matita/matita/contribs/lambdadelta/ground/lib/subset.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 include "ground/lib/relations.ma".
  16 include "ground/notation/functions/powerclass_1.ma".
  17 include "ground/notation/relations/epsilon_3.ma".
  18 include "ground/notation/relations/not_epsilon_3.ma".
  19
  20 (* SUBSETS ******************************************************************)
  21
  22 definition subset (A): Type[0] ≝
  23            predicate A.
  24
  25 interpretation
  26   "power class (subset)"
  27   'PowerClass A = (subset A).
  28
  29 definition subset_in (A): 𝒫❨A❩ → 𝒫❨A❩ ≝
  30            λu.u.
  31
  32 interpretation
  33   "membership (subset)"
  34   'Epsilon A a u = (subset_in A u a).
  35
  36 interpretation
  37   "negated membership (subset)"
  38   'NotEpsilon A a u = (negation (subset_in A u a)).