definition Subset \def \lambda (D:Domain). D \to Prop.
-(* subset inclusion *)
-definition ssub: \forall D. Subset D \to Subset D \to Prop \def
- \lambda D,U1,U2. \forall d. U1 d \to U2 d.
+(* subset membership (epsilon) *)
+definition sin : \forall D. Subset D \to D \to Prop \def
+ \lambda (D:Domain). \lambda U,d. cin D d \and U d.
+
+(* subset top (full subset) *)
+definition stop \def \lambda (D:Domain). true_f D.
+
+(* subset bottom (empty subset) *)
+definition sbot \def \lambda (D:Domain). false_f D.
+
+(* subset and (binary intersection) *)
+definition sand: \forall D. Subset D \to Subset D \to Subset D \def
+ \lambda D,U1,U2,d. U1 d \land U2 d.
+(* subset or (binary union) *)
+definition sor: \forall D. Subset D \to Subset D \to Subset D \def
+ \lambda D,U1,U2,d. U1 d \lor U2 d.
+(* subset less or equal (inclusion) *)
+definition sle: \forall D. Subset D \to Subset D \to Prop \def
+ \lambda D,U1,U2. \forall d. U1 d \to U2 d.
+(*
(* subset overlap *)
definition sover: \forall D. Subset D \to Subset D \to Prop \def
\lambda D,U1,U2. \forall d. U1 d \to U2 d.
+*)
+
+(* coercions **************************************************************)
+(* the class of the subsets of a domain (not an implicit coercion) *)
+definition class_of_subsets_of \def
+ \lambda D. mk_Class (Subset D) (true_f ?) (sle ?).
+(* the domain built upon a subset *)
+definition domain_of_subset: \forall D. (Subset D) \to Domain \def
+ \lambda (D:Domain). \lambda U.
+ mk_Domain (mk_Class D (sin D U) (cle1 D)).
-(* full subset: "subset top" *)
-definition stop \def \lambda (D:Domain). \lambda (_:D). True.
+coercion domain_of_subset.
+(* the full subset of a domain *)
coercion stop.
-(* empty subset: "subset bottom" *)
-definition sbot \def \lambda (D:Domain). \lambda (_:D). False.
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