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[helm.git] / 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                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* STATO: COMPILA *)
+
+(* Project started Wed Oct 12, 2005 ***************************************)
+
+set "baseuri" "cic:/matita/PREDICATIVE-TOPOLOGY/class_defs".
+
+include "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;
+   ceq: class \to class \to Prop;
+   cin_repl: \forall c1,c2. cin c1 \to ceq c1 c2 \to cin c2;
+   ceq_repl: \forall c1,c2,d1,d2. cin c1 \to
+      ceq c1 c2 \to ceq c1 d1 \to ceq c2 d2 \to ceq d1 d2;
+   ceq_refl: \forall c. cin c \to ceq c c
+}.
+
+(* external universal quantification *)
+inductive call (C:Class) (P:C \to Prop) : Prop \def
+   | call_intro: (\forall c. cin ? c \to P c) \to call C P.
+
+inductive call2 (C1,C2:Class) (P:C1 \to C2 \to Prop) : Prop \def
+   | call2_intro: 
+      (\forall c1,c2. cin ? c1 \to cin ? c2 \to P c1 c2) \to call2 C1 C2 P.
+
+(* notations **************************************************************)
+
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation "external for all" 'xforall \eta.x =
+  (cic:/matita/PREDICATIVE-TOPOLOGY/class_defs/call.ind#xpointer(1/1) _ x).
+
+notation > "hvbox(\xforall ident i opt (: ty) break . p)"
+  right associative with precedence 20
+for @{ 'xforall ${default
+  @{\lambda ${ident i} : $ty. $p}
+  @{\lambda ${ident i} . $p}}}.
+
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation "external for all 2" 'xforall2 \eta.x \eta.y =
+  (cic:/matita/PREDICATIVE-TOPOLOGY/class_defs/call2.ind#xpointer(1/1) _ _ x y).
+
+notation > "hvbox(\xforall ident i1 opt (: ty1) ident i2 opt (: ty2) break . p)"
+  right associative with precedence 20
+for @{ 'xforall2 ${default
+  @{\lambda ${ident i1} : $ty1. \lambda ${ident i2} : $ty2. $p}
+  @{\lambda ${ident i1}, ${ident i2}. $p}}}.