Notation for equality used everywhere.

[helm.git] / helm / matita / library / logic / equality.ma

diff --git a/helm/matita/library/logic/equality.ma b/helm/matita/library/logic/equality.ma
index fb7e66edfb5fd84071751997cc9ff8a3812567fd..0be69f9b7e3054e93cc5fc72270739a46ebcd06d 100644 (file)
--- a/helm/matita/library/logic/equality.ma
+++ b/helm/matita/library/logic/equality.ma
@@ -18,6 +18,11 @@ include "higher_order_defs/relations.ma".
 
 inductive eq (A:Type) (x:A) : A \to Prop \def
     refl_eq : eq A x x.
+
+interpretation "leibnitz's equality"
+   'eq x y = (cic:/matita/logic/equality/eq.ind#xpointer(1/1) _ x y).
+alias symbol "eq" (instance 0) = "leibnitz's equality".
+
     
 theorem reflexive_eq : \forall A:Type. reflexive A (eq A).
 simplify.intros.apply refl_eq.
@@ -27,19 +32,19 @@ theorem symmetric_eq: \forall A:Type. symmetric A (eq A).
 simplify.intros.elim H. apply refl_eq.
 qed.
 
-theorem sym_eq : \forall A:Type.\forall x,y:A. eq A x y  \to eq A y x
+theorem sym_eq : \forall A:Type.\forall x,y:A. x=y  \to y=x
 \def symmetric_eq.
 
 theorem transitive_eq : \forall A:Type. transitive A (eq A).
 simplify.intros.elim H1.assumption.
 qed.
 
-theorem trans_eq : \forall A:Type.\forall x,y,z:A. eq A x y  \to eq A y z \to eq A x z
+theorem trans_eq : \forall A:Type.\forall x,y,z:A. x=y  \to y=z \to x=z
 \def transitive_eq.
 
 theorem eq_elim_r:
  \forall A:Type.\forall x:A. \forall P: A \to Prop.
-   P x \to \forall y:A. eq A y x \to P y.
+   P x \to \forall y:A. y=x \to P y.
 intros. elim sym_eq ? ? ? H1.assumption.
 qed.
 
@@ -51,12 +56,12 @@ default "equality"
  cic:/matita/logic/equality/eq_elim_r.con. 
  
 theorem eq_f: \forall  A,B:Type.\forall f:A\to B.
-\forall x,y:A. eq A x y \to eq B (f x) (f y).
+\forall x,y:A. x=y \to (f x)=(f y).
 intros.elim H.reflexivity.
 qed.
 
 theorem eq_f2: \forall  A,B,C:Type.\forall f:A\to B \to C.
 \forall x1,x2:A. \forall y1,y2:B.
-eq A x1 x2\to eq B y1 y2\to eq C (f x1 y1) (f x2 y2).
+x1=x2 \to y1=y2 \to (f x1 y1)=(f x2 y2).
 intros.elim H1.elim H.reflexivity.
-qed.
\ No newline at end of file
+qed.