--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/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.
+
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