1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 set "baseuri" "cic:/matita/PREDICATIVE-TOPOLOGY/subset_defs".
17 include "domain_defs.ma".
20 - We use predicative subsets coded as propositional functions
21 according to G.Sambin and S.Valentini "Toolbox"
24 definition Subset \def \lambda (D:Domain). D \to Prop.
26 (* subset membership (epsilon) *)
27 definition sin : \forall D. Subset D \to D \to Prop \def
28 \lambda (D:Domain). \lambda U,d. cin D d \and U d.
30 (* subset top (full subset) *)
31 definition stop \def \lambda (D:Domain). true_f D.
33 (* subset bottom (empty subset) *)
34 definition sbot \def \lambda (D:Domain). false_f D.
36 (* subset and (binary intersection) *)
37 definition sand: \forall D. Subset D \to Subset D \to Subset D \def
38 \lambda D,U1,U2,d. U1 d \land U2 d.
40 (* subset or (binary union) *)
41 definition sor: \forall D. Subset D \to Subset D \to Subset D \def
42 \lambda D,U1,U2,d. U1 d \lor U2 d.
44 (* subset less or equal (inclusion) *)
45 definition sle: \forall D. Subset D \to Subset D \to Prop \def
46 \lambda D,U1,U2. \iforall d. U1 d \to U2 d.
49 definition sover: \forall D. Subset D \to Subset D \to Prop \def
50 \lambda D,U1,U2. \iexists d. U1 d \land U2 d.
52 (* coercions **************************************************************)
55 (* the class of the subsets of a domain (not an implicit coercion) *)
56 definition class_of_subsets_of \def
57 \lambda D. mk_Class (Subset D) (true_f ?) (sle ?).
60 (* the domain built upon a subset (not an implicit coercion) *)
61 definition domain_of_subset: \forall D. Subset D \to Domain \def
62 \lambda (D:Domain). \lambda U.
63 mk_Domain (mk_Class D (sin D U) (cle1 D)).
65 (* the full subset of a domain *)