### Shorter names for common math notions.

`Also do some minor clean up.`
parent ba6d9390
 ... @@ -59,9 +59,9 @@ Program Instance agree_op : Op (agree A) := λ x y, ... @@ -59,9 +59,9 @@ Program Instance agree_op : Op (agree A) := λ x y, Next Obligation. naive_solver eauto using agree_valid_S, dist_S. Qed. Next Obligation. naive_solver eauto using agree_valid_S, dist_S. Qed. Instance agree_unit : Unit (agree A) := id. Instance agree_unit : Unit (agree A) := id. Instance agree_minus : Minus (agree A) := λ x y, x. Instance agree_minus : Minus (agree A) := λ x y, x. Instance: Commutative (≡) (@op (agree A) _). Instance: Comm (≡) (@op (agree A) _). Proof. intros x y; split; [naive_solver|by intros n (?&?&Hxy); apply Hxy]. Qed. Proof. intros x y; split; [naive_solver|by intros n (?&?&Hxy); apply Hxy]. Qed. Definition agree_idempotent (x : agree A) : x ⋅ x ≡ x. Definition agree_idemp (x : agree A) : x ⋅ x ≡ x. Proof. split; naive_solver. Qed. Proof. split; naive_solver. Qed. Instance: ∀ n : nat, Proper (dist n ==> impl) (@validN (agree A) _ n). Instance: ∀ n : nat, Proper (dist n ==> impl) (@validN (agree A) _ n). Proof. Proof. ... @@ -79,18 +79,18 @@ Proof. ... @@ -79,18 +79,18 @@ Proof. eauto using agree_valid_le. eauto using agree_valid_le. Qed. Qed. Instance: Proper (dist n ==> dist n ==> dist n) (@op (agree A) _). Instance: Proper (dist n ==> dist n ==> dist n) (@op (agree A) _). Proof. by intros n x1 x2 Hx y1 y2 Hy; rewrite Hy !(commutative _ _ y2) Hx. Qed. Proof. by intros n x1 x2 Hx y1 y2 Hy; rewrite Hy !(comm _ _ y2) Hx. Qed. Instance: Proper ((≡) ==> (≡) ==> (≡)) op := ne_proper_2 _. Instance: Proper ((≡) ==> (≡) ==> (≡)) op := ne_proper_2 _. Instance: Associative (≡) (@op (agree A) _). Instance: Assoc (≡) (@op (agree A) _). Proof. Proof. intros x y z; split; simpl; intuition; intros x y z; split; simpl; intuition; repeat match goal with H : agree_is_valid _ _ |- _ => clear H end; repeat match goal with H : agree_is_valid _ _ |- _ => clear H end; by cofe_subst; rewrite !agree_idempotent. by cofe_subst; rewrite !agree_idemp. Qed. Qed. Lemma agree_includedN (x y : agree A) n : x ≼{n} y ↔ y ≡{n}≡ x ⋅ y. Lemma agree_includedN (x y : agree A) n : x ≼{n} y ↔ y ≡{n}≡ x ⋅ y. Proof. Proof. split; [|by intros ?; exists y]. split; [|by intros ?; exists y]. by intros [z Hz]; rewrite Hz (associative _) agree_idempotent. by intros [z Hz]; rewrite Hz assoc agree_idemp. Qed. Qed. Definition agree_cmra_mixin : CMRAMixin (agree A). Definition agree_cmra_mixin : CMRAMixin (agree A). Proof. Proof. ... @@ -99,7 +99,7 @@ Proof. ... @@ -99,7 +99,7 @@ Proof. * intros n x [? Hx]; split; [by apply agree_valid_S|intros n' ?]. * intros n x [? Hx]; split; [by apply agree_valid_S|intros n' ?]. rewrite (Hx n'); last auto. rewrite (Hx n'); last auto. symmetry; apply dist_le with n; try apply Hx; auto. symmetry; apply dist_le with n; try apply Hx; auto. * intros x; apply agree_idempotent. * intros x; apply agree_idemp. * by intros x y n [(?&?&?) ?]. * by intros x y n [(?&?&?) ?]. * by intros x y n; rewrite agree_includedN. * by intros x y n; rewrite agree_includedN. Qed. Qed. ... @@ -108,13 +108,13 @@ Proof. intros Hxy; apply Hxy. Qed. ... @@ -108,13 +108,13 @@ Proof. intros Hxy; apply Hxy. Qed. Lemma agree_valid_includedN (x y : agree A) n : ✓{n} y → x ≼{n} y → x ≡{n}≡ y. Lemma agree_valid_includedN (x y : agree A) n : ✓{n} y → x ≼{n} y → x ≡{n}≡ y. Proof. Proof. move=> Hval [z Hy]; move: Hval; rewrite Hy. move=> Hval [z Hy]; move: Hval; rewrite Hy. by move=> /agree_op_inv->; rewrite agree_idempotent. by move=> /agree_op_inv->; rewrite agree_idemp. Qed. Qed. Definition agree_cmra_extend_mixin : CMRAExtendMixin (agree A). Definition agree_cmra_extend_mixin : CMRAExtendMixin (agree A). Proof. Proof. intros n x y1 y2 Hval Hx; exists (x,x); simpl; split. intros n x y1 y2 Hval Hx; exists (x,x); simpl; split. * by rewrite agree_idempotent. * by rewrite agree_idemp. * by move: Hval; rewrite Hx; move=> /agree_op_inv->; rewrite agree_idempotent. * by move: Hval; rewrite Hx; move=> /agree_op_inv->; rewrite agree_idemp. Qed. Qed. Canonical Structure agreeRA : cmraT := Canonical Structure agreeRA : cmraT := CMRAT agree_cofe_mixin agree_cmra_mixin agree_cmra_extend_mixin. CMRAT agree_cofe_mixin agree_cmra_mixin agree_cmra_extend_mixin. ... @@ -125,7 +125,7 @@ Solve Obligations with done. ... @@ -125,7 +125,7 @@ Solve Obligations with done. Global Instance to_agree_ne n : Proper (dist n ==> dist n) to_agree. 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. 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_proper : Proper ((≡) ==> (≡)) to_agree := ne_proper _. Global Instance to_agree_inj n : Injective (dist n) (dist n) (to_agree). Global Instance to_agree_inj n : Inj (dist n) (dist n) (to_agree). Proof. by intros x y [_ Hxy]; apply Hxy. Qed. 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. 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. Proof. intros [??]; split; naive_solver eauto using agree_valid_le. Qed. ... ...
 ... @@ -106,10 +106,10 @@ Proof. ... @@ -106,10 +106,10 @@ Proof. * by intros n x1 x2 [Hx Hx'] y1 y2 [Hy Hy']; * by intros n x1 x2 [Hx Hx'] y1 y2 [Hy Hy']; split; simpl; rewrite ?Hy ?Hy' ?Hx ?Hx'. split; simpl; rewrite ?Hy ?Hy' ?Hx ?Hx'. * intros n [[] ?] ?; naive_solver eauto using cmra_includedN_S, cmra_validN_S. * intros n [[] ?] ?; naive_solver eauto using cmra_includedN_S, cmra_validN_S. * by split; simpl; rewrite associative. * by split; simpl; rewrite assoc. * by split; simpl; rewrite commutative. * by split; simpl; rewrite comm. * by split; simpl; rewrite ?cmra_unit_l. * by split; simpl; rewrite ?cmra_unit_l. * by split; simpl; rewrite ?cmra_unit_idempotent. * by split; simpl; rewrite ?cmra_unit_idemp. * intros n ??; rewrite! auth_includedN; intros [??]. * intros n ??; rewrite! auth_includedN; intros [??]. by split; simpl; apply cmra_unit_preservingN. by split; simpl; apply cmra_unit_preservingN. * assert (∀ n (a b1 b2 : A), b1 ⋅ b2 ≼{n} a → b1 ≼{n} a). * assert (∀ n (a b1 b2 : A), b1 ⋅ b2 ≼{n} a → b1 ≼{n} a). ... @@ -153,8 +153,8 @@ Lemma auth_update a a' b b' : ... @@ -153,8 +153,8 @@ Lemma auth_update a a' b b' : Proof. Proof. move=> Hab [[?| |] bf1] n // =>-[[bf2 Ha] ?]; do 2 red; simpl in *. move=> Hab [[?| |] bf1] n // =>-[[bf2 Ha] ?]; do 2 red; simpl in *. destruct (Hab n (bf1 ⋅ bf2)) as [Ha' ?]; auto. destruct (Hab n (bf1 ⋅ bf2)) as [Ha' ?]; auto. { by rewrite Ha left_id associative. } { by rewrite Ha left_id assoc. } split; [by rewrite Ha' left_id associative; apply cmra_includedN_l|done]. split; [by rewrite Ha' left_id assoc; apply cmra_includedN_l|done]. Qed. Qed. Lemma auth_local_update L `{!LocalUpdate Lv L} a a' : Lemma auth_local_update L `{!LocalUpdate Lv L} a a' : ... @@ -170,7 +170,7 @@ Lemma auth_update_op_l a a' b : ... @@ -170,7 +170,7 @@ Lemma auth_update_op_l a a' b : Proof. by intros; apply (auth_local_update _). Qed. Proof. by intros; apply (auth_local_update _). Qed. Lemma auth_update_op_r a a' b : Lemma auth_update_op_r a a' b : ✓ (a ⋅ b) → ● a ⋅ ◯ a' ~~> ● (a ⋅ b) ⋅ ◯ (a' ⋅ b). ✓ (a ⋅ b) → ● a ⋅ ◯ a' ~~> ● (a ⋅ b) ⋅ ◯ (a' ⋅ b). Proof. rewrite -!(commutative _ b); apply auth_update_op_l. Qed. Proof. rewrite -!(comm _ b); apply auth_update_op_l. Qed. (* This does not seem to follow from auth_local_update. (* This does not seem to follow from auth_local_update. The trouble is that given ✓ (L a ⋅ a'), Lv a The trouble is that given ✓ (L a ⋅ a'), Lv a ... ...
 ... @@ -43,10 +43,10 @@ Record CMRAMixin A `{Dist A, Equiv A, Unit A, Op A, ValidN A, Minus A} := { ... @@ -43,10 +43,10 @@ Record CMRAMixin A `{Dist A, Equiv A, Unit A, Op A, ValidN A, Minus A} := { (* valid *) (* valid *) mixin_cmra_validN_S n x : ✓{S n} x → ✓{n} x; mixin_cmra_validN_S n x : ✓{S n} x → ✓{n} x; (* monoid *) (* monoid *) mixin_cmra_associative : Associative (≡) (⋅); mixin_cmra_assoc : Assoc (≡) (⋅); mixin_cmra_commutative : Commutative (≡) (⋅); mixin_cmra_comm : Comm (≡) (⋅); mixin_cmra_unit_l x : unit x ⋅ x ≡ x; mixin_cmra_unit_l x : unit x ⋅ x ≡ x; mixin_cmra_unit_idempotent x : unit (unit x) ≡ unit x; mixin_cmra_unit_idemp x : unit (unit x) ≡ unit x; mixin_cmra_unit_preservingN n x y : x ≼{n} y → unit x ≼{n} unit y; mixin_cmra_unit_preservingN n x y : x ≼{n} y → unit x ≼{n} unit y; mixin_cmra_validN_op_l n x y : ✓{n} (x ⋅ y) → ✓{n} x; mixin_cmra_validN_op_l n x y : ✓{n} (x ⋅ y) → ✓{n} x; mixin_cmra_op_minus n x y : x ≼{n} y → x ⋅ y ⩪ x ≡{n}≡ y mixin_cmra_op_minus n x y : x ≼{n} y → x ⋅ y ⩪ x ≡{n}≡ y ... @@ -101,14 +101,14 @@ Section cmra_mixin. ... @@ -101,14 +101,14 @@ Section cmra_mixin. Proof. apply (mixin_cmra_minus_ne _ (cmra_mixin A)). Qed. Proof. apply (mixin_cmra_minus_ne _ (cmra_mixin A)). Qed. Lemma cmra_validN_S n x : ✓{S n} x → ✓{n} x. Lemma cmra_validN_S n x : ✓{S n} x → ✓{n} x. Proof. apply (mixin_cmra_validN_S _ (cmra_mixin A)). Qed. Proof. apply (mixin_cmra_validN_S _ (cmra_mixin A)). Qed. Global Instance cmra_associative : Associative (≡) (@op A _). Global Instance cmra_assoc : Assoc (≡) (@op A _). Proof. apply (mixin_cmra_associative _ (cmra_mixin A)). Qed. Proof. apply (mixin_cmra_assoc _ (cmra_mixin A)). Qed. Global Instance cmra_commutative : Commutative (≡) (@op A _). Global Instance cmra_comm : Comm (≡) (@op A _). Proof. apply (mixin_cmra_commutative _ (cmra_mixin A)). Qed. Proof. apply (mixin_cmra_comm _ (cmra_mixin A)). Qed. Lemma cmra_unit_l x : unit x ⋅ x ≡ x. Lemma cmra_unit_l x : unit x ⋅ x ≡ x. Proof. apply (mixin_cmra_unit_l _ (cmra_mixin A)). Qed. Proof. apply (mixin_cmra_unit_l _ (cmra_mixin A)). Qed. Lemma cmra_unit_idempotent x : unit (unit x) ≡ unit x. Lemma cmra_unit_idemp x : unit (unit x) ≡ unit x. Proof. apply (mixin_cmra_unit_idempotent _ (cmra_mixin A)). Qed. Proof. apply (mixin_cmra_unit_idemp _ (cmra_mixin A)). Qed. Lemma cmra_unit_preservingN n x y : x ≼{n} y → unit x ≼{n} unit y. Lemma cmra_unit_preservingN n x y : x ≼{n} y → unit x ≼{n} unit y. Proof. apply (mixin_cmra_unit_preservingN _ (cmra_mixin A)). Qed. Proof. apply (mixin_cmra_unit_preservingN _ (cmra_mixin A)). Qed. Lemma cmra_validN_op_l n x y : ✓{n} (x ⋅ y) → ✓{n} x. Lemma cmra_validN_op_l n x y : ✓{n} (x ⋅ y) → ✓{n} x. ... @@ -166,7 +166,7 @@ Proof. apply (ne_proper _). Qed. ... @@ -166,7 +166,7 @@ Proof. apply (ne_proper _). Qed. Global Instance cmra_op_ne' n : Proper (dist n ==> dist n ==> dist n) (@op A _). Global Instance cmra_op_ne' n : Proper (dist n ==> dist n ==> dist n) (@op A _). Proof. Proof. intros x1 x2 Hx y1 y2 Hy. intros x1 x2 Hx y1 y2 Hy. by rewrite Hy (commutative _ x1) Hx (commutative _ y2). by rewrite Hy (comm _ x1) Hx (comm _ y2). Qed. Qed. Global Instance ra_op_proper' : Proper ((≡) ==> (≡) ==> (≡)) (@op A _). Global Instance ra_op_proper' : Proper ((≡) ==> (≡) ==> (≡)) (@op A _). Proof. apply (ne_proper_2 _). Qed. Proof. apply (ne_proper_2 _). Qed. ... @@ -217,15 +217,15 @@ Proof. induction 2; eauto using cmra_validN_S. Qed. ... @@ -217,15 +217,15 @@ Proof. induction 2; eauto using cmra_validN_S. Qed. Lemma cmra_valid_op_l x y : ✓ (x ⋅ y) → ✓ x. Lemma cmra_valid_op_l x y : ✓ (x ⋅ y) → ✓ x. Proof. rewrite !cmra_valid_validN; eauto using cmra_validN_op_l. Qed. Proof. rewrite !cmra_valid_validN; eauto using cmra_validN_op_l. Qed. Lemma cmra_validN_op_r x y n : ✓{n} (x ⋅ y) → ✓{n} y. Lemma cmra_validN_op_r x y n : ✓{n} (x ⋅ y) → ✓{n} y. Proof. rewrite (commutative _ x); apply cmra_validN_op_l. Qed. Proof. rewrite (comm _ x); apply cmra_validN_op_l. Qed. Lemma cmra_valid_op_r x y : ✓ (x ⋅ y) → ✓ y. Lemma cmra_valid_op_r x y : ✓ (x ⋅ y) → ✓ y. Proof. rewrite !cmra_valid_validN; eauto using cmra_validN_op_r. Qed. Proof. rewrite !cmra_valid_validN; eauto using cmra_validN_op_r. Qed. (** ** Units *) (** ** Units *) Lemma cmra_unit_r x : x ⋅ unit x ≡ x. Lemma cmra_unit_r x : x ⋅ unit x ≡ x. Proof. by rewrite (commutative _ x) cmra_unit_l. Qed. Proof. by rewrite (comm _ x) cmra_unit_l. Qed. Lemma cmra_unit_unit x : unit x ⋅ unit x ≡ unit x. Lemma cmra_unit_unit x : unit x ⋅ unit x ≡ unit x. Proof. by rewrite -{2}(cmra_unit_idempotent x) cmra_unit_r. Qed. Proof. by rewrite -{2}(cmra_unit_idemp x) cmra_unit_r. Qed. Lemma cmra_unit_validN x n : ✓{n} x → ✓{n} unit x. Lemma cmra_unit_validN x n : ✓{n} x → ✓{n} unit x. Proof. rewrite -{1}(cmra_unit_l x); apply cmra_validN_op_l. Qed. Proof. rewrite -{1}(cmra_unit_l x); apply cmra_validN_op_l. Qed. Lemma cmra_unit_valid x : ✓ x → ✓ unit x. Lemma cmra_unit_valid x : ✓ x → ✓ unit x. ... @@ -243,7 +243,7 @@ Proof. ... @@ -243,7 +243,7 @@ Proof. split. split. * by intros x; exists (unit x); rewrite cmra_unit_r. * by intros x; exists (unit x); rewrite cmra_unit_r. * intros x y z [z1 Hy] [z2 Hz]; exists (z1 ⋅ z2). * intros x y z [z1 Hy] [z2 Hz]; exists (z1 ⋅ z2). by rewrite (associative _) -Hy -Hz. by rewrite assoc -Hy -Hz. Qed. Qed. Global Instance cmra_included_preorder: PreOrder (@included A _ _). Global Instance cmra_included_preorder: PreOrder (@included A _ _). Proof. Proof. ... @@ -265,22 +265,22 @@ Proof. by exists y. Qed. ... @@ -265,22 +265,22 @@ Proof. by exists y. Qed. Lemma cmra_included_l x y : x ≼ x ⋅ y. Lemma cmra_included_l x y : x ≼ x ⋅ y. Proof. by exists y. Qed. Proof. by exists y. Qed. Lemma cmra_includedN_r n x y : y ≼{n} x ⋅ y. Lemma cmra_includedN_r n x y : y ≼{n} x ⋅ y. Proof. rewrite (commutative op); apply cmra_includedN_l. Qed. Proof. rewrite (comm op); apply cmra_includedN_l. Qed. Lemma cmra_included_r x y : y ≼ x ⋅ y. Lemma cmra_included_r x y : y ≼ x ⋅ y. Proof. rewrite (commutative op); apply cmra_included_l. Qed. Proof. rewrite (comm op); apply cmra_included_l. Qed. Lemma cmra_unit_preserving x y : x ≼ y → unit x ≼ unit y. Lemma cmra_unit_preserving x y : x ≼ y → unit x ≼ unit y. Proof. rewrite !cmra_included_includedN; eauto using cmra_unit_preservingN. Qed. Proof. rewrite !cmra_included_includedN; eauto using cmra_unit_preservingN. Qed. Lemma cmra_included_unit x : unit x ≼ x. Lemma cmra_included_unit x : unit x ≼ x. Proof. by exists x; rewrite cmra_unit_l. Qed. Proof. by exists x; rewrite cmra_unit_l. Qed. Lemma cmra_preservingN_l n x y z : x ≼{n} y → z ⋅ x ≼{n} z ⋅ y. Lemma cmra_preservingN_l n x y z : x ≼{n} y → z ⋅ x ≼{n} z ⋅ y. Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1 (associative op). Qed. Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1 (assoc op). Qed. Lemma cmra_preserving_l x y z : x ≼ y → z ⋅ x ≼ z ⋅ y. Lemma cmra_preserving_l x y z : x ≼ y → z ⋅ x ≼ z ⋅ y. Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1 (associative op). Qed. Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1 (assoc op). Qed. Lemma cmra_preservingN_r n x y z : x ≼{n} y → x ⋅ z ≼{n} y ⋅ z. Lemma cmra_preservingN_r n x y z : x ≼{n} y → x ⋅ z ≼{n} y ⋅ z. Proof. by intros; rewrite -!(commutative _ z); apply cmra_preservingN_l. Qed. Proof. by intros; rewrite -!(comm _ z); apply cmra_preservingN_l. Qed. Lemma cmra_preserving_r x y z : x ≼ y → x ⋅ z ≼ y ⋅ z. Lemma cmra_preserving_r x y z : x ≼ y → x ⋅ z ≼ y ⋅ z. Proof. by intros; rewrite -!(commutative _ z); apply cmra_preserving_l. Qed. Proof. by intros; rewrite -!(comm _ z); apply cmra_preserving_l. Qed. Lemma cmra_included_dist_l x1 x2 x1' n : Lemma cmra_included_dist_l x1 x2 x1' n : x1 ≼ x2 → x1' ≡{n}≡ x1 → ∃ x2', x1' ≼ x2' ∧ x2' ≡{n}≡ x2. x1 ≼ x2 → x1' ≡{n}≡ x1 → ∃ x2', x1' ≼ x2' ∧ x2' ≡{n}≡ x2. ... @@ -321,7 +321,7 @@ Section identity. ... @@ -321,7 +321,7 @@ Section identity. Lemma cmra_empty_least x : ∅ ≼ x. Lemma cmra_empty_least x : ∅ ≼ x. Proof. by exists x; rewrite left_id. Qed. Proof. by exists x; rewrite left_id. Qed. Global Instance cmra_empty_right_id : RightId (≡) ∅ (⋅). Global Instance cmra_empty_right_id : RightId (≡) ∅ (⋅). Proof. by intros x; rewrite (commutative op) left_id. Qed. Proof. by intros x; rewrite (comm op) left_id. Qed. Lemma cmra_unit_empty : unit ∅ ≡ ∅. Lemma cmra_unit_empty : unit ∅ ≡ ∅. Proof. by rewrite -{2}(cmra_unit_l ∅) right_id. Qed. Proof. by rewrite -{2}(cmra_unit_l ∅) right_id. Qed. End identity. End identity. ... @@ -336,7 +336,7 @@ Lemma local_update L `{!LocalUpdate Lv L} x y : ... @@ -336,7 +336,7 @@ Lemma local_update L `{!LocalUpdate Lv L} x y : Proof. by rewrite equiv_dist=>?? n; apply (local_updateN L). Qed. Proof. by rewrite equiv_dist=>?? n; apply (local_updateN L). Qed. Global Instance local_update_op x : LocalUpdate (λ _, True) (op x). Global Instance local_update_op x : LocalUpdate (λ _, True) (op x). Proof. split. apply _. by intros n y1 y2 _ _; rewrite associative. Qed. Proof. split. apply _. by intros n y1 y2 _ _; rewrite assoc. Qed. (** ** Updates *) (** ** Updates *) Global Instance cmra_update_preorder : PreOrder (@cmra_update A). Global Instance cmra_update_preorder : PreOrder (@cmra_update A). ... @@ -366,10 +366,10 @@ Lemma cmra_updateP_op (P1 P2 Q : A → Prop) x1 x2 : ... @@ -366,10 +366,10 @@ Lemma cmra_updateP_op (P1 P2 Q : A → Prop) x1 x2 : x1 ~~>: P1 → x2 ~~>: P2 → (∀ y1 y2, P1 y1 → P2 y2 → Q (y1 ⋅ y2)) → x1 ⋅ x2 ~~>: Q. x1 ~~>: P1 → x2 ~~>: P2 → (∀ y1 y2, P1 y1 → P2 y2 → Q (y1 ⋅ y2)) → x1 ⋅ x2 ~~>: Q. Proof. Proof. intros Hx1 Hx2 Hy z n ?. intros Hx1 Hx2 Hy z n ?. destruct (Hx1 (x2 ⋅ z) n) as (y1&?&?); first by rewrite associative. destruct (Hx1 (x2 ⋅ z) n) as (y1&?&?); first by rewrite assoc. destruct (Hx2 (y1 ⋅ z) n) as (y2&?&?); destruct (Hx2 (y1 ⋅ z) n) as (y2&?&?); first by rewrite associative (commutative _ x2) -associative. first by rewrite assoc (comm _ x2) -assoc. exists (y1 ⋅ y2); split; last rewrite (commutative _ y1) -associative; auto. exists (y1 ⋅ y2); split; last rewrite (comm _ y1) -assoc; auto. Qed. Qed. Lemma cmra_updateP_op' (P1 P2 : A → Prop) x1 x2 : Lemma cmra_updateP_op' (P1 P2 : A → Prop) x1 x2 : x1 ~~>: P1 → x2 ~~>: P2 → x1 ⋅ x2 ~~>: λ y, ∃ y1 y2, y = y1 ⋅ y2 ∧ P1 y1 ∧ P2 y2. x1 ~~>: P1 → x2 ~~>: P2 → x1 ⋅ x2 ~~>: λ y, ∃ y1 y2, y = y1 ⋅ y2 ∧ P1 y1 ∧ P2 y2. ... @@ -448,10 +448,10 @@ Class RA A `{Equiv A, Unit A, Op A, Valid A, Minus A} := { ... @@ -448,10 +448,10 @@ Class RA A `{Equiv A, Unit A, Op A, Valid A, Minus A} := { ra_validN_ne :> Proper ((≡) ==> impl) valid; ra_validN_ne :> Proper ((≡) ==> impl) valid; ra_minus_ne :> Proper ((≡) ==> (≡) ==> (≡)) minus; ra_minus_ne :> Proper ((≡) ==> (≡) ==> (≡)) minus; (* monoid *) (* monoid *) ra_associative :> Associative (≡) (⋅); ra_assoc :> Assoc (≡) (⋅); ra_commutative :> Commutative (≡) (⋅); ra_comm :> Comm (≡) (⋅); ra_unit_l x : unit x ⋅ x ≡ x; ra_unit_l x : unit x ⋅ x ≡ x; ra_unit_idempotent x : unit (unit x) ≡ unit x; ra_unit_idemp x : unit (unit x) ≡ unit x; ra_unit_preserving x y : x ≼ y → unit x ≼ unit y; ra_unit_preserving x y : x ≼ y → unit x ≼ unit y; ra_valid_op_l x y : ✓ (x ⋅ y) → ✓ x; ra_valid_op_l x y : ✓ (x ⋅ y) → ✓ x; ra_op_minus x y : x ≼ y → x ⋅ y ⩪ x ≡ y ra_op_minus x y : x ≼ y → x ⋅ y ⩪ x ≡ y ... @@ -524,10 +524,10 @@ Section prod. ... @@ -524,10 +524,10 @@ Section prod. * by intros n x1 x2 [Hx1 Hx2] y1 y2 [Hy1 Hy2]; * by intros n x1 x2 [Hx1 Hx2] y1 y2 [Hy1 Hy2]; split; rewrite /= ?Hx1 ?Hx2 ?Hy1 ?Hy2. split; rewrite /= ?Hx1 ?Hx2 ?Hy1 ?Hy2. * by intros n x [??]; split; apply cmra_validN_S. * by intros n x [??]; split; apply cmra_validN_S. * split; simpl; apply (associative _). * by split; rewrite /= assoc. * split; simpl; apply (commutative _). * by split; rewrite /= comm. * split; simpl; apply cmra_unit_l. * by split; rewrite /= cmra_unit_l. * split; simpl; apply cmra_unit_idempotent. * by split; rewrite /= cmra_unit_idemp. * intros n x y; rewrite !prod_includedN. * intros n x y; rewrite !prod_includedN. by intros [??]; split; apply cmra_unit_preservingN. by intros [??]; split; apply cmra_unit_preservingN. * intros n x y [??]; split; simpl in *; eauto using cmra_validN_op_l. * intros n x y [??]; split; simpl in *; eauto using cmra_validN_op_l. ... ...
 ... @@ -22,21 +22,21 @@ Global Instance big_op_permutation : Proper ((≡ₚ) ==> (≡)) big_op. ... @@ -22,21 +22,21 @@ Global Instance big_op_permutation : Proper ((≡ₚ) ==> (≡)) big_op. Proof. Proof. induction 1 as [|x xs1 xs2 ? IH|x y xs|xs1 xs2 xs3]; simpl; auto. induction 1 as [|x xs1 xs2 ? IH|x y xs|xs1 xs2 xs3]; simpl; auto. * by rewrite IH. * by rewrite IH. * by rewrite !(associative _) (commutative _ x). * by rewrite !assoc (comm _ x). * by transitivity (big_op xs2). * by transitivity (big_op xs2). Qed. Qed. Global Instance big_op_proper : Proper ((≡) ==> (≡)) big_op. Global Instance big_op_proper : Proper ((≡) ==> (≡)) big_op. Proof. by induction 1; simpl; repeat apply (_ : Proper (_ ==> _ ==> _) op). Qed. Proof. by induction 1; simpl; repeat apply (_ : Proper (_ ==> _ ==> _) op). Qed. Lemma big_op_app xs ys : big_op (xs ++ ys) ≡ big_op xs ⋅ big_op ys. Lemma big_op_app xs ys : big_op (xs ++ ys) ≡ big_op xs ⋅ big_op ys. Proof. Proof. induction xs as [|x xs IH]; simpl; first by rewrite ?(left_id _ _). induction xs as [|x xs IH]; simpl; first by rewrite ?left_id. by rewrite IH (associative _). by rewrite IH assoc. Qed. Qed. Lemma big_op_contains xs ys : xs `contains` ys → big_op xs ≼ big_op ys. Lemma big_op_contains xs ys : xs `contains` ys → big_op xs ≼ big_op ys. Proof. Proof. induction 1 as [|x xs ys|x y xs|x xs ys|xs ys zs]; rewrite //=. induction 1 as [|x xs ys|x y xs|x xs ys|xs ys zs]; rewrite //=. * by apply cmra_preserving_l. * by apply cmra_preserving_l. * by rewrite !associative (commutative _ y). * by rewrite !assoc (comm _ y). * by transitivity (big_op ys); last apply cmra_included_r. * by transitivity (big_op ys); last apply cmra_included_r. * by transitivity (big_op ys). * by transitivity (big_op ys). Qed. Qed. ... @@ -58,7 +58,7 @@ Qed. ... @@ -58,7 +58,7 @@ Qed. Lemma big_opM_singleton i x : big_opM ({[i ↦ x]} : M A) ≡ x. Lemma big_opM_singleton i x : big_opM ({[i ↦ x]} : M A) ≡ x. Proof. Proof. rewrite -insert_empty big_opM_insert /=; last auto using lookup_empty. rewrite -insert_empty big_opM_insert /=; last auto using lookup_empty. by rewrite big_opM_empty (right_id _ _). by rewrite big_opM_empty right_id. Qed. Qed. Global Instance big_opM_proper : Proper ((≡) ==> (≡)) (big_opM : M A → _). Global Instance big_opM_proper : Proper ((≡) ==> (≡)) (big_opM : M A → _). Proof. Proof. ... ...
 ... @@ -25,7 +25,7 @@ Module ra_reflection. Section ra_reflection. ... @@ -25,7 +25,7 @@ Module ra_reflection. Section ra_reflection. eval Σ e ≡ big_op ((λ n, from_option ∅ (Σ !! n)) <\$> flatten e). eval Σ e ≡ big_op ((λ n, from_option ∅ (Σ !! n)) <\$> flatten e). Proof. Proof. by induction e as [| |e1 IH1 e2 IH2]; by induction e as [| |e1 IH1 e2 IH2]; rewrite /= ?(right_id _ _) ?fmap_app ?big_op_app ?IH1 ?IH2. rewrite /= ?right_id ?fmap_app ?big_op_app ?IH1 ?IH2. Qed. Qed. Lemma flatten_correct Σ e1 e2 : Lemma flatten_correct Σ e1 e2 : flatten e1 `contains` flatten e2 → eval Σ e1 ≼ eval Σ e2. flatten e1 `contains` flatten e2 → eval Σ e1 ≼ eval Σ e2. ... ...
 ... @@ -337,7 +337,7 @@ Section later. ... @@ -337,7 +337,7 @@ Section later. Canonical Structure laterC : cofeT := CofeT later_cofe_mixin. Canonical Structure laterC : cofeT := CofeT later_cofe_mixin. Global Instance Next_contractive : Contractive (@Next A). Global Instance Next_contractive : Contractive (@Next A). Proof. intros [|n] x y Hxy; [done|]; apply Hxy; lia. Qed. Proof. intros [|n] x y Hxy; [done|]; apply Hxy; lia. Qed. Global Instance Later_inj n : Injective (dist n) (dist (S n)) (@Next A). Global Instance Later_inj n : Inj (dist n) (dist (S n)) (@Next A). Proof. by intros x y. Qed. Proof. by intros x y. Qed. End later. End later. ... ...
 ... @@ -46,14 +46,14 @@ Class DRA A `{Equiv A, Valid A, Unit A, Disjoint A, Op A, Minus A} := { ... @@ -46,14 +46,14 @@ Class DRA A `{Equiv A, Valid A, Unit A, Disjoint A, Op A, Minus A} := { dra_unit_valid x : ✓ x → ✓ unit x; dra_unit_valid x : ✓ x → ✓ unit x; dra_minus_valid x y : ✓ x → ✓ y → x ≼ y → ✓ (y ⩪ x); dra_minus_valid x y : ✓ x → ✓ y → x ≼ y → ✓ (y ⩪ x); (* monoid *) (* monoid *) dra_associative :> Associative (≡) (⋅); dra_assoc :> Assoc (≡) (⋅); dra_disjoint_ll x y z : ✓ x → ✓ y → ✓ z → x ⊥ y → x ⋅ y ⊥ z → x ⊥ z; dra_disjoint_ll x y z : ✓ x → ✓ y → ✓ z → x ⊥ y → x ⋅ y ⊥ z → x ⊥ z; dra_disjoint_move_l x y z : ✓ x → ✓ y → ✓ z → x ⊥ y → x ⋅ y ⊥ z → x ⊥ y ⋅ z; dra_disjoint_move_l x y z : ✓ x → ✓ y → ✓ z → x ⊥ y → x ⋅ y ⊥ z → x ⊥ y ⋅ z; dra_symmetric :> Symmetric (@disjoint A _); dra_symmetric :> Symmetric (@disjoint A _); dra_commutative x y : ✓ x → ✓ y → x ⊥ y → x ⋅ y ≡ y ⋅ x; dra_comm x y : ✓ x → ✓ y → x ⊥ y → x ⋅ y ≡ y ⋅ x; dra_unit_disjoint_l x : ✓ x → unit x ⊥ x; dra_unit_disjoint_l x : ✓ x → unit x ⊥ x; dra_unit_l x : ✓ x → unit x ⋅ x ≡ x; dra_unit_l x : ✓ x → unit x ⋅ x ≡ x; dra_unit_idempotent x : ✓ x → unit (unit x) ≡ unit x; dra_unit_idemp x : ✓ x → unit (unit x) ≡ unit x; dra_unit_preserving x y : ✓ x → ✓ y → x ≼ y → unit x ≼ unit y; dra_unit_preserving x y : ✓ x → ✓ y → x ≼ y → unit x ≼ unit y; dra_disjoint_minus x y : ✓ x → ✓ y → x ≼ y → x ⊥ y ⩪ x; dra_disjoint_minus x y : ✓ x → ✓ y → x ≼ y → x ⊥ y ⩪ x; dra_op_minus x y : ✓ x → ✓ y → x ≼ y → x ⋅ y ⩪ x ≡ y dra_op_minus x y : ✓ x → ✓ y → x ≼ y → x ⋅ y ⩪ x ≡ y ... @@ -73,12 +73,12 @@ Qed. ... @@ -73,12 +73,12 @@ Qed. Lemma dra_disjoint_rl x y z : ✓ x → ✓ y → ✓ z → y ⊥ z → x ⊥ y ⋅ z → x ⊥ y. Lemma dra_disjoint_rl x y z : ✓ x → ✓ y → ✓ z → y ⊥ z → x ⊥ y ⋅ z → x ⊥ y. Proof. intros ???. rewrite !(symmetry_iff _ x). by apply dra_disjoint_ll. Qed. Proof. intros ???. rewrite !(symmetry_iff _ x). by apply dra_disjoint_ll. Qed. Lemma dra_disjoint_lr x y z : ✓ x → ✓ y → ✓ z → x ⊥ y → x ⋅ y ⊥ z → y ⊥ z. Lemma dra_disjoint_lr x y z : ✓ x → ✓ y → ✓ z → x ⊥ y → x ⋅ y ⊥ z → y ⊥ z. Proof. intros ????. rewrite dra_commutative //. by apply dra_disjoint_ll. Qed. Proof. intros ????. rewrite dra_comm //. by apply dra_disjoint_ll. Qed. Lemma dra_disjoint_move_r x y z : Lemma dra_disjoint_move_r x y z : ✓ x → ✓ y → ✓ z → y ⊥ z → x ⊥ y ⋅ z → x ⋅ y ⊥ z. ✓ x → ✓ y → ✓ z → y ⊥ z → x ⊥ y ⋅ z → x ⋅ y ⊥ z. Proof. Proof. intros; symmetry; rewrite dra_commutative; eauto using dra_disjoint_rl. intros; symmetry; rewrite dra_comm; eauto using dra_disjoint_rl. apply dra_disjoint_move_l; auto; by rewrite dra_commutative. apply dra_disjoint_move_l; auto; by rewrite dra_comm. Qed. Qed. Hint Immediate dra_disjoint_move_l dra_disjoint_move_r. Hint Immediate dra_disjoint_move_l dra_disjoint_move_r. Hint Unfold dra_included. Hint Unfold dra_included. ... @@ -114,11 +114,11 @@ Proof. ... @@ -114,11 +114,11 @@ Proof. + exists z. by rewrite Hx ?Hy; tauto. + exists z. by rewrite Hx ?Hy; tauto. * intros [x px ?] [y py ?] [z pz ?]; split; simpl; * intros [x px ?] [y py ?] [z pz ?]; split; simpl; [intuition eauto 2 using dra_disjoint_lr, dra_disjoint_rl [intuition eauto 2 using dra_disjoint_lr, dra_disjoint_rl |intros; apply (associative _)]. |by intros; rewrite assoc]. * intros [x px ?] [y py ?]; split; naive_solver eauto using dra_commutative. * intros [x px ?] [y py ?]; split; naive_solver eauto using dra_comm. * intros [x px ?]; split; * intros [x px ?]; split; naive_solver eauto using dra_unit_l, dra_unit_disjoint_l. naive_solver eauto using dra_unit_l, dra_unit_disjoint_l. * intros [x px ?]; split; naive_solver eauto using dra_unit_idempotent. * intros [x px ?]; split; naive_solver eauto using dra_unit_idemp. * intros x y Hxy; exists (unit y ⩪ unit x). * intros x y Hxy; exists (unit y ⩪ unit x). destruct x as [x px ?], y as [y py ?], Hxy as [[z pz ?] [??]]; simpl in *. destruct x as [x px ?], y as [y py ?], Hxy as [[z pz ?] [??]]; simpl in *. assert (py → unit x ≼ unit y) assert (py → unit x ≼ unit y) ... ...
 ... @@ -31,9 +31,9 @@ Global Instance Excl_ne : Proper (dist n ==> dist n) (@Excl A). ... @@ -31,9 +31,9 @@ Global Instance Excl_ne : Proper (dist n ==> dist n) (@Excl A). Proof. by constructor. Qed. Proof. by constructor. Qed. Global Instance Excl_proper : Proper ((≡) ==> (≡)) (@Excl A). Global Instance Excl_proper : Proper ((≡) ==> (≡)) (@Excl A). Proof. by constructor. Qed. Proof. by constructor. Qed. Global Instance Excl_inj : Injective (≡) (≡) (@Excl A). Global Instance Excl_inj : Inj (≡) (≡) (@Excl A). Proof. by inversion_clear 1. Qed. Proof. by inversion_clear 1. Qed. Global Instance Excl_dist_inj n : Injective (dist n) (dist n) (@Excl A). Global Instance Excl_dist_inj n : Inj (dist n) (dist n) (@Excl A). Proof. by inversion_clear 1. Qed. Proof. by inversion_clear 1. Qed. Program Definition excl_chain Program Definition excl_chain (c : chain (excl A)) (x : A) (H : maybe Excl (c 1) = Some x) : chain A := (c : chain (excl A)) (x : A) (H : maybe Excl (c 1) = Some x) : chain A := ... ...
 ... @@ -121,10 +121,10 @@ Proof. ... @@ -121,10 +121,10 @@ Proof. * by intros n m1 m2 Hm ? i; rewrite -(Hm i). * by intros n m1 m2 Hm ? i; rewrite -(Hm i). * by intros n m1 m1' Hm1 m2 m2' Hm2 i; rewrite !lookup_minus (Hm1 i) (Hm2 i). * by intros n m1 m1' Hm1 m2 m2' Hm2 i; rewrite !lookup_minus (Hm1 i) (Hm2 i). * intros n m Hm i; apply cmra_validN_S, Hm. * intros n m Hm i; apply cmra_validN_S, Hm. * by intros m1 m2 m3 i; rewrite !lookup_op associative. * by intros m1 m2 m3 i; rewrite !lookup_op assoc. * by intros m1 m2 i; rewrite !lookup_op commutative.