matita/matita/contribs/lambdadelta/ground/lib/subset_inclusion.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/subset.ma".
  16
  17 (* INCLUSION FOR SUBSETS ****************************************************)
  18
  19 definition subset_le (A): relation2 𝒫❨A❩ 𝒫❨A❩ ≝
  20            λu1,u2. ∀p. p ϵ u1 → p ϵ u2.
  21
  22 interpretation
  23   "inclusion (subset)"
  24   'subseteq u1 u2 = (subset_le ? u1 u2).
  25
  26 (* Basic constructions ******************************************************)
  27
  28 lemma subset_le_refl (A):
  29       reflexive … (subset_le A).
  30 // qed.
  31
  32 (* Main constructions *******************************************************)
  33
  34 theorem subset_le_trans (A):
  35         Transitive … (subset_le A).
  36 /3 width=1 by/ qed-.