bool.ma

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   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
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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 "basics/bool.ma".
  16 include "ground_2/lib/relations.ma".
  17 include "ground_2/notation/constructors/no_0.ma".
  18 include "ground_2/notation/constructors/yes_0.ma".
  19
  20 (* BOOLEAN PROPERTIES *******************************************************)
  21
  22 interpretation "boolean false" 'no = false.
  23
  24 interpretation "boolean true" 'yes = true.
  25
  26 (* Basic properties *********************************************************)
  27
  28 lemma commutative_orb: commutative … orb.
  29 * * // qed.
  30
  31 lemma orb_true_dx: ∀b. (b ∨ Ⓣ) = Ⓣ.
  32 * // qed.
  33
  34 lemma orb_true_sn: ∀b. (Ⓣ ∨ b) = Ⓣ.
  35 // qed.
  36
  37 lemma commutative_andb: commutative … andb.
  38 * * // qed.
  39
  40 lemma andb_false_dx: ∀b. (b ∧ Ⓕ) = Ⓕ.
  41 * // qed.
  42
  43 lemma andb_false_sn: ∀b. (Ⓕ ∧ b) = Ⓕ.
  44 // qed.
  45
  46 lemma eq_bool_dec: ∀b1,b2:bool. Decidable (b1 = b2).
  47 * * /2 width=1 by or_introl/
  48 @or_intror #H destruct
  49 qed-.
  50
  51 (* Basic inversion lemmas ***************************************************)
  52
  53 lemma orb_inv_false_dx: ∀b1,b2:bool. (b1 ∨ b2) = Ⓕ → b1 = Ⓕ ∧ b2 = Ⓕ.
  54 * normalize /2 width=1 by conj/ #b2 #H destruct
  55 qed-.
  56
  57 lemma andb_inv_true_dx: ∀b1,b2:bool. (b1 ∧ b2) = Ⓣ → b1 = Ⓣ ∧ b2 = Ⓣ.
  58 * normalize /2 width=1 by conj/ #b2 #H destruct
  59 qed-.