matita/matita/contribs/lambdadelta/ground/lib/bool_and.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/bool.ma".
  16
  17 (* CONJUNCTION FOR BOOLEANS *************************************************)
  18
  19 (* Advanced constructions ***************************************************)
  20
  21 lemma commutative_andb:
  22       commutative … andb.
  23 * * // qed.
  24
  25 lemma andb_false_dx (b):
  26       Ⓕ = (b ∧ Ⓕ).
  27 * // qed.
  28
  29 lemma andb_false_sn (b):
  30       Ⓕ = (Ⓕ ∧ b).
  31 // qed.
  32
  33 lemma andb_true_dx (b):
  34       b = (b ∧ Ⓣ).
  35 * // qed.
  36
  37 lemma andb_true_sn (b):
  38       b = (Ⓣ ∧ b).
  39 // qed.
  40
  41 (* Advanced inversions ******************************************************)
  42
  43 lemma andb_inv_true_sn (b1) (b2):
  44       Ⓣ = (b1 ∧ b2) → ∧∧ Ⓣ = b1 & Ⓣ = b2.
  45 * normalize
  46 [ /2 width=1 by conj/
  47 | #b2 #H destruct
  48 ]
  49 qed-.