From a1dbfbc43a9f01eaf64d4f2cc66d6be1ac6943ce Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Sat, 29 Oct 2005 18:03:03 +0000
Subject: [PATCH] Aczel categories finished quantification domains started

---
 .../contribs/PREDICATIVE-TOPOLOGY/ac_defs.ma  | 49 +++++++++++++++++--
 .../contribs/PREDICATIVE-TOPOLOGY/qd_defs.ma  | 26 ++++++++++
 2 files changed, 71 insertions(+), 4 deletions(-)
 create mode 100644 helm/matita/contribs/PREDICATIVE-TOPOLOGY/qd_defs.ma

diff --git a/helm/matita/contribs/PREDICATIVE-TOPOLOGY/ac_defs.ma b/helm/matita/contribs/PREDICATIVE-TOPOLOGY/ac_defs.ma
index 6a5b86cf1..e4ee22311 100644
--- 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
@@ -41,5 +41,46 @@ record AC: Type \def {
 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 a b \to aceq A b c \to eq A a c. *)
+   | 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.
diff --git a/helm/matita/contribs/PREDICATIVE-TOPOLOGY/qd_defs.ma b/helm/matita/contribs/PREDICATIVE-TOPOLOGY/qd_defs.ma
new file mode 100644
index 000000000..cd019ccfc
--- /dev/null
+++ b/helm/matita/contribs/PREDICATIVE-TOPOLOGY/qd_defs.ma
@@ -0,0 +1,26 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/PREDICATIVE-TOPOLOGY/qd_defs".
+
+include "ac_defs.ma".
+
+(* QUANTIFICATION DOMAINS
+*)
+
+record QD: Type \def {
+   qd: AC
+}.
+
+coercion qd.
-- 
2.39.2