helm/matita/contribs/PREDICATIVE-TOPOLOGY/subset_defs.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 set "baseuri" "cic:/matita/PREDICATIVE-TOPOLOGY/subset_defs".
  16
  17 include "domain_defs.ma".
  18
  19 (* SUBSETS
  20    - We use predicative subsets coded as propositional functions
  21      according to G.Sambin and S.Valentini "Toolbox"
  22 *)
  23
  24 definition Subset \def \lambda (D:Domain). D \to Prop.
  25
  26 (* subset inclusion *)
  27 definition ssub: \forall D. Subset D \to Subset D \to Prop \def
  28    \lambda D,U1,U2. \forall d. U1 d \to U2 d.
  29
  30
  31 (* subset overlap *)
  32 definition sover: \forall D. Subset D \to Subset D \to Prop \def
  33    \lambda D,U1,U2. \forall d. U1 d \to U2 d.
  34
  35
  36
  37 (* full subset: "subset top" *)
  38 definition stop \def \lambda (D:Domain). \lambda (_:D). True.
  39
  40 coercion stop.
  41
  42 (* empty subset: "subset bottom" *)
  43 definition sbot \def \lambda (D:Domain). \lambda (_:D). False.