Coercions are now inserted also around the sources of lambda abstractions.

[helm.git] / helm / matita / contribs / PREDICATIVE-TOPOLOGY / ac_defs.ma

diff --git a/helm/matita/contribs/PREDICATIVE-TOPOLOGY/ac_defs.ma b/helm/matita/contribs/PREDICATIVE-TOPOLOGY/ac_defs.ma
index f2cd4860fb95effe7b5f054cad9362f56a27c769..b375c69ada732dd06bc7926a945f3e59baada69b 100644 (file)
--- a/helm/matita/contribs/PREDICATIVE-TOPOLOGY/ac_defs.ma
+++ b/helm/matita/contribs/PREDICATIVE-TOPOLOGY/ac_defs.ma
@@ -12,11 +12,11 @@
 (*                                                                        *)
 (**************************************************************************)
 
-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
@@ -40,5 +40,47 @@ record AC: Type \def {
 
 coercion ac. 
 
-inductive eq (A:AC) : A \to A \to Prop \def
-   | eq_refl: \forall a. acin ? a \to eq A a a.
+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.
+
+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.
+
+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.