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From heap_lang Require Export lifting.
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From algebra Require Import upred_big_op frac dec_agree.
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From program_logic Require Export invariants ghost_ownership.
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From program_logic Require Import ownership auth.
Import uPred.
(* TODO: The entire construction could be generalized to arbitrary languages that have
   a finmap as their state. Or maybe even beyond "as their state", i.e. arbitrary
   predicates over finmaps instead of just ownP. *)

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Definition heapR : cmraT := mapR loc (fracR (dec_agreeR val)).
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(** The CMRA we need. *)
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Class heapG Σ := HeapG {
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  heap_inG :> authG heap_lang Σ heapR;
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  heap_name : gname
}.
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(** The Functor we need. *)
Definition heapGF : rFunctor := authGF heapR.
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Definition to_heap : state  heapR := fmap (λ v, Frac 1 (DecAgree v)).
Definition of_heap : heapR  state :=
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  omap (mbind (maybe DecAgree  snd)  maybe2 Frac).
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Section definitions.
  Context `{i : heapG Σ}.
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  Definition heap_mapsto (l : loc) (q : Qp) (v: val) : iPropG heap_lang Σ :=
    auth_own heap_name {[ l := Frac q (DecAgree v) ]}.
  Definition heap_inv (h : heapR) : iPropG heap_lang Σ :=
    ownP (of_heap h).
  Definition heap_ctx (N : namespace) : iPropG heap_lang Σ :=
    auth_ctx heap_name N heap_inv.

  Global Instance heap_inv_proper : Proper (() ==> ()) heap_inv.
  Proof. solve_proper. Qed.
  Global Instance heap_ctx_always_stable N : AlwaysStable (heap_ctx N).
  Proof. apply _. Qed.
End definitions.
Typeclasses Opaque heap_ctx heap_mapsto.
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Notation "l ↦{ q } v" := (heap_mapsto l q v)
  (at level 20, q at level 50, format "l  ↦{ q }  v") : uPred_scope.
Notation "l ↦ v" := (heap_mapsto l 1 v) (at level 20) : uPred_scope.
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Section heap.
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  Context {Σ : rFunctorG}.
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  Implicit Types N : namespace.
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  Implicit Types P Q : iPropG heap_lang Σ.
  Implicit Types Φ : val  iPropG heap_lang Σ.
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  Implicit Types σ : state.
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  Implicit Types h g : heapR.
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  (** Conversion to heaps and back *)
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  Global Instance of_heap_proper : Proper (() ==> (=)) of_heap.
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  Proof. solve_proper. Qed.
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  Lemma from_to_heap σ : of_heap (to_heap σ) = σ.
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  Proof.
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    apply map_eq=>l. rewrite lookup_omap lookup_fmap. by case (σ !! l).
  Qed.
  Lemma to_heap_valid σ :  to_heap σ.
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  Proof. intros l. rewrite lookup_fmap. by case (σ !! l). Qed.
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  Lemma of_heap_insert l v h :
    of_heap (<[l:=Frac 1 (DecAgree v)]> h) = <[l:=v]> (of_heap h).
  Proof. by rewrite /of_heap -(omap_insert _ _ _ (Frac 1 (DecAgree v))). Qed.
  Lemma of_heap_singleton_op l q v h :
     ({[l := Frac q (DecAgree v)]}  h) 
    of_heap ({[l := Frac q (DecAgree v)]}  h) = <[l:=v]> (of_heap h).
  Proof.
    intros Hv. apply map_eq=> l'; destruct (decide (l' = l)) as [->|].
    - move: (Hv l). rewrite /of_heap lookup_insert
        lookup_omap (lookup_op _ h) lookup_singleton.
      case _:(h !! l)=>[[q' [v'|]|]|] //=; last by move=> [??].
      move=> [? /dec_agree_op_inv [->]]. by rewrite dec_agree_idemp.
    - rewrite /of_heap lookup_insert_ne // !lookup_omap.
      by rewrite (lookup_op _ h) lookup_singleton_ne // left_id.
  Qed.
  Lemma to_heap_insert l v σ :
    to_heap (<[l:=v]> σ) = <[l:=Frac 1 (DecAgree v)]> (to_heap σ).
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  Proof. by rewrite /to_heap -fmap_insert. Qed.
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  Lemma of_heap_None h l :
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     h  of_heap h !! l = None  h !! l = None  h !! l  Some FracUnit.
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  Proof.
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    move=> /(_ l). rewrite /of_heap lookup_omap.
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    by case: (h !! l)=> [[q [v|]|]|] //=; destruct 1; auto.
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  Qed.
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  Lemma heap_store_valid l h v1 v2 :
     ({[l := Frac 1 (DecAgree v1)]}  h) 
     ({[l := Frac 1 (DecAgree v2)]}  h).
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  Proof.
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    intros Hv l'; move: (Hv l'). destruct (decide (l' = l)) as [->|].
    - rewrite !lookup_op !lookup_singleton.
      case: (h !! l)=>[x|]; [|done]=> /frac_valid_inv_l=>-> //.
    - by rewrite !lookup_op !lookup_singleton_ne.
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  Qed.
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  Hint Resolve heap_store_valid.
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  (** Allocation *)
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  Lemma heap_alloc N E σ :
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    authG heap_lang Σ heapR  nclose N  E 
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    ownP σ  (|={E}=>  _ : heapG Σ, heap_ctx N  Π★{map σ} (λ l v, l  v)).
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  Proof.
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    intros. rewrite -{1}(from_to_heap σ). etrans.
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    { rewrite [ownP _]later_intro.
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      apply (auth_alloc (ownP  of_heap) N E (to_heap σ)); last done.
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      apply to_heap_valid. }
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    apply pvs_mono, exist_elim=> γ.
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    rewrite -(exist_intro (HeapG _ _ γ)) /heap_ctx; apply and_mono_r.
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    rewrite /heap_mapsto /heap_name.
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    induction σ as [|l v σ Hl IH] using map_ind.
    { rewrite big_sepM_empty; apply True_intro. }
    rewrite to_heap_insert big_sepM_insert //.
    rewrite (map_insert_singleton_op (to_heap σ));
      last rewrite lookup_fmap Hl; auto.
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    (* FIXME: investigate why we have to unfold auth_own here. *)
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    by rewrite auth_own_op IH. 
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  Qed.
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  Context `{heapG Σ}.

  (** General properties of mapsto *)
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  Lemma heap_mapsto_op_eq l q1 q2 v :
    (l {q1} v  l {q2} v)%I  (l {q1+q2} v)%I.
  Proof. by rewrite -auth_own_op map_op_singleton Frac_op dec_agree_idemp. Qed.

  Lemma heap_mapsto_op l q1 q2 v1 v2 :
    (l {q1} v1  l {q2} v2)%I  (v1 = v2  l {q1+q2} v1)%I.
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  Proof.
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    destruct (decide (v1 = v2)) as [->|].
    { by rewrite heap_mapsto_op_eq const_equiv // left_id. }
    rewrite -auth_own_op map_op_singleton Frac_op dec_agree_ne //.
    apply (anti_symm ()); last by apply const_elim_l.
    rewrite auth_own_valid map_validI (forall_elim l) lookup_singleton.
    rewrite option_validI frac_validI discrete_valid. by apply const_elim_r.
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  Qed.

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  Lemma heap_mapsto_op_split l q v :
    (l {q} v)%I  (l {q/2} v  l {q/2} v)%I.
  Proof. by rewrite heap_mapsto_op_eq Qp_div_2. Qed.

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  (** Weakest precondition *)
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  Lemma wp_alloc N E e v P Φ :
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    to_val e = Some v 
    P  heap_ctx N  nclose N  E 
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    P  (  l, l  v - Φ (LocV l)) 
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    P  #> Alloc e @ E {{ Φ }}.
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  Proof.
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    rewrite /heap_ctx /heap_inv=> ??? HP.
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    trans (|={E}=> auth_own heap_name   P)%I.
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    { by rewrite -pvs_frame_r -(auth_empty _ E) left_id. }
    apply wp_strip_pvs, (auth_fsa heap_inv (wp_fsa (Alloc e)))
      with N heap_name ; simpl; eauto with I.
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    rewrite -later_intro. apply sep_mono_r,forall_intro=> h; apply wand_intro_l.
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    rewrite -assoc left_id; apply const_elim_sep_l=> ?.
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    rewrite -(wp_alloc_pst _ (of_heap h)) //.
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    apply sep_mono_r; rewrite HP; apply later_mono.
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    apply forall_mono=> l; apply wand_intro_l.
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    rewrite always_and_sep_l -assoc; apply const_elim_sep_l=> ?.
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    rewrite -(exist_intro (op {[ l := Frac 1 (DecAgree v) ]})).
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    repeat erewrite <-exist_intro by apply _; simpl.
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    rewrite -of_heap_insert left_id right_id.
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    rewrite /heap_mapsto. ecancel [_ - Φ _]%I.
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    rewrite -(map_insert_singleton_op h); last by apply of_heap_None.
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    rewrite const_equiv; last by apply (map_insert_valid h).
    by rewrite left_id -later_intro.
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  Qed.

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  Lemma wp_load N E l q v P Φ :
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    P  heap_ctx N  nclose N  E 
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    P  ( l {q} v   (l {q} v - Φ v)) 
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    P  #> Load (Loc l) @ E {{ Φ }}.
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  Proof.
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    rewrite /heap_ctx /heap_inv=> ?? HPΦ.
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    apply (auth_fsa' heap_inv (wp_fsa _) id)
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      with N heap_name {[ l := Frac q (DecAgree v) ]}; simpl; eauto with I.
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    rewrite HPΦ{HPΦ}; apply sep_mono_r, forall_intro=> h; apply wand_intro_l.
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    rewrite -assoc; apply const_elim_sep_l=> ?.
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    rewrite -(wp_load_pst _ (<[l:=v]>(of_heap h))) ?lookup_insert //.
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    rewrite const_equiv // left_id.
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    rewrite /heap_inv of_heap_singleton_op //.
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    apply sep_mono_r, later_mono, wand_intro_l. by rewrite -later_intro.
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  Qed.

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  Lemma wp_store N E l v' e v P Φ :
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    to_val e = Some v 
    P  heap_ctx N  nclose N  E 
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    P  ( l  v'   (l  v - Φ (LitV LitUnit))) 
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    P  #> Store (Loc l) e @ E {{ Φ }}.
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  Proof.
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    rewrite /heap_ctx /heap_inv=> ??? HPΦ.
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    apply (auth_fsa' heap_inv (wp_fsa _) (alter (λ _, Frac 1 (DecAgree v)) l))
      with N heap_name {[ l := Frac 1 (DecAgree v') ]}; simpl; eauto with I.
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    rewrite HPΦ{HPΦ}; apply sep_mono_r, forall_intro=> h; apply wand_intro_l.
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    rewrite -assoc; apply const_elim_sep_l=> ?.
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    rewrite -(wp_store_pst _ (<[l:=v']>(of_heap h))) ?lookup_insert //.
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    rewrite /heap_inv alter_singleton insert_insert !of_heap_singleton_op; eauto.
    rewrite const_equiv; last naive_solver.
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    apply sep_mono_r, later_mono, wand_intro_l. by rewrite left_id -later_intro.
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  Qed.

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  Lemma wp_cas_fail N E l q v' e1 v1 e2 v2 P Φ :
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    to_val e1 = Some v1  to_val e2 = Some v2  v'  v1 
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    P  heap_ctx N  nclose N  E 
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    P  ( l {q} v'   (l {q} v' - Φ (LitV (LitBool false)))) 
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    P  #> CAS (Loc l) e1 e2 @ E {{ Φ }}.
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  Proof.
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    rewrite /heap_ctx /heap_inv=>????? HPΦ.
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    apply (auth_fsa' heap_inv (wp_fsa _) id)
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      with N heap_name {[ l := Frac q (DecAgree v') ]}; simpl; eauto 10 with I.
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    rewrite HPΦ{HPΦ}; apply sep_mono_r, forall_intro=> h; apply wand_intro_l.
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    rewrite -assoc; apply const_elim_sep_l=> ?.
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    rewrite -(wp_cas_fail_pst _ (<[l:=v']>(of_heap h))) ?lookup_insert //.
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    rewrite const_equiv // left_id.
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    rewrite /heap_inv !of_heap_singleton_op //.
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    apply sep_mono_r, later_mono, wand_intro_l. by rewrite -later_intro.
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  Qed.
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  Lemma wp_cas_suc N E l e1 v1 e2 v2 P Φ :
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    to_val e1 = Some v1  to_val e2 = Some v2 
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    P  heap_ctx N  nclose N  E 
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    P  ( l  v1   (l  v2 - Φ (LitV (LitBool true)))) 
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    P  #> CAS (Loc l) e1 e2 @ E {{ Φ }}.
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  Proof.
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    rewrite /heap_ctx /heap_inv=> ???? HPΦ.
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    apply (auth_fsa' heap_inv (wp_fsa _) (alter (λ _, Frac 1 (DecAgree v2)) l))
      with N heap_name {[ l := Frac 1 (DecAgree v1) ]}; simpl; eauto 10 with I.
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    rewrite HPΦ{HPΦ}; apply sep_mono_r, forall_intro=> h; apply wand_intro_l.
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    rewrite -assoc; apply const_elim_sep_l=> ?.
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    rewrite -(wp_cas_suc_pst _ (<[l:=v1]>(of_heap h))) //;
      last by rewrite lookup_insert.
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    rewrite /heap_inv alter_singleton insert_insert !of_heap_singleton_op; eauto.
    rewrite lookup_insert const_equiv; last naive_solver.
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    apply sep_mono_r, later_mono, wand_intro_l. by rewrite left_id -later_intro.
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  Qed.
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End heap.