par.v 1.95 KB
Newer Older
1 2
From heap_lang Require Export heap spawn.
From heap_lang Require Import wp_tactics notation.
Ralf Jung's avatar
Ralf Jung committed
3
Import uPred.
4 5

Definition par : val :=
Robbert Krebbers's avatar
Robbert Krebbers committed
6 7 8 9 10
  λ: "fs",
    let: "handle" := ^spawn (Fst '"fs") in
    let: "v2" := Snd '"fs" #() in
    let: "v1" := ^join '"handle" in
    Pair '"v1" '"v2".
11 12
Notation Par e1 e2 := (^par (Pair (λ: <>, e1) (λ: <>, e2)))%E.
Notation ParV e1 e2 := (par (Pair (λ: <>, e1) (λ: <>, e2)))%E.
13 14
(* We want both par and par^ to print like this. *)
Infix "||" := ParV : expr_scope.
Ralf Jung's avatar
Ralf Jung committed
15
Infix "||" := Par : expr_scope.
16 17 18 19 20 21

Section proof.
Context {Σ : rFunctorG} `{!heapG Σ, !spawnG Σ}.
Context (heapN N : namespace).
Local Notation iProp := (iPropG heap_lang Σ).

22
Lemma par_spec (Ψ1 Ψ2 : val  iProp) e (f1 f2 : val) (Φ : val  iProp) :
Robbert Krebbers's avatar
Robbert Krebbers committed
23
  heapN  N  to_val e = Some (f1,f2)%V 
24
  (heap_ctx heapN  #> f1 #() {{ Ψ1 }}  #> f2 #() {{ Ψ2 }} 
25
    v1 v2, Ψ1 v1  Ψ2 v2 -  Φ (v1,v2)%V)
26
   #> par e {{ Φ }}.
27
Proof.
Ralf Jung's avatar
Ralf Jung committed
28
  intros. rewrite /par.
29 30 31
  wp_focus e. etransitivity; last by eapply wp_value. wp_let.
  (* FIXME: wp_proj should not spawn these goals. *)
  wp_proj; eauto using to_of_val.
Robbert Krebbers's avatar
Robbert Krebbers committed
32
  ewp eapply spawn_spec; eauto using to_of_val.
33
  apply sep_mono_r. apply sep_mono_r.
Ralf Jung's avatar
Ralf Jung committed
34
  apply forall_intro=>h. apply wand_intro_l. wp_let.
35 36
  wp_proj; eauto using to_of_val.
  wp_focus (f2 _). rewrite wp_frame_r wp_frame_l. apply wp_mono=>v2. wp_let.
Robbert Krebbers's avatar
Robbert Krebbers committed
37
  ewp eapply join_spec; eauto using to_of_val. apply sep_mono_r.
38 39 40
  apply forall_intro=>v1. apply wand_intro_l. 
  rewrite (forall_elim v1) (forall_elim v2). rewrite assoc wand_elim_r.
  wp_let. eapply wp_value, to_val_Pair; eapply to_of_val.
Ralf Jung's avatar
Ralf Jung committed
41
Qed.
42

Ralf Jung's avatar
Ralf Jung committed
43 44 45
Lemma wp_par (Ψ1 Ψ2 : val  iProp) (e1 e2 : expr []) (Φ : val  iProp) :
  heapN  N 
  (heap_ctx heapN  #> e1 {{ Ψ1 }}  #> e2 {{ Ψ2 }} 
46
    v1 v2, Ψ1 v1  Ψ2 v2 -  Φ (v1,v2)%V)
47
   #> ParV e1 e2 {{ Φ }}.
Ralf Jung's avatar
Ralf Jung committed
48
Proof.
49
  intros. rewrite -par_spec //. apply sep_mono_r.
Ralf Jung's avatar
Ralf Jung committed
50 51 52 53
  apply sep_mono; last apply sep_mono_l; by wp_seq.
Qed.

End proof.