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.