matita/matita/contribs/lambdadelta/ground/lib/subset_ext_equivalence.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                  *)
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  13 (**************************************************************************)
  14
  15 include "ground/lib/subset_ext_inclusion.ma".
  16 include "ground/lib/subset_equivalence.ma".
  17
  18 (* EXTENSIONS FOR SUBSETS ***************************************************)
  19
  20 (* Constructions with subset_equivalence ************************************)
  21
  22 lemma subset_equivalence_ext_f1_exteq (A1) (A0) (f1) (f2) (u):
  23       f1 ⊜ f2 → subset_ext_f1 A1 A0 f1 u ⇔ subset_ext_f1 A1 A0 f2 u.
  24 /3 width=3 by subset_inclusion_ext_f1_exteq, conj/
  25 qed.
  26
  27 lemma subset_equivalence_ext_f1_bi (A1) (A0) (f) (u1) (v1):
  28       u1 ⇔ v1 → subset_ext_f1 A1 A0 f u1 ⇔ subset_ext_f1 A1 A0 f v1.
  29 #A1 #A0 #f #u1 #v1 * #Huv1 #Hvu1
  30 /3 width=3 by subset_inclusion_ext_f1_bi, conj/
  31 qed.
  32
  33 lemma subset_equivalence_ext_f1_1_bi (A11) (A21) (A0) (f1) (f2) (u11) (u21) (v11) (v21):
  34       u11 ⇔ v11 → u21 ⇔ v21 →
  35       subset_ext_f1_1 A11 A21 A0 f1 f2 u11 u21 ⇔ subset_ext_f1_1 A11 A21 A0 f1 f2 v11 v21.
  36 #A11 #A21 #A0 #f1 #f2 #u11 #u21 #v11 #v21 * #Huv11 #Hvu11 * #Huv21 #Hvu21
  37 /3 width=5 by subset_inclusion_ext_f1_1_bi, conj/
  38 qed.
  39
  40 lemma subset_inclusion_ext_f1_compose (A0) (A1) (A2) (f1) (f2) (u):
  41       subset_ext_f1 A1 A2 f2 (subset_ext_f1 A0 A1 f1 u) ⇔ subset_ext_f1 A0 A2 (f2∘f1) u.
  42 /3 width=1 by subset_inclusion_ext_f1_compose_dx, subset_inclusion_ext_f1_compose_sn, conj/
  43 qed.