From algebra Require Export cmra option. From prelude Require Export gmap. From algebra Require Import functor upred. Section cofe. Context `{Countable K} {A : cofeT}. Implicit Types m : gmap K A. Instance map_dist : Dist (gmap K A) := λ n m1 m2, ∀ i, m1 !! i ≡{n}≡ m2 !! i. Program Definition map_chain (c : chain (gmap K A)) (k : K) : chain (option A) := {| chain_car n := c n !! k |}. Next Obligation. by intros c k n i ?; apply (chain_cauchy c). Qed. Instance map_compl : Compl (gmap K A) := λ c, map_imap (λ i _, compl (map_chain c i)) (c 1). Definition map_cofe_mixin : CofeMixin (gmap K A). Proof. split. - intros m1 m2; split. + by intros Hm n k; apply equiv_dist. + intros Hm k; apply equiv_dist; intros n; apply Hm. - intros n; split. + by intros m k. + by intros m1 m2 ? k. + by intros m1 m2 m3 ?? k; trans (m2 !! k). - by intros n m1 m2 ? k; apply dist_S. - intros n c k; rewrite /compl /map_compl lookup_imap. feed inversion (λ H, chain_cauchy c 0 (S n) H k); simpl; auto with lia. by rewrite conv_compl /=; apply reflexive_eq. Qed. Canonical Structure mapC : cofeT := CofeT map_cofe_mixin. (* why doesn't this go automatic? *) Global Instance mapC_leibniz: LeibnizEquiv A → LeibnizEquiv mapC. Proof. intros; change (LeibnizEquiv (gmap K A)); apply _. Qed. Global Instance lookup_ne n k : Proper (dist n ==> dist n) (lookup k : gmap K A → option A). Proof. by intros m1 m2. Qed. Global Instance lookup_proper k : Proper ((≡) ==> (≡)) (lookup k : gmap K A → option A) := _. Global Instance alter_ne f k n : Proper (dist n ==> dist n) f → Proper (dist n ==> dist n) (alter f k). Proof. intros ? m m' Hm k'. by destruct (decide (k = k')); simplify_map_eq; rewrite (Hm k'). Qed. Global Instance insert_ne i n : Proper (dist n ==> dist n ==> dist n) (insert (M:=gmap K A) i). Proof. intros x y ? m m' ? j; destruct (decide (i = j)); simplify_map_eq; [by constructor|by apply lookup_ne]. Qed. Global Instance singleton_ne i n : Proper (dist n ==> dist n) (singletonM i : A → gmap K A). Proof. by intros ???; apply insert_ne. Qed. Global Instance delete_ne i n : Proper (dist n ==> dist n) (delete (M:=gmap K A) i). Proof. intros m m' ? j; destruct (decide (i = j)); simplify_map_eq; [by constructor|by apply lookup_ne]. Qed. Global Instance map_timeless `{∀ a : A, Timeless a} m : Timeless m. Proof. by intros m' ? i; apply: timeless. Qed. Instance map_empty_timeless : Timeless (∅ : gmap K A). Proof. intros m Hm i; specialize (Hm i); rewrite lookup_empty in Hm |- *. inversion_clear Hm; constructor. Qed. Global Instance map_lookup_timeless m i : Timeless m → Timeless (m !! i). Proof. intros ? [x|] Hx; [|by symmetry; apply: timeless]. assert (m ≡{0}≡ <[i:=x]> m) by (by symmetry in Hx; inversion Hx; cofe_subst; rewrite insert_id). by rewrite (timeless m (<[i:=x]>m)) // lookup_insert. Qed. Global Instance map_insert_timeless m i x : Timeless x → Timeless m → Timeless (<[i:=x]>m). Proof. intros ?? m' Hm j; destruct (decide (i = j)); simplify_map_eq. { by apply: timeless; rewrite -Hm lookup_insert. } by apply: timeless; rewrite -Hm lookup_insert_ne. Qed. Global Instance map_singleton_timeless i x : Timeless x → Timeless ({[ i := x ]} : gmap K A) := _. End cofe. Arguments mapC _ {_ _} _. (* CMRA *) Section cmra. Context `{Countable K} {A : cmraT}. Implicit Types m : gmap K A. Instance map_op : Op (gmap K A) := merge op. Instance map_unit : Unit (gmap K A) := fmap unit. Instance map_validN : ValidN (gmap K A) := λ n m, ∀ i, ✓{n} (m !! i). Instance map_minus : Minus (gmap K A) := merge minus. Lemma lookup_op m1 m2 i : (m1 ⋅ m2) !! i = m1 !! i ⋅ m2 !! i. Proof. by apply lookup_merge. Qed. Lemma lookup_minus m1 m2 i : (m1 ⩪ m2) !! i = m1 !! i ⩪ m2 !! i. Proof. by apply lookup_merge. Qed. Lemma lookup_unit m i : unit m !! i = unit (m !! i). Proof. by apply lookup_fmap. Qed. Lemma map_valid_spec m : ✓ m ↔ ∀ i, ✓ (m !! i). Proof. split; intros Hm ??; apply Hm. Qed. Lemma map_included_spec (m1 m2 : gmap K A) : m1 ≼ m2 ↔ ∀ i, m1 !! i ≼ m2 !! i. Proof. split. - by intros [m Hm]; intros i; exists (m !! i); rewrite -lookup_op Hm. - intros Hm; exists (m2 ⩪ m1); intros i. by rewrite lookup_op lookup_minus cmra_op_minus'. Qed. Lemma map_includedN_spec (m1 m2 : gmap K A) n : m1 ≼{n} m2 ↔ ∀ i, m1 !! i ≼{n} m2 !! i. Proof. split. - by intros [m Hm]; intros i; exists (m !! i); rewrite -lookup_op Hm. - intros Hm; exists (m2 ⩪ m1); intros i. by rewrite lookup_op lookup_minus cmra_op_minus. Qed. Definition map_cmra_mixin : CMRAMixin (gmap K A). Proof. split. - by intros n m1 m2 m3 Hm i; rewrite !lookup_op (Hm i). - by intros n m1 m2 Hm i; rewrite !lookup_unit (Hm i). - by intros n m1 m2 Hm ? i; rewrite -(Hm i). - by intros n m1 m1' Hm1 m2 m2' Hm2 i; rewrite !lookup_minus (Hm1 i) (Hm2 i). - intros n m Hm i; apply cmra_validN_S, Hm. - by intros m1 m2 m3 i; rewrite !lookup_op assoc. - by intros m1 m2 i; rewrite !lookup_op comm. - by intros m i; rewrite lookup_op !lookup_unit cmra_unit_l. - by intros m i; rewrite !lookup_unit cmra_unit_idemp. - intros n x y; rewrite !map_includedN_spec; intros Hm i. by rewrite !lookup_unit; apply cmra_unit_preservingN. - intros n m1 m2 Hm i; apply cmra_validN_op_l with (m2 !! i). by rewrite -lookup_op. - intros n x y; rewrite map_includedN_spec=> ? i. by rewrite lookup_op lookup_minus cmra_op_minus. Qed. Definition map_cmra_extend_mixin : CMRAExtendMixin (gmap K A). Proof. intros n m m1 m2 Hm Hm12. assert (∀ i, m !! i ≡{n}≡ m1 !! i ⋅ m2 !! i) as Hm12' by (by intros i; rewrite -lookup_op). set (f i := cmra_extend_op n (m !! i) (m1 !! i) (m2 !! i) (Hm i) (Hm12' i)). set (f_proj i := proj1_sig (f i)). exists (map_imap (λ i _, (f_proj i).1) m, map_imap (λ i _, (f_proj i).2) m); repeat split; intros i; rewrite /= ?lookup_op !lookup_imap. - destruct (m !! i) as [x|] eqn:Hx; rewrite !Hx /=; [|constructor]. rewrite -Hx; apply (proj2_sig (f i)). - destruct (m !! i) as [x|] eqn:Hx; rewrite /=; [apply (proj2_sig (f i))|]. pose proof (Hm12' i) as Hm12''; rewrite Hx in Hm12''. by symmetry; apply option_op_positive_dist_l with (m2 !! i). - destruct (m !! i) as [x|] eqn:Hx; simpl; [apply (proj2_sig (f i))|]. pose proof (Hm12' i) as Hm12''; rewrite Hx in Hm12''. by symmetry; apply option_op_positive_dist_r with (m1 !! i). Qed. Canonical Structure mapRA : cmraT := CMRAT map_cofe_mixin map_cmra_mixin map_cmra_extend_mixin. Global Instance map_cmra_identity : CMRAIdentity mapRA. Proof. split. - by intros ? n; rewrite lookup_empty. - by intros m i; rewrite /= lookup_op lookup_empty (left_id_L None _). - apply map_empty_timeless. Qed. Global Instance mapRA_leibniz : LeibnizEquiv A → LeibnizEquiv mapRA. Proof. intros; change (LeibnizEquiv (gmap K A)); apply _. Qed. (** Internalized properties *) Lemma map_equivI {M} m1 m2 : (m1 ≡ m2)%I ≡ (∀ i, m1 !! i ≡ m2 !! i : uPred M)%I. Proof. done. Qed. Lemma map_validI {M} m : (✓ m)%I ≡ (∀ i, ✓ (m !! i) : uPred M)%I. Proof. done. Qed. End cmra. Arguments mapRA _ {_ _} _. Section properties. Context `{Countable K} {A : cmraT}. Implicit Types m : gmap K A. Implicit Types i : K. Implicit Types a : A. Lemma map_lookup_validN n m i x : ✓{n} m → m !! i ≡{n}≡ Some x → ✓{n} x. Proof. by move=> /(_ i) Hm Hi; move:Hm; rewrite Hi. Qed. Lemma map_lookup_valid m i x : ✓ m → m !! i ≡ Some x → ✓ x. Proof. move=>Hm Hi n. move:(Hm n i). by rewrite Hi. Qed. Lemma map_insert_validN n m i x : ✓{n} x → ✓{n} m → ✓{n} <[i:=x]>m. Proof. by intros ?? j; destruct (decide (i = j)); simplify_map_eq. Qed. Lemma map_insert_valid m i x : ✓ x → ✓ m → ✓ <[i:=x]>m. Proof. intros ?? n j; apply map_insert_validN; auto. Qed. Lemma map_singleton_validN n i x : ✓{n} ({[ i := x ]} : gmap K A) ↔ ✓{n} x. Proof. split; [|by intros; apply map_insert_validN, cmra_empty_valid]. by move=>/(_ i); simplify_map_eq. Qed. Lemma map_singleton_valid i x : ✓ ({[ i := x ]} : gmap K A) ↔ ✓ x. Proof. split; intros ? n; eapply map_singleton_validN; eauto. Qed. Lemma map_insert_singleton_opN n m i x : m !! i = None ∨ m !! i ≡{n}≡ Some (unit x) → <[i:=x]> m ≡{n}≡ {[ i := x ]} ⋅ m. Proof. intros Hi j; destruct (decide (i = j)) as [->|]; [|by rewrite lookup_op lookup_insert_ne // lookup_singleton_ne // left_id]. rewrite lookup_op lookup_insert lookup_singleton. by destruct Hi as [->| ->]; constructor; rewrite ?cmra_unit_r. Qed. Lemma map_insert_singleton_op m i x : m !! i = None ∨ m !! i ≡ Some (unit x) → <[i:=x]> m ≡ {[ i := x ]} ⋅ m. Proof. rewrite !equiv_dist; naive_solver eauto using map_insert_singleton_opN. Qed. Lemma map_unit_singleton (i : K) (x : A) : unit ({[ i := x ]} : gmap K A) = {[ i := unit x ]}. Proof. apply map_fmap_singleton. Qed. Lemma map_op_singleton (i : K) (x y : A) : {[ i := x ]} ⋅ {[ i := y ]} = ({[ i := x ⋅ y ]} : gmap K A). Proof. by apply (merge_singleton _ _ _ x y). Qed. Lemma singleton_includedN n m i x : {[ i := x ]} ≼{n} m ↔ ∃ y, m !! i ≡{n}≡ Some y ∧ x ≼ y. (* not m !! i = Some y ∧ x ≼{n} y to deal with n = 0 *) Proof. split. - move=> [m' /(_ i)]; rewrite lookup_op lookup_singleton=> Hm. destruct (m' !! i) as [y|]; [exists (x ⋅ y)|exists x]; eauto using cmra_included_l. - intros (y&Hi&?); rewrite map_includedN_spec=>j. destruct (decide (i = j)); simplify_map_eq. + by rewrite Hi; apply Some_Some_includedN, cmra_included_includedN. + apply None_includedN. Qed. Lemma map_dom_op m1 m2 : dom (gset K) (m1 ⋅ m2) ≡ dom _ m1 ∪ dom _ m2. Proof. apply elem_of_equiv; intros i; rewrite elem_of_union !elem_of_dom. unfold is_Some; setoid_rewrite lookup_op. destruct (m1 !! i), (m2 !! i); naive_solver. Qed. Lemma map_insert_updateP (P : A → Prop) (Q : gmap K A → Prop) m i x : x ~~>: P → (∀ y, P y → Q (<[i:=y]>m)) → <[i:=x]>m ~~>: Q. Proof. intros Hx%option_updateP' HP n mf Hm. destruct (Hx n (mf !! i)) as ([y|]&?&?); try done. { by generalize (Hm i); rewrite lookup_op; simplify_map_eq. } exists (<[i:=y]> m); split; first by auto. intros j; move: (Hm j)=>{Hm}; rewrite !lookup_op=>Hm. destruct (decide (i = j)); simplify_map_eq/=; auto. Qed. Lemma map_insert_updateP' (P : A → Prop) m i x : x ~~>: P → <[i:=x]>m ~~>: λ m', ∃ y, m' = <[i:=y]>m ∧ P y. Proof. eauto using map_insert_updateP. Qed. Lemma map_insert_update m i x y : x ~~> y → <[i:=x]>m ~~> <[i:=y]>m. Proof. rewrite !cmra_update_updateP; eauto using map_insert_updateP with subst. Qed. Lemma map_singleton_updateP (P : A → Prop) (Q : gmap K A → Prop) i x : x ~~>: P → (∀ y, P y → Q {[ i := y ]}) → {[ i := x ]} ~~>: Q. Proof. apply map_insert_updateP. Qed. Lemma map_singleton_updateP' (P : A → Prop) i x : x ~~>: P → {[ i := x ]} ~~>: λ m, ∃ y, m = {[ i := y ]} ∧ P y. Proof. apply map_insert_updateP'. Qed. Lemma map_singleton_update i (x y : A) : x ~~> y → {[ i := x ]} ~~> {[ i := y ]}. Proof. apply map_insert_update. Qed. Lemma map_singleton_updateP_empty `{Empty A, !CMRAIdentity A} (P : A → Prop) (Q : gmap K A → Prop) i : ∅ ~~>: P → (∀ y, P y → Q {[ i := y ]}) → ∅ ~~>: Q. Proof. intros Hx HQ n gf Hg. destruct (Hx n (from_option ∅ (gf !! i))) as (y&?&Hy). { move:(Hg i). rewrite !left_id. case _: (gf !! i); simpl; auto using cmra_empty_valid. } exists {[ i := y ]}; split; first by auto. intros i'; destruct (decide (i' = i)) as [->|]. - rewrite lookup_op lookup_singleton. move:Hy; case _: (gf !! i); first done. by rewrite right_id. - move:(Hg i'). by rewrite !lookup_op lookup_singleton_ne // !left_id. Qed. Lemma map_singleton_updateP_empty' `{Empty A, !CMRAIdentity A} (P: A → Prop) i : ∅ ~~>: P → ∅ ~~>: λ m, ∃ y, m = {[ i := y ]} ∧ P y. Proof. eauto using map_singleton_updateP_empty. Qed. Section freshness. Context `{Fresh K (gset K), !FreshSpec K (gset K)}. Lemma map_updateP_alloc_strong (Q : gmap K A → Prop) (I : gset K) m x : ✓ x → (∀ i, m !! i = None → i ∉ I → Q (<[i:=x]>m)) → m ~~>: Q. Proof. intros ? HQ n mf Hm. set (i := fresh (I ∪ dom (gset K) (m ⋅ mf))). assert (i ∉ I ∧ i ∉ dom (gset K) m ∧ i ∉ dom (gset K) mf) as [?[??]]. { rewrite -not_elem_of_union -map_dom_op -not_elem_of_union; apply is_fresh. } exists (<[i:=x]>m); split. { by apply HQ; last done; apply not_elem_of_dom. } rewrite map_insert_singleton_opN; last by left; apply not_elem_of_dom. rewrite -assoc -map_insert_singleton_opN; last by left; apply not_elem_of_dom; rewrite map_dom_op not_elem_of_union. by apply map_insert_validN; [apply cmra_valid_validN|]. Qed. Lemma map_updateP_alloc (Q : gmap K A → Prop) m x : ✓ x → (∀ i, m !! i = None → Q (<[i:=x]>m)) → m ~~>: Q. Proof. move=>??. eapply map_updateP_alloc_strong with (I:=∅); by eauto. Qed. Lemma map_updateP_alloc_strong' m x (I : gset K) : ✓ x → m ~~>: λ m', ∃ i, i ∉ I ∧ m' = <[i:=x]>m ∧ m !! i = None. Proof. eauto using map_updateP_alloc_strong. Qed. Lemma map_updateP_alloc' m x : ✓ x → m ~~>: λ m', ∃ i, m' = <[i:=x]>m ∧ m !! i = None. Proof. eauto using map_updateP_alloc. Qed. End freshness. (* Allocation is a local update: Just use composition with a singleton map. *) (* Deallocation is *not* a local update. The trouble is that if we own {[ i ↦ x ]}, then the frame could always own "unit x", and prevent deallocation. *) (* Applying a local update at a position we own is a local update. *) Global Instance map_alter_update `{!LocalUpdate Lv L} i : LocalUpdate (λ m, ∃ x, m !! i = Some x ∧ Lv x) (alter L i). Proof. split; first apply _. intros n m1 m2 (x&Hix&?) Hm j; destruct (decide (i = j)) as [->|]. - rewrite lookup_alter !lookup_op lookup_alter Hix /=. move: (Hm j); rewrite lookup_op Hix. case: (m2 !! j)=>[y|] //=; constructor. by apply (local_updateN L). - by rewrite lookup_op !lookup_alter_ne // lookup_op. Qed. End properties. (** Functor *) Instance map_fmap_ne `{Countable K} {A B : cofeT} (f : A → B) n : Proper (dist n ==> dist n) f → Proper (dist n ==>dist n) (fmap (M:=gmap K) f). Proof. by intros ? m m' Hm k; rewrite !lookup_fmap; apply option_fmap_ne. Qed. Instance map_fmap_cmra_monotone `{Countable K} {A B : cmraT} (f : A → B) `{!CMRAMonotone f} : CMRAMonotone (fmap f : gmap K A → gmap K B). Proof. split. - intros m1 m2 n; rewrite !map_includedN_spec; intros Hm i. by rewrite !lookup_fmap; apply: includedN_preserving. - by intros n m ? i; rewrite lookup_fmap; apply validN_preserving. Qed. Definition mapC_map `{Countable K} {A B} (f: A -n> B) : mapC K A -n> mapC K B := CofeMor (fmap f : mapC K A → mapC K B). Instance mapC_map_ne `{Countable K} {A B} n : Proper (dist n ==> dist n) (@mapC_map K _ _ A B). Proof. intros f g Hf m k; rewrite /= !lookup_fmap. destruct (_ !! k) eqn:?; simpl; constructor; apply Hf. Qed. Program Definition mapF K `{Countable K} (Σ : iFunctor) : iFunctor := {| ifunctor_car := mapRA K ∘ Σ; ifunctor_map A B := mapC_map ∘ ifunctor_map Σ |}. Next Obligation. by intros K ?? Σ A B n f g Hfg; apply mapC_map_ne, ifunctor_map_ne. Qed. Next Obligation. intros K ?? Σ A x. rewrite /= -{2}(map_fmap_id x). apply map_fmap_setoid_ext=> ? y _; apply ifunctor_map_id. Qed. Next Obligation. intros K ?? Σ A B C f g x. rewrite /= -map_fmap_compose. apply map_fmap_setoid_ext=> ? y _; apply ifunctor_map_compose. Qed.