Require Export algebra.excl. Require Import algebra.functor. Local Arguments validN _ _ _ !_ /. Record auth (A : Type) : Type := Auth { authoritative : excl A ; own : A }. Add Printing Constructor auth. Arguments Auth {_} _ _. Arguments authoritative {_} _. Arguments own {_} _. Notation "◯ a" := (Auth ExclUnit a) (at level 20). Notation "● a" := (Auth (Excl a) ∅) (at level 20). (* COFE *) Section cofe. Context {A : cofeT}. Implicit Types a 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; etransitivity; eauto. * by intros ? [??] [??] [??]; split; apply dist_S. * intros c n; split. apply (conv_compl (chain_map authoritative c) n). apply (conv_compl (chain_map own c) n). Qed. Canonical Structure authC := CofeT auth_cofe_mixin. Global Instance auth_timeless (x : auth A) : Timeless (authoritative x) → Timeless (own x) → Timeless x. Proof. by intros ?? [??] [??]; split; simpl in *; apply (timeless _). 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 : cmraT}. Implicit Types a b : A. Implicit Types x y : auth A. Global Instance auth_empty `{Empty A} : Empty (auth A) := Auth ∅ ∅. Instance auth_validN : ValidN (auth A) := λ n x, match authoritative x with | Excl a => own x ≼{n} a ∧ ✓{n} a | ExclUnit => ✓{n} own x | ExclBot => False end. Global Arguments auth_validN _ !_ /. Instance auth_unit : Unit (auth A) := λ x, Auth (unit (authoritative x)) (unit (own x)). Instance auth_op : Op (auth A) := λ x y, Auth (authoritative x ⋅ authoritative y) (own x ⋅ own y). Instance auth_minus : Minus (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 auth_includedN n (x y : auth A) : x ≼{n} y ↔ authoritative x ≼{n} authoritative y ∧ own x ≼{n} 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. Definition auth_cmra_mixin : CMRAMixin (auth A). Proof. split. * 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; intros ?; cofe_subst; auto. * by intros n x1 x2 [Hx Hx'] y1 y2 [Hy Hy']; split; simpl; rewrite ?Hy ?Hy' ?Hx ?Hx'. * 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_unit_l. * by split; simpl; rewrite ?cmra_unit_idemp. * intros n ??; rewrite! auth_includedN; intros [??]. by split; simpl; apply cmra_unit_preservingN. * 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. * by intros n ??; rewrite auth_includedN; intros [??]; split; simpl; apply cmra_op_minus. Qed. Definition auth_cmra_extend_mixin : CMRAExtendMixin (auth A). Proof. intros n x y1 y2 ? [??]; simpl in *. destruct (cmra_extend_op n (authoritative x) (authoritative y1) (authoritative y2)) as (ea&?&?&?); auto using authoritative_validN. destruct (cmra_extend_op 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 authRA : cmraT := CMRAT auth_cofe_mixin auth_cmra_mixin auth_cmra_extend_mixin. (** The notations ◯ and ● only work for CMRAs with an empty element. So, in what follows, we assume we have an empty element. *) Context `{Empty A, !CMRAIdentity A}. Global Instance auth_cmra_identity : CMRAIdentity authRA. Proof. split; simpl. * by apply (@cmra_empty_valid A _). * by intros x; constructor; rewrite /= left_id. * apply _. 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. move=> Hab [[?| |] bf1] n // =>-[[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 done. 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 done. by rewrite -(local_updateN L) // EQ -(local_updateN L) // -EQ. Qed. End cmra. Arguments authRA : 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; rewrite /auth_map excl_map_id. 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; rewrite /auth_map excl_map_compose. 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 using excl_map_ext. Qed. Instance auth_map_cmra_ne {A B : cofeT} n : Proper ((dist n ==> dist n) ==> dist n ==> dist n) (@auth_map A B). Proof. intros f g Hf [??] [??] [??]; split; [by apply excl_map_cmra_ne|by apply Hf]. Qed. Instance auth_map_cmra_monotone {A B : cmraT} (f : A → B) : (∀ n, Proper (dist n ==> dist n) f) → CMRAMonotone f → CMRAMonotone (auth_map f). Proof. split. * by intros n [x a] [y b]; rewrite !auth_includedN /=; intros [??]; split; simpl; apply: includedN_preserving. * intros n [[a| |] b]; rewrite /= /cmra_validN; naive_solver eauto using @includedN_preserving, @validN_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 authF (Σ : iFunctor) : iFunctor := {| ifunctor_car := authRA ∘ Σ; ifunctor_map A B := authC_map ∘ ifunctor_map Σ |}. Next Obligation. by intros Σ A B n f g Hfg; apply authC_map_ne, ifunctor_map_ne. Qed. Next Obligation. intros Σ A x. rewrite /= -{2}(auth_map_id x). apply auth_map_ext=>y; apply ifunctor_map_id. Qed. Next Obligation. intros Σ A B C f g x. rewrite /= -auth_map_compose. apply auth_map_ext=>y; apply ifunctor_map_compose. Qed.