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[helm.git] / matita / contribs / PREDICATIVE-TOPOLOGY / coa_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: NON COMPILA: attendo che l'oggetto "pippo" venga accettato *) 
+
+set "baseuri" "cic:/matita/PREDICATIVE-TOPOLOGY/coa_defs".
+
+include "iff.ma".
+include "domain_data.ma".
+
+(* COMPLETE OVERLAP ALGEBRAS 
+*)
+
+record COA: Type \def {
+   coa:> Class;                                  (* carrier *)
+        le: coa \to coa \to Prop;                     (* inclusion *)
+        ov: coa \to coa \to Prop;                     (* overlap *)
+        sup: \forall (D:Domain). (D \to coa) \to coa; (* supremum *)
+        inf: \forall (D:Domain). (D \to coa) \to coa; (* infimum *)
+        le_refl: \forall p. le p p;
+        le_trans: \forall p,r. le p r \to \forall q. le r q \to le p q; 
+        le_antysym: \forall q,p. le q p \to le p q \to ceq ? p q;
+        ov_sym: \forall q,p. ov q p \to ov p q;
+        sup_le: \forall D,ps,q. le (sup D ps) q \liff \iforall d. le (ps d) q;
+        inf_le: \forall D,p,qs. le p (inf D qs) \liff \iforall d. le p (qs d);
+        sup_ov: \forall D,ps,q. ov (sup D ps) q \liff \iexists d. ov (ps d) q;
+        density: \forall p,q. (\forall r. ov p r \to ov q r) \to le p q
+}.
+
+definition zero: \forall (P:COA). P \def
+   \lambda (P:COA). inf P ? (dvoid_ixfam P).
+
+definition one: \forall (P:COA). P \def
+   \lambda (P:COA). sup P ? (dvoid_ixfam P).
+
+definition binf: \forall (P:COA). P \to P \to P \def
+   \lambda (P:COA). \lambda p0,p1.
+   inf P ? (dbool_ixfam P p0 p1).
+
+definition bsup: \forall (P:COA). P \to P \to P \def
+   \lambda (P:COA). \lambda p0,p1.
+   sup P ? (dbool_ixfam P p0 p1).
+
+(*                          
+   inf_ov: forall p q, ov p q -> ov p (inf QDBool (bool_family _ p q))
+        properness: ov zero zero -> False;
+        distributivity: forall I p q, id _ (inf QDBool (bool_family _ (sup I p) q)) (sup I (fun i => (inf QDBool (bool_family _ (p i) q))));
+*)
+
+inductive pippo : Prop \def
+   | Pippo: let x \def zero in zero = x \to pippo.
+