From iris.algebra Require Export excl local_updates. From iris.base_logic Require Import base_logic proofmode_classes. Set Default Proof Using "Type". Record auth (A : Type) := Auth { authoritative : excl' A; auth_own : A }. Add Printing Constructor auth. Arguments Auth {_} _ _. Arguments authoritative {_} _. Arguments auth_own {_} _. Instance: Params (@Auth) 1. Instance: Params (@authoritative) 1. Instance: Params (@auth_own) 1. Notation "◯ a" := (Auth None a) (at level 20). Notation "● a" := (Auth (Excl' a) ε) (at level 20). (* COFE *) Section cofe. Context {A : ofeT}. Implicit Types a : excl' A. Implicit Types b : A. Implicit Types x y : auth A. Instance auth_equiv : Equiv (auth A) := λ x y, authoritative x ≡ authoritative y ∧ auth_own x ≡ auth_own y. Instance auth_dist : Dist (auth A) := λ n x y, authoritative x ≡{n}≡ authoritative y ∧ auth_own x ≡{n}≡ auth_own y. Global Instance Auth_ne : NonExpansive2 (@Auth A). Proof. by split. Qed. Global Instance Auth_proper : Proper ((≡) ==> (≡) ==> (≡)) (@Auth A). Proof. by split. Qed. Global Instance authoritative_ne: NonExpansive (@authoritative A). Proof. by destruct 1. Qed. Global Instance authoritative_proper : Proper ((≡) ==> (≡)) (@authoritative A). Proof. by destruct 1. Qed. Global Instance own_ne : NonExpansive (@auth_own A). Proof. by destruct 1. Qed. Global Instance own_proper : Proper ((≡) ==> (≡)) (@auth_own A). Proof. by destruct 1. Qed. Definition auth_ofe_mixin : OfeMixin (auth A). Proof. by apply (iso_ofe_mixin (λ x, (authoritative x, auth_own x))). Qed. Canonical Structure authC := OfeT (auth A) auth_ofe_mixin. Global Instance auth_cofe `{Cofe A} : Cofe authC. Proof. apply (iso_cofe (λ y : _ * _, Auth (y.1) (y.2)) (λ x, (authoritative x, auth_own x))); by repeat intro. Qed. Global Instance Auth_discrete a b : Discrete a → Discrete b → Discrete (Auth a b). Proof. by intros ?? [??] [??]; split; apply: discrete. Qed. Global Instance auth_ofe_discrete : OfeDiscrete A → OfeDiscrete authC. Proof. intros ? [??]; apply _. Qed. Global Instance auth_leibniz : LeibnizEquiv A → LeibnizEquiv (auth A). Proof. by intros ? [??] [??] [??]; f_equal/=; apply leibniz_equiv. Qed. End cofe. Arguments authC : clear implicits. (* CMRA *) Section cmra. Context {A : ucmraT}. Implicit Types a b : A. Implicit Types x y : auth A. Instance auth_valid : Valid (auth A) := λ x, match authoritative x with | Excl' a => (∀ n, auth_own x ≼{n} a) ∧ ✓ a | None => ✓ auth_own x | ExclBot' => False end. Global Arguments auth_valid !_ /. Instance auth_validN : ValidN (auth A) := λ n x, match authoritative x with | Excl' a => auth_own x ≼{n} a ∧ ✓{n} a | None => ✓{n} auth_own x | ExclBot' => False end. Global Arguments auth_validN _ !_ /. Instance auth_pcore : PCore (auth A) := λ x, Some (Auth (core (authoritative x)) (core (auth_own x))). Instance auth_op : Op (auth A) := λ x y, Auth (authoritative x ⋅ authoritative y) (auth_own x ⋅ auth_own y). Definition auth_valid_eq : valid = λ x, match authoritative x with | Excl' a => (∀ n, auth_own x ≼{n} a) ∧ ✓ a | None => ✓ auth_own x | ExclBot' => False end := eq_refl _. Definition auth_validN_eq : validN = λ n x, match authoritative x with | Excl' a => auth_own x ≼{n} a ∧ ✓{n} a | None => ✓{n} auth_own x | ExclBot' => False end := eq_refl _. Lemma auth_included (x y : auth A) : x ≼ y ↔ authoritative x ≼ authoritative y ∧ auth_own x ≼ auth_own y. Proof. split; [intros [[z1 z2] Hz]; split; [exists z1|exists z2]; apply Hz|]. intros [[z1 Hz1] [z2 Hz2]]; exists (Auth z1 z2); split; auto. Qed. Lemma authoritative_validN n x : ✓{n} x → ✓{n} authoritative x. Proof. by destruct x as [[[]|]]. Qed. Lemma auth_own_validN n x : ✓{n} x → ✓{n} auth_own x. Proof. rewrite auth_validN_eq. destruct x as [[[]|]]; naive_solver eauto using cmra_validN_includedN. Qed. Lemma auth_valid_discrete `{CmraDiscrete A} x : ✓ x ↔ match authoritative x with | Excl' a => auth_own x ≼ a ∧ ✓ a | None => ✓ auth_own x | ExclBot' => False end. Proof. rewrite auth_valid_eq. destruct x as [[[?|]|] ?]; simpl; try done. setoid_rewrite <-cmra_discrete_included_iff; naive_solver eauto using 0. Qed. Lemma auth_validN_2 n a b : ✓{n} (● a ⋅ ◯ b) ↔ b ≼{n} a ∧ ✓{n} a. Proof. by rewrite auth_validN_eq /= left_id. Qed. Lemma auth_valid_discrete_2 `{CmraDiscrete A} a b : ✓ (● a ⋅ ◯ b) ↔ b ≼ a ∧ ✓ a. Proof. by rewrite auth_valid_discrete /= left_id. Qed. Lemma authoritative_valid x : ✓ x → ✓ authoritative x. Proof. by destruct x as [[[]|]]. Qed. Lemma auth_own_valid `{CmraDiscrete A} x : ✓ x → ✓ auth_own x. Proof. rewrite auth_valid_discrete. destruct x as [[[]|]]; naive_solver eauto using cmra_valid_included. Qed. Lemma auth_cmra_mixin : CmraMixin (auth A). Proof. apply cmra_total_mixin. - eauto. - by intros n x y1 y2 [Hy Hy']; split; simpl; rewrite ?Hy ?Hy'. - by intros n y1 y2 [Hy Hy']; split; simpl; rewrite ?Hy ?Hy'. - intros n [x a] [y b] [Hx Ha]; simpl in *. rewrite !auth_validN_eq. destruct Hx as [?? Hx|]; first destruct Hx; intros ?; ofe_subst; auto. - intros [[[?|]|] ?]; rewrite /= ?auth_valid_eq ?auth_validN_eq /= ?cmra_included_includedN ?cmra_valid_validN; naive_solver eauto using O. - intros n [[[]|] ?]; rewrite !auth_validN_eq /=; naive_solver eauto using cmra_includedN_S, cmra_validN_S. - by split; simpl; rewrite assoc. - by split; simpl; rewrite comm. - by split; simpl; rewrite ?cmra_core_l. - by split; simpl; rewrite ?cmra_core_idemp. - intros ??; rewrite! auth_included; intros [??]. by split; simpl; apply cmra_core_mono. - assert (∀ n (a b1 b2 : A), b1 ⋅ b2 ≼{n} a → b1 ≼{n} a). { intros n a b1 b2 <-; apply cmra_includedN_l. } intros n [[[a1|]|] b1] [[[a2|]|] b2]; rewrite auth_validN_eq; naive_solver eauto using cmra_validN_op_l, cmra_validN_includedN. - intros n x y1 y2 ? [??]; simpl in *. destruct (cmra_extend n (authoritative x) (authoritative y1) (authoritative y2)) as (ea1&ea2&?&?&?); auto using authoritative_validN. destruct (cmra_extend n (auth_own x) (auth_own y1) (auth_own y2)) as (b1&b2&?&?&?); auto using auth_own_validN. by exists (Auth ea1 b1), (Auth ea2 b2). Qed. Canonical Structure authR := CmraT (auth A) auth_cmra_mixin. Global Instance auth_cmra_discrete : CmraDiscrete A → CmraDiscrete authR. Proof. split; first apply _. intros [[[?|]|] ?]; rewrite auth_valid_eq auth_validN_eq /=; auto. - setoid_rewrite <-cmra_discrete_included_iff. rewrite -cmra_discrete_valid_iff. tauto. - by rewrite -cmra_discrete_valid_iff. Qed. Instance auth_empty : Unit (auth A) := Auth ε ε. Lemma auth_ucmra_mixin : UcmraMixin (auth A). Proof. split; simpl. - rewrite auth_valid_eq /=. apply ucmra_unit_valid. - by intros x; constructor; rewrite /= left_id. - do 2 constructor; simpl; apply (core_id_core _). Qed. Canonical Structure authUR := UcmraT (auth A) auth_ucmra_mixin. Global Instance auth_frag_core_id a : CoreId a → CoreId (◯ a). Proof. do 2 constructor; simpl; auto. by apply core_id_core. Qed. (** Internalized properties *) Lemma auth_equivI {M} (x y : auth A) : x ≡ y ⊣⊢ (authoritative x ≡ authoritative y ∧ auth_own x ≡ auth_own y : uPred M). Proof. by uPred.unseal. Qed. Lemma auth_validI {M} (x : auth A) : ✓ x ⊣⊢ (match authoritative x with | Excl' a => (∃ b, a ≡ auth_own x ⋅ b) ∧ ✓ a | None => ✓ auth_own x | ExclBot' => False end : uPred M). Proof. uPred.unseal. by destruct x as [[[]|]]. Qed. Lemma auth_frag_op a b : ◯ (a ⋅ b) = ◯ a ⋅ ◯ b. Proof. done. Qed. Lemma auth_frag_mono a b : a ≼ b → ◯ a ≼ ◯ b. Proof. intros [c ->]. rewrite auth_frag_op. apply cmra_included_l. Qed. Global Instance auth_frag_sep_homomorphism : MonoidHomomorphism op op (≡) (Auth None). Proof. by split; [split; try apply _|]. Qed. Lemma auth_both_op a b : Auth (Excl' a) b ≡ ● a ⋅ ◯ b. Proof. by rewrite /op /auth_op /= left_id. Qed. Lemma auth_auth_valid a : ✓ a → ✓ (● a). Proof. intros; split; simpl; auto using ucmra_unit_leastN. Qed. Lemma auth_update a b a' b' : (a,b) ~l~> (a',b') → ● a ⋅ ◯ b ~~> ● a' ⋅ ◯ b'. Proof. intros Hup; apply cmra_total_update. move=> n [[[?|]|] bf1] // [[bf2 Ha] ?]; do 2 red; simpl in *. move: Ha; rewrite !left_id -assoc=> Ha. destruct (Hup n (Some (bf1 ⋅ bf2))); auto. split; last done. exists bf2. by rewrite -assoc. Qed. Lemma auth_update_alloc a a' b' : (a,ε) ~l~> (a',b') → ● a ~~> ● a' ⋅ ◯ b'. Proof. intros. rewrite -(right_id _ _ (● a)). by apply auth_update. Qed. Lemma auth_update_dealloc a b a' : (a,b) ~l~> (a',ε) → ● a ⋅ ◯ b ~~> ● a'. Proof. intros. rewrite -(right_id _ _ (● a')). by apply auth_update. Qed. Lemma auth_local_update (a b0 b1 a' b0' b1': A) : (b0, b1) ~l~> (b0', b1') → b0' ≼ a' → ✓ a' → (● a ⋅ ◯ b0, ● a ⋅ ◯ b1) ~l~> (● a' ⋅ ◯ b0', ● a' ⋅ ◯ b1'). Proof. rewrite !local_update_unital=> Hup ? ? n /=. move=> [[[ac|]|] bc] /auth_validN_2 [Le Val] [] /=; inversion_clear 1 as [?? Ha|]; inversion_clear Ha. (* need setoid_discriminate! *) rewrite !left_id=> ?. destruct (Hup n bc) as [Hval' Heq]; eauto using cmra_validN_includedN. rewrite -!auth_both_op auth_validN_eq /=. split_and!; [by apply cmra_included_includedN|by apply cmra_valid_validN|done]. Qed. End cmra. Arguments authR : clear implicits. Arguments authUR : clear implicits. (* Proof mode class instances *) Instance is_op_auth_frag {A : ucmraT} (a b1 b2 : A) : IsOp a b1 b2 → IsOp' (◯ a) (◯ b1) (◯ b2). Proof. done. Qed. (* Functor *) Definition auth_map {A B} (f : A → B) (x : auth A) : auth B := Auth (excl_map f <\$> authoritative x) (f (auth_own x)). Lemma auth_map_id {A} (x : auth A) : auth_map id x = x. Proof. by destruct x as [[[]|]]. Qed. Lemma auth_map_compose {A B C} (f : A → B) (g : B → C) (x : auth A) : auth_map (g ∘ f) x = auth_map g (auth_map f x). Proof. by destruct x as [[[]|]]. Qed. Lemma auth_map_ext {A B : ofeT} (f g : A → B) x : (∀ x, f x ≡ g x) → auth_map f x ≡ auth_map g x. Proof. constructor; simpl; auto. apply option_fmap_equiv_ext=> a; by apply excl_map_ext. Qed. Instance auth_map_ne {A B : ofeT} n : Proper ((dist n ==> dist n) ==> dist n ==> dist n) (@auth_map A B). Proof. intros f g Hf [??] [??] [??]; split; simpl in *; [|by apply Hf]. apply option_fmap_ne; [|done]=> x y ?; by apply excl_map_ne. Qed. Instance auth_map_cmra_morphism {A B : ucmraT} (f : A → B) : CmraMorphism f → CmraMorphism (auth_map f). Proof. split; try apply _. - intros n [[[a|]|] b]; rewrite !auth_validN_eq; try naive_solver eauto using cmra_morphism_monotoneN, cmra_morphism_validN. - intros [??]. apply Some_proper; rewrite /auth_map /=. by f_equiv; rewrite /= cmra_morphism_core. - intros [[?|]?] [[?|]?]; try apply Auth_proper=>//=; by rewrite cmra_morphism_op. Qed. Definition authC_map {A B} (f : A -n> B) : authC A -n> authC B := CofeMor (auth_map f). Lemma authC_map_ne A B : NonExpansive (@authC_map A B). Proof. intros n f f' Hf [[[a|]|] b]; repeat constructor; apply Hf. Qed. Program Definition authRF (F : urFunctor) : rFunctor := {| rFunctor_car A B := authR (urFunctor_car F A B); rFunctor_map A1 A2 B1 B2 fg := authC_map (urFunctor_map F fg) |}. Next Obligation. by intros F A1 A2 B1 B2 n f g Hfg; apply authC_map_ne, urFunctor_ne. Qed. Next Obligation. intros F A B x. rewrite /= -{2}(auth_map_id x). apply auth_map_ext=>y; apply urFunctor_id. Qed. Next Obligation. intros F A1 A2 A3 B1 B2 B3 f g f' g' x. rewrite /= -auth_map_compose. apply auth_map_ext=>y; apply urFunctor_compose. Qed. Instance authRF_contractive F : urFunctorContractive F → rFunctorContractive (authRF F). Proof. by intros ? A1 A2 B1 B2 n f g Hfg; apply authC_map_ne, urFunctor_contractive. Qed. Program Definition authURF (F : urFunctor) : urFunctor := {| urFunctor_car A B := authUR (urFunctor_car F A B); urFunctor_map A1 A2 B1 B2 fg := authC_map (urFunctor_map F fg) |}. Next Obligation. by intros F A1 A2 B1 B2 n f g Hfg; apply authC_map_ne, urFunctor_ne. Qed. Next Obligation. intros F A B x. rewrite /= -{2}(auth_map_id x). apply auth_map_ext=>y; apply urFunctor_id. Qed. Next Obligation. intros F A1 A2 A3 B1 B2 B3 f g f' g' x. rewrite /= -auth_map_compose. apply auth_map_ext=>y; apply urFunctor_compose. Qed. Instance authURF_contractive F : urFunctorContractive F → urFunctorContractive (authURF F). Proof. by intros ? A1 A2 B1 B2 n f g Hfg; apply authC_map_ne, urFunctor_contractive. Qed.