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

algebra/agree.v

@@ -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.
 

algebra/auth.v

@@ -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 *.

algebra/cmra.v

@@ -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.

algebra/cofe.v

@@ -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.

algebra/cofe_solver.v

@@ -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).

algebra/excl.v

@@ -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).

algebra/fin_maps.v

@@ -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.

algebra/iprod.v

@@ -21,7 +21,7 @@ Section iprod_cofe.