X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fmatita%2Fcontribs%2FPREDICATIVE-TOPOLOGY%2Fac_defs.ma;h=b375c69ada732dd06bc7926a945f3e59baada69b;hb=ca7c474381445e17d15ced6bf42da16946335598;hp=89e67c2f9dce96adce6cd4016658060d40060566;hpb=01856fd0fbc93c845b2f8fb3cec1e7a966f23210;p=helm.git

diff --git a/helm/matita/contribs/PREDICATIVE-TOPOLOGY/ac_defs.ma b/helm/matita/contribs/PREDICATIVE-TOPOLOGY/ac_defs.ma
index 89e67c2f9..b375c69ad 100644
--- a/helm/matita/contribs/PREDICATIVE-TOPOLOGY/ac_defs.ma
+++ b/helm/matita/contribs/PREDICATIVE-TOPOLOGY/ac_defs.ma
@@ -12,20 +12,25 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/PREDICATIVE-TOPOLOGY/ac_defs".
-
 (* Project started Wed Oct 12, 2005 ***************************************)
 
+set "baseuri" "cic:/matita/PREDICATIVE-TOPOLOGY/ac_defs".
 
+include "coq.ma".
 
-(* ACZEL CATEGORIES:                                                      *)
-(* We use typoids with a compatible membership relation                   *)
-(* The category is intended to be the domain of the membership relation   *)
-(* The membership relation is necessary because we need to regard the     *)
-(* domain of a propositional function (ie a subset in the predicative     *)
-(* setting) as a quantification domain and therefore as a category, but   *)
-(* there is no type in CIC representing the domain of a propositional     *)
-(* function                                                               *)
+(* ACZEL CATEGORIES:
+   - We use typoids with a compatible membership relation
+   - The category is intended to be the domain of the membership relation
+   - The membership relation is necessary because we need to regard the
+     domain of a propositional function (ie a predicative subset) as a
+     quantification domain and therefore as a category, but there is no
+     type in CIC representing the domain of a propositional function
+   - We set up a single equality predicate, parametric on the category,
+     defined as the reflexive, symmetic, transitive and compatible closure
+     of the aceq predicate given inside the category. Then we prove the 
+     properties of the equality that usually are axiomatized inside the 
+     category structure. This makes categories easier to use
+*) 
 
 record AC: Type \def {
    ac: Type;
@@ -33,9 +38,49 @@ record AC: Type \def {
    aceq: ac \to ac \to Prop
 }.
 
-definition a \def \lambda A. ac A.
+coercion ac. 
+
+inductive eq (A:AC) (a:A): A \to Prop \def
+   | eq_refl:   acin ? a \to eq ? a a
+   | eq_sing_r: \forall b,c. 
+                eq ? a b \to acin ? c \to aceq ? b c \to eq ? a c
+   | eq_sing_l: \forall b,c. 
+                eq ? a b \to acin ? c \to aceq ? c b \to eq ? a c.
+
+theorem eq_cl: \forall A,a,b. eq ? a b \to acin A a \land acin A b.
+intros; elim H; clear H; clear b; 
+   [ auto | decompose H2; auto | decompose H2; auto ].
+qed.
+
+theorem eq_trans: \forall A,b,a,c.
+                  eq A b c \to eq ? a b \to eq ? a c.
+intros 5; elim H; clear H; clear c;
+   [ auto 
+   | apply eq_sing_r; [||| apply H4 ]; auto
+   | apply eq_sing_l; [||| apply H4 ]; auto
+   ].
+qed.
+
+theorem eq_conf_rev: \forall A,b,a,c.
+                     eq A c b \to eq ? a b \to eq ? a c.
+intros 5; elim H; clear H; clear b;
+   [ auto 
+   | lapply eq_cl; [ decompose Hletin |||| apply H1 ].
+     apply H2; apply eq_sing_l; [||| apply H4 ]; auto
+   | lapply eq_cl; [ decompose Hletin |||| apply H1 ].
+     apply H2; apply eq_sing_r; [||| apply H4 ]; auto
+   ].
+qed.
 
-coercion a. 
+theorem eq_sym: \forall A,a,b. eq A a b \to eq A b a.
+intros;
+lapply eq_cl; [ decompose Hletin |||| apply H ].
+auto.
+qed.
 
-inductive eq (A:AC) : ac A \to ac A \to Prop \def
-   | eq_refl: \forall a. acin ? a \to eq A a a.
+theorem eq_conf: \forall A,b,a,c.
+                 eq A a b \to eq ? a c \to eq ? b c.
+intros.
+lapply eq_sym; [|||| apply H ].
+apply eq_trans; [| auto | auto ].
+qed.