proto_channel.v 37.9 KB
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(** This file defines the core of the Actris logic:

- It defines dependent separation protocols and the various operations on it
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  like dual, append, and branching.
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- It defines the connective [c ↣ prot] for ownership of channel endpoints.
- It proves Actris's specifications of [send] and [receive] w.r.t. dependent
  separation protocols.

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Dependent separation protocols are defined by instantiating the parameterized
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version in [proto_model] with type of values [val] of HeapLang and the
propositions [iProp] of Iris.
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In doing so we define ways of constructing instances of the instantiated type
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via two constructors:
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- [iProto_end], which is identical to [proto_end].
- [iProto_message], which takes an action and a continuation to construct
  the corresponding message protocols.

For convenience sake, we provide the following notations:
- [END], which is simply [iProto_end].
- [<!> x1 .. xn, MSG v; {{ P }}; prot] and [<?> x1 .. xn, MSG v; {{ P }}; prot],
  which construct an instance of [iProto_message] with the appropriate
  continuation.

Futhermore, we define the following operations:
- [iProto_dual], which turns all [Send] of a protocol into [Recv] and vice-versa
- [iProto_app], which appends two protocols as described in proto_model.v

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An encoding of the usual branching connectives [prot1 <{Q1}+{Q2}> prot2] and
[prot1 <{Q1}&{Q2}> prot2], inspired by session types, is also included in this
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file.

The logical connective for protocol ownership is denoted as [c ↣ prot]. It
describes that channel endpoint [c] adheres to protocol [prot]. This connective
is modeled using Iris invariants and ghost state along with the logical
connectives of the channel encodings [is_chan] and [chan_own].

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Lastly, relevant type classes instances are defined for each of the above
notions, such as contractiveness and non-expansiveness, after which the
specifications of the message-passing primitives are defined in terms of the
protocol connectives. *)
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From actris.channel Require Export channel. 
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From actris.channel Require Import proto_model.
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From iris.base_logic.lib Require Import invariants.
From iris.heap_lang Require Import proofmode notation.
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From iris.algebra Require Import excl_auth.
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Export action.
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Definition start_chan : val := λ: "f",
  let: "cc" := new_chan #() in
  Fork ("f" (Snd "cc"));; Fst "cc".

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(** * Setup of Iris's cameras *)
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Class proto_chanG Σ := {
  proto_chanG_chanG :> chanG Σ;
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  proto_chanG_authG :> inG Σ (excl_authR (laterO (proto val (iPrePropO Σ) (iPrePropO Σ))));
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}.

Definition proto_chanΣ := #[
  chanΣ;
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  GFunctor (authRF (optionURF (exclRF (laterOF (protoOF val idOF idOF)))))
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].
Instance subG_chanΣ {Σ} : subG proto_chanΣ Σ  proto_chanG Σ.
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Proof. intros [??%subG_inG]%subG_inv. constructor; apply _. Qed.
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(** * Types *)
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Definition iProto Σ := proto val (iPropO Σ) (iPropO Σ).
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Delimit Scope proto_scope with proto.
Bind Scope proto_scope with iProto.

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(** * Operators *)
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Definition iProto_end_def {Σ} : iProto Σ := proto_end.
Definition iProto_end_aux : seal (@iProto_end_def). by eexists. Qed.
Definition iProto_end := iProto_end_aux.(unseal).
Definition iProto_end_eq : @iProto_end = @iProto_end_def := iProto_end_aux.(seal_eq).
Arguments iProto_end {_}.

Program Definition iProto_message_def {Σ} {TT : tele} (a : action)
    (pc : TT  val * iProp Σ * iProto Σ) : iProto Σ :=
  proto_message a (λ v, λne f,  x : TT,
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    (** We need the later to make [iProto_message] contractive *)
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     v = (pc x).1.1  
     (pc x).1.2 
    f (Next (pc x).2))%I.
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Next Obligation. solve_proper. Qed.
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Definition iProto_message_aux : seal (@iProto_message_def). by eexists. Qed.
Definition iProto_message := iProto_message_aux.(unseal).
Definition iProto_message_eq : @iProto_message = @iProto_message_def := iProto_message_aux.(seal_eq).
Arguments iProto_message {_ _} _ _%proto.
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Instance: Params (@iProto_message) 3 := {}.
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Notation "< a > x1 .. xn , 'MSG' v {{ P } } ; p" :=
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  (iProto_message
    a
    (tele_app (TT:=TeleS (λ x1, .. (TeleS (λ xn, TeleO)) .. )) $
                       λ x1, .. (λ xn, (v%V,P%I,p%proto)) ..))
  (at level 20, a at level 10, x1 binder, xn binder, v at level 20, P, p at level 200) : proto_scope.
Notation "< a > x1 .. xn , 'MSG' v ; p" :=
  (iProto_message
    a
    (tele_app (TT:=TeleS (λ x1, .. (TeleS (λ xn, TeleO)) .. )) $
                       λ x1, .. (λ xn, (v%V,True%I,p%proto)) ..))
  (at level 20, a at level 10, x1 binder, xn binder, v at level 20, p at level 200) : proto_scope.
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Notation "< a > 'MSG' v {{ P } } ; p" :=
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  (iProto_message
    a
    (tele_app (TT:=TeleO) (v%V,P%I,p%proto)))
  (at level 20, a at level 10, v at level 20, P, p at level 200) : proto_scope.
Notation "< a > 'MSG' v ; p" :=
  (iProto_message
    a
    (tele_app (TT:=TeleO) (v%V,True%I,p%proto)))
  (at level 20, a at level 10, v at level 20, p at level 200) : proto_scope.
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Notation "<!> x1 .. xn , 'MSG' v {{ P } } ; p" :=
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  (iProto_message
    Send
    (tele_app (TT:=TeleS (λ x1, .. (TeleS (λ xn, TeleO)) .. )) $
                       λ x1, .. (λ xn, (v%V,P%I,p%proto)) ..))
  (at level 20, x1 binder, xn binder, v at level 20, P, p at level 200) : proto_scope.
Notation "<!> x1 .. xn , 'MSG' v ; p" :=
  (iProto_message
    Send
    (tele_app (TT:=TeleS (λ x1, .. (TeleS (λ xn, TeleO)) .. )) $
                       λ x1, .. (λ xn, (v%V,True%I,p%proto)) ..))
  (at level 20, x1 binder, xn binder, v at level 20, p at level 200) : proto_scope.
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Notation "<!> 'MSG' v {{ P } } ; p" :=
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  (iProto_message
    (TT:=TeleO)
    Send
    (tele_app (TT:=TeleO) (v%V,P%I,p%proto)))
  (at level 20, v at level 20, P, p at level 200) : proto_scope.
Notation "<!> 'MSG' v ; p" :=
  (iProto_message
    (TT:=TeleO)
    Send
    (tele_app (TT:=TeleO) (v%V,True%I,p%proto)))
  (at level 20, v at level 20, p at level 200) : proto_scope.
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Notation "<?> x1 .. xn , 'MSG' v {{ P } } ; p" :=
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  (iProto_message
    Receive
    (tele_app (TT:=TeleS (λ x1, .. (TeleS (λ xn, TeleO)) .. )) $
                       λ x1, .. (λ xn, (v%V,P%I,p%proto)) ..))
  (at level 20, x1 binder, xn binder, v at level 20, P, p at level 200) : proto_scope.
Notation "<?> x1 .. xn , 'MSG' v ; p" :=
  (iProto_message
    Receive
    (tele_app (TT:=TeleS (λ x1, .. (TeleS (λ xn, TeleO)) .. )) $
                       λ x1, .. (λ xn, (v%V,True%I,p%proto)) ..))
  (at level 20, x1 binder, xn binder, v at level 20, p at level 200) : proto_scope.
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Notation "<?> 'MSG' v {{ P } } ; p" :=
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  (iProto_message
    Receive
    (tele_app (TT:=TeleO) (v%V,P%I,p%proto)))
  (at level 20, v at level 20, P, p at level 200) : proto_scope.
Notation "<?> 'MSG' v ; p" :=
  (iProto_message
    Receive
    (tele_app (TT:=TeleO) (v%V,True%I,p%proto)))
  (at level 20, v at level 20, p at level 200) : proto_scope.
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Notation "'END'" := iProto_end : proto_scope.
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(** * Operations *)
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Definition iProto_dual {Σ} (p : iProto Σ) : iProto Σ :=
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  proto_map action_dual cid cid p.
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Arguments iProto_dual {_} _%proto.
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Instance: Params (@iProto_dual) 1 := {}.
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Definition iProto_dual_if {Σ} (d : bool) (p : iProto Σ) : iProto Σ :=
  if d then iProto_dual p else p.
Arguments iProto_dual_if {_} _ _%proto.
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Instance: Params (@iProto_dual_if) 2 := {}.
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Definition iProto_branch {Σ} (a : action) (P1 P2 : iProp Σ)
    (p1 p2 : iProto Σ) : iProto Σ :=
  (<a> (b : bool), MSG #b {{ if b then P1 else P2 }}; if b then p1 else p2)%proto.
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Typeclasses Opaque iProto_branch.
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Arguments iProto_branch {_} _ _%I _%I _%proto _%proto.
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Instance: Params (@iProto_branch) 2 := {}.
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Infix "<{ P1 }+{ P2 }>" := (iProto_branch Send P1 P2) (at level 85) : proto_scope.
Infix "<{ P1 }&{ P2 }>" := (iProto_branch Receive P1 P2) (at level 85) : proto_scope.
Infix "<+{ P2 }>" := (iProto_branch Send True P2) (at level 85) : proto_scope.
Infix "<&{ P2 }>" := (iProto_branch Receive True P2) (at level 85) : proto_scope.
Infix "<{ P1 }+>" := (iProto_branch Send P1 True) (at level 85) : proto_scope.
Infix "<{ P1 }&>" := (iProto_branch Receive P1 True) (at level 85) : proto_scope.
Infix "<+>" := (iProto_branch Send True True) (at level 85) : proto_scope.
Infix "<&>" := (iProto_branch Receive True True) (at level 85) : proto_scope.
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Definition iProto_app {Σ} (p1 p2 : iProto Σ) : iProto Σ := proto_app p1 p2.
Arguments iProto_app {_} _%proto _%proto.
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Instance: Params (@iProto_app) 1 := {}.
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Infix "<++>" := iProto_app (at level 60) : proto_scope.

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(** * Auxiliary definitions and invariants *)
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Definition proto_eq_next {Σ} (p : iProto Σ) : laterO (iProto Σ) -n> iPropO Σ :=
  OfeMor (sbi_internal_eq (Next p)).

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Program Definition iProto_le_aux `{invG Σ} (rec : iProto Σ -n> iProto Σ -n> iPropO Σ) :
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    iProto Σ -n> iProto Σ -n> iPropO Σ := λne p1 p2,
  ((p1  proto_end  p2  proto_end) 
   ( pc1 pc2,
     p1  proto_message Send pc1  p2  proto_message Send pc2 
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      v p2', pc2 v (proto_eq_next p2') ={}=
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        p1',  rec p1' p2'  pc1 v (proto_eq_next p1')) 
   ( pc1 pc2,
     p1  proto_message Receive pc1  p2  proto_message Receive pc2 
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      v p1', pc1 v (proto_eq_next p1') ={}=
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        p2',  rec p1' p2'  pc2 v (proto_eq_next p2')))%I.
Solve Obligations with solve_proper.
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Local Instance iProto_le_aux_contractive `{invG Σ} : Contractive (@iProto_le_aux Σ _).
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Proof. solve_contractive. Qed.
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Definition iProto_le `{invG Σ} (p1 p2 : iProto Σ) : iProp Σ :=
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  fixpoint iProto_le_aux p1 p2.
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Arguments iProto_le {_ _} _%proto _%proto.
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Fixpoint proto_interp {Σ} (vs : list val) (p1 p2 : iProto Σ) : iProp Σ :=
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  match vs with
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  | [] => iProto_dual p1  p2
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  | v :: vs =>  pc p2',
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     p2  proto_message Receive pc 
     pc v (proto_eq_next p2') 
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     proto_interp vs p1 p2'
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  end%I.
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Arguments proto_interp {_} _ _%proto _%proto : simpl nomatch.
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Record proto_name := ProtName {
  proto_c_name : chan_name;
  proto_l_name : gname;
  proto_r_name : gname
}.

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Definition to_pre_proto {Σ} (p : iProto Σ) :
    proto val (iPrePropO Σ) (iPrePropO Σ) :=
  proto_map id iProp_fold iProp_unfold p.
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Definition proto_own_frag `{!proto_chanG Σ} (γ : proto_name) (s : side)
    (p : iProto Σ) : iProp Σ :=
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  own (side_elim s proto_l_name proto_r_name γ) (E (Next (to_pre_proto p))).
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Definition proto_own_auth `{!proto_chanG Σ} (γ : proto_name) (s : side)
    (p : iProto Σ) : iProp Σ :=
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  own (side_elim s proto_l_name proto_r_name γ) (E (Next (to_pre_proto p))).
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Definition proto_inv `{!proto_chanG Σ} (γ : proto_name) : iProp Σ :=
  ( l r pl pr,
    chan_own (proto_c_name γ) Left l 
    chan_own (proto_c_name γ) Right r 
    proto_own_auth γ Left pl 
    proto_own_auth γ Right pr 
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     ((r = []  proto_interp l pl pr) 
       (l = []  proto_interp r pr pl)))%I.
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Definition protoN := nroot .@ "proto".

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(** * The connective for ownership of channel ends *)
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Definition mapsto_proto_def `{!proto_chanG Σ, !heapG Σ}
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    (c : val) (p : iProto Σ) : iProp Σ :=
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  ( s (c1 c2 : val) γ p',
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     c = side_elim s c1 c2  
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    iProto_le p' p 
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    proto_own_frag γ s p' 
    is_chan protoN (proto_c_name γ) c1 c2 
    inv protoN (proto_inv γ))%I.
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Definition mapsto_proto_aux : seal (@mapsto_proto_def). by eexists. Qed.
Definition mapsto_proto {Σ pΣ hΣ} := mapsto_proto_aux.(unseal) Σ pΣ hΣ.
Definition mapsto_proto_eq : @mapsto_proto = @mapsto_proto_def := mapsto_proto_aux.(seal_eq).
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Arguments mapsto_proto {_ _ _} _ _%proto.
Instance: Params (@mapsto_proto) 4 := {}.
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Notation "c ↣ p" := (mapsto_proto c p)
  (at level 20, format "c  ↣  p").
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(** * Proofs *)
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Section proto.
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  Context `{!proto_chanG Σ, !heapG Σ}.
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  Implicit Types p : iProto Σ.
  Implicit Types TT : tele.

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  (** ** Non-expansiveness of operators *)
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  Lemma iProto_message_contractive {TT} a
      (pc1 pc2 : TT  val * iProp Σ * iProto Σ) n :
    (.. x, (pc1 x).1.1 = (pc2 x).1.1) 
    (.. x, dist_later n ((pc1 x).1.2) ((pc2 x).1.2)) 
    (.. x, dist_later n ((pc1 x).2) ((pc2 x).2)) 
    iProto_message a pc1 {n} iProto_message a pc2.
  Proof.
    rewrite !tforall_forall=> Hv HP Hp.
    rewrite iProto_message_eq /iProto_message_def.
    f_equiv=> v f /=. apply bi.exist_ne=> x.
    repeat (apply Hv || apply HP || apply Hp || f_contractive || f_equiv).
  Qed.
  Lemma iProto_message_ne {TT} a
      (pc1 pc2 : TT  val * iProp Σ * iProto Σ) n :
    (.. x, (pc1 x).1.1 = (pc2 x).1.1) 
    (.. x, (pc1 x).1.2 {n} (pc2 x).1.2) 
    (.. x, (pc1 x).2 {n} (pc2 x).2) 
    iProto_message a pc1 {n} iProto_message a pc2.
  Proof.
    rewrite !tforall_forall=> Hv HP Hp.
    apply iProto_message_contractive; apply tforall_forall; eauto using dist_dist_later.
  Qed.
  Lemma iProto_message_proper {TT} a
      (pc1 pc2 : TT  val * iProp Σ * iProto Σ) :
    (.. x, (pc1 x).1.1 = (pc2 x).1.1) 
    (.. x, (pc1 x).1.2  (pc2 x).1.2) 
    (.. x, (pc1 x).2  (pc2 x).2) 
    iProto_message a pc1  iProto_message a pc2.
  Proof.
    rewrite !tforall_forall=> Hv HP Hp. apply equiv_dist => n.
    apply iProto_message_ne; apply tforall_forall=> x; by try apply equiv_dist.
  Qed.
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  Global Instance iProto_branch_contractive n a :
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    Proper (dist_later n ==> dist_later n ==>
            dist_later n ==> dist_later n ==> dist n) (@iProto_branch Σ a).
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  Proof.
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    intros p1 p1' Hp1 p2 p2' Hp2 P1 P1' HP1 P2 P2' HP2.
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    apply iProto_message_contractive=> /= -[] //.
  Qed.
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  Global Instance iProto_branch_ne n a :
    Proper (dist n ==> dist n ==> dist n ==> dist n ==> dist n) (@iProto_branch Σ a).
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  Proof.
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    intros p1 p1' Hp1 p2 p2' Hp2 P1 P1' HP1 P2 P2' HP2.
    by apply iProto_message_ne=> /= -[].
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  Qed.
  Global Instance iProto_branch_proper a :
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    Proper (() ==> () ==> () ==> () ==> ()) (@iProto_branch Σ a).
  Proof.
    intros p1 p1' Hp1 p2 p2' Hp2 P1 P1' HP1 P2 P2' HP2.
    by apply iProto_message_proper=> /= -[].
  Qed.
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  (** ** Dual *)
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  Global Instance iProto_dual_ne : NonExpansive (@iProto_dual Σ).
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  Proof. solve_proper. Qed.
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  Global Instance iProto_dual_proper : Proper (() ==> ()) (@iProto_dual Σ).
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  Proof. apply (ne_proper _). Qed.
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  Global Instance iProto_dual_if_ne d : NonExpansive (@iProto_dual_if Σ d).
  Proof. solve_proper. Qed.
  Global Instance iProto_dual_if_proper d : Proper (() ==> ()) (@iProto_dual_if Σ d).
  Proof. apply (ne_proper _). Qed.
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  Global Instance iProto_dual_involutive : Involutive () (@iProto_dual Σ).
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  Proof.
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    intros p. rewrite /iProto_dual -proto_map_compose -{2}(proto_map_id p).
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    apply: proto_map_ext=> //. by intros [].
  Qed.
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  Lemma iProto_dual_end : iProto_dual (Σ:=Σ) END  END%proto.
  Proof. by rewrite iProto_end_eq /iProto_dual proto_map_end. Qed.
  Lemma iProto_dual_message {TT} a (pc : TT  val * iProp Σ * iProto Σ) :
    iProto_dual (iProto_message a pc)
     iProto_message (action_dual a) (prod_map id iProto_dual  pc).
  Proof.
    rewrite /iProto_dual iProto_message_eq /iProto_message_def proto_map_message.
    by f_equiv=> v f /=.
  Qed.

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  Lemma iProto_dual_branch a P1 P2 p1 p2 :
    iProto_dual (iProto_branch a P1 P2 p1 p2)
     iProto_branch (action_dual a) P1 P2 (iProto_dual p1) (iProto_dual p2).
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  Proof.
    rewrite /iProto_branch iProto_dual_message /=.
    by apply iProto_message_proper=> /= -[].
  Qed.

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  (** ** Append *)
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  Global Instance iProto_app_ne : NonExpansive2 (@iProto_app Σ).
  Proof. apply _. Qed.
  Global Instance iProto_app_proper : Proper (() ==> () ==> ()) (@iProto_app Σ).
  Proof. apply (ne_proper_2 _). Qed.

  Lemma iProto_app_message {TT} a (pc : TT  val * iProp Σ * iProto Σ) p2 :
    (iProto_message a pc <++> p2)%proto  iProto_message a (prod_map id (flip iProto_app p2)  pc).
  Proof.
    rewrite /iProto_app iProto_message_eq /iProto_message_def proto_app_message.
    by f_equiv=> v f /=.
  Qed.

  Global Instance iProto_app_end_l : LeftId () END%proto (@iProto_app Σ).
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  Proof.
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    intros p. by rewrite iProto_end_eq /iProto_end_def /iProto_app proto_app_end_l.
  Qed.
  Global Instance iProto_app_end_r : RightId () END%proto (@iProto_app Σ).
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  Proof.
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    intros p. by rewrite iProto_end_eq /iProto_end_def /iProto_app proto_app_end_r.
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  Qed.
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  Global Instance iProto_app_assoc : Assoc () (@iProto_app Σ).
  Proof. intros p1 p2 p3. by rewrite /iProto_app proto_app_assoc. Qed.

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  Lemma iProto_app_branch a P1 P2 p1 p2 q :
    (iProto_branch a P1 P2 p1 p2 <++> q)%proto
     (iProto_branch a P1 P2 (p1 <++> q) (p2 <++> q))%proto.
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  Proof.
    rewrite /iProto_branch iProto_app_message.
    by apply iProto_message_proper=> /= -[].
  Qed.

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  Lemma iProto_dual_app p1 p2 :
    iProto_dual (p1 <++> p2)  (iProto_dual p1 <++> iProto_dual p2)%proto.
  Proof. by rewrite /iProto_dual /iProto_app proto_map_app. Qed.

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  (** ** Protocol entailment **)
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  Global Instance iProto_le_ne : NonExpansive2 (@iProto_le Σ _).
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  Proof. solve_proper. Qed.
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  Global Instance iProto_le_proper : Proper (() ==> () ==> ()) (@iProto_le Σ _).
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  Proof. solve_proper. Qed.

  Lemma iProto_le_unfold p1 p2 :
    iProto_le p1 p2  iProto_le_aux (fixpoint iProto_le_aux) p1 p2.
  Proof. apply: (fixpoint_unfold iProto_le_aux). Qed.

  Lemma iProto_le_refl p : iProto_le p p.
  Proof.
    iLöb as "IH" forall (p). destruct (proto_case p) as [->|([]&pc&->)].
    - rewrite iProto_le_unfold. iLeft; by auto.
    - rewrite iProto_le_unfold. iRight; iLeft. iExists _, _; do 2 (iSplit; [done|]).
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      iIntros (v p') "Hpc". iExists p'. iIntros "{$Hpc} !> !>". iApply "IH".
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    - rewrite iProto_le_unfold. iRight; iRight. iExists _, _; do 2 (iSplit; [done|]).
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      iIntros (v p') "Hpc". iExists p'. iIntros "{$Hpc} !> !>". iApply "IH".
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  Qed.

  Lemma iProto_le_end_inv p : iProto_le p proto_end - p  proto_end.
  Proof.
    rewrite iProto_le_unfold. iIntros "[[Hp _]|H] //".
    iDestruct "H" as "[H|H]"; iDestruct "H" as (pc1 pc2) "(_ & Heq & _)";
      by rewrite proto_end_message_equivI.
  Qed.

  Lemma iProto_le_send_inv p1 pc2 :
    iProto_le p1 (proto_message Send pc2) -  pc1,
      p1  proto_message Send pc1 
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       v p2', pc2 v (proto_eq_next p2') ={}=
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         p1',  iProto_le p1' p2'  pc1 v (proto_eq_next p1').
  Proof.
    rewrite iProto_le_unfold. iIntros "[[_ Heq]|[H|H]]".
    - by rewrite proto_message_end_equivI.
    - iDestruct "H" as (pc1 pc2') "(Hp1 & Heq & H)".
      iDestruct (proto_message_equivI with "Heq") as "[_ #Heq]".
      iExists pc1. iIntros "{$Hp1}" (v p2') "Hpc".
      iSpecialize ("Heq" $! v). iDestruct (bi.ofe_morO_equivI with "Heq") as "Heq'".
      iRewrite ("Heq'" $! (proto_eq_next p2')) in "Hpc". by iApply "H".
    - iDestruct "H" as (pc1 pc2') "(_ & Heq & _)".
      by iDestruct (proto_message_equivI with "Heq") as "[% ?]".
  Qed.

  Lemma iProto_le_recv_inv p1 pc2 :
    iProto_le p1 (proto_message Receive pc2) -  pc1,
      p1  proto_message Receive pc1 
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       v p1', pc1 v (proto_eq_next p1') ={}=
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         p2',  iProto_le p1' p2'  pc2 v (proto_eq_next p2').
  Proof.
    rewrite iProto_le_unfold. iIntros "[[_ Heq]|[H|H]]".
    - by rewrite proto_message_end_equivI.
    - iDestruct "H" as (pc1 pc2') "(_ & Heq & _)".
      by iDestruct (proto_message_equivI with "Heq") as "[% ?]".
    - iDestruct "H" as (pc1 pc2') "(Hp1 & Heq & H)".
      iDestruct (proto_message_equivI with "Heq") as "[_ #Heq]".
      iExists pc1. iIntros "{$Hp1}" (v p1') "Hpc".
      iSpecialize ("Heq" $! v). iDestruct (bi.ofe_morO_equivI with "Heq") as "Heq'".
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      iMod ("H" with "Hpc") as (p2') "[Hle Hpc]". iModIntro.
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      iExists p2'. iFrame "Hle". by iRewrite ("Heq'" $! (proto_eq_next p2')).
  Qed.

  Lemma iProto_le_trans p1 p2 p3 :
    iProto_le p1 p2 - iProto_le p2 p3 - iProto_le p1 p3.
  Proof.
    iIntros "H1 H2". iLöb as "IH" forall (p1 p2 p3).
    destruct (proto_case p3) as [->|([]&pc3&->)].
    - rewrite iProto_le_end_inv. by iRewrite "H2" in "H1".
    - iDestruct (iProto_le_send_inv with "H2") as (pc2) "[Hp2 H3]".
      iRewrite "Hp2" in "H1".
      iDestruct (iProto_le_send_inv with "H1") as (pc1) "[Hp1 H2]".
      iRewrite "Hp1". rewrite iProto_le_unfold; iRight; iLeft.
      iExists _, _; do 2 (iSplit; [done|]).
      iIntros (v p3') "Hpc".
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      iMod ("H3" with "Hpc") as (p2') "[Hle Hpc]".
      iMod ("H2" with "Hpc") as (p1') "[Hle' Hpc]".
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      iExists p1'. iIntros "{$Hpc} !>". by iApply ("IH" with "Hle'").
    - iDestruct (iProto_le_recv_inv with "H2") as (pc2) "[Hp2 H3]".
      iRewrite "Hp2" in "H1".
      iDestruct (iProto_le_recv_inv with "H1") as (pc1) "[Hp1 H2]".
      iRewrite "Hp1". rewrite iProto_le_unfold; iRight; iRight.
      iExists _, _; do 2 (iSplit; [done|]).
      iIntros (v p1') "Hpc".
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      iMod ("H2" with "Hpc") as (p2') "[Hle Hpc]".
      iMod ("H3" with "Hpc") as (p3') "[Hle' Hpc]".
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      iExists p3'. iIntros "{$Hpc} !>". by iApply ("IH" with "Hle").
  Qed.

  Lemma iProto_send_le {TT1 TT2} (pc1 : TT1  val * iProp Σ * iProto Σ)
      (pc2 : TT2  val * iProp Σ * iProto Σ) :
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    (.. x2 : TT2,  (pc2 x2).1.2 ={}= .. x1 : TT1,
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      (pc1 x1).1.1 = (pc2 x2).1.1 
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       (pc1 x1).1.2 
       iProto_le (pc1 x1).2 (pc2 x2).2) -
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    iProto_le (iProto_message Send pc1) (iProto_message Send pc2).
  Proof.
    iIntros "H". rewrite iProto_le_unfold iProto_message_eq. iRight; iLeft.
    iExists _, _; do 2 (iSplit; [done|]).
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    iIntros (v p2') "/=". iDestruct 1 as (x2 ->) "[Hpc #Heq]".
    iMod ("H" with "Hpc") as (x1 ?) "[Hpc Hle]".
    iExists (pc1 x1).2. iSplitL "Hle".
    { iIntros "!> !>". change (fixpoint iProto_le_aux ?p1 ?p2) with (iProto_le p1 p2).
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      by iRewrite "Heq". }
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    iModIntro. iExists x1. iSplit; [done|]. iSplit; [by iApply "Hpc"|done].
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  Qed.

  Lemma iProto_recv_le {TT1 TT2} (pc1 : TT1  val * iProp Σ * iProto Σ)
      (pc2 : TT2  val * iProp Σ * iProto Σ) :
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    (.. x1 : TT1,  (pc1 x1).1.2 ={}= .. x2 : TT2,
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      (pc1 x1).1.1 = (pc2 x2).1.1 
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       (pc2 x2).1.2 
       iProto_le (pc1 x1).2 (pc2 x2).2) -
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    iProto_le (iProto_message Receive pc1) (iProto_message Receive pc2).
  Proof.
    iIntros "H". rewrite iProto_le_unfold iProto_message_eq. iRight; iRight.
    iExists _, _; do 2 (iSplit; [done|]).
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    iIntros (v p1') "/=". iDestruct 1 as (x1 ->) "[Hpc #Heq]".
    iMod ("H" with "Hpc") as (x2 ?) "[Hpc Hle]". iExists (pc2 x2).2. iSplitL "Hle".
    { iIntros "!> !>". change (fixpoint iProto_le_aux ?p1 ?p2) with (iProto_le p1 p2).
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      by iRewrite "Heq". }
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    iModIntro. iExists x2. iSplit; [done|]. iSplit; [by iApply "Hpc"|done].
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  Qed.

  Lemma iProto_mapsto_le c p1 p2 : c  p1 - iProto_le p1 p2 - c  p2.
  Proof.
    rewrite mapsto_proto_eq. iDestruct 1 as (s c1 c2 γ p1' ->) "[Hle H]".
    iIntros "Hle'". iExists s, c1, c2, γ, p1'. iSplit; first done. iFrame "H".
    by iApply (iProto_le_trans with "Hle").
  Qed.

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  (** ** Lemmas about the auxiliary definitions and invariants *)
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  Global Instance proto_interp_ne : NonExpansive2 (proto_interp (Σ:=Σ) vs).
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  Proof. induction vs; solve_proper. Qed.
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  Global Instance proto_interp_proper vs :
    Proper (() ==> () ==> ()) (proto_interp (Σ:=Σ) vs).
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  Proof. apply (ne_proper_2 _). Qed.

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  Global Instance to_pre_proto_ne : NonExpansive (to_pre_proto (Σ:=Σ)).
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  Proof. solve_proper. Qed.
  Global Instance proto_own_ne γ s : NonExpansive (proto_own_frag γ s).
  Proof. solve_proper. Qed.
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  Global Instance mapsto_proto_ne c : NonExpansive (mapsto_proto c).
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  Proof. rewrite mapsto_proto_eq. solve_proper. Qed.
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  Global Instance mapsto_proto_proper c : Proper (() ==> ()) (mapsto_proto c).
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  Proof. apply (ne_proper _). Qed.

  Lemma proto_own_auth_agree γ s p p' :
    proto_own_auth γ s p - proto_own_frag γ s p' -  (p  p').
  Proof.
    iIntros "Hauth Hfrag".
    iDestruct (own_valid_2 with "Hauth Hfrag") as "Hvalid".
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    iDestruct (excl_auth_agreeI with "Hvalid") as "Hvalid".
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    iDestruct (bi.later_eq_1 with "Hvalid") as "Hvalid"; iNext.
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    rewrite /to_pre_proto. assert ( p,
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      proto_map id iProp_unfold iProp_fold (proto_map id iProp_fold iProp_unfold p)  p) as help.
    { intros p''. rewrite -proto_map_compose -{2}(proto_map_id p'').
      apply proto_map_ext=> // pc /=; by rewrite iProp_fold_unfold. }
    rewrite -{2}(help p). iRewrite "Hvalid". by rewrite help.
  Qed.

  Lemma proto_own_auth_update γ s p p' p'' :
    proto_own_auth γ s p - proto_own_frag γ s p' ==
    proto_own_auth γ s p''  proto_own_frag γ s p''.
  Proof.
    iIntros "Hauth Hfrag".
    iDestruct (own_update_2 with "Hauth Hfrag") as "H".
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    { eapply (excl_auth_update _ _ (Next (to_pre_proto p''))). }
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    by rewrite own_op.
  Qed.

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  Lemma proto_eq_next_dual p :
    ofe_mor_map (laterO_map (proto_map action_dual cid cid)) cid (proto_eq_next (iProto_dual p)) 
    proto_eq_next p.
  Proof.
    intros lp. iSplit; iIntros "Hlp /="; last by iRewrite -"Hlp".
    destruct (Next_uninj lp) as [p' ->].
    rewrite /later_map /= !bi.later_equivI. iNext.
    rewrite -{2}(involutive iProto_dual p) -{2}(involutive iProto_dual p').
    by iRewrite "Hlp".
  Qed.

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  Lemma proto_interp_send v vs pc p1 p2 :
    proto_interp vs (proto_message Send pc) p2 -
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    pc v (proto_eq_next p1) -
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    proto_interp (vs ++ [v]) p1 p2.
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  Proof.
    iIntros "Heval Hc". iInduction vs as [|v' vs] "IH" forall (p2); simpl.
    - iDestruct "Heval" as "#Heval".
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      iExists _, (iProto_dual p1). iSplit.
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      { iRewrite -"Heval". by rewrite /iProto_dual proto_map_message. }
      rewrite /= proto_eq_next_dual; auto.
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    - iDestruct "Heval" as (pc' p2') "(Heq & Hc' & Heval)".
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      iExists pc', p2'. iFrame "Heq Hc'". iApply ("IH" with "Heval Hc").
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  Qed.

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  Lemma proto_interp_recv v vs p1 pc :
     proto_interp (v :: vs) p1 (proto_message Receive pc) -  p2,
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       pc v (proto_eq_next p2) 
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        proto_interp vs p1 p2.
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  Proof.
    simpl. iDestruct 1 as (pc' p2) "(Heq & Hc & Hp2)". iExists p2. iFrame "Hp2".
    iDestruct (@proto_message_equivI with "Heq") as "[_ Heq]".
    iSpecialize ("Heq" $! v). rewrite bi.ofe_morO_equivI.
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    by iRewrite ("Heq" $! (proto_eq_next p2)).
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  Qed.

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  Lemma proto_interp_False p pc v vs :
    proto_interp (v :: vs) p (proto_message Send pc) - False.
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  Proof.
    simpl. iDestruct 1 as (pc' p2') "[Heq _]".
    by iDestruct (@proto_message_equivI with "Heq") as "[% _]".
  Qed.

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  Lemma proto_interp_nil p1 p2 : proto_interp [] p1 p2 - p1  iProto_dual p2.
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  Proof. iIntros "#H /=". iRewrite -"H". by rewrite involutive. Qed.
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  Arguments proto_interp : simpl never.
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  (** ** Helper lemma for initialization of a channel *)
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  Lemma proto_init E cγ c1 c2 p :
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    is_chan protoN cγ c1 c2 -
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    chan_own cγ Left [] - chan_own cγ Right [] ={E}=
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    c1  p  c2  iProto_dual p.
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  Proof.
    iIntros "#Hcctx Hcol Hcor".
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    iMod (own_alloc (E (Next (to_pre_proto p)) 
                     E (Next (to_pre_proto p)))) as (lγ) "[Hlsta Hlstf]".
    { by apply excl_auth_valid. }
    iMod (own_alloc (E (Next (to_pre_proto (iProto_dual p))) 
                     E (Next (to_pre_proto (iProto_dual p))))) as (rγ) "[Hrsta Hrstf]".
    { by apply excl_auth_valid. }
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    pose (ProtName cγ lγ rγ) as pγ.
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    iMod (inv_alloc protoN _ (proto_inv pγ) with "[-Hlstf Hrstf Hcctx]") as "#Hinv".
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    { iNext. rewrite /proto_inv. eauto 10 with iFrame. }
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    iModIntro. rewrite mapsto_proto_eq. iSplitL "Hlstf".
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    - iExists Left, c1, c2, pγ, p.
      iFrame "Hlstf Hinv Hcctx". iSplit; [done|]. iApply iProto_le_refl.
    - iExists Right, c1, c2, pγ, (iProto_dual p).
      iFrame "Hrstf Hinv Hcctx". iSplit; [done|]. iApply iProto_le_refl.
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  Qed.

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  (** ** Accessor style lemmas, used as helpers to prove the specifications of
  [send] and [recv]. *)
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  Lemma proto_send_acc {TT} c (pc : TT  val * iProp Σ * iProto Σ) (x : TT) :
    c  iProto_message Send pc -
    (pc x).1.2 -
     s c1 c2 γ,
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       c = side_elim s c1 c2  
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      is_chan protoN (proto_c_name γ) c1 c2  |={,∖↑protoN}=>  vs,
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        chan_own (proto_c_name γ) s vs 
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         (chan_own (proto_c_name γ) s (vs ++ [(pc x).1.1]) ={∖↑protoN,}=
           c  (pc x).2).
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  Proof.
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    rewrite {1}mapsto_proto_eq iProto_message_eq. iIntros "Hc HP".
    iDestruct "Hc" as (s c1 c2 γ p ->) "(Hle & Hst & #[Hcctx Hinv])".
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    iExists s, c1, c2, γ. iSplit; first done. iFrame "Hcctx".
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    iDestruct (iProto_le_send_inv with "Hle") as (pc') "[Hp H] /=".
    iRewrite "Hp" in "Hst"; clear p.
    iMod ("H" with "[HP]") as (p1') "[Hle HP]".
    { iExists _. iFrame "HP". by iSplit. }
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    iInv protoN as (l r pl pr) "(>Hcl & >Hcr & Hstla & Hstra & Hinv')" "Hclose".
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    (* TODO: refactor to avoid twice nearly the same proof *)
    iModIntro. destruct s.
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    - iExists _. iIntros "{$Hcl} !> Hcl".
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      iDestruct (proto_own_auth_agree with "Hstla Hst") as "#Heq".
      iMod (proto_own_auth_update _ _ _ _ p1' with "Hstla Hst") as "[Hstla Hst]".
      iMod ("Hclose" with "[-Hst Hle]") as "_".
      { iNext. iExists _,_,_,_. iFrame "Hcr Hstra Hstla Hcl". iLeft.
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        iRewrite "Heq" in "Hinv'".
        iDestruct "Hinv'" as "[[-> Heval]|[-> Heval]]".
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        { iSplit=> //. by iApply (proto_interp_send with "Heval [HP]"). }
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        destruct r as [|vr r]; last first.
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        { iDestruct (proto_interp_False with "Heval") as %[]. }
        iSplit; first done; simpl. iRewrite (proto_interp_nil with "Heval").
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        iApply (proto_interp_send _ [] with "[//] HP"). }
      iModIntro. rewrite mapsto_proto_eq. iExists Left, c1, c2, γ, p1'.
      by iFrame "Hcctx Hinv Hst Hle".
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    - iExists _. iIntros "{$Hcr} !> Hcr".
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      iDestruct (proto_own_auth_agree with "Hstra Hst") as "#Heq".
      iMod (proto_own_auth_update _ _ _ _ p1' with "Hstra Hst") as "[Hstra Hst]".
      iMod ("Hclose" with "[-Hst Hle]") as "_".
      { iNext. iExists _, _, _, _. iFrame "Hcl Hstra Hstla Hcr". iRight.
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        iRewrite "Heq" in "Hinv'".
        iDestruct "Hinv'" as "[[-> Heval]|[-> Heval]]"; last first.
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        { iSplit=> //. by iApply (proto_interp_send with "Heval [HP]"). }
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        destruct l as [|vl l]; last first.
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        { iDestruct (proto_interp_False with "Heval") as %[]. }
        iSplit; first done; simpl. iRewrite (proto_interp_nil with "Heval").
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        iApply (proto_interp_send _ [] with "[//] HP"). }
      iModIntro. rewrite mapsto_proto_eq. iExists Right, c1, c2, γ, p1'.
      by iFrame "Hcctx Hinv Hst Hle".
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  Qed.

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  Lemma proto_recv_acc {TT} c (pc : TT  val * iProp Σ * iProto Σ) :
    c  iProto_message Receive pc -
     s c1 c2 γ,
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       c = side_elim s c2 c1  
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      is_chan protoN (proto_c_name γ) c1 c2  |={,∖↑protoN}=>  vs,
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        chan_own (proto_c_name γ) s vs 
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         ((chan_own (proto_c_name γ) s vs ={∖↑protoN,}=
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             c  iProto_message Receive pc) 
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           ( v vs',
              vs = v :: vs'  -
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             chan_own (proto_c_name γ) s vs' ={∖↑protoN,,}=   x : TT,
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              v = (pc x).1.1   c  (pc x).2  (pc x).1.2)).
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  Proof.
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    rewrite {1}mapsto_proto_eq iProto_message_eq.
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    iDestruct 1 as (s c1 c2 γ p ->) "(Hle & Hst & #[Hcctx Hinv])".
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    iDestruct (iProto_le_recv_inv with "Hle") as (pc') "[Hp Hle] /=".
    iRewrite "Hp" in "Hst". clear p.
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    iExists (side_elim s Right Left), c1, c2, γ. iSplit; first by destruct s.
    iFrame "Hcctx".
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    iInv protoN as (l r pl pr) "(>Hcl & >Hcr & Hstla & Hstra & Hinv')" "Hclose".
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    iExists (side_elim s r l). iModIntro.
    (* TODO: refactor to avoid twice nearly the same proof *)
    destruct s; simpl.
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    - iIntros "{$Hcr} !>". 
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      iDestruct (proto_own_auth_agree with "Hstla Hst") as "#Heq".
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      iSplit.
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      + iIntros "Hcr".
        iMod ("Hclose" with "[-Hst Hle]") as "_".
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        { iNext. iExists l, r, _, _. iFrame. }
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        iModIntro. rewrite mapsto_proto_eq.
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        iExists Left, c1, c2, γ, (proto_message Receive pc').
        iFrame "Hcctx Hinv Hst". iSplit; first done.
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        rewrite iProto_le_unfold. iRight; auto 10.
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      + iIntros (v vs ->) "Hcr".
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        iDestruct "Hinv'" as "[[>% _]|[>-> Heval]]"; first done.
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        iAssert ( proto_interp (v :: vs) pr (proto_message Receive pc'))%I
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          with "[Heval]" as "Heval".
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        { iNext. by iRewrite "Heq" in "Heval". }
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        iDestruct (proto_interp_recv with "Heval") as (q) "[Hpc Heval]".
        iMod (proto_own_auth_update _ _ _ _ q with "Hstla Hst") as "[Hstla Hst]".
        iMod ("Hclose" with "[-Hst Hpc Hle]") as %_.
        { iExists _, _,_ ,_; iFrame; eauto. }
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        iIntros "!> !>". iMod ("Hle" with "Hpc") as (q') "[Hle H]".
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        iDestruct "H" as (x) "(Hv & HP & #Hf) /=".
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        iIntros "!> !>". iExists x. iFrame "Hv HP". iRewrite -"Hf".
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