weakestpre.v 11.3 KB
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From program_logic Require Export pviewshifts.
From program_logic Require Import wsat.
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Local Hint Extern 10 (_  _) => omega.
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Local Hint Extern 100 (@eq coPset _ _) => eassumption || set_solver.
Local Hint Extern 100 (_  _) => set_solver.
Local Hint Extern 100 (@subseteq coPset _ _ _) => set_solver.
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Local Hint Extern 10 ({_} _) =>
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  repeat match goal with
  | H : wsat _ _ _ _ |- _ => apply wsat_valid in H; last omega
  end; solve_validN.
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Record wp_go {Λ Σ} (E : coPset) (Φ Φfork : expr Λ  nat  iRes Λ Σ  Prop)
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    (k : nat) (rf : iRes Λ Σ) (e1 : expr Λ) (σ1 : state Λ) := {
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  wf_safe : reducible e1 σ1;
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  wp_step e2 σ2 ef :
    prim_step e1 σ1 e2 σ2 ef 
     r2 r2',
      wsat k E σ2 (r2  r2'  rf) 
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      Φ e2 k r2 
       e', ef = Some e'  Φfork e' k r2'
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}.
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CoInductive wp_pre {Λ Σ} (E : coPset)
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     (Φ : val Λ  iProp Λ Σ) : expr Λ  nat  iRes Λ Σ  Prop :=
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  | wp_pre_value n r v : (|={E}=> Φ v)%I n r  wp_pre E Φ (of_val v) n r
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  | wp_pre_step n r1 e1 :
     to_val e1 = None 
     ( rf k Ef σ1,
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       0 < k < n  E  Ef =  
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       wsat (S k) (E  Ef) σ1 (r1  rf) 
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       wp_go (E  Ef) (wp_pre E Φ)
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                      (wp_pre  (λ _, True%I)) k rf e1 σ1) 
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     wp_pre E Φ e1 n r1.
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(* TODO: Consider sealing this, like all the definitions in upred.v. *)
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Program Definition wp {Λ Σ} (E : coPset) (e : expr Λ)
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  (Φ : val Λ  iProp Λ Σ) : iProp Λ Σ := {| uPred_holds := wp_pre E Φ e |}.
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Next Obligation.
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  intros Λ Σ E e Φ n r1 r2 Hwp Hr.
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  destruct Hwp as [|n r1 e2 ? Hgo]; constructor; rewrite -?Hr; auto.
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  intros rf k Ef σ1 ?; rewrite -(dist_le _ _ _ _ Hr); naive_solver.
Qed.
Next Obligation.
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  intros Λ Σ E e Φ n1 n2 r1 r2; revert Φ E e n2 r1 r2.
  induction n1 as [n1 IH] using lt_wf_ind; intros Φ E e n2 r1 r1'.
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  destruct 1 as [|n1 r1 e1 ? Hgo].
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  - constructor; eauto using uPred_weaken.
  - intros [rf' Hr] ??; constructor; [done|intros rf k Ef σ1 ???].
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    destruct (Hgo (rf'  rf) k Ef σ1) as [Hsafe Hstep];
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      rewrite ?assoc -?Hr; auto; constructor; [done|].
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    intros e2 σ2 ef ?; destruct (Hstep e2 σ2 ef) as (r2&r2'&?&?&?); auto.
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    exists r2, (r2'  rf'); split_and?; eauto 10 using (IH k), cmra_included_l.
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    by rewrite -!assoc (assoc _ r2).
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Qed.
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Instance: Params (@wp) 4.
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Notation "|| e @ E {{ Φ } }" := (wp E e Φ)
  (at level 20, e, Φ at level 200,
   format "||  e  @  E  {{  Φ  } }") : uPred_scope.
Notation "|| e {{ Φ } }" := (wp  e Φ)
  (at level 20, e, Φ at level 200,
   format "||  e   {{  Φ  } }") : uPred_scope.

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Section wp.
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Context {Λ : language} {Σ : iFunctor}.
Implicit Types P : iProp Λ Σ.
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Implicit Types Φ : val Λ  iProp Λ Σ.
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Implicit Types v : val Λ.
Implicit Types e : expr Λ.
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Transparent uPred_holds.
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Global Instance wp_ne E e n :
  Proper (pointwise_relation _ (dist n) ==> dist n) (@wp Λ Σ E e).
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Proof.
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  cut ( Φ Ψ, ( v, Φ v {n} Ψ v) 
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     n' r, n'  n  {n'} r  wp E e Φ n' r  wp E e Ψ n' r).
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  { intros help Φ Ψ HΦΨ. by do 2 split; apply help. }
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  intros Φ Ψ HΦΨ n' r; revert e r.
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  induction n' as [n' IH] using lt_wf_ind=> e r.
  destruct 3 as [n' r v HpvsQ|n' r e1 ? Hgo].
  { constructor. by eapply pvs_ne, HpvsQ; eauto. }
  constructor; [done|]=> rf k Ef σ1 ???.
  destruct (Hgo rf k Ef σ1) as [Hsafe Hstep]; auto.
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  split; [done|intros e2 σ2 ef ?].
  destruct (Hstep e2 σ2 ef) as (r2&r2'&?&?&?); auto.
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  exists r2, r2'; split_and?; [|eapply IH|]; eauto.
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Qed.
Global Instance wp_proper E e :
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  Proper (pointwise_relation _ () ==> ()) (@wp Λ Σ E e).
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Proof.
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  by intros Φ Φ' ?; apply equiv_dist=>n; apply wp_ne=>v; apply equiv_dist.
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Qed.
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Lemma wp_mask_frame_mono E1 E2 e Φ Ψ :
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  E1  E2  ( v, Φ v  Ψ v)  || e @ E1 {{ Φ }}  || e @ E2 {{ Ψ }}.
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Proof.
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  intros HE HΦ; split=> n r.
  revert e r; induction n as [n IH] using lt_wf_ind=> e r.
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  destruct 2 as [n' r v HpvsQ|n' r e1 ? Hgo].
  { constructor; eapply pvs_mask_frame_mono, HpvsQ; eauto. }
  constructor; [done|]=> rf k Ef σ1 ???.
  assert (E2  Ef = E1  (E2  E1  Ef)) as HE'.
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  { by rewrite assoc_L -union_difference_L. }
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  destruct (Hgo rf k ((E2  E1)  Ef) σ1) as [Hsafe Hstep]; rewrite -?HE'; auto.
  split; [done|intros e2 σ2 ef ?].
  destruct (Hstep e2 σ2 ef) as (r2&r2'&?&?&?); auto.
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  exists r2, r2'; split_and?; [rewrite HE'|eapply IH|]; eauto.
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Qed.
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Lemma wp_value_inv E Φ v n r :
  || of_val v @ E {{ Φ }}%I n r  (|={E}=> Φ v)%I n r.
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Proof.
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  by inversion 1 as [|??? He]; [|rewrite ?to_of_val in He]; simplify_eq.
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Qed.
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Lemma wp_step_inv E Ef Φ e k n σ r rf :
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  to_val e = None  0 < k < n  E  Ef =  
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  || e @ E {{ Φ }}%I n r  wsat (S k) (E  Ef) σ (r  rf) 
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  wp_go (E  Ef) (λ e, wp E e Φ) (λ e, wp  e (λ _, True%I)) k rf e σ.
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Proof. intros He; destruct 3; [by rewrite ?to_of_val in He|eauto]. Qed.
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Lemma wp_value' E Φ v : Φ v  || of_val v @ E {{ Φ }}.
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Proof. split=> n r; constructor; by apply pvs_intro. Qed.
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Lemma pvs_wp E e Φ : (|={E}=> || e @ E {{ Φ }})  || e @ E {{ Φ }}.
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Proof.
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  split=> n r ? Hvs.
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  destruct (to_val e) as [v|] eqn:He; [apply of_to_val in He; subst|].
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  { constructor; eapply pvs_trans', pvs_mono, Hvs; eauto.
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    split=> ???; apply wp_value_inv. }
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  constructor; [done|]=> rf k Ef σ1 ???.
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  destruct (Hvs rf (S k) Ef σ1) as (r'&Hwp&?); auto.
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  eapply wp_step_inv with (S k) r'; eauto.
Qed.
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Lemma wp_pvs E e Φ : || e @  E {{ λ v, |={E}=> Φ v }}  || e @ E {{ Φ }}.
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Proof.
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  split=> n r; revert e r; induction n as [n IH] using lt_wf_ind=> e r Hr HΦ.
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  destruct (to_val e) as [v|] eqn:He; [apply of_to_val in He; subst|].
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  { constructor; apply pvs_trans', (wp_value_inv _ (pvs E E  Φ)); auto. }
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  constructor; [done|]=> rf k Ef σ1 ???.
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  destruct (wp_step_inv E Ef (pvs E E  Φ) e k n σ1 r rf) as [? Hstep]; auto.
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  split; [done|intros e2 σ2 ef ?].
  destruct (Hstep e2 σ2 ef) as (r2&r2'&?&Hwp'&?); auto.
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  exists r2, r2'; split_and?; [|apply (IH k)|]; auto.
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Qed.
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Lemma wp_atomic E1 E2 e Φ :
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  E2  E1  atomic e 
  (|={E1,E2}=> || e @ E2 {{ λ v, |={E2,E1}=> Φ v }})  || e @ E1 {{ Φ }}.
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Proof.
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  intros ? He; split=> n r ? Hvs; constructor; eauto using atomic_not_val.
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  intros rf k Ef σ1 ???.
  destruct (Hvs rf (S k) Ef σ1) as (r'&Hwp&?); auto.
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  destruct (wp_step_inv E2 Ef (pvs E2 E1  Φ) e k (S k) σ1 r' rf)
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    as [Hsafe Hstep]; auto using atomic_not_val.
  split; [done|]=> e2 σ2 ef ?.
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  destruct (Hstep e2 σ2 ef) as (r2&r2'&?&Hwp'&?); clear Hsafe Hstep; auto.
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  destruct Hwp' as [k r2 v Hvs'|k r2 e2 Hgo];
    [|destruct (atomic_step e σ1 e2 σ2 ef); naive_solver].
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  apply pvs_trans in Hvs'; auto.
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  destruct (Hvs' (r2'  rf) k Ef σ2) as (r3&[]); rewrite ?assoc; auto.
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  exists r3, r2'; split_and?; last done.
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  - by rewrite -assoc.
  - constructor; apply pvs_intro; auto.
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Qed.
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Lemma wp_frame_r E e Φ R : (|| e @ E {{ Φ }}  R)  || e @ E {{ λ v, Φ v  R }}.
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Proof.
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  uPred.unseal; split; intros n r' Hvalid (r&rR&Hr&Hwp&?); revert Hvalid.
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  rewrite Hr; clear Hr; revert e r Hwp.
  induction n as [n IH] using lt_wf_ind; intros e r1.
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  destruct 1 as [|n r e ? Hgo]=>?.
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  { constructor. rewrite -uPred_sep_eq; apply pvs_frame_r; auto.
    uPred.unseal; exists r, rR; eauto. }
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  constructor; [done|]=> rf k Ef σ1 ???.
  destruct (Hgo (rRrf) k Ef σ1) as [Hsafe Hstep]; auto.
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  { by rewrite assoc. }
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  split; [done|intros e2 σ2 ef ?].
  destruct (Hstep e2 σ2 ef) as (r2&r2'&?&?&?); auto.
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  exists (r2  rR), r2'; split_and?; auto.
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  - by rewrite -(assoc _ r2)
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      (comm _ rR) !assoc -(assoc _ _ rR).
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  - apply IH; eauto using uPred_weaken.
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Qed.
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Lemma wp_frame_later_r E e Φ R :
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  to_val e = None  (|| e @ E {{ Φ }}   R)  || e @ E {{ λ v, Φ v  R }}.
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Proof.
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  intros He; uPred.unseal; split; intros n r' Hvalid (r&rR&Hr&Hwp&?).
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  revert Hvalid; rewrite Hr; clear Hr.
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  destruct Hwp as [|n r e ? Hgo]; [by rewrite to_of_val in He|].
  constructor; [done|]=>rf k Ef σ1 ???; destruct n as [|n]; first omega.
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  destruct (Hgo (rRrf) k Ef σ1) as [Hsafe Hstep];rewrite ?assoc;auto.
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  split; [done|intros e2 σ2 ef ?].
  destruct (Hstep e2 σ2 ef) as (r2&r2'&?&?&?); auto.
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  exists (r2  rR), r2'; split_and?; auto.
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  - by rewrite -(assoc _ r2) (comm _ rR) !assoc -(assoc _ _ rR).
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  - rewrite -uPred_sep_eq.
    apply wp_frame_r; [auto|uPred.unseal; exists r2, rR; split_and?; auto].
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    eapply uPred_weaken with n rR; eauto.
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Qed.
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Lemma wp_bind `{LanguageCtx Λ K} E e Φ :
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  || e @ E {{ λ v, || K (of_val v) @ E {{ Φ }} }}  || K e @ E {{ Φ }}.
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Proof.
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  split=> n r; revert e r; induction n as [n IH] using lt_wf_ind=> e r ?.
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  destruct 1 as [|n r e ? Hgo]; [by apply pvs_wp|].
  constructor; auto using fill_not_val=> rf k Ef σ1 ???.
  destruct (Hgo rf k Ef σ1) as [Hsafe Hstep]; auto.
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  split.
  { destruct Hsafe as (e2&σ2&ef&?).
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    by exists (K e2), σ2, ef; apply fill_step. }
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  intros e2 σ2 ef ?.
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  destruct (fill_step_inv e σ1 e2 σ2 ef) as (e2'&->&?); auto.
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  destruct (Hstep e2' σ2 ef) as (r2&r2'&?&?&?); auto.
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  exists r2, r2'; split_and?; try eapply IH; eauto.
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Qed.

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(** * Derived rules *)
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Opaque uPred_holds.
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Import uPred.
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Lemma wp_mono E e Φ Ψ : ( v, Φ v  Ψ v)  || e @ E {{ Φ }}  || e @ E {{ Ψ }}.
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Proof. by apply wp_mask_frame_mono. Qed.
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Global Instance wp_mono' E e :
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  Proper (pointwise_relation _ () ==> ()) (@wp Λ Σ E e).
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Proof. by intros Φ Φ' ?; apply wp_mono. Qed.
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Lemma wp_strip_pvs E e P Φ :
  P  || e @ E {{ Φ }}  (|={E}=> P)  || e @ E {{ Φ }}.
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Proof. move=>->. by rewrite pvs_wp. Qed.
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Lemma wp_value E Φ e v : to_val e = Some v  Φ v  || e @ E {{ Φ }}.
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Proof. intros; rewrite -(of_to_val e v) //; by apply wp_value'. Qed.
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Lemma wp_value_pvs E Φ e v :
  to_val e = Some v  (|={E}=> Φ v)  || e @ E {{ Φ }}.
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Proof. intros. rewrite -wp_pvs. rewrite -wp_value //. Qed.
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Lemma wp_frame_l E e Φ R : (R  || e @ E {{ Φ }})  || e @ E {{ λ v, R  Φ v }}.
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Proof. setoid_rewrite (comm _ R); apply wp_frame_r. Qed.
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Lemma wp_frame_later_l E e Φ R :
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  to_val e = None  ( R  || e @ E {{ Φ }})  || e @ E {{ λ v, R  Φ v }}.
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Proof.
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  rewrite (comm _ ( R)%I); setoid_rewrite (comm _ R).
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  apply wp_frame_later_r.
Qed.
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Lemma wp_always_l E e Φ R `{!AlwaysStable R} :
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  (R  || e @ E {{ Φ }})  || e @ E {{ λ v, R  Φ v }}.
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Proof. by setoid_rewrite (always_and_sep_l _ _); rewrite wp_frame_l. Qed.
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Lemma wp_always_r E e Φ R `{!AlwaysStable R} :
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  (|| e @ E {{ Φ }}  R)  || e @ E {{ λ v, Φ v  R }}.
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Proof. by setoid_rewrite (always_and_sep_r _ _); rewrite wp_frame_r. Qed.
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Lemma wp_impl_l E e Φ Ψ :
  ((  v, Φ v  Ψ v)  || e @ E {{ Φ }})  || e @ E {{ Ψ }}.
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Proof.
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  rewrite wp_always_l; apply wp_mono=> // v.
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  by rewrite always_elim (forall_elim v) impl_elim_l.
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Qed.
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Lemma wp_impl_r E e Φ Ψ :
  (|| e @ E {{ Φ }}   ( v, Φ v  Ψ v))  || e @ E {{ Ψ }}.
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Proof. by rewrite comm wp_impl_l. Qed.
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Lemma wp_mask_weaken E1 E2 e Φ :
  E1  E2  || e @ E1 {{ Φ }}  || e @ E2 {{ Φ }}.
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Proof. auto using wp_mask_frame_mono. Qed.

(** * Weakest-pre is a FSA. *)
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Definition wp_fsa (e : expr Λ) : FSA Λ Σ (val Λ) := λ E, wp E e.
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Global Arguments wp_fsa _ _ / _.
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Global Instance wp_fsa_prf : FrameShiftAssertion (atomic e) (wp_fsa e).
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Proof.
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  rewrite /wp_fsa; split; auto using wp_mask_frame_mono, wp_atomic, wp_frame_r.
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  intros E Φ. by rewrite -(pvs_wp E e Φ) -(wp_pvs E e Φ).
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Qed.
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End wp.