Commit b13debed by Robbert Krebbers

### Cleanup agree and prove some additional properties.

parent 69407372
 ... ... @@ -51,6 +51,11 @@ Proof. 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 |}. ... ... @@ -62,6 +67,12 @@ Instance: Commutative (≡) (@op (agree A) _). Proof. intros x y; split; [naive_solver|by intros n (?&?&Hxy); apply Hxy]. Qed. Definition agree_idempotent (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]. ... ... @@ -88,9 +99,6 @@ Qed. Definition agree_cmra_mixin : CMRAMixin (agree A). Proof. split; try (apply _ || done). * 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. * by intros n x1 x2 Hx y1 y2 Hy. * intros x; split; [apply agree_valid_0|]. by intros n'; rewrite Nat.le_0_r; intros ->. ... ... @@ -101,14 +109,18 @@ Proof. * by intros x y n [(?&?&?) ?]. * by intros x y n; rewrite agree_includedN. Qed. Lemma agree_op_inv (x y1 y2 : agree A) n : ✓{n} x → x ={n}= y1 ⋅ y2 → y1 ={n}= y2. Proof. by intros [??] Hxy; apply Hxy. 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_idempotent. Qed. Definition agree_cmra_extend_mixin : CMRAExtendMixin (agree A). Proof. intros n x y1 y2 ? Hx; exists (x,x); simpl; split. intros n x y1 y2 Hval Hx; exists (x,x); simpl; split. * by rewrite agree_idempotent. * by rewrite Hx (agree_op_inv x y1 y2) // agree_idempotent. * by move: Hval; rewrite Hx; move=> /agree_op_inv->; rewrite agree_idempotent. Qed. Canonical Structure agreeRA : cmraT := CMRAT agree_cofe_mixin agree_cmra_mixin agree_cmra_extend_mixin. ... ... @@ -118,15 +130,9 @@ Program Definition to_agree (x : A) : agree A := 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. 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 ≤ i → ✓{i} x → x n ={n}= x i. Proof. by intros ? [? Hx]; apply Hx. Qed. Lemma agree_to_agree_inj (x y : agree A) a n : ✓{n} x → x ={n}= to_agree a ⋅ y → x n ={n}= a. Proof. by intros; transitivity ((to_agree a ⋅ y) n); first apply agree_car_ne. Qed. Global Instance to_agree_proper : Proper ((≡) ==> (≡)) to_agree := ne_proper _. Global Instance to_agree_inj n : Injective (dist n) (dist n) (to_agree). Proof. by intros x y [_ Hxy]; apply Hxy. Qed. End agree. Arguments agreeC : clear implicits. ... ... @@ -137,8 +143,8 @@ Program Definition agree_map {A B} (f : A → B) (x : agree A) : agree B := Solve Obligations with auto using agree_valid_0, 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). 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. ... ...
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