Initial Iris commit

theories/base.v

@@ -23,6 +23,7 @@ Arguments compose _ _ _ _ _ _ /.
 Arguments flip _ _ _ _ _ _ /.
 Arguments const _ _ _ _ /.
 Typeclasses Transparent id compose flip const.
+Instance: Params (@pair) 2.
 
 (** Change [True] and [False] into notations in order to enable overloading.
 We will use this in the file [assertions] to give [True] and [False] a
@@ -792,6 +793,11 @@ Instance pointwise_transitive {A} `{R : relation B} :
   Transitive R → Transitive (pointwise_relation A R) | 9.
 Proof. firstorder. Qed.
 
+(** ** Unit *)
+Instance unit_equiv : Equiv unit := λ _ _, True.
+Instance unit_equivalence : Equivalence (@equiv unit _).
+Proof. repeat split. Qed.
+
 (** ** Products *)
 Instance prod_map_injective {A A' B B'} (f : A → A') (g : B → B') :
   Injective (=) (=) f → Injective (=) (=) g →
@@ -825,6 +831,15 @@ Section prod_relation.
   Proof. firstorder eauto. Qed.
 End prod_relation.
 
+Instance prod_equiv `{Equiv A,Equiv B} : Equiv (A * B) := prod_relation (≡) (≡).
+Instance pair_proper `{Equiv A, Equiv B} :
+  Proper ((≡) ==> (≡) ==> (≡)) (@pair A B) | 0 := _.
+Instance fst_proper `{Equiv A, Equiv B} :
+  Proper ((≡) ==> (≡)) (@fst A B) | 0 := _.
+Instance snd_proper `{Equiv A, Equiv B} :
+  Proper ((≡) ==> (≡)) (@snd A B) | 0 := _.
+Typeclasses Opaque prod_equiv.
+
 (** ** Other *)
 Lemma or_l P Q : ¬Q → P ∨ Q ↔ P.
 Proof. tauto. Qed.

theories/collections.v

@@ -249,7 +249,7 @@ Ltac unfold_elem_of :=
 For goals that do not involve [≡], [⊆], [map], or quantifiers this tactic is
 generally powerful enough. This tactic either fails or proves the goal. *)
 Tactic Notation "solve_elem_of" tactic3(tac) :=
-  simpl in *;
+  setoid_subst;
   decompose_empty;
   unfold_elem_of;
   solve [intuition (simplify_equality; tac)].
@@ -261,7 +261,7 @@ fails or loops on very small goals generated by [solve_elem_of] already. We
 use the [naive_solver] tactic as a substitute. This tactic either fails or
 proves the goal. *)
 Tactic Notation "esolve_elem_of" tactic3(tac) :=
-  simpl in *;
+  setoid_subst;
   decompose_empty;
   unfold_elem_of;
   naive_solver tac.
@@ -273,6 +273,11 @@ Section collection.
 
   Global Instance: Lattice C.
   Proof. split. apply _. firstorder auto. solve_elem_of. Qed.
+  Global Instance difference_proper : Proper ((≡) ==> (≡) ==> (≡)) (∖).
+  Proof.
+    intros X1 X2 HX Y1 Y2 HY; apply elem_of_equiv; intros x.
+    by rewrite !elem_of_difference, HX, HY.
+  Qed.
   Lemma intersection_singletons x : {[x]} ∩ {[x]} ≡ {[x]}.
   Proof. esolve_elem_of. Qed.
   Lemma difference_twice X Y : (X ∖ Y) ∖ Y ≡ X ∖ Y.
@@ -283,6 +288,8 @@ Section collection.
   Proof. esolve_elem_of. Qed.
   Lemma difference_union_distr_l X Y Z : (X ∪ Y) ∖ Z ≡ X ∖ Z ∪ Y ∖ Z.
   Proof. esolve_elem_of. Qed.
+  Lemma difference_union_distr_r X Y Z : Z ∖ (X ∪ Y) ≡ (Z ∖ X) ∩ (Z ∖ Y).
+  Proof. esolve_elem_of. Qed.
   Lemma difference_intersection_distr_l X Y Z : (X ∩ Y) ∖ Z ≡ X ∖ Z ∩ Y ∖ Z.
   Proof. esolve_elem_of. Qed.
 
@@ -298,6 +305,8 @@ Section collection.
     Proof. unfold_leibniz. apply difference_diag. Qed.
     Lemma difference_union_distr_l_L X Y Z : (X ∪ Y) ∖ Z = X ∖ Z ∪ Y ∖ Z.
     Proof. unfold_leibniz. apply difference_union_distr_l. Qed.
+    Lemma difference_union_distr_r_L X Y Z : Z ∖ (X ∪ Y) = (Z ∖ X) ∩ (Z ∖ Y).
+    Proof. unfold_leibniz. apply difference_union_distr_r. Qed.
     Lemma difference_intersection_distr_l_L X Y Z :
       (X ∩ Y) ∖ Z = X ∖ Z ∩ Y ∖ Z.
     Proof. unfold_leibniz. apply difference_intersection_distr_l. Qed.
@@ -518,16 +527,13 @@ Section collection_monad.
 
   Global Instance collection_fmap_proper {A B} (f : A → B) :
     Proper ((≡) ==> (≡)) (fmap f).
-  Proof. intros X Y E. esolve_elem_of. Qed.
-  Global Instance collection_ret_proper {A} :
-    Proper ((=) ==> (≡)) (@mret M _ A).
-  Proof. intros X Y E. esolve_elem_of. Qed.
+  Proof. intros X Y [??]; split; esolve_elem_of. Qed.
   Global Instance collection_bind_proper {A B} (f : A → M B) :
     Proper ((≡) ==> (≡)) (mbind f).
-  Proof. intros X Y E. esolve_elem_of. Qed.
+  Proof. intros X Y [??]; split; esolve_elem_of. Qed.
   Global Instance collection_join_proper {A} :
     Proper ((≡) ==> (≡)) (@mjoin M _ A).
-  Proof. intros X Y E. esolve_elem_of. Qed.
+  Proof. intros X Y [??]; split; esolve_elem_of. Qed.
 
   Lemma collection_bind_singleton {A B} (f : A → M B) x : {[ x ]} ≫= f ≡ f x.
   Proof. esolve_elem_of. Qed.

theories/fin_collections.v

@@ -3,7 +3,7 @@
 (** This file collects definitions and theorems on finite collections. Most
 importantly, it implements a fold and size function and some useful induction
 principles on finite collections . *)
-Require Import Permutation ars listset.
+Require Import Permutation relations listset.
 Require Export numbers collections.
 
 Instance collection_size `{Elements A C} : Size C := length ∘ elements.

theories/fin_map_dom.v

@@ -77,15 +77,15 @@ Proof. rewrite not_elem_of_dom. apply delete_partial_alter. Qed.
 Lemma delete_insert_dom {A} (m : M A) i x :
   i ∉ dom D m → delete i (<[i:=x]>m) = m.
 Proof. rewrite not_elem_of_dom. apply delete_insert. Qed.
-Lemma map_disjoint_dom {A} (m1 m2 : M A) : m1 ⊥ m2 ↔ dom D m1 ∩ dom D m2 ≡ ∅.
+Lemma map_disjoint_dom {A} (m1 m2 : M A) : m1 ⊥ₘ m2 ↔ dom D m1 ∩ dom D m2 ≡ ∅.
 Proof.
   rewrite map_disjoint_spec, elem_of_equiv_empty.
   setoid_rewrite elem_of_intersection.
   setoid_rewrite elem_of_dom. unfold is_Some. naive_solver.
 Qed.
-Lemma map_disjoint_dom_1 {A} (m1 m2 : M A) : m1 ⊥ m2 → dom D m1 ∩ dom D m2 ≡ ∅.
+Lemma map_disjoint_dom_1 {A} (m1 m2 : M A) : m1 ⊥ₘ m2 → dom D m1 ∩ dom D m2 ≡ ∅.
 Proof. apply map_disjoint_dom. Qed.
-Lemma map_disjoint_dom_2 {A} (m1 m2 : M A) : dom D m1 ∩ dom D m2 ≡ ∅ → m1 ⊥ m2.
+Lemma map_disjoint_dom_2 {A} (m1 m2 : M A) : dom D m1 ∩ dom D m2 ≡ ∅ → m1 ⊥ₘ m2.
 Proof. apply map_disjoint_dom. Qed.
 Lemma dom_union {A} (m1 m2 : M A) : dom D (m1 ∪ m2) ≡ dom D m1 ∪ dom D m2.
 Proof.

theories/fin_maps.v

@@ -5,7 +5,7 @@ finite maps and collects some theory on it. Most importantly, it proves useful
 induction principles for finite maps and implements the tactic
 [simplify_map_equality] to simplify goals involving finite maps. *)
 Require Import Permutation.
-Require Export ars vector orders.
+Require Export relations vector orders.
 
 (** * Axiomatization of finite maps *)
 (** We require Leibniz equality to be extensional on finite maps. This of
@@ -71,25 +71,26 @@ Instance map_intersection_with `{Merge M} {A} : IntersectionWith A (M A) :=
 Instance map_difference_with `{Merge M} {A} : DifferenceWith A (M A) :=
   λ f, merge (difference_with f).
 
+Instance map_equiv `{∀ A, Lookup K A (M A), Equiv A} : Equiv (M A) | 1 := λ m1 m2,
+  ∀ i, m1 !! i ≡ m2 !! i.
+
 (** The relation [intersection_forall R] on finite maps describes that the
 relation [R] holds for each pair in the intersection. *)
 Definition map_Forall `{Lookup K A M} (P : K → A → Prop) : M → Prop :=
   λ m, ∀ i x, m !! i = Some x → P i x.
-Definition map_Forall2 `{∀ A, Lookup K A (M A)} {A B}
-    (R : A → B → Prop) (P : A → Prop) (Q : B → Prop)
-    (m1 : M A) (m2 : M B) : Prop := ∀ i,
-  match m1 !! i, m2 !! i with
-  | Some x, Some y => R x y
-  | Some x, None => P x
-  | None, Some y => Q y
-  | None, None => True
-  end.
+Definition map_relation `{∀ A, Lookup K A (M A)} {A B} (R : A → B → Prop)
+    (P : A → Prop) (Q : B → Prop) (m1 : M A) (m2 : M B) : Prop := ∀ i,
+  option_relation R P Q (m1 !! i) (m2 !! i).
 Definition map_included `{∀ A, Lookup K A (M A)} {A}
-  (R : relation A) : relation (M A) := map_Forall2 R (λ _, False) (λ _, True).
-Instance map_disjoint `{∀ A, Lookup K A (M A)} {A} : Disjoint (M A) :=
-  map_Forall2 (λ _ _, False) (λ _, True) (λ _, True).
+  (R : relation A) : relation (M A) := map_relation R (λ _, False) (λ _, True).
+Definition map_disjoint `{∀ A, Lookup K A (M A)} {A} : relation (M A) :=
+  map_relation (λ _ _, False) (λ _, True) (λ _, True).
+Infix "⊥ₘ" := map_disjoint (at level 70) : C_scope.
+Hint Extern 0 (_ ⊥ₘ _) => symmetry; eassumption.
+Notation "( m ⊥ₘ.)" := (map_disjoint m) (only parsing) : C_scope.
+Notation "(.⊥ₘ m )" := (λ m2, m2 ⊥ₘ m) (only parsing) : C_scope.
 Instance map_subseteq `{∀ A, Lookup K A (M A)} {A} : SubsetEq (M A) :=
-  map_Forall2 (=) (λ _, False) (λ _, True).
+  map_included (=).
 
 (** The union of two finite maps only has a meaningful definition for maps
 that are disjoint. However, as working with partial functions is inconvenient
@@ -114,12 +115,67 @@ Definition map_imap `{∀ A, Insert K A (M A), ∀ A, Empty (M A),
 Section theorems.
 Context `{FinMap K M}.
 
+(** ** Setoids *)
+Section setoid.
+  Context `{Equiv A}.
+  Global Instance map_equivalence `{!Equivalence ((≡) : relation A)} :
+    Equivalence ((≡) : relation (M A)).
+  Proof.
+    split.
+    * by intros m i.
+    * by intros m1 m2 ? i.
+    * by intros m1 m2 m3 ?? i; transitivity (m2 !! i).
+  Qed.
+  Global Instance lookup_proper (i : K) :
+    Proper ((≡) ==> (≡)) (lookup (M:=M A) i).
+  Proof. by intros m1 m2 Hm. Qed.
+  Global Instance partial_alter_proper :
+    Proper (((≡) ==> (≡)) ==> (=) ==> (≡) ==> (≡)) partial_alter.
+  Proof.
+    by intros f1 f2 Hf i ? <- m1 m2 Hm j; destruct (decide (i = j)) as [->|];
+      rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne by done;
+      try apply Hf; apply lookup_proper.
+  Qed.
+  Global Instance insert_proper (i : K) :
+    Proper ((≡) ==> (≡) ==> (≡)) (insert (M:=M A) i).
+  Proof. by intros ???; apply partial_alter_proper; [constructor|]. Qed.
+  Global Instance delete_proper (i : K) :
+    Proper ((≡) ==> (≡)) (delete (M:=M A) i).
+  Proof. by apply partial_alter_proper; [constructor|]. Qed.
+  Global Instance alter_proper :
+    Proper (((≡) ==> (≡)) ==> (=) ==> (≡) ==> (≡)) (alter (A:=A) (M:=M A)).
+  Proof.
+    intros ?? Hf; apply partial_alter_proper.
+    by destruct 1; constructor; apply Hf.
+  Qed.
+  Lemma merge_ext f g
+      `{!PropHolds (f None None = None), !PropHolds (g None None = None)} :
+    ((≡) ==> (≡) ==> (≡))%signature f g →
+    ((≡) ==> (≡) ==> (≡))%signature (merge f) (merge g).
+  Proof.
+    by intros Hf ?? Hm1 ?? Hm2 i; rewrite !lookup_merge by done; apply Hf.
+  Qed.
+  Global Instance union_with_proper :
+    Proper (((≡) ==> (≡) ==> (≡)) ==> (≡) ==> (≡) ==> (≡)) union_with.
+  Proof.
+    intros ?? Hf ?? Hm1 ?? Hm2 i; apply (merge_ext _ _); auto.
+    by do 2 destruct 1; first [apply Hf | constructor].
+  Qed.    
+  Global Instance map_leibniz `{!LeibnizEquiv A} : LeibnizEquiv (M A).
+  Proof.
+    intros m1 m2; split.
+    * by intros Hm; apply map_eq; intros i; unfold_leibniz; apply lookup_proper.
+    * by intros <-; intros i; fold_leibniz.
+  Qed.
+End setoid.
+
+(** ** General properties *)
 Lemma map_eq_iff {A} (m1 m2 : M A) : m1 = m2 ↔ ∀ i, m1 !! i = m2 !! i.
 Proof. split. by intros ->. apply map_eq. Qed.
 Lemma map_subseteq_spec {A} (m1 m2 : M A) :
   m1 ⊆ m2 ↔ ∀ i x, m1 !! i = Some x → m2 !! i = Some x.
 Proof.
-  unfold subseteq, map_subseteq, map_Forall2. split; intros Hm i;
+  unfold subseteq, map_subseteq, map_relation. split; intros Hm i;
     specialize (Hm i); destruct (m1 !! i), (m2 !! i); naive_solver.
 Qed.
 Global Instance: EmptySpec (M A).
@@ -129,9 +185,10 @@ Proof.
 Qed.
 Global Instance: ∀ {A} (R : relation A), PreOrder R → PreOrder (map_included R).
 Proof.
-  split; [intros m i; by destruct (m !! i)|].
+  split; [intros m i; by destruct (m !! i); simpl|].
   intros m1 m2 m3 Hm12 Hm23 i; specialize (Hm12 i); specialize (Hm23 i).
-  destruct (m1 !! i), (m2 !! i), (m3 !! i); try done; etransitivity; eauto.
+  destruct (m1 !! i), (m2 !! i), (m3 !! i); simplify_equality';
+    done || etransitivity; eauto.
 Qed.
 Global Instance: PartialOrder ((⊆) : relation (M A)).
 Proof.
@@ -357,8 +414,8 @@ Lemma insert_included {A} R `{!Reflexive R} (m : M A) i x :
   (∀ y, m !! i = Some y → R y x) → map_included R m (<[i:=x]>m).
 Proof.
   intros ? j; destruct (decide (i = j)) as [->|].
-  * rewrite lookup_insert. destruct (m !! j); eauto.
-  * rewrite lookup_insert_ne by done. by destruct (m !! j).
+  * rewrite lookup_insert. destruct (m !! j); simpl; eauto.
+  * rewrite lookup_insert_ne by done. by destruct (m !! j); simpl.
 Qed.
 Lemma insert_subseteq {A} (m : M A) i x : m !! i = None → m ⊆ <[i:=x]>m.
 Proof. apply partial_alter_subseteq. Qed.
@@ -465,6 +522,17 @@ Proof.
   * by rewrite lookup_omap, !lookup_singleton.
   * by rewrite lookup_omap, !lookup_singleton_ne.
 Qed.
+Lemma map_fmap_id {A} (m : M A) : id <$> m = m.
+Proof. apply map_eq; intros i; by rewrite lookup_fmap, option_fmap_id. Qed.
+Lemma map_fmap_compose {A B C} (f : A → B) (g : B → C) (m : M A) :
+  g ∘ f <$> m = g <$> f <$> m.
+Proof. apply map_eq; intros i; by rewrite !lookup_fmap,option_fmap_compose. Qed.
+Lemma map_fmap_ext {A B} (f1 f2 : A → B) m :
+  (∀ i x, m !! i = Some x → f1 x = f2 x) → f1 <$> m = f2 <$> m.
+Proof.
+  intros Hi; apply map_eq; intros i; rewrite !lookup_fmap.
+  by destruct (m !! i) eqn:?; simpl; erewrite ?Hi by eauto.
+Qed.
 
 (** ** Properties of conversion to lists *)
 Lemma map_to_list_unique {A} (m : M A) i x y :
@@ -807,7 +875,7 @@ Lemma insert_merge_r m1 m2 i x z :
 Proof. by intros; apply partial_alter_merge_r. Qed.
 End more_merge.
 
-(** ** Properties on the [map_Forall2] relation *)
+(** ** Properties on the [map_relation] relation *)
 Section Forall2.
 Context {A B} (R : A → B → Prop) (P : A → Prop) (Q : B → Prop).
 Context `{∀ x y, Decision (R x y), ∀ x, Decision (P x), ∀ y, Decision (Q y)}.
@@ -819,8 +887,8 @@ Let f (mx : option A) (my : option B) : option bool :=
   | None, Some y => Some (bool_decide (Q y))
   | None, None => None
   end.
-Lemma map_Forall2_alt (m1 : M A) (m2 : M B) :
-  map_Forall2 R P Q m1 m2 ↔ map_Forall (λ _, Is_true) (merge f m1 m2).
+Lemma map_relation_alt (m1 : M A) (m2 : M B) :
+  map_relation R P Q m1 m2 ↔ map_Forall (λ _, Is_true) (merge f m1 m2).
 Proof.
   split.
   * intros Hm i P'; rewrite lookup_merge by done; intros.
@@ -830,71 +898,72 @@ Proof.
     destruct (m1 !! i), (m2 !! i); simplify_equality'; auto;
       by eapply bool_decide_unpack, Hm.
 Qed.
-Global Instance map_Forall2_dec `{∀ x y, Decision (R x y), ∀ x, Decision (P x),
-  ∀ y, Decision (Q y)} m1 m2 : Decision (map_Forall2 R P Q m1 m2).
+Global Instance map_relation_dec `{∀ x y, Decision (R x y), ∀ x, Decision (P x),
+  ∀ y, Decision (Q y)} m1 m2 : Decision (map_relation R P Q m1 m2).
 Proof.
   refine (cast_if (decide (map_Forall (λ _, Is_true) (merge f m1 m2))));
-    abstract by rewrite map_Forall2_alt.
+    abstract by rewrite map_relation_alt.
 Defined.
 (** Due to the finiteness of finite maps, we can extract a witness if the
 relation does not hold. *)
 Lemma map_not_Forall2 (m1 : M A) (m2 : M B) :
-  ¬map_Forall2 R P Q m1 m2 ↔ ∃ i,
+  ¬map_relation R P Q m1 m2 ↔ ∃ i,
     (∃ x y, m1 !! i = Some x ∧ m2 !! i = Some y ∧ ¬R x y)
     ∨ (∃ x, m1 !! i = Some x ∧ m2 !! i = None ∧ ¬P x)
     ∨ (∃ y, m1 !! i = None ∧ m2 !! i = Some y ∧ ¬Q y).
 Proof.
   split.
-  * rewrite map_Forall2_alt, (map_not_Forall _). intros (i&?&Hm&?); exists i.
+  * rewrite map_relation_alt, (map_not_Forall _). intros (i&?&Hm&?); exists i.
     rewrite lookup_merge in Hm by done.
     destruct (m1 !! i), (m2 !! i); naive_solver auto 2 using bool_decide_pack.
-  * by intros [i[(x&y&?&?&?)|[(x&?&?&?)|(y&?&?&?)]]] Hm;
+  * unfold map_relation, option_relation.
+    by intros [i[(x&y&?&?&?)|[(x&?&?&?)|(y&?&?&?)]]] Hm;
       specialize (Hm i); simplify_option_equality.
 Qed.
 End Forall2.
 
 (** ** Properties on the disjoint maps *)
 Lemma map_disjoint_spec {A} (m1 m2 : M A) :
-  m1 ⊥ m2 ↔ ∀ i x y, m1 !! i = Some x → m2 !! i = Some y → False.
+  m1 ⊥ₘ m2 ↔ ∀ i x y, m1 !! i = Some x → m2 !! i = Some y → False.
 Proof.
   split; intros Hm i; specialize (Hm i);
     destruct (m1 !! i), (m2 !! i); naive_solver.
 Qed.
 Lemma map_disjoint_alt {A} (m1 m2 : M A) :
-  m1 ⊥ m2 ↔ ∀ i, m1 !! i = None ∨ m2 !! i = None.
+  m1 ⊥ₘ m2 ↔ ∀ i, m1 !! i = None ∨ m2 !! i = None.
 Proof.
   split; intros Hm1m2 i; specialize (Hm1m2 i);
     destruct (m1 !! i), (m2 !! i); naive_solver.
 Qed.
 Lemma map_not_disjoint {A} (m1 m2 : M A) :
-  ¬m1 ⊥ m2 ↔ ∃ i x1 x2, m1 !! i = Some x1 ∧ m2 !! i = Some x2.
+  ¬m1 ⊥ₘ m2 ↔ ∃ i x1 x2, m1 !! i = Some x1 ∧ m2 !! i = Some x2.
 Proof.
   unfold disjoint, map_disjoint. rewrite map_not_Forall2 by solve_decision.
   split; [|naive_solver].
   intros [i[(x&y&?&?&?)|[(x&?&?&[])|(y&?&?&[])]]]; naive_solver.
 Qed.
-Global Instance: Symmetric (@disjoint (M A) _).
+Global Instance: Symmetric (map_disjoint : relation (M A)).
 Proof. intros A m1 m2. rewrite !map_disjoint_spec. naive_solver. Qed.
-Lemma map_disjoint_empty_l {A} (m : M A) : ∅ ⊥ m.
+Lemma map_disjoint_empty_l {A} (m : M A) : ∅ ⊥ₘ m.
 Proof. rewrite !map_disjoint_spec. intros i x y. by rewrite lookup_empty. Qed.
-Lemma map_disjoint_empty_r {A} (m : M A) : m ⊥ ∅.
+Lemma map_disjoint_empty_r {A} (m : M A) : m ⊥ₘ ∅.
 Proof. rewrite !map_disjoint_spec. intros i x y. by rewrite lookup_empty. Qed.
 Lemma map_disjoint_weaken {A} (m1 m1' m2 m2' : M A) :
-  m1' ⊥ m2' → m1 ⊆ m1' → m2 ⊆ m2' → m1 ⊥ m2.
+  m1' ⊥ₘ m2' → m1 ⊆ m1' → m2 ⊆ m2' → m1 ⊥ₘ m2.
 Proof. rewrite !map_subseteq_spec, !map_disjoint_spec. eauto. Qed.
 Lemma map_disjoint_weaken_l {A} (m1 m1' m2  : M A) :
-  m1' ⊥ m2 → m1 ⊆ m1' → m1 ⊥ m2.
+  m1' ⊥ₘ m2 → m1 ⊆ m1' → m1 ⊥ₘ m2.
 Proof. eauto using map_disjoint_weaken. Qed.
 Lemma map_disjoint_weaken_r {A} (m1 m2 m2' : M A) :
-  m1 ⊥ m2' → m2 ⊆ m2' → m1 ⊥ m2.
+  m1 ⊥ₘ m2' → m2 ⊆ m2' → m1 ⊥ₘ m2.
 Proof. eauto using map_disjoint_weaken. Qed.
 Lemma map_disjoint_Some_l {A} (m1 m2 : M A) i x:
-  m1 ⊥ m2 → m1 !! i = Some x → m2 !! i = None.
+  m1 ⊥ₘ m2 → m1 !! i = Some x → m2 !! i = None.
 Proof. rewrite map_disjoint_spec, eq_None_not_Some. intros ?? [??]; eauto. Qed.
 Lemma map_disjoint_Some_r {A} (m1 m2 : M A) i x:
-  m1 ⊥ m2 → m2 !! i = Some x → m1 !! i = None.
+  m1 ⊥ₘ m2 → m2 !! i