matita/matita/contribs/lambdadelta/ground/lib/subset_ext_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_inclusion.ma".
  16 include "ground/lib/subset_ext.ma".
  17 include "ground/lib/exteq.ma".
  18
  19 (* EXTENSIONS FOR SUBSETS ***************************************************)
  20
  21 (* Constructions with subset_inclusion **************************************)
  22
  23 lemma subset_inclusion_ext_f1_exteq (A1) (A0) (f1) (f2) (u):
  24       f1 ⊜ f2 → subset_ext_f1 A1 A0 f1 u ⊆ subset_ext_f1 A1 A0 f2 u.
  25 #A1 #A0 #f1 #f2 #u #Hf #a0 * #a1 #Hau1 #H destruct
  26 /2 width=1 by subset_in_ext_f1_dx/
  27 qed.
  28
  29 lemma subset_inclusion_ext_f1_bi (A1) (A0) (f) (u1) (v1):
  30       u1 ⊆ v1 → subset_ext_f1 A1 A0 f u1 ⊆ subset_ext_f1 A1 A0 f v1.
  31 #A1 #A0 #f #u1 #v1 #Huv1 #a0 * #a1 #Hau1 #H destruct
  32 /3 width=3 by subset_in_ext_f1_dx/
  33 qed.
  34
  35 lemma subset_inclusion_ext_f1_compose_sn (A0) (A1) (A2) (f1) (f2) (u):
  36       subset_ext_f1 A1 A2 f2 (subset_ext_f1 A0 A1 f1 u) ⊆ subset_ext_f1 A0 A2 (f2∘f1) u.
  37 #A0 #A1 #A2 #f1 #f2 #u #a2 * #a1 * #a0 #Ha0 #H1 #H2 destruct
  38 /2 width=1 by subset_in_ext_f1_dx/
  39 qed.
  40
  41 lemma subset_inclusion_ext_f1_compose_dx (A0) (A1) (A2) (f1) (f2) (u):
  42       subset_ext_f1 A0 A2 (f2∘f1) u ⊆ subset_ext_f1 A1 A2 f2 (subset_ext_f1 A0 A1 f1 u).
  43 #A0 #A1 #A2 #f1 #f2 #u #a2 * #a0 #Ha0 #H0 destruct
  44 /3 width=1 by subset_in_ext_f1_dx/
  45 qed.
  46
  47 lemma subset_inclusion_ext_f1_1_bi (A11) (A21) (A0) (f1) (f2) (u11) (u21) (v11) (v21):
  48       u11 ⊆ v11 → u21 ⊆ v21 →
  49       subset_ext_f1_1 A11 A21 A0 f1 f2 u11 u21 ⊆ subset_ext_f1_1 A11 A21 A0 f1 f2 v11 v21.
  50 #A11 #A21 #A0 #f1 #f2 #u11 #u21 #v11 #v21 #Huv11 #Huv21 #a0 *
  51 /3 width=3 by subset_inclusion_ext_f1_bi, or_introl, or_intror/
  52 qed.
  53
  54 lemma subset_inclusion_ext_p1_trans (A1) (Q) (u1) (v1):
  55       u1 ⊆ v1 → subset_ext_p1 A1 Q v1 → subset_ext_p1 A1 Q u1.
  56 #A1 #Q #u1 #v1 #Huv1 #Hv1
  57 /3 width=1 by/
  58 qed-.