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Commit 2e5cc41f authored by Jacques-Henri Jourdan's avatar Jacques-Henri Jourdan
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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
parents b99b916f 4b5d254e
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theories/base.v
View file @ 2e5cc41f
......@@ -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
View file @ 2e5cc41f
......@@ -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
View file @ 2e5cc41f
......@@ -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
View file @ 2e5cc41f
......@@ -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.
......
theories/fin_maps.v
View file @ 2e5cc41f
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