matita/matita/contribs/lambdadelta/ground/lib/subset_ext.ma

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   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
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  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 (* EXTENSIONS FOR SUBSETS ***************************************************)
  18
  19 definition subset_ext_f1 (A1) (A0) (f:A1→A0): 𝒫❨A1❩ → 𝒫❨A0❩ ≝
  20            λu1,a0. ∃∃a1. a1 ϵ u1 & f a1 = a0.
  21
  22 definition subset_ext_f1_1 (A11) (A21) (A0) (f1:A11→A0) (f2:A21→A0): 𝒫❨A11❩ → 𝒫❨A21❩ → 𝒫❨A0❩ ≝
  23            λu11,u21,a0.
  24            ∨∨ subset_ext_f1 A11 A0 f1 u11 a0
  25             | subset_ext_f1 A21 A0 f2 u21 a0.
  26
  27 definition subset_ext_p1 (A1) (Q:predicate A1): predicate (𝒫❨A1❩) ≝
  28            λu1. ∀a1. a1 ϵ u1 → Q a1.
  29
  30 (* Basic constructions ******************************************************)
  31
  32 lemma subset_in_ext_f1_dx (A1) (A0) (f) (u1) (a1):
  33       a1 ϵ u1 → f a1 ϵ subset_ext_f1 A1 A0 f u1.
  34 /2 width=3 by ex2_intro/ qed.
  35
  36 lemma subset_in_ext_f1_1_dx_1 (A11) (A21) (A0) (f1) (f2) (u11) (u21) (a11):
  37       a11 ϵ u11 → f1 a11 ϵ subset_ext_f1_1 A11 A21 A0 f1 f2 u11 u21.
  38 /3 width=3 by subset_in_ext_f1_dx, or_introl/ qed.
  39
  40 lemma subset_in_ext_f1_1_dx_2 (A11) (A21) (A0) (f1) (f2) (u11) (u21) (a21):
  41       a21 ϵ u21 → f2 a21 ϵ subset_ext_f1_1 A11 A21 A0 f1 f2 u11 u21.
  42 /3 width=3 by subset_in_ext_f1_dx, or_intror/ qed.
  43
  44 (* Basic inversions *********************************************************)
  45
  46 lemma subset_in_inv_ext_p1_dx (A1) (Q) (u1) (a1):
  47       a1 ϵ u1 → subset_ext_p1 A1 Q u1 → Q a1.
  48 /2 width=1 by/ qed-.