heap.v 9.16 KB
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From iris.heap_lang Require Export lifting.
From iris.algebra Require Import upred_big_op frac dec_agree.
From iris.program_logic Require Export invariants ghost_ownership.
From iris.program_logic Require Import ownership auth.
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From iris.proofmode Require Import weakestpre.
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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 heapUR : ucmraT := gmapUR loc (prodR 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 heapUR;
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  heap_name : gname
}.
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(** The Functor we need. *)
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Definition heapGF : gFunctor := authGF heapUR.
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Definition to_heap : state  heapUR := fmap (λ v, (1%Qp, DecAgree v)).
Definition of_heap : heapUR  state := omap (maybe DecAgree  snd).
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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 :=
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    auth_own heap_name {[ l := (q, DecAgree v) ]}.
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  Definition heap_inv (h : heapUR) : iPropG heap_lang Σ :=
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    ownP (of_heap h).
  Definition heap_ctx (N : namespace) : iPropG heap_lang Σ :=
    auth_ctx heap_name N heap_inv.

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  Global Instance heap_inv_proper : Proper (() ==> ()) heap_inv.
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  Proof. solve_proper. Qed.
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  Global Instance heap_ctx_persistent N : PersistentP (heap_ctx N).
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  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 : gFunctors}.
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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 : heapUR.
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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 :
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    of_heap (<[l:=(1%Qp, DecAgree v)]> h) = <[l:=v]> (of_heap h).
  Proof. by rewrite /of_heap -(omap_insert _ _ _ (1%Qp, DecAgree v)). Qed.
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  Lemma of_heap_singleton_op l q v h :
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     ({[l := (q, DecAgree v)]}  h) 
    of_heap ({[l := (q, DecAgree v)]}  h) = <[l:=v]> (of_heap h).
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  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.
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      case _:(h !! l)=>[[q' [v'|]]|] //=; last by move=> [??].
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      move=> [? /dec_agree_op_inv [->]]. by rewrite dec_agree_idemp.
    - rewrite /of_heap lookup_insert_ne // !lookup_omap.
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      by rewrite (lookup_op _ h) lookup_singleton_ne // left_id_L.
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  Qed.
  Lemma to_heap_insert l v σ :
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    to_heap (<[l:=v]> σ) = <[l:=(1%Qp, 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 :  h  of_heap h !! l = None  h !! l = None.
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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 :
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     ({[l := (1%Qp, DecAgree v1)]}  h) 
     ({[l := (1%Qp, 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.
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      by case: (h !! l)=> [x|] // /Some_valid/exclusive_l.
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    - 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 heapUR  nclose N  E 
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    ownP σ ={E}=>  _ : heapG Σ, heap_ctx N  [ map] lv  σ, 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); auto using 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 //.
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    rewrite (insert_singleton_op (to_heap σ));
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      last by rewrite lookup_fmap Hl; auto.
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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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  Global Instance heap_mapsto_timeless l q v : TimelessP (l {q} v).
  Proof. rewrite /heap_mapsto. apply _. Qed.

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  Lemma heap_mapsto_op_eq l q1 q2 v : l {q1} v  l {q2} v  l {q1+q2} v.
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  Proof. by rewrite -auth_own_op op_singleton pair_op dec_agree_idemp. Qed.
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  Lemma heap_mapsto_op l q1 q2 v1 v2 :
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    l {q1} v1  l {q2} v2  v1 = v2  l {q1+q2} v1.
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  Proof.
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    destruct (decide (v1 = v2)) as [->|].
    { by rewrite heap_mapsto_op_eq const_equiv // left_id. }
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    rewrite -auth_own_op op_singleton pair_op dec_agree_ne //.
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    apply (anti_symm ()); last by apply const_elim_l.
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    rewrite auth_own_valid gmap_validI (forall_elim l) lookup_singleton.
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    rewrite option_validI prod_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  (l {q/2} v  l {q/2} v).
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  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 Φ :
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    to_val e = Some v  nclose N  E 
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    heap_ctx N   ( l, l  v - Φ (LitV (LitLoc l)))  WP Alloc e @ E {{ Φ }}.
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  Proof.
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    iIntros {??} "[#Hinv HΦ]". rewrite /heap_ctx.
    iPvs (auth_empty heap_name) as "Hheap".
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    iApply (auth_fsa heap_inv (wp_fsa _)); simpl; eauto.
    iFrame "Hinv Hheap". iIntros {h}. rewrite left_id.
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    iIntros "[% Hheap]". rewrite /heap_inv.
    iApply wp_alloc_pst; first done. iFrame "Hheap". iNext.
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    iIntros {l} "[% Hheap]". iExists {[ l := (1%Qp, DecAgree v) ]}.
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    rewrite -of_heap_insert -(insert_singleton_op h); last by apply of_heap_None.
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    iFrame "Hheap". iSplit; first iPureIntro.
    { by apply alloc_singleton_local_update; first apply of_heap_None. }
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    iIntros "Hheap". iApply "HΦ". by rewrite /heap_mapsto.
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  Qed.

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  Lemma wp_load N E l q v Φ :
    nclose N  E 
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    heap_ctx N   l {q} v   (l {q} v - Φ v)
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     WP Load (Lit (LitLoc l)) @ E {{ Φ }}.
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  Proof.
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    iIntros {?} "[#Hh [Hl HΦ]]". rewrite /heap_ctx /heap_mapsto.
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    iApply (auth_fsa heap_inv (wp_fsa _)); simpl; eauto.
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    iFrame "Hh Hl". iIntros {h} "[% Hl]". rewrite /heap_inv.
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    iApply (wp_load_pst _ (<[l:=v]>(of_heap h)));first by rewrite lookup_insert.
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    rewrite of_heap_singleton_op //. iFrame "Hl".
    iIntros "> Hown". iExists _; iSplit; first done.
    rewrite of_heap_singleton_op //. by iFrame.
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  Qed.

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  Lemma wp_store N E l v' e v Φ :
    to_val e = Some v  nclose N  E 
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    heap_ctx N   l  v'   (l  v - Φ (LitV LitUnit))
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     WP Store (Lit (LitLoc l)) e @ E {{ Φ }}.
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  Proof.
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    iIntros {??} "[#Hh [Hl HΦ]]". rewrite /heap_ctx /heap_mapsto.
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    iApply (auth_fsa heap_inv (wp_fsa _)); simpl; eauto.
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    iFrame "Hh Hl". iIntros {h} "[% Hl]". rewrite /heap_inv.
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    iApply (wp_store_pst _ (<[l:=v']>(of_heap h))); rewrite ?lookup_insert //.
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    rewrite insert_insert !of_heap_singleton_op; eauto. iFrame "Hl".
    iIntros "> Hown". iExists {[l := (1%Qp, DecAgree v)]}; iSplit.
    { iPureIntro; by apply singleton_local_update, exclusive_local_update. }
    rewrite of_heap_singleton_op //; eauto. by iFrame.
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  Qed.

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  Lemma wp_cas_fail N E l q v' e1 v1 e2 v2 Φ :
    to_val e1 = Some v1  to_val e2 = Some v2  v'  v1  nclose N  E 
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    heap_ctx N   l {q} v'   (l {q} v' - Φ (LitV (LitBool false)))
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     WP CAS (Lit (LitLoc l)) e1 e2 @ E {{ Φ }}.
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  Proof.
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    iIntros {????} "[#Hh [Hl HΦ]]". rewrite /heap_ctx /heap_mapsto.
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    iApply (auth_fsa heap_inv (wp_fsa _)); simpl; eauto 10.
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    iFrame "Hh Hl". iIntros {h} "[% Hl]". rewrite /heap_inv.
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    iApply (wp_cas_fail_pst _ (<[l:=v']>(of_heap h))); rewrite ?lookup_insert //.
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    rewrite of_heap_singleton_op //. iFrame "Hl".
    iIntros "> Hown". iExists _; iSplit; first done.
    rewrite of_heap_singleton_op //. by iFrame.
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  Qed.
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  Lemma wp_cas_suc N E l e1 v1 e2 v2 Φ :
    to_val e1 = Some v1  to_val e2 = Some v2  nclose N  E 
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    heap_ctx N   l  v1   (l  v2 - Φ (LitV (LitBool true)))
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     WP CAS (Lit (LitLoc l)) e1 e2 @ E {{ Φ }}.
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  Proof.
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    iIntros {???} "[#Hh [Hl HΦ]]". rewrite /heap_ctx /heap_mapsto.
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    iApply (auth_fsa heap_inv (wp_fsa _)); simpl; eauto 10.
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    iFrame "Hh Hl". iIntros {h} "[% Hl]". rewrite /heap_inv.
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    iApply (wp_cas_suc_pst _ (<[l:=v1]>(of_heap h))); rewrite ?lookup_insert //.
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    rewrite insert_insert !of_heap_singleton_op; eauto. iFrame "Hl".
    iIntros "> Hown". iExists {[l := (1%Qp, DecAgree v2)]}; iSplit.
    { iPureIntro; by apply singleton_local_update, exclusive_local_update. }
    rewrite of_heap_singleton_op //; eauto. by iFrame.
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  Qed.
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End heap.