From iris.algebra Require Import auth gmap frac agree. From iris.base_logic.lib Require Export own. From iris.bi.lib Require Import fractional. From iris.proofmode Require Import tactics. Set Default Proof Using "Type". Import uPred. Definition gen_heapUR (L V : Type) `{Countable L} : ucmraT := gmapUR L (prodR fracR (agreeR (leibnizC V))). Definition to_gen_heap {L V} `{Countable L} : gmap L V → gen_heapUR L V := fmap (λ v, (1%Qp, to_agree (v : leibnizC V))). (** The CMRA we need. *) Class gen_heapG (L V : Type) (Σ : gFunctors) `{Countable L} := GenHeapG { gen_heap_inG :> inG Σ (authR (gen_heapUR L V)); gen_heap_name : gname }. Arguments gen_heap_name {_ _ _ _ _} _ : assert. Class gen_heapPreG (L V : Type) (Σ : gFunctors) `{Countable L} := { gen_heap_preG_inG :> inG Σ (authR (gen_heapUR L V)) }. Definition gen_heapΣ (L V : Type) `{Countable L} : gFunctors := #[GFunctor (authR (gen_heapUR L V))]. Instance subG_gen_heapPreG {Σ L V} `{Countable L} : subG (gen_heapΣ L V) Σ → gen_heapPreG L V Σ. Proof. solve_inG. Qed. Section definitions. Context `{hG : gen_heapG L V Σ}. Definition gen_heap_ctx (σ : gmap L V) : iProp Σ := own (gen_heap_name hG) (● (to_gen_heap σ)). Definition mapsto_def (l : L) (q : Qp) (v: V) : iProp Σ := own (gen_heap_name hG) (◯ {[ l := (q, to_agree (v : leibnizC V)) ]}). Definition mapsto_aux : seal (@mapsto_def). by eexists. Qed. Definition mapsto := mapsto_aux.(unseal). Definition mapsto_eq : @mapsto = @mapsto_def := mapsto_aux.(seal_eq). End definitions. Local Notation "l ↦{ q } v" := (mapsto l q v) (at level 20, q at level 50, format "l ↦{ q } v") : bi_scope. Local Notation "l ↦ v" := (mapsto l 1 v) (at level 20) : bi_scope. Local Notation "l ↦{ q } -" := (∃ v, l ↦{q} v)%I (at level 20, q at level 50, format "l ↦{ q } -") : bi_scope. Local Notation "l ↦ -" := (l ↦{1} -)%I (at level 20) : bi_scope. Section to_gen_heap. Context (L V : Type) `{Countable L}. Implicit Types σ : gmap L V. (** Conversion to heaps and back *) Lemma to_gen_heap_valid σ : ✓ to_gen_heap σ. Proof. intros l. rewrite lookup_fmap. by case (σ !! l). Qed. Lemma lookup_to_gen_heap_None σ l : σ !! l = None → to_gen_heap σ !! l = None. Proof. by rewrite /to_gen_heap lookup_fmap=> ->. Qed. Lemma gen_heap_singleton_included σ l q v : {[l := (q, to_agree v)]} ≼ to_gen_heap σ → σ !! l = Some v. Proof. rewrite singleton_included=> -[[q' av] []]. rewrite /to_gen_heap lookup_fmap fmap_Some_equiv => -[v' [Hl [/= -> ->]]]. move=> /Some_pair_included_total_2 [_] /to_agree_included /leibniz_equiv_iff -> //. Qed. Lemma to_gen_heap_insert l v σ : to_gen_heap (<[l:=v]> σ) = <[l:=(1%Qp, to_agree (v:leibnizC V))]> (to_gen_heap σ). Proof. by rewrite /to_gen_heap fmap_insert. Qed. Lemma to_gen_heap_delete l σ : to_gen_heap (delete l σ) = delete l (to_gen_heap σ). Proof. by rewrite /to_gen_heap fmap_delete. Qed. End to_gen_heap. Lemma gen_heap_init `{gen_heapPreG L V Σ} σ : (|==> ∃ _ : gen_heapG L V Σ, gen_heap_ctx σ)%I. Proof. iMod (own_alloc (● to_gen_heap σ)) as (γ) "Hh". { apply: auth_auth_valid. exact: to_gen_heap_valid. } iModIntro. by iExists (GenHeapG L V Σ _ _ _ γ). Qed. Section gen_heap. Context `{gen_heapG L V Σ}. Implicit Types P Q : iProp Σ. Implicit Types Φ : V → iProp Σ. Implicit Types σ : gmap L V. Implicit Types h g : gen_heapUR L V. Implicit Types l : L. Implicit Types v : V. (** General properties of mapsto *) Global Instance mapsto_timeless l q v : Timeless (l ↦{q} v). Proof. rewrite mapsto_eq /mapsto_def. apply _. Qed. Global Instance mapsto_fractional l v : Fractional (λ q, l ↦{q} v)%I. Proof. intros p q. by rewrite mapsto_eq /mapsto_def -own_op -auth_frag_op op_singleton pair_op agree_idemp. Qed. Global Instance mapsto_as_fractional l q v : AsFractional (l ↦{q} v) (λ q, l ↦{q} v)%I q. Proof. split. done. apply _. Qed. Lemma mapsto_agree l q1 q2 v1 v2 : l ↦{q1} v1 -∗ l ↦{q2} v2 -∗ ⌜v1 = v2⌝. Proof. apply wand_intro_r. rewrite mapsto_eq /mapsto_def -own_op -auth_frag_op own_valid discrete_valid. f_equiv=> /auth_own_valid /=. rewrite op_singleton singleton_valid pair_op. by intros [_ ?%agree_op_invL']. Qed. Global Instance ex_mapsto_fractional l : Fractional (λ q, l ↦{q} -)%I. Proof. intros p q. iSplit. - iDestruct 1 as (v) "[H1 H2]". iSplitL "H1"; eauto. - iIntros "[H1 H2]". iDestruct "H1" as (v1) "H1". iDestruct "H2" as (v2) "H2". iDestruct (mapsto_agree with "H1 H2") as %->. iExists v2. by iFrame. Qed. Global Instance ex_mapsto_as_fractional l q : AsFractional (l ↦{q} -) (λ q, l ↦{q} -)%I q. Proof. split. done. apply _. Qed. Lemma mapsto_valid l q v : l ↦{q} v -∗ ✓ q. Proof. rewrite mapsto_eq /mapsto_def own_valid !discrete_valid. by apply pure_mono=> /auth_own_valid /singleton_valid [??]. Qed. Lemma mapsto_valid_2 l q1 q2 v1 v2 : l ↦{q1} v1 -∗ l ↦{q2} v2 -∗ ✓ (q1 + q2)%Qp. Proof. iIntros "H1 H2". iDestruct (mapsto_agree with "H1 H2") as %->. iApply (mapsto_valid l _ v2). by iFrame. Qed. Lemma gen_heap_alloc σ l v : σ !! l = None → gen_heap_ctx σ ==∗ gen_heap_ctx (<[l:=v]>σ) ∗ l ↦ v. Proof. iIntros (?) "Hσ". rewrite /gen_heap_ctx mapsto_eq /mapsto_def. iMod (own_update with "Hσ") as "[Hσ Hl]". { eapply auth_update_alloc, (alloc_singleton_local_update _ _ (1%Qp, to_agree (v:leibnizC _)))=> //. by apply lookup_to_gen_heap_None. } iModIntro. rewrite to_gen_heap_insert. iFrame. Qed. Lemma gen_heap_dealloc σ l v : gen_heap_ctx σ -∗ l ↦ v ==∗ gen_heap_ctx (delete l σ). Proof. iIntros "Hσ Hl". rewrite /gen_heap_ctx mapsto_eq /mapsto_def. rewrite to_gen_heap_delete. iApply (own_update_2 with "Hσ Hl"). eapply auth_update_dealloc, (delete_singleton_local_update _ _ _). Qed. Lemma gen_heap_valid σ l q v : gen_heap_ctx σ -∗ l ↦{q} v -∗ ⌜σ !! l = Some v⌝. Proof. iIntros "Hσ Hl". rewrite /gen_heap_ctx mapsto_eq /mapsto_def. iDestruct (own_valid_2 with "Hσ Hl") as %[Hl%gen_heap_singleton_included _]%auth_valid_discrete_2; auto. Qed. Lemma gen_heap_update σ l v1 v2 : gen_heap_ctx σ -∗ l ↦ v1 ==∗ gen_heap_ctx (<[l:=v2]>σ) ∗ l ↦ v2. Proof. iIntros "Hσ Hl". rewrite /gen_heap_ctx mapsto_eq /mapsto_def. iDestruct (own_valid_2 with "Hσ Hl") as %[Hl%gen_heap_singleton_included _]%auth_valid_discrete_2. iMod (own_update_2 with "Hσ Hl") as "[Hσ Hl]". { eapply auth_update, singleton_local_update, (exclusive_local_update _ (1%Qp, to_agree (v2:leibnizC _)))=> //. by rewrite /to_gen_heap lookup_fmap Hl. } iModIntro. rewrite to_gen_heap_insert. iFrame. Qed. End gen_heap.