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

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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* Project started Wed Oct 12, 2005 ***************************************)
+
+set "baseuri" "cic:/matita/PREDICATIVE-TOPOLOGY/class_defs".
+
+include "../../library/logic/connectives.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 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 cle1 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
+*) 
+
+definition true_f \def \lambda (X:Type). \lambda (_:X). True.
+
+definition false_f \def \lambda (X:Type). \lambda (_:X). False.
+
+record Class: Type \def {
+   class:> Type;
+   cin: class \to Prop;
+   cle1: class \to class \to Prop
+}.
+
+inductive cle (C:Class) (c1:C): C \to Prop \def
+   | cle_refl: cin ? c1 \to cle ? c1 c1
+   | ceq_sing: \forall c2,c3. 
+               cle ? c1 c2 \to cin ? c3 \to cle1 ? c2 c3 \to cle ? c1 c3.
+
+inductive ceq (C:Class) (c1:C) (c2:C): Prop \def
+   | ceq_intro: cle ? c1 c2 \to cle ? c2 c1 \to ceq ? c1 c2.