From iris.program_logic Require Export weakestpre. From iris.base_logic.lib Require Export invariants. From iris.heap_lang Require Export lang. From iris.proofmode Require Import tactics. From iris.heap_lang Require Import proofmode notation. From iris.algebra Require Import excl. Set Default Proof Using "Type". Definition spawn : val := λ: "f", let: "c" := ref NONE in Fork ("c" <- SOME ("f" #())) ;; "c". Definition join : val := rec: "join" "c" := match: !"c" with SOME "x" => "x" | NONE => "join" "c" end. (** The CMRA & functor we need. *) (* Not bundling heapG, as it may be shared with other users. *) Class spawnG Σ := SpawnG { spawn_tokG :> inG Σ (exclR unitC) }. Definition spawnΣ : gFunctors := #[GFunctor (exclR unitC)]. Instance subG_spawnΣ {Σ} : subG spawnΣ Σ → spawnG Σ. Proof. solve_inG. Qed. (** Now we come to the Iris part of the proof. *) Section proof. Context `{!heapG Σ, !spawnG Σ} (N : namespace). Definition spawn_inv (γ : gname) (l : loc) (Ψ : val → iProp Σ) : iProp Σ := (∃ lv, l ↦ lv ∗ (⌜lv = NONEV⌝ ∨ ∃ v, ⌜lv = SOMEV v⌝ ∗ (Ψ v ∨ own γ (Excl ()))))%I. Definition join_handle (l : loc) (Ψ : val → iProp Σ) : iProp Σ := (∃ γ, own γ (Excl ()) ∗ inv N (spawn_inv γ l Ψ))%I. Global Instance spawn_inv_ne n γ l : Proper (pointwise_relation val (dist n) ==> dist n) (spawn_inv γ l). Proof. solve_proper. Qed. Global Instance join_handle_ne n l : Proper (pointwise_relation val (dist n) ==> dist n) (join_handle l). Proof. solve_proper. Qed. (** The main proofs. *) Lemma spawn_spec (Ψ : val → iProp Σ) e (f : val) `{Hef : !IntoVal e f} : {{{ WP f #() {{ Ψ }} }}} spawn e {{{ l, RET #l; join_handle l Ψ }}}. Proof. apply of_to_val in Hef as <-. iIntros (Φ) "Hf HΦ". rewrite /spawn /=. wp_let. wp_alloc l as "Hl". wp_let. iMod (own_alloc (Excl ())) as (γ) "Hγ"; first done. iMod (inv_alloc N _ (spawn_inv γ l Ψ) with "[Hl]") as "#?". { iNext. iExists NONEV. iFrame; eauto. } wp_apply wp_fork; simpl. iSplitR "Hf". - wp_seq. iApply "HΦ". rewrite /join_handle. eauto. - wp_bind (f _). iApply (wp_wand with "Hf"); iIntros (v) "Hv". iInv N as (v') "[Hl _]" "Hclose". wp_store. iApply "Hclose". iNext. iExists (SOMEV v). iFrame. eauto. Qed. Lemma join_spec (Ψ : val → iProp Σ) l : {{{ join_handle l Ψ }}} join #l {{{ v, RET v; Ψ v }}}. Proof. iIntros (Φ) "H HΦ". iDestruct "H" as (γ) "[Hγ #?]". iLöb as "IH". wp_rec. wp_bind (! _)%E. iInv N as (v) "[Hl Hinv]" "Hclose". wp_load. iDestruct "Hinv" as "[%|Hinv]"; subst. - iMod ("Hclose" with "[Hl]"); [iNext; iExists _; iFrame; eauto|]. iModIntro. wp_match. iApply ("IH" with "Hγ [HΦ]"). auto. - iDestruct "Hinv" as (v') "[% [HΨ|Hγ']]"; simplify_eq/=. + iMod ("Hclose" with "[Hl Hγ]"); [iNext; iExists _; iFrame; eauto|]. iModIntro. wp_match. by iApply "HΦ". + iDestruct (own_valid_2 with "Hγ Hγ'") as %[]. Qed. End proof. Typeclasses Opaque join_handle.