Require Export algebra.cmra prelude.gmap algebra.option. Require Import algebra.functor. Section cofe. Context `{Countable K} {A : cofeT}. (* COFE *) 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; transitivity (m2 !! k). * by intros n m1 m2 ? k; apply dist_S. * by intros m1 m2 k. * intros c n k; unfold compl, map_compl; rewrite lookup_imap. destruct (decide (n = 0)) as [->|]; [constructor|]. feed inversion (λ H, chain_cauchy c 1 n H k); simpl; auto with lia. by rewrite conv_compl; simpl; apply reflexive_eq. Qed. Canonical Structure mapC : cofeT := CofeT map_cofe_mixin. 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 insert_ne (i : K) 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_equality; [by constructor|by apply lookup_ne]. Qed. Global Instance singleton_ne (i : K) n : Proper (dist n ==> dist n) (singletonM i : A → gmap K A). Proof. by intros ???; apply insert_ne. Qed. Global Instance delete_ne (i : K) n : Proper (dist n ==> dist n) (delete (M:=gmap K A) i). Proof. intros m m' ? j; destruct (decide (i = j)); simplify_map_equality; [by constructor|by apply lookup_ne]. 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 : gmap K A) i : Timeless m → Timeless (m !! i). Proof. intros ? [x|] Hx; [|by symmetry; apply (timeless _)]. assert (m ={1}= <[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 : gmap K A) i x : Timeless x → Timeless m → Timeless (<[i:=x]>m). Proof. intros ?? m' Hm j; destruct (decide (i = j)); simplify_map_equality. { by apply (timeless _); rewrite -Hm lookup_insert. } by apply (timeless _); rewrite -Hm lookup_insert_ne. Qed. Global Instance map_singleton_timeless (i : K) (x : A) : Timeless x → Timeless ({[ i ↦ x ]} : gmap K A) := _. End cofe. Arguments mapC _ {_ _} _. (* CMRA *) Section cmra. Context `{Countable K} {A : cmraT}. 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_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). * by intros m i. * intros n m Hm i; apply cmra_validN_S, Hm. * by intros m1 m2 m3 i; rewrite !lookup_op associative. * by intros m1 m2 i; rewrite !lookup_op commutative. * by intros m i; rewrite lookup_op !lookup_unit cmra_unit_l. * by intros m i; rewrite !lookup_unit cmra_unit_idempotent. * 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 x y n; 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. End cmra. Arguments mapRA _ {_ _} _. Section properties. Context `{Countable K} {A: cmraT}. Implicit Types m : gmap K 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_insert_validN n m i x : ✓{n} x → ✓{n} m → ✓{n} (<[i:=x]>m). Proof. by intros ?? j; destruct (decide (i = j)); simplify_map_equality. Qed. Lemma map_insert_op m1 m2 i x : m2 !! i = None → <[i:=x]>(m1 ⋅ m2) = <[i:=x]>m1 ⋅ m2. Proof. by intros Hi; apply (insert_merge_l _ m1 m2); rewrite Hi. Qed. Lemma map_validN_singleton n (i : K) (x : A) : ✓{n} x <-> ✓{n} ({[ i ↦ x ]} : gmap K A). Proof. split. - move=>Hx j. destruct (decide (i = j)); simplify_map_equality; done. - move=>Hm. move: (Hm i). by simplify_map_equality. 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_equality. + 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 mf n Hm. destruct (Hx (mf !! i) n) as ([y|]&?&?); try done. { by generalize (Hm i); rewrite lookup_op; simplify_map_equality. } exists (<[i:=y]> m); split; first by auto. intros j; move: (Hm j)=>{Hm}; rewrite !lookup_op=>Hm. destruct (decide (i = j)); simplify_map_equality'; 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 congruence. 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. eauto using map_singleton_updateP. Qed. Lemma map_singleton_update i (x y : A) : x ~~> y → {[ i ↦ x ]} ~~> {[ i ↦ y ]}. Proof. rewrite !cmra_update_updateP=>?. eapply map_singleton_updateP; first eassumption. by move=>? ->. Qed. Context `{Fresh K (gset K), !FreshSpec K (gset K)}. Lemma map_updateP_alloc (Q : gmap K A → Prop) m x : ✓ x → (∀ i, m !! i = None → Q (<[i:=x]>m)) → m ~~>: Q. Proof. intros ? HQ mf n Hm. set (i := fresh (dom (gset K) (m ⋅ mf))). assert (i ∉ dom (gset K) m ∧ i ∉ dom (gset K) mf) as [??]. { rewrite -not_elem_of_union -map_dom_op; apply is_fresh. } exists (<[i:=x]>m); split; first by apply HQ, not_elem_of_dom. rewrite -map_insert_op; last by apply not_elem_of_dom. by apply map_insert_validN; [apply cmra_valid_validN|]. 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 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.