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From iris.program_logic Require Export weakestpre.
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From iris.algebra Require Import gmap auth agree gset coPset.
From iris.base_logic Require Import big_op soundness.
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From iris.program_logic Require Import wsat.
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From iris.proofmode Require Import tactics.
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Import uPred.
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Record adequate {Λ} (e1 : expr Λ) (σ1 : state Λ) (φ : val Λ  Prop) := {
  adequate_result t2 σ2 v2 :
   rtc step ([e1], σ1) (of_val v2 :: t2, σ2)  φ v2;
  adequate_safe t2 σ2 e2 :
   rtc step ([e1], σ1) (t2, σ2) 
   e2  t2  (is_Some (to_val e2)  reducible e2 σ2)
}.

Theorem adequate_tp_safe {Λ} (e1 : expr Λ) t2 σ1 σ2 φ :
  adequate e1 σ1 φ 
  rtc step ([e1], σ1) (t2, σ2) 
  Forall (λ e, is_Some (to_val e)) t2   t3 σ3, step (t2, σ2) (t3, σ3).
Proof.
  intros Had ?.
  destruct (decide (Forall (λ e, is_Some (to_val e)) t2)) as [|Ht2]; [by left|].
  apply (not_Forall_Exists _), Exists_exists in Ht2; destruct Ht2 as (e2&?&He2).
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  destruct (adequate_safe e1 σ1 φ Had t2 σ2 e2) as [?|(e3&σ3&efs&?)];
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    rewrite ?eq_None_not_Some; auto.
  { exfalso. eauto. }
  destruct (elem_of_list_split t2 e2) as (t2'&t2''&->); auto.
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  right; exists (t2' ++ e3 :: t2'' ++ efs), σ3; econstructor; eauto.
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Qed.

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Section adequacy.
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Context `{irisG Λ Σ}.
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Implicit Types e : expr Λ.
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Implicit Types P Q : iProp Σ.
Implicit Types Φ : val Λ  iProp Σ.
Implicit Types Φs : list (val Λ  iProp Σ).
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Notation world σ := (wsat  ownE   ownP_auth σ)%I.

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Notation wptp t := ([ list] ef  t, WP ef {{ _, True }})%I.
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Lemma wp_step e1 σ1 e2 σ2 efs Φ :
  prim_step e1 σ1 e2 σ2 efs 
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  world σ1  WP e1 {{ Φ }} ==  |==>  (world σ2  WP e2 {{ Φ }}  wptp efs).
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Proof.
  rewrite {1}wp_unfold /wp_pre. iIntros (Hstep) "[(Hw & HE & Hσ) [H|[_ H]]]".
  { iDestruct "H" as (v) "[% _]". apply val_stuck in Hstep; simplify_eq. }
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  rewrite fupd_eq /fupd_def.
  iUpd ("H" $! σ1 with "Hσ [Hw HE]") as ">(Hw & HE & _ & H)"; first by iFrame.
  iUpdIntro; iNext.
  iUpd ("H" $! e2 σ2 efs with "[%] [Hw HE]")
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    as ">($ & $ & $ & $)"; try iFrame; eauto.
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Qed.

Lemma wptp_step e1 t1 t2 σ1 σ2 Φ :
  step (e1 :: t1,σ1) (t2, σ2) 
  world σ1  WP e1 {{ Φ }}  wptp t1
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  ==  e2 t2', t2 = e2 :: t2'   |==>  (world σ2  WP e2 {{ Φ }}  wptp t2').
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Proof.
  iIntros (Hstep) "(HW & He & Ht)".
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  destruct Hstep as [e1' σ1' e2' σ2' efs [|? t1'] t2' ?? Hstep]; simplify_eq/=.
  - iExists e2', (t2' ++ efs); iSplitR; first eauto.
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    rewrite big_sepL_app. iFrame "Ht". iApply wp_step; try iFrame; eauto.
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  - iExists e, (t1' ++ e2' :: t2' ++ efs); iSplitR; first eauto.
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    rewrite !big_sepL_app !big_sepL_cons big_sepL_app.
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    iDestruct "Ht" as "($ & He' & $)"; iFrame "He".
    iApply wp_step; try iFrame; eauto.
Qed.

Lemma wptp_steps n e1 t1 t2 σ1 σ2 Φ :
  nsteps step n (e1 :: t1, σ1) (t2, σ2) 
  world σ1  WP e1 {{ Φ }}  wptp t1 
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  Nat.iter (S n) (λ P, |==>  P) ( e2 t2',
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    t2 = e2 :: t2'  world σ2  WP e2 {{ Φ }}  wptp t2').
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Proof.
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  revert e1 t1 t2 σ1 σ2; simpl; induction n as [|n IH]=> e1 t1 t2 σ1 σ2 /=.
  { inversion_clear 1; iIntros "?"; eauto 10. }
  iIntros (Hsteps) "H". inversion_clear Hsteps as [|?? [t1' σ1']].
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  iUpd (wptp_step with "H") as (e1' t1'') "[% H]"; first eauto; simplify_eq.
  iUpdIntro; iNext; iUpd "H" as ">?". by iApply IH.
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Qed.
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Instance bupd_iter_mono n : Proper (() ==> ()) (Nat.iter n (λ P, |==>  P)%I).
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Proof. intros P Q HP. induction n; simpl; do 2?f_equiv; auto. Qed.

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Lemma bupd_iter_frame_l n R Q :
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  R  Nat.iter n (λ P, |==>  P) Q  Nat.iter n (λ P, |==>  P) (R  Q).
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Proof.
  induction n as [|n IH]; simpl; [done|].
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  by rewrite bupd_frame_l {1}(later_intro R) -later_sep IH.
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Qed.

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Lemma wptp_result n e1 t1 v2 t2 σ1 σ2 φ :
  nsteps step n (e1 :: t1, σ1) (of_val v2 :: t2, σ2) 
  world σ1  WP e1 {{ v,  φ v }}  wptp t1 
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  Nat.iter (S (S n)) (λ P, |==>  P) ( φ v2).
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Proof.
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  intros. rewrite wptp_steps //.
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  rewrite (Nat_iter_S_r (S n)). apply bupd_iter_mono.
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  iDestruct 1 as (e2 t2') "(% & (Hw & HE & _) & H & _)"; simplify_eq.
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  iDestruct (wp_value_inv with "H") as "H". rewrite fupd_eq /fupd_def.
  iUpd ("H" with "[Hw HE]") as ">(_ & _ & $)"; iFrame; auto.
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Qed.
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Lemma wp_safe e σ Φ :
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  world σ  WP e {{ Φ }} ==   (is_Some (to_val e)  reducible e σ).
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Proof.
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  rewrite wp_unfold /wp_pre. iIntros "[(Hw&HE&Hσ) [H|[_ H]]]".
  { iDestruct "H" as (v) "[% _]"; eauto 10. }
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  rewrite fupd_eq. iUpd ("H" with "* Hσ [-]") as ">(?&?&%&?)"; first by iFrame.
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  eauto 10.
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Qed.
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Lemma wptp_safe n e1 e2 t1 t2 σ1 σ2 Φ :
  nsteps step n (e1 :: t1, σ1) (t2, σ2)  e2  t2 
  world σ1  WP e1 {{ Φ }}  wptp t1 
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  Nat.iter (S (S n)) (λ P, |==>  P) ( (is_Some (to_val e2)  reducible e2 σ2)).
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Proof.
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  intros ? He2. rewrite wptp_steps //; rewrite (Nat_iter_S_r (S n)). apply bupd_iter_mono.
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  iDestruct 1 as (e2' t2') "(% & Hw & H & Htp)"; simplify_eq.
  apply elem_of_cons in He2 as [<-|?]; first (iApply wp_safe; by iFrame "Hw H").
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  iApply wp_safe. iFrame "Hw". by iApply (big_sepL_elem_of with "Htp").
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Qed.
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Lemma wptp_invariance n e1 e2 t1 t2 σ1 σ2 I φ Φ :
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  nsteps step n (e1 :: t1, σ1) (t2, σ2) 
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  (I ={,}=  σ', ownP σ'   φ σ') 
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  I  world σ1  WP e1 {{ Φ }}  wptp t1 
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  Nat.iter (S (S n)) (λ P, |==>  P) ( φ σ2).
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Proof.
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  intros ? HI. rewrite wptp_steps //.
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  rewrite (Nat_iter_S_r (S n)) bupd_iter_frame_l. apply bupd_iter_mono.
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  iIntros "[HI H]".
  iDestruct "H" as (e2' t2') "(% & (Hw&HE&Hσ) & _)"; subst.
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  rewrite fupd_eq in HI;
    iUpd (HI with "HI [Hw HE]") as "> (_ & _ & H)"; first by iFrame.
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  iDestruct "H" as (σ2') "[Hσf %]".
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  iDestruct (ownP_agree σ2 σ2' with "[-]") as %<-. by iFrame. eauto.
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Qed.
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End adequacy.
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Theorem wp_adequacy Σ `{irisPreG Λ Σ} e σ φ :
  ( `{irisG Λ Σ}, ownP σ  WP e {{ v,  φ v }}) 
  adequate e σ φ.
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Proof.
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  intros Hwp; split.
  - intros t2 σ2 v2 [n ?]%rtc_nsteps.
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    eapply (soundness (M:=iResUR Σ) _ (S (S (S n)))); iIntros "".
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    rewrite Nat_iter_S. iUpd (iris_alloc σ) as (?) "(?&?&?&Hσ)".
    iUpdIntro. iNext. iApply wptp_result; eauto.
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    iFrame. iSplitL. by iApply Hwp. by iApply big_sepL_nil.
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  - intros t2 σ2 e2 [n ?]%rtc_nsteps ?.
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    eapply (soundness (M:=iResUR Σ) _ (S (S (S n)))); iIntros "".
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    rewrite Nat_iter_S. iUpd (iris_alloc σ) as (?) "(Hw & HE & Hσ & Hσf)".
    iUpdIntro. iNext. iApply wptp_safe; eauto.
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    iFrame "Hw HE Hσ". iSplitL. by iApply Hwp. by iApply big_sepL_nil.
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Qed.
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Theorem wp_invariance Σ `{irisPreG Λ Σ} e σ1 t2 σ2 I φ Φ :
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  ( `{irisG Λ Σ}, ownP σ1 ={}= I  WP e {{ Φ }}) 
  ( `{irisG Λ Σ}, I ={,}=  σ', ownP σ'   φ σ') 
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  rtc step ([e], σ1) (t2, σ2) 
  φ σ2.
Proof.
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  intros Hwp HI [n ?]%rtc_nsteps.
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  eapply (soundness (M:=iResUR Σ) _ (S (S (S n)))); iIntros "".
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  rewrite Nat_iter_S. iUpd (iris_alloc σ1) as (?) "(Hw & HE & ? & Hσ)".
  rewrite fupd_eq in Hwp.
  iUpd (Hwp _ with "Hσ [Hw HE]") as ">(? & ? & ? & ?)"; first by iFrame.
  iUpdIntro. iNext. iApply wptp_invariance; eauto. iFrame. by iApply big_sepL_nil.
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Qed.