptor X,
       proper_descriptor X dfirst bs
       ∧ existing_descriptor X dfinal
           (finite_trace_last bs
              (bprefix ++
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |} :: bsuffix))
         ∧ equivocator_vlsm_trace_project X
             (bprefix ++
              {|
                l := l;
                input := iom;
                destination := s;
                output := oom
              |} :: bsuffix) dfinal =
           Some (tr, dfirst))
  simpl in Htr.message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs
  (finite_trace_last bs bprefix) bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
  (finite_trace_last bs bprefix) bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X
    (bprefix ++
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |} :: bsuffix) (Existing j) =
  Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Ht: transition l (finite_trace_last bs bprefix, iom) =
(s, oom)
Hv: valid l (finite_trace_last bs bprefix, iom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X
  ([{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}] ++ bsuffix) (Existing i) =
Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
s_i: state X
_Hi: equivocator_state_project s i = Some s_i
lst_i': state X
id: equivocator_state_project
  (finite_trace_last bs bprefix) idl =
Some lst_i'
Hex: existing_equivocator_label X l
_Hitemx: itemx =
{|
  l :=
    existing_equivocator_label_extract X l
      Hex;
  input := iom;
  destination := s_i;
  output := oom
|}
Hvs': valid
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom)
Hts': transition
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom) = (s_i, oom)
prefix: list transition_item
dfirst: MachineDescriptor X
Hprefix: equivocator_vlsm_trace_project X bprefix
  (Existing idl) = Some (prefix, dfirst)
Htr: equivocator_vlsm_trace_project X
  (bprefix ++
   {|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix) (Existing i) =
Some (prefix ++ itemx :: suffix, dfirst)
∃ tr : list transition_item,
  itemx ∈ tr
  ∧ (∃ dfinal dfirst : MachineDescriptor X,
       proper_descriptor X dfirst bs
       ∧ existing_descriptor X dfinal
           (finite_trace_last bs
              (bprefix ++
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |} :: bsuffix))
         ∧ equivocator_vlsm_trace_project X
             (bprefix ++
              {|
                l := l;
                input := iom;
                destination := s;
                output := oom
              |} :: bsuffix) dfinal =
           Some (tr, dfirst))
  exists (prefix ++ itemx :: suffix).message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs
  (finite_trace_last bs bprefix) bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
  (finite_trace_last bs bprefix) bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hinv2: ∀ (j : nat) (di : MachineDescriptor X) 
  (trX : list transition_item),
  equivocator_vlsm_trace_project X
    (bprefix ++
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |} :: bsuffix) (Existing j) =
  Some (trX, di)
  → ∃ fsj : state X,
      equivocator_state_project bf j =
      Some fsj
      ∧ match di with
        | NewMachine sn =>
            finite_constrained_trace_init_to X
              sn fsj trX
        | Existing i =>
            ∃ isi : state X,
              equivocator_state_project bs i =
              Some isi
              ∧ finite_constrained_trace_from_to
                  X isi fsj trX
                ∧ (initial_state_prop bs
                   → initial_state_prop isi)
        end
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Ht: transition l (finite_trace_last bs bprefix, iom) =
(s, oom)
Hv: valid l (finite_trace_last bs bprefix, iom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X
  ([{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}] ++ bsuffix) (Existing i) =
Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
s_i: state X
_Hi: equivocator_state_project s i = Some s_i
lst_i': state X
id: equivocator_state_project
  (finite_trace_last bs bprefix) idl =
Some lst_i'
Hex: existing_equivocator_label X l
_Hitemx: itemx =
{|
  l :=
    existing_equivocator_label_extract X l
      Hex;
  input := iom;
  destination := s_i;
  output := oom
|}
Hvs': valid
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom)
Hts': transition
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom) = (s_i, oom)
prefix: list transition_item
dfirst: MachineDescriptor X
Hprefix: equivocator_vlsm_trace_project X bprefix
  (Existing idl) = Some (prefix, dfirst)
Htr: equivocator_vlsm_trace_project X
  (bprefix ++
   {|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix) (Existing i) =
Some (prefix ++ itemx :: suffix, dfirst)
itemx ∈ prefix ++ itemx :: suffix
∧ (∃ dfinal dfirst : MachineDescriptor X,
     proper_descriptor X dfirst bs
     ∧ existing_descriptor X dfinal
         (finite_trace_last bs
            (bprefix ++
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} :: bsuffix))
       ∧ equivocator_vlsm_trace_project X
           (bprefix ++
            {|
              l := l;
              input := iom;
              destination := s;
              output := oom
            |} :: bsuffix) dfinal =
         Some (prefix ++ itemx :: suffix, dfirst))
  specialize (Hinv2 _ _ _ Htr) as [bfi [Hdfinal Hdinitial]].message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs
  (finite_trace_last bs bprefix) bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
  (finite_trace_last bs bprefix) bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Ht: transition l (finite_trace_last bs bprefix, iom) =
(s, oom)
Hv: valid l (finite_trace_last bs bprefix, iom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X
  ([{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}] ++ bsuffix) (Existing i) =
Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
s_i: state X
_Hi: equivocator_state_project s i = Some s_i
lst_i': state X
id: equivocator_state_project
  (finite_trace_last bs bprefix) idl =
Some lst_i'
Hex: existing_equivocator_label X l
_Hitemx: itemx =
{|
  l :=
    existing_equivocator_label_extract X l
      Hex;
  input := iom;
  destination := s_i;
  output := oom
|}
Hvs': valid
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom)
Hts': transition
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom) = (s_i, oom)
prefix: list transition_item
dfirst: MachineDescriptor X
Hprefix: equivocator_vlsm_trace_project X bprefix
  (Existing idl) = Some (prefix, dfirst)
Htr: equivocator_vlsm_trace_project X
  (bprefix ++
   {|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix) (Existing i) =
Some (prefix ++ itemx :: suffix, dfirst)
bfi: state X
Hdfinal: equivocator_state_project bf i = Some bfi
Hdinitial: match dfirst with
| NewMachine sn =>
    finite_constrained_trace_init_to X sn
      bfi (prefix ++ itemx :: suffix)
| Existing i =>
    ∃ isi : state X,
      equivocator_state_project bs i =
      Some isi
      ∧ finite_constrained_trace_from_to X
          isi bfi
          (prefix ++ itemx :: suffix)
        ∧ (initial_state_prop bs
           → initial_state_prop isi)
end
itemx ∈ prefix ++ itemx :: suffix
∧ (∃ dfinal dfirst : MachineDescriptor X,
     proper_descriptor X dfirst bs
     ∧ existing_descriptor X dfinal
         (finite_trace_last bs
            (bprefix ++
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} :: bsuffix))
       ∧ equivocator_vlsm_trace_project X
           (bprefix ++
            {|
              l := l;
              input := iom;
              destination := s;
              output := oom
            |} :: bsuffix) dfinal =
         Some (prefix ++ itemx :: suffix, dfirst))
  split.message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs
  (finite_trace_last bs bprefix) bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
  (finite_trace_last bs bprefix) bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Ht: transition l (finite_trace_last bs bprefix, iom) =
(s, oom)
Hv: valid l (finite_trace_last bs bprefix, iom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X
  ([{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}] ++ bsuffix) (Existing i) =
Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
s_i: state X
_Hi: equivocator_state_project s i = Some s_i
lst_i': state X
id: equivocator_state_project
  (finite_trace_last bs bprefix) idl =
Some lst_i'
Hex: existing_equivocator_label X l
_Hitemx: itemx =
{|
  l :=
    existing_equivocator_label_extract X l
      Hex;
  input := iom;
  destination := s_i;
  output := oom
|}
Hvs': valid
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom)
Hts': transition
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom) = (s_i, oom)
prefix: list transition_item
dfirst: MachineDescriptor X
Hprefix: equivocator_vlsm_trace_project X bprefix
  (Existing idl) = Some (prefix, dfirst)
Htr: equivocator_vlsm_trace_project X
  (bprefix ++
   {|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix) (Existing i) =
Some (prefix ++ itemx :: suffix, dfirst)
bfi: state X
Hdfinal: equivocator_state_project bf i = Some bfi
Hdinitial: match dfirst with
| NewMachine sn =>
    finite_constrained_trace_init_to X sn
      bfi (prefix ++ itemx :: suffix)
| Existing i =>
    ∃ isi : state X,
      equivocator_state_project bs i =
      Some isi
      ∧ finite_constrained_trace_from_to X
          isi bfi
          (prefix ++ itemx :: suffix)
        ∧ (initial_state_prop bs
           → initial_state_prop isi)
end
itemx ∈ prefix ++ itemx :: suffix
message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs
  (finite_trace_last bs bprefix) bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
  (finite_trace_last bs bprefix) bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Ht: transition l (finite_trace_last bs bprefix, iom) =
(s, oom)
Hv: valid l (finite_trace_last bs bprefix, iom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X
  ([{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}] ++ bsuffix) (Existing i) =
Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
s_i: state X
_Hi: equivocator_state_project s i = Some s_i
lst_i': state X
id: equivocator_state_project
  (finite_trace_last bs bprefix) idl =
Some lst_i'
Hex: existing_equivocator_label X l
_Hitemx: itemx =
{|
  l :=
    existing_equivocator_label_extract X l
      Hex;
  input := iom;
  destination := s_i;
  output := oom
|}
Hvs': valid
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom)
Hts': transition
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom) = (s_i, oom)
prefix: list transition_item
dfirst: MachineDescriptor X
Hprefix: equivocator_vlsm_trace_project X bprefix
  (Existing idl) = Some (prefix, dfirst)
Htr: equivocator_vlsm_trace_project X
  (bprefix ++
   {|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix) (Existing i) =
Some (prefix ++ itemx :: suffix, dfirst)
bfi: state X
Hdfinal: equivocator_state_project bf i = Some bfi
Hdinitial: match dfirst with
| NewMachine sn =>
    finite_constrained_trace_init_to X sn
      bfi (prefix ++ itemx :: suffix)
| Existing i =>
    ∃ isi : state X,
      equivocator_state_project bs i =
      Some isi
      ∧ finite_constrained_trace_from_to X
          isi bfi
          (prefix ++ itemx :: suffix)
        ∧ (initial_state_prop bs
           → initial_state_prop isi)
end
∃ dfinal dfirst : MachineDescriptor X,
  proper_descriptor X dfirst bs
  ∧ existing_descriptor X dfinal
      (finite_trace_last bs
         (bprefix ++
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} :: bsuffix))
    ∧ equivocator_vlsm_trace_project X
        (bprefix ++
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |} :: bsuffix) dfinal =
      Some (prefix ++ itemx :: suffix, dfirst)
  -message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs
  (finite_trace_last bs bprefix) bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
  (finite_trace_last bs bprefix) bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Ht: transition l (finite_trace_last bs bprefix, iom) =
(s, oom)
Hv: valid l (finite_trace_last bs bprefix, iom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X
  ([{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}] ++ bsuffix) (Existing i) =
Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
s_i: state X
_Hi: equivocator_state_project s i = Some s_i
lst_i': state X
id: equivocator_state_project
  (finite_trace_last bs bprefix) idl =
Some lst_i'
Hex: existing_equivocator_label X l
_Hitemx: itemx =
{|
  l :=
    existing_equivocator_label_extract X l
      Hex;
  input := iom;
  destination := s_i;
  output := oom
|}
Hvs': valid
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom)
Hts': transition
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom) = (s_i, oom)
prefix: list transition_item
dfirst: MachineDescriptor X
Hprefix: equivocator_vlsm_trace_project X bprefix
  (Existing idl) = Some (prefix, dfirst)
Htr: equivocator_vlsm_trace_project X
  (bprefix ++
   {|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix) (Existing i) =
Some (prefix ++ itemx :: suffix, dfirst)
bfi: state X
Hdfinal: equivocator_state_project bf i = Some bfi
Hdinitial: match dfirst with
| NewMachine sn =>
    finite_constrained_trace_init_to X sn
      bfi (prefix ++ itemx :: suffix)
| Existing i =>
    ∃ isi : state X,
      equivocator_state_project bs i =
      Some isi
      ∧ finite_constrained_trace_from_to X
          isi bfi
          (prefix ++ itemx :: suffix)
        ∧ (initial_state_prop bs
           → initial_state_prop isi)
end
itemx ∈ prefix ++ itemx :: suffix apply elem_of_app.message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs
  (finite_trace_last bs bprefix) bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
  (finite_trace_last bs bprefix) bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Ht: transition l (finite_trace_last bs bprefix, iom) =
(s, oom)
Hv: valid l (finite_trace_last bs bprefix, iom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X
  ([{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}] ++ bsuffix) (Existing i) =
Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
s_i: state X
_Hi: equivocator_state_project s i = Some s_i
lst_i': state X
id: equivocator_state_project
  (finite_trace_last bs bprefix) idl =
Some lst_i'
Hex: existing_equivocator_label X l
_Hitemx: itemx =
{|
  l :=
    existing_equivocator_label_extract X l
      Hex;
  input := iom;
  destination := s_i;
  output := oom
|}
Hvs': valid
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom)
Hts': transition
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom) = (s_i, oom)
prefix: list transition_item
dfirst: MachineDescriptor X
Hprefix: equivocator_vlsm_trace_project X bprefix
  (Existing idl) = Some (prefix, dfirst)
Htr: equivocator_vlsm_trace_project X
  (bprefix ++
   {|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix) (Existing i) =
Some (prefix ++ itemx :: suffix, dfirst)
bfi: state X
Hdfinal: equivocator_state_project bf i = Some bfi
Hdinitial: match dfirst with
| NewMachine sn =>
    finite_constrained_trace_init_to X sn
      bfi (prefix ++ itemx :: suffix)
| Existing i =>
    ∃ isi : state X,
      equivocator_state_project bs i =
      Some isi
      ∧ finite_constrained_trace_from_to X
          isi bfi
          (prefix ++ itemx :: suffix)
        ∧ (initial_state_prop bs
           → initial_state_prop isi)
end
itemx ∈ prefix ∨ itemx ∈ itemx :: suffix right.message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs
  (finite_trace_last bs bprefix) bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
  (finite_trace_last bs bprefix) bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Ht: transition l (finite_trace_last bs bprefix, iom) =
(s, oom)
Hv: valid l (finite_trace_last bs bprefix, iom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X
  ([{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}] ++ bsuffix) (Existing i) =
Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
s_i: state X
_Hi: equivocator_state_project s i = Some s_i
lst_i': state X
id: equivocator_state_project
  (finite_trace_last bs bprefix) idl =
Some lst_i'
Hex: existing_equivocator_label X l
_Hitemx: itemx =
{|
  l :=
    existing_equivocator_label_extract X l
      Hex;
  input := iom;
  destination := s_i;
  output := oom
|}
Hvs': valid
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom)
Hts': transition
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom) = (s_i, oom)
prefix: list transition_item
dfirst: MachineDescriptor X
Hprefix: equivocator_vlsm_trace_project X bprefix
  (Existing idl) = Some (prefix, dfirst)
Htr: equivocator_vlsm_trace_project X
  (bprefix ++
   {|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix) (Existing i) =
Some (prefix ++ itemx :: suffix, dfirst)
bfi: state X
Hdfinal: equivocator_state_project bf i = Some bfi
Hdinitial: match dfirst with
| NewMachine sn =>
    finite_constrained_trace_init_to X sn
      bfi (prefix ++ itemx :: suffix)
| Existing i =>
    ∃ isi : state X,
      equivocator_state_project bs i =
      Some isi
      ∧ finite_constrained_trace_from_to X
          isi bfi
          (prefix ++ itemx :: suffix)
        ∧ (initial_state_prop bs
           → initial_state_prop isi)
end
itemx ∈ itemx :: suffix left.
  -message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs
  (finite_trace_last bs bprefix) bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
  (finite_trace_last bs bprefix) bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Ht: transition l (finite_trace_last bs bprefix, iom) =
(s, oom)
Hv: valid l (finite_trace_last bs bprefix, iom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X
  ([{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}] ++ bsuffix) (Existing i) =
Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
s_i: state X
_Hi: equivocator_state_project s i = Some s_i
lst_i': state X
id: equivocator_state_project
  (finite_trace_last bs bprefix) idl =
Some lst_i'
Hex: existing_equivocator_label X l
_Hitemx: itemx =
{|
  l :=
    existing_equivocator_label_extract X l
      Hex;
  input := iom;
  destination := s_i;
  output := oom
|}
Hvs': valid
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom)
Hts': transition
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom) = (s_i, oom)
prefix: list transition_item
dfirst: MachineDescriptor X
Hprefix: equivocator_vlsm_trace_project X bprefix
  (Existing idl) = Some (prefix, dfirst)
Htr: equivocator_vlsm_trace_project X
  (bprefix ++
   {|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix) (Existing i) =
Some (prefix ++ itemx :: suffix, dfirst)
bfi: state X
Hdfinal: equivocator_state_project bf i = Some bfi
Hdinitial: match dfirst with
| NewMachine sn =>
    finite_constrained_trace_init_to X sn
      bfi (prefix ++ itemx :: suffix)
| Existing i =>
    ∃ isi : state X,
      equivocator_state_project bs i =
      Some isi
      ∧ finite_constrained_trace_from_to X
          isi bfi
          (prefix ++ itemx :: suffix)
        ∧ (initial_state_prop bs
           → initial_state_prop isi)
end
∃ dfinal dfirst : MachineDescriptor X,
  proper_descriptor X dfirst bs
  ∧ existing_descriptor X dfinal
      (finite_trace_last bs
         (bprefix ++
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} :: bsuffix))
    ∧ equivocator_vlsm_trace_project X
        (bprefix ++
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |} :: bsuffix) dfinal =
      Some (prefix ++ itemx :: suffix, dfirst) eexists _, _.message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs
  (finite_trace_last bs bprefix) bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
  (finite_trace_last bs bprefix) bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Ht: transition l (finite_trace_last bs bprefix, iom) =
(s, oom)
Hv: valid l (finite_trace_last bs bprefix, iom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X
  ([{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}] ++ bsuffix) (Existing i) =
Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
s_i: state X
_Hi: equivocator_state_project s i = Some s_i
lst_i': state X
id: equivocator_state_project
  (finite_trace_last bs bprefix) idl =
Some lst_i'
Hex: existing_equivocator_label X l
_Hitemx: itemx =
{|
  l :=
    existing_equivocator_label_extract X l
      Hex;
  input := iom;
  destination := s_i;
  output := oom
|}
Hvs': valid
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom)
Hts': transition
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom) = (s_i, oom)
prefix: list transition_item
dfirst: MachineDescriptor X
Hprefix: equivocator_vlsm_trace_project X bprefix
  (Existing idl) = Some (prefix, dfirst)
Htr: equivocator_vlsm_trace_project X
  (bprefix ++
   {|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix) (Existing i) =
Some (prefix ++ itemx :: suffix, dfirst)
bfi: state X
Hdfinal: equivocator_state_project bf i = Some bfi
Hdinitial: match dfirst with
| NewMachine sn =>
    finite_constrained_trace_init_to X sn
      bfi (prefix ++ itemx :: suffix)
| Existing i =>
    ∃ isi : state X,
      equivocator_state_project bs i =
      Some isi
      ∧ finite_constrained_trace_from_to X
          isi bfi
          (prefix ++ itemx :: suffix)
        ∧ (initial_state_prop bs
           → initial_state_prop isi)
end
proper_descriptor X ?Goal0 bs
∧ existing_descriptor X ?Goal
    (finite_trace_last bs
       (bprefix ++
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} :: bsuffix))
  ∧ equivocator_vlsm_trace_project X
      (bprefix ++
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |} :: bsuffix) ?Goal =
    Some (prefix ++ itemx :: suffix, ?Goal0) repeat split; [.. | by apply Htr].message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs
  (finite_trace_last bs bprefix) bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
  (finite_trace_last bs bprefix) bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Ht: transition l (finite_trace_last bs bprefix, iom) =
(s, oom)
Hv: valid l (finite_trace_last bs bprefix, iom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X
  ([{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}] ++ bsuffix) (Existing i) =
Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
s_i: state X
_Hi: equivocator_state_project s i = Some s_i
lst_i': state X
id: equivocator_state_project
  (finite_trace_last bs bprefix) idl =
Some lst_i'
Hex: existing_equivocator_label X l
_Hitemx: itemx =
{|
  l :=
    existing_equivocator_label_extract X l
      Hex;
  input := iom;
  destination := s_i;
  output := oom
|}
Hvs': valid
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom)
Hts': transition
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom) = (s_i, oom)
prefix: list transition_item
dfirst: MachineDescriptor X
Hprefix: equivocator_vlsm_trace_project X bprefix
  (Existing idl) = Some (prefix, dfirst)
Htr: equivocator_vlsm_trace_project X
  (bprefix ++
   {|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix) (Existing i) =
Some (prefix ++ itemx :: suffix, dfirst)
bfi: state X
Hdfinal: equivocator_state_project bf i = Some bfi
Hdinitial: match dfirst with
| NewMachine sn =>
    finite_constrained_trace_init_to X sn
      bfi (prefix ++ itemx :: suffix)
| Existing i =>
    ∃ isi : state X,
      equivocator_state_project bs i =
      Some isi
      ∧ finite_constrained_trace_from_to X
          isi bfi
          (prefix ++ itemx :: suffix)
        ∧ (initial_state_prop bs
           → initial_state_prop isi)
end
proper_descriptor X dfirst bs
message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs
  (finite_trace_last bs bprefix) bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
  (finite_trace_last bs bprefix) bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Ht: transition l (finite_trace_last bs bprefix, iom) =
(s, oom)
Hv: valid l (finite_trace_last bs bprefix, iom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X
  ([{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}] ++ bsuffix) (Existing i) =
Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
s_i: state X
_Hi: equivocator_state_project s i = Some s_i
lst_i': state X
id: equivocator_state_project
  (finite_trace_last bs bprefix) idl =
Some lst_i'
Hex: existing_equivocator_label X l
_Hitemx: itemx =
{|
  l :=
    existing_equivocator_label_extract X l
      Hex;
  input := iom;
  destination := s_i;
  output := oom
|}
Hvs': valid
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom)
Hts': transition
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom) = (s_i, oom)
prefix: list transition_item
dfirst: MachineDescriptor X
Hprefix: equivocator_vlsm_trace_project X bprefix
  (Existing idl) = Some (prefix, dfirst)
Htr: equivocator_vlsm_trace_project X
  (bprefix ++
   {|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix) (Existing i) =
Some (prefix ++ itemx :: suffix, dfirst)
bfi: state X
Hdfinal: equivocator_state_project bf i = Some bfi
Hdinitial: match dfirst with
| NewMachine sn =>
    finite_constrained_trace_init_to X sn
      bfi (prefix ++ itemx :: suffix)
| Existing i =>
    ∃ isi : state X,
      equivocator_state_project bs i =
      Some isi
      ∧ finite_constrained_trace_from_to X
          isi bfi
          (prefix ++ itemx :: suffix)
        ∧ (initial_state_prop bs
           → initial_state_prop isi)
end
existing_descriptor X (Existing i)
  (finite_trace_last bs
     (bprefix ++
      {|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |} :: bsuffix))
    +message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs
  (finite_trace_last bs bprefix) bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
  (finite_trace_last bs bprefix) bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Ht: transition l (finite_trace_last bs bprefix, iom) =
(s, oom)
Hv: valid l (finite_trace_last bs bprefix, iom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X
  ([{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}] ++ bsuffix) (Existing i) =
Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
s_i: state X
_Hi: equivocator_state_project s i = Some s_i
lst_i': state X
id: equivocator_state_project
  (finite_trace_last bs bprefix) idl =
Some lst_i'
Hex: existing_equivocator_label X l
_Hitemx: itemx =
{|
  l :=
    existing_equivocator_label_extract X l
      Hex;
  input := iom;
  destination := s_i;
  output := oom
|}
Hvs': valid
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom)
Hts': transition
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom) = (s_i, oom)
prefix: list transition_item
dfirst: MachineDescriptor X
Hprefix: equivocator_vlsm_trace_project X bprefix
  (Existing idl) = Some (prefix, dfirst)
Htr: equivocator_vlsm_trace_project X
  (bprefix ++
   {|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix) (Existing i) =
Some (prefix ++ itemx :: suffix, dfirst)
bfi: state X
Hdfinal: equivocator_state_project bf i = Some bfi
Hdinitial: match dfirst with
| NewMachine sn =>
    finite_constrained_trace_init_to X sn
      bfi (prefix ++ itemx :: suffix)
| Existing i =>
    ∃ isi : state X,
      equivocator_state_project bs i =
      Some isi
      ∧ finite_constrained_trace_from_to X
          isi bfi
          (prefix ++ itemx :: suffix)
        ∧ (initial_state_prop bs
           → initial_state_prop isi)
end
proper_descriptor X dfirst bs clear -Hdinitial.message: Type
X: VLSM message
bs: state (equivocator_vlsm X)
itemx: transition_item
suffix, prefix: list transition_item
dfirst: MachineDescriptor X
bfi: state X
Hdinitial: match dfirst with
| NewMachine sn =>
    finite_constrained_trace_init_to X sn
      bfi (prefix ++ itemx :: suffix)
| Existing i =>
    ∃ isi : state X,
      equivocator_state_project bs i =
      Some isi
      ∧ finite_constrained_trace_from_to X
          isi bfi
          (prefix ++ itemx :: suffix)
        ∧ (initial_state_prop bs
           → initial_state_prop isi)
end
proper_descriptor X dfirst bs
      destruct dfirst as [sn | j].message: Type
X: VLSM message
bs: state (equivocator_vlsm X)
itemx: transition_item
suffix, prefix: list transition_item
sn, bfi: state X
Hdinitial: finite_constrained_trace_init_to X sn bfi
  (prefix ++ itemx :: suffix)
proper_descriptor X (NewMachine sn) bs
message: Type
X: VLSM message
bs: state (equivocator_vlsm X)
itemx: transition_item
suffix, prefix: list transition_item
j: nat
bfi: state X
Hdinitial: ∃ isi : state X,
  equivocator_state_project bs j =
  Some isi
  ∧ finite_constrained_trace_from_to X isi
      bfi (prefix ++ itemx :: suffix)
    ∧ (initial_state_prop bs
       → initial_state_prop isi)
proper_descriptor X (Existing j) bs
      *message: Type
X: VLSM message
bs: state (equivocator_vlsm X)
itemx: transition_item
suffix, prefix: list transition_item
sn, bfi: state X
Hdinitial: finite_constrained_trace_init_to X sn bfi
  (prefix ++ itemx :: suffix)
proper_descriptor X (NewMachine sn) bs by destruct Hdinitial.
      *message: Type
X: VLSM message
bs: state (equivocator_vlsm X)
itemx: transition_item
suffix, prefix: list transition_item
j: nat
bfi: state X
Hdinitial: ∃ isi : state X,
  equivocator_state_project bs j =
  Some isi
  ∧ finite_constrained_trace_from_to X isi
      bfi (prefix ++ itemx :: suffix)
    ∧ (initial_state_prop bs
       → initial_state_prop isi)
proper_descriptor X (Existing j) bs by destruct Hdinitial as [bsj [Hdinitial _]]; exists bsj.
    +message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs
  (finite_trace_last bs bprefix) bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
  (finite_trace_last bs bprefix) bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Ht: transition l (finite_trace_last bs bprefix, iom) =
(s, oom)
Hv: valid l (finite_trace_last bs bprefix, iom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X
  ([{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}] ++ bsuffix) (Existing i) =
Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
s_i: state X
_Hi: equivocator_state_project s i = Some s_i
lst_i': state X
id: equivocator_state_project
  (finite_trace_last bs bprefix) idl =
Some lst_i'
Hex: existing_equivocator_label X l
_Hitemx: itemx =
{|
  l :=
    existing_equivocator_label_extract X l
      Hex;
  input := iom;
  destination := s_i;
  output := oom
|}
Hvs': valid
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom)
Hts': transition
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom) = (s_i, oom)
prefix: list transition_item
dfirst: MachineDescriptor X
Hprefix: equivocator_vlsm_trace_project X bprefix
  (Existing idl) = Some (prefix, dfirst)
Htr: equivocator_vlsm_trace_project X
  (bprefix ++
   {|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix) (Existing i) =
Some (prefix ++ itemx :: suffix, dfirst)
bfi: state X
Hdfinal: equivocator_state_project bf i = Some bfi
Hdinitial: match dfirst with
| NewMachine sn =>
    finite_constrained_trace_init_to X sn
      bfi (prefix ++ itemx :: suffix)
| Existing i =>
    ∃ isi : state X,
      equivocator_state_project bs i =
      Some isi
      ∧ finite_constrained_trace_from_to X
          isi bfi
          (prefix ++ itemx :: suffix)
        ∧ (initial_state_prop bs
           → initial_state_prop isi)
end
existing_descriptor X (Existing i)
  (finite_trace_last bs
     (bprefix ++
      {|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |} :: bsuffix)) remember (bprefix ++ _) as btr.message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs
  (finite_trace_last bs bprefix) bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
  (finite_trace_last bs bprefix) bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Ht: transition l (finite_trace_last bs bprefix, iom) =
(s, oom)
Hv: valid l (finite_trace_last bs bprefix, iom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X
  ([{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}] ++ bsuffix) (Existing i) =
Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
s_i: state X
_Hi: equivocator_state_project s i = Some s_i
lst_i': state X
id: equivocator_state_project
  (finite_trace_last bs bprefix) idl =
Some lst_i'
Hex: existing_equivocator_label X l
_Hitemx: itemx =
{|
  l :=
    existing_equivocator_label_extract X l
      Hex;
  input := iom;
  destination := s_i;
  output := oom
|}
Hvs': valid
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom)
Hts': transition
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom) = (s_i, oom)
prefix: list transition_item
dfirst: MachineDescriptor X
Hprefix: equivocator_vlsm_trace_project X bprefix
  (Existing idl) = Some (prefix, dfirst)
Htr: equivocator_vlsm_trace_project X
  (bprefix ++
   {|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix) (Existing i) =
Some (prefix ++ itemx :: suffix, dfirst)
bfi: state X
Hdfinal: equivocator_state_project bf i = Some bfi
Hdinitial: match dfirst with
| NewMachine sn =>
    finite_constrained_trace_init_to X sn
      bfi (prefix ++ itemx :: suffix)
| Existing i =>
    ∃ isi : state X,
      equivocator_state_project bs i =
      Some isi
      ∧ finite_constrained_trace_from_to X
          isi bfi
          (prefix ++ itemx :: suffix)
        ∧ (initial_state_prop bs
           → initial_state_prop isi)
end
btr: list transition_item
Heqbtr: btr =
bprefix ++
{|
  l := l;
  input := iom;
  destination := s;
  output := oom
|} :: bsuffix
existing_descriptor X (Existing i)
  (finite_trace_last bs btr)
      specialize (equivocator_vlsm_trace_project_inv X btr) as Hinv.message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs
  (finite_trace_last bs bprefix) bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
  (finite_trace_last bs bprefix) bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Ht: transition l (finite_trace_last bs bprefix, iom) =
(s, oom)
Hv: valid l (finite_trace_last bs bprefix, iom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X
  ([{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}] ++ bsuffix) (Existing i) =
Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
s_i: state X
_Hi: equivocator_state_project s i = Some s_i
lst_i': state X
id: equivocator_state_project
  (finite_trace_last bs bprefix) idl =
Some lst_i'
Hex: existing_equivocator_label X l
_Hitemx: itemx =
{|
  l :=
    existing_equivocator_label_extract X l
      Hex;
  input := iom;
  destination := s_i;
  output := oom
|}
Hvs': valid
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom)
Hts': transition
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom) = (s_i, oom)
prefix: list transition_item
dfirst: MachineDescriptor X
Hprefix: equivocator_vlsm_trace_project X bprefix
  (Existing idl) = Some (prefix, dfirst)
Htr: equivocator_vlsm_trace_project X
  (bprefix ++
   {|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix) (Existing i) =
Some (prefix ++ itemx :: suffix, dfirst)
bfi: state X
Hdfinal: equivocator_state_project bf i = Some bfi
Hdinitial: match dfirst with
| NewMachine sn =>
    finite_constrained_trace_init_to X sn
      bfi (prefix ++ itemx :: suffix)
| Existing i =>
    ∃ isi : state X,
      equivocator_state_project bs i =
      Some isi
      ∧ finite_constrained_trace_from_to X
          isi bfi
          (prefix ++ itemx :: suffix)
        ∧ (initial_state_prop bs
           → initial_state_prop isi)
end
btr: list transition_item
Heqbtr: btr =
bprefix ++
{|
  l := l;
  input := iom;
  destination := s;
  output := oom
|} :: bsuffix
Hinv: btr ≠ []
→ ∀ j : nat,
    is_Some
      (equivocator_vlsm_trace_project X btr
         (Existing j))
    → ∀ is : state (equivocator_vlsm X),
        ∃ sj : state X,
          equivocator_state_project
            (finite_trace_last is btr) j =
          Some sj
existing_descriptor X (Existing i)
  (finite_trace_last bs btr)
      spec Hinv; [by destruct bprefix; subst |].message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs
  (finite_trace_last bs bprefix) bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
  (finite_trace_last bs bprefix) bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Ht: transition l (finite_trace_last bs bprefix, iom) =
(s, oom)
Hv: valid l (finite_trace_last bs bprefix, iom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X
  ([{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}] ++ bsuffix) (Existing i) =
Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
s_i: state X
_Hi: equivocator_state_project s i = Some s_i
lst_i': state X
id: equivocator_state_project
  (finite_trace_last bs bprefix) idl =
Some lst_i'
Hex: existing_equivocator_label X l
_Hitemx: itemx =
{|
  l :=
    existing_equivocator_label_extract X l
      Hex;
  input := iom;
  destination := s_i;
  output := oom
|}
Hvs': valid
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom)
Hts': transition
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom) = (s_i, oom)
prefix: list transition_item
dfirst: MachineDescriptor X
Hprefix: equivocator_vlsm_trace_project X bprefix
  (Existing idl) = Some (prefix, dfirst)
Htr: equivocator_vlsm_trace_project X
  (bprefix ++
   {|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix) (Existing i) =
Some (prefix ++ itemx :: suffix, dfirst)
bfi: state X
Hdfinal: equivocator_state_project bf i = Some bfi
Hdinitial: match dfirst with
| NewMachine sn =>
    finite_constrained_trace_init_to X sn
      bfi (prefix ++ itemx :: suffix)
| Existing i =>
    ∃ isi : state X,
      equivocator_state_project bs i =
      Some isi
      ∧ finite_constrained_trace_from_to X
          isi bfi
          (prefix ++ itemx :: suffix)
        ∧ (initial_state_prop bs
           → initial_state_prop isi)
end
btr: list transition_item
Heqbtr: btr =
bprefix ++
{|
  l := l;
  input := iom;
  destination := s;
  output := oom
|} :: bsuffix
Hinv: ∀ j : nat,
  is_Some
    (equivocator_vlsm_trace_project X btr
       (Existing j))
  → ∀ is : state (equivocator_vlsm X),
      ∃ sj : state X,
        equivocator_state_project
          (finite_trace_last is btr) j =
        Some sj
existing_descriptor X (Existing i)
  (finite_trace_last bs btr)
      specialize (Hinv i).message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs
  (finite_trace_last bs bprefix) bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
  (finite_trace_last bs bprefix) bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Ht: transition l (finite_trace_last bs bprefix, iom) =
(s, oom)
Hv: valid l (finite_trace_last bs bprefix, iom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X
  ([{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}] ++ bsuffix) (Existing i) =
Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
s_i: state X
_Hi: equivocator_state_project s i = Some s_i
lst_i': state X
id: equivocator_state_project
  (finite_trace_last bs bprefix) idl =
Some lst_i'
Hex: existing_equivocator_label X l
_Hitemx: itemx =
{|
  l :=
    existing_equivocator_label_extract X l
      Hex;
  input := iom;
  destination := s_i;
  output := oom
|}
Hvs': valid
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom)
Hts': transition
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom) = (s_i, oom)
prefix: list transition_item
dfirst: MachineDescriptor X
Hprefix: equivocator_vlsm_trace_project X bprefix
  (Existing idl) = Some (prefix, dfirst)
Htr: equivocator_vlsm_trace_project X
  (bprefix ++
   {|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix) (Existing i) =
Some (prefix ++ itemx :: suffix, dfirst)
bfi: state X
Hdfinal: equivocator_state_project bf i = Some bfi
Hdinitial: match dfirst with
| NewMachine sn =>
    finite_constrained_trace_init_to X sn
      bfi (prefix ++ itemx :: suffix)
| Existing i =>
    ∃ isi : state X,
      equivocator_state_project bs i =
      Some isi
      ∧ finite_constrained_trace_from_to X
          isi bfi
          (prefix ++ itemx :: suffix)
        ∧ (initial_state_prop bs
           → initial_state_prop isi)
end
btr: list transition_item
Heqbtr: btr =
bprefix ++
{|
  l := l;
  input := iom;
  destination := s;
  output := oom
|} :: bsuffix
Hinv: is_Some
  (equivocator_vlsm_trace_project X btr
     (Existing i))
→ ∀ is : state (equivocator_vlsm X),
    ∃ sj : state X,
      equivocator_state_project
        (finite_trace_last is btr) i = 
      Some sj
existing_descriptor X (Existing i)
  (finite_trace_last bs btr)
      spec Hinv; [by subst; eexists |].message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs
  (finite_trace_last bs bprefix) bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
  (finite_trace_last bs bprefix) bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Ht: transition l (finite_trace_last bs bprefix, iom) =
(s, oom)
Hv: valid l (finite_trace_last bs bprefix, iom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X
  ([{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}] ++ bsuffix) (Existing i) =
Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
s_i: state X
_Hi: equivocator_state_project s i = Some s_i
lst_i': state X
id: equivocator_state_project
  (finite_trace_last bs bprefix) idl =
Some lst_i'
Hex: existing_equivocator_label X l
_Hitemx: itemx =
{|
  l :=
    existing_equivocator_label_extract X l
      Hex;
  input := iom;
  destination := s_i;
  output := oom
|}
Hvs': valid
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom)
Hts': transition
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom) = (s_i, oom)
prefix: list transition_item
dfirst: MachineDescriptor X
Hprefix: equivocator_vlsm_trace_project X bprefix
  (Existing idl) = Some (prefix, dfirst)
Htr: equivocator_vlsm_trace_project X
  (bprefix ++
   {|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix) (Existing i) =
Some (prefix ++ itemx :: suffix, dfirst)
bfi: state X
Hdfinal: equivocator_state_project bf i = Some bfi
Hdinitial: match dfirst with
| NewMachine sn =>
    finite_constrained_trace_init_to X sn
      bfi (prefix ++ itemx :: suffix)
| Existing i =>
    ∃ isi : state X,
      equivocator_state_project bs i =
      Some isi
      ∧ finite_constrained_trace_from_to X
          isi bfi
          (prefix ++ itemx :: suffix)
        ∧ (initial_state_prop bs
           → initial_state_prop isi)
end
btr: list transition_item
Heqbtr: btr =
bprefix ++
{|
  l := l;
  input := iom;
  destination := s;
  output := oom
|} :: bsuffix
Hinv: ∀ is : state (equivocator_vlsm X),
  ∃ sj : state X,
    equivocator_state_project
      (finite_trace_last is btr) i = 
    Some sj
existing_descriptor X (Existing i)
  (finite_trace_last bs btr)
      specialize (Hinv bs) as [lst_i Hlst_i].message: Type
X: VLSM message
bs, bf: state (equivocator_vlsm X)
bprefix, bsuffix: list transition_item
Hbprefix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) bs
  (finite_trace_last bs bprefix) bprefix
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hbsuffix: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
  (finite_trace_last bs bprefix) bf
  ({|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix)
idl: nat
Hlbitem: equivocator_label_descriptor l =
Existing idl
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s bsuffix
Ht: transition l (finite_trace_last bs bprefix, iom) =
(s, oom)
Hv: valid l (finite_trace_last bs bprefix, iom)
Hpitem: ∀ _i : nat,
  equivocator_label_descriptor l =
  Existing _i
  → ∃ (i : nat) (si : state X),
      equivocator_state_project s i = Some si
      ∧ (∃ itemx : transition_item,
           equivocator_vlsm_transition_item_project
             X
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |} (Existing i) =
           Some (Some itemx, Existing _i))
i: nat
si: state X
Hi: equivocator_state_project s i = Some si
itemx: transition_item
Hitemx: equivocator_vlsm_transition_item_project X
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} (Existing i) =
Some (Some itemx, Existing idl)
suffix: list transition_item
Hsuffix': equivocator_vlsm_trace_project X
  ([{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}] ++ bsuffix) (Existing i) =
Some ([itemx] ++ suffix, Existing idl)
Hsuffix: equivocator_vlsm_trace_project X bsuffix
  (Existing i) = Some (suffix, Existing i)
s_i: state X
_Hi: equivocator_state_project s i = Some s_i
lst_i': state X
id: equivocator_state_project
  (finite_trace_last bs bprefix) idl =
Some lst_i'
Hex: existing_equivocator_label X l
_Hitemx: itemx =
{|
  l :=
    existing_equivocator_label_extract X l
      Hex;
  input := iom;
  destination := s_i;
  output := oom
|}
Hvs': valid
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom)
Hts': transition
  (existing_equivocator_label_extract X l Hex)
  (lst_i', iom) = (s_i, oom)
prefix: list transition_item
dfirst: MachineDescriptor X
Hprefix: equivocator_vlsm_trace_project X bprefix
  (Existing idl) = Some (prefix, dfirst)
Htr: equivocator_vlsm_trace_project X
  (bprefix ++
   {|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |} :: bsuffix) (Existing i) =
Some (prefix ++ itemx :: suffix, dfirst)
bfi: state X
Hdfinal: equivocator_state_project bf i = Some bfi
Hdinitial: match dfirst with
| NewMachine sn =>
    finite_constrained_trace_init_to X sn
      bfi (prefix ++ itemx :: suffix)
| Existing i =>
    ∃ isi : state X,
      equivocator_state_project bs i =
      Some isi
      ∧ finite_constrained_trace_from_to X
          isi bfi
          (prefix ++ itemx :: suffix)
        ∧ (initial_state_prop bs
           → initial_state_prop isi)
end
btr: list transition_item
Heqbtr: btr =
bprefix ++
{|
  l := l;
  input := iom;
  destination := s;
  output := oom
|} :: bsuffix
lst_i: state X
Hlst_i: equivocator_state_project
  (finite_trace_last bs btr) i = 
Some lst_i
existing_descriptor X (Existing i)
  (finite_trace_last bs btr)
      by eexists.
Qed.

If an equivocator_vlsm's valid trace segment outputs a message, then one of its projections must do so too.

Lemma equivocator_vlsm_trace_project_output_reflecting_inv
  (is : state (equivocator_vlsm X))
  (tr : list (transition_item (equivocator_vlsm X)))
  (Htr : finite_constrained_trace_from (equivocator_vlsm X) is tr)
  (m : message)
  (Hbbs : Exists (field_selector output m) tr)
  : exists
    (j i : MachineDescriptor X)
    (Hi : proper_descriptor X i is)
    (Hj : existing_descriptor X j (finite_trace_last is tr))
    (trX : list (transition_item X))
    (HtrX : equivocator_vlsm_trace_project _ tr j = Some (trX, i))
    ,
    Exists (field_selector output m) trX.message: Type
X: VLSM message
is: state (equivocator_vlsm X)
tr: list transition_item
Htr: finite_constrained_trace_from
  (equivocator_vlsm X) is tr
m: message
Hbbs: Exists (field_selector output m) tr
∃ (j i : MachineDescriptor X) (_ : proper_descriptor X
                                     i is) (_ : existing_descriptor
                                                 X j
                                                 (finite_trace_last
                                                 is tr)) 
  (trX : list transition_item) (_ : equivocator_vlsm_trace_project
                                      X tr j =
                                    Some (trX, i)),
  Exists (field_selector output m) trX
Proof.message: Type
X: VLSM message
is: state (equivocator_vlsm X)
tr: list transition_item
Htr: finite_constrained_trace_from
  (equivocator_vlsm X) is tr
m: message
Hbbs: Exists (field_selector output m) tr
∃ (j i : MachineDescriptor X) (_ : proper_descriptor X
                                     i is) (_ : existing_descriptor
                                                 X j
                                                 (finite_trace_last
                                                 is tr)) 
  (trX : list transition_item) (_ : equivocator_vlsm_trace_project
                                      X tr j =
                                    Some (trX, i)),
  Exists (field_selector output m) trX
  apply Exists_exists in Hbbs.message: Type
X: VLSM message
is: state (equivocator_vlsm X)
tr: list transition_item
Htr: finite_constrained_trace_from
  (equivocator_vlsm X) is tr
m: message
Hbbs: ∃ x : transition_item,
  x ∈ tr ∧ field_selector output m x
∃ (j i : MachineDescriptor X) (_ : proper_descriptor X
                                     i is) (_ : existing_descriptor
                                                 X j
                                                 (finite_trace_last
                                                 is tr)) 
  (trX : list transition_item) (_ : equivocator_vlsm_trace_project
                                      X tr j =
                                    Some (trX, i)),
  Exists (field_selector output m) trX
  destruct Hbbs as [item [Hin Houtput]].message: Type
X: VLSM message
is: state (equivocator_vlsm X)
tr: list transition_item
Htr: finite_constrained_trace_from
  (equivocator_vlsm X) is tr
m: message
item: transition_item
Hin: item ∈ tr
Houtput: field_selector output m item
∃ (j i : MachineDescriptor X) (_ : proper_descriptor X
                                     i is) (_ : existing_descriptor
                                                 X j
                                                 (finite_trace_last
                                                 is tr)) 
  (trX : list transition_item) (_ : equivocator_vlsm_trace_project
                                      X tr j =
                                    Some (trX, i)),
  Exists (field_selector output m) trX
  destruct (equivocator_label_descriptor (l item)) as [sn | i] eqn: Hsndl.message: Type
X: VLSM message
is: state (equivocator_vlsm X)
tr: list transition_item
Htr: finite_constrained_trace_from
  (equivocator_vlsm X) is tr
m: message
item: transition_item
Hin: item ∈ tr
Houtput: field_selector output m item
sn: state X
Hsndl: equivocator_label_descriptor (l item) =
NewMachine sn
∃ (j i : MachineDescriptor X) (_ : proper_descriptor X
                                     i is) (_ : existing_descriptor
                                                 X j
                                                 (finite_trace_last
                                                 is tr)) 
  (trX : list transition_item) (_ : equivocator_vlsm_trace_project
                                      X tr j =
                                    Some (trX, i)),
  Exists (field_selector output m) trX
message: Type
X: VLSM message
is: state (equivocator_vlsm X)
tr: list transition_item
Htr: finite_constrained_trace_from
  (equivocator_vlsm X) is tr
m: message
item: transition_item
Hin: item ∈ tr
Houtput: field_selector output m item
i: nat
Hsndl: equivocator_label_descriptor (l item) =
Existing i
∃ (j i : MachineDescriptor X) 
  (_ : proper_descriptor X i is) 
  (_ : existing_descriptor X j
         (finite_trace_last is tr)) 
  (trX : list transition_item) 
  (_ : equivocator_vlsm_trace_project X tr j =
       Some (trX, i)),
  Exists (field_selector output m) trX
  -message: Type
X: VLSM message
is: state (equivocator_vlsm X)
tr: list transition_item
Htr: finite_constrained_trace_from
  (equivocator_vlsm X) is tr
m: message
item: transition_item
Hin: item ∈ tr
Houtput: field_selector output m item
sn: state X
Hsndl: equivocator_label_descriptor (l item) =
NewMachine sn
∃ (j i : MachineDescriptor X) (_ : proper_descriptor X
                                     i is) (_ : existing_descriptor
                                                 X j
                                                 (finite_trace_last
                                                 is tr)) 
  (trX : list transition_item) (_ : equivocator_vlsm_trace_project
                                      X tr j =
                                    Some (trX, i)),
  Exists (field_selector output m) trX destruct item.message: Type
X: VLSM message
is: state (equivocator_vlsm X)
tr: list transition_item
Htr: finite_constrained_trace_from
  (equivocator_vlsm X) is tr
m: message
l: label (equivocator_vlsm X)
input: option message
destination: state (equivocator_vlsm X)
output: option message
Hin: {|
  l := l;
  input := input;
  destination := destination;
  output := output
|} ∈ tr
Houtput: field_selector VLSM.output m
  {|
    l := l;
    input := input;
    destination := destination;
    output := output
  |}
sn: state X
Hsndl: equivocator_label_descriptor
  (VLSM.l
     {|
       l := l;
       input := input;
       destination := destination;
       output := output
     |}) = NewMachine sn
∃ (j i : MachineDescriptor X) (_ : proper_descriptor X
                                     i is) (_ : existing_descriptor
                                                 X j
                                                 (finite_trace_last
                                                 is tr)) 
  (trX : list transition_item) (_ : equivocator_vlsm_trace_project
                                      X tr j =
                                    Some (trX, i)),
  Exists (field_selector VLSM.output m) trX destruct l; inversion Hsndl.message: Type
X: VLSM message
is: state (equivocator_vlsm X)
tr: list transition_item
Htr: finite_constrained_trace_from
  (equivocator_vlsm X) is tr
m: message
s: state X
input: option message
destination: state (equivocator_vlsm X)
output: option message
Hin: {|
  l := Spawn s;
  input := input;
  destination := destination;
  output := output
|} ∈ tr
Houtput: field_selector VLSM.output m
  {|
    l := Spawn s;
    input := input;
    destination := destination;
    output := output
  |}
sn: state X
Hsndl: equivocator_label_descriptor
  (l
     {|
       l := Spawn s;
       input := input;
       destination := destination;
       output := output
     |}) = NewMachine sn
H0: s = sn
∃ (j i : MachineDescriptor X) (_ : proper_descriptor X
                                     i is) (_ : existing_descriptor
                                                 X j
                                                 (finite_trace_last
                                                 is tr)) 
  (trX : list transition_item) (_ : equivocator_vlsm_trace_project
                                      X tr j =
                                    Some (trX, i)),
  Exists (field_selector VLSM.output m) trX
    subst.message: Type
X: VLSM message
is: state (equivocator_vlsm X)
tr: list transition_item
Htr: finite_constrained_trace_from
  (equivocator_vlsm X) is tr
m: message
input: option message
destination: state (equivocator_vlsm X)
output: option message
sn: state X
Hsndl: equivocator_label_descriptor
  (l
     {|
       l := Spawn sn;
       input := input;
       destination := destination;
       output := output
     |}) = NewMachine sn
Houtput: field_selector VLSM.output m
  {|
    l := Spawn sn;
    input := input;
    destination := destination;
    output := output
  |}
Hin: {|
  l := Spawn sn;
  input := input;
  destination := destination;
  output := output
|} ∈ tr
∃ (j i : MachineDescriptor X) (_ : proper_descriptor X
                                     i is) (_ : existing_descriptor
                                                 X j
                                                 (finite_trace_last
                                                 is tr)) 
  (trX : list transition_item) (_ : equivocator_vlsm_trace_project
                                      X tr j =
                                    Some (trX, i)),
  Exists (field_selector VLSM.output m) trX simpl in *.message: Type
X: VLSM message
is: Equivocators.bounded_state_copies X
tr: list transition_item
Htr: finite_constrained_trace_from
  (equivocator_vlsm X) is tr
m: message
input: option message
destination: Equivocators.bounded_state_copies X
output: option message
sn: state X
Hsndl: NewMachine sn = NewMachine sn
Houtput: output = Some m
Hin: {|
  l := Spawn sn;
  input := input;
  destination := destination;
  output := output
|} ∈ tr
∃ (j i : MachineDescriptor X) (_ : proper_descriptor X
                                     i is) (_ : existing_descriptor
                                                 X j
                                                 (finite_trace_last
                                                 is tr)) 
  (trX : list transition_item) (_ : equivocator_vlsm_trace_project
                                      X tr j =
                                    Some (trX, i)),
  Exists (field_selector VLSM.output m) trX
    by destruct (preloaded_equivocator_vlsm_trace_project_valid_item_new_machine
      _ _ Htr _ Hin _ eq_refl)
      as  [_ [Hcontra _]]; cbn in *; congruence.
  -message: Type
X: VLSM message
is: state (equivocator_vlsm X)
tr: list transition_item
Htr: finite_constrained_trace_from
  (equivocator_vlsm X) is tr
m: message
item: transition_item
Hin: item ∈ tr
Houtput: field_selector output m item
i: nat
Hsndl: equivocator_label_descriptor (l item) =
Existing i
∃ (j i : MachineDescriptor X) (_ : proper_descriptor X
                                     i is) (_ : existing_descriptor
                                                 X j
                                                 (finite_trace_last
                                                 is tr)) 
  (trX : list transition_item) (_ : equivocator_vlsm_trace_project
                                      X tr j =
                                    Some (trX, i)),
  Exists (field_selector output m) trX red in Htr; apply valid_trace_add_default_last in Htr.message: Type
X: VLSM message
is: state (equivocator_vlsm X)
tr: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) is
  (finite_trace_last is tr) tr
m: message
item: transition_item
Hin: item ∈ tr
Houtput: field_selector output m item
i: nat
Hsndl: equivocator_label_descriptor (l item) =
Existing i
∃ (j i : MachineDescriptor X) (_ : proper_descriptor X
                                     i is) (_ : existing_descriptor
                                                 X j
                                                 (finite_trace_last
                                                 is tr)) 
  (trX : list transition_item) (_ : equivocator_vlsm_trace_project
                                      X tr j =
                                    Some (trX, i)),
  Exists (field_selector output m) trX
    specialize
    (preloaded_equivocator_vlsm_trace_project_valid_item
       _ _ _ Htr _ Hin _ Hsndl)
      as [itemx [[d Hitemx] [trx [Hinx [ifinal [ifirst [Hifirst [Hifinal Htrx]]]]]]]].message: Type
X: VLSM message
is: state (equivocator_vlsm X)
tr: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) is
  (finite_trace_last is tr) tr
m: message
item: transition_item
Hin: item ∈ tr
Houtput: field_selector output m item
i: nat
Hsndl: equivocator_label_descriptor (l item) =
Existing i
itemx: transition_item
d: MachineDescriptor X
Hitemx: equivocator_vlsm_transition_item_project X
  item d = Some (Some itemx, Existing i)
trx: list transition_item
Hinx: itemx ∈ trx
ifinal, ifirst: MachineDescriptor X
Hifirst: proper_descriptor X ifirst is
Hifinal: existing_descriptor X ifinal
  (finite_trace_last is tr)
Htrx: equivocator_vlsm_trace_project X tr ifinal =
Some (trx, ifirst)
∃ (j i : MachineDescriptor X) (_ : proper_descriptor X
                                     i is) (_ : existing_descriptor
                                                 X j
                                                 (finite_trace_last
                                                 is tr)) 
  (trX : list transition_item) (_ : equivocator_vlsm_trace_project
                                      X tr j =
                                    Some (trX, i)),
  Exists (field_selector output m) trX
  exists ifinal.message: Type
X: VLSM message
is: state (equivocator_vlsm X)
tr: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) is
  (finite_trace_last is tr) tr
m: message
item: transition_item
Hin: item ∈ tr
Houtput: field_selector output m item
i: nat
Hsndl: equivocator_label_descriptor (l item) =
Existing i
itemx: transition_item
d: MachineDescriptor X
Hitemx: equivocator_vlsm_transition_item_project X
  item d = Some (Some itemx, Existing i)
trx: list transition_item
Hinx: itemx ∈ trx
ifinal, ifirst: MachineDescriptor X
Hifirst: proper_descriptor X ifirst is
Hifinal: existing_descriptor X ifinal
  (finite_trace_last is tr)
Htrx: equivocator_vlsm_trace_project X tr ifinal =
Some (trx, ifirst)
∃ (i : MachineDescriptor X) (_ : proper_descriptor X i
                                   is) (_ : existing_descriptor
                                              X ifinal
                                              (finite_trace_last
                                                 is tr)) 
  (trX : list transition_item) (_ : equivocator_vlsm_trace_project
                                      X tr ifinal =
                                    Some (trX, i)),
  Exists (field_selector output m) trX exists ifirst.message: Type
X: VLSM message
is: state (equivocator_vlsm X)
tr: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) is
  (finite_trace_last is tr) tr
m: message
item: transition_item
Hin: item ∈ tr
Houtput: field_selector output m item
i: nat
Hsndl: equivocator_label_descriptor (l item) =
Existing i
itemx: transition_item
d: MachineDescriptor X
Hitemx: equivocator_vlsm_transition_item_project X
  item d = Some (Some itemx, Existing i)
trx: list transition_item
Hinx: itemx ∈ trx
ifinal, ifirst: MachineDescriptor X
Hifirst: proper_descriptor X ifirst is
Hifinal: existing_descriptor X ifinal
  (finite_trace_last is tr)
Htrx: equivocator_vlsm_trace_project X tr ifinal =
Some (trx, ifirst)
∃ (_ : proper_descriptor X ifirst is) (_ : existing_descriptor
                                             X ifinal
                                             (finite_trace_last
                                                is tr)) 
  (trX : list transition_item) (_ : equivocator_vlsm_trace_project
                                      X tr ifinal =
                                    Some (trX, ifirst)),
  Exists (field_selector output m) trX split; [done |].message: Type
X: VLSM message
is: state (equivocator_vlsm X)
tr: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) is
  (finite_trace_last is tr) tr
m: message
item: transition_item
Hin: item ∈ tr
Houtput: field_selector output m item
i: nat
Hsndl: equivocator_label_descriptor (l item) =
Existing i
itemx: transition_item
d: MachineDescriptor X
Hitemx: equivocator_vlsm_transition_item_project X
  item d = Some (Some itemx, Existing i)
trx: list transition_item
Hinx: itemx ∈ trx
ifinal, ifirst: MachineDescriptor X
Hifirst: proper_descriptor X ifirst is
Hifinal: existing_descriptor X ifinal
  (finite_trace_last is tr)
Htrx: equivocator_vlsm_trace_project X tr ifinal =
Some (trx, ifirst)
∃ (_ : existing_descriptor X ifinal
         (finite_trace_last is tr)) (trX : list
                                             transition_item) 
  (_ : equivocator_vlsm_trace_project X tr ifinal =
       Some (trX, ifirst)),
  Exists (field_selector output m) trX
  split; [done |].message: Type
X: VLSM message
is: state (equivocator_vlsm X)
tr: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) is
  (finite_trace_last is tr) tr
m: message
item: transition_item
Hin: item ∈ tr
Houtput: field_selector output m item
i: nat
Hsndl: equivocator_label_descriptor (l item) =
Existing i
itemx: transition_item
d: MachineDescriptor X
Hitemx: equivocator_vlsm_transition_item_project X
  item d = Some (Some itemx, Existing i)
trx: list transition_item
Hinx: itemx ∈ trx
ifinal, ifirst: MachineDescriptor X
Hifirst: proper_descriptor X ifirst is
Hifinal: existing_descriptor X ifinal
  (finite_trace_last is tr)
Htrx: equivocator_vlsm_trace_project X tr ifinal =
Some (trx, ifirst)
∃ (trX : list transition_item) (_ : equivocator_vlsm_trace_project
                                      X tr ifinal =
                                    Some (trX, ifirst)),
  Exists (field_selector output m) trX
  exists trx.message: Type
X: VLSM message
is: state (equivocator_vlsm X)
tr: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) is
  (finite_trace_last is tr) tr
m: message
item: transition_item
Hin: item ∈ tr
Houtput: field_selector output m item
i: nat
Hsndl: equivocator_label_descriptor (l item) =
Existing i
itemx: transition_item
d: MachineDescriptor X
Hitemx: equivocator_vlsm_transition_item_project X
  item d = Some (Some itemx, Existing i)
trx: list transition_item
Hinx: itemx ∈ trx
ifinal, ifirst: MachineDescriptor X
Hifirst: proper_descriptor X ifirst is
Hifinal: existing_descriptor X ifinal
  (finite_trace_last is tr)
Htrx: equivocator_vlsm_trace_project X tr ifinal =
Some (trx, ifirst)
∃ _ : equivocator_vlsm_trace_project X tr ifinal =
      Some (trx, ifirst),
  Exists (field_selector output m) trx exists Htrx.message: Type
X: VLSM message
is: state (equivocator_vlsm X)
tr: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) is
  (finite_trace_last is tr) tr
m: message
item: transition_item
Hin: item ∈ tr
Houtput: field_selector output m item
i: nat
Hsndl: equivocator_label_descriptor (l item) =
Existing i
itemx: transition_item
d: MachineDescriptor X
Hitemx: equivocator_vlsm_transition_item_project X
  item d = Some (Some itemx, Existing i)
trx: list transition_item
Hinx: itemx ∈ trx
ifinal, ifirst: MachineDescriptor X
Hifirst: proper_descriptor X ifirst is
Hifinal: existing_descriptor X ifinal
  (finite_trace_last is tr)
Htrx: equivocator_vlsm_trace_project X tr ifinal =
Some (trx, ifirst)
Exists (field_selector output m) trx
  apply Exists_exists.message: Type
X: VLSM message
is: state (equivocator_vlsm X)
tr: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) is
  (finite_trace_last is tr) tr
m: message
item: transition_item
Hin: item ∈ tr
Houtput: field_selector output m item
i: nat
Hsndl: equivocator_label_descriptor (l item) =
Existing i
itemx: transition_item
d: MachineDescriptor X
Hitemx: equivocator_vlsm_transition_item_project X
  item d = Some (Some itemx, Existing i)
trx: list transition_item
Hinx: itemx ∈ trx
ifinal, ifirst: MachineDescriptor X
Hifirst: proper_descriptor X ifirst is
Hifinal: existing_descriptor X ifinal
  (finite_trace_last is tr)
Htrx: equivocator_vlsm_trace_project X tr ifinal =
Some (trx, ifirst)
∃ x : transition_item,
  x ∈ trx ∧ field_selector output m x exists itemx.message: Type
X: VLSM message
is: state (equivocator_vlsm X)
tr: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) is
  (finite_trace_last is tr) tr
m: message
item: transition_item
Hin: item ∈ tr
Houtput: field_selector output m item
i: nat
Hsndl: equivocator_label_descriptor (l item) =
Existing i
itemx: transition_item
d: MachineDescriptor X
Hitemx: equivocator_vlsm_transition_item_project X
  item d = Some (Some itemx, Existing i)
trx: list transition_item
Hinx: itemx ∈ trx
ifinal, ifirst: MachineDescriptor X
Hifirst: proper_descriptor X ifirst is
Hifinal: existing_descriptor X ifinal
  (finite_trace_last is tr)
Htrx: equivocator_vlsm_trace_project X tr ifinal =
Some (trx, ifirst)
itemx ∈ trx ∧ field_selector output m itemx split; [done |].message: Type
X: VLSM message
is: state (equivocator_vlsm X)
tr: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) is
  (finite_trace_last is tr) tr
m: message
item: transition_item
Hin: item ∈ tr
Houtput: field_selector output m item
i: nat
Hsndl: equivocator_label_descriptor (l item) =
Existing i
itemx: transition_item
d: MachineDescriptor X
Hitemx: equivocator_vlsm_transition_item_project X
  item d = Some (Some itemx, Existing i)
trx: list transition_item
Hinx: itemx ∈ trx
ifinal, ifirst: MachineDescriptor X
Hifirst: proper_descriptor X ifirst is
Hifinal: existing_descriptor X ifinal
  (finite_trace_last is tr)
Htrx: equivocator_vlsm_trace_project X tr ifinal =
Some (trx, ifirst)
field_selector output m itemx
  apply equivocator_transition_item_project_inv_messages in Hitemx.message: Type
X: VLSM message
is: state (equivocator_vlsm X)
tr: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) is
  (finite_trace_last is tr) tr
m: message
item: transition_item
Hin: item ∈ tr
Houtput: field_selector output m item
i: nat
Hsndl: equivocator_label_descriptor (l item) =
Existing i
itemx: transition_item
d: MachineDescriptor X
Hitemx: ∃ i : nat,
  d = Existing i
  ∧ proper_descriptor X d (destination item)
    ∧ input item = input itemx
      ∧ output item = output itemx
trx: list transition_item
Hinx: itemx ∈ trx
ifinal, ifirst: MachineDescriptor X
Hifirst: proper_descriptor X ifirst is
Hifinal: existing_descriptor X ifinal
  (finite_trace_last is tr)
Htrx: equivocator_vlsm_trace_project X tr ifinal =
Some (trx, ifirst)
field_selector output m itemx
  destruct Hitemx as [_ [_ [_ [_ Hitemx]]]].message: Type
X: VLSM message
is: state (equivocator_vlsm X)
tr: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) is
  (finite_trace_last is tr) tr
m: message
item: transition_item
Hin: item ∈ tr
Houtput: field_selector output m item
i: nat
Hsndl: equivocator_label_descriptor (l item) =
Existing i
itemx: transition_item
d: MachineDescriptor X
Hitemx: output item = output itemx
trx: list transition_item
Hinx: itemx ∈ trx
ifinal, ifirst: MachineDescriptor X
Hifirst: proper_descriptor X ifirst is
Hifinal: existing_descriptor X ifinal
  (finite_trace_last is tr)
Htrx: equivocator_vlsm_trace_project X tr ifinal =
Some (trx, ifirst)
field_selector output m itemx
  simpl in *.message: Type
X: VLSM message
is: Equivocators.bounded_state_copies X
tr: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) is
  (finite_trace_last is tr) tr
m: message
item: transition_item
Hin: item ∈ tr
Houtput: output item = Some m
i: nat
Hsndl: equivocator_label_descriptor (l item) =
Existing i
itemx: transition_item
d: MachineDescriptor X
Hitemx: output item = output itemx
trx: list transition_item
Hinx: itemx ∈ trx
ifinal, ifirst: MachineDescriptor X
Hifirst: proper_descriptor X ifirst is
Hifinal: existing_descriptor X ifinal
  (finite_trace_last is tr)
Htrx: equivocator_vlsm_trace_project X tr ifinal =
Some (trx, ifirst)
output itemx = Some m
  by congruence.
Qed.

Section sec_oracle_lifting.

Context
  (selector : message -> transition_item X -> Prop)
  (Hselector_io :
    forall l1 l2 s1 s2 im om m,
      selector m  {| l := l1; input := im; destination := s1; output := om |} <->
      selector m  {| l := l2; input := im; destination := s2; output := om |})
  (oracle : state X -> message -> Prop)
  (Hdec : RelDecision oracle)
  (Hstepwise : oracle_stepwise_props (vlsm := X) selector oracle).

Definition equivocator_selector
  (m : message)
  (item : transition_item (equivocator_vlsm X))
  : Prop
  :=
  match (l item) with
  | Spawn _ => False
  | ContinueWith _ l | ForkWith _ l =>
    selector m
      {| l := l
       ; input := input item
       ; destination := equivocator_state_zero (destination item)
       ; output := output item
      |}
  end.

We define equivocator_oracle for the equivocator_vlsm as being the oracle for any of the internal machines.

Definition equivocator_oracle
  (s : state (equivocator_vlsm X))
  (m : message)
  : Prop
  :=
  exists i si, equivocator_state_project s i = Some si /\
    oracle si m.

Lemma equivocator_oracle_dec
  : RelDecision equivocator_oracle.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
RelDecision equivocator_oracle
Proof.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
RelDecision equivocator_oracle
  intros s m.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
s: state (equivocator_vlsm X)
m: message
Decision (equivocator_oracle s m)
  eapply
    (Decision_iff
      (P := Exists
        (fun i =>
          from_option (fun si => oracle si m) False
            (equivocator_state_project s i)) _)).message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
s: state (equivocator_vlsm X)
m: message
Exists
  (λ i : nat,
     from_option (λ si : state X, oracle si m) False
       (equivocator_state_project s i)) ?Goal
↔ equivocator_oracle s m
message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
s: state (equivocator_vlsm X)
m: message
Decision
  (Exists
     (λ i : nat,
        from_option (λ si : state X, oracle si m)
          False (equivocator_state_project s i)) 
     ?Goal)
  -message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
s: state (equivocator_vlsm X)
m: message
Exists
  (λ i : nat,
     from_option (λ si : state X, oracle si m) False
       (equivocator_state_project s i)) ?Goal
↔ equivocator_oracle s m split; unfold equivocator_oracle; rewrite Exists_exists; cycle 1.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
s: state (equivocator_vlsm X)
m: message
(∃ (i : nat) (si : state X),
   equivocator_state_project s i = Some si
   ∧ oracle si m)
→ ∃ x : nat,
    x ∈ ?Goal
    ∧ from_option (λ si : state X, oracle si m) False
        (equivocator_state_project s x)
message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
s: state (equivocator_vlsm X)
m: message
(∃ x : nat,
   x ∈ ?Goal
   ∧ from_option (λ si : state X, oracle si m) False
       (equivocator_state_project s x))
→ ∃ (i : nat) (si : state X),
    equivocator_state_project s i = Some si
    ∧ oracle si m
    +message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
s: state (equivocator_vlsm X)
m: message
(∃ (i : nat) (si : state X),
   equivocator_state_project s i = Some si
   ∧ oracle si m)
→ ∃ x : nat,
    x ∈ ?Goal
    ∧ from_option (λ si : state X, oracle si m) False
        (equivocator_state_project s x) intros (i & si & Hsi & Hi); exists i.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
s: state (equivocator_vlsm X)
m: message
i: nat
si: state X
Hsi: equivocator_state_project s i = Some si
Hi: oracle si m
i ∈ ?Goal
∧ from_option (λ si : state X, oracle si m) False
    (equivocator_state_project s i)
      split; [| by rewrite Hsi].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
s: state (equivocator_vlsm X)
m: message
i: nat
si: state X
Hsi: equivocator_state_project s i = Some si
Hi: oracle si m
i ∈ ?Goal
      by eapply up_to_n_full, equivocator_state_project_Some_rev.
    +message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
s: state (equivocator_vlsm X)
m: message
(∃ x : nat,
   x ∈ up_to_n_listing (equivocator_state_n s)
   ∧ from_option (λ si : state X, oracle si m) False
       (equivocator_state_project s x))
→ ∃ (i : nat) (si : state X),
    equivocator_state_project s i = Some si
    ∧ oracle si m intros (i & _ & Hi); exists i.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
s: state (equivocator_vlsm X)
m: message
i: nat
Hi: from_option (λ si : state X, oracle si m) False
  (equivocator_state_project s i)
∃ si : state X,
  equivocator_state_project s i = Some si
  ∧ oracle si m
      by destruct (equivocator_state_project s i); [eexists |].
  -message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
s: state (equivocator_vlsm X)
m: message
Decision
  (Exists
     (λ i : nat,
        from_option (λ si : state X, oracle si m)
          False (equivocator_state_project s i))
     (up_to_n_listing (equivocator_state_n s))) apply Exists_dec; intro i.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
s: state (equivocator_vlsm X)
m: message
i: nat
Decision
  (from_option (λ si : state X, oracle si m) False
     (equivocator_state_project s i))
    by destruct (equivocator_state_project s i); [apply Hdec | typeclasses eauto].
Qed.

Lemma equivocator_oracle_stepwise_props
  : oracle_stepwise_props (vlsm := equivocator_vlsm X) equivocator_selector equivocator_oracle.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_stepwise_props equivocator_selector
  equivocator_oracle
Proof.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_stepwise_props equivocator_selector
  equivocator_oracle
  destruct Hstepwise.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
oracle_stepwise_props equivocator_selector
  equivocator_oracle
  split; intros.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
s: state (equivocator_vlsm X)
H: initial_state_prop s
m: message
¬ equivocator_oracle s m
message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
im: option message
s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
H: input_constrained_transition 
  (equivocator_vlsm X) l (
  s, im) (s', om)
msg: message
equivocator_oracle s' msg
↔ equivocator_selector msg
    {|
      l := l;
      input := im;
      destination := s';
      output := om
    |} ∨ equivocator_oracle s msg
  -message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
s: state (equivocator_vlsm X)
H: initial_state_prop s
m: message
¬ equivocator_oracle s m destruct H as [Hn His].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
s: state (equivocator_vlsm X)
Hn: is_singleton_state X s
His: initial_state_prop (equivocator_state_zero s)
m: message
¬ equivocator_oracle s m unfold is_singleton_state in Hn.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
s: state (equivocator_vlsm X)
Hn: equivocator_state_n s = 1
His: initial_state_prop (equivocator_state_zero s)
m: message
¬ equivocator_oracle s m
    intros [j [sj [Hsj Hmbrj]]].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
s: state (equivocator_vlsm X)
Hn: equivocator_state_n s = 1
His: initial_state_prop (equivocator_state_zero s)
m: message
j: nat
sj: state X
Hsj: equivocator_state_project s j = Some sj
Hmbrj: oracle sj m
False
    apply equivocator_state_project_Some_rev in Hsj as Hltj.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
s: state (equivocator_vlsm X)
Hn: equivocator_state_n s = 1
His: initial_state_prop (equivocator_state_zero s)
m: message
j: nat
sj: state X
Hsj: equivocator_state_project s j = Some sj
Hmbrj: oracle sj m
Hltj: j < equivocator_state_n s
False
    assert (j = 0) by lia.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
s: state (equivocator_vlsm X)
Hn: equivocator_state_n s = 1
His: initial_state_prop (equivocator_state_zero s)
m: message
j: nat
sj: state X
Hsj: equivocator_state_project s j = Some sj
Hmbrj: oracle sj m
Hltj: j < equivocator_state_n s
H: j = 0
False subst.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
s: state (equivocator_vlsm X)
Hn: equivocator_state_n s = 1
His: initial_state_prop (equivocator_state_zero s)
m: message
sj: state X
Hsj: equivocator_state_project s 0 = Some sj
Hmbrj: oracle sj m
Hltj: 0 < equivocator_state_n s
False
    rewrite equivocator_state_project_zero in Hsj.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
s: state (equivocator_vlsm X)
Hn: equivocator_state_n s = 1
His: initial_state_prop (equivocator_state_zero s)
m: message
sj: state X
Hsj: Some (equivocator_state_zero s) = Some sj
Hmbrj: oracle sj m
Hltj: 0 < equivocator_state_n s
False
    inversion Hsj.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
s: state (equivocator_vlsm X)
Hn: equivocator_state_n s = 1
His: initial_state_prop (equivocator_state_zero s)
m: message
sj: state X
Hsj: Some (equivocator_state_zero s) = Some sj
Hmbrj: oracle sj m
Hltj: 0 < equivocator_state_n s
H0: equivocator_state_zero s = sj
False subst.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
s: state (equivocator_vlsm X)
Hn: equivocator_state_n s = 1
His: initial_state_prop (equivocator_state_zero s)
m: message
Hmbrj: oracle (equivocator_state_zero s) m
Hsj: Some (equivocator_state_zero s) =
Some (equivocator_state_zero s)
Hltj: 0 < equivocator_state_n s
False
    by elim (oracle_no_inits  _ His m).
  -message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
im: option message
s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
H: input_constrained_transition 
  (equivocator_vlsm X) l (
  s, im) (s', om)
msg: message
equivocator_oracle s' msg
↔ equivocator_selector msg
    {|
      l := l;
      input := im;
      destination := s';
      output := om
    |} ∨ equivocator_oracle s msg unfold equivocator_oracle.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
im: option message
s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
H: input_constrained_transition 
  (equivocator_vlsm X) l (
  s, im) (s', om)
msg: message
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := l;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg)
    destruct H as [[Hs [_ Hv]] Ht].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
im: option message
s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hv: valid l (s, im)
Ht: transition l (s, im) = (s', om)
msg: message
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := l;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg)
    destruct l as [sdesc | idesc l | idesc l].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
sdesc: state X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
im: option message
s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hv: valid (Spawn sdesc) (s, im)
Ht: transition (Spawn sdesc) (s, im) = (s', om)
msg: message
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := Spawn sdesc;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg)
message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
idesc: nat
l: label X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
im: option message
s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hv: valid (ContinueWith idesc l) (s, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := ContinueWith idesc l;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg)
message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
idesc: nat
l: label X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
im: option message
s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hv: valid (ForkWith idesc l) (s, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := ForkWith idesc l;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg)
    +message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
sdesc: state X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
im: option message
s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hv: valid (Spawn sdesc) (s, im)
Ht: transition (Spawn sdesc) (s, im) = (s', om)
msg: message
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := Spawn sdesc;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg) inversion Ht.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
sdesc: state X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
im: option message
s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hv: valid (Spawn sdesc) (s, im)
Ht: transition (Spawn sdesc) (s, im) = (s', om)
msg: message
H0: equivocator_state_extend s sdesc = s'
H1: None = om
(∃ (i : nat) (si : state X),
   equivocator_state_project
     (equivocator_state_extend s sdesc) i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := Spawn sdesc;
      input := im;
      destination := equivocator_state_extend s sdesc;
      output := None
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg) subst.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
sdesc: state X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
im: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hv: valid (Spawn sdesc) (s, im)
Ht: transition (Spawn sdesc) (s, im) =
(equivocator_state_extend s sdesc, None)
msg: message
(∃ (i : nat) (si : state X),
   equivocator_state_project
     (equivocator_state_extend s sdesc) i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := Spawn sdesc;
      input := im;
      destination := equivocator_state_extend s sdesc;
      output := None
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg) clear Ht.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
sdesc: state X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
im: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hv: valid (Spawn sdesc) (s, im)
msg: message
(∃ (i : nat) (si : state X),
   equivocator_state_project
     (equivocator_state_extend s sdesc) i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := Spawn sdesc;
      input := im;
      destination := equivocator_state_extend s sdesc;
      output := None
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg)
      destruct Hv as [Hisdesc Him].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
sdesc: state X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
im: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hisdesc: initial_state_prop sdesc
Him: (s, im).2 = None
msg: message
(∃ (i : nat) (si : state X),
   equivocator_state_project
     (equivocator_state_extend s sdesc) i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := Spawn sdesc;
      input := im;
      destination := equivocator_state_extend s sdesc;
      output := None
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg) simpl in Him.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
sdesc: state X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
im: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hisdesc: initial_state_prop sdesc
Him: im = None
msg: message
(∃ (i : nat) (si : state X),
   equivocator_state_project
     (equivocator_state_extend s sdesc) i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := Spawn sdesc;
      input := im;
      destination := equivocator_state_extend s sdesc;
      output := None
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg) subst im.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
sdesc: state X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hisdesc: initial_state_prop sdesc
msg: message
(∃ (i : nat) (si : state X),
   equivocator_state_project
     (equivocator_state_extend s sdesc) i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := Spawn sdesc;
      input := None;
      destination := equivocator_state_extend s sdesc;
      output := None
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg)
      split.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
sdesc: state X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hisdesc: initial_state_prop sdesc
msg: message
(∃ (i : nat) (si : state X),
   equivocator_state_project
     (equivocator_state_extend s sdesc) i = Some si
   ∧ oracle si msg)
→ equivocator_selector msg
    {|
      l := Spawn sdesc;
      input := None;
      destination := equivocator_state_extend s sdesc;
      output := None
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg)
message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
sdesc: state X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hisdesc: initial_state_prop sdesc
msg: message
equivocator_selector msg
  {|
    l := Spawn sdesc;
    input := None;
    destination := equivocator_state_extend s sdesc;
    output := None
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg)
→ ∃ (i : nat) (si : state X),
    equivocator_state_project
      (equivocator_state_extend s sdesc) i = 
    Some si ∧ oracle si msg
      *message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
sdesc: state X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hisdesc: initial_state_prop sdesc
msg: message
(∃ (i : nat) (si : state X),
   equivocator_state_project
     (equivocator_state_extend s sdesc) i = Some si
   ∧ oracle si msg)
→ equivocator_selector msg
    {|
      l := Spawn sdesc;
      input := None;
      destination := equivocator_state_extend s sdesc;
      output := None
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg) intros [ins [sins [Hsins Hir]]].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
sdesc: state X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hisdesc: initial_state_prop sdesc
msg: message
ins: nat
sins: state X
Hsins: equivocator_state_project
  (equivocator_state_extend s sdesc) ins =
Some sins
Hir: oracle sins msg
equivocator_selector msg
  {|
    l := Spawn sdesc;
    input := None;
    destination := equivocator_state_extend s sdesc;
    output := None
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg) right.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
sdesc: state X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hisdesc: initial_state_prop sdesc
msg: message
ins: nat
sins: state X
Hsins: equivocator_state_project
  (equivocator_state_extend s sdesc) ins =
Some sins
Hir: oracle sins msg
∃ (i : nat) (si : state X),
  equivocator_state_project s i = Some si
  ∧ oracle si msg
        destruct_equivocator_state_extend_project s sdesc ins Hins
        ; [by exists ins, sins | | done].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
sdesc: state X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hisdesc: initial_state_prop sdesc
msg: message
ins: nat
sins: state X
Hsins: Some sdesc = Some sins
Hir: oracle sins msg
Hni: ¬ ins < equivocator_state_n s
Hins: ins = equivocator_state_n s
∃ (i : nat) (si : state X),
  equivocator_state_project s i = Some si
  ∧ oracle si msg
        inversion Hsins.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
sdesc: state X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hisdesc: initial_state_prop sdesc
msg: message
ins: nat
sins: state X
Hsins: Some sdesc = Some sins
Hir: oracle sins msg
Hni: ¬ ins < equivocator_state_n s
Hins: ins = equivocator_state_n s
H0: sdesc = sins
∃ (i : nat) (si : state X),
  equivocator_state_project s i = Some si
  ∧ oracle si msg subst.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
sins: state X
Hsins: Some sins = Some sins
Hisdesc: initial_state_prop sins
msg: message
Hir: oracle sins msg
Hni: ¬ equivocator_state_n s < equivocator_state_n s
∃ (i : nat) (si : state X),
  equivocator_state_project s i = Some si
  ∧ oracle si msg
        by elim (oracle_no_inits _ Hisdesc msg).
      *message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
sdesc: state X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hisdesc: initial_state_prop sdesc
msg: message
equivocator_selector msg
  {|
    l := Spawn sdesc;
    input := None;
    destination := equivocator_state_extend s sdesc;
    output := None
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg)
→ ∃ (i : nat) (si : state X),
    equivocator_state_project
      (equivocator_state_extend s sdesc) i = Some si
    ∧ oracle si msg intros [H | [ins [sins [Hsins Hir]]]]; [done |].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
sdesc: state X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hisdesc: initial_state_prop sdesc
msg: message
ins: nat
sins: state X
Hsins: equivocator_state_project s ins = Some sins
Hir: oracle sins msg
∃ (i : nat) (si : state X),
  equivocator_state_project
    (equivocator_state_extend s sdesc) i = Some si
  ∧ oracle si msg
        exists ins, sins.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
sdesc: state X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hisdesc: initial_state_prop sdesc
msg: message
ins: nat
sins: state X
Hsins: equivocator_state_project s ins = Some sins
Hir: oracle sins msg
equivocator_state_project
  (equivocator_state_extend s sdesc) ins = Some sins
∧ oracle sins msg split; [| done].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
sdesc: state X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hisdesc: initial_state_prop sdesc
msg: message
ins: nat
sins: state X
Hsins: equivocator_state_project s ins = Some sins
Hir: oracle sins msg
equivocator_state_project
  (equivocator_state_extend s sdesc) ins = Some sins
        rewrite equivocator_state_extend_project_1; [done |].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
sdesc: state X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hisdesc: initial_state_prop sdesc
msg: message
ins: nat
sins: state X
Hsins: equivocator_state_project s ins = Some sins
Hir: oracle sins msg
ins < equivocator_state_n s
        by apply equivocator_state_project_Some_rev in Hsins.
    +message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
idesc: nat
l: label X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
im: option message
s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hv: valid (ContinueWith idesc l) (s, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := ContinueWith idesc l;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg) cbn in Hv.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
idesc: nat
l: label X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
im: option message
s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hv: match equivocator_state_project s idesc with
| Some si => valid l (si, im)
| None => False
end
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := ContinueWith idesc l;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg)
      destruct (equivocator_state_project s idesc) as [sidesc |] eqn: Hidesc; [| done].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
idesc: nat
l: label X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
im: option message
s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
sidesc: state X
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := ContinueWith idesc l;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg)
      destruct (transition X l (sidesc, im)) as (sidesc', om') eqn: Htx.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
idesc: nat
l: label X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
im: option message
s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
sidesc: state X
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
sidesc': state X
om': option message
Htx: transition l (sidesc, im) = (sidesc', om')
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := ContinueWith idesc l;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg)
      specialize
        (oracle_step_update l sidesc im sidesc' om').message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
om': option message
oracle_step_update: input_constrained_transition X l
  (sidesc, im) (sidesc', om')
→ ∀ msg : message,
    oracle sidesc' msg
    ↔ selector msg
        {|
          l := l;
          input := im;
          destination := sidesc';
          output := om'
        |} ∨ oracle sidesc msg
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
Htx: transition l (sidesc, im) = (sidesc', om')
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := ContinueWith idesc l;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg)
      spec oracle_step_update.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
om': option message
oracle_step_update: input_constrained_transition X l
  (sidesc, im) (sidesc', om')
→ ∀ msg : message,
    oracle sidesc' msg
    ↔ selector msg
        {|
          l := l;
          input := im;
          destination := sidesc';
          output := om'
        |} ∨ oracle sidesc msg
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
Htx: transition l (sidesc, im) = (sidesc', om')
input_constrained_transition X l (sidesc, im)
  (sidesc', om')
message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
om': option message
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
Htx: transition l (sidesc, im) = (sidesc', om')
oracle_step_update: ∀ msg : message,
  oracle sidesc' msg
  ↔ selector msg
      {|
        l := l;
        input := im;
        destination := sidesc';
        output := om'
      |} ∨ oracle sidesc msg
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := ContinueWith idesc l;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg)
      {message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
om': option message
oracle_step_update: input_constrained_transition X l
  (sidesc, im) (sidesc', om')
→ ∀ msg : message,
    oracle sidesc' msg
    ↔ selector msg
        {|
          l := l;
          input := im;
          destination := sidesc';
          output := om'
        |} ∨ oracle sidesc msg
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
Htx: transition l (sidesc, im) = (sidesc', om')
input_constrained_transition X l (sidesc, im)
  (sidesc', om')
        repeat split; [| | done..].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
om': option message
oracle_step_update: input_constrained_transition X l
  (sidesc, im) (sidesc', om')
→ ∀ msg : message,
    oracle sidesc' msg
    ↔ selector msg
        {|
          l := l;
          input := im;
          destination := sidesc';
          output := om'
        |} ∨ oracle sidesc msg
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
Htx: transition l (sidesc, im) = (sidesc', om')
valid_state_prop (preloaded_with_all_messages_vlsm X)
  sidesc
message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
om': option message
oracle_step_update: input_constrained_transition X l
  (sidesc, im) (sidesc', om')
→ ∀ msg : message,
    oracle sidesc' msg
    ↔ selector msg
        {|
          l := l;
          input := im;
          destination := sidesc';
          output := om'
        |} ∨ oracle sidesc msg
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
Htx: transition l (sidesc, im) = (sidesc', om')
option_valid_message_prop
  (preloaded_with_all_messages_vlsm X) im
        -message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
om': option message
oracle_step_update: input_constrained_transition X l
  (sidesc, im) (sidesc', om')
→ ∀ msg : message,
    oracle sidesc' msg
    ↔ selector msg
        {|
          l := l;
          input := im;
          destination := sidesc';
          output := om'
        |} ∨ oracle sidesc msg
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
Htx: transition l (sidesc, im) = (sidesc', om')
valid_state_prop (preloaded_with_all_messages_vlsm X)
  sidesc by apply (preloaded_equivocator_state_project_valid_state X _ Hs _ _ Hidesc).
        -message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
om': option message
oracle_step_update: input_constrained_transition X l
  (sidesc, im) (sidesc', om')
→ ∀ msg : message,
    oracle sidesc' msg
    ↔ selector msg
        {|
          l := l;
          input := im;
          destination := sidesc';
          output := om'
        |} ∨ oracle sidesc msg
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
Htx: transition l (sidesc, im) = (sidesc', om')
option_valid_message_prop
  (preloaded_with_all_messages_vlsm X) im by eexists _; apply (preloaded_with_all_messages_message_valid_initial_state_message X).
      }message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
om': option message
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
Htx: transition l (sidesc, im) = (sidesc', om')
oracle_step_update: ∀ msg : message,
  oracle sidesc' msg
  ↔ selector msg
      {|
        l := l;
        input := im;
        destination := sidesc';
        output := om'
      |} ∨ oracle sidesc msg
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := ContinueWith idesc l;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg)
      specialize (existing_false_label_equivocator_state_project_not_same X Ht _ Hidesc)
        as Hnot_same.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
om': option message
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
Htx: transition l (sidesc, im) = (sidesc', om')
oracle_step_update: ∀ msg : message,
  oracle sidesc' msg
  ↔ selector msg
      {|
        l := l;
        input := im;
        destination := sidesc';
        output := om'
      |} ∨ oracle sidesc msg
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := ContinueWith idesc l;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg)
      specialize (existing_false_label_equivocator_state_project_same X Ht _ Hidesc _ _ Htx)
        as [Hom Hsame].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
om': option message
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
Htx: transition l (sidesc, im) = (sidesc', om')
oracle_step_update: ∀ msg : message,
  oracle sidesc' msg
  ↔ selector msg
      {|
        l := l;
        input := im;
        destination := sidesc';
        output := om'
      |} ∨ oracle sidesc msg
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hom: om' = om
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := ContinueWith idesc l;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg)
      subst om'.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: ∀ msg : message,
  oracle sidesc' msg
  ↔ selector msg
      {|
        l := l;
        input := im;
        destination := sidesc';
        output := om
      |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := ContinueWith idesc l;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg)
      specialize (existing_false_label_equivocator_transition_size X Ht _ Hidesc) as Ht_size.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: ∀ msg : message,
  oracle sidesc' msg
  ↔ selector msg
      {|
        l := l;
        input := im;
        destination := sidesc';
        output := om
      |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := ContinueWith idesc l;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg)
      specialize (oracle_step_update msg).message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := ContinueWith idesc l;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg)
      split.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
→ equivocator_selector msg
    {|
      l := ContinueWith idesc l;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg)
message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
equivocator_selector msg
  {|
    l := ContinueWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg)
→ ∃ (i : nat) (si : state X),
    equivocator_state_project s' i = Some si
    ∧ oracle si msg
      *message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
→ equivocator_selector msg
    {|
      l := ContinueWith idesc l;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg) intros [i [s'i [Hs'i Hbri]]].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
i: nat
s'i: state X
Hs'i: equivocator_state_project s' i = Some s'i
Hbri: oracle s'i msg
equivocator_selector msg
  {|
    l := ContinueWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg)
        apply equivocator_state_project_Some_rev in Hs'i as Hlti.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
i: nat
s'i: state X
Hs'i: equivocator_state_project s' i = Some s'i
Hbri: oracle s'i msg
Hlti: i < equivocator_state_n s'
equivocator_selector msg
  {|
    l := ContinueWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg)
        destruct (decide (idesc = i)).message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
i: nat
s'i: state X
Hs'i: equivocator_state_project s' i = Some s'i
Hbri: oracle s'i msg
Hlti: i < equivocator_state_n s'
e: idesc = i
equivocator_selector msg
  {|
    l := ContinueWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg)
message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
i: nat
s'i: state X
Hs'i: equivocator_state_project s' i = Some s'i
Hbri: oracle s'i msg
Hlti: i < equivocator_state_n s'
n: idesc ≠ i
equivocator_selector msg
  {|
    l := ContinueWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg)
        --message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
i: nat
s'i: state X
Hs'i: equivocator_state_project s' i = Some s'i
Hbri: oracle s'i msg
Hlti: i < equivocator_state_n s'
e: idesc = i
equivocator_selector msg
  {|
    l := ContinueWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg) subst i.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
s'i: state X
Hs'i: equivocator_state_project s' idesc = Some s'i
Hbri: oracle s'i msg
Hlti: idesc < equivocator_state_n s'
equivocator_selector msg
  {|
    l := ContinueWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg) simpl in Hsame.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: default (equivocator_state_zero s')
  (equivocator_state_project s' idesc) =
sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
s'i: state X
Hs'i: equivocator_state_project s' idesc = Some s'i
Hbri: oracle s'i msg
Hlti: idesc < equivocator_state_n s'
equivocator_selector msg
  {|
    l := ContinueWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg) rewrite Hs'i in Hsame.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
s'i: state X
Hsame: default (equivocator_state_zero s') (Some s'i) =
sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hs'i: equivocator_state_project s' idesc = Some s'i
Hbri: oracle s'i msg
Hlti: idesc < equivocator_state_n s'
equivocator_selector msg
  {|
    l := ContinueWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg)
           simpl in Hsame.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
s'i: state X
Hsame: s'i = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hs'i: equivocator_state_project s' idesc = Some s'i
Hbri: oracle s'i msg
Hlti: idesc < equivocator_state_n s'
equivocator_selector msg
  {|
    l := ContinueWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg) subst s'i.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hbri: oracle sidesc' msg
Hs'i: equivocator_state_project s' idesc =
Some sidesc'
Hlti: idesc < equivocator_state_n s'
equivocator_selector msg
  {|
    l := ContinueWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg)
           apply oracle_step_update in Hbri.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hbri: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
Hs'i: equivocator_state_project s' idesc =
Some sidesc'
Hlti: idesc < equivocator_state_n s'
equivocator_selector msg
  {|
    l := ContinueWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg)
           destruct Hbri as [H | Hbri]; [| by right; eexists _, _].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Ht_size: equivocator_state_n s' =
equivocator_state_n s
H: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |}
Hs'i: equivocator_state_project s' idesc =
Some sidesc'
Hlti: idesc < equivocator_state_n s'
equivocator_selector msg
  {|
    l := ContinueWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg)
           by left; revert H; apply Hselector_io.
        --message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
i: nat
s'i: state X
Hs'i: equivocator_state_project s' i = Some s'i
Hbri: oracle s'i msg
Hlti: i < equivocator_state_n s'
n: idesc ≠ i
equivocator_selector msg
  {|
    l := ContinueWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg) right.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
i: nat
s'i: state X
Hs'i: equivocator_state_project s' i = Some s'i
Hbri: oracle s'i msg
Hlti: i < equivocator_state_n s'
n: idesc ≠ i
∃ (i : nat) (si : state X),
  equivocator_state_project s i = Some si
  ∧ oracle si msg exists i, s'i.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
i: nat
s'i: state X
Hs'i: equivocator_state_project s' i = Some s'i
Hbri: oracle s'i msg
Hlti: i < equivocator_state_n s'
n: idesc ≠ i
equivocator_state_project s i = Some s'i
∧ oracle s'i msg split; [| done].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
i: nat
s'i: state X
Hs'i: equivocator_state_project s' i = Some s'i
Hbri: oracle s'i msg
Hlti: i < equivocator_state_n s'
n: idesc ≠ i
equivocator_state_project s i = Some s'i
           specialize (Hnot_same i).message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
i: nat
Hnot_same: i < equivocator_state_n s
→ idesc ≠ i
  → equivocator_state_descriptor_project
      s' (Existing i) =
    equivocator_state_descriptor_project s
      (Existing i)
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
s'i: state X
Hs'i: equivocator_state_project s' i = Some s'i
Hbri: oracle s'i msg
Hlti: i < equivocator_state_n s'
n: idesc ≠ i
equivocator_state_project s i = Some s'i
           spec Hnot_same; [lia |].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
i: nat
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
s'i: state X
Hs'i: equivocator_state_project s' i = Some s'i
Hbri: oracle s'i msg
Hlti: i < equivocator_state_n s'
n: idesc ≠ i
Hnot_same: idesc ≠ i
→ equivocator_state_descriptor_project s'
    (Existing i) =
  equivocator_state_descriptor_project s
    (Existing i)
equivocator_state_project s i = Some s'i specialize (Hnot_same n).message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
i: nat
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
s'i: state X
Hs'i: equivocator_state_project s' i = Some s'i
Hbri: oracle s'i msg
Hlti: i < equivocator_state_n s'
n: idesc ≠ i
Hnot_same: equivocator_state_descriptor_project s'
  (Existing i) =
equivocator_state_descriptor_project s
  (Existing i)
equivocator_state_project s i = Some s'i
           simpl in Hnot_same.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
i: nat
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
s'i: state X
Hs'i: equivocator_state_project s' i = Some s'i
Hbri: oracle s'i msg
Hlti: i < equivocator_state_n s'
n: idesc ≠ i
Hnot_same: default (equivocator_state_zero s')
  (equivocator_state_project s' i) =
default (equivocator_state_zero s)
  (equivocator_state_project s i)
equivocator_state_project s i = Some s'i rewrite Hs'i in Hnot_same.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
i: nat
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
s'i: state X
Hs'i: equivocator_state_project s' i = Some s'i
Hbri: oracle s'i msg
Hlti: i < equivocator_state_n s'
n: idesc ≠ i
Hnot_same: default (equivocator_state_zero s')
  (Some s'i) =
default (equivocator_state_zero s)
  (equivocator_state_project s i)
equivocator_state_project s i = Some s'i
           destruct_equivocator_state_project s i si Hlti'; [| lia].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
i: nat
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
s'i: state X
Hs'i: equivocator_state_project s' i = Some s'i
Hbri: oracle s'i msg
Hlti: i < equivocator_state_n s'
n: idesc ≠ i
si: state X
Hnot_same: default (equivocator_state_zero s')
  (Some s'i) =
default (equivocator_state_zero s)
  (Some si)
Hlti': i < equivocator_state_n s
Some si = Some s'i
           by cbn in Hnot_same; congruence.
      *message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
equivocator_selector msg
  {|
    l := ContinueWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg)
→ ∃ (i : nat) (si : state X),
    equivocator_state_project s' i = Some si
    ∧ oracle si msg apply proj2 in oracle_step_update.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
equivocator_selector msg
  {|
    l := ContinueWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg)
→ ∃ (i : nat) (si : state X),
    equivocator_state_project s' i = Some si
    ∧ oracle si msg
        apply equivocator_state_project_Some_rev in Hidesc as Hltidesc.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: idesc < equivocator_state_n s
equivocator_selector msg
  {|
    l := ContinueWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg)
→ ∃ (i : nat) (si : state X),
    equivocator_state_project s' i = Some si
    ∧ oracle si msg
        intros [Heq_im | [ins [sins [Hsins Hbri]]]].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: idesc < equivocator_state_n s
Heq_im: equivocator_selector msg
  {|
    l := ContinueWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∃ (i : nat) (si : state X),
  equivocator_state_project s' i = Some si
  ∧ oracle si msg
message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: idesc < equivocator_state_n s
ins: nat
sins: state X
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
∃ (i : nat) (si : state X),
  equivocator_state_project s' i = Some si
  ∧ oracle si msg
        --message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: idesc < equivocator_state_n s
Heq_im: equivocator_selector msg
  {|
    l := ContinueWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∃ (i : nat) (si : state X),
  equivocator_state_project s' i = Some si
  ∧ oracle si msg unfold equivocator_selector in Heq_im.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: idesc < equivocator_state_n s
Heq_im: match
  VLSM.l
    {|
      l := ContinueWith idesc l;
      input := im;
      destination := s';
      output := om
    |}
with
| Spawn _ => False
| ContinueWith _ l0 | ForkWith _ l0 =>
    selector msg
      {|
        l := l0;
        input :=
          input
            {|
              l := ContinueWith idesc l;
              input := im;
              destination := s';
              output := om
            |};
        destination :=
          equivocator_state_zero
            (destination
               {|
                 l := ContinueWith idesc l;
                 input := im;
                 destination := s';
                 output := om
               |});
        output :=
          output
            {|
              l := ContinueWith idesc l;
              input := im;
              destination := s';
              output := om
            |}
      |}
end
∃ (i : nat) (si : state X),
  equivocator_state_project s' i = Some si
  ∧ oracle si msg simpl in Heq_im.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: idesc < equivocator_state_n s
Heq_im: selector msg
  {|
    l := l;
    input := im;
    destination := equivocator_state_zero s';
    output := om
  |}
∃ (i : nat) (si : state X),
  equivocator_state_project s' i = Some si
  ∧ oracle si msg
           rewrite Hselector_io with (l2 := l) (s2 := sidesc') in Heq_im.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: idesc < equivocator_state_n s
Heq_im: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |}
∃ (i : nat) (si : state X),
  equivocator_state_project s' i = Some si
  ∧ oracle si msg
           specialize (oracle_step_update (or_introl Heq_im)).message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: idesc < equivocator_state_n s
Heq_im: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |}
∃ (i : nat) (si : state X),
  equivocator_state_project s' i = Some si
  ∧ oracle si msg
           exists idesc, sidesc'.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: idesc < equivocator_state_n s
Heq_im: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |}
equivocator_state_project s' idesc = Some sidesc'
∧ oracle sidesc' msg
           simpl in Hsame.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: default (equivocator_state_zero s')
  (equivocator_state_project s' idesc) =
sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: idesc < equivocator_state_n s
Heq_im: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |}
equivocator_state_project s' idesc = Some sidesc'
∧ oracle sidesc' msg
           destruct_equivocator_state_project  s' idesc _sidesc' Hlst; [| lia].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
_sidesc': state X
Hsame: default (equivocator_state_zero s')
  (Some _sidesc') = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: idesc < equivocator_state_n s
Heq_im: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |}
Hlst: idesc < equivocator_state_n s'
Some _sidesc' = Some sidesc' ∧ oracle sidesc' msg
           simpl in Hsame.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
_sidesc': state X
Hsame: _sidesc' = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: idesc < equivocator_state_n s
Heq_im: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |}
Hlst: idesc < equivocator_state_n s'
Some _sidesc' = Some sidesc' ∧ oracle sidesc' msg
           by subst.
        --message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: idesc < equivocator_state_n s
ins: nat
sins: state X
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
∃ (i : nat) (si : state X),
  equivocator_state_project s' i = Some si
  ∧ oracle si msg apply equivocator_state_project_Some_rev in Hsins as Hltins.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: equivocator_state_descriptor_project s'
  (Existing idesc) = sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: idesc < equivocator_state_n s
ins: nat
sins: state X
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
Hltins: ins < equivocator_state_n s
∃ (i : nat) (si : state X),
  equivocator_state_project s' i = Some si
  ∧ oracle si msg
           simpl in Hsame.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: default (equivocator_state_zero s')
  (equivocator_state_project s' idesc) =
sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: idesc < equivocator_state_n s
ins: nat
sins: state X
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
Hltins: ins < equivocator_state_n s
∃ (i : nat) (si : state X),
  equivocator_state_project s' i = Some si
  ∧ oracle si msg
           destruct (decide (idesc = ins)).message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: default (equivocator_state_zero s')
  (equivocator_state_project s' idesc) =
sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: idesc < equivocator_state_n s
ins: nat
sins: state X
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
Hltins: ins < equivocator_state_n s
e: idesc = ins
∃ (i : nat) (si : state X),
  equivocator_state_project s' i = Some si
  ∧ oracle si msg
message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: default (equivocator_state_zero s')
  (equivocator_state_project s' idesc) =
sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: idesc < equivocator_state_n s
ins: nat
sins: state X
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
Hltins: ins < equivocator_state_n s
n: idesc ≠ ins
∃ (i : nat) (si : state X),
  equivocator_state_project s' i = Some si
  ∧ oracle si msg subst idesc.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
ins: nat
Hidesc: equivocator_state_project s ins = Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith ins l) (s, im) =
(s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hsame: default (equivocator_state_zero s')
  (equivocator_state_project s' ins) = sidesc'
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → ins ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: ins < equivocator_state_n s
sins: state X
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
Hltins: ins < equivocator_state_n s
∃ (i : nat) (si : state X),
  equivocator_state_project s' i = Some si
  ∧ oracle si msg
message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: default (equivocator_state_zero s')
  (equivocator_state_project s' idesc) =
sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: idesc < equivocator_state_n s
ins: nat
sins: state X
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
Hltins: ins < equivocator_state_n s
n: idesc ≠ ins
∃ (i : nat) (si : state X),
  equivocator_state_project s' i = Some si
  ∧ oracle si msg
           ++message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
ins: nat
Hidesc: equivocator_state_project s ins = Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith ins l) (s, im) =
(s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hsame: default (equivocator_state_zero s')
  (equivocator_state_project s' ins) = sidesc'
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → ins ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: ins < equivocator_state_n s
sins: state X
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
Hltins: ins < equivocator_state_n s
∃ (i : nat) (si : state X),
  equivocator_state_project s' i = Some si
  ∧ oracle si msg rewrite Hsins in Hidesc.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
ins: nat
sins: state X
Hidesc: Some sins = Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith ins l) (s, im) =
(s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hsame: default (equivocator_state_zero s')
  (equivocator_state_project s' ins) = sidesc'
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → ins ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: ins < equivocator_state_n s
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
Hltins: ins < equivocator_state_n s
∃ (i : nat) (si : state X),
  equivocator_state_project s' i = Some si
  ∧ oracle si msg inversion Hidesc.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
ins: nat
sins: state X
Hidesc: Some sins = Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith ins l) (s, im) =
(s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hsame: default (equivocator_state_zero s')
  (equivocator_state_project s' ins) = sidesc'
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → ins ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: ins < equivocator_state_n s
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
Hltins: ins < equivocator_state_n s
H0: sins = sidesc
∃ (i : nat) (si : state X),
  equivocator_state_project s' i = Some si
  ∧ oracle si msg subst sidesc.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc': state X
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
ins: nat
sins: state X
Hv: valid l (sins, im)
Hidesc: Some sins = Some sins
Ht: transition (ContinueWith ins l) (s, im) =
(s', om)
msg: message
Htx: transition l (sins, im) = (sidesc', om)
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sins msg
→ oracle sidesc' msg
Hsame: default (equivocator_state_zero s')
  (equivocator_state_project s' ins) = sidesc'
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → ins ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: ins < equivocator_state_n s
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
Hltins: ins < equivocator_state_n s
∃ (i : nat) (si : state X),
  equivocator_state_project s' i = Some si
  ∧ oracle si msg
              specialize (oracle_step_update (or_intror Hbri)).message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc': state X
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
ins: nat
sins: state X
Hv: valid l (sins, im)
Hidesc: Some sins = Some sins
Ht: transition (ContinueWith ins l) (s, im) =
(s', om)
msg: message
Htx: transition l (sins, im) = (sidesc', om)
oracle_step_update: oracle sidesc' msg
Hsame: default (equivocator_state_zero s')
  (equivocator_state_project s' ins) = sidesc'
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → ins ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: ins < equivocator_state_n s
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
Hltins: ins < equivocator_state_n s
∃ (i : nat) (si : state X),
  equivocator_state_project s' i = Some si
  ∧ oracle si msg
              exists ins, sidesc'.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc': state X
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
ins: nat
sins: state X
Hv: valid l (sins, im)
Hidesc: Some sins = Some sins
Ht: transition (ContinueWith ins l) (s, im) =
(s', om)
msg: message
Htx: transition l (sins, im) = (sidesc', om)
oracle_step_update: oracle sidesc' msg
Hsame: default (equivocator_state_zero s')
  (equivocator_state_project s' ins) = sidesc'
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → ins ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: ins < equivocator_state_n s
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
Hltins: ins < equivocator_state_n s
equivocator_state_project s' ins = Some sidesc'
∧ oracle sidesc' msg split; [| done].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc': state X
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
ins: nat
sins: state X
Hv: valid l (sins, im)
Hidesc: Some sins = Some sins
Ht: transition (ContinueWith ins l) (s, im) =
(s', om)
msg: message
Htx: transition l (sins, im) = (sidesc', om)
oracle_step_update: oracle sidesc' msg
Hsame: default (equivocator_state_zero s')
  (equivocator_state_project s' ins) = sidesc'
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → ins ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: ins < equivocator_state_n s
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
Hltins: ins < equivocator_state_n s
equivocator_state_project s' ins = Some sidesc'
              simpl in Hsame.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc': state X
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
ins: nat
sins: state X
Hv: valid l (sins, im)
Hidesc: Some sins = Some sins
Ht: transition (ContinueWith ins l) (s, im) =
(s', om)
msg: message
Htx: transition l (sins, im) = (sidesc', om)
oracle_step_update: oracle sidesc' msg
Hsame: default (equivocator_state_zero s')
  (equivocator_state_project s' ins) = sidesc'
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → ins ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: ins < equivocator_state_n s
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
Hltins: ins < equivocator_state_n s
equivocator_state_project s' ins = Some sidesc'
              by destruct_equivocator_state_project s' ins _sidesc' Hins; [subst | lia].
           ++message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: default (equivocator_state_zero s')
  (equivocator_state_project s' idesc) =
sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: idesc < equivocator_state_n s
ins: nat
sins: state X
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
Hltins: ins < equivocator_state_n s
n: idesc ≠ ins
∃ (i : nat) (si : state X),
  equivocator_state_project s' i = Some si
  ∧ oracle si msg exists ins, sins.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: default (equivocator_state_zero s')
  (equivocator_state_project s' idesc) =
sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: idesc < equivocator_state_n s
ins: nat
sins: state X
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
Hltins: ins < equivocator_state_n s
n: idesc ≠ ins
equivocator_state_project s' ins = Some sins
∧ oracle sins msg split; [| done].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_same: ∀ ni : nat,
  ni < equivocator_state_n s
  → idesc ≠ ni
    → equivocator_state_descriptor_project
        s' (Existing ni) =
      equivocator_state_descriptor_project
        s (Existing ni)
Hsame: default (equivocator_state_zero s')
  (equivocator_state_project s' idesc) =
sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: idesc < equivocator_state_n s
ins: nat
sins: state X
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
Hltins: ins < equivocator_state_n s
n: idesc ≠ ins
equivocator_state_project s' ins = Some sins
              specialize (Hnot_same ins).message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
ins: nat
Hnot_same: ins < equivocator_state_n s
→ idesc ≠ ins
  → equivocator_state_descriptor_project
      s' (Existing ins) =
    equivocator_state_descriptor_project s
      (Existing ins)
Hsame: default (equivocator_state_zero s')
  (equivocator_state_project s' idesc) =
sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: idesc < equivocator_state_n s
sins: state X
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
Hltins: ins < equivocator_state_n s
n: idesc ≠ ins
equivocator_state_project s' ins = Some sins spec Hnot_same; [lia |].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
ins: nat
Hsame: default (equivocator_state_zero s')
  (equivocator_state_project s' idesc) =
sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: idesc < equivocator_state_n s
sins: state X
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
Hltins: ins < equivocator_state_n s
n: idesc ≠ ins
Hnot_same: idesc ≠ ins
→ equivocator_state_descriptor_project s'
    (Existing ins) =
  equivocator_state_descriptor_project s
    (Existing ins)
equivocator_state_project s' ins = Some sins
              specialize (Hnot_same n).message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
ins: nat
Hsame: default (equivocator_state_zero s')
  (equivocator_state_project s' idesc) =
sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: idesc < equivocator_state_n s
sins: state X
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
Hltins: ins < equivocator_state_n s
n: idesc ≠ ins
Hnot_same: equivocator_state_descriptor_project s'
  (Existing ins) =
equivocator_state_descriptor_project s
  (Existing ins)
equivocator_state_project s' ins = Some sins
              simpl in Hnot_same.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
ins: nat
Hsame: default (equivocator_state_zero s')
  (equivocator_state_project s' idesc) =
sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: idesc < equivocator_state_n s
sins: state X
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
Hltins: ins < equivocator_state_n s
n: idesc ≠ ins
Hnot_same: default (equivocator_state_zero s')
  (equivocator_state_project s' ins) =
default (equivocator_state_zero s)
  (equivocator_state_project s ins)
equivocator_state_project s' ins = Some sins rewrite Hsins in Hnot_same.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
ins: nat
Hsame: default (equivocator_state_zero s')
  (equivocator_state_project s' idesc) =
sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: idesc < equivocator_state_n s
sins: state X
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
Hltins: ins < equivocator_state_n s
n: idesc ≠ ins
Hnot_same: default (equivocator_state_zero s')
  (equivocator_state_project s' ins) =
default (equivocator_state_zero s)
  (Some sins)
equivocator_state_project s' ins = Some sins
              destruct_equivocator_state_project s' ins _sins Hins; [| lia].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ContinueWith idesc l) (s, im) =
(s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
ins: nat
Hsame: default (equivocator_state_zero s')
  (equivocator_state_project s' idesc) =
sidesc'
Ht_size: equivocator_state_n s' =
equivocator_state_n s
Hltidesc: idesc < equivocator_state_n s
sins: state X
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
Hltins: ins < equivocator_state_n s
n: idesc ≠ ins
_sins: state X
Hnot_same: default (equivocator_state_zero s')
  (Some _sins) =
default (equivocator_state_zero s)
  (Some sins)
Hins: ins < equivocator_state_n s'
Some _sins = Some sins
              cbn in Hnot_same; congruence.
    +message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
idesc: nat
l: label X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
im: option message
s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hv: valid (ForkWith idesc l) (s, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := ForkWith idesc l;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg) cbn in Hv.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
idesc: nat
l: label X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
im: option message
s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hv: match equivocator_state_project s idesc with
| Some si => valid l (si, im)
| None => False
end
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := ForkWith idesc l;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg)
      destruct (equivocator_state_project s idesc) as [sidesc |] eqn: Hidesc; [| done].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
idesc: nat
l: label X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
im: option message
s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
sidesc: state X
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := ForkWith idesc l;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg)
      destruct (transition X l (sidesc, im)) as (sidesc', om') eqn: Htx.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
oracle_step_update: ∀ (l : label
         (preloaded_with_all_messages_vlsm
            X)) (s : state
                      (preloaded_with_all_messages_vlsm
                      X)) 
  (im : option message) 
  (s' : state
          (preloaded_with_all_messages_vlsm
             X)) (om : 
                  option message),
  input_constrained_transition X
    l (s, im) (s', om)
  → ∀ msg : message,
      oracle s' msg
      ↔ selector msg
          {|
            l := l;
            input := im;
            destination := s';
            output := om
          |} ∨ oracle s msg
idesc: nat
l: label X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
im: option message
s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
sidesc: state X
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
sidesc': state X
om': option message
Htx: transition l (sidesc, im) = (sidesc', om')
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := ForkWith idesc l;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg)
      specialize
        (oracle_step_update l sidesc im sidesc' om').message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
om': option message
oracle_step_update: input_constrained_transition X l
  (sidesc, im) (sidesc', om')
→ ∀ msg : message,
    oracle sidesc' msg
    ↔ selector msg
        {|
          l := l;
          input := im;
          destination := sidesc';
          output := om'
        |} ∨ oracle sidesc msg
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
Htx: transition l (sidesc, im) = (sidesc', om')
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := ForkWith idesc l;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg)
      spec oracle_step_update.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
om': option message
oracle_step_update: input_constrained_transition X l
  (sidesc, im) (sidesc', om')
→ ∀ msg : message,
    oracle sidesc' msg
    ↔ selector msg
        {|
          l := l;
          input := im;
          destination := sidesc';
          output := om'
        |} ∨ oracle sidesc msg
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
Htx: transition l (sidesc, im) = (sidesc', om')
input_constrained_transition X l (sidesc, im)
  (sidesc', om')
message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
om': option message
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
Htx: transition l (sidesc, im) = (sidesc', om')
oracle_step_update: ∀ msg : message,
  oracle sidesc' msg
  ↔ selector msg
      {|
        l := l;
        input := im;
        destination := sidesc';
        output := om'
      |} ∨ oracle sidesc msg
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := ForkWith idesc l;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg)
      {message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
om': option message
oracle_step_update: input_constrained_transition X l
  (sidesc, im) (sidesc', om')
→ ∀ msg : message,
    oracle sidesc' msg
    ↔ selector msg
        {|
          l := l;
          input := im;
          destination := sidesc';
          output := om'
        |} ∨ oracle sidesc msg
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
Htx: transition l (sidesc, im) = (sidesc', om')
input_constrained_transition X l (sidesc, im)
  (sidesc', om')
        repeat split; [| | done..].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
om': option message
oracle_step_update: input_constrained_transition X l
  (sidesc, im) (sidesc', om')
→ ∀ msg : message,
    oracle sidesc' msg
    ↔ selector msg
        {|
          l := l;
          input := im;
          destination := sidesc';
          output := om'
        |} ∨ oracle sidesc msg
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
Htx: transition l (sidesc, im) = (sidesc', om')
valid_state_prop (preloaded_with_all_messages_vlsm X)
  sidesc
message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
om': option message
oracle_step_update: input_constrained_transition X l
  (sidesc, im) (sidesc', om')
→ ∀ msg : message,
    oracle sidesc' msg
    ↔ selector msg
        {|
          l := l;
          input := im;
          destination := sidesc';
          output := om'
        |} ∨ oracle sidesc msg
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
Htx: transition l (sidesc, im) = (sidesc', om')
option_valid_message_prop
  (preloaded_with_all_messages_vlsm X) im
        -message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
om': option message
oracle_step_update: input_constrained_transition X l
  (sidesc, im) (sidesc', om')
→ ∀ msg : message,
    oracle sidesc' msg
    ↔ selector msg
        {|
          l := l;
          input := im;
          destination := sidesc';
          output := om'
        |} ∨ oracle sidesc msg
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
Htx: transition l (sidesc, im) = (sidesc', om')
valid_state_prop (preloaded_with_all_messages_vlsm X)
  sidesc by apply (preloaded_equivocator_state_project_valid_state X _ Hs _ _ Hidesc).
        -message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
om': option message
oracle_step_update: input_constrained_transition X l
  (sidesc, im) (sidesc', om')
→ ∀ msg : message,
    oracle sidesc' msg
    ↔ selector msg
        {|
          l := l;
          input := im;
          destination := sidesc';
          output := om'
        |} ∨ oracle sidesc msg
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
Htx: transition l (sidesc, im) = (sidesc', om')
option_valid_message_prop
  (preloaded_with_all_messages_vlsm X) im by eexists _; apply (preloaded_with_all_messages_message_valid_initial_state_message X).
      }message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
om': option message
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
Htx: transition l (sidesc, im) = (sidesc', om')
oracle_step_update: ∀ msg : message,
  oracle sidesc' msg
  ↔ selector msg
      {|
        l := l;
        input := im;
        destination := sidesc';
        output := om'
      |} ∨ oracle sidesc msg
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := ForkWith idesc l;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg)
      specialize (existing_true_label_equivocator_state_project_not_last X Ht _ Hidesc)
        as Hnot_last.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
om': option message
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
Htx: transition l (sidesc, im) = (sidesc', om')
oracle_step_update: ∀ msg : message,
  oracle sidesc' msg
  ↔ selector msg
      {|
        l := l;
        input := im;
        destination := sidesc';
        output := om'
      |} ∨ oracle sidesc msg
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := ForkWith idesc l;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg)
       specialize (existing_true_label_equivocator_state_project_last X Ht _ Hidesc _ _ Htx)
        as [Hom Hlast].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
om': option message
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
Htx: transition l (sidesc, im) = (sidesc', om')
oracle_step_update: ∀ msg : message,
  oracle sidesc' msg
  ↔ selector msg
      {|
        l := l;
        input := im;
        destination := sidesc';
        output := om'
      |} ∨ oracle sidesc msg
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Hom: om' = om
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := ForkWith idesc l;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg)
      subst om'.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: ∀ msg : message,
  oracle sidesc' msg
  ↔ selector msg
      {|
        l := l;
        input := im;
        destination := sidesc';
        output := om
      |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := ForkWith idesc l;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg)
      specialize (existing_true_label_equivocator_transition_size X Ht _ Hidesc) as Ht_size.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: ∀ msg : message,
  oracle sidesc' msg
  ↔ selector msg
      {|
        l := l;
        input := im;
        destination := sidesc';
        output := om
      |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := ForkWith idesc l;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg)
      specialize (oracle_step_update msg).message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
↔ equivocator_selector msg
    {|
      l := ForkWith idesc l;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg)
      split.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
→ equivocator_selector msg
    {|
      l := ForkWith idesc l;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg)
message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
equivocator_selector msg
  {|
    l := ForkWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg)
→ ∃ (i : nat) (si : state X),
    equivocator_state_project s' i = Some si
    ∧ oracle si msg
      *message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
(∃ (i : nat) (si : state X),
   equivocator_state_project s' i = Some si
   ∧ oracle si msg)
→ equivocator_selector msg
    {|
      l := ForkWith idesc l;
      input := im;
      destination := s';
      output := om
    |}
  ∨ (∃ (i : nat) (si : state X),
       equivocator_state_project s i = Some si
       ∧ oracle si msg) intros [i [s'i [Hs'i Hbri]]].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
i: nat
s'i: state X
Hs'i: equivocator_state_project s' i = Some s'i
Hbri: oracle s'i msg
equivocator_selector msg
  {|
    l := ForkWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg)
        apply equivocator_state_project_Some_rev in Hs'i as Hlti.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
i: nat
s'i: state X
Hs'i: equivocator_state_project s' i = Some s'i
Hbri: oracle s'i msg
Hlti: i < equivocator_state_n s'
equivocator_selector msg
  {|
    l := ForkWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg)
        destruct (decide (i = equivocator_state_n s)).message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
i: nat
s'i: state X
Hs'i: equivocator_state_project s' i = Some s'i
Hbri: oracle s'i msg
Hlti: i < equivocator_state_n s'
e: i = equivocator_state_n s
equivocator_selector msg
  {|
    l := ForkWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg)
message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
i: nat
s'i: state X
Hs'i: equivocator_state_project s' i = Some s'i
Hbri: oracle s'i msg
Hlti: i < equivocator_state_n s'
n: i ≠ equivocator_state_n s
equivocator_selector msg
  {|
    l := ForkWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg)
        --message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
i: nat
s'i: state X
Hs'i: equivocator_state_project s' i = Some s'i
Hbri: oracle s'i msg
Hlti: i < equivocator_state_n s'
e: i = equivocator_state_n s
equivocator_selector msg
  {|
    l := ForkWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg) subst i.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
s'i: state X
Hs'i: equivocator_state_project s'
  (equivocator_state_n s) = 
Some s'i
Hbri: oracle s'i msg
Hlti: equivocator_state_n s < equivocator_state_n s'
equivocator_selector msg
  {|
    l := ForkWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg) simpl in Hlast.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Hlast: default (equivocator_state_zero s')
  (equivocator_state_project s'
     (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
s'i: state X
Hs'i: equivocator_state_project s'
  (equivocator_state_n s) = 
Some s'i
Hbri: oracle s'i msg
Hlti: equivocator_state_n s < equivocator_state_n s'
equivocator_selector msg
  {|
    l := ForkWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg) rewrite Hs'i in Hlast.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
s'i: state X
Hlast: default (equivocator_state_zero s') (Some s'i) =
sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
Hs'i: equivocator_state_project s'
  (equivocator_state_n s) = 
Some s'i
Hbri: oracle s'i msg
Hlti: equivocator_state_n s < equivocator_state_n s'
equivocator_selector msg
  {|
    l := ForkWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg)
           simpl in Hlast.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
s'i: state X
Hlast: s'i = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
Hs'i: equivocator_state_project s'
  (equivocator_state_n s) = 
Some s'i
Hbri: oracle s'i msg
Hlti: equivocator_state_n s < equivocator_state_n s'
equivocator_selector msg
  {|
    l := ForkWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg) subst s'i.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
Hbri: oracle sidesc' msg
Hs'i: equivocator_state_project s'
  (equivocator_state_n s) = 
Some sidesc'
Hlti: equivocator_state_n s < equivocator_state_n s'
equivocator_selector msg
  {|
    l := ForkWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg)
           apply oracle_step_update in Hbri.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
Hbri: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
Hs'i: equivocator_state_project s'
  (equivocator_state_n s) = 
Some sidesc'
Hlti: equivocator_state_n s < equivocator_state_n s'
equivocator_selector msg
  {|
    l := ForkWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg)
           destruct Hbri as [H | Hbri]; [| by right; eexists _, _].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
H: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |}
Hs'i: equivocator_state_project s'
  (equivocator_state_n s) = 
Some sidesc'
Hlti: equivocator_state_n s < equivocator_state_n s'
equivocator_selector msg
  {|
    l := ForkWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg)
           by left; revert H; apply Hselector_io.
        --message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
i: nat
s'i: state X
Hs'i: equivocator_state_project s' i = Some s'i
Hbri: oracle s'i msg
Hlti: i < equivocator_state_n s'
n: i ≠ equivocator_state_n s
equivocator_selector msg
  {|
    l := ForkWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg) right.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
i: nat
s'i: state X
Hs'i: equivocator_state_project s' i = Some s'i
Hbri: oracle s'i msg
Hlti: i < equivocator_state_n s'
n: i ≠ equivocator_state_n s
∃ (i : nat) (si : state X),
  equivocator_state_project s i = Some si
  ∧ oracle si msg exists i, s'i.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
i: nat
s'i: state X
Hs'i: equivocator_state_project s' i = Some s'i
Hbri: oracle s'i msg
Hlti: i < equivocator_state_n s'
n: i ≠ equivocator_state_n s
equivocator_state_project s i = Some s'i
∧ oracle s'i msg split; [| done].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
i: nat
s'i: state X
Hs'i: equivocator_state_project s' i = Some s'i
Hbri: oracle s'i msg
Hlti: i < equivocator_state_n s'
n: i ≠ equivocator_state_n s
equivocator_state_project s i = Some s'i
           specialize (Hnot_last i).message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
i: nat
Hnot_last: i < equivocator_state_n s
→ equivocator_state_descriptor_project s'
    (Existing i) =
  equivocator_state_descriptor_project s
    (Existing i)
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
s'i: state X
Hs'i: equivocator_state_project s' i = Some s'i
Hbri: oracle s'i msg
Hlti: i < equivocator_state_n s'
n: i ≠ equivocator_state_n s
equivocator_state_project s i = Some s'i
           spec Hnot_last; [lia |].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
i: nat
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
s'i: state X
Hs'i: equivocator_state_project s' i = Some s'i
Hbri: oracle s'i msg
Hlti: i < equivocator_state_n s'
n: i ≠ equivocator_state_n s
Hnot_last: equivocator_state_descriptor_project s'
  (Existing i) =
equivocator_state_descriptor_project s
  (Existing i)
equivocator_state_project s i = Some s'i
           simpl in Hnot_last.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
i: nat
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
s'i: state X
Hs'i: equivocator_state_project s' i = Some s'i
Hbri: oracle s'i msg
Hlti: i < equivocator_state_n s'
n: i ≠ equivocator_state_n s
Hnot_last: default (equivocator_state_zero s')
  (equivocator_state_project s' i) =
default (equivocator_state_zero s)
  (equivocator_state_project s i)
equivocator_state_project s i = Some s'i rewrite Hs'i in Hnot_last.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
i: nat
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
s'i: state X
Hs'i: equivocator_state_project s' i = Some s'i
Hbri: oracle s'i msg
Hlti: i < equivocator_state_n s'
n: i ≠ equivocator_state_n s
Hnot_last: default (equivocator_state_zero s')
  (Some s'i) =
default (equivocator_state_zero s)
  (equivocator_state_project s i)
equivocator_state_project s i = Some s'i
           destruct_equivocator_state_project s i si Hlti'; [| lia].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
i: nat
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
s'i: state X
Hs'i: equivocator_state_project s' i = Some s'i
Hbri: oracle s'i msg
Hlti: i < equivocator_state_n s'
n: i ≠ equivocator_state_n s
si: state X
Hnot_last: default (equivocator_state_zero s')
  (Some s'i) =
default (equivocator_state_zero s)
  (Some si)
Hlti': i < equivocator_state_n s
Some si = Some s'i
           by cbn in Hnot_last; congruence.
      *message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: oracle sidesc' msg
↔ selector msg
    {|
      l := l;
      input := im;
      destination := sidesc';
      output := om
    |} ∨ oracle sidesc msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
equivocator_selector msg
  {|
    l := ForkWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg)
→ ∃ (i : nat) (si : state X),
    equivocator_state_project s' i = Some si
    ∧ oracle si msg apply proj2 in oracle_step_update.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
equivocator_selector msg
  {|
    l := ForkWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∨ (∃ (i : nat) (si : state X),
     equivocator_state_project s i = Some si
     ∧ oracle si msg)
→ ∃ (i : nat) (si : state X),
    equivocator_state_project s' i = Some si
    ∧ oracle si msg
        intros [Heq_im | [ins [sins [Hsins Hbri]]]].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
Heq_im: equivocator_selector msg
  {|
    l := ForkWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∃ (i : nat) (si : state X),
  equivocator_state_project s' i = Some si
  ∧ oracle si msg
message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
ins: nat
sins: state X
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
∃ (i : nat) (si : state X),
  equivocator_state_project s' i = Some si
  ∧ oracle si msg
        --message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
Heq_im: equivocator_selector msg
  {|
    l := ForkWith idesc l;
    input := im;
    destination := s';
    output := om
  |}
∃ (i : nat) (si : state X),
  equivocator_state_project s' i = Some si
  ∧ oracle si msg unfold equivocator_selector in Heq_im.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
Heq_im: match
  VLSM.l
    {|
      l := ForkWith idesc l;
      input := im;
      destination := s';
      output := om
    |}
with
| Spawn _ => False
| ContinueWith _ l0 | ForkWith _ l0 =>
    selector msg
      {|
        l := l0;
        input :=
          input
            {|
              l := ForkWith idesc l;
              input := im;
              destination := s';
              output := om
            |};
        destination :=
          equivocator_state_zero
            (destination
               {|
                 l := ForkWith idesc l;
                 input := im;
                 destination := s';
                 output := om
               |});
        output :=
          output
            {|
              l := ForkWith idesc l;
              input := im;
              destination := s';
              output := om
            |}
      |}
end
∃ (i : nat) (si : state X),
  equivocator_state_project s' i = Some si
  ∧ oracle si msg simpl in Heq_im.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
Heq_im: selector msg
  {|
    l := l;
    input := im;
    destination := equivocator_state_zero s';
    output := om
  |}
∃ (i : nat) (si : state X),
  equivocator_state_project s' i = Some si
  ∧ oracle si msg
           rewrite Hselector_io with (l2 := l) (s2 := sidesc') in Heq_im.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
Heq_im: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |}
∃ (i : nat) (si : state X),
  equivocator_state_project s' i = Some si
  ∧ oracle si msg
           specialize (oracle_step_update (or_introl Heq_im)).message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
Heq_im: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |}
∃ (i : nat) (si : state X),
  equivocator_state_project s' i = Some si
  ∧ oracle si msg
           exists (equivocator_state_n s), sidesc'.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
Heq_im: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |}
equivocator_state_project s' (equivocator_state_n s) =
Some sidesc' ∧ oracle sidesc' msg
           simpl in Hlast.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Hlast: default (equivocator_state_zero s')
  (equivocator_state_project s'
     (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
Heq_im: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |}
equivocator_state_project s' (equivocator_state_n s) =
Some sidesc' ∧ oracle sidesc' msg
           destruct_equivocator_state_project  s' (equivocator_state_n s) _sidesc' Hlst; [| lia].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
_sidesc': state X
Hlast: default (equivocator_state_zero s')
  (Some _sidesc') = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
Heq_im: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |}
Hlst: equivocator_state_n s < equivocator_state_n s'
Some _sidesc' = Some sidesc' ∧ oracle sidesc' msg
           by subst.
        --message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
ins: nat
sins: state X
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
∃ (i : nat) (si : state X),
  equivocator_state_project s' i = Some si
  ∧ oracle si msg apply equivocator_state_project_Some_rev in Hsins as Hltins.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
Hnot_last: ∀ ni : nat,
  ni < equivocator_state_n s
  → equivocator_state_descriptor_project
      s' (Existing ni) =
    equivocator_state_descriptor_project s
      (Existing ni)
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
ins: nat
sins: state X
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
Hltins: ins < equivocator_state_n s
∃ (i : nat) (si : state X),
  equivocator_state_project s' i = Some si
  ∧ oracle si msg
           specialize (Hnot_last ins).message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
ins: nat
Hnot_last: ins < equivocator_state_n s
→ equivocator_state_descriptor_project s'
    (Existing ins) =
  equivocator_state_descriptor_project s
    (Existing ins)
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
sins: state X
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
Hltins: ins < equivocator_state_n s
∃ (i : nat) (si : state X),
  equivocator_state_project s' i = Some si
  ∧ oracle si msg spec Hnot_last; [lia |].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
ins: nat
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
sins: state X
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
Hltins: ins < equivocator_state_n s
Hnot_last: equivocator_state_descriptor_project s'
  (Existing ins) =
equivocator_state_descriptor_project s
  (Existing ins)
∃ (i : nat) (si : state X),
  equivocator_state_project s' i = Some si
  ∧ oracle si msg
           simpl in Hnot_last.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
ins: nat
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
sins: state X
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
Hltins: ins < equivocator_state_n s
Hnot_last: default (equivocator_state_zero s')
  (equivocator_state_project s' ins) =
default (equivocator_state_zero s)
  (equivocator_state_project s ins)
∃ (i : nat) (si : state X),
  equivocator_state_project s' i = Some si
  ∧ oracle si msg rewrite Hsins in Hnot_last.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
ins: nat
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
sins: state X
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
Hltins: ins < equivocator_state_n s
Hnot_last: default (equivocator_state_zero s')
  (equivocator_state_project s' ins) =
default (equivocator_state_zero s)
  (Some sins)
∃ (i : nat) (si : state X),
  equivocator_state_project s' i = Some si
  ∧ oracle si msg
           exists ins, sins.message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
ins: nat
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
sins: state X
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
Hltins: ins < equivocator_state_n s
Hnot_last: default (equivocator_state_zero s')
  (equivocator_state_project s' ins) =
default (equivocator_state_zero s)
  (Some sins)
equivocator_state_project s' ins = Some sins
∧ oracle sins msg split; [| done].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
ins: nat
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
sins: state X
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
Hltins: ins < equivocator_state_n s
Hnot_last: default (equivocator_state_zero s')
  (equivocator_state_project s' ins) =
default (equivocator_state_zero s)
  (Some sins)
equivocator_state_project s' ins = Some sins
           destruct_equivocator_state_project s' ins _sins Hltins'; [| lia].message: Type
X: VLSM message
selector: message → transition_item → Prop
Hselector_io: ∀ (l1 l2 : label X) (s1 s2 : state X) 
  (im om : option message) 
  (m : message),
  selector m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ selector m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
oracle: state X → message → Prop
Hdec: RelDecision oracle
Hstepwise: oracle_stepwise_props selector oracle
oracle_no_inits: ∀ s : state X,
  initial_state_prop s
  → ∀ m : message, ¬ oracle s m
l: label X
im: option message
sidesc, sidesc': state X
idesc: nat
s, s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
Hs: valid_state_prop
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X)) s
Hidesc: equivocator_state_project s idesc =
Some sidesc
Hv: valid l (sidesc, im)
Ht: transition (ForkWith idesc l) (s, im) = (s', om)
msg: message
oracle_step_update: selector msg
  {|
    l := l;
    input := im;
    destination := sidesc';
    output := om
  |} ∨ oracle sidesc msg
→ oracle sidesc' msg
Htx: transition l (sidesc, im) = (sidesc', om)
ins: nat
Hlast: equivocator_state_descriptor_project s'
  (Existing (equivocator_state_n s)) = sidesc'
Ht_size: equivocator_state_n s' =
S (equivocator_state_n s)
sins: state X
Hsins: equivocator_state_project s ins = Some sins
Hbri: oracle sins msg
Hltins: ins < equivocator_state_n s
_sins: state X
Hnot_last: default (equivocator_state_zero s')
  (Some _sins) =
default (equivocator_state_zero s)
  (Some sins)
Hltins': ins < equivocator_state_n s'
Some _sins = Some sins
           by cbn in Hnot_last; congruence.
Qed.

End sec_oracle_lifting.

Section sec_has_been_received_lifting.

Lifting the HasBeenReceivedCapability

Context
  `{HasBeenReceivedCapability message X}
  .

We define has_been_received for the equivocator_vlsm as being received by any of the internal machines.

Definition equivocator_has_been_received  := equivocator_oracle (has_been_received X).

#[export] Instance equivocator_has_been_received_dec
  : RelDecision equivocator_has_been_received
  := equivocator_oracle_dec (has_been_received X) _.

Lemma equivocator_has_been_received_stepwise_props
  : has_been_received_stepwise_prop (vlsm := equivocator_vlsm X) equivocator_has_been_received.message: Type
X: VLSM message
H: HasBeenReceivedCapability X
has_been_received_stepwise_prop
  equivocator_has_been_received
Proof.message: Type
X: VLSM message
H: HasBeenReceivedCapability X
has_been_received_stepwise_prop
  equivocator_has_been_received
  eapply oracle_stepwise_props_change_selector.message: Type
X: VLSM message
H: HasBeenReceivedCapability X
oracle_stepwise_props ?selector
  equivocator_has_been_received
message: Type
X: VLSM message
H: HasBeenReceivedCapability X
∀ (s : state
         (preloaded_with_all_messages_vlsm
            (equivocator_vlsm X))) 
  (item : transition_item),
  input_constrained_transition_item
    (equivocator_vlsm X) s item
  → ∀ m : message,
      ?selector m item ↔ field_selector input m item
  -message: Type
X: VLSM message
H: HasBeenReceivedCapability X
oracle_stepwise_props ?selector
  equivocator_has_been_received apply equivocator_oracle_stepwise_props
    ; [| by apply has_been_received_stepwise_props].message: Type
X: VLSM message
H: HasBeenReceivedCapability X
∀ (l1 l2 : label X) (s1 s2 : state X) (im
                                       om : option
                                              message) 
  (m : message),
  field_selector input m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ field_selector input m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
    by cbv; itauto.
  -message: Type
X: VLSM message
H: HasBeenReceivedCapability X
∀ (s : state
         (preloaded_with_all_messages_vlsm
            (equivocator_vlsm X))) (item : transition_item),
  input_constrained_transition_item
    (equivocator_vlsm X) s item
  → ∀ m : message,
      equivocator_selector (field_selector input) m
        item ↔ field_selector input m item intros s item; destruct item, l; cbn; [| by itauto..].message: Type
X: VLSM message
H: HasBeenReceivedCapability X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
s0: state X
input: option message
destination: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
output: option message
input_constrained_transition_item (equivocator_vlsm X)
  s
  {|
    l := Spawn s0;
    input := input;
    destination := destination;
    output := output
  |} → ∀ m : message, False ↔ input = Some m
    intros [(_ & _ & _ & Him) _]; simpl in Him; subst.message: Type
X: VLSM message
H: HasBeenReceivedCapability X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
s0: state X
destination: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
output: option message
∀ m : message, False ↔ None = Some m
    by itauto congruence.
Qed.

Finally we define the HasBeenReceivedCapability for the equivocator_vlsm.

#[export] Instance equivocator_HasBeenReceivedCapability
  : HasBeenReceivedCapability (equivocator_vlsm X)
  := Build_HasBeenReceivedCapability (equivocator_vlsm X)
    equivocator_has_been_received
    equivocator_has_been_received_dec
    equivocator_has_been_received_stepwise_props.

End sec_has_been_received_lifting.

Section sec_has_been_sent_lifting.

Lifting the HasBeenSentCapability

Context
  `{HasBeenSentCapability message X}
  .

We define has_been_sent for the equivocator_vlsm as being sent by any of the internal machines.

Definition equivocator_has_been_sent  := equivocator_oracle (has_been_sent X).

#[export] Instance equivocator_has_been_sent_dec
  : RelDecision equivocator_has_been_sent
  := equivocator_oracle_dec (has_been_sent X) _.

Lemma equivocator_has_been_sent_stepwise_props
  : has_been_sent_stepwise_prop (vlsm := equivocator_vlsm X) equivocator_has_been_sent.message: Type
X: VLSM message
H: HasBeenSentCapability X
has_been_sent_stepwise_prop equivocator_has_been_sent
Proof.message: Type
X: VLSM message
H: HasBeenSentCapability X
has_been_sent_stepwise_prop equivocator_has_been_sent
  eapply oracle_stepwise_props_change_selector.message: Type
X: VLSM message
H: HasBeenSentCapability X
oracle_stepwise_props ?selector
  equivocator_has_been_sent
message: Type
X: VLSM message
H: HasBeenSentCapability X
∀ (s : state
         (preloaded_with_all_messages_vlsm
            (equivocator_vlsm X))) 
  (item : transition_item),
  input_constrained_transition_item
    (equivocator_vlsm X) s item
  → ∀ m : message,
      ?selector m item ↔ field_selector output m item
  -message: Type
X: VLSM message
H: HasBeenSentCapability X
oracle_stepwise_props ?selector
  equivocator_has_been_sent apply equivocator_oracle_stepwise_props
    ; [| by apply has_been_sent_stepwise_props].message: Type
X: VLSM message
H: HasBeenSentCapability X
∀ (l1 l2 : label X) (s1 s2 : state X) (im
                                       om : option
                                              message) 
  (m : message),
  field_selector output m
    {|
      l := l1;
      input := im;
      destination := s1;
      output := om
    |}
  ↔ field_selector output m
      {|
        l := l2;
        input := im;
        destination := s2;
        output := om
      |}
    by cbv; itauto.
  -message: Type
X: VLSM message
H: HasBeenSentCapability X
∀ (s : state
         (preloaded_with_all_messages_vlsm
            (equivocator_vlsm X))) (item : transition_item),
  input_constrained_transition_item
    (equivocator_vlsm X) s item
  → ∀ m : message,
      equivocator_selector (field_selector output) m
        item ↔ field_selector output m item intros s item; destruct item, l; cbn; [| by itauto..].message: Type
X: VLSM message
H: HasBeenSentCapability X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
s0: state X
input: option message
destination: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
output: option message
input_constrained_transition_item (equivocator_vlsm X)
  s
  {|
    l := Spawn s0;
    input := input;
    destination := destination;
    output := output
  |} → ∀ m : message, False ↔ output = Some m
    intros [_ Ht]; inversion_clear Ht.message: Type
X: VLSM message
H: HasBeenSentCapability X
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
s0: state X
input: option message
destination: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
output: option message
∀ m : message, False ↔ None = Some m
    by itauto congruence.
Qed.

Finally we define the HasBeenSentCapability for the equivocator_vlsm.

#[export] Instance equivocator_HasBeenSentCapability
  : HasBeenSentCapability (equivocator_vlsm X)
  := Build_HasBeenSentCapability
    (equivocator_vlsm X)
    equivocator_has_been_sent
    equivocator_has_been_sent_dec
    equivocator_has_been_sent_stepwise_props.

End sec_has_been_sent_lifting.

Section sec_ComputableSentMessages_lifting.

Lifting the ComputableSentMessages property

Context
  `{!ComputableSentMessages X}
  `(EqDecision message)
  .

We define the sent_messages_set for the equivocator_vlsm as the union of all sent_messages_set for its internal machines.

Definition equivocator_sent_messages_set
  (s : state (equivocator_vlsm X))
  : set message
  :=
  fold_right set_union []
    (map
      (fun i =>
        match equivocator_state_project s i with
        | None => []
        | Some si => sent_messages_set si
        end)
      (up_to_n_listing (equivocator_state_n s))).

Lemma equivocator_elem_of_sent_messages_set :
  forall (s : state (equivocator_vlsm X)) (m : message),
    equivocator_has_been_sent s m
      <->
    m ∈ equivocator_sent_messages_set s.message: Type
X: VLSM message
ComputableSentMessages0: ComputableSentMessages X
EqDecision0: EqDecision message
∀ (s : state (equivocator_vlsm X)) (m : message),
  equivocator_has_been_sent s m
  ↔ m ∈ equivocator_sent_messages_set s
Proof.message: Type
X: VLSM message
ComputableSentMessages0: ComputableSentMessages X
EqDecision0: EqDecision message
∀ (s : state (equivocator_vlsm X)) (m : message),
  equivocator_has_been_sent s m
  ↔ m ∈ equivocator_sent_messages_set s
  split.message: Type
X: VLSM message
ComputableSentMessages0: ComputableSentMessages X
EqDecision0: EqDecision message
s: state (equivocator_vlsm X)
m: message
equivocator_has_been_sent s m
→ m ∈ equivocator_sent_messages_set s
message: Type
X: VLSM message
ComputableSentMessages0: ComputableSentMessages X
EqDecision0: EqDecision message
s: state (equivocator_vlsm X)
m: message
m ∈ equivocator_sent_messages_set s
→ equivocator_has_been_sent s m
  -message: Type
X: VLSM message
ComputableSentMessages0: ComputableSentMessages X
EqDecision0: EqDecision message
s: state (equivocator_vlsm X)
m: message
equivocator_has_been_sent s m
→ m ∈ equivocator_sent_messages_set s intros (i & si & Hsi & Hbs).message: Type
X: VLSM message
ComputableSentMessages0: ComputableSentMessages X
EqDecision0: EqDecision message
s: state (equivocator_vlsm X)
m: message
i: nat
si: state X
Hsi: equivocator_state_project s i = Some si
Hbs: has_been_sent X si m
m ∈ equivocator_sent_messages_set s
    apply set_union_in_iterated, Exists_exists.message: Type
X: VLSM message
ComputableSentMessages0: ComputableSentMessages X
EqDecision0: EqDecision message
s: state (equivocator_vlsm X)
m: message
i: nat
si: state X
Hsi: equivocator_state_project s i = Some si
Hbs: has_been_sent X si m
∃ x : set message,
  x
  ∈ map
      (λ i : nat,
         match equivocator_state_project s i with
         | Some si => sent_messages_set si
         | None => []
         end)
      (up_to_n_listing (equivocator_state_n s))
  ∧ m ∈ x
    eexists; split; [| by eapply elem_of_sent_messages_set].message: Type
X: VLSM message
ComputableSentMessages0: ComputableSentMessages X
EqDecision0: EqDecision message
s: state (equivocator_vlsm X)
m: message
i: nat
si: state X
Hsi: equivocator_state_project s i = Some si
Hbs: has_been_sent X si m
sent_messages_set si
∈ map
    (λ i : nat,
       match equivocator_state_project s i with
       | Some si => sent_messages_set si
       | None => []
       end) (up_to_n_listing (equivocator_state_n s))
    apply elem_of_list_fmap; exists i; split; [by rewrite Hsi |].message: Type
X: VLSM message
ComputableSentMessages0: ComputableSentMessages X
EqDecision0: EqDecision message
s: state (equivocator_vlsm X)
m: message
i: nat
si: state X
Hsi: equivocator_state_project s i = Some si
Hbs: has_been_sent X si m
i ∈ up_to_n_listing (equivocator_state_n s)
    by eapply up_to_n_full, equivocator_state_project_Some_rev.
  -message: Type
X: VLSM message
ComputableSentMessages0: ComputableSentMessages X
EqDecision0: EqDecision message
s: state (equivocator_vlsm X)
m: message
m ∈ equivocator_sent_messages_set s
→ equivocator_has_been_sent s m intros Hm.message: Type
X: VLSM message
ComputableSentMessages0: ComputableSentMessages X
EqDecision0: EqDecision message
s: state (equivocator_vlsm X)
m: message
Hm: m ∈ equivocator_sent_messages_set s
equivocator_has_been_sent s m
    apply set_union_in_iterated, Exists_exists in Hm as (senti & Hsenti & Hm).message: Type
X: VLSM message
ComputableSentMessages0: ComputableSentMessages X
EqDecision0: EqDecision message
s: state (equivocator_vlsm X)
m: message
senti: set message
Hsenti: senti
∈ map
    (λ i : nat,
       match
         equivocator_state_project s i
       with
       | Some si => sent_messages_set si
       | None => []
       end)
    (up_to_n_listing (equivocator_state_n s))
Hm: m ∈ senti
equivocator_has_been_sent s m
    apply elem_of_list_fmap in Hsenti as (i & -> & Hi).message: Type
X: VLSM message
ComputableSentMessages0: ComputableSentMessages X
EqDecision0: EqDecision message
s: state (equivocator_vlsm X)
m: message
i: nat
Hi: i ∈ up_to_n_listing (equivocator_state_n s)
Hm: m
∈ match equivocator_state_project s i with
  | Some si => sent_messages_set si
  | None => []
  end
equivocator_has_been_sent s m
    apply up_to_n_full in Hi.message: Type
X: VLSM message
ComputableSentMessages0: ComputableSentMessages X
EqDecision0: EqDecision message
s: state (equivocator_vlsm X)
m: message
i: nat
Hi: i < equivocator_state_n s
Hm: m
∈ match equivocator_state_project s i with
  | Some si => sent_messages_set si
  | None => []
  end
equivocator_has_been_sent s m
    destruct (equivocator_state_project s i) as [si |] eqn: Hsi; [| inversion Hm].message: Type
X: VLSM message
ComputableSentMessages0: ComputableSentMessages X
EqDecision0: EqDecision message
s: state (equivocator_vlsm X)
m: message
i: nat
Hi: i < equivocator_state_n s
si: state X
Hsi: equivocator_state_project s i = Some si
Hm: m ∈ sent_messages_set si
equivocator_has_been_sent s m
    by exists i, si; split; [| apply elem_of_sent_messages_set].
Qed.

Lemma equivocator_ComputableSentMessages :
  ComputableSentMessages (equivocator_vlsm X).message: Type
X: VLSM message
ComputableSentMessages0: ComputableSentMessages X
EqDecision0: EqDecision message
ComputableSentMessages (equivocator_vlsm X)
Proof.message: Type
X: VLSM message
ComputableSentMessages0: ComputableSentMessages X
EqDecision0: EqDecision message
ComputableSentMessages (equivocator_vlsm X)
  constructor 1 with equivocator_sent_messages_set; constructor; intros.message: Type
X: VLSM message
ComputableSentMessages0: ComputableSentMessages X
EqDecision0: EqDecision message
s: state (equivocator_vlsm X)
H: initial_state_prop s
m: message
m ∉ equivocator_sent_messages_set s
message: Type
X: VLSM message
ComputableSentMessages0: ComputableSentMessages X
EqDecision0: EqDecision message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
im: option message
s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
H: input_constrained_transition 
  (equivocator_vlsm X) l (
  s, im) (s', om)
msg: message
msg ∈ equivocator_sent_messages_set s'
↔ field_selector output msg
    {|
      l := l;
      input := im;
      destination := s';
      output := om
    |} ∨ msg ∈ equivocator_sent_messages_set s
  -message: Type
X: VLSM message
ComputableSentMessages0: ComputableSentMessages X
EqDecision0: EqDecision message
s: state (equivocator_vlsm X)
H: initial_state_prop s
m: message
m ∉ equivocator_sent_messages_set s rewrite <- equivocator_elem_of_sent_messages_set.message: Type
X: VLSM message
ComputableSentMessages0: ComputableSentMessages X
EqDecision0: EqDecision message
s: state (equivocator_vlsm X)
H: initial_state_prop s
m: message
¬ equivocator_has_been_sent s m
    by eapply equivocator_has_been_sent_stepwise_props.
  -message: Type
X: VLSM message
ComputableSentMessages0: ComputableSentMessages X
EqDecision0: EqDecision message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
im: option message
s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
H: input_constrained_transition 
  (equivocator_vlsm X) l (
  s, im) (s', om)
msg: message
msg ∈ equivocator_sent_messages_set s'
↔ field_selector output msg
    {|
      l := l;
      input := im;
      destination := s';
      output := om
    |} ∨ msg ∈ equivocator_sent_messages_set s rewrite <- !equivocator_elem_of_sent_messages_set.message: Type
X: VLSM message
ComputableSentMessages0: ComputableSentMessages X
EqDecision0: EqDecision message
l: label
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
s: state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
im: option message
s': state
  (preloaded_with_all_messages_vlsm
     (equivocator_vlsm X))
om: option message
H: input_constrained_transition 
  (equivocator_vlsm X) l (
  s, im) (s', om)
msg: message
equivocator_has_been_sent s' msg
↔ field_selector output msg
    {|
      l := l;
      input := im;
      destination := s';
      output := om
    |} ∨ equivocator_has_been_sent s msg
    by eapply equivocator_has_been_sent_stepwise_props.
Qed.

End sec_ComputableSentMessages_lifting.

End sec_equivocator_vlsm_message_properties.