From 92bc50d79f2cb7c3f94c71e967f62480533cc5b4 Mon Sep 17 00:00:00 2001
From: Niels <nnmvdw@gmail.com>
Date: Tue, 8 Aug 2017 15:30:12 +0200
Subject: [PATCH] Improved notation to unicode

---
 FiniteSets/notation.v | 89 +++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 89 insertions(+)
 create mode 100644 FiniteSets/notation.v

diff --git a/FiniteSets/notation.v b/FiniteSets/notation.v
new file mode 100644
index 0000000..76d66d5
--- /dev/null
+++ b/FiniteSets/notation.v
@@ -0,0 +1,89 @@
+Require Import HoTT.
+
+Section binary_operation.
+  Variable A : Type.
+  
+  Definition operation := A -> A -> A.
+End binary_operation.
+
+Section Defs.
+  Variable A : Type.
+  Variable f : A -> A -> A.
+
+  Class Commutative :=
+    commutative : forall x y, f x y = f y x.
+
+  Class Associative :=
+    associativity : forall x y z, f (f x y) z = f x (f y z).
+
+  Class Idempotent :=
+    idempotency : forall x, f x x = x.
+
+  Variable g : operation A.
+
+  Class Absorption :=
+    absorb : forall x y, f x (g x y) = x.
+
+  Variable n : A.
+
+  Class NeutralL :=
+    neutralityL : forall x, f n x = x.
+
+  Class NeutralR :=
+    neutralityR : forall x, f x n = x.
+
+End Defs.
+
+Arguments Commutative {_} _.
+Arguments Associative {_} _.
+Arguments Idempotent {_} _.
+Arguments NeutralL {_} _ _.
+Arguments NeutralR {_} _ _.
+Arguments Absorption {_} _ _.
+Arguments commutative {_} {_} {_} _ _.
+Arguments associativity {_} {_} {_} _ _ _.
+Arguments idempotency {_} {_} {_} _.
+Arguments neutralityL {_} {_} {_} {_} _.
+Arguments neutralityR {_} {_} {_} {_} _.
+Arguments absorb {_} {_} {_} {_} _ _.
+
+Section structure.
+  Variable (T : Type -> Type).
+  
+  Class hasMembership : Type :=
+    member : forall A : Type, A -> T A -> hProp.
+
+  Class hasEmpty : Type :=
+    empty : forall A, T A.
+
+  Class hasSingleton : Type :=
+    singleton : forall A, A -> T A.
+  
+  Class hasUnion : Type :=
+    union : forall A, T A -> T A -> T A.
+
+  Class hasIntersection : Type :=
+    intersection : forall A, T A -> T A -> T A.
+
+  Class hasComprehension : Type :=
+    filter : forall A, (A -> Bool) -> T A -> T A.
+End structure.
+
+Arguments member {_} {_} {_} _ _.
+Arguments empty {_} {_} {_}.
+Arguments singleton {_} {_} {_} _.
+Arguments union {_} {_} {_} _ _.
+Arguments filter {_} {_} {_} _ _.
+
+Notation "∅" := empty.
+Notation "{| x |}" :=  (singleton x).
+Infix "∪" := union (at level 8, right associativity).
+Notation "(∪)" := union (only parsing).
+Notation "( X ∪)" := (union X) (only parsing).
+Notation "( ∪ Y )" := (fun X => X ∪ Y) (only parsing).
+Infix "∩" := intersection (at level 8, right associativity).
+Notation "( ∩ )" := intersection (only parsing).
+Notation "( X ∩ )" := (intersection X) (only parsing).
+Notation "( ∩ Y )" := (fun X => X ∩ Y) (only parsing).
+Notation "{| X & ϕ |}" := (filter ϕ X).
+Infix "∈" := member (at level 9, right associativity).
\ No newline at end of file