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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 actris.channel Require Import multi_proto_model multi_proto.
Set Default Proof Using "Type".
Export action.
Definition iProto_example1 {Σ V} : gmap nat (iProto Σ V) :=
∅.
Lemma iProto_example1_consistent {Σ V} :
⊢ iProto_consistent (@iProto_example1 Σ V).
Proof.
rewrite iProto_consistent_unfold.
iIntros (i j m1 m2) "Hi Hj".
rewrite lookup_total_empty.
by rewrite iProto_end_message_equivI.
Qed.
Definition iProto_example2 `{!invGS Σ} (P : iProp Σ) : gmap nat (iProto Σ Z) :=
<[0 := (<(Send 1) @ (x:Z)> MSG x {{ P }} ; END)%proto ]>
(<[1 := (<(Recv 0) @ (x:Z)> MSG x {{ P }} ; END)%proto ]>
∅).
Lemma iProto_example2_consistent `{!invGS Σ} (P : iProp Σ) :
⊢ iProto_consistent (@iProto_example2 _ Σ invGS0 P).
Proof.
rewrite iProto_consistent_unfold.
rewrite /iProto_example2.
iIntros (i j m1 m2) "Hm1 Hm2".
destruct i, j.
- rewrite !lookup_total_insert.
rewrite !iProto_message_equivI.
iDestruct "Hm1" as (Hieq) "#Hm1".
iDestruct "Hm2" as (Hjeq) "#Hm2".
done.
- destruct j; last first.
{ rewrite !lookup_total_insert.
rewrite !iProto_message_equivI.
iDestruct "Hm1" as (Hieq) "#Hm1". done. }
rewrite !lookup_total_insert.
rewrite lookup_total_insert_ne; [|done].
rewrite !lookup_total_insert.
rewrite !iProto_message_equivI.
iDestruct "Hm1" as (Hieq) "#Hm1".
iDestruct "Hm2" as (Hjeq) "#Hm2".
iIntros (v p1) "Hm1'".
iSpecialize ("Hm1" $!v (Next p1)).
rewrite iMsg_base_eq. simpl.
rewrite iMsg_exist_eq. simpl.
iRewrite -"Hm1" in "Hm1'".
iExists END%proto.
iSpecialize ("Hm2" $!v (Next END%proto)).
iRewrite -"Hm2".
simpl.
iDestruct "Hm1'" as (x Heq) "[#Heq HP]".
iSplitL.
{ iExists x. iSplit; [done|]. iFrame. done. }
iNext. iRewrite -"Heq".
rewrite insert_commute; [|done].
rewrite insert_insert.
rewrite insert_commute; [|done].
rewrite insert_insert.
rewrite iProto_consistent_unfold.
iIntros (i' j' m1' m2') "Hm1' Hm2'".
destruct i'.
{ rewrite lookup_total_insert.
iDestruct (iProto_end_message_equivI with "Hm1'") as "H".
done. }
destruct i'.
{ rewrite lookup_total_insert_ne; [|done].
rewrite lookup_total_insert.
iDestruct (iProto_end_message_equivI with "Hm1'") as "H".
done. }
rewrite lookup_total_insert_ne; [|done].
rewrite lookup_total_insert_ne; [|done].
rewrite lookup_total_alt. simpl.
iDestruct (iProto_end_message_equivI with "Hm1'") as "H".
done.
- destruct i; last first.
{ rewrite lookup_total_insert_ne; [|done].
rewrite lookup_total_insert_ne; [|done].
rewrite lookup_total_alt. simpl.
iDestruct (iProto_end_message_equivI with "Hm1") as "H".
done. }
rewrite !lookup_total_insert.
rewrite lookup_total_insert_ne; [|done].
rewrite !lookup_total_insert.
rewrite !iProto_message_equivI.
iDestruct "Hm1" as (Hieq) "#Hm1". done.
- destruct i.
{ rewrite lookup_total_insert_ne; [|done].
rewrite lookup_total_insert.
rewrite !iProto_message_equivI.
iDestruct "Hm1" as (Hieq) "#Hm1". done. }
rewrite lookup_total_insert_ne; [|done].
rewrite lookup_total_insert_ne; [|done].
rewrite lookup_total_alt. simpl.
iDestruct (iProto_end_message_equivI with "Hm1") as "H".
done.
Qed.
Definition iProto_example3 `{!invGS Σ} : gmap nat (iProto Σ Z) :=
<[0 := (<(Send 1) @ (x:Z)> MSG x ; <(Recv 2)> MSG x; END)%proto ]>
(<[1 := (<(Recv 0) @ (x:Z)> MSG x ; <(Send 2)> MSG x; END)%proto ]>
(<[2 := (<(Recv 1) @ (x:Z)> MSG x ; <(Send 0)> MSG x; END)%proto ]>
∅)).
Lemma iProto_example3_consistent `{!invGS Σ} :
⊢ iProto_consistent (@iProto_example3 _ Σ invGS0).
Proof.
rewrite iProto_consistent_unfold.
rewrite /iProto_example3.
iIntros (i j m1 m2) "Hm1 Hm2".
destruct i; last first.
{ destruct i.
{ rewrite lookup_total_insert_ne; [|done].
rewrite !lookup_total_insert.
rewrite !iProto_message_equivI.
by iDestruct "Hm1" as (Hieq) "#Hm1". }
destruct i.
{ rewrite lookup_total_insert_ne; [|done].
rewrite lookup_total_insert_ne; [|done].
rewrite !iProto_message_equivI.
by iDestruct "Hm1" as (Hieq) "#Hm1". }
rewrite lookup_total_insert_ne; [|done].
rewrite lookup_total_insert_ne; [|done].
rewrite lookup_total_insert_ne; [|done].
rewrite lookup_total_empty=> /=.
by rewrite iProto_end_message_equivI. }
destruct j.
{ rewrite !lookup_total_insert.
rewrite !iProto_message_equivI.
by iDestruct "Hm2" as (Hjeq) "#Hm2". }
destruct j; last first.
{ rewrite !lookup_total_insert.
rewrite lookup_total_insert_ne; [|done].
rewrite lookup_total_insert_ne; [|done].
destruct j.
{ rewrite !lookup_total_insert.
rewrite !iProto_message_equivI.
by iDestruct "Hm2" as (Hjeq) "#Hm2". }
rewrite lookup_total_insert_ne; [|done].
rewrite lookup_total_empty=> /=.
by rewrite iProto_end_message_equivI. }
rewrite !lookup_total_insert.
rewrite lookup_total_insert_ne; [|done].
rewrite !lookup_total_insert.
rewrite !iProto_message_equivI.
iDestruct "Hm1" as (Hieq) "#Hm1".
iDestruct "Hm2" as (Hjeq) "#Hm2".
iIntros (v p1) "Hm1'".
iSpecialize ("Hm1" $!v (Next p1)).
rewrite !iMsg_base_eq.
rewrite !iMsg_exist_eq.
iRewrite -"Hm1" in "Hm1'".
iDestruct "Hm1'" as (x Heq) "[#Hm1' _]".
iSpecialize ("Hm2" $!v (Next (<Send 2> iMsg_base_def x True END)))%proto.
iExists (<Send 2> iMsg_base_def x True END)%proto.
iRewrite -"Hm2".
simpl.
iSplitL.
{ iExists x. iSplit; [done|]. iFrame. done. }
iNext.
rewrite iProto_consistent_unfold.
rewrite insert_commute; [|done].
rewrite insert_insert.
rewrite insert_commute; [|done].
rewrite insert_insert.
iRewrite -"Hm1'".
iClear (m1 m2 Hieq Hjeq p1 v Heq) "Hm1 Hm2 Hm1'".
iIntros (i j m1 m2) "Hm1 Hm2".
destruct i.
{ rewrite !lookup_total_insert.
rewrite !iProto_message_equivI.
by iDestruct "Hm1" as (Hieq) "#Hm1". }
rewrite lookup_total_insert_ne; [|done].
destruct i; last first.
{ destruct i.
{ rewrite lookup_total_insert_ne; [|done].
rewrite lookup_total_insert.
rewrite !iProto_message_equivI.
by iDestruct "Hm1" as (Hieq) "#Hm1". }
rewrite lookup_total_insert_ne; [|done].
rewrite lookup_total_insert_ne; [|done].
rewrite lookup_total_empty=> /=.
by rewrite iProto_end_message_equivI. }
rewrite lookup_total_insert.
destruct j.
{ rewrite lookup_total_insert.
rewrite !iProto_message_equivI.
by iDestruct "Hm2" as (Hjeq) "#Hm2". }
rewrite lookup_total_insert_ne; [|done].
destruct j.
{ rewrite lookup_total_insert.
rewrite !iProto_message_equivI.
by iDestruct "Hm2" as (Hjeq) "#Hm2". }
rewrite lookup_total_insert_ne; [|done].
destruct j; last first.
{ rewrite lookup_total_insert_ne; [|done].
rewrite lookup_total_empty.
by rewrite iProto_end_message_equivI. }
rewrite lookup_total_insert.
rewrite !iProto_message_equivI.
iDestruct "Hm1" as (Hieq) "#Hm1".
iDestruct "Hm2" as (Hjeq) "#Hm2".
iIntros (v p1) "Hm1'".
iSpecialize ("Hm1" $!v (Next p1)).
iRewrite -"Hm1" in "Hm1'".
iDestruct "Hm1'" as (Heq) "[#Hm1' _]".
iSpecialize ("Hm2" $!v (Next (<Send 0> iMsg_base_def x True END)))%proto.
iExists (<Send 0> iMsg_base_def x True END)%proto.
iRewrite -"Hm2".
simpl.
iSplitL.
{ iExists x. iSplit; [done|]. iFrame. done. }
iNext.
rewrite iProto_consistent_unfold.
rewrite (insert_commute _ 1 2); [|done].
rewrite (insert_commute _ 1 0); [|done].
rewrite insert_insert.
rewrite (insert_commute _ 2 0); [|done].
rewrite (insert_commute _ 2 1); [|done].
rewrite insert_insert.
iRewrite -"Hm1'".
iClear (m1 m2 Hieq Hjeq p1 v Heq) "Hm1 Hm2 Hm1'".
iIntros (i j m1 m2) "Hm1 Hm2".
destruct i.
{ rewrite !lookup_total_insert.
rewrite !iProto_message_equivI.
by iDestruct "Hm1" as (Hieq) "#Hm1". }
rewrite lookup_total_insert_ne; [|done].
destruct i.
{ rewrite lookup_total_insert.
by rewrite iProto_end_message_equivI. }
rewrite lookup_total_insert_ne; [|done].
destruct i; last first.
{ rewrite lookup_total_insert_ne; [|done].
rewrite lookup_total_empty.
by rewrite iProto_end_message_equivI. }
rewrite lookup_total_insert.
destruct j; last first.
{ rewrite lookup_total_insert_ne; [|done].
destruct j.
{ rewrite lookup_total_insert.
by rewrite iProto_end_message_equivI. }
rewrite lookup_total_insert_ne; [|done].
destruct j.
{ rewrite lookup_total_insert.
rewrite !iProto_message_equivI.
by iDestruct "Hm2" as (Hjeq) "#Hm2". }
rewrite lookup_total_insert_ne; [|done].
rewrite lookup_total_empty.
by rewrite iProto_end_message_equivI. }
rewrite lookup_total_insert.
rewrite !iProto_message_equivI.
iDestruct "Hm1" as (Hieq) "#Hm1".
iDestruct "Hm2" as (Hjeq) "#Hm2".
iIntros (v p1) "Hm1'".
iSpecialize ("Hm1" $!v (Next p1)).
iRewrite -"Hm1" in "Hm1'".
iDestruct "Hm1'" as (Heq) "[#Hm1' _]".
iSpecialize ("Hm2" $!v (Next END))%proto.
iExists END%proto.
iRewrite -"Hm2".
simpl.
iSplitL; [done|].
rewrite insert_insert.
rewrite insert_commute; [|done].
rewrite (insert_commute _ 2 1); [|done].
rewrite insert_insert.
iNext.
iRewrite -"Hm1'".
rewrite iProto_consistent_unfold.
iClear (m1 m2 Hieq Hjeq p1 v Heq) "Hm1 Hm2 Hm1'".
iIntros (i j m1 m2) "Hm1 Hm2".
destruct i.
{ rewrite lookup_total_insert.
by rewrite !iProto_end_message_equivI. }
rewrite lookup_total_insert_ne; [|done].
destruct i.
{ rewrite lookup_total_insert.
by rewrite !iProto_end_message_equivI. }
rewrite lookup_total_insert_ne; [|done].
destruct i.
{ rewrite lookup_total_insert.
by rewrite !iProto_end_message_equivI. }
rewrite lookup_total_insert_ne; [|done].
rewrite lookup_total_empty.
by rewrite iProto_end_message_equivI.
Qed.