From algebra Require Export cmra. From algebra Require Import functor upred. Local Hint Extern 10 (_ ≤ _) => omega. Record agree (A : Type) : Type := Agree { agree_car :> nat → A; agree_is_valid : nat → Prop; agree_valid_S n : agree_is_valid (S n) → agree_is_valid n }. Arguments Agree {_} _ _ _. Arguments agree_car {_} _ _. Arguments agree_is_valid {_} _ _. Section agree. Context {A : cofeT}. Instance agree_validN : ValidN (agree A) := λ n x, agree_is_valid x n ∧ ∀ n', n' ≤ n → x n' ≡{n'}≡ x n. Lemma agree_valid_le (x : agree A) n n' : agree_is_valid x n → n' ≤ n → agree_is_valid x n'. Proof. induction 2; eauto using agree_valid_S. Qed. Instance agree_equiv : Equiv (agree A) := λ x y, (∀ n, agree_is_valid x n ↔ agree_is_valid y n) ∧ (∀ n, agree_is_valid x n → x n ≡{n}≡ y n). Instance agree_dist : Dist (agree A) := λ n x y, (∀ n', n' ≤ n → agree_is_valid x n' ↔ agree_is_valid y n') ∧ (∀ n', n' ≤ n → agree_is_valid x n' → x n' ≡{n'}≡ y n'). Program Instance agree_compl : Compl (agree A) := λ c, {| agree_car n := c (S n) n; agree_is_valid n := agree_is_valid (c (S n)) n |}. Next Obligation. intros c n ?. apply (chain_cauchy c n (S (S n))), agree_valid_S; auto. Qed. Definition agree_cofe_mixin : CofeMixin (agree A). Proof. split. * intros x y; split. + by intros Hxy n; split; intros; apply Hxy. + by intros Hxy; split; intros; apply Hxy with n. * split. + by split. + by intros x y Hxy; split; intros; symmetry; apply Hxy; auto; apply Hxy. + intros x y z Hxy Hyz; split; intros n'; intros. - transitivity (agree_is_valid y n'). by apply Hxy. by apply Hyz. - transitivity (y n'). by apply Hxy. by apply Hyz, Hxy. * intros n x y Hxy; split; intros; apply Hxy; auto. * intros c n; apply and_wlog_r; intros; symmetry; apply (chain_cauchy c); naive_solver. Qed. Canonical Structure agreeC := CofeT agree_cofe_mixin. Lemma agree_car_ne (x y : agree A) n : ✓{n} x → x ≡{n}≡ y → x n ≡{n}≡ y n. Proof. by intros [??] Hxy; apply Hxy. Qed. Lemma agree_cauchy (x : agree A) n i : ✓{n} x → i ≤ n → x i ≡{i}≡ x n. Proof. by intros [? Hx]; apply Hx. Qed. Program Instance agree_op : Op (agree A) := λ x y, {| agree_car := x; agree_is_valid n := agree_is_valid x n ∧ agree_is_valid y n ∧ x ≡{n}≡ y |}. Next Obligation. naive_solver eauto using agree_valid_S, dist_S. Qed. Instance agree_unit : Unit (agree A) := id. Instance agree_minus : Minus (agree A) := λ x y, x. Instance: Comm (≡) (@op (agree A) _). Proof. intros x y; split; [naive_solver|by intros n (?&?&Hxy); apply Hxy]. Qed. Definition agree_idemp (x : agree A) : x ⋅ x ≡ x. Proof. split; naive_solver. Qed. Instance: ∀ n : nat, Proper (dist n ==> impl) (@validN (agree A) _ n). Proof. intros n x y Hxy [? Hx]; split; [by apply Hxy|intros n' ?]. rewrite -(proj2 Hxy n') 1?(Hx n'); eauto using agree_valid_le. by apply dist_le with n; try apply Hxy. Qed. Instance: ∀ x : agree A, Proper (dist n ==> dist n) (op x). Proof. intros n x y1 y2 [Hy' Hy]; split; [|done]. split; intros (?&?&Hxy); repeat (intro || split); try apply Hy'; eauto using agree_valid_le. * etransitivity; [apply Hxy|apply Hy]; eauto using agree_valid_le. * etransitivity; [apply Hxy|symmetry; apply Hy, Hy']; eauto using agree_valid_le. Qed. Instance: Proper (dist n ==> dist n ==> dist n) (@op (agree A) _). Proof. by intros n x1 x2 Hx y1 y2 Hy; rewrite Hy !(comm _ _ y2) Hx. Qed. Instance: Proper ((≡) ==> (≡) ==> (≡)) op := ne_proper_2 _. Instance: Assoc (≡) (@op (agree A) _). Proof. intros x y z; split; simpl; intuition; repeat match goal with H : agree_is_valid _ _ |- _ => clear H end; by cofe_subst; rewrite !agree_idemp. Qed. Lemma agree_includedN (x y : agree A) n : x ≼{n} y ↔ y ≡{n}≡ x ⋅ y. Proof. split; [|by intros ?; exists y]. by intros [z Hz]; rewrite Hz assoc agree_idemp. Qed. Definition agree_cmra_mixin : CMRAMixin (agree A). Proof. split; try (apply _ || done). * by intros n x1 x2 Hx y1 y2 Hy. * intros n x [? Hx]; split; [by apply agree_valid_S|intros n' ?]. rewrite (Hx n'); last auto. symmetry; apply dist_le with n; try apply Hx; auto. * intros x; apply agree_idemp. * by intros x y n [(?&?&?) ?]. * by intros x y n; rewrite agree_includedN. Qed. Lemma agree_op_inv (x1 x2 : agree A) n : ✓{n} (x1 ⋅ x2) → x1 ≡{n}≡ x2. Proof. intros Hxy; apply Hxy. Qed. Lemma agree_valid_includedN (x y : agree A) n : ✓{n} y → x ≼{n} y → x ≡{n}≡ y. Proof. move=> Hval [z Hy]; move: Hval; rewrite Hy. by move=> /agree_op_inv->; rewrite agree_idemp. Qed. Definition agree_cmra_extend_mixin : CMRAExtendMixin (agree A). Proof. intros n x y1 y2 Hval Hx; exists (x,x); simpl; split. * by rewrite agree_idemp. * by move: Hval; rewrite Hx; move=> /agree_op_inv->; rewrite agree_idemp. Qed. Canonical Structure agreeRA : cmraT := CMRAT agree_cofe_mixin agree_cmra_mixin agree_cmra_extend_mixin. Program Definition to_agree (x : A) : agree A := {| agree_car n := x; agree_is_valid n := True |}. Solve Obligations with done. Global Instance to_agree_ne n : Proper (dist n ==> dist n) to_agree. Proof. intros x1 x2 Hx; split; naive_solver eauto using @dist_le. Qed. Global Instance to_agree_proper : Proper ((≡) ==> (≡)) to_agree := ne_proper _. Global Instance to_agree_inj n : Inj (dist n) (dist n) (to_agree). Proof. by intros x y [_ Hxy]; apply Hxy. Qed. Lemma to_agree_car n (x : agree A) : ✓{n} x → to_agree (x n) ≡{n}≡ x. Proof. intros [??]; split; naive_solver eauto using agree_valid_le. Qed. (** Internalized properties *) Lemma agree_validI {M} x y : ✓ (x ⋅ y) ⊑ (x ≡ y : uPred M). Proof. by intros r n _ ?; apply: agree_op_inv. Qed. End agree. Arguments agreeC : clear implicits. Arguments agreeRA : clear implicits. Program Definition agree_map {A B} (f : A → B) (x : agree A) : agree B := {| agree_car n := f (x n); agree_is_valid := agree_is_valid x |}. Solve Obligations with auto using agree_valid_S. Lemma agree_map_id {A} (x : agree A) : agree_map id x = x. Proof. by destruct x. Qed. Lemma agree_map_compose {A B C} (f : A → B) (g : B → C) (x : agree A) : agree_map (g ∘ f) x = agree_map g (agree_map f x). Proof. done. Qed. Section agree_map. Context {A B : cofeT} (f : A → B) `{Hf: ∀ n, Proper (dist n ==> dist n) f}. Global Instance agree_map_ne n : Proper (dist n ==> dist n) (agree_map f). Proof. by intros x1 x2 Hx; split; simpl; intros; [apply Hx|apply Hf, Hx]. Qed. Global Instance agree_map_proper : Proper ((≡) ==> (≡)) (agree_map f) := ne_proper _. Lemma agree_map_ext (g : A → B) x : (∀ x, f x ≡ g x) → agree_map f x ≡ agree_map g x. Proof. by intros Hfg; split; simpl; intros; rewrite ?Hfg. Qed. Global Instance agree_map_monotone : CMRAMonotone (agree_map f). Proof. split; [|by intros n x [? Hx]; split; simpl; [|by intros n' ?; rewrite Hx]]. intros x y n; rewrite !agree_includedN; intros Hy; rewrite Hy. split; last done; split; simpl; last tauto. by intros (?&?&Hxy); repeat split; intros; try apply Hxy; try apply Hf; eauto using @agree_valid_le. Qed. End agree_map. Definition agreeC_map {A B} (f : A -n> B) : agreeC A -n> agreeC B := CofeMor (agree_map f : agreeC A → agreeC B). Instance agreeC_map_ne A B n : Proper (dist n ==> dist n) (@agreeC_map A B). Proof. intros f g Hfg x; split; simpl; intros; first done. by apply dist_le with n; try apply Hfg. Qed. Program Definition agreeF : iFunctor := {| ifunctor_car := agreeRA; ifunctor_map := @agreeC_map |}. Solve Obligations with done.