From iris.algebra Require Export excl. From iris.algebra Require Import upred. Local Arguments valid _ _ !_ /. Local Arguments validN _ _ _ !_ /. Record auth (A : Type) := Auth { authoritative : option (excl A); own : A }. Add Printing Constructor auth. Arguments Auth {_} _ _. Arguments authoritative {_} _. Arguments own {_} _. Notation "◯ a" := (Auth None a) (at level 20). Notation "● a" := (Auth (Excl' a) ∅) (at level 20). (* COFE *) Section cofe. Context {A : cofeT}. Implicit Types a : option (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 ∧ own x ≡ own y. Instance auth_dist : Dist (auth A) := λ n x y, authoritative x ≡{n}≡ authoritative y ∧ own x ≡{n}≡ own y. Global Instance Auth_ne : Proper (dist n ==> dist n ==> dist n) (@Auth A). Proof. by split. Qed. Global Instance Auth_proper : Proper ((≡) ==> (≡) ==> (≡)) (@Auth A). Proof. by split. Qed. Global Instance authoritative_ne: Proper (dist n ==> dist n) (@authoritative A). Proof. by destruct 1. Qed. Global Instance authoritative_proper : Proper ((≡) ==> (≡)) (@authoritative A). Proof. by destruct 1. Qed. Global Instance own_ne : Proper (dist n ==> dist n) (@own A). Proof. by destruct 1. Qed. Global Instance own_proper : Proper ((≡) ==> (≡)) (@own A). Proof. by destruct 1. Qed. Instance auth_compl : Compl (auth A) := λ c, Auth (compl (chain_map authoritative c)) (compl (chain_map own c)). Definition auth_cofe_mixin : CofeMixin (auth A). Proof. split. - intros x y; unfold dist, auth_dist, equiv, auth_equiv. rewrite !equiv_dist; naive_solver. - intros n; split. + by intros ?; split. + by intros ?? [??]; split; symmetry. + intros ??? [??] [??]; split; etrans; eauto. - by intros ? [??] [??] [??]; split; apply dist_S. - intros n c; split. apply (conv_compl n (chain_map authoritative c)). apply (conv_compl n (chain_map own c)). Qed. Canonical Structure authC := CofeT (auth A) auth_cofe_mixin. Global Instance Auth_timeless a b : Timeless a → Timeless b → Timeless (Auth a b). Proof. by intros ?? [??] [??]; split; apply: timeless. Qed. Global Instance auth_discrete : Discrete A → Discrete 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, own x ≼{n} a) ∧ ✓ a | None => ✓ own x | ExclBot' => False end. Global Arguments auth_valid !_ /. Instance auth_validN : ValidN (auth A) := λ n x, match authoritative x with | Excl' a => own x ≼{n} a ∧ ✓{n} a | None => ✓{n} own x | ExclBot' => False end. Global Arguments auth_validN _ !_ /. Instance auth_pcore : PCore (auth A) := λ x, Some (Auth (core (authoritative x)) (core (own x))). Instance auth_op : Op (auth A) := λ x y, Auth (authoritative x ⋅ authoritative y) (own x ⋅ own y). Lemma auth_included (x y : auth A) : x ≼ y ↔ authoritative x ≼ authoritative y ∧ own x ≼ 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 : auth A) : ✓{n} x → ✓{n} authoritative x. Proof. by destruct x as [[[]|]]. Qed. Lemma own_validN n (x : auth A) : ✓{n} x → ✓{n} own x. Proof. 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 => own x ≼ a ∧ ✓ a | None => ✓ own x | ExclBot' => False end. Proof. destruct x as [[[?|]|] ?]; simpl; try done. setoid_rewrite <-cmra_discrete_included_iff; naive_solver eauto using 0. 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 *. destruct Hx as [?? Hx|]; first destruct Hx; intros ?; cofe_subst; auto. - intros [[[?|]|] ?]; rewrite /= ?cmra_included_includedN ?cmra_valid_validN; naive_solver eauto using O. - intros n [[[]|] ?] ?; 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_preserving. - 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]; 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 (ea&?&?&?); auto using authoritative_validN. destruct (cmra_extend n (own x) (own y1) (own y2)) as (b&?&?&?); auto using own_validN. by exists (Auth (ea.1) (b.1), Auth (ea.2) (b.2)). Qed. Canonical Structure authR := CMRAT (auth A) auth_cofe_mixin auth_cmra_mixin. Global Instance auth_cmra_discrete : CMRADiscrete A → CMRADiscrete authR. Proof. split; first apply _. intros [[[?|]|] ?]; rewrite /= /cmra_valid /cmra_validN /=; auto. - setoid_rewrite <-cmra_discrete_included_iff. rewrite -cmra_discrete_valid_iff. tauto. - by rewrite -cmra_discrete_valid_iff. Qed. Instance auth_empty : Empty (auth A) := Auth ∅ ∅. Lemma auth_ucmra_mixin : UCMRAMixin (auth A). Proof. split; simpl. - apply (@ucmra_unit_valid A). - by intros x; constructor; rewrite /= left_id. - apply _. - do 2 constructor; simpl; apply (persistent_core _). Qed. Canonical Structure authUR := UCMRAT (auth A) auth_cofe_mixin auth_cmra_mixin auth_ucmra_mixin. (** Internalized properties *) Lemma auth_equivI {M} (x y : auth A) : x ≡ y ⊣⊢ (authoritative x ≡ authoritative y ∧ own x ≡ 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 ≡ own x ⋅ b) ∧ ✓ a | None => ✓ 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_both_op a b : Auth (Excl' a) b ≡ ● a ⋅ ◯ b. Proof. by rewrite /op /auth_op /= left_id. Qed. Lemma auth_update a a' b b' : (∀ n af, ✓{n} a → a ≡{n}≡ a' ⋅ af → b ≡{n}≡ b' ⋅ af ∧ ✓{n} b) → ● a ⋅ ◯ a' ~~> ● b ⋅ ◯ b'. Proof. intros Hab; apply cmra_total_update. move=> n [[[?|]|] bf1] // =>-[[bf2 Ha] ?]; do 2 red; simpl in *. destruct (Hab n (bf1 ⋅ bf2)) as [Ha' ?]; auto. { by rewrite Ha left_id assoc. } split; [by rewrite Ha' left_id assoc; apply cmra_includedN_l|done]. Qed. Lemma auth_local_update L `{!LocalUpdate Lv L} a a' : Lv a → ✓ L a' → ● a' ⋅ ◯ a ~~> ● L a' ⋅ ◯ L a. Proof. intros. apply auth_update=>n af ? EQ; split; last by apply cmra_valid_validN. by rewrite EQ (local_updateN L) // -EQ. Qed. Lemma auth_update_op_l a a' b : ✓ (b ⋅ a) → ● a ⋅ ◯ a' ~~> ● (b ⋅ a) ⋅ ◯ (b ⋅ a'). Proof. by intros; apply (auth_local_update _). Qed. Lemma auth_update_op_r a a' b : ✓ (a ⋅ b) → ● a ⋅ ◯ a' ~~> ● (a ⋅ b) ⋅ ◯ (a' ⋅ b). Proof. rewrite -!(comm _ b); apply auth_update_op_l. Qed. (* This does not seem to follow from auth_local_update. The trouble is that given ✓ (L a ⋅ a'), Lv a we need ✓ (a ⋅ a'). I think this should hold for every local update, but adding an extra axiom to local updates just for this is silly. *) Lemma auth_local_update_l L `{!LocalUpdate Lv L} a a' : Lv a → ✓ (L a ⋅ a') → ● (a ⋅ a') ⋅ ◯ a ~~> ● (L a ⋅ a') ⋅ ◯ L a. Proof. intros. apply auth_update=>n af ? EQ; split; last by apply cmra_valid_validN. by rewrite -(local_updateN L) // EQ -(local_updateN L) // -EQ. Qed. End cmra. Arguments authR : clear implicits. Arguments authUR : clear implicits. (* Functor *) Definition auth_map {A B} (f : A → B) (x : auth A) : auth B := Auth (excl_map f <\$> authoritative x) (f (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 : cofeT} (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_setoid_ext=> a; by apply excl_map_ext. Qed. Instance auth_map_ne {A B : cofeT} 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_monotone {A B : ucmraT} (f : A → B) : CMRAMonotone f → CMRAMonotone (auth_map f). Proof. split; try apply _. - intros n [[[a|]|] b]; rewrite /= /cmra_validN /=; try naive_solver eauto using includedN_preserving, validN_preserving. - by intros [x a] [y b]; rewrite !auth_included /=; intros [??]; split; simpl; apply: included_preserving. 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 n : Proper (dist n ==> dist n) (@authC_map A B). Proof. intros f f' Hf [[[a|]|] b]; repeat constructor; apply Hf. 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.