From iris.heap_lang Require Export spawn.
From iris.heap_lang Require Import proofmode notation.
Set Default Proof Using "Type".
Import uPred.

Definition parN : namespace := nroot .@ "par".

Definition par : val :=
  λ: "e1" "e2",
    let: "handle" := spawn "e1" in
    let: "v2" := "e2" #() in
    let: "v1" := join "handle" in
    ("v1", "v2").
Notation "e1 ||| e2" := (par (λ: <>, e1)%E (λ: <>, e2)%E) : expr_scope.

Section proof.
Local Set Default Proof Using "Type*".
Context `{!heapG Σ, !spawnG Σ}.

(* Notice that this allows us to strip a later *after* the two Ψ have been
   brought together.  That is strictly stronger than first stripping a later
   and then merging them, as demonstrated by [tests/joining_existentials.v].
   This is why these are not Texan triples. *)
Lemma par_spec (Ψ1 Ψ2 : val → iProp Σ) (f1 f2 : val) (Φ : val → iProp Σ) :
  WP f1 #() {{ Ψ1 }} -∗ WP f2 #() {{ Ψ2 }} -∗
  (▷ ∀ v1 v2, Ψ1 v1 ∗ Ψ2 v2 -∗ ▷ Φ (v1,v2)%V) -∗
  WP par f1 f2 {{ Φ }}.
Proof.
  iIntros "Hf1 Hf2 HΦ". wp_lam. wp_let.
  wp_apply (spawn_spec parN with "Hf1").
  iIntros (l) "Hl". wp_let. wp_bind (f2 _).
  wp_apply (wp_wand with "Hf2"); iIntros (v) "H2". wp_let.
  wp_apply (join_spec with "[$Hl]"). iIntros (w) "H1".
  iSpecialize ("HΦ" with "[$H1 $H2]"). by wp_pures.
Qed.

Lemma wp_par (Ψ1 Ψ2 : val → iProp Σ) (e1 e2 : expr) (Φ : val → iProp Σ) :
  WP e1 {{ Ψ1 }} -∗ WP e2 {{ Ψ2 }} -∗
  (∀ v1 v2, Ψ1 v1 ∗ Ψ2 v2 -∗ ▷ Φ (v1,v2)%V) -∗
  WP par (LamV BAnon e1) (LamV BAnon e2) {{ Φ }}.
Proof.
  iIntros "H1 H2 H".
  wp_apply (par_spec Ψ1 Ψ2 with "[H1] [H2] [H]"); [by wp_lam..|auto].
Qed.
End proof.