set "baseuri" "cic:/matita/PREDICATIVE-TOPOLOGY/coa_defs".
include "iff.ma".
-include "domain_defs.ma".
+include "domain_data.ma".
(* COMPLETE OVERLAP ALGEBRAS
*)
record COA: Type \def {
- coa: Class; (* carrier *)
+ 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; (* suprimum *)
+ 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;
density: \forall p,q. (\forall r. ov p r \to ov q r) \to le p q
}.
-coercion coa.
+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))));
-*)
\ No newline at end of file
+*)
+
+inductive pippo : Prop \def
+ | Pippo: let x \def zero in zero = x \to pippo.
+
\ No newline at end of file