--- /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 *)
+(* *)
+(**************************************************************************)
+
+include "Domain/defs.ma".
+
+(* SUBSETS
+ - We use predicative subsets coded as propositional functions
+ according to G.Sambin and S.Valentini "Toolbox".
+*)
+
+definition Subset ≝ λD:Domain. D → Prop.
+
+(* subset membership (epsilon) *)
+definition sin : ∀D. Subset D → D → Prop ≝
+ λD:Domain. λU,d. cin D d ∧ U d.
+
+(* subset top (full subset) *)
+definition stop ≝ λD:Domain. true_f D.
+
+(* subset bottom (empty subset) *)
+definition sbot ≝ λD:Domain. false_f D.
+
+(* subset and (binary intersection) *)
+definition sand: ∀D. Subset D → Subset D → Subset D ≝
+ λD,U1,U2,d. U1 d ∧ U2 d.
+
+(* subset or (binary union) *)
+definition sor: ∀D. Subset D → Subset D → Subset D ≝
+ λD,U1,U2,d. U1 d ∨ U2 d.
+
+(* subset less or equal (inclusion) *)
+definition sle: ∀D. Subset D → Subset D → Prop ≝
+ λD,U1,U2. \iforall d. U1 d → U2 d.
+
+(* subset overlap *)
+definition sover: ∀D. Subset D → Subset D → Prop ≝
+ λD,U1,U2. \iexists d. U1 d ∧ 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 (not an implicit coercion) *)
+definition domain_of_subset: ∀D. Subset D \to Domain ≝
+ λD:Domain. λU. mk_Domain (mk_Class D (sin D U) (ces D)).
+
+(* the full subset of a domain *)
+coercion stop.