helm/software/matita/contribs/ng_assembly/num/oct_lemmas.ma

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   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
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   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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  14
  15 (* ********************************************************************** *)
  16 (*                          Progetto FreeScale                            *)
  17 (*                                                                        *)
  18 (*   Sviluppato da: Ing. Cosimo Oliboni, oliboni@cs.unibo.it              *)
  19 (*   Ultima modifica: 05/08/2009                                          *)
  20 (*                                                                        *)
  21 (* ********************************************************************** *)
  22
  23 include "num/oct.ma".
  24 include "num/bool_lemmas.ma".
  25
  26 (* ****** *)
  27 (* OTTALI *)
  28 (* ****** *)
  29
  30 ndefinition oct_destruct_aux ≝
  31 Πn1,n2:oct.ΠP:Prop.n1 = n2 →
  32  match eq_oct n1 n2 with [ true ⇒ P → P | false ⇒ P ].
  33
  34 ndefinition oct_destruct : oct_destruct_aux.
  35  #n1; #n2; #P; #H;
  36  nrewrite < H;
  37  nelim n1;
  38  nnormalize;
  39  napply (λx.x).
  40 nqed.
  41
  42 nlemma symmetric_eqoct : symmetricT oct bool eq_oct.
  43  #n1; #n2;
  44  nelim n1;
  45  nelim n2;
  46  nnormalize;
  47  napply refl_eq.
  48 nqed.
  49
  50 nlemma eqoct_to_eq : ∀n1,n2.eq_oct n1 n2 = true → n1 = n2.
  51  #n1; #n2;
  52  ncases n1;
  53  ncases n2;
  54  nnormalize;
  55  ##[ ##1,10,19,28,37,46,55,64: #H; napply refl_eq
  56  ##| ##*: #H; napply (bool_destruct … H)
  57  ##]
  58 nqed.
  59
  60 nlemma eq_to_eqoct : ∀n1,n2.n1 = n2 → eq_oct n1 n2 = true.
  61  #n1; #n2;
  62  ncases n1;
  63  ncases n2;
  64  nnormalize;
  65  ##[ ##1,10,19,28,37,46,55,64: #H; napply refl_eq
  66  ##| ##*: #H; napply (oct_destruct … H)
  67  ##]
  68 nqed.
  69
  70 nlemma decidable_oct : ∀x,y:oct.decidable (x = y).
  71  #x; #y;
  72  nnormalize;
  73  nelim x;
  74  nelim y;
  75  ##[ ##1,10,19,28,37,46,55,64: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
  76  ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …);
  77           nnormalize; #H;
  78           napply False_ind;
  79           napply (oct_destruct … H)
  80  ##]
  81 nqed.
  82
  83 nlemma neqoct_to_neq : ∀n1,n2.eq_oct n1 n2 = false → n1 ≠ n2.
  84  #n1; #n2;
  85  ncases n1;
  86  ncases n2;
  87  nnormalize;
  88  ##[ ##1,10,19,28,37,46,55,64: #H; napply (bool_destruct … H)
  89  ##| ##*: #H; #H1; napply (oct_destruct … H1)
  90  ##]
  91 nqed.
  92
  93 nlemma neq_to_neqoct : ∀n1,n2.n1 ≠ n2 → eq_oct n1 n2 = false.
  94  #n1; #n2;
  95  ncases n1;
  96  ncases n2;
  97  nnormalize;
  98  ##[ ##1,10,19,28,37,46,55,64: #H; nelim (H (refl_eq …))
  99  ##| ##*: #H; napply refl_eq
 100  ##]
 101 nqed.