barrier.v 24.7 KB
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From prelude Require Export functions.
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From algebra Require Export upred_big_op upred_tactics.
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From program_logic Require Export sts saved_prop.
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From program_logic Require Import hoare.
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From heap_lang Require Export derived heap wp_tactics notation.
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Import uPred.
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Definition newchan := (λ: "", ref '0)%L.
Definition signal := (λ: "x", "x" <- '1)%L.
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Definition wait := (rec: "wait" "x" :=if: !"x" = '1 then '() else "wait" "x")%L.
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(** The STS describing the main barrier protocol. Every state has an index-set
    associated with it. These indices are actually [gname], because we use them
    with saved propositions. *)
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Module barrier_proto.
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  Inductive phase := Low | High.
  Record stateT := State { state_phase : phase; state_I : gset gname }.
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  Inductive token := Change (i : gname) | Send.

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  Global Instance stateT_inhabited: Inhabited stateT.
  Proof. split. exact (State Low ). Qed.

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  Definition change_tokens (I : gset gname) : set token :=
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    mkSet (λ t, match t with Change i => i  I | Send => False end).
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  Inductive trans : relation stateT :=
  | ChangeI p I2 I1 : trans (State p I1) (State p I2)
  | ChangePhase I : trans (State Low I) (State High I).
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  Definition tok (s : stateT) : set token :=
      change_tokens (state_I s)
     match state_phase s with Low =>  | High => {[ Send ]} end.
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  Canonical Structure sts := sts.STS trans tok.
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  (* The set of states containing some particular i *)
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  Definition i_states (i : gname) : set stateT :=
    mkSet (λ s, i  state_I s).

  Lemma i_states_closed i :
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    sts.closed (i_states i) {[ Change i ]}.
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  Proof.
    split.
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    - move=>[p I]. rewrite /= /tok !mkSet_elem_of /= =>HI.
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      destruct p; set_solver eauto.
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    - (* If we do the destruct of the states early, and then inversion
         on the proof of a transition, it doesn't work - we do not obtain
         the equalities we need. So we destruct the states late, because this
         means we can use "destruct" instead of "inversion". *)
      move=>s1 s2. rewrite !mkSet_elem_of /==> Hs1 Hstep.
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      inversion_clear Hstep as [T1 T2 Hdisj Hstep'].
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      inversion_clear Hstep' as [? ? ? ? Htrans _ _ Htok].
      destruct Htrans; last done; move:Hs1 Hdisj Htok.
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      rewrite /= /tok /=. 
      (* TODO: Can this be done better? *)
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      intros. apply dec_stable. 
      assert (Change i  change_tokens I1) as HI1
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        by (rewrite mkSet_not_elem_of; set_solver +Hs1).
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      assert (Change i  change_tokens I2) as HI2.
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      { destruct p; set_solver +Htok Hdisj HI1. }
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      done.
  Qed.
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  (* The set of low states *)
  Definition low_states : set stateT :=
    mkSet (λ s, if state_phase s is Low then True else False).
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  Lemma low_states_closed : sts.closed low_states {[ Send ]}.
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  Proof.
    split.
    - move=>[p I]. rewrite /= /tok !mkSet_elem_of /= =>HI.
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      destruct p; set_solver.
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    - move=>s1 s2. rewrite !mkSet_elem_of /==> Hs1 Hstep.
      inversion_clear Hstep as [T1 T2 Hdisj Hstep'].
      inversion_clear Hstep' as [? ? ? ? Htrans _ _ Htok].
      destruct Htrans; move:Hs1 Hdisj Htok =>/=;
                                first by destruct p.
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      rewrite /= /tok /=. intros. set_solver +Hdisj Htok.
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  Qed.

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  (* Proof that we can take the steps we need. *)
  Lemma signal_step I:
    sts.steps (State Low I, {[Send]}) (State High I, ).
  Proof.
    apply rtc_once. constructor; first constructor;
                        rewrite /= /tok /=; set_solver.
  Qed.

  Lemma wait_step i I :
    i  I  sts.steps (State High I, {[ Change i ]}) (State High (I  {[ i ]}), ).
  Proof.
    intros. apply rtc_once.
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    constructor; first constructor; rewrite /= /tok /=; [set_solver eauto..|].
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    (* TODO this proof is rather annoying. *)
    apply elem_of_equiv=>t. rewrite !elem_of_union.
    rewrite !mkSet_elem_of /change_tokens /=.
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    destruct t as [j|]; last set_solver.
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    rewrite elem_of_difference elem_of_singleton.
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    destruct (decide (i = j)); set_solver.
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  Qed.
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  Lemma split_step p i i1 i2 I :
    i  I  i1  I  i2  I  i1  i2 
    sts.steps (State p I, {[ Change i ]})
        (State p ({[i1]}  ({[i2]}  (I  {[i]}))), {[ Change i1; Change i2 ]}).
  Proof.
    intros. apply rtc_once.
    constructor; first constructor; rewrite /= /tok /=; first (destruct p; set_solver).
    (* This gets annoying... and I think I can see a pattern with all these proofs. Automatable? *)
    - apply elem_of_equiv=>t. destruct t; last set_solver.
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      rewrite !mkSet_elem_of. destruct p; set_solver.
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    - apply elem_of_equiv=>t. destruct t as [j|]; last set_solver.
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      rewrite !mkSet_elem_of.
      destruct (decide (i1 = j)); first set_solver. 
      destruct (decide (i2 = j)); first set_solver.
      destruct (decide (i = j)); set_solver.
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  Qed.

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End barrier_proto.
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(* I am too lazy to type the full module name all the time. But then
   why did we even put this into a module? Because some of the names 
   are so general.
   What we'd really like here is to import *some* of the names from
   the module into our namespaces. But Coq doesn't seem to support that...?? *)
Import barrier_proto.
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(* The functors we need. *)
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Definition barrierGF : iFunctors := [stsGF sts; agreeF].
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(** Now we come to the Iris part of the proof. *)
Section proof.
  Context {Σ : iFunctorG} (N : namespace).
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  Context `{heapG Σ} (heapN : namespace).
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  (* These are exactly the elements of barrierGF *)
  Context `{inGF heap_lang Σ (stsGF sts)} `{inGF heap_lang Σ agreeF}.
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  Local Hint Immediate i_states_closed low_states_closed : sts.
  Local Hint Resolve signal_step wait_step split_step : sts.
  Local Hint Resolve sts.closed_op : sts.

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  Hint Extern 50 (_  _) => try rewrite !mkSet_elem_of; set_solver : sts.
  Hint Extern 50 (_  _) => try rewrite !mkSet_elem_of; set_solver : sts.
  Hint Extern 50 (_  _) => try rewrite !mkSet_elem_of; set_solver : sts.
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  Local Notation iProp := (iPropG heap_lang Σ).
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  Definition waiting (P : iProp) (I : gset gname) : iProp :=
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    ( Ψ : gname  iProp, (P - Π★{set I} (λ i, Ψ i)) 
                             Π★{set I} (λ i, saved_prop_own i (Ψ i)))%I.
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  Definition ress (I : gset gname) : iProp :=
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    (Π★{set I} (λ i,  R, saved_prop_own i R  R))%I.
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  Local Notation state_to_val s :=
    (match s with State Low _ => 0 | State High _ => 1 end).
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  Definition barrier_inv (l : loc) (P : iProp) (s : stateT) : iProp :=
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    (l  '(state_to_val s) 
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     match s with State Low I' => waiting P I' | State High I' => ress I' end
    )%I.
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  Definition barrier_ctx (γ : gname) (l : loc) (P : iProp) : iProp :=
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    ( (heapN  N)  heap_ctx heapN  sts_ctx γ N (barrier_inv l P))%I.
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  Global Instance barrier_ctx_ne n γ l : Proper (dist n ==> dist n) (barrier_ctx γ l).
  Proof.
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    move=>? ? EQ. rewrite /barrier_ctx. apply sep_ne; first done.
    apply sep_ne; first done. apply sts_ctx_ne.
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    move=>[p I]. rewrite /barrier_inv. destruct p; last done.
    rewrite /waiting. by setoid_rewrite EQ.
  Qed.

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  Definition send (l : loc) (P : iProp) : iProp :=
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    ( γ, barrier_ctx γ l P  sts_ownS γ low_states {[ Send ]})%I.
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  Global Instance send_ne n l : Proper (dist n ==> dist n) (send l).
  Proof. (* TODO: This really ought to be doable by an automatic tactic. it is just application of already regostered congruence lemmas. *)
    move=>? ? EQ. rewrite /send. apply exist_ne=>γ. by rewrite EQ.
  Qed.

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  Definition recv (l : loc) (R : iProp) : iProp :=
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    ( γ P Q i, barrier_ctx γ l P  sts_ownS γ (i_states i) {[ Change i ]} 
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        saved_prop_own i Q  (Q - R))%I.

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  Global Instance recv_ne n l : Proper (dist n ==> dist n) (recv l).
  Proof.
    move=>? ? EQ. rewrite /send. do 4 apply exist_ne=>?. by rewrite EQ.
  Qed.

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  Lemma waiting_split i i1 i2 Q R1 R2 P I :
    i  I  i1  I  i2  I  i1  i2 
    (saved_prop_own i2 R2  saved_prop_own i1 R1  saved_prop_own i Q 
     (Q - R1  R2)  waiting P I)
     waiting P ({[i1]}  ({[i2]}  (I  {[i]}))).
  Proof.
    intros. rewrite /waiting !sep_exist_l. apply exist_elim=>Ψ.
    rewrite -(exist_intro (<[i1:=R1]> (<[i2:=R2]> Ψ))).
    rewrite [(Π★{set _} (λ _, saved_prop_own _ _))%I](big_sepS_delete _ I i) //.
    rewrite !assoc [(_  (_ - _))%I]comm !assoc [(_   _)%I]comm.
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    rewrite !assoc [(_  _ i _)%I]comm !assoc [(_  _ i _)%I]comm -!assoc.
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    do 4 (rewrite big_sepS_insert; last set_solver).
    rewrite !fn_lookup_insert fn_lookup_insert_ne // !fn_lookup_insert.
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    rewrite 3!assoc. apply sep_mono.
    - rewrite saved_prop_agree. u_strip_later.
      apply wand_intro_l. rewrite [(_  (_ - Π★{set _} _))%I]comm !assoc wand_elim_r.
      rewrite (big_sepS_delete _ I i) //.
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      rewrite [(_  Π★{set _} _)%I]comm [(_  Π★{set _} _)%I]comm -!assoc.
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      apply sep_mono.
      + apply big_sepS_mono; first done. intros j.
        rewrite elem_of_difference not_elem_of_singleton. intros.
        rewrite fn_lookup_insert_ne; last naive_solver.
        rewrite fn_lookup_insert_ne; last naive_solver.
        done.
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      + rewrite !assoc.
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        eapply wand_apply_r'; first done.
        apply: (eq_rewrite (Ψ i) Q (λ x, x)%I); last by eauto with I.
        rewrite eq_sym. eauto with I.
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    - rewrite !assoc. apply sep_mono.
      + by rewrite comm.
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      + apply big_sepS_mono; first done. intros j.
        rewrite elem_of_difference not_elem_of_singleton. intros.
        rewrite fn_lookup_insert_ne; last naive_solver.
        rewrite fn_lookup_insert_ne; last naive_solver.
        done.
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  Qed. 

  Lemma ress_split i i1 i2 Q R1 R2 I :
    i  I  i1  I  i2  I  i1  i2 
    (saved_prop_own i2 R2  saved_prop_own i1 R1  saved_prop_own i Q 
     (Q - R1  R2)  ress I)
     ress ({[i1]}  ({[i2]}  (I  {[i]}))).
  Proof.
    intros. rewrite /ress.
    rewrite [(Π★{set _} _)%I](big_sepS_delete _ I i) // !assoc !sep_exist_l !sep_exist_r.
    apply exist_elim=>R.
    rewrite big_sepS_insert; last set_solver.
    rewrite big_sepS_insert; last set_solver.
    rewrite -(exist_intro R1) -(exist_intro R2) [(_ i2 _  _)%I]comm -!assoc.
    apply sep_mono_r. rewrite !assoc. apply sep_mono_l.
    rewrite [( _  _ i2 _)%I]comm -!assoc. apply sep_mono_r.
    rewrite !assoc [(_  _ i R)%I]comm !assoc saved_prop_agree.
    rewrite [( _  _)%I]comm -!assoc. eapply wand_apply_l.
    { rewrite <-later_wand, <-later_intro. done. }
    { by rewrite later_sep. }
    u_strip_later.
    apply: (eq_rewrite R Q (λ x, x)%I); eauto with I.
  Qed.
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  Lemma newchan_spec (P : iProp) (Φ : val  iProp) :
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    heapN  N 
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    (heap_ctx heapN   l, recv l P  send l P - Φ (LocV l))
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     || newchan '() {{ Φ }}.
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  Proof.
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    intros HN. rewrite /newchan. wp_seq.
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    rewrite -wp_pvs. wp eapply wp_alloc; eauto with I ndisj.
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    apply forall_intro=>l. rewrite (forall_elim l). apply wand_intro_l.
    rewrite !assoc. apply pvs_wand_r.
    (* The core of this proof: Allocating the STS and the saved prop. *)
    eapply sep_elim_True_r.
    { by eapply (saved_prop_alloc _ P). }
    rewrite pvs_frame_l. apply pvs_strip_pvs. rewrite sep_exist_l.
    apply exist_elim=>i.
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    trans (pvs   (heap_ctx heapN   (barrier_inv l P (State Low {[ i ]}))   saved_prop_own i P)).
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    - rewrite -pvs_intro. rewrite [(_  heap_ctx _)%I]comm -!assoc. apply sep_mono_r.
      rewrite {1}[saved_prop_own _ _]always_sep_dup !assoc. apply sep_mono_l.
      rewrite /barrier_inv /waiting -later_intro. apply sep_mono_r.
      rewrite -(exist_intro (const P)) /=. rewrite -[saved_prop_own _ _](left_id True%I ()%I).
      apply sep_mono.
      + rewrite -later_intro. apply wand_intro_l. rewrite right_id.
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        by rewrite big_sepS_singleton.
      + by rewrite big_sepS_singleton.
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    - rewrite (sts_alloc (barrier_inv l P)  N); last by eauto.
      rewrite !pvs_frame_r !pvs_frame_l. 
      rewrite pvs_trans'. apply pvs_strip_pvs. rewrite sep_exist_r sep_exist_l.
      apply exist_elim=>γ.
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      (* TODO: The record notation is rather annoying here *)
      rewrite /recv /send. rewrite -(exist_intro γ) -(exist_intro P).
      rewrite -(exist_intro P) -(exist_intro i) -(exist_intro γ).
      (* This is even more annoying than usually, since rewrite sometimes unfolds stuff... *)
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      rewrite [barrier_ctx _ _ _]lock !assoc [(_ locked _)%I]comm !assoc -lock.
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      rewrite -always_sep_dup.
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      rewrite [barrier_ctx _ _ _]lock always_and_sep_l -!assoc assoc -lock.
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      rewrite -pvs_frame_l. rewrite /barrier_ctx const_equiv // left_id. apply sep_mono_r.
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      rewrite [(saved_prop_own _ _  _)%I]comm !assoc. rewrite -pvs_frame_r.
      apply sep_mono_l.
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      rewrite -assoc [( _  _)%I]comm assoc -pvs_frame_r.
      eapply sep_elim_True_r; last eapply sep_mono_l.
      { rewrite -later_intro. apply wand_intro_l. by rewrite right_id. }
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      rewrite (sts_own_weaken  _ _ (i_states i  low_states) _ 
                              ({[ Change i ]}  {[ Send ]})).
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      + apply pvs_mono. rewrite sts_ownS_op; eauto with sts.
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      + rewrite /= /tok /=  =>t. rewrite !mkSet_elem_of.
        move=>[[?]|?]; set_solver. 
      + eauto with sts.
      + eauto with sts.
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  Qed.
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  Lemma signal_spec l P (Φ : val  iProp) :
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    (send l P  P  Φ '())  || signal (LocV l) {{ Φ }}.
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  Proof.
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    rewrite /signal /send /barrier_ctx. rewrite sep_exist_r.
    apply exist_elim=>γ. rewrite -!assoc. apply const_elim_sep_l=>?. wp_let.
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    (* I think some evars here are better than repeating *everything* *)
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    eapply (sts_fsaS _ (wp_fsa _)) with (N0:=N) (γ0:=γ); simpl;
      eauto with I ndisj.
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    rewrite !assoc [(_  sts_ownS _ _ _)%I]comm -!assoc. apply sep_mono_r.
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    apply forall_intro=>-[p I]. apply wand_intro_l. rewrite -!assoc.
    apply const_elim_sep_l=>Hs. destruct p; last done.
    rewrite {1}/barrier_inv =>/={Hs}. rewrite later_sep.
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    eapply wp_store; eauto with I ndisj. 
    rewrite -!assoc. apply sep_mono_r. u_strip_later.
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    apply wand_intro_l. rewrite -(exist_intro (State High I)).
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    rewrite -(exist_intro ). rewrite const_equiv /=; last by eauto with sts.
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    rewrite left_id -later_intro {2}/barrier_inv -!assoc. apply sep_mono_r.
    rewrite !assoc [(_  P)%I]comm !assoc -2!assoc.
    apply sep_mono; last first.
    { apply wand_intro_l. eauto with I. }
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    (* Now we come to the core of the proof: Updating from waiting to ress. *)
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    rewrite /waiting /ress sep_exist_l. apply exist_elim=>{Φ} Φ.
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    rewrite later_wand {1}(later_intro P) !assoc wand_elim_r.
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    rewrite big_sepS_later -big_sepS_sepS. apply big_sepS_mono'=>i.
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    rewrite -(exist_intro (Φ i)) comm. done.
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  Qed.
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  Lemma wait_spec l P (Φ : val  iProp) :
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    (recv l P  (P - Φ '()))  || wait (LocV l) {{ Φ }}.
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  Proof.
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    rename P into R. wp_rec.
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    rewrite {1}/recv /barrier_ctx. rewrite !sep_exist_r.
    apply exist_elim=>γ. rewrite !sep_exist_r. apply exist_elim=>P.
    rewrite !sep_exist_r. apply exist_elim=>Q. rewrite !sep_exist_r.
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    apply exist_elim=>i. rewrite -!assoc. apply const_elim_sep_l=>?.
    wp_focus (! _)%L.
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    (* I think some evars here are better than repeating *everything* *)
    eapply (sts_fsaS _ (wp_fsa _)) with (N0:=N) (γ0:=γ); simpl;
      eauto with I ndisj.
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    rewrite !assoc [(_  sts_ownS _ _ _)%I]comm -!assoc. apply sep_mono_r.
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    apply forall_intro=>-[p I]. apply wand_intro_l. rewrite -!assoc.
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    apply const_elim_sep_l=>Hs.
    rewrite {1}/barrier_inv =>/=. rewrite later_sep.
    eapply wp_load; eauto with I ndisj.
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    rewrite -!assoc. apply sep_mono_r. u_strip_later.
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    apply wand_intro_l. destruct p.
    { (* a Low state. The comparison fails, and we recurse. *)
      rewrite -(exist_intro (State Low I)) -(exist_intro {[ Change i ]}).
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      rewrite [( sts.steps _ _ )%I]const_equiv /=; last by apply rtc_refl.
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      rewrite left_id -[( barrier_inv _ _ _)%I]later_intro {3}/barrier_inv.
      rewrite -!assoc. apply sep_mono_r, sep_mono_r, wand_intro_l.
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      wp_op; first done. intros _. wp_if. rewrite !assoc.
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      rewrite -always_wand_impl always_elim.
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      rewrite -{2}pvs_wp. apply pvs_wand_r.
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      rewrite -(exist_intro γ) -(exist_intro P) -(exist_intro Q) -(exist_intro i).
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      rewrite !assoc.
      do 3 (rewrite -pvs_frame_r; apply sep_mono; last (try apply later_intro; reflexivity)).
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      rewrite [(_  heap_ctx _)%I]comm -!assoc.
      rewrite const_equiv // left_id -pvs_frame_l. apply sep_mono_r.
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      rewrite comm -pvs_frame_l. apply sep_mono_r.
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      apply sts_ownS_weaken; eauto using sts.up_subseteq with sts. }
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    (* a High state: the comparison succeeds, and we perform a transition and
       return to the client *)
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    rewrite [(_   (_  _ ))%I]sep_elim_l.
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    rewrite -(exist_intro (State High (I  {[ i ]}))) -(exist_intro ).
    change (i  I) in Hs.
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    rewrite const_equiv /=; last by eauto with sts.
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    rewrite left_id -[( barrier_inv _ _ _)%I]later_intro {2}/barrier_inv.
    rewrite -!assoc. apply sep_mono_r. rewrite /ress.
    rewrite (big_sepS_delete _ I i) // [(_  Π★{set _} _)%I]comm -!assoc.
    apply sep_mono_r. rewrite !sep_exist_r. apply exist_elim=>Q'.
    apply wand_intro_l. rewrite [(heap_ctx _  _)%I]sep_elim_r.
    rewrite [(sts_own _ _ _  _)%I]sep_elim_r [(sts_ctx _ _ _  _)%I]sep_elim_r.
    rewrite !assoc [(_  saved_prop_own i Q)%I]comm !assoc saved_prop_agree.
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    wp_op>; last done. intros _. u_strip_later.
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    wp_if. 
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    eapply wand_apply_r; [done..|]. eapply wand_apply_r; [done..|].
    apply: (eq_rewrite Q' Q (λ x, x)%I); last by eauto with I.
    rewrite eq_sym. eauto with I.
  Qed.
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  Lemma recv_split l P1 P2 Φ :
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    (recv l (P1  P2)  (recv l P1  recv l P2 - Φ '()))  || Skip {{ Φ }}.
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  Proof.
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    rename P1 into R1. rename P2 into R2.
    rewrite {1}/recv /barrier_ctx. rewrite sep_exist_r.
    apply exist_elim=>γ. rewrite sep_exist_r.  apply exist_elim=>P. 
    rewrite sep_exist_r.  apply exist_elim=>Q. rewrite sep_exist_r.
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    apply exist_elim=>i. rewrite -!assoc. apply const_elim_sep_l=>?. rewrite -wp_pvs.
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    (* I think some evars here are better than repeating *everything* *)
    eapply (sts_fsaS _ (wp_fsa _)) with (N0:=N) (γ0:=γ); simpl;
      eauto with I ndisj.
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    rewrite !assoc [(_  sts_ownS _ _ _)%I]comm -!assoc. apply sep_mono_r.
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    apply forall_intro=>-[p I]. apply wand_intro_l. rewrite -!assoc.
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    apply const_elim_sep_l=>Hs. rewrite -wp_pvs. wp_seq.
    eapply sep_elim_True_l.
    { eapply saved_prop_alloc_strong with (P0 := R1) (G := I). }
    rewrite pvs_frame_r. apply pvs_strip_pvs. rewrite sep_exist_r.
    apply exist_elim=>i1. rewrite always_and_sep_l. rewrite -assoc.
    apply const_elim_sep_l=>Hi1. eapply sep_elim_True_l.
    { eapply saved_prop_alloc_strong with (P0 := R2) (G := I  {[ i1 ]}). }
    rewrite pvs_frame_r. apply pvs_mono. rewrite sep_exist_r.
    apply exist_elim=>i2. rewrite always_and_sep_l. rewrite -assoc.
    apply const_elim_sep_l=>Hi2.
    rewrite ->not_elem_of_union, elem_of_singleton in Hi2.
    destruct Hi2 as [Hi2 Hi12]. change (i  I) in Hs. destruct p.
    (* Case I: Low state. *)
    - rewrite -(exist_intro (State Low ({[i1]}  ({[i2]}  (I  {[i]}))))).
      rewrite -(exist_intro ({[Change i1 ]}  {[ Change i2 ]})).
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      rewrite [( sts.steps _ _)%I]const_equiv; last by eauto with sts.
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      rewrite left_id -later_intro {1 3}/barrier_inv.
      (* FIXME ssreflect rewrite fails if there are evars around. Also, this is very slow because we don't have a proof mode. *)
      rewrite -(waiting_split _ _ _ Q R1 R2); [|done..].
      rewrite {1}[saved_prop_own i1 _]always_sep_dup.
      rewrite {1}[saved_prop_own i2 _]always_sep_dup.
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      cancel (saved_prop_own i1 R1).
      cancel (saved_prop_own i2 R2).
      cancel (l  '0)%I.
      cancel (waiting P I).
      cancel (Q - R1  R2)%I.
      cancel (saved_prop_own i Q).
      apply wand_intro_l. rewrite !assoc. eapply pvs_wand_r. rewrite /recv.
      rewrite -(exist_intro γ) -(exist_intro P) -(exist_intro R1) -(exist_intro i1).
      rewrite -(exist_intro γ) -(exist_intro P) -(exist_intro R2) -(exist_intro i2).
      do 2 rewrite !(assoc ()%I) [(_  sts_ownS _ _ _)%I]comm.
      rewrite -!assoc. rewrite [(sts_ownS _ _ _  _  _)%I]assoc -pvs_frame_r.
      apply sep_mono.
      * rewrite -sts_ownS_op; by eauto using sts_own_weaken with sts.
      * rewrite const_equiv // !left_id.
        rewrite {1}[heap_ctx _]always_sep_dup !assoc [(_  heap_ctx _)%I]comm -!assoc. apply sep_mono_r.
        rewrite !assoc ![(_  heap_ctx _)%I]comm -!assoc. apply sep_mono_r.
        rewrite {1}[sts_ctx _ _ _]always_sep_dup !assoc [(_  sts_ctx _ _ _)%I]comm -!assoc. apply sep_mono_r.
        rewrite !assoc ![(_  sts_ctx _ _ _)%I]comm -!assoc. apply sep_mono_r.
        rewrite comm. apply sep_mono_r. apply sep_intro_True_l.
        { rewrite -later_intro. apply wand_intro_l. by rewrite right_id. }
        apply sep_intro_True_r; first done.
        { rewrite -later_intro. apply wand_intro_l. by rewrite right_id. }
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(* Case II: High state. TODO: Lots of this script is just copy-n-paste of the previous one.
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   Most of that is because the goals are fairly similar in structure, and the proof scripts
   are mostly concerned with manually managaing the structure (assoc, comm, dup) of
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   the context. *)
    - rewrite -(exist_intro (State High ({[i1]}  ({[i2]}  (I  {[i]}))))).
      rewrite -(exist_intro ({[Change i1 ]}  {[ Change i2 ]})).
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      rewrite const_equiv; last by eauto with sts.
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      rewrite left_id -later_intro {1 3}/barrier_inv.
      rewrite -(ress_split _ _ _ Q R1 R2); [|done..].
      rewrite {1}[saved_prop_own i1 _]always_sep_dup.
      rewrite {1}[saved_prop_own i2 _]always_sep_dup.
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      cancel (saved_prop_own i1 R1).
      cancel (saved_prop_own i2 R2).
      cancel (l  '1)%I.
      cancel (Q - R1  R2)%I.
      cancel (saved_prop_own i Q).
      cancel (ress I).
      apply wand_intro_l. rewrite !assoc. eapply pvs_wand_r. rewrite /recv.
      rewrite -(exist_intro γ) -(exist_intro P) -(exist_intro R1) -(exist_intro i1).
      rewrite -(exist_intro γ) -(exist_intro P) -(exist_intro R2) -(exist_intro i2).
      do 2 rewrite !(assoc ()%I) [(_  sts_ownS _ _ _)%I]comm.
      rewrite -!assoc. rewrite [(sts_ownS _ _ _  _  _)%I]assoc -pvs_frame_r.
      apply sep_mono.
      * rewrite -sts_ownS_op; by eauto using sts_own_weaken with sts.
      * rewrite const_equiv // !left_id.
        rewrite {1}[heap_ctx _]always_sep_dup !assoc [(_  heap_ctx _)%I]comm -!assoc. apply sep_mono_r.
        rewrite !assoc ![(_  heap_ctx _)%I]comm -!assoc. apply sep_mono_r.
        rewrite {1}[sts_ctx _ _ _]always_sep_dup !assoc [(_  sts_ctx _ _ _)%I]comm -!assoc. apply sep_mono_r.
        rewrite !assoc ![(_  sts_ctx _ _ _)%I]comm -!assoc. apply sep_mono_r.
        rewrite comm. apply sep_mono_r. apply sep_intro_True_l.
        { rewrite -later_intro. apply wand_intro_l. by rewrite right_id. }
        apply sep_intro_True_r; first done.
        { rewrite -later_intro. apply wand_intro_l. by rewrite right_id. }
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  Qed.
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  Lemma recv_strengthen l P1 P2 :
    (P1 - P2)  (recv l P1 - recv l P2).
  Proof.
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    apply wand_intro_l. rewrite /recv. rewrite sep_exist_r. apply exist_mono=>γ.
    rewrite sep_exist_r. apply exist_mono=>P. rewrite sep_exist_r.
    apply exist_mono=>Q. rewrite sep_exist_r. apply exist_mono=>i.
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    rewrite -!assoc. apply sep_mono_r, sep_mono_r, sep_mono_r, sep_mono_r, sep_mono_r.
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    rewrite (later_intro (P1 - _)%I) -later_sep. apply later_mono.
    apply wand_intro_l. rewrite !assoc wand_elim_r wand_elim_r. done.
  Qed.
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End proof.
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Section spec.
  Context {Σ : iFunctorG}.
  Context `{heapG Σ}.
  Context `{stsG heap_lang Σ barrier_proto.sts}.
  Context `{savedPropG heap_lang Σ}.

  Local Notation iProp := (iPropG heap_lang Σ).

  (* TODO: Maybe notation for LocV (and Loc)? *)
  Lemma barrier_spec (heapN N : namespace) :
    heapN  N 
     (recv send : loc -> iProp -n> iProp),
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      ( P, heap_ctx heapN  ({{ True }} newchan '() {{ λ v,  l, v = LocV l  recv l P  send l P }})) 
      ( l P, {{ send l P  P }} signal (LocV l) {{ λ _, True }}) 
      ( l P, {{ recv l P }} wait (LocV l) {{ λ _, P }}) 
      ( l P Q, {{ recv l (P  Q) }} Skip {{ λ _, recv l P  recv l Q }}) 
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      ( l P Q, (P - Q)  (recv l P - recv l Q)).
  Proof.
    intros HN. exists (λ l, CofeMor (recv N heapN l)). exists (λ l, CofeMor (send N heapN l)).
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    split_and?; cbn.
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    - intros. apply: always_intro. apply impl_intro_l. rewrite -newchan_spec //.
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      rewrite comm always_and_sep_r. apply sep_mono_r. apply forall_intro=>l.
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      apply wand_intro_l. rewrite right_id -(exist_intro l) const_equiv // left_id;
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      done.
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    - intros. apply ht_alt. rewrite -signal_spec. by rewrite right_id.
    - intros. apply ht_alt. rewrite -wait_spec.
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      apply sep_intro_True_r; first done. apply wand_intro_l. eauto with I.
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    - intros. apply ht_alt. rewrite -recv_split.
      apply sep_intro_True_r; first done. apply wand_intro_l. eauto with I.
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    - intros. apply recv_strengthen.
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  Qed.
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End spec.