par.v 1.59 KB
Newer Older
1
From iris.heap_lang Require Export spawn.
Robbert Krebbers's avatar
Robbert Krebbers committed
2
From iris.heap_lang Require Import proofmode notation.
Ralf Jung's avatar
Ralf Jung committed
3
Import uPred.
4

5
Definition par : val :=
Robbert Krebbers's avatar
Robbert Krebbers committed
6
  λ: "fs",
7 8 9 10
    let: "handle" := spawn (Fst "fs") in
    let: "v2" := Snd "fs" #() in
    let: "v1" := join "handle" in
    Pair "v1" "v2".
11
Notation Par e1 e2 := (par (Pair (λ: <>, e1) (λ: <>, e2)))%E.
Ralf Jung's avatar
Ralf Jung committed
12
Infix "||" := Par : expr_scope.
13
Global Opaque par.
14

15
Section proof.
16
Context {Σ : gFunctors} `{!heapG Σ, !spawnG Σ}.
17 18 19
Context (heapN N : namespace).
Local Notation iProp := (iPropG heap_lang Σ).

20
Lemma par_spec (Ψ1 Ψ2 : val  iProp) e (f1 f2 : val) (Φ : val  iProp) :
Robbert Krebbers's avatar
Robbert Krebbers committed
21
  heapN  N  to_val e = Some (f1,f2)%V 
22
  (heap_ctx heapN  WP f1 #() {{ Ψ1 }}  WP f2 #() {{ Ψ2 }} 
23
    v1 v2, Ψ1 v1  Ψ2 v2 -  Φ (v1,v2)%V)
24
   WP par e {{ Φ }}.
25
Proof.
26
  iIntros (??) "(#Hh&Hf1&Hf2&HΦ)".
27
  rewrite /par. wp_value. iPvsIntro. wp_let. wp_proj.
Robbert Krebbers's avatar
Robbert Krebbers committed
28
  wp_apply spawn_spec; try wp_done. iFrame "Hf1 Hh".
29 30 31
  iIntros (l) "Hl". wp_let. wp_proj. wp_focus (f2 _).
  iApply wp_wand_l; iFrame "Hf2"; iIntros (v) "H2". wp_let.
  wp_apply join_spec; iFrame "Hl". iIntros (w) "H1".
32
  iSpecialize ("HΦ" with "* [-]"); first by iSplitL "H1". by wp_let.
Ralf Jung's avatar
Ralf Jung committed
33
Qed.
34

35 36
Lemma wp_par (Ψ1 Ψ2 : val  iProp) (e1 e2 : expr) `{!Closed [] e1, Closed [] e2}
    (Φ : val  iProp) :
Ralf Jung's avatar
Ralf Jung committed
37
  heapN  N 
38
  (heap_ctx heapN  WP e1 {{ Ψ1 }}  WP e2 {{ Ψ2 }} 
39
    v1 v2, Ψ1 v1  Ψ2 v2 -  Φ (v1,v2)%V)
40
   WP e1 || e2 {{ Φ }}.
Ralf Jung's avatar
Ralf Jung committed
41
Proof.
42
  iIntros (?) "(#Hh&H1&H2&H)". iApply (par_spec Ψ1 Ψ2); auto. apply is_value.
Robbert Krebbers's avatar
Robbert Krebbers committed
43
  iFrame "Hh H". iSplitL "H1"; by wp_let.
Ralf Jung's avatar
Ralf Jung committed
44 45
Qed.
End proof.