Not is now inductive.

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

diff --git a/helm/software/matita/library/logic/equality.ma b/helm/software/matita/library/logic/equality.ma
index 510f0c5c6fe3144b395840770c1f8dbe15084163..f4f12ec6c103a8290765517c95b151985e04cbab 100644 (file)
--- a/helm/software/matita/library/logic/equality.ma
+++ b/helm/software/matita/library/logic/equality.ma
@@ -1,5 +1,5 @@
 (**************************************************************************)
-(*       ___                                                             *)
+(*       ___                                                               *)
 (*      ||M||                                                             *)
 (*      ||A||       A project by Andrea Asperti                           *)
 (*      ||T||                                                             *)
@@ -12,20 +12,14 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/logic/equality/".
-
 include "higher_order_defs/relations.ma".
 
 inductive eq (A:Type) (x:A) : A \to Prop \def
     refl_eq : eq A x x.
 
-(*CSC: the URI must disappear: there is a bug now *)
-interpretation "leibnitz's equality"
-   'eq x y = (cic:/matita/logic/equality/eq.ind#xpointer(1/1) _ x y).
-(*CSC: the URI must disappear: there is a bug now *)
-interpretation "leibnitz's non-equality"
-  'neq x y = (cic:/matita/logic/connectives/Not.con
-    (cic:/matita/logic/equality/eq.ind#xpointer(1/1) _ x y)).
+interpretation "leibnitz's equality" 'eq t x y = (eq t x y).
+
+interpretation "leibnitz's non-equality" 'neq t x y = (Not (eq t x y)).
 
 theorem eq_rect':
  \forall A. \forall x:A. \forall P: \forall y:A. x=y \to Type.
@@ -54,12 +48,31 @@ qed.
 variant trans_eq : \forall A:Type.\forall x,y,z:A. x=y  \to y=z \to x=z
 \def transitive_eq.
 
+theorem symmetric_not_eq: \forall A:Type. symmetric A (λx,y.x ≠ y).
+unfold symmetric.simplify.intros.unfold.intro.apply H.apply sym_eq.assumption.
+qed.
+
+variant sym_neq: ∀A:Type.∀x,y.x ≠ y →y ≠ x
+≝ symmetric_not_eq.
+
 theorem eq_elim_r:
  \forall A:Type.\forall x:A. \forall P: A \to Prop.
    P x \to \forall y:A. y=x \to P y.
 intros. elim (sym_eq ? ? ? H1).assumption.
 qed.
 
+theorem eq_elim_r':
+ \forall A:Type.\forall x:A. \forall P: A \to Set.
+   P x \to \forall y:A. y=x \to P y.
+intros. elim (sym_eq ? ? ? H).assumption.
+qed.
+
+theorem eq_elim_r'':
+ \forall A:Type.\forall x:A. \forall P: A \to Type.
+   P x \to \forall y:A. y=x \to P y.
+intros. elim (sym_eq ? ? ? H).assumption.
+qed.
+
 theorem eq_f: \forall  A,B:Type.\forall f:A\to B.
 \forall x,y:A. x=y \to f x = f y.
 intros.elim H.apply refl_eq.
@@ -71,8 +84,8 @@ intros.elim H.apply refl_eq.
 qed.
 
 (*  *)
-coercion cic:/matita/logic/equality/sym_eq.con.
-coercion cic:/matita/logic/equality/eq_f.con.
+coercion sym_eq.
+coercion eq_f.
 (* *)
 
 default "equality"
@@ -81,6 +94,10 @@ default "equality"
  cic:/matita/logic/equality/transitive_eq.con
  cic:/matita/logic/equality/eq_ind.con
  cic:/matita/logic/equality/eq_elim_r.con
+ cic:/matita/logic/equality/eq_rec.con
+ cic:/matita/logic/equality/eq_elim_r'.con
+ cic:/matita/logic/equality/eq_rect.con
+ cic:/matita/logic/equality/eq_elim_r''.con
  cic:/matita/logic/equality/eq_f.con
 (* *)
  cic:/matita/logic/equality/eq_OF_eq.con.