From iris.heap_lang Require Export spawn.
From iris.heap_lang Require Import proofmode notation.
Import uPred.

Definition par {X} : expr X :=
  λ: "fs",
    let: "handle" := ^spawn (Fst '"fs") in
    let: "v2" := Snd '"fs" #() in
    let: "v1" := ^join '"handle" in
    Pair '"v1" '"v2".
Notation Par e1 e2 := (par (Pair (λ: <>, e1) (λ: <>, e2)))%E.
Infix "||" := Par : expr_scope.

Instance do_wexpr_par {X Y} (H : X `included` Y) : WExpr H par par := _.
Instance do_wsubst_par {X Y} x es (H : X `included` x :: Y) :
  WSubst x es H par par := do_wsubst_closed _ x es H _.
Typeclasses Opaque par.

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

Lemma par_spec (Ψ1 Ψ2 : val → iProp) e (f1 f2 : val) (Φ : val → iProp) :
  heapN ⊥ N → to_val e = Some (f1,f2)%V →
  (heap_ctx heapN ★ WP f1 #() {{ Ψ1 }} ★ WP f2 #() {{ Ψ2 }} ★
   ∀ v1 v2, Ψ1 v1 ★ Ψ2 v2 -★ ▷ Φ (v1,v2)%V)
  ⊢ WP par e {{ Φ }}.
Proof.
  iIntros {??} "(#Hh&Hf1&Hf2&HΦ)". rewrite /par. wp_value. wp_let. wp_proj.
  wp_apply spawn_spec; try wp_done. iFrame "Hf1 Hh".
  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".
  iSpecialize "HΦ" "-"; first by iSplitL "H1". by wp_let.
Qed.

Lemma wp_par (Ψ1 Ψ2 : val → iProp) (e1 e2 : expr []) (Φ : val → iProp) :
  heapN ⊥ N →
  (heap_ctx heapN ★ WP e1 {{ Ψ1 }} ★ WP e2 {{ Ψ2 }} ★
   ∀ v1 v2, Ψ1 v1 ★ Ψ2 v2 -★ ▷ Φ (v1,v2)%V)
  ⊢ WP e1 || e2 {{ Φ }}.
Proof.
  iIntros {?} "(#Hh&H1&H2&H)". iApply (par_spec Ψ1 Ψ2); auto.
  iFrame "Hh H". iSplitL "H1"; by wp_let.
Qed.
End proof.