Merge branch 'jh/bottom_notation' into 'master'

Notation for disjointness: replace ⊥ with ##, so that ⊥ can be used for bottom.

See merge request robbertkrebbers/coq-stdpp!12

theories/base.v

@@ -681,7 +681,7 @@ Arguments sig_map _ _ _ _ _ _ !_ / : assert.
 (** We define operational type classes for the traditional operations and
 relations on collections: the empty collection [∅], the union [(∪)],
 intersection [(∩)], and difference [(∖)], the singleton [{[_]}], the subset
-[(⊆)] and element of [(∈)] relation, and disjointess [(⊥)]. *)
+[(⊆)] and element of [(∈)] relation, and disjointess [(##)]. *)
 Class Empty A := empty: A.
 Hint Mode Empty ! : typeclass_instances.
 Notation "∅" := empty : C_scope.
@@ -800,56 +800,56 @@ Notation "( x ∉)" := (λ X, x ∉ X) (only parsing) : C_scope.
 Notation "(∉ X )" := (λ x, x ∉ X) (only parsing) : C_scope.
 
 Class Disjoint A := disjoint : A → A → Prop.
-Hint Mode Disjoint ! : typeclass_instances.
+ Hint Mode Disjoint ! : typeclass_instances.
 Instance: Params (@disjoint) 2.
-Infix "⊥" := disjoint (at level 70) : C_scope.
-Notation "(⊥)" := disjoint (only parsing) : C_scope.
-Notation "( X ⊥.)" := (disjoint X) (only parsing) : C_scope.
-Notation "(.⊥ X )" := (λ Y, Y ⊥ X) (only parsing) : C_scope.
-Infix "⊥*" := (Forall2 (⊥)) (at level 70) : C_scope.
-Notation "(⊥*)" := (Forall2 (⊥)) (only parsing) : C_scope.
-Infix "⊥**" := (Forall2 (⊥*)) (at level 70) : C_scope.
-Infix "⊥1*" := (Forall2 (λ p q, p.1 ⊥ q.1)) (at level 70) : C_scope.
-Infix "⊥2*" := (Forall2 (λ p q, p.2 ⊥ q.2)) (at level 70) : C_scope.
-Infix "⊥1**" := (Forall2 (λ p q, p.1 ⊥* q.1)) (at level 70) : C_scope.
-Infix "⊥2**" := (Forall2 (λ p q, p.2 ⊥* q.2)) (at level 70) : C_scope.
-Hint Extern 0 (_ ⊥ _) => symmetry; eassumption.
-Hint Extern 0 (_ ⊥* _) => symmetry; eassumption.
+Infix "##" := disjoint (at level 70) : C_scope.
+Notation "(##)" := disjoint (only parsing) : C_scope.
+Notation "( X ##.)" := (disjoint X) (only parsing) : C_scope.
+Notation "(.## X )" := (λ Y, Y ## X) (only parsing) : C_scope.
+Infix "##*" := (Forall2 (##)) (at level 70) : C_scope.
+Notation "(##*)" := (Forall2 (##)) (only parsing) : C_scope.
+Infix "##**" := (Forall2 (##*)) (at level 70) : C_scope.
+Infix "##1*" := (Forall2 (λ p q, p.1 ## q.1)) (at level 70) : C_scope.
+Infix "##2*" := (Forall2 (λ p q, p.2 ## q.2)) (at level 70) : C_scope.
+Infix "##1**" := (Forall2 (λ p q, p.1 ##* q.1)) (at level 70) : C_scope.
+Infix "##2**" := (Forall2 (λ p q, p.2 ##* q.2)) (at level 70) : C_scope.
+Hint Extern 0 (_ ## _) => symmetry; eassumption.
+Hint Extern 0 (_ ##* _) => symmetry; eassumption.
 
 Class DisjointE E A := disjointE : E → A → A → Prop.
 Hint Mode DisjointE - ! : typeclass_instances.
 Instance: Params (@disjointE) 4.
-Notation "X ⊥{ Γ } Y" := (disjointE Γ X Y)
-  (at level 70, format "X  ⊥{ Γ }  Y") : C_scope.
-Notation "(⊥{ Γ } )" := (disjointE Γ) (only parsing, Γ at level 1) : C_scope.
-Notation "Xs ⊥{ Γ }* Ys" := (Forall2 (⊥{Γ}) Xs Ys)
-  (at level 70, format "Xs  ⊥{ Γ }*  Ys") : C_scope.
-Notation "(⊥{ Γ }* )" := (Forall2 (⊥{Γ}))
+Notation "X ##{ Γ } Y" := (disjointE Γ X Y)
+  (at level 70, format "X  ##{ Γ }  Y") : C_scope.
+Notation "(##{ Γ } )" := (disjointE Γ) (only parsing, Γ at level 1) : C_scope.
+Notation "Xs ##{ Γ }* Ys" := (Forall2 (##{Γ}) Xs Ys)
+  (at level 70, format "Xs  ##{ Γ }*  Ys") : C_scope.
+Notation "(##{ Γ }* )" := (Forall2 (##{Γ}))
   (only parsing, Γ at level 1) : C_scope.
-Notation "X ⊥{ Γ1 , Γ2 , .. , Γ3 } Y" := (disjoint (pair .. (Γ1, Γ2) .. Γ3) X Y)
-  (at level 70, format "X  ⊥{ Γ1 , Γ2 , .. , Γ3 }  Y") : C_scope.
-Notation "Xs ⊥{ Γ1 , Γ2 , .. , Γ3 }* Ys" :=
+Notation "X ##{ Γ1 , Γ2 , .. , Γ3 } Y" := (disjoint (pair .. (Γ1, Γ2) .. Γ3) X Y)
+  (at level 70, format "X  ##{ Γ1 , Γ2 , .. , Γ3 }  Y") : C_scope.
+Notation "Xs ##{ Γ1 , Γ2 , .. , Γ3 }* Ys" :=
   (Forall2 (disjoint (pair .. (Γ1, Γ2) .. Γ3)) Xs Ys)
-  (at level 70, format "Xs  ⊥{ Γ1 ,  Γ2 , .. , Γ3 }*  Ys") : C_scope.
-Hint Extern 0 (_ ⊥{_} _) => symmetry; eassumption.
+  (at level 70, format "Xs  ##{ Γ1 ,  Γ2 , .. , Γ3 }*  Ys") : C_scope.
+Hint Extern 0 (_ ##{_} _) => symmetry; eassumption.
 
 Class DisjointList A := disjoint_list : list A → Prop.
 Hint Mode DisjointList ! : typeclass_instances.
 Instance: Params (@disjoint_list) 2.
-Notation "⊥ Xs" := (disjoint_list Xs) (at level 20, format "⊥  Xs") : C_scope.
+Notation "## Xs" := (disjoint_list Xs) (at level 20, format "##  Xs") : C_scope.
 
 Section disjoint_list.
   Context `{Disjoint A, Union A, Empty A}.
   Implicit Types X : A.
 
   Inductive disjoint_list_default : DisjointList A :=
-    | disjoint_nil_2 : ⊥ (@nil A)
-    | disjoint_cons_2 (X : A) (Xs : list A) : X ⊥ ⋃ Xs → ⊥ Xs → ⊥ (X :: Xs).
+    | disjoint_nil_2 : ## (@nil A)
+    | disjoint_cons_2 (X : A) (Xs : list A) : X ## ⋃ Xs → ## Xs → ## (X :: Xs).
   Global Existing Instance disjoint_list_default.
 
-  Lemma disjoint_list_nil  : ⊥ @nil A ↔ True.
+  Lemma disjoint_list_nil  : ## @nil A ↔ True.
   Proof. split; constructor. Qed.
-  Lemma disjoint_list_cons X Xs : ⊥ (X :: Xs) ↔ X ⊥ ⋃ Xs ∧ ⊥ Xs.
+  Lemma disjoint_list_cons X Xs : ## (X :: Xs) ↔ X ## ⋃ Xs ∧ ## Xs.
   Proof. split. inversion_clear 1; auto. intros [??]. constructor; auto. Qed.
 End disjoint_list.
 

theories/collections.v

@@ -186,7 +186,7 @@ Section set_unfold_simple.
   Qed.
   Global Instance set_unfold_disjoint (P Q : A → Prop) X Y :
     (∀ x, SetUnfold (x ∈ X) (P x)) → (∀ x, SetUnfold (x ∈ Y) (Q x)) →
-    SetUnfold (X ⊥ Y) (∀ x, P x → Q x → False).
+    SetUnfold (X ## Y) (∀ x, P x → Q x → False).
   Proof. constructor. unfold disjoint, collection_disjoint. naive_solver. Qed.
 
   Context `{!LeibnizEquiv C}.
@@ -371,9 +371,9 @@ Section simple_collection.
   Lemma empty_union X Y : X ∪ Y ≡ ∅ ↔ X ≡ ∅ ∧ Y ≡ ∅.
   Proof. set_solver. Qed.
 
-  Lemma union_cancel_l X Y Z : Z ⊥ X → Z ⊥ Y → Z ∪ X ≡ Z ∪ Y → X ≡ Y.
+  Lemma union_cancel_l X Y Z : Z ## X → Z ## Y → Z ∪ X ≡ Z ∪ Y → X ≡ Y.
   Proof. set_solver. Qed.
-  Lemma union_cancel_r X Y Z : X ⊥ Z → Y ⊥ Z → X ∪ Z ≡ Y ∪ Z → X ≡ Y.
+  Lemma union_cancel_r X Y Z : X ## Z → Y ## Z → X ∪ Z ≡ Y ∪ Z → X ≡ Y.
   Proof. set_solver. Qed.
 
   (** Empty *)
@@ -405,22 +405,22 @@ Section simple_collection.
   Proof. by rewrite elem_of_singleton. Qed.
 
   (** Disjointness *)
-  Lemma elem_of_disjoint X Y : X ⊥ Y ↔ ∀ x, x ∈ X → x ∈ Y → False.
+  Lemma elem_of_disjoint X Y : X ## Y ↔ ∀ x, x ∈ X → x ∈ Y → False.
   Proof. done. Qed.
 
   Global Instance disjoint_sym : Symmetric (@disjoint C _).
   Proof. intros X Y. set_solver. Qed.
-  Lemma disjoint_empty_l Y : ∅ ⊥ Y.
+  Lemma disjoint_empty_l Y : ∅ ## Y.
   Proof. set_solver. Qed.
-  Lemma disjoint_empty_r X : X ⊥ ∅.
+  Lemma disjoint_empty_r X : X ## ∅.
   Proof. set_solver. Qed.
-  Lemma disjoint_singleton_l x Y : {[ x ]} ⊥ Y ↔ x ∉ Y.
+  Lemma disjoint_singleton_l x Y : {[ x ]} ## Y ↔ x ∉ Y.
   Proof. set_solver. Qed.
-  Lemma disjoint_singleton_r y X : X ⊥ {[ y ]} ↔ y ∉ X.
+  Lemma disjoint_singleton_r y X : X ## {[ y ]} ↔ y ∉ X.
   Proof. set_solver. Qed.
-  Lemma disjoint_union_l X1 X2 Y : X1 ∪ X2 ⊥ Y ↔ X1 ⊥ Y ∧ X2 ⊥ Y.
+  Lemma disjoint_union_l X1 X2 Y : X1 ∪ X2 ## Y ↔ X1 ## Y ∧ X2 ## Y.
   Proof. set_solver. Qed.
-  Lemma disjoint_union_r X Y1 Y2 : X ⊥ Y1 ∪ Y2 ↔ X ⊥ Y1 ∧ X ⊥ Y2.
+  Lemma disjoint_union_r X Y1 Y2 : X ## Y1 ∪ Y2 ↔ X ## Y1 ∧ X ## Y2.
   Proof. set_solver. Qed.
 
   (** Big unions *)
@@ -494,9 +494,9 @@ Section simple_collection.
     Lemma empty_union_L X Y : X ∪ Y = ∅ ↔ X = ∅ ∧ Y = ∅.
     Proof. unfold_leibniz. apply empty_union. Qed.
 
-    Lemma union_cancel_l_L X Y Z : Z ⊥ X → Z ⊥ Y → Z ∪ X = Z ∪ Y → X = Y.
+    Lemma union_cancel_l_L X Y Z : Z ## X → Z ## Y → Z ∪ X = Z ∪ Y → X = Y.
     Proof. unfold_leibniz. apply union_cancel_l. Qed.
-    Lemma union_cancel_r_L X Y Z : X ⊥ Z → Y ⊥ Z → X ∪ Z = Y ∪ Z → X = Y.
+    Lemma union_cancel_r_L X Y Z : X ## Z → Y ## Z → X ∪ Z = Y ∪ Z → X = Y.
     Proof. unfold_leibniz. apply union_cancel_r. Qed.
 
     (** Empty *)
@@ -618,7 +618,7 @@ Section collection.
   Proof. set_solver. Qed.
   Lemma difference_intersection_distr_l X Y Z : (X ∩ Y) ∖ Z ≡ X ∖ Z ∩ Y ∖ Z.
   Proof. set_solver. Qed.
-  Lemma difference_disjoint X Y : X ⊥ Y → X ∖ Y ≡ X.
+  Lemma difference_disjoint X Y : X ## Y → X ∖ Y ≡ X.
   Proof. set_solver. Qed.
 
   Lemma difference_mono X1 X2 Y1 Y2 :
@@ -630,7 +630,7 @@ Section collection.
   Proof. set_solver. Qed.
 
   (** Disjointness *)
-  Lemma disjoint_intersection X Y : X ⊥ Y ↔ X ∩ Y ≡ ∅.
+  Lemma disjoint_intersection X Y : X ## Y ↔ X ∩ Y ≡ ∅.
   Proof. set_solver. Qed.
 
   Section leibniz.
@@ -683,11 +683,11 @@ Section collection.
     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.
-    Lemma difference_disjoint_L X Y : X ⊥ Y → X ∖ Y = X.
+    Lemma difference_disjoint_L X Y : X ## Y → X ∖ Y = X.
     Proof. unfold_leibniz. apply difference_disjoint. Qed.
 
     (** Disjointness *)
-    Lemma disjoint_intersection_L X Y : X ⊥ Y ↔ X ∩ Y = ∅.
+    Lemma disjoint_intersection_L X Y : X ## Y ↔ X ∩ Y = ∅.
     Proof. unfold_leibniz. apply disjoint_intersection. Qed.
   End leibniz.
 
@@ -707,7 +707,7 @@ Section collection.
       intros x. rewrite !elem_of_union; rewrite elem_of_difference.
       split; [ | destruct (decide (x ∈ Y)) ]; intuition.
     Qed.
-    Lemma subseteq_disjoint_union X Y : X ⊆ Y ↔ ∃ Z, Y ≡ X ∪ Z ∧ X ⊥ Z.
+    Lemma subseteq_disjoint_union X Y : X ⊆ Y ↔ ∃ Z, Y ≡ X ∪ Z ∧ X ## Z.
     Proof.
       split; [|set_solver].
       exists (Y ∖ X); split; [auto using union_difference|set_solver].
@@ -732,7 +732,7 @@ Section collection.
     Proof. unfold_leibniz. apply non_empty_difference. Qed.
     Lemma empty_difference_subseteq_L X Y : X ∖ Y = ∅ → X ⊆ Y.
     Proof. unfold_leibniz. apply empty_difference_subseteq. Qed.
-    Lemma subseteq_disjoint_union_L X Y : X ⊆ Y ↔ ∃ Z, Y = X ∪ Z ∧ X ⊥ Z.
+    Lemma subseteq_disjoint_union_L X Y : X ⊆ Y ↔ ∃ Z, Y = X ∪ Z ∧ X ## Z.
     Proof. unfold_leibniz. apply subseteq_disjoint_union. Qed.
     Lemma singleton_union_difference_L X Y x :
       {[x]} ∪ (X ∖ Y) = ({[x]} ∪ X) ∖ (Y ∖ {[x]}).
@@ -1052,7 +1052,7 @@ Section seq_set.
   Qed.
 
   Lemma seq_set_S_disjoint start len :
-    {[ start + len ]} ⊥ seq_set (C:=C) start len.
+    {[ start + len ]} ## seq_set (C:=C) start len.
   Proof. intros x. rewrite elem_of_singleton, elem_of_seq_set. omega. Qed.
 
   Lemma seq_set_S_union start len :

theories/fin_collections.v

@@ -122,7 +122,7 @@ Proof.
   - set_solver.
 Qed.
 
-Lemma size_union X Y : X ⊥ Y → size (X ∪ Y) = size X + size Y.
+Lemma size_union X Y : X ## Y → size (X ∪ Y) = size X + size Y.
 Proof.
   intros. unfold size, collection_size. simpl. rewrite <-app_length.
   apply Permutation_length, NoDup_Permutation.

theories/fin_map_dom.v

@@ -81,14 +81,14 @@ 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 ⊥ₘ 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_disjoint.
   setoid_rewrite elem_of_dom. unfold is_Some. naive_solver.
 Qed.
-Lemma map_disjoint_dom_1 {A} (m1 m2 : M A) : m1 ⊥ₘ 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 ⊥ₘ 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.