Commit f6909092 by Ralf Jung

change notation of step-indexed equality to ≡{n}≡

parent fbedbd17
 ... ... @@ -16,16 +16,16 @@ 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. 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). (∀ 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'). (∀ 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 n n; agree_is_valid n := agree_is_valid (c n) n |}. Next Obligation. intros; apply agree_valid_0. Qed. ... ... @@ -51,14 +51,14 @@ 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. 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. 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 |}. agree_is_valid n := agree_is_valid x n ∧ agree_is_valid y n ∧ x ≡{n}≡ y |}. Next Obligation. by intros; simpl; split_ands; try apply agree_valid_0. Qed. Next Obligation. naive_solver eauto using agree_valid_S, dist_S. Qed. Instance agree_unit : Unit (agree A) := id. ... ... @@ -91,7 +91,7 @@ Proof. repeat match goal with H : agree_is_valid _ _ |- _ => clear H end; by cofe_subst; rewrite !agree_idempotent. 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. split; [|by intros ?; exists y]. by intros [z Hz]; rewrite Hz (associative _) agree_idempotent. ... ... @@ -109,9 +109,9 @@ Proof. * 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. 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. 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. ... ... @@ -133,7 +133,7 @@ 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 : Injective (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. 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. End agree. ... ...
 ... ... @@ -19,7 +19,7 @@ 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. 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. ... ... @@ -148,7 +148,7 @@ Lemma auth_frag_op a b : ◯ (a ⋅ b) ≡ ◯ a ⋅ ◯ b. Proof. done. Qed. Lemma auth_update a a' b b' : (∀ n af, ✓{S n} a → a ={S n}= a' ⋅ af → b ={S n}= b' ⋅ af ∧ ✓{S n} b) → (∀ n af, ✓{S n} a → a ≡{S n}≡ a' ⋅ af → b ≡{S n}≡ b' ⋅ af ∧ ✓{S n} b) → ● a ⋅ ◯ a' ~~> ● b ⋅ ◯ b'. Proof. move=> Hab [[?| |] bf1] n // =>-[[bf2 Ha] ?]; do 2 red; simpl in *. ... ...
 ... ... @@ -27,7 +27,7 @@ Instance: Params (@valid) 2. Notation "✓" := valid (at level 1). Instance validN_valid `{ValidN A} : Valid A := λ x, ∀ n, ✓{n} x. Definition includedN `{Dist A, Op A} (n : nat) (x y : A) := ∃ z, y ={n}= x ⋅ z. Definition includedN `{Dist A, Op A} (n : nat) (x y : A) := ∃ z, y ≡{n}≡ x ⋅ z. Notation "x ≼{ n } y" := (includedN n x y) (at level 70, format "x ≼{ n } y") : C_scope. Instance: Params (@includedN) 4. ... ... @@ -49,11 +49,11 @@ Record CMRAMixin A `{Dist A, Equiv A, Unit A, Op A, ValidN A, Minus A} := { mixin_cmra_unit_idempotent x : unit (unit x) ≡ unit x; 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_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 }. Definition CMRAExtendMixin A `{Equiv A, Dist A, Op A, ValidN A} := ∀ n x y1 y2, ✓{n} x → x ={n}= y1 ⋅ y2 → { z | x ≡ z.1 ⋅ z.2 ∧ z.1 ={n}= y1 ∧ z.2 ={n}= y2 }. ✓{n} x → x ≡{n}≡ y1 ⋅ y2 → { z | x ≡ z.1 ⋅ z.2 ∧ z.1 ≡{n}≡ y1 ∧ z.2 ≡{n}≡ y2 }. (** Bundeled version *) Structure cmraT := CMRAT { ... ... @@ -115,11 +115,11 @@ Section cmra_mixin. Proof. apply (mixin_cmra_unit_preservingN _ (cmra_mixin A)). Qed. Lemma cmra_validN_op_l n x y : ✓{n} (x ⋅ y) → ✓{n} x. Proof. apply (mixin_cmra_validN_op_l _ (cmra_mixin A)). Qed. Lemma cmra_op_minus n x y : x ≼{n} y → x ⋅ y ⩪ x ={n}= y. Lemma cmra_op_minus n x y : x ≼{n} y → x ⋅ y ⩪ x ≡{n}≡ y. Proof. apply (mixin_cmra_op_minus _ (cmra_mixin A)). Qed. Lemma cmra_extend_op n x y1 y2 : ✓{n} x → x ={n}= y1 ⋅ y2 → { z | x ≡ z.1 ⋅ z.2 ∧ z.1 ={n}= y1 ∧ z.2 ={n}= y2 }. ✓{n} x → x ≡{n}≡ y1 ⋅ y2 → { z | x ≡ z.1 ⋅ z.2 ∧ z.1 ≡{n}≡ y1 ∧ z.2 ≡{n}≡ y2 }. Proof. apply (cmra_extend_mixin A). Qed. End cmra_mixin. ... ... @@ -277,7 +277,7 @@ Lemma cmra_preserving_r x y z : x ≼ y → x ⋅ z ≼ y ⋅ z. Proof. by intros; rewrite -!(commutative _ z); apply cmra_preserving_l. Qed. 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. Proof. intros [z Hx2] Hx1; exists (x1' ⋅ z); split; auto using cmra_included_l. by rewrite Hx1 Hx2. ... ...
 ... ... @@ -3,10 +3,10 @@ Require Export algebra.base. (** Unbundeled version *) Class Dist A := dist : nat → relation A. Instance: Params (@dist) 3. Notation "x ={ n }= y" := (dist n x y) (at level 70, n at next level, format "x ={ n }= y"). Hint Extern 0 (?x ={_}= ?y) => reflexivity. Hint Extern 0 (_ ={_}= _) => symmetry; assumption. Notation "x ≡{ n }≡ y" := (dist n x y) (at level 70, n at next level, format "x ≡{ n }≡ y"). Hint Extern 0 (?x ≡{_}≡ ?y) => reflexivity. Hint Extern 0 (_ ≡{_}≡ _) => symmetry; assumption. Tactic Notation "cofe_subst" ident(x) := repeat match goal with ... ... @@ -23,18 +23,18 @@ Tactic Notation "cofe_subst" := Record chain (A : Type) `{Dist A} := { chain_car :> nat → A; chain_cauchy n i : n ≤ i → chain_car n ={n}= chain_car i chain_cauchy n i : n ≤ i → chain_car n ≡{n}≡ chain_car i }. Arguments chain_car {_ _} _ _. Arguments chain_cauchy {_ _} _ _ _ _. Class Compl A `{Dist A} := compl : chain A → A. Record CofeMixin A `{Equiv A, Compl A} := { mixin_equiv_dist x y : x ≡ y ↔ ∀ n, x ={n}= y; mixin_equiv_dist x y : x ≡ y ↔ ∀ n, x ≡{n}≡ y; mixin_dist_equivalence n : Equivalence (dist n); mixin_dist_S n x y : x ={S n}= y → x ={n}= y; mixin_dist_0 x y : x ={0}= y; mixin_conv_compl (c : chain A) n : compl c ={n}= c n mixin_dist_S n x y : x ≡{S n}≡ y → x ≡{n}≡ y; mixin_dist_0 x y : x ≡{0}≡ y; mixin_conv_compl (c : chain A) n : compl c ≡{n}≡ c n }. Class Contractive `{Dist A, Dist B} (f : A -> B) := contractive n : Proper (dist n ==> dist (S n)) f. ... ... @@ -60,19 +60,19 @@ Arguments cofe_mixin : simpl never. Section cofe_mixin. Context {A : cofeT}. Implicit Types x y : A. Lemma equiv_dist x y : x ≡ y ↔ ∀ n, x ={n}= y. Lemma equiv_dist x y : x ≡ y ↔ ∀ n, x ≡{n}≡ y. Proof. apply (mixin_equiv_dist _ (cofe_mixin A)). Qed. Global Instance dist_equivalence n : Equivalence (@dist A _ n). Proof. apply (mixin_dist_equivalence _ (cofe_mixin A)). Qed. Lemma dist_S n x y : x ={S n}= y → x ={n}= y. Lemma dist_S n x y : x ≡{S n}≡ y → x ≡{n}≡ y. Proof. apply (mixin_dist_S _ (cofe_mixin A)). Qed. Lemma dist_0 x y : x ={0}= y. Lemma dist_0 x y : x ≡{0}≡ y. Proof. apply (mixin_dist_0 _ (cofe_mixin A)). Qed. Lemma conv_compl (c : chain A) n : compl c ={n}= c n. Lemma conv_compl (c : chain A) n : compl c ≡{n}≡ c n. Proof. apply (mixin_conv_compl _ (cofe_mixin A)). Qed. End cofe_mixin. Hint Extern 0 (_ ={0}= _) => apply dist_0. Hint Extern 0 (_ ≡{0}≡ _) => apply dist_0. (** General properties *) Section cofe. ... ... @@ -97,7 +97,7 @@ Section cofe. Qed. Global Instance dist_proper_2 n x : Proper ((≡) ==> iff) (dist n x). Proof. by apply dist_proper. Qed. Lemma dist_le (x y : A) n n' : x ={n}= y → n' ≤ n → x ={n'}= y. Lemma dist_le (x y : A) n n' : x ≡{n}≡ y → n' ≤ n → x ≡{n'}≡ y. Proof. induction 2; eauto using dist_S. Qed. Instance ne_proper {B : cofeT} (f : A → B) `{!∀ n, Proper (dist n ==> dist n) f} : Proper ((≡) ==> (≡)) f | 100. ... ... @@ -109,7 +109,7 @@ Section cofe. unfold Proper, respectful; setoid_rewrite equiv_dist. by intros x1 x2 Hx y1 y2 Hy n; rewrite (Hx n) (Hy n). Qed. Lemma compl_ne (c1 c2: chain A) n : c1 n ={n}= c2 n → compl c1 ={n}= compl c2. Lemma compl_ne (c1 c2: chain A) n : c1 n ≡{n}≡ c2 n → compl c1 ≡{n}≡ compl c2. Proof. intros. by rewrite (conv_compl c1 n) (conv_compl c2 n). Qed. Lemma compl_ext (c1 c2 : chain A) : (∀ i, c1 i ≡ c2 i) → compl c1 ≡ compl c2. Proof. setoid_rewrite equiv_dist; naive_solver eauto using compl_ne. Qed. ... ... @@ -127,9 +127,9 @@ Program Definition chain_map `{Dist A, Dist B} (f : A → B) Next Obligation. by intros ? A ? B f Hf c n i ?; apply Hf, chain_cauchy. Qed. (** Timeless elements *) Class Timeless {A : cofeT} (x : A) := timeless y : x ={1}= y → x ≡ y. Class Timeless {A : cofeT} (x : A) := timeless y : x ≡{1}≡ y → x ≡ y. Arguments timeless {_} _ {_} _ _. Lemma timeless_S {A : cofeT} (x y : A) n : Timeless x → x ≡ y ↔ x ={S n}= y. Lemma timeless_S {A : cofeT} (x y : A) n : Timeless x → x ≡ y ↔ x ≡{S n}≡ y. Proof. split; intros; [by apply equiv_dist|]. apply (timeless _), dist_le with (S n); auto with lia. ... ... @@ -154,7 +154,7 @@ Section fixpoint. by rewrite {1}(chain_cauchy (fixpoint_chain f) n (S n)); last lia. Qed. Lemma fixpoint_ne (g : A → A) `{!Contractive g} n : (∀ z, f z ={n}= g z) → fixpoint f ={n}= fixpoint g. (∀ z, f z ≡{n}≡ g z) → fixpoint f ≡{n}≡ fixpoint g. Proof. intros Hfg; unfold fixpoint. rewrite (conv_compl (fixpoint_chain f) n) (conv_compl (fixpoint_chain g) n). ... ... @@ -181,7 +181,7 @@ Section cofe_mor. Global Instance cofe_mor_proper (f : cofeMor A B) : Proper ((≡) ==> (≡)) f. Proof. apply ne_proper, cofe_mor_ne. Qed. Instance cofe_mor_equiv : Equiv (cofeMor A B) := λ f g, ∀ x, f x ≡ g x. Instance cofe_mor_dist : Dist (cofeMor A B) := λ n f g, ∀ x, f x ={n}= g x. Instance cofe_mor_dist : Dist (cofeMor A B) := λ n f g, ∀ x, f x ≡{n}≡ g x. Program Definition fun_chain `(c : chain (cofeMor A B)) (x : A) : chain B := {| chain_car n := c n x |}. Next Obligation. intros c x n i ?. by apply (chain_cauchy c). Qed. ... ... @@ -230,7 +230,7 @@ Definition ccompose {A B C} Instance: Params (@ccompose) 3. Infix "◎" := ccompose (at level 40, left associativity). Lemma ccompose_ne {A B C} (f1 f2 : B -n> C) (g1 g2 : A -n> B) n : f1 ={n}= f2 → g1 ={n}= g2 → f1 ◎ g1 ={n}= f2 ◎ g2. f1 ≡{n}≡ f2 → g1 ≡{n}≡ g2 → f1 ◎ g1 ≡{n}≡ f2 ◎ g2. Proof. by intros Hf Hg x; rewrite /= (Hg x) (Hf (g2 x)). Qed. (** unit *) ... ... @@ -325,7 +325,7 @@ Section later. Context {A : cofeT}. Instance later_equiv : Equiv (later A) := λ x y, later_car x ≡ later_car y. Instance later_dist : Dist (later A) := λ n x y, match n with 0 => True | S n => later_car x ={n}= later_car y end. match n with 0 => True | S n => later_car x ≡{n}≡ later_car y end. Program Definition later_chain (c : chain (later A)) : chain A := {| chain_car n := later_car (c (S n)) |}. Next Obligation. intros c n i ?; apply (chain_cauchy c (S n)); lia. Qed. ... ...
 ... ... @@ -42,7 +42,7 @@ Proof. induction k as [|k IH]; simpl in *; [by destruct x|]. rewrite -map_comp -{2}(map_id _ _ x); by apply map_ext. Qed. Lemma fg {n k} (x : A (S k)) : n ≤ k → f (g x) ={n}= x. Lemma fg {n k} (x : A (S k)) : n ≤ k → f (g x) ≡{n}≡ x. Proof. intros Hnk; apply dist_le with k; auto; clear Hnk. induction k as [|k IH]; simpl; [apply dist_0|]. ... ... @@ -57,7 +57,7 @@ Record tower := { g_tower k : g (tower_car (S k)) ≡ tower_car k }. Instance tower_equiv : Equiv tower := λ X Y, ∀ k, X k ≡ Y k. Instance tower_dist : Dist tower := λ n X Y, ∀ k, X k ={n}= Y k. Instance tower_dist : Dist tower := λ n X Y, ∀ k, X k ≡{n}≡ Y k. Program Definition tower_chain (c : chain tower) (k : nat) : chain (A k) := {| chain_car i := c i k |}. Next Obligation. intros c k n i ?; apply (chain_cauchy c n); lia. Qed. ... ... @@ -91,9 +91,9 @@ Fixpoint gg {k} (i : nat) : A (i + k) -n> A k := match i with 0 => cid | S i => gg i ◎ g end. Lemma ggff {k i} (x : A k) : gg i (ff i x) ≡ x. Proof. induction i as [|i IH]; simpl; [done|by rewrite (gf (ff i x)) IH]. Qed. Lemma f_tower {n k} (X : tower) : n ≤ k → f (X k) ={n}= X (S k). Lemma f_tower {n k} (X : tower) : n ≤ k → f (X k) ≡{n}≡ X (S k). Proof. intros. by rewrite -(fg (X (S k))) // -(g_tower X). Qed. Lemma ff_tower {n} k i (X : tower) : n ≤ k → ff i (X k) ={n}= X (i + k). Lemma ff_tower {n} k i (X : tower) : n ≤ k → ff i (X k) ≡{n}≡ X (i + k). Proof. intros; induction i as [|i IH]; simpl; [done|]. by rewrite IH (f_tower X); last lia. ... ... @@ -170,7 +170,7 @@ Proof. * assert (H : (i - S k) + (1 + k) = i) by lia; rewrite (ff_ff _ H) /=. by erewrite coerce_proper by done. Qed. Lemma embed_tower j n (X : T) : n ≤ j → embed j (X j) ={n}= X. Lemma embed_tower j n (X : T) : n ≤ j → embed j (X j) ≡{n}≡ X. Proof. move=> Hn i; rewrite /= /embed'; destruct (le_lt_dec i j) as [H|H]; simpl. * rewrite -(gg_tower i (j - i) X). ... ...
 ... ... @@ -23,10 +23,10 @@ Inductive excl_equiv : Equiv (excl A) := | ExclBot_equiv : ExclBot ≡ ExclBot. Existing Instance excl_equiv. Inductive excl_dist `{Dist A} : Dist (excl A) := | excl_dist_0 (x y : excl A) : x ={0}= y | Excl_dist (x y : A) n : x ={n}= y → Excl x ={n}= Excl y | ExclUnit_dist n : ExclUnit ={n}= ExclUnit | ExclBot_dist n : ExclBot ={n}= ExclBot. | excl_dist_0 (x y : excl A) : x ≡{0}≡ y | Excl_dist (x y : A) n : x ≡{n}≡ y → Excl x ≡{n}≡ Excl y | ExclUnit_dist n : ExclUnit ≡{n}≡ ExclUnit | ExclBot_dist n : ExclBot ≡{n}≡ ExclBot. Existing Instance excl_dist. Global Instance Excl_ne : Proper (dist n ==> dist n) (@Excl A). Proof. by constructor. Qed. ... ... @@ -138,7 +138,7 @@ Lemma excl_validN_inv_l n x y : ✓{S n} (Excl x ⋅ y) → y = ∅. Proof. by destruct y. Qed. Lemma excl_validN_inv_r n x y : ✓{S n} (x ⋅ Excl y) → x = ∅. Proof. by destruct x. Qed. Lemma Excl_includedN n x y : ✓{n} y → Excl x ≼{n} y ↔ y ={n}= Excl x. Lemma Excl_includedN n x y : ✓{n} y → Excl x ≼{n} y ↔ y ≡{n}≡ Excl x. Proof. intros Hvalid; split; [destruct n as [|n]; [done|]|by intros ->]. by intros [z ?]; cofe_subst; rewrite (excl_validN_inv_l n x z). ... ...
 ... ... @@ -6,7 +6,7 @@ Context `{Countable K} {A : cofeT}. Implicit Types m : gmap K A. Instance map_dist : Dist (gmap K A) := λ n m1 m2, ∀ i, m1 !! i ={n}= m2 !! i. ∀ i, m1 !! i ≡{n}≡ m2 !! i. Program Definition map_chain (c : chain (gmap K A)) (k : K) : chain (option A) := {| chain_car n := c n !! k |}. Next Obligation. by intros c k n i ?; apply (chain_cauchy c). Qed. ... ... @@ -60,7 +60,7 @@ Qed. Global Instance map_lookup_timeless m i : Timeless m → Timeless (m !! i). Proof. intros ? [x|] Hx; [|by symmetry; apply (timeless _)]. assert (m ={1}= <[i:=x]> m) assert (m ≡{1}≡ <[i:=x]> m) by (by symmetry in Hx; inversion Hx; cofe_subst; rewrite insert_id). by rewrite (timeless m (<[i:=x]>m)) // lookup_insert. Qed. ... ... @@ -132,7 +132,7 @@ Qed. Definition map_cmra_extend_mixin : CMRAExtendMixin (gmap K A). Proof. intros n m m1 m2 Hm Hm12. assert (∀ i, m !! i ={n}= m1 !! i ⋅ m2 !! i) as Hm12' assert (∀ i, m !! i ≡{n}≡ m1 !! i ⋅ m2 !! i) as Hm12' by (by intros i; rewrite -lookup_op). set (f i := cmra_extend_op n (m !! i) (m1 !! i) (m2 !! i) (Hm i) (Hm12' i)). set (f_proj i := proj1_sig (f i)). ... ... @@ -166,7 +166,7 @@ Implicit Types m : gmap K A. Implicit Types i : K. Implicit Types a : A. Lemma map_lookup_validN n m i x : ✓{n} m → m !! i ={n}= Some x → ✓{n} x. Lemma map_lookup_validN n m i x : ✓{n} m → m !! i ≡{n}≡ Some x → ✓{n} x. Proof. by move=> /(_ i) Hm Hi; move:Hm; rewrite Hi. Qed. Lemma map_insert_validN n m i x : ✓{n} x → ✓{n} m → ✓{n} (<[i:=x]>m). Proof. by intros ?? j; destruct (decide (i = j)); simplify_map_equality. Qed. ... ... @@ -201,7 +201,7 @@ Lemma map_op_singleton (i : K) (x y : A) : Proof. by apply (merge_singleton _ _ _ x y). Qed. Lemma singleton_includedN n m i x : {[ i ↦ x ]} ≼{n} m ↔ ∃ y, m !! i ={n}= Some y ∧ x ≼ y. {[ i ↦ x ]} ≼{n} m ↔ ∃ y, m !! i ≡{n}≡ Some y ∧ x ≼ y. (* not m !! i = Some y ∧ x ≼{n} y to deal with n = 0 *) Proof. split. ... ...
 ... ... @@ -21,7 +21,7 @@ Section iprod_cofe. Implicit Types f g : iprod B. Instance iprod_equiv : Equiv (iprod B) := λ f g, ∀ x, f x ≡ g x. Instance iprod_dist : Dist (iprod B) := λ n f g, ∀ x, f x ={n}= g x. Instance iprod_dist : Dist (iprod B) := λ n f g, ∀ x, f x ≡{n}≡ g x. Program Definition iprod_chain (c : chain (iprod B)) (x : A) : chain (B x) := {| chain_car n := c n x |}. Next Obligation. by intros c x n i ?; apply (chain_cauchy c). Qed. ... ...
 ... ... @@ -5,9 +5,9 @@ Require Import algebra.functor. Section cofe. Context {A : cofeT}. Inductive option_dist : Dist (option A) := | option_0_dist (x y : option A) : x ={0}= y | Some_dist n x y : x ={n}= y → Some x ={n}= Some y | None_dist n : None ={n}= None. | option_0_dist (x y : option A) : x ≡{0}≡ y | Some_dist n x y : x ≡{n}≡ y → Some x ≡{n}≡ Some y | None_dist n : None ≡{n}≡ None. Existing Instance option_dist. Program Definition option_chain (c : chain (option A)) (x : A) (H : c 1 = Some x) : chain A := ... ... @@ -134,9 +134,9 @@ Lemma op_is_Some mx my : is_Some (mx ⋅ my) ↔ is_Some mx ∨ is_Some my. Proof. destruct mx, my; rewrite /op /option_op /= -!not_eq_None_Some; naive_solver. Qed. Lemma option_op_positive_dist_l n mx my : mx ⋅ my ={n}= None → mx ={n}= None. Lemma option_op_positive_dist_l n mx my : mx ⋅ my ≡{n}≡ None → mx ≡{n}≡ None. Proof. by destruct mx, my; inversion_clear 1. Qed. Lemma option_op_positive_dist_r n mx my : mx ⋅ my ={n}= None → my ={n}= None. Lemma option_op_positive_dist_r n mx my : mx ⋅ my ≡{n}≡ None → my ≡{n}≡ None. Proof. by destruct mx, my; inversion_clear 1. Qed. Lemma option_updateP (P : A → Prop) (Q : option A → Prop) x : ... ...
 ... ... @@ -5,7 +5,7 @@ Local Hint Extern 10 (_ ≤ _) => omega. Record uPred (M : cmraT) : Type := IProp { uPred_holds :> nat → M → Prop; uPred_ne x1 x2 n : uPred_holds n x1 → x1 ={n}= x2 → uPred_holds n x2; uPred_ne x1 x2 n : uPred_holds n x1 → x1 ≡{n}≡ x2 → uPred_holds n x2; uPred_0 x : uPred_holds 0 x; uPred_weaken x1 x2 n1 n2 : uPred_holds n1 x1 → x1 ≼ x2 → n2 ≤ n1 → ✓{n2} x2 → uPred_holds n2 x2 ... ... @@ -54,7 +54,7 @@ Instance uPred_proper {M} (P : uPred M) n : Proper ((≡) ==> iff) (P n). Proof. by intros x1 x2 Hx; apply uPred_ne', equiv_dist. Qed. Lemma uPred_holds_ne {M} (P1 P2 : uPred M) n x : P1 ={n}= P2 → ✓{n} x → P1 n x → P2 n x. P1 ≡{n}≡ P2 → ✓{n} x → P1 n x → P2 n x. Proof. intros HP ?; apply HP; auto. Qed. Lemma uPred_weaken' {M} (P : uPred M) x1 x2 n1 n2 : x1 ≼ x2 → n2 ≤ n1 → ✓{n2} x2 → P n1 x1 → P n2 x2. ... ... @@ -90,7 +90,7 @@ Definition uPredC_map {M1 M2 : cmraT} (f : M2 -n> M1) `{!CMRAMonotone f} : uPredC M1 -n> uPredC M2 := CofeMor (uPred_map f : uPredC M1 → uPredC M2). Lemma upredC_map_ne {M1 M2 : cmraT} (f g : M2 -n> M1) `{!CMRAMonotone f, !CMRAMonotone g} n : f ={n}= g → uPredC_map f ={n}= uPredC_map g. f ≡{n}≡ g → uPredC_map f ≡{n}≡ uPredC_map g. Proof. by intros Hfg P y n' ??; rewrite /uPred_holds /= (dist_le _ _ _ _(Hfg y)); last lia. ... ... @@ -120,7 +120,7 @@ Program Definition uPred_impl {M} (P Q : uPred M) : uPred M := Next Obligation. intros M P Q x1' x1 n1 HPQ Hx1 x2 n2 ????. destruct (cmra_included_dist_l x1 x2 x1' n1) as (x2'&?&Hx2); auto. assert (x2' ={n2}= x2) as Hx2' by (by apply dist_le with n1). assert (x2' ≡{n2}≡ x2) as Hx2' by (by apply dist_le with n1). assert (✓{n2} x2') by (by rewrite Hx2'); rewrite -Hx2'. eauto using uPred_weaken, uPred_ne. Qed. ... ... @@ -140,18 +140,18 @@ Next Obligation. Qed. Program Definition uPred_eq {M} {A : cofeT} (a1 a2 : A) : uPred M := {| uPred_holds n x := a1 ={n}= a2 |}. {| uPred_holds n x := a1 ≡{n}≡ a2 |}. Solve Obligations with naive_solver eauto 2 using (dist_le (A:=A)). Program Definition uPred_sep {M} (P Q : uPred M) : uPred M := {| uPred_holds n x := ∃ x1 x2, x ={n}= x1 ⋅ x2 ∧ P n x1 ∧ Q n x2 |}. {| uPred_holds n x := ∃ x1 x2, x ≡{n}≡ x1 ⋅ x2 ∧ P n x1 ∧ Q n x2 |}. Next Obligation. by intros M P Q x y n (x1&x2&?&?&?) Hxy; exists x1, x2; rewrite -Hxy. Qed. Next Obligation. by intros M P Q x; exists x, x. Qed. Next Obligation. intros M P Q x y n1 n2 (x1&x2&Hx&?&?) Hxy ??. assert (∃ x2', y ={n2}= x1 ⋅ x2' ∧ x2 ≼ x2') as (x2'&Hy&?). assert (∃ x2', y ≡{n2}≡ x1 ⋅ x2' ∧ x2 ≼ x2') as (x2'&Hy&?). { destruct Hxy as [z Hy]; exists (x2 ⋅ z); split; eauto using cmra_included_l. apply dist_le with n1; auto. by rewrite (associative op) -Hx Hy. } clear Hxy; cofe_subst y; exists x1, x2'; split_ands; [done| |]. ... ...
 ... ... @@ -65,7 +65,7 @@ Proof. rewrite /ownG; apply _. Qed. (* inversion lemmas *) Lemma ownI_spec r n i P : ✓{n} r → (ownI i P) n r ↔ wld r !! i ={n}= Some (to_agree (Later (iProp_unfold P))). (ownI i P) n r ↔ wld r !! i ≡{n}≡ Some (to_agree (Later (iProp_unfold P))). Proof. intros [??]; rewrite /uPred_holds/=res_includedN/=singleton_includedN; split. * intros [(P'&Hi&HP) _]; rewrite Hi. ... ... @@ -73,7 +73,7 @@ Proof. (cmra_included_includedN _ P'),HP; apply map_lookup_validN with (wld r) i. * intros ?; split_ands; try apply cmra_empty_leastN; eauto. Qed. Lemma ownP_spec r n σ : ✓{n} r → (ownP σ) n r ↔ pst r ={n}= Excl σ. Lemma ownP_spec r n σ : ✓{n} r → (ownP σ) n r ↔ pst r ≡{n}≡ Excl σ. Proof. intros (?&?&?); rewrite /uPred_holds /= res_includedN /= Excl_includedN //. naive_solver (apply cmra_empty_leastN). ... ...
 ... ... @@ -24,7 +24,7 @@ Inductive res_equiv' (r1 r2 : res Λ Σ A) := Res_equiv : wld r1 ≡ wld r2 → pst r1 ≡ pst r2 → gst r1 ≡ gst r2 → res_equiv' r1 r2. Instance res_equiv : Equiv (res Λ Σ A) := res_equiv'. Inductive res_dist' (n : nat) (r1 r2 : res Λ Σ A) := Res_dist : wld r1 ={n}= wld r2 → pst r1 ={n}= pst r2 → gst r1 ={n}= gst r2 → wld r1 ≡{n}≡ wld r2 → pst r1 ≡{n}≡ pst r2 → gst r1 ≡{n}≡ gst r2 → res_dist' n r1 r2. Instance res_dist : Dist (res Λ Σ A) := res_dist'. Global Instance Res_ne n : ... ... @@ -148,14 +148,14 @@ Proof. done. Qed. Lemma Res_unit w σ m : unit (Res w σ m) = Res (unit w) (unit σ) (unit m). Proof. done. Qed. Lemma lookup_wld_op_l n r1 r2 i P : ✓{n} (r1⋅r2) → wld r1 !! i ={n}= Some P → (wld r1 ⋅ wld r2) !! i ={n}= Some P. ✓{n} (r1⋅r2) → wld r1 !! i ≡{n}≡ Some P → (wld r1 ⋅ wld r2) !! i ≡{n}≡ Some P. Proof. move=>/wld_validN /(_ i) Hval Hi1P; move: Hi1P Hval; rewrite lookup_op. destruct (wld r2 !! i) as [P'|] eqn:Hi; rewrite !Hi ?right_id // =>-> ?. by constructor; rewrite (agree_op_inv P P') // agree_idempotent. Qed. Lemma lookup_wld_op_r n r1 r2 i P :