From c914ef04e2c79d3f52d40ace2a9c6faea0ad63e5 Mon Sep 17 00:00:00 2001
From: Jonas Kastberg Hinrichsen <jihgfee@gmail.com>
Date: Thu, 21 Mar 2024 20:27:47 +0100
Subject: [PATCH] WIP: Computable consistency
---
multi_actris/channel/proto_alt.v | 1408 ++++++++++++++++++++++++++++++
1 file changed, 1408 insertions(+)
create mode 100644 multi_actris/channel/proto_alt.v
diff --git a/multi_actris/channel/proto_alt.v b/multi_actris/channel/proto_alt.v
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+(** This file defines the core of the Actris logic: It defines dependent
+separation protocols and the various operations on it like dual, append, and
+branching.
+
+Dependent separation protocols [iProto] are defined by instantiating the
+parameterized version in [proto_model] with the type of propositions [iProp] of Iris.
+We define ways of constructing instances of the instantiated type via two
+constructors:
+- [iProto_end], which is identical to [proto_end].
+- [iProto_message], which takes an [action] and an [iMsg]. The type [iMsg] is a
+ sequence of binders [iMsg_exist], terminated by the payload constructed with
+ [iMsg_base] based on arguments [v], [P] and [prot], which are the value, the
+ carried proposition and the [iProto] tail, respectively.
+
+For convenience sake, we provide the following notations:
+- [END], which is simply [iProto_end].
+- [∃ x, m], which is [iMsg_exist] with argument [m].
+- [MSG v {{ P }}; prot], which is [iMsg_Base] with arguments [v], [P] and [prot].
+- [<a> m], which is [iProto_message] with arguments [a] and [m].
+
+We also include custom notation to more easily construct complete constructions:
+- [<a x1 .. xn> m], which is [<a> ∃ x1, .. ∃ xn, m].
+- [<a x1 .. xn> MSG v; {{ P }}; prot], which constructs a full protocol.
+
+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.
+
+In addition we define the subprotocol relation [iProto_le] [⊑], which generalises
+the following inductive definition for asynchronous subtyping on session types:
+
+ p1 <: p2 p1 <: p2 p1 <: !B.p3 ?A.p3 <: p2
+---------- ---------------- ---------------- ----------------------------
+end <: end !A.p1 <: !A.p2 ?A.p1 <: ?A.p2 ?A.p1 <: !B.p2
+
+Example:
+
+!R <: !R ?Q <: ?Q ?Q <: ?Q
+------------------- --------------
+?Q.!R <: !R.?Q ?P.?Q <: ?P.?Q
+------------------------------------
+ ?P.?Q.!R <: !R.?P.?Q
+
+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. *)
+From iris.algebra Require Import gmap excl_auth gmap_view.
+From iris.proofmode Require Import proofmode.
+From iris.base_logic Require Export lib.iprop.
+From iris.base_logic Require Import lib.own.
+From iris.program_logic Require Import language.
+From multi_actris.channel Require Import proto_model.
+Set Default Proof Using "Type".
+Export action.
+
+(** * Setup of Iris's cameras *)
+Class protoG Σ V :=
+ protoG_authG ::
+ inG Σ (gmap_viewR natO
+ (optionUR (exclR (laterO (proto (leibnizO V) (iPropO Σ) (iPropO Σ)))))).
+
+Definition protoΣ V := #[
+ GFunctor ((gmap_viewRF natO (optionRF (exclRF (laterOF (protoOF (leibnizO V) idOF idOF))))))
+].
+Global Instance subG_chanΣ {Σ V} : subG (protoΣ V) Σ → protoG Σ V.
+Proof. solve_inG. Qed.
+
+(** * Types *)
+Definition iProto Σ V := proto V (iPropO Σ) (iPropO Σ).
+Declare Scope proto_scope.
+Delimit Scope proto_scope with proto.
+Bind Scope proto_scope with iProto.
+Local Open Scope proto.
+
+(** * Messages *)
+Section iMsg.
+ Set Primitive Projections.
+ Record iMsg Σ V := IMsg { iMsg_car : V → laterO (iProto Σ V) -n> iPropO Σ }.
+End iMsg.
+Arguments IMsg {_ _} _.
+Arguments iMsg_car {_ _} _.
+
+Declare Scope msg_scope.
+Delimit Scope msg_scope with msg.
+Bind Scope msg_scope with iMsg.
+Global Instance iMsg_inhabited {Σ V} : Inhabited (iMsg Σ V) := populate (IMsg inhabitant).
+
+Section imsg_ofe.
+ Context {Σ : gFunctors} {V : Type}.
+
+ Instance iMsg_equiv : Equiv (iMsg Σ V) := λ m1 m2,
+ ∀ w p, iMsg_car m1 w p ≡ iMsg_car m2 w p.
+ Instance iMsg_dist : Dist (iMsg Σ V) := λ n m1 m2,
+ ∀ w p, iMsg_car m1 w p ≡{n}≡ iMsg_car m2 w p.
+
+ Lemma iMsg_ofe_mixin : OfeMixin (iMsg Σ V).
+ Proof. by apply (iso_ofe_mixin (iMsg_car : _ → V -d> _ -n> _)). Qed.
+ Canonical Structure iMsgO := Ofe (iMsg Σ V) iMsg_ofe_mixin.
+
+ Global Instance iMsg_cofe : Cofe iMsgO.
+ Proof. by apply (iso_cofe (IMsg : (V -d> _ -n> _) → _) iMsg_car). Qed.
+End imsg_ofe.
+
+Program Definition iMsg_base_def {Σ V}
+ (v : V) (P : iProp Σ) (p : iProto Σ V) : iMsg Σ V :=
+ IMsg (λ v', λne p', ⌜ v = v' ⌠∗ Next p ≡ p' ∗ P)%I.
+Next Obligation. solve_proper. Qed.
+Definition iMsg_base_aux : seal (@iMsg_base_def). by eexists. Qed.
+Definition iMsg_base := iMsg_base_aux.(unseal).
+Definition iMsg_base_eq : @iMsg_base = @iMsg_base_def := iMsg_base_aux.(seal_eq).
+Arguments iMsg_base {_ _} _%V _%I _%proto.
+Global Instance: Params (@iMsg_base) 3 := {}.
+
+Program Definition iMsg_exist_def {Σ V A} (m : A → iMsg Σ V) : iMsg Σ V :=
+ IMsg (λ v', λne p', ∃ x, iMsg_car (m x) v' p')%I.
+Next Obligation. solve_proper. Qed.
+Definition iMsg_exist_aux : seal (@iMsg_exist_def). by eexists. Qed.
+Definition iMsg_exist := iMsg_exist_aux.(unseal).
+Definition iMsg_exist_eq : @iMsg_exist = @iMsg_exist_def := iMsg_exist_aux.(seal_eq).
+Arguments iMsg_exist {_ _ _} _%msg.
+Global Instance: Params (@iMsg_exist) 3 := {}.
+
+Definition iMsg_texist {Σ V} {TT : tele} (m : TT → iMsg Σ V) : iMsg Σ V :=
+ tele_fold (@iMsg_exist Σ V) (λ x, x) (tele_bind m).
+Arguments iMsg_texist {_ _ !_} _%msg /.
+
+Notation "'MSG' v {{ P } } ; p" := (iMsg_base v P p)
+ (at level 200, v at level 20, right associativity,
+ format "MSG v {{ P } } ; p") : msg_scope.
+Notation "'MSG' v ; p" := (iMsg_base v True p)
+ (at level 200, v at level 20, right associativity,
+ format "MSG v ; p") : msg_scope.
+Notation "∃ x .. y , m" :=
+ (iMsg_exist (λ x, .. (iMsg_exist (λ y, m)) ..)%msg) : msg_scope.
+Notation "'∃..' x .. y , m" :=
+ (iMsg_texist (λ x, .. (iMsg_texist (λ y, m)) .. )%msg)
+ (at level 200, x binder, y binder, right associativity,
+ format "∃.. x .. y , m") : msg_scope.
+
+Lemma iMsg_texist_exist {Σ V} {TT : tele} w lp (m : TT → iMsg Σ V) :
+ iMsg_car (∃.. x, m x)%msg w lp ⊣⊢ (∃.. x, iMsg_car (m x) w lp).
+Proof.
+ rewrite /iMsg_texist iMsg_exist_eq.
+ induction TT as [|T TT IH]; simpl; [done|]. f_equiv=> x. apply IH.
+Qed.
+
+(** * Operators *)
+Definition iProto_end_def {Σ V} : iProto Σ V := 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 {_ _}.
+
+Definition iProto_message_def {Σ V} (a : action) (m : iMsg Σ V) : iProto Σ V :=
+ proto_message a (iMsg_car m).
+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 {_ _} _ _%msg.
+Global Instance: Params (@iProto_message) 3 := {}.
+
+Notation "'END'" := iProto_end : proto_scope.
+
+Notation "< a > m" := (iProto_message a m)
+ (at level 200, a at level 10, m at level 200,
+ format "< a > m") : proto_scope.
+Notation "< a @ x1 .. xn > m" := (iProto_message a (∃ x1, .. (∃ xn, m) ..))
+ (at level 200, a at level 10, x1 closed binder, xn closed binder, m at level 200,
+ format "< a @ x1 .. xn > m") : proto_scope.
+Notation "< a @.. x1 .. xn > m" := (iProto_message a (∃.. x1, .. (∃.. xn, m) ..))
+ (at level 200, a at level 10, x1 closed binder, xn closed binder, m at level 200,
+ format "< a @.. x1 .. xn > m") : proto_scope.
+
+Class MsgTele {Σ V} {TT : tele} (m : iMsg Σ V)
+ (tv : TT -t> V) (tP : TT -t> iProp Σ) (tp : TT -t> iProto Σ V) :=
+ msg_tele : m ≡ (∃.. x, MSG tele_app tv x {{ tele_app tP x }}; tele_app tp x)%msg.
+Global Hint Mode MsgTele ! ! - ! - - - : typeclass_instances.
+
+(** * Operations *)
+Program Definition iMsg_map {Σ V}
+ (rec : iProto Σ V → iProto Σ V) (m : iMsg Σ V) : iMsg Σ V :=
+ IMsg (λ v, λne p1', ∃ p1, iMsg_car m v (Next p1) ∗ p1' ≡ Next (rec p1))%I.
+Next Obligation. solve_proper. Qed.
+
+Program Definition iProto_map_app_aux {Σ V}
+ (f : action → action) (p2 : iProto Σ V)
+ (rec : iProto Σ V -n> iProto Σ V) : iProto Σ V -n> iProto Σ V := λne p,
+ proto_elim p2 (λ a m,
+ proto_message (f a) (iMsg_car (iMsg_map rec (IMsg m)))) p.
+Next Obligation.
+ intros Σ V f p2 rec n p1 p1' Hp. apply proto_elim_ne=> // a m1 m2 Hm.
+ apply proto_message_ne=> v p' /=. by repeat f_equiv.
+Qed.
+
+Global Instance iProto_map_app_aux_contractive {Σ V} f (p2 : iProto Σ V) :
+ Contractive (iProto_map_app_aux f p2).
+Proof.
+ intros n rec1 rec2 Hrec p1; simpl. apply proto_elim_ne=> // a m1 m2 Hm.
+ apply proto_message_ne=> v p' /=. by repeat (f_contractive || f_equiv).
+Qed.
+
+Definition iProto_map_app {Σ V} (f : action → action)
+ (p2 : iProto Σ V) : iProto Σ V -n> iProto Σ V :=
+ fixpoint (iProto_map_app_aux f p2).
+
+Definition iProto_app_def {Σ V} (p1 p2 : iProto Σ V) : iProto Σ V :=
+ iProto_map_app id p2 p1.
+Definition iProto_app_aux : seal (@iProto_app_def). Proof. by eexists. Qed.
+Definition iProto_app := iProto_app_aux.(unseal).
+Definition iProto_app_eq : @iProto_app = @iProto_app_def := iProto_app_aux.(seal_eq).
+Arguments iProto_app {_ _} _%proto _%proto.
+Global Instance: Params (@iProto_app) 2 := {}.
+Infix "<++>" := iProto_app (at level 60) : proto_scope.
+Notation "m <++> p" := (iMsg_map (flip iProto_app p) m) : msg_scope.
+
+Definition iProto_dual_def {Σ V} (p : iProto Σ V) : iProto Σ V :=
+ iProto_map_app action_dual proto_end p.
+Definition iProto_dual_aux : seal (@iProto_dual_def). Proof. by eexists. Qed.
+Definition iProto_dual := iProto_dual_aux.(unseal).
+Definition iProto_dual_eq :
+ @iProto_dual = @iProto_dual_def := iProto_dual_aux.(seal_eq).
+Arguments iProto_dual {_ _} _%proto.
+Global Instance: Params (@iProto_dual) 2 := {}.
+Notation iMsg_dual := (iMsg_map iProto_dual).
+
+Definition iProto_dual_if {Σ V} (d : bool) (p : iProto Σ V) : iProto Σ V :=
+ if d then iProto_dual p else p.
+Arguments iProto_dual_if {_ _} _ _%proto.
+Global Instance: Params (@iProto_dual_if) 3 := {}.
+
+(** * Proofs *)
+Section proto.
+ Context `{!protoG Σ V}.
+ Implicit Types v : V.
+ Implicit Types p pl pr : iProto Σ V.
+ Implicit Types m : iMsg Σ V.
+
+ (** ** Equality *)
+ Lemma iProto_case p : p ≡ END ∨ ∃ t n m, p ≡ <(t,n)> m.
+ Proof.
+ rewrite iProto_message_eq iProto_end_eq.
+ destruct (proto_case p) as [|([a n]&m&?)]; [by left|right].
+ by exists a, n, (IMsg m).
+ Qed.
+ Lemma iProto_message_equivI `{!BiInternalEq SPROP} a1 a2 m1 m2 :
+ (<a1> m1) ≡ (<a2> m2) ⊣⊢@{SPROP} ⌜ a1 = a2 ⌠∧
+ (∀ v lp, iMsg_car m1 v lp ≡ iMsg_car m2 v lp).
+ Proof. rewrite iProto_message_eq. apply proto_message_equivI. Qed.
+
+ Lemma iProto_message_end_equivI `{!BiInternalEq SPROP} a m :
+ (<a> m) ≡ END ⊢@{SPROP} False.
+ Proof. rewrite iProto_message_eq iProto_end_eq. apply proto_message_end_equivI. Qed.
+ Lemma iProto_end_message_equivI `{!BiInternalEq SPROP} a m :
+ END ≡ (<a> m) ⊢@{SPROP} False.
+ Proof. by rewrite internal_eq_sym iProto_message_end_equivI. Qed.
+
+ (** ** Non-expansiveness of operators *)
+ Global Instance iMsg_car_proper :
+ Proper (iMsg_equiv ==> (=) ==> (≡) ==> (≡)) (iMsg_car (Σ:=Σ) (V:=V)).
+ Proof.
+ intros m1 m2 meq v1 v2 veq p1 p2 peq. rewrite meq.
+ f_equiv; [ by f_equiv | done ].
+ Qed.
+ Global Instance iMsg_car_ne n :
+ Proper (iMsg_dist n ==> (=) ==> (dist n) ==> (dist n)) (iMsg_car (Σ:=Σ) (V:=V)).
+ Proof.
+ intros m1 m2 meq v1 v2 veq p1 p2 peq. rewrite meq.
+ f_equiv; [ by f_equiv | done ].
+ Qed.
+
+ Global Instance iMsg_contractive v n :
+ Proper (dist n ==> dist_later n ==> dist n) (iMsg_base (Σ:=Σ) (V:=V) v).
+ Proof. rewrite iMsg_base_eq=> P1 P2 HP p1 p2 Hp w q /=. solve_contractive. Qed.
+ Global Instance iMsg_ne v : NonExpansive2 (iMsg_base (Σ:=Σ) (V:=V) v).
+ Proof. rewrite iMsg_base_eq=> P1 P2 HP p1 p2 Hp w q /=. solve_proper. Qed.
+ Global Instance iMsg_proper v :
+ Proper ((≡) ==> (≡) ==> (≡)) (iMsg_base (Σ:=Σ) (V:=V) v).
+ Proof. apply (ne_proper_2 _). Qed.
+
+ Global Instance iMsg_exist_ne A n :
+ Proper (pointwise_relation _ (dist n) ==> (dist n)) (@iMsg_exist Σ V A).
+ Proof. rewrite iMsg_exist_eq=> m1 m2 Hm v p /=. f_equiv=> x. apply Hm. Qed.
+ Global Instance iMsg_exist_proper A :
+ Proper (pointwise_relation _ (≡) ==> (≡)) (@iMsg_exist Σ V A).
+ Proof. rewrite iMsg_exist_eq=> m1 m2 Hm v p /=. f_equiv=> x. apply Hm. Qed.
+
+ Global Instance msg_tele_base (v:V) (P : iProp Σ) (p : iProto Σ V) :
+ MsgTele (TT:=TeleO) (MSG v {{ P }}; p) v P p.
+ Proof. done. Qed.
+ Global Instance msg_tele_exist {A} {TT : A → tele} (m : A → iMsg Σ V) tv tP tp :
+ (∀ x, MsgTele (TT:=TT x) (m x) (tv x) (tP x) (tp x)) →
+ MsgTele (TT:=TeleS TT) (∃ x, m x) tv tP tp.
+ Proof. intros Hm. rewrite /MsgTele /=. f_equiv=> x. apply Hm. Qed.
+
+ Global Instance iProto_message_ne a :
+ NonExpansive (iProto_message (Σ:=Σ) (V:=V) a).
+ Proof. rewrite iProto_message_eq. solve_proper. Qed.
+ Global Instance iProto_message_proper a :
+ Proper ((≡) ==> (≡)) (iProto_message (Σ:=Σ) (V:=V) a).
+ Proof. apply (ne_proper _). Qed.
+
+ Lemma iProto_message_equiv {TT1 TT2 : tele} a1 a2
+ (m1 m2 : iMsg Σ V)
+ (v1 : TT1 -t> V) (v2 : TT2 -t> V)
+ (P1 : TT1 -t> iProp Σ) (P2 : TT2 -t> iProp Σ)
+ (prot1 : TT1 -t> iProto Σ V) (prot2 : TT2 -t> iProto Σ V) :
+ MsgTele m1 v1 P1 prot1 →
+ MsgTele m2 v2 P2 prot2 →
+ ⌜ a1 = a2 ⌠-∗
+ (■∀.. (xs1 : TT1), tele_app P1 xs1 -∗
+ ∃.. (xs2 : TT2), ⌜tele_app v1 xs1 = tele_app v2 xs2⌠∗
+ ▷ (tele_app prot1 xs1 ≡ tele_app prot2 xs2) ∗
+ tele_app P2 xs2) -∗
+ (■∀.. (xs2 : TT2), tele_app P2 xs2 -∗
+ ∃.. (xs1 : TT1), ⌜tele_app v1 xs1 = tele_app v2 xs2⌠∗
+ ▷ (tele_app prot1 xs1 ≡ tele_app prot2 xs2) ∗
+ tele_app P1 xs1) -∗
+ (<a1> m1) ≡ (<a2> m2).
+ Proof.
+ iIntros (Hm1 Hm2 Heq) "#Heq1 #Heq2".
+ unfold MsgTele in Hm1. rewrite Hm1. clear Hm1.
+ unfold MsgTele in Hm2. rewrite Hm2. clear Hm2.
+ rewrite iProto_message_eq proto_message_equivI.
+ iSplit; [ done | ].
+ iIntros (v p').
+ do 2 rewrite iMsg_texist_exist.
+ rewrite iMsg_base_eq /=.
+ iApply prop_ext.
+ iIntros "!>". iSplit.
+ - iDestruct 1 as (xs1 Hveq1) "[Hrec1 HP1]".
+ iDestruct ("Heq1" with "HP1") as (xs2 Hveq2) "[Hrec2 HP2]".
+ iExists xs2. rewrite -Hveq1 Hveq2.
+ iSplitR; [ done | ]. iSplitR "HP2"; [ | done ].
+ iRewrite -"Hrec1". iApply later_equivI. iIntros "!>". by iRewrite "Hrec2".
+ - iDestruct 1 as (xs2 Hveq2) "[Hrec2 HP2]".
+ iDestruct ("Heq2" with "HP2") as (xs1 Hveq1) "[Hrec1 HP1]".
+ iExists xs1. rewrite -Hveq2 Hveq1.
+ iSplitR; [ done | ]. iSplitR "HP1"; [ | done ].
+ iRewrite -"Hrec2". iApply later_equivI. iIntros "!>". by iRewrite "Hrec1".
+ Qed.
+
+ (** Helpers *)
+ Lemma iMsg_map_base f v P p :
+ NonExpansive f →
+ iMsg_map f (MSG v {{ P }}; p) ≡ (MSG v {{ P }}; f p)%msg.
+ Proof.
+ rewrite iMsg_base_eq. intros ? v' p'; simpl. iSplit.
+ - iDestruct 1 as (p'') "[(->&Hp&$) Hp']". iSplit; [done|].
+ iRewrite "Hp'". iIntros "!>". by iRewrite "Hp".
+ - iIntros "(->&Hp'&$)". iExists p. iRewrite -"Hp'". auto.
+ Qed.
+ Lemma iMsg_map_exist {A} f (m : A → iMsg Σ V) :
+ iMsg_map f (∃ x, m x) ≡ (∃ x, iMsg_map f (m x))%msg.
+ Proof.
+ rewrite iMsg_exist_eq. intros v' p'; simpl. iSplit.
+ - iDestruct 1 as (p'') "[H Hp']". iDestruct "H" as (x) "H"; auto.
+ - iDestruct 1 as (x p'') "[Hm Hp']". auto.
+ Qed.
+
+ (** ** Dual *)
+ Global Instance iProto_dual_ne : NonExpansive (@iProto_dual Σ V).
+ Proof. rewrite iProto_dual_eq. solve_proper. Qed.
+ Global Instance iProto_dual_proper : Proper ((≡) ==> (≡)) (@iProto_dual Σ V).
+ Proof. apply (ne_proper _). Qed.
+ Global Instance iProto_dual_if_ne d : NonExpansive (@iProto_dual_if Σ V d).
+ Proof. solve_proper. Qed.
+ Global Instance iProto_dual_if_proper d :
+ Proper ((≡) ==> (≡)) (@iProto_dual_if Σ V d).
+ Proof. apply (ne_proper _). Qed.
+
+ Lemma iProto_dual_end : iProto_dual (Σ:=Σ) (V:=V) END ≡ END.
+ Proof.
+ rewrite iProto_end_eq iProto_dual_eq /iProto_dual_def /iProto_map_app.
+ etrans; [apply (fixpoint_unfold (iProto_map_app_aux _ _))|]; simpl.
+ by rewrite proto_elim_end.
+ Qed.
+ Lemma iProto_dual_message a m :
+ iProto_dual (<a> m) ≡ <action_dual a> iMsg_dual m.
+ Proof.
+ rewrite iProto_message_eq iProto_dual_eq /iProto_dual_def /iProto_map_app.
+ etrans; [apply (fixpoint_unfold (iProto_map_app_aux _ _))|]; simpl.
+ rewrite /iProto_message_def. rewrite ->proto_elim_message; [done|].
+ intros a' m1 m2 Hm; f_equiv; solve_proper.
+ Qed.
+ Lemma iMsg_dual_base v P p :
+ iMsg_dual (MSG v {{ P }}; p) ≡ (MSG v {{ P }}; iProto_dual p)%msg.
+ Proof. apply iMsg_map_base, _. Qed.
+ Lemma iMsg_dual_exist {A} (m : A → iMsg Σ V) :
+ iMsg_dual (∃ x, m x) ≡ (∃ x, iMsg_dual (m x))%msg.
+ Proof. apply iMsg_map_exist. Qed.
+
+ Global Instance iProto_dual_involutive : Involutive (≡) (@iProto_dual Σ V).
+ Proof.
+ intros p. apply (uPred.internal_eq_soundness (M:=iResUR Σ)).
+ iLöb as "IH" forall (p). destruct (iProto_case p) as [->|(a&n&m&->)].
+ { by rewrite !iProto_dual_end. }
+ rewrite !iProto_dual_message involutive.
+ iApply iProto_message_equivI; iSplit; [done|]; iIntros (v p') "/=".
+ iApply prop_ext; iIntros "!>"; iSplit.
+ - iDestruct 1 as (pd) "[H Hp']". iRewrite "Hp'".
+ iDestruct "H" as (pdd) "[H #Hpd]".
+ iApply (internal_eq_rewrite); [|done]; iIntros "!>".
+ iRewrite "Hpd". by iRewrite ("IH" $! pdd).
+ - iIntros "H". destruct (Next_uninj p') as [p'' Hp']. iExists _.
+ rewrite Hp'. iSplitL; [by auto|]. iIntros "!>". by iRewrite ("IH" $! p'').
+ Qed.
+
+ (** ** Append *)
+ Global Instance iProto_app_end_l : LeftId (≡) END (@iProto_app Σ V).
+ Proof.
+ intros p. rewrite iProto_end_eq iProto_app_eq /iProto_app_def /iProto_map_app.
+ etrans; [apply (fixpoint_unfold (iProto_map_app_aux _ _))|]; simpl.
+ by rewrite proto_elim_end.
+ Qed.
+ Lemma iProto_app_message a m p2 : (<a> m) <++> p2 ≡ <a> m <++> p2.
+ Proof.
+ rewrite iProto_message_eq iProto_app_eq /iProto_app_def /iProto_map_app.
+ etrans; [apply (fixpoint_unfold (iProto_map_app_aux _ _))|]; simpl.
+ rewrite /iProto_message_def. rewrite ->proto_elim_message; [done|].
+ intros a' m1 m2 Hm. f_equiv; solve_proper.
+ Qed.
+
+ Global Instance iProto_app_ne : NonExpansive2 (@iProto_app Σ V).
+ Proof.
+ assert (∀ n, Proper (dist n ==> (=) ==> dist n) (@iProto_app Σ V)).
+ { intros n p1 p1' Hp1 p2 p2' <-. by rewrite iProto_app_eq /iProto_app_def Hp1. }
+ assert (Proper ((≡) ==> (=) ==> (≡)) (@iProto_app Σ V)).
+ { intros p1 p1' Hp1 p2 p2' <-. by rewrite iProto_app_eq /iProto_app_def Hp1. }
+ intros n p1 p1' Hp1 p2 p2' Hp2. rewrite Hp1. clear p1 Hp1.
+ revert p1'. induction (lt_wf n) as [n _ IH]; intros p1.
+ destruct (iProto_case p1) as [->|(a&i&m&->)].
+ { by rewrite !left_id. }
+ rewrite !iProto_app_message. f_equiv=> v p' /=. do 4 f_equiv.
+ f_contractive. apply IH; eauto using dist_le with lia.
+ Qed.
+ Global Instance iProto_app_proper : Proper ((≡) ==> (≡) ==> (≡)) (@iProto_app Σ V).
+ Proof. apply (ne_proper_2 _). Qed.
+
+ Lemma iMsg_app_base v P p1 p2 :
+ ((MSG v {{ P }}; p1) <++> p2)%msg ≡ (MSG v {{ P }}; p1 <++> p2)%msg.
+ Proof. apply: iMsg_map_base. Qed.
+ Lemma iMsg_app_exist {A} (m : A → iMsg Σ V) p2 :
+ ((∃ x, m x) <++> p2)%msg ≡ (∃ x, m x <++> p2)%msg.
+ Proof. apply iMsg_map_exist. Qed.
+
+ Global Instance iProto_app_end_r : RightId (≡) END (@iProto_app Σ V).
+ Proof.
+ intros p. apply (uPred.internal_eq_soundness (M:=iResUR Σ)).
+ iLöb as "IH" forall (p). destruct (iProto_case p) as [->|(a&i&m&->)].
+ { by rewrite left_id. }
+ rewrite iProto_app_message.
+ iApply iProto_message_equivI; iSplit; [done|]; iIntros (v p') "/=".
+ iApply prop_ext; iIntros "!>". iSplit.
+ - iDestruct 1 as (p1') "[H Hp']". iRewrite "Hp'".
+ iApply (internal_eq_rewrite); [|done]; iIntros "!>".
+ by iRewrite ("IH" $! p1').
+ - iIntros "H". destruct (Next_uninj p') as [p'' Hp']. iExists p''.
+ rewrite Hp'. iSplitL; [by auto|]. iIntros "!>". by iRewrite ("IH" $! p'').
+ Qed.
+ Global Instance iProto_app_assoc : Assoc (≡) (@iProto_app Σ V).
+ Proof.
+ intros p1 p2 p3. apply (uPred.internal_eq_soundness (M:=iResUR Σ)).
+ iLöb as "IH" forall (p1). destruct (iProto_case p1) as [->|(a&i&m&->)].
+ { by rewrite !left_id. }
+ rewrite !iProto_app_message.
+ iApply iProto_message_equivI; iSplit; [done|]; iIntros (v p123) "/=".
+ iApply prop_ext; iIntros "!>". iSplit.
+ - iDestruct 1 as (p1') "[H #Hp']".
+ iExists (p1' <++> p2). iSplitL; [by auto|].
+ iRewrite "Hp'". iIntros "!>". iApply "IH".
+ - iDestruct 1 as (p12) "[H #Hp123]". iDestruct "H" as (p1') "[H #Hp12]".
+ iExists p1'. iFrame "H". iRewrite "Hp123".
+ iIntros "!>". iRewrite "Hp12". by iRewrite ("IH" $! p1').
+ Qed.
+
+ Lemma iProto_dual_app p1 p2 :
+ iProto_dual (p1 <++> p2) ≡ iProto_dual p1 <++> iProto_dual p2.
+ Proof.
+ apply (uPred.internal_eq_soundness (M:=iResUR Σ)).
+ iLöb as "IH" forall (p1 p2). destruct (iProto_case p1) as [->|(a&i&m&->)].
+ { by rewrite iProto_dual_end !left_id. }
+ rewrite iProto_dual_message !iProto_app_message iProto_dual_message /=.
+ iApply iProto_message_equivI; iSplit; [done|]; iIntros (v p12) "/=".
+ iApply prop_ext; iIntros "!>". iSplit.
+ - iDestruct 1 as (p12d) "[H #Hp12]". iDestruct "H" as (p1') "[H #Hp12d]".
+ iExists (iProto_dual p1'). iSplitL; [by auto|].
+ iRewrite "Hp12". iIntros "!>". iRewrite "Hp12d". iApply "IH".
+ - iDestruct 1 as (p1') "[H #Hp12]". iDestruct "H" as (p1d) "[H #Hp1']".
+ iExists (p1d <++> p2). iSplitL; [by auto|].
+ iRewrite "Hp12". iIntros "!>". iRewrite "Hp1'". by iRewrite ("IH" $! p1d p2).
+ Qed.
+
+End proto.
+
+Global Instance iProto_inhabited {Σ V} : Inhabited (iProto Σ V) := populate END.
+
+Program Definition iProto_elim {Σ V A}
+ (x : A) (f : action → iMsg Σ V -> A) (p : iProto Σ V) : A :=
+ proto_elim x (λ a m, f a (IMsg (λ v, λne p, m v p)))%I p.
+Next Obligation. solve_proper. Qed.
+
+Lemma iProto_elim_message {Σ V} {A:ofe}
+ (x : A) (f : action → iMsg Σ V -> A) a m
+ (* `{Hf : ∀ a, Proper ((≡) ==> (≡)) (f a)} : *)
+ :
+ iProto_elim x f (iProto_message a m) ≡ f a m.
+Proof.
+ rewrite /iProto_elim.
+ rewrite iProto_message_eq /iProto_message_def. simpl.
+ setoid_rewrite proto_elim_message.
+ { f_equiv. destruct m. f_equiv. simpl.
+ admit. }
+ intros a'.
+ intros f1 f2 Hf'. f_equiv. f_equiv.
+Admitted.
+
+Definition nat_beq := Eval compute in Nat.eqb.
+
+Definition find_recv {Σ V} (i:nat) (j:nat) (ps : list (iProto Σ V)) :
+ option $ iMsg Σ V :=
+ iProto_elim None (λ a m,
+ match a with
+ | (Recv, i') => if nat_beq i i' then Some m else None
+ | (Send, _) => None
+ end) (ps !!! j).
+
+Fixpoint sync_pairs_aux {Σ V} (i : nat) (ps_full ps : list (iProto Σ V)) :
+ list (nat * nat * iMsg Σ V * iMsg Σ V) :=
+ match ps with
+ | [] => []
+ | p :: ps =>
+ iProto_elim (sync_pairs_aux (S i) ps_full ps)
+ (λ a mi, match a with
+ | (Recv,_) => sync_pairs_aux (S i) ps_full ps
+ | (Send,j) => match find_recv i j ps_full with
+ | None => sync_pairs_aux (S i) ps_full ps
+ | Some mj => (i,j,mi,mj) ::
+ sync_pairs_aux (S i) ps_full ps
+ end
+ end) p
+ end.
+
+Notation sync_pairs ps := (sync_pairs_aux 0 ps ps).
+
+Definition can_step {Σ V} (rec : list (iProto Σ V) → iProp Σ)
+ (ps : list (iProto Σ V)) : iProp Σ :=
+ [∧ list] '(i,j,m1,m2) ∈ sync_pairs ps,
+ ∀ v p1, iMsg_car m1 v (Next p1) -∗
+ ∃ p2, iMsg_car m2 v (Next p2) ∗
+ â–· (rec (<[i:=p1]>(<[j:=p2]>ps))).
+
+From iris.heap_lang Require Import notation.
+
+Definition iProto_binary `{!heapGS Σ} : list (iProto Σ val) :=
+ [(<(Send, 1) @ (x:Z)> MSG #x ; END)%proto;
+ (<(Recv, 0) @ (x:Z)> MSG #x ; END)%proto].
+
+Lemma iProto_binary_consistent `{!heapGS Σ} :
+ ⊢ can_step (λ _, True) (@iProto_binary _ Σ heapGS).
+Proof. rewrite /iProto_binary /can_step /iProto_elim. simpl.
+ rewrite /find_recv. simpl.
+ Fail rewrite iProto_elim_message.
+ (* OBS: Break here *)
+
+
+Definition valid_target {Σ V} (ps : list (iProto Σ V)) (i j : nat) : iProp Σ :=
+ ∀ a m, (ps !!! i ≡ <(a, j)> m) -∗ ⌜is_Some (ps !! j)âŒ.
+
+Definition iProto_consistent_pre {Σ V} (rec : list (iProto Σ V) → iProp Σ)
+ (ps : list (iProto Σ V)) : iProp Σ :=
+ (∀ i j, valid_target ps i j) ∗ (can_step rec ps).
+
+Global Instance iProto_consistent_pre_ne {Σ V}
+ (rec : listO (iProto Σ V) -n> iPropO Σ) :
+ NonExpansive (iProto_consistent_pre rec).
+Proof. rewrite /iProto_consistent_pre /can_step /valid_target. solve_proper. Qed.
+
+Program Definition iProto_consistent_pre' {Σ V}
+ (rec : listO (iProto Σ V) -n> iPropO Σ) :
+ listO (iProto Σ V) -n> iPropO Σ :=
+ λne ps, iProto_consistent_pre (λ ps, rec ps) ps.
+
+Local Instance iProto_consistent_pre_contractive {Σ V} : Contractive (@iProto_consistent_pre' Σ V).
+Proof.
+ rewrite /iProto_consistent_pre' /iProto_consistent_pre /can_step.
+ solve_contractive.
+Qed.
+
+Definition iProto_consistent {Σ V} (ps : list (iProto Σ V)) : iProp Σ :=
+ fixpoint iProto_consistent_pre' ps.
+
+Arguments iProto_consistent {_ _} _%proto.
+Global Instance: Params (@iProto_consistent) 1 := {}.
+
+Global Instance iProto_consistent_ne {Σ V} : NonExpansive (@iProto_consistent Σ V).
+Proof. solve_proper. Qed.
+Global Instance iProto_consistent_proper {Σ V} : Proper ((≡) ==> (⊣⊢)) (@iProto_consistent Σ V).
+Proof. solve_proper. Qed.
+
+Lemma iProto_consistent_unfold {Σ V} (ps : list (iProto Σ V)) :
+ iProto_consistent ps ≡ (iProto_consistent_pre iProto_consistent) ps.
+Proof.
+ apply: (fixpoint_unfold iProto_consistent_pre').
+Qed.
+
+(** * Protocol entailment *)
+Definition iProto_le_pre {Σ V}
+ (rec : iProto Σ V → iProto Σ V → iProp Σ) (p1 p2 : iProto Σ V) : iProp Σ :=
+ (p1 ≡ END ∗ p2 ≡ END) ∨
+ ∃ a1 a2 m1 m2,
+ (p1 ≡ <a1> m1) ∗ (p2 ≡ <a2> m2) ∗
+ match a1, a2 with
+ | (Recv,i), (Recv,j) => ⌜i = j⌠∗ ∀ v p1',
+ iMsg_car m1 v (Next p1') -∗ ∃ p2', ▷ rec p1' p2' ∗ iMsg_car m2 v (Next p2')
+ | (Send,i), (Send,j) => ⌜i = j⌠∗ ∀ v p2',
+ iMsg_car m2 v (Next p2') -∗ ∃ p1', ▷ rec p1' p2' ∗ iMsg_car m1 v (Next p1')
+ | _, _ => False
+ end.
+Global Instance iProto_le_pre_ne {Σ V} (rec : iProto Σ V → iProto Σ V → iProp Σ) :
+ NonExpansive2 (iProto_le_pre rec).
+Proof. solve_proper. Qed.
+
+Program Definition iProto_le_pre' {Σ V}
+ (rec : iProto Σ V -n> iProto Σ V -n> iPropO Σ) :
+ iProto Σ V -n> iProto Σ V -n> iPropO Σ := λne p1 p2,
+ iProto_le_pre (λ p1' p2', rec p1' p2') p1 p2.
+Solve Obligations with solve_proper.
+Local Instance iProto_le_pre_contractive {Σ V} : Contractive (@iProto_le_pre' Σ V).
+Proof.
+ intros n rec1 rec2 Hrec p1 p2. rewrite /iProto_le_pre' /iProto_le_pre /=.
+ by repeat (f_contractive || f_equiv).
+Qed.
+Definition iProto_le {Σ V} (p1 p2 : iProto Σ V) : iProp Σ :=
+ fixpoint iProto_le_pre' p1 p2.
+Arguments iProto_le {_ _} _%proto _%proto.
+Global Instance: Params (@iProto_le) 2 := {}.
+Notation "p ⊑ q" := (iProto_le p q) : bi_scope.
+
+Global Instance iProto_le_ne {Σ V} : NonExpansive2 (@iProto_le Σ V).
+Proof. solve_proper. Qed.
+Global Instance iProto_le_proper {Σ V} : Proper ((≡) ==> (≡) ==> (⊣⊢)) (@iProto_le Σ V).
+Proof. solve_proper. Qed.
+
+Record proto_name := ProtName { proto_names : gmap nat gname }.
+Global Instance proto_name_inhabited : Inhabited proto_name :=
+ populate (ProtName inhabitant).
+Global Instance proto_name_eq_dec : EqDecision proto_name.
+Proof. solve_decision. Qed.
+Global Instance proto_name_countable : Countable proto_name.
+Proof.
+ refine (inj_countable (λ '(ProtName γs), (γs))
+ (λ '(γs), Some (ProtName γs)) _); by intros [].
+Qed.
+
+Definition iProto_own_frag `{!protoG Σ V} (γ : gname)
+ (i : nat) (p : iProto Σ V) : iProp Σ :=
+ own γ (gmap_view_frag i (DfracOwn 1) (Excl' (Next p))).
+
+Definition iProto_own_auth `{!protoG Σ V} (γ : gname)
+ (ps : list (iProto Σ V)) : iProp Σ :=
+ own γ (gmap_view_auth (DfracOwn 1) (((λ p, Excl' (Next p)) <$>
+ (list_to_map (zip (seq 0 (length ps)) ps))) : gmap _ _)).
+
+Definition iProto_ctx `{protoG Σ V}
+ (γ : gname) (ps_len : nat) : iProp Σ :=
+ ∃ ps, ⌜length ps = ps_len⌠∗ iProto_own_auth γ ps ∗ ▷ iProto_consistent ps.
+
+(** * The connective for ownership of channel ends *)
+Definition iProto_own `{!protoG Σ V}
+ (γ : gname) (i : nat) (p : iProto Σ V) : iProp Σ :=
+ ∃ p', ▷ (p' ⊑ p) ∗ iProto_own_frag γ i p'.
+Arguments iProto_own {_ _ _} _ _ _%proto.
+Global Instance: Params (@iProto_own) 3 := {}.
+
+Global Instance iProto_own_frag_contractive `{protoG Σ V} γ i :
+ Contractive (iProto_own_frag γ i).
+Proof. solve_contractive. Qed.
+
+Global Instance iProto_own_contractive `{protoG Σ V} γ i :
+ Contractive (iProto_own γ i).
+Proof. solve_contractive. Qed.
+Global Instance iProto_own_ne `{protoG Σ V} γ s : NonExpansive (iProto_own γ s).
+Proof. solve_proper. Qed.
+Global Instance iProto_own_proper `{protoG Σ V} γ s :
+ Proper ((≡) ==> (≡)) (iProto_own γ s).
+Proof. apply (ne_proper _). Qed.
+
+(** * Proofs *)
+Section proto.
+ Context `{!protoG Σ V}.
+ Implicit Types v : V.
+ Implicit Types p pl pr : iProto Σ V.
+ Implicit Types m : iMsg Σ V.
+
+ Lemma own_prot_idx γ i j (p1 p2 : iProto Σ V) :
+ own γ (gmap_view_frag i (DfracOwn 1) (Excl' (Next p1))) -∗
+ own γ (gmap_view_frag j (DfracOwn 1) (Excl' (Next p2))) -∗
+ ⌜i ≠jâŒ.
+ Proof.
+ iIntros "Hown Hown'" (->).
+ iDestruct (own_valid_2 with "Hown Hown'") as "H".
+ rewrite uPred.cmra_valid_elim.
+ by iDestruct "H" as %[]%gmap_view_frag_op_validN.
+ Qed.
+
+ Lemma own_prot_excl γ i (p1 p2 : iProto Σ V) :
+ own γ (gmap_view_frag i (DfracOwn 1) (Excl' (Next p1))) -∗
+ own γ (gmap_view_frag i (DfracOwn 1) (Excl' (Next p2))) -∗
+ False.
+ Proof. iIntros "Hi Hj". by iDestruct (own_prot_idx with "Hi Hj") as %?. Qed.
+
+ (** ** Protocol entailment **)
+ Lemma iProto_le_unfold p1 p2 : iProto_le p1 p2 ≡ iProto_le_pre iProto_le p1 p2.
+ Proof. apply: (fixpoint_unfold iProto_le_pre'). Qed.
+
+ Lemma iProto_le_end : ⊢ END ⊑ (END : iProto Σ V).
+ Proof. rewrite iProto_le_unfold. iLeft. auto 10. Qed.
+
+ Lemma iProto_le_end_inv_r p : p ⊑ END -∗ (p ≡ END).
+ Proof.
+ rewrite iProto_le_unfold. iIntros "[[Hp _]|H] //".
+ iDestruct "H" as (a1 a2 m1 m2) "(_ & Heq & _)".
+ by iDestruct (iProto_end_message_equivI with "Heq") as %[].
+ Qed.
+
+ Lemma iProto_le_end_inv_l p : END ⊑ p -∗ (p ≡ END).
+ Proof.
+ rewrite iProto_le_unfold. iIntros "[[_ Hp]|H] //".
+ iDestruct "H" as (a1 a2 m1 m2) "(Heq & _ & _)".
+ iDestruct (iProto_end_message_equivI with "Heq") as %[].
+ Qed.
+
+ Lemma iProto_le_send_inv i p1 m2 :
+ p1 ⊑ (<(Send,i)> m2) -∗ ∃ m1,
+ (p1 ≡ <(Send,i)> m1) ∗
+ ∀ v p2', iMsg_car m2 v (Next p2') -∗
+ ∃ p1', ▷ (p1' ⊑ p2') ∗ iMsg_car m1 v (Next p1').
+ Proof.
+ rewrite iProto_le_unfold.
+ iIntros "[[_ Heq]|H]".
+ { by iDestruct (iProto_message_end_equivI with "Heq") as %[]. }
+ iDestruct "H" as (a1 a2 m1 m2') "(Hp1 & Hp2 & H)".
+ rewrite iProto_message_equivI. iDestruct "Hp2" as "[%Heq Hm2]".
+ simplify_eq.
+ destruct a1 as [[]]; [|done].
+ iDestruct "H" as (->) "H". iExists m1. iFrame "Hp1".
+ iIntros (v p2). iSpecialize ("Hm2" $! v (Next p2)). by iRewrite "Hm2".
+ Qed.
+
+ Lemma iProto_le_send_send_inv i m1 m2 v p2' :
+ (<(Send,i)> m1) ⊑ (<(Send,i)> m2) -∗
+ iMsg_car m2 v (Next p2') -∗ ∃ p1', ▷ (p1' ⊑ p2') ∗ iMsg_car m1 v (Next p1').
+ Proof.
+ iIntros "H Hm2". iDestruct (iProto_le_send_inv with "H") as (m1') "[Hm1 H]".
+ iDestruct (iProto_message_equivI with "Hm1") as (Heq) "Hm1".
+ iDestruct ("H" with "Hm2") as (p1') "[Hle Hm]".
+ iRewrite -("Hm1" $! v (Next p1')) in "Hm". auto with iFrame.
+ Qed.
+
+ Lemma iProto_le_recv_inv_l i m1 p2 :
+ (<(Recv,i)> m1) ⊑ p2 -∗ ∃ m2,
+ (p2 ≡ <(Recv,i)> m2) ∗
+ ∀ v p1', iMsg_car m1 v (Next p1') -∗
+ ∃ p2', ▷ (p1' ⊑ p2') ∗ iMsg_car m2 v (Next p2').
+ Proof.
+ rewrite iProto_le_unfold.
+ iIntros "[[Heq _]|H]".
+ { iDestruct (iProto_message_end_equivI with "Heq") as %[]. }
+ iDestruct "H" as (a1 a2 m1' m2) "(Hp1 & Hp2 & H)".
+ rewrite iProto_message_equivI. iDestruct "Hp1" as "[%Heq Hm1]".
+ simplify_eq.
+ destruct a2 as [[]]; [done|].
+ iDestruct "H" as (->) "H". iExists m2. iFrame "Hp2".
+ iIntros (v p1). iSpecialize ("Hm1" $! v (Next p1)). by iRewrite "Hm1".
+ Qed.
+
+ Lemma iProto_le_recv_inv_r i p1 m2 :
+ (p1 ⊑ <(Recv,i)> m2) -∗ ∃ m1,
+ (p1 ≡ <(Recv,i)> m1) ∗
+ ∀ v p1', iMsg_car m1 v (Next p1') -∗
+ ∃ p2', ▷ (p1' ⊑ p2') ∗ iMsg_car m2 v (Next p2').
+ Proof.
+ rewrite iProto_le_unfold.
+ iIntros "[[_ Heq]|H]".
+ { iDestruct (iProto_message_end_equivI with "Heq") as %[]. }
+ iDestruct "H" as (a1 a2 m1 m2') "(Hp1 & Hp2 & H)".
+ rewrite iProto_message_equivI.
+ iDestruct "Hp2" as "[%Heq Hm2]".
+ simplify_eq.
+ destruct a1 as [[]]; [done|].
+ iDestruct "H" as (->) "H".
+ iExists m1. iFrame.
+ iIntros (v p2).
+ iIntros "Hm1". iDestruct ("H" with "Hm1") as (p2') "[Hle H]".
+ iSpecialize ("Hm2" $! v (Next p2')).
+ iExists p2'. iFrame.
+ iRewrite "Hm2". iApply "H".
+ Qed.
+
+ Lemma iProto_le_recv_recv_inv i m1 m2 v p1' :
+ (<(Recv, i)> m1) ⊑ (<(Recv, i)> m2) -∗
+ iMsg_car m1 v (Next p1') -∗ ∃ p2', ▷ (p1' ⊑ p2') ∗ iMsg_car m2 v (Next p2').
+ Proof.
+ iIntros "H Hm2". iDestruct (iProto_le_recv_inv_r with "H") as (m1') "[Hm1 H]".
+ iApply "H". iDestruct (iProto_message_equivI with "Hm1") as (_) "Hm1".
+ by iRewrite -("Hm1" $! v (Next p1')).
+ Qed.
+
+ Lemma iProto_le_msg_inv_l i a m1 p2 :
+ (<(a,i)> m1) ⊑ p2 -∗ ∃ m2, p2 ≡ <(a,i)> m2.
+ Proof.
+ rewrite iProto_le_unfold /iProto_le_pre.
+ iIntros "[[Heq _]|H]".
+ { iDestruct (iProto_message_end_equivI with "Heq") as %[]. }
+ iDestruct "H" as (a1 a2 m1' m2) "(Hp1 & Hp2 & H)".
+ destruct a1 as [t1 ?], a2 as [t2 ?].
+ destruct t1,t2; [|done|done|].
+ - rewrite iProto_message_equivI.
+ iDestruct "Hp1" as (Heq) "Hp1". simplify_eq.
+ iDestruct "H" as (->) "H". by iExists _.
+ - rewrite iProto_message_equivI.
+ iDestruct "Hp1" as (Heq) "Hp1". simplify_eq.
+ iDestruct "H" as (->) "H". by iExists _.
+ Qed.
+
+ Lemma iProto_le_msg_inv_r j a p1 m2 :
+ (p1 ⊑ <(a,j)> m2) -∗ ∃ m1, p1 ≡ <(a,j)> m1.
+ Proof.
+ rewrite iProto_le_unfold /iProto_le_pre.
+ iIntros "[[_ Heq]|H]".
+ { iDestruct (iProto_message_end_equivI with "Heq") as %[]. }
+ iDestruct "H" as (a1 a2 m1 m2') "(Hp1 & Hp2 & H)".
+ destruct a1 as [t1 ?], a2 as [t2 ?].
+ destruct t1,t2; [|done|done|].
+ - rewrite iProto_message_equivI.
+ iDestruct "Hp2" as (Heq) "Hp2". simplify_eq.
+ iDestruct "H" as (->) "H". by iExists _.
+ - rewrite iProto_message_equivI.
+ iDestruct "Hp2" as (Heq) "Hp2". simplify_eq.
+ iDestruct "H" as (->) "H". by iExists _.
+ Qed.
+
+ Lemma valid_target_le ps i p1 p2 :
+ (∀ i' j', valid_target ps i' j') -∗
+ ps !!! i ≡ p1 -∗
+ p1 ⊑ p2 -∗
+ (∀ i' j', valid_target (<[i := p2]>ps) i' j') ∗ p1 ⊑ p2.
+ Proof. Admitted.
+ (* iIntros "Hprot #HSome Hle". *)
+ (* pose proof (iProto_case p1) as [Hend|Hmsg]. *)
+ (* { rewrite Hend. iDestruct (iProto_le_end_inv_l with "Hle") as "#H". *)
+ (* iFrame "Hle". *)
+ (* iIntros (i' j' a m) "Hm". *)
+ (* destruct (decide (i = j')) as [->|Hneqj]. *)
+ (* { Search list_lookup_total insert. rewrite list_lookup_total_insert. ; [done|]. lia. done. } *)
+ (* rewrite lookup_insert_ne; [|done]. *)
+ (* destruct (decide (i = i')) as [->|Hneqi]. *)
+ (* { rewrite lookup_total_insert. iRewrite "H" in "Hm". *)
+ (* by iDestruct (iProto_end_message_equivI with "Hm") as "Hm". } *)
+ (* rewrite lookup_total_insert_ne; [|done]. *)
+ (* by iApply "Hprot". } *)
+ (* destruct Hmsg as (t & n & m & Hmsg). *)
+ (* setoid_rewrite Hmsg. *)
+ (* iDestruct (iProto_le_msg_inv_l with "Hle") as (m2) "#Heq". iFrame "Hle". *)
+ (* iIntros (i' j' a m') "Hm". *)
+ (* destruct (decide (i = j')) as [->|Hneqj]. *)
+ (* { rewrite lookup_insert. done. } *)
+ (* rewrite lookup_insert_ne; [|done]. *)
+ (* destruct (decide (i = i')) as [->|Hneqi]. *)
+ (* { rewrite lookup_total_insert. iRewrite "Heq" in "Hm". *)
+ (* iDestruct (iProto_message_equivI with "Hm") as (Heq) "Hm". *)
+ (* simplify_eq. by iApply "Hprot". } *)
+ (* rewrite lookup_total_insert_ne; [|done]. *)
+ (* by iApply "Hprot". *)
+ (* Qed. *)
+
+ Lemma iProto_consistent_le ps i p1 p2 :
+ iProto_consistent ps -∗
+ ps !!! i ≡ p1 -∗
+ p1 ⊑ p2 -∗
+ iProto_consistent (<[i := p2]>ps).
+ Proof.
+ iIntros "Hprot #HSome Hle".
+ iRevert "HSome".
+ iLöb as "IH" forall (p1 p2 ps).
+ iIntros "#HSome".
+ rewrite !iProto_consistent_unfold.
+ iDestruct "Hprot" as "(Htar & Hprot)".
+ iDestruct (valid_target_le with "Htar HSome Hle") as "[Htar Hle]".
+ iFrame.
+ iIntros (i' j' m1 m2) "#Hm1 #Hm2".
+ destruct (decide (i = i')) as [<-|Hneq].
+ { rewrite list_lookup_total_insert; [|admit].
+ pose proof (iProto_case p2) as [Hend|Hmsg].
+ { setoid_rewrite Hend. rewrite iProto_end_message_equivI. done. }
+ destruct Hmsg as (a&?&m&Hmsg).
+ setoid_rewrite Hmsg.
+ destruct a; last first.
+ { rewrite iProto_message_equivI.
+ iDestruct "Hm1" as "[%Htag Hm1]". done. }
+ rewrite iProto_message_equivI.
+ iDestruct "Hm1" as "[%Htag Hm1]".
+ inversion Htag. simplify_eq.
+ iIntros (v p) "Hm1'".
+ iSpecialize ("Hm1" $! v (Next p)).
+ iDestruct (iProto_le_send_inv with "Hle") as "Hle".
+ iRewrite -"Hm1" in "Hm1'".
+ iDestruct "Hle" as (m') "[#Heq H]".
+ iDestruct ("H" with "Hm1'") as (p') "[Hle H]".
+ destruct (decide (i = j')) as [<-|Hneq].
+ { rewrite list_lookup_total_insert. rewrite iProto_message_equivI.
+ iDestruct "Hm2" as "[%Heq _]". done. admit. }
+ iDestruct ("Hprot" $!i j' with "[] [] H") as "Hprot".
+ { iRewrite -"Heq". rewrite !list_lookup_total_alt. iRewrite "HSome". done. }
+ { rewrite list_lookup_total_insert_ne; [|done]. done. }
+ iDestruct "Hprot" as (p'') "[Hm Hprot]".
+ iExists p''. iFrame.
+ iNext.
+ iDestruct ("IH" with "Hprot Hle [HSome]") as "HI".
+ { rewrite list_lookup_total_insert; [done|]. admit. }
+ iClear "IH Hm1 Hm2 Heq".
+ rewrite list_insert_insert.
+ rewrite (list_insert_commute _ j' i); [|done].
+ rewrite list_insert_insert. done. }
+ rewrite list_lookup_total_insert_ne; [|done].
+ destruct (decide (i = j')) as [<-|Hneq'].
+ { rewrite list_lookup_total_insert.
+ pose proof (iProto_case p2) as [Hend|Hmsg].
+ { setoid_rewrite Hend. rewrite iProto_end_message_equivI. done. }
+ destruct Hmsg as (a&?&m&Hmsg).
+ setoid_rewrite Hmsg.
+ destruct a.
+ { rewrite iProto_message_equivI.
+ iDestruct "Hm2" as "[%Htag Hm2]". done. }
+ rewrite iProto_message_equivI.
+ iDestruct "Hm2" as "[%Htag Hm2]".
+ inversion Htag. simplify_eq.
+ iIntros (v p) "Hm1'".
+ iDestruct (iProto_le_recv_inv_r with "Hle") as "Hle".
+ iDestruct "Hle" as (m') "[#Heq Hle]".
+ iDestruct ("Hprot" $!i' with "[] [] Hm1'") as "Hprot".
+ { done. }
+ { rewrite !list_lookup_total_alt. iRewrite "HSome". done. }
+ iDestruct ("Hprot") as (p') "[Hm1' Hprot]".
+ iDestruct ("Hle" with "Hm1'") as (p2') "[Hle Hm']".
+ iSpecialize ("Hm2" $! v (Next p2')).
+ iExists p2'.
+ iRewrite -"Hm2". iFrame.
+ iDestruct ("IH" with "Hprot Hle []") as "HI".
+ { iPureIntro. rewrite list_lookup_total_insert_ne; [|done].
+ rewrite list_lookup_total_insert. done. admit. }
+ rewrite list_insert_commute; [|done].
+ rewrite !list_insert_insert. done. admit. }
+ rewrite list_lookup_total_insert_ne; [|done].
+ iIntros (v p) "Hm1'".
+ iDestruct ("Hprot" $!i' j' with "[//] [//] Hm1'") as "Hprot".
+ iDestruct "Hprot" as (p') "[Hm2' Hprot]".
+ iExists p'. iFrame.
+ iNext.
+ rewrite (list_insert_commute _ j' i); [|done].
+ rewrite (list_insert_commute _ i' i); [|done].
+ iApply ("IH" with "Hprot Hle []").
+ rewrite list_lookup_total_insert_ne; [|done].
+ rewrite list_lookup_total_insert_ne; [|done].
+ done.
+ Admitted.
+
+ Lemma iProto_le_send i m1 m2 :
+ (∀ v p2', iMsg_car m2 v (Next p2') -∗ ∃ p1',
+ ▷ (p1' ⊑ p2') ∗ iMsg_car m1 v (Next p1')) -∗
+ (<(Send,i)> m1) ⊑ (<(Send,i)> m2).
+ Proof.
+ iIntros "Hle". rewrite iProto_le_unfold.
+ iRight. iExists (Send, i), (Send, i), m1, m2. by eauto.
+ Qed.
+
+ Lemma iProto_le_recv i m1 m2 :
+ (∀ v p1', iMsg_car m1 v (Next p1') -∗ ∃ p2',
+ ▷ (p1' ⊑ p2') ∗ iMsg_car m2 v (Next p2')) -∗
+ (<(Recv,i)> m1) ⊑ (<(Recv,i)> m2).
+ Proof.
+ iIntros "Hle". rewrite iProto_le_unfold.
+ iRight. iExists (Recv, i), (Recv, i), m1, m2. by eauto.
+ Qed.
+
+ Lemma iProto_le_base a v P p1 p2 :
+ ▷ (p1 ⊑ p2) -∗
+ (<a> MSG v {{ P }}; p1) ⊑ (<a> MSG v {{ P }}; p2).
+ Proof.
+ rewrite iMsg_base_eq. iIntros "H". destruct a as [[]].
+ - iApply iProto_le_send. iIntros (v' p') "(->&Hp&$)".
+ iExists p1. iSplit; [|by auto]. iIntros "!>". by iRewrite -"Hp".
+ - iApply iProto_le_recv. iIntros (v' p') "(->&Hp&$)".
+ iExists p2. iSplit; [|by auto]. iIntros "!>". by iRewrite -"Hp".
+ Qed.
+
+ Lemma iProto_le_trans p1 p2 p3 : p1 ⊑ p2 -∗ p2 ⊑ p3 -∗ p1 ⊑ p3.
+ Proof.
+ iIntros "H1 H2". iLöb as "IH" forall (p1 p2 p3).
+ destruct (iProto_case p3) as [->|([]&i&m3&->)].
+ - iDestruct (iProto_le_end_inv_r with "H2") as "H2". by iRewrite "H2" in "H1".
+ - iDestruct (iProto_le_send_inv with "H2") as (m2) "[Hp2 H2]".
+ iRewrite "Hp2" in "H1"; clear p2.
+ iDestruct (iProto_le_send_inv with "H1") as (m1) "[Hp1 H1]".
+ iRewrite "Hp1"; clear p1.
+ iApply iProto_le_send. iIntros (v p3') "Hm3".
+ iDestruct ("H2" with "Hm3") as (p2') "[Hle Hm2]".
+ iDestruct ("H1" with "Hm2") as (p1') "[Hle' Hm1]".
+ iExists p1'. iIntros "{$Hm1} !>". by iApply ("IH" with "Hle'").
+ - iDestruct (iProto_le_recv_inv_r with "H2") as (m2) "[Hp2 H3]".
+ iRewrite "Hp2" in "H1".
+ iDestruct (iProto_le_recv_inv_r with "H1") as (m1) "[Hp1 H2]".
+ iRewrite "Hp1". iApply iProto_le_recv. iIntros (v p1') "Hm1".
+ iDestruct ("H2" with "Hm1") as (p2') "[Hle Hm2]".
+ iDestruct ("H3" with "Hm2") as (p3') "[Hle' Hm3]".
+ iExists p3'. iIntros "{$Hm3} !>". by iApply ("IH" with "Hle").
+ Qed.
+
+ Lemma iProto_le_refl p : ⊢ p ⊑ p.
+ Proof.
+ iLöb as "IH" forall (p). destruct (iProto_case p) as [->|([]&i&m&->)].
+ - iApply iProto_le_end.
+ - iApply iProto_le_send. auto 10 with iFrame.
+ - iApply iProto_le_recv. auto 10 with iFrame.
+ Qed.
+
+ Lemma iProto_le_dual p1 p2 : p2 ⊑ p1 -∗ iProto_dual p1 ⊑ iProto_dual p2.
+ Proof.
+ iIntros "H". iLöb as "IH" forall (p1 p2).
+ destruct (iProto_case p1) as [->|([]&i&m1&->)].
+ - iDestruct (iProto_le_end_inv_r with "H") as "H".
+ iRewrite "H". iApply iProto_le_refl.
+ - iDestruct (iProto_le_send_inv with "H") as (m2) "[Hp2 H]".
+ iRewrite "Hp2"; clear p2. iEval (rewrite !iProto_dual_message).
+ iApply iProto_le_recv. iIntros (v p1d).
+ iDestruct 1 as (p1') "[Hm1 #Hp1d]".
+ iDestruct ("H" with "Hm1") as (p2') "[H Hm2]".
+ iDestruct ("IH" with "H") as "H". iExists (iProto_dual p2').
+ iSplitL "H"; [iIntros "!>"; by iRewrite "Hp1d"|]. simpl; auto.
+ - iDestruct (iProto_le_recv_inv_r with "H") as (m2) "[Hp2 H]".
+ iRewrite "Hp2"; clear p2. iEval (rewrite !iProto_dual_message /=).
+ iApply iProto_le_send. iIntros (v p2d).
+ iDestruct 1 as (p2') "[Hm2 #Hp2d]".
+ iDestruct ("H" with "Hm2") as (p1') "[H Hm1]".
+ iDestruct ("IH" with "H") as "H". iExists (iProto_dual p1').
+ iSplitL "H"; [iIntros "!>"; by iRewrite "Hp2d"|]. simpl; auto.
+ Qed.
+
+ Lemma iProto_le_dual_l p1 p2 : iProto_dual p2 ⊑ p1 ⊢ iProto_dual p1 ⊑ p2.
+ Proof.
+ iIntros "H". iEval (rewrite -(involutive iProto_dual p2)).
+ by iApply iProto_le_dual.
+ Qed.
+ Lemma iProto_le_dual_r p1 p2 : p2 ⊑ iProto_dual p1 ⊢ p1 ⊑ iProto_dual p2.
+ Proof.
+ iIntros "H". iEval (rewrite -(involutive iProto_dual p1)).
+ by iApply iProto_le_dual.
+ Qed.
+
+ Lemma iProto_le_app p1 p2 p3 p4 :
+ p1 ⊑ p2 -∗ p3 ⊑ p4 -∗ p1 <++> p3 ⊑ p2 <++> p4.
+ Proof.
+ iIntros "H1 H2". iLöb as "IH" forall (p1 p2 p3 p4).
+ destruct (iProto_case p2) as [->|([]&i&m2&->)].
+ - iDestruct (iProto_le_end_inv_r with "H1") as "H1".
+ iRewrite "H1". by rewrite !left_id.
+ - iDestruct (iProto_le_send_inv with "H1") as (m1) "[Hp1 H1]".
+ iRewrite "Hp1"; clear p1. rewrite !iProto_app_message.
+ iApply iProto_le_send. iIntros (v p24).
+ iDestruct 1 as (p2') "[Hm2 #Hp24]".
+ iDestruct ("H1" with "Hm2") as (p1') "[H1 Hm1]".
+ iExists (p1' <++> p3). iSplitR "Hm1"; [|by simpl; eauto].
+ iIntros "!>". iRewrite "Hp24". by iApply ("IH" with "H1").
+ - iDestruct (iProto_le_recv_inv_r with "H1") as (m1) "[Hp1 H1]".
+ iRewrite "Hp1"; clear p1. rewrite !iProto_app_message.
+ iApply iProto_le_recv.
+ iIntros (v p13). iDestruct 1 as (p1') "[Hm1 #Hp13]".
+ iDestruct ("H1" with "Hm1") as (p2'') "[H1 Hm2]".
+ iExists (p2'' <++> p4). iSplitR "Hm2"; [|by simpl; eauto].
+ iIntros "!>". iRewrite "Hp13". by iApply ("IH" with "H1").
+ Qed.
+
+ Lemma iProto_le_payload_elim_l i m v P p :
+ (P -∗ (<(Recv,i)> MSG v; p) ⊑ (<(Recv,i)> m)) ⊢
+ (<(Recv,i)> MSG v {{ P }}; p) ⊑ <(Recv,i)> m.
+ Proof.
+ rewrite iMsg_base_eq. iIntros "H".
+ iApply iProto_le_recv. iIntros (v' p') "(->&Hp&HP)".
+ iApply (iProto_le_recv_recv_inv with "(H HP)"); simpl; auto.
+ Qed.
+ Lemma iProto_le_payload_elim_r i m v P p :
+ (P -∗ (<(Send, i)> m) ⊑ (<(Send, i)> MSG v; p)) ⊢
+ (<(Send,i)> m) ⊑ (<(Send,i)> MSG v {{ P }}; p).
+ Proof.
+ rewrite iMsg_base_eq. iIntros "H".
+ iApply iProto_le_send. iIntros (v' p') "(->&Hp&HP)".
+ iApply (iProto_le_send_send_inv with "(H HP)"); simpl; auto.
+ Qed.
+ Lemma iProto_le_payload_intro_l i v P p :
+ P -∗ (<(Send,i)> MSG v {{ P }}; p) ⊑ (<(Send,i)> MSG v; p).
+ Proof.
+ rewrite iMsg_base_eq.
+ iIntros "HP". iApply iProto_le_send. iIntros (v' p') "(->&Hp&_) /=".
+ iExists p'. iSplitR; [iApply iProto_le_refl|]. auto.
+ Qed.
+ Lemma iProto_le_payload_intro_r i v P p :
+ P -∗ (<(Recv,i)> MSG v; p) ⊑ (<(Recv,i)> MSG v {{ P }}; p).
+ Proof.
+ rewrite iMsg_base_eq.
+ iIntros "HP". iApply iProto_le_recv. iIntros (v' p') "(->&Hp&_) /=".
+ iExists p'. iSplitR; [iApply iProto_le_refl|]. auto.
+ Qed.
+ Lemma iProto_le_exist_elim_l {A} i (m1 : A → iMsg Σ V) m2 :
+ (∀ x, (<(Recv,i)> m1 x) ⊑ (<(Recv,i)> m2)) ⊢
+ (<(Recv,i) @ x> m1 x) ⊑ (<(Recv,i)> m2).
+ Proof.
+ rewrite iMsg_exist_eq. iIntros "H".
+ iApply iProto_le_recv. iIntros (v p1') "/=". iDestruct 1 as (x) "Hm".
+ by iApply (iProto_le_recv_recv_inv with "H").
+ Qed.
+ Lemma iProto_le_exist_elim_r {A} i m1 (m2 : A → iMsg Σ V) :
+ (∀ x, (<(Send,i)> m1) ⊑ (<(Send,i)> m2 x)) ⊢
+ (<(Send,i)> m1) ⊑ (<(Send,i) @ x> m2 x).
+ Proof.
+ rewrite iMsg_exist_eq. iIntros "H".
+ iApply iProto_le_send. iIntros (v p2'). iDestruct 1 as (x) "Hm".
+ by iApply (iProto_le_send_send_inv with "H").
+ Qed.
+ Lemma iProto_le_exist_intro_l {A} i (m : A → iMsg Σ V) a :
+ ⊢ (<(Send,i) @ x> m x) ⊑ (<(Send,i)> m a).
+ Proof.
+ rewrite iMsg_exist_eq. iApply iProto_le_send. iIntros (v p') "Hm /=".
+ iExists p'. iSplitR; last by auto. iApply iProto_le_refl.
+ Qed.
+ Lemma iProto_le_exist_intro_r {A} i (m : A → iMsg Σ V) a :
+ ⊢ (<(Recv,i)> m a) ⊑ (<(Recv,i) @ x> m x).
+ Proof.
+ rewrite iMsg_exist_eq. iApply iProto_le_recv. iIntros (v p') "Hm /=".
+ iExists p'. iSplitR; last by auto. iApply iProto_le_refl.
+ Qed.
+
+ Lemma iProto_le_texist_elim_l {TT : tele} i (m1 : TT → iMsg Σ V) m2 :
+ (∀ x, (<(Recv,i)> m1 x) ⊑ (<(Recv,i)> m2)) ⊢
+ (<(Recv,i) @.. x> m1 x) ⊑ (<(Recv,i)> m2).
+ Proof.
+ iIntros "H". iInduction TT as [|T TT] "IH"; simpl; [done|].
+ iApply iProto_le_exist_elim_l; iIntros (x).
+ iApply "IH". iIntros (xs). iApply "H".
+ Qed.
+ Lemma iProto_le_texist_elim_r {TT : tele} i m1 (m2 : TT → iMsg Σ V) :
+ (∀ x, (<(Send,i)> m1) ⊑ (<(Send,i)> m2 x)) -∗
+ (<(Send,i)> m1) ⊑ (<(Send,i) @.. x> m2 x).
+ Proof.
+ iIntros "H". iInduction TT as [|T TT] "IH"; simpl; [done|].
+ iApply iProto_le_exist_elim_r; iIntros (x).
+ iApply "IH". iIntros (xs). iApply "H".
+ Qed.
+
+ Lemma iProto_le_texist_intro_l {TT : tele} i (m : TT → iMsg Σ V) x :
+ ⊢ (<(Send,i) @.. x> m x) ⊑ (<(Send,i)> m x).
+ Proof.
+ induction x as [|T TT x xs IH] using tele_arg_ind; simpl.
+ { iApply iProto_le_refl. }
+ iApply iProto_le_trans; [by iApply iProto_le_exist_intro_l|]. iApply IH.
+ Qed.
+ Lemma iProto_le_texist_intro_r {TT : tele} i (m : TT → iMsg Σ V) x :
+ ⊢ (<(Recv,i)> m x) ⊑ (<(Recv,i) @.. x> m x).
+ Proof.
+ induction x as [|T TT x xs IH] using tele_arg_ind; simpl.
+ { iApply iProto_le_refl. }
+ iApply iProto_le_trans; [|by iApply iProto_le_exist_intro_r]. iApply IH.
+ Qed.
+
+ Lemma iProto_consistent_target ps m a i j :
+ iProto_consistent ps -∗
+ ps !!! i ≡ (<(a, j)> m) -∗
+ ⌜is_Some (ps !! j)âŒ.
+ Proof.
+ rewrite iProto_consistent_unfold. iDestruct 1 as "[Htar _]". iApply "Htar".
+ Qed.
+
+ Lemma iProto_consistent_step ps m1 m2 i j v p1 :
+ iProto_consistent ps -∗
+ ps !!! i ≡ (<(Send, j)> m1) -∗
+ ps !!! j ≡ (<(Recv, i)> m2) -∗
+ iMsg_car m1 v (Next p1) -∗
+ ∃ p2, iMsg_car m2 v (Next p2) ∗
+ â–· iProto_consistent (<[i := p1]>(<[j := p2]>ps)).
+ Proof.
+ iIntros "Hprot #Hi #Hj Hm1".
+ rewrite iProto_consistent_unfold /iProto_consistent_pre.
+ iDestruct "Hprot" as "[_ Hprot]".
+ iDestruct ("Hprot" with "Hi Hj Hm1") as (p2) "[Hm2 Hprot]".
+ iExists p2. iFrame.
+ Qed.
+
+ Global Instance iProto_own_frag_ne γ s : NonExpansive (iProto_own_frag γ s).
+ Proof. solve_proper. Qed.
+
+ Lemma iProto_own_auth_agree γ ps i p :
+ iProto_own_auth γ ps -∗ iProto_own_frag γ i p -∗ ▷ (ps !!! i ≡ p).
+ Proof. Admitted.
+ (* iIntros "Hâ— Hâ—¯". *)
+ (* iDestruct (own_valid_2 with "H◠H◯") as "H✓". *)
+ (* rewrite gmap_view_both_validI. *)
+ (* iDestruct "H✓" as "[_ [H1 H2]]". *)
+ (* rewrite list_lookup_total_alt lookup_fmap. *)
+ (* destruct (ps !! i); last first. *)
+ (* { rewrite !option_equivI. } *)
+ (* simpl. rewrite !option_equivI excl_equivI. by iNext. *)
+ (* Qed. *)
+
+ Lemma iProto_own_auth_update γ ps i p p' :
+ iProto_own_auth γ ps -∗ iProto_own_frag γ i p ==∗
+ iProto_own_auth γ (<[i := p']>ps) ∗ iProto_own_frag γ i p'.
+ Proof.
+ iIntros "Hâ— Hâ—¯".
+ iMod (own_update_2 with "Hâ— Hâ—¯") as "[H1 H2]"; [|iModIntro].
+ { eapply (gmap_view_replace _ _ _ (Excl' (Next p'))). done. }
+ iFrame. rewrite -fmap_insert. Admitted.
+
+ Lemma iProto_own_auth_alloc ps :
+ ⊢ |==> ∃ γ, iProto_own_auth γ ps ∗ [∗ list] i ↦p ∈ ps, iProto_own γ i p.
+ Proof. Admitted.
+ (* iMod (own_alloc (gmap_view_auth (DfracOwn 1) ∅)) as (γ) "Hauth". *)
+ (* { apply gmap_view_auth_valid. } *)
+ (* iExists γ. *)
+ (* iInduction ps as [|i p ps Hin] "IH" using map_ind. *)
+ (* { iModIntro. iFrame. by iApply big_sepM_empty. } *)
+ (* iMod ("IH" with "Hauth") as "[Hauth Hfrags]". *)
+ (* rewrite big_sepM_insert; [|done]. iFrame "Hfrags". *)
+ (* iMod (own_update with "Hauth") as "[Hauth Hfrag]". *)
+ (* { apply (gmap_view_alloc _ i (DfracOwn 1) (Excl' (Next p))); [|done|done]. *)
+ (* by rewrite lookup_fmap Hin. } *)
+ (* iModIntro. rewrite -fmap_insert. iFrame. *)
+ (* iExists _. iFrame. iApply iProto_le_refl. *)
+ (* Qed. *)
+
+ Lemma iProto_own_le γ s p1 p2 :
+ iProto_own γ s p1 -∗ ▷ (p1 ⊑ p2) -∗ iProto_own γ s p2.
+ Proof.
+ iDestruct 1 as (p1') "[Hle H]". iIntros "Hle'".
+ iExists p1'. iFrame "H". by iApply (iProto_le_trans with "Hle").
+ Qed.
+
+ Lemma iProto_init ps :
+ ▷ iProto_consistent ps -∗
+ |==> ∃ γ, iProto_ctx γ (length ps) ∗ [∗ list] i ↦p ∈ ps, iProto_own γ i p.
+ Proof.
+ iIntros "Hconsistent".
+ iMod iProto_own_auth_alloc as (γ) "[Hauth Hfrags]".
+ iExists γ. iFrame. iExists _. by iFrame.
+ Qed.
+
+
+ Lemma iProto_step γ ps_dom i j m1 m2 p1 v :
+ iProto_ctx γ ps_dom -∗
+ iProto_own γ i (<(Send, j)> m1) -∗
+ iProto_own γ j (<(Recv, i)> m2) -∗
+ iMsg_car m1 v (Next p1) ==∗
+ ▷ ∃ p2, iMsg_car m2 v (Next p2) ∗ iProto_ctx γ ps_dom ∗
+ iProto_own γ i p1 ∗ iProto_own γ j p2.
+ Proof.
+ iIntros "Hctx Hi Hj Hm".
+ iDestruct "Hi" as (pi) "[Hile Hi]".
+ iDestruct "Hj" as (pj) "[Hjle Hj]".
+ iDestruct "Hctx" as (ps Hdom) "[Hauth Hconsistent]".
+ iDestruct (iProto_own_auth_agree with "Hauth Hi") as "#Hpi".
+ iDestruct (iProto_own_auth_agree with "Hauth Hj") as "#Hpj".
+ iDestruct (own_prot_idx with "Hi Hj") as %Hneq.
+ iAssert (▷ (<[i:=<(Send, j)> m1]>ps !!! j ≡ pj))%I as "Hpj'".
+ { by rewrite list_lookup_total_insert_ne. }
+ iAssert (▷ (⌜is_Some (ps !! i)⌠∗ (pi ⊑ (<(Send, j)> m1))))%I with "[Hile]"
+ as "[Hi' Hile]".
+ { iNext. iDestruct (iProto_le_msg_inv_r with "Hile") as (m) "#Heq".
+ iFrame. iRewrite "Heq" in "Hpi". rewrite list_lookup_total_alt.
+ destruct (ps !! i); [done|].
+ iDestruct (iProto_end_message_equivI with "Hpi") as "[]". }
+ iAssert (▷ (⌜is_Some (ps !! j)⌠∗ (pj ⊑ (<(Recv, i)> m2))))%I with "[Hjle]"
+ as "[Hj' Hjle]".
+ { iNext. iDestruct (iProto_le_msg_inv_r with "Hjle") as (m) "#Heq".
+ iFrame. iRewrite "Heq" in "Hpj". rewrite !list_lookup_total_alt.
+ destruct (ps !! j); [done|].
+ iDestruct (iProto_end_message_equivI with "Hpj") as "[]". }
+ iDestruct (iProto_consistent_le with "Hconsistent Hpi Hile") as "Hconsistent".
+ iDestruct (iProto_consistent_le with "Hconsistent Hpj' Hjle") as "Hconsistent".
+ iDestruct (iProto_consistent_step _ _ _ i j with "Hconsistent [] [] [Hm //]") as
+ (p2) "[Hm2 Hconsistent]".
+ { rewrite list_lookup_total_insert_ne; [|done].
+ rewrite list_lookup_total_insert_ne; [|done].
+ rewrite list_lookup_total_insert; [done|]. admit. }
+ { rewrite list_lookup_total_insert_ne; [|done].
+ rewrite list_lookup_total_insert; [done|]. admit. }
+ iMod (iProto_own_auth_update _ _ _ _ p2 with "Hauth Hj") as "[Hauth Hj]".
+ iMod (iProto_own_auth_update _ _ _ _ p1 with "Hauth Hi") as "[Hauth Hi]".
+ iIntros "!>!>". iExists p2. iFrame "Hm2".
+ iDestruct "Hi'" as %Hi. iDestruct "Hj'" as %Hj.
+ iSplitL "Hconsistent Hauth".
+ { iExists (<[i:=p1]> (<[j:=p2]> ps)).
+ iSplit.
+ { admit.
+ (* rewrite !dom_insert_lookup_L; [done..|by rewrite lookup_insert_ne]. *)}
+ iFrame. rewrite list_insert_insert.
+ rewrite list_insert_commute; [|done]. rewrite list_insert_insert.
+ by rewrite list_insert_commute; [|done]. }
+ iSplitL "Hi"; iExists _; iFrame; iApply iProto_le_refl.
+ Admitted.
+
+ Lemma iProto_target γ ps_dom i a j m :
+ iProto_ctx γ ps_dom -∗
+ iProto_own γ i (<(a, j)> m) -∗
+ â–· (⌜j < ps_domâŒ) ∗ iProto_ctx γ ps_dom ∗ iProto_own γ i (<(a, j)> m).
+ Proof.
+ iIntros "Hctx Hown".
+ rewrite /iProto_ctx /iProto_own.
+ iDestruct "Hctx" as (ps Hdom) "[Hauth Hps]".
+ iDestruct "Hown" as (p') "[Hle Hown]".
+ iDestruct (iProto_own_auth_agree with "Hauth Hown") as "#Hi".
+ iDestruct (iProto_le_msg_inv_r with "Hle") as (m') "#Heq".
+ iAssert (▷ (⌜is_Some (ps !! j)⌠∗ iProto_consistent ps))%I
+ with "[Hps]" as "[HSome Hps]".
+ { iNext. iRewrite "Heq" in "Hi".
+ iDestruct (iProto_consistent_target with "Hps Hi") as "#$". by iFrame. }
+ iSplitL "HSome".
+ { iNext. iDestruct "HSome" as %Heq.
+ iPureIntro. simplify_eq. admit. }
+ iSplitL "Hauth Hps"; iExists _; by iFrame.
+ Admitted.
+
+ (* (** The instances below make it possible to use the tactics [iIntros], *)
+ (* [iExist], [iSplitL]/[iSplitR], [iFrame] and [iModIntro] on [iProto_le] goals. *) *)
+ Global Instance iProto_le_from_forall_l {A} i (m1 : A → iMsg Σ V) m2 name :
+ AsIdentName m1 name →
+ FromForall (iProto_message (Recv,i) (iMsg_exist m1) ⊑ (<(Recv,i)> m2))
+ (λ x, (<(Recv, i)> m1 x) ⊑ (<(Recv, i)> m2))%I name | 10.
+ Proof. intros _. apply iProto_le_exist_elim_l. Qed.
+ Global Instance iProto_le_from_forall_r {A} i m1 (m2 : A → iMsg Σ V) name :
+ AsIdentName m2 name →
+ FromForall ((<(Send,i)> m1) ⊑ iProto_message (Send,i) (iMsg_exist m2))
+ (λ x, (<(Send,i)> m1) ⊑ (<(Send,i)> m2 x))%I name | 11.
+ Proof. intros _. apply iProto_le_exist_elim_r. Qed.
+
+ Global Instance iProto_le_from_wand_l i m v P p :
+ TCIf (TCEq P True%I) False TCTrue →
+ FromWand ((<(Recv,i)> MSG v {{ P }}; p) ⊑ (<(Recv,i)> m)) P ((<(Recv,i)> MSG v; p) ⊑ (<(Recv,i)> m)) | 10.
+ Proof. intros _. apply iProto_le_payload_elim_l. Qed.
+ Global Instance iProto_le_from_wand_r i m v P p :
+ FromWand ((<(Send,i)> m) ⊑ (<(Send,i)> MSG v {{ P }}; p)) P ((<(Send,i)> m) ⊑ (<(Send,i)> MSG v; p)) | 11.
+ Proof. apply iProto_le_payload_elim_r. Qed.
+
+ Global Instance iProto_le_from_exist_l {A} i (m : A → iMsg Σ V) p :
+ FromExist ((<(Send,i) @ x> m x) ⊑ p) (λ a, (<(Send,i)> m a) ⊑ p)%I | 10.
+ Proof.
+ rewrite /FromExist. iDestruct 1 as (x) "H".
+ iApply (iProto_le_trans with "[] H"). iApply iProto_le_exist_intro_l.
+ Qed.
+ Global Instance iProto_le_from_exist_r {A} i (m : A → iMsg Σ V) p :
+ FromExist (p ⊑ <(Recv,i) @ x> m x) (λ a, p ⊑ (<(Recv,i)> m a))%I | 11.
+ Proof.
+ rewrite /FromExist. iDestruct 1 as (x) "H".
+ iApply (iProto_le_trans with "H"). iApply iProto_le_exist_intro_r.
+ Qed.
+
+ Global Instance iProto_le_from_sep_l i m v P p :
+ FromSep ((<(Send,i)> MSG v {{ P }}; p) ⊑ (<(Send,i)> m)) P ((<(Send,i)> MSG v; p) ⊑ (<(Send,i)> m)) | 10.
+ Proof.
+ rewrite /FromSep. iIntros "[HP H]".
+ iApply (iProto_le_trans with "[HP] H"). by iApply iProto_le_payload_intro_l.
+ Qed.
+ Global Instance iProto_le_from_sep_r i m v P p :
+ FromSep ((<(Recv,i)> m) ⊑ (<(Recv,i)> MSG v {{ P }}; p)) P ((<(Recv,i)> m) ⊑ (<(Recv,i)> MSG v; p)) | 11.
+ Proof.
+ rewrite /FromSep. iIntros "[HP H]".
+ iApply (iProto_le_trans with "H"). by iApply iProto_le_payload_intro_r.
+ Qed.
+
+ Global Instance iProto_le_frame_l i q m v R P Q p :
+ Frame q R P Q →
+ Frame q R ((<(Send,i)> MSG v {{ P }}; p) ⊑ (<(Send,i)> m))
+ ((<(Send,i)> MSG v {{ Q }}; p) ⊑ (<(Send,i)> m)) | 10.
+ Proof.
+ rewrite /Frame /=. iIntros (HP) "[HR H]".
+ iApply (iProto_le_trans with "[HR] H"). iApply iProto_le_payload_elim_r.
+ iIntros "HQ". iApply iProto_le_payload_intro_l. iApply HP; iFrame.
+ Qed.
+ Global Instance iProto_le_frame_r i q m v R P Q p :
+ Frame q R P Q →
+ Frame q R ((<(Recv,i)> m) ⊑ (<(Recv,i)> MSG v {{ P }}; p))
+ ((<(Recv,i)> m) ⊑ (<(Recv,i)> MSG v {{ Q }}; p)) | 11.
+ Proof.
+ rewrite /Frame /=. iIntros (HP) "[HR H]".
+ iApply (iProto_le_trans with "H"). iApply iProto_le_payload_elim_l.
+ iIntros "HQ". iApply iProto_le_payload_intro_r. iApply HP; iFrame.
+ Qed.
+
+ Global Instance iProto_le_from_modal a v p1 p2 :
+ FromModal True (modality_instances.modality_laterN 1) (p1 ⊑ p2)
+ ((<a> MSG v; p1) ⊑ (<a> MSG v; p2)) (p1 ⊑ p2).
+ Proof. intros _. iApply iProto_le_base. Qed.
+
+End proto.
+
+Typeclasses Opaque iProto_ctx iProto_own.
+
+Global Hint Extern 0 (environments.envs_entails _ (?x ⊑ ?y)) =>
+ first [is_evar x; fail 1 | is_evar y; fail 1|iApply iProto_le_refl] : core.
--
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