helm/software/matita/contribs/ng_assembly/freescale/nat_lemmas.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                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 (* ********************************************************************** *)
  16 (*                           Progetto FreeScale                           *)
  17 (*                                                                        *)
  18 (* Sviluppato da:                                                         *)
  19 (*   Cosimo Oliboni, oliboni@cs.unibo.it                                  *)
  20 (*                                                                        *)
  21 (* Questo materiale fa parte della tesi:                                  *)
  22 (*   "Formalizzazione Interattiva dei Microcontroller a 8bit FreeScale"   *)
  23 (*                                                                        *)
  24 (*                    data ultima modifica 15/11/2007                     *)
  25 (* ********************************************************************** *)
  26
  27 include "freescale/bool_lemmas.ma".
  28 include "freescale/nat.ma".
  29
  30 (* ******** *)
  31 (* NATURALI *)
  32 (* ******** *)
  33
  34 nlemma nat_destruct_S_S : ∀n1,n2:nat.S n1 = S n2 → n1 = n2.
  35  #n1; #n2; #H;
  36  nchange with (match S n2 with [ O ⇒ False | S a ⇒ n1 = a ]);
  37  nrewrite < H;
  38  nnormalize;
  39  napply (refl_eq ??).
  40 nqed.
  41
  42 nlemma nat_destruct_0_S : ∀n:nat.O = S n → False.
  43  #n; #H;
  44  nchange with (match S n with [ O ⇒ True | S a ⇒ False ]);
  45  nrewrite < H;
  46  nnormalize;
  47  napply I.
  48 nqed.
  49
  50 nlemma nat_destruct_S_0 : ∀n:nat.S n = O → False.
  51  #n; #H;
  52  nchange with (match S n with [ O ⇒ True | S a ⇒ False ]);
  53  nrewrite > H;
  54  nnormalize;
  55  napply I.
  56 nqed.
  57
  58 nlemma symmetric_eqnat : symmetricT nat bool eq_nat.
  59  nnormalize;
  60  #n1;
  61  nelim n1;
  62  ##[ ##1: #n2;
  63           nelim n2;
  64           nnormalize;
  65           ##[ ##1: napply (refl_eq ??)
  66           ##| ##2: #n3; #H; napply (refl_eq ??)
  67           ##]
  68  ##| ##2: #n2; #H; #n3;
  69           nnormalize;
  70           ncases n3;
  71           nnormalize;
  72           ##[ ##1: napply (refl_eq ??)
  73           ##| ##2: #n4; napply (H n4)
  74           ##]
  75  ##]
  76 nqed.
  77
  78 nlemma eq_to_eqnat : ∀n1,n2:nat.n1 = n2 → eq_nat n1 n2 = true.
  79  #n1;
  80  nelim n1;
  81  ##[ ##1: #n2;
  82           nelim n2;
  83           nnormalize;
  84           ##[ ##1: #H; napply (refl_eq ??)
  85           ##| ##2: #n3; #H; #H1; nelim (nat_destruct_0_S ? H1)
  86           ##]
  87  ##| ##2: #n2; #H; #n3; #H1;
  88           ncases n3 in H1:(%) ⊢ %;
  89           nnormalize;
  90           ##[ ##1: #H1; nelim (nat_destruct_S_0 ? H1)
  91           ##| ##2: #n4; #H1;
  92                    napply (H n4 (nat_destruct_S_S ?? H1))
  93           ##]
  94  ##]
  95 nqed.
  96
  97 nlemma eqnat_to_eq : ∀n1,n2:nat.(eq_nat n1 n2 = true → n1 = n2).
  98  #n1;
  99  nelim n1;
 100  ##[ ##1: #n2;
 101           nelim n2;
 102           nnormalize;
 103           ##[ ##1: #H; napply (refl_eq ??)
 104           ##| ##2: #n3; #H; #H1; napply (bool_destruct ?? (O = S n3) H1)
 105           ##]
 106  ##| ##2: #n2; #H; #n3; #H1;
 107           ncases n3 in H1:(%) ⊢ %;
 108           nnormalize;
 109           ##[ ##1: #H1; napply (bool_destruct ?? (S n2 = O) H1)
 110           ##| ##2: #n4; #H1;
 111                    nrewrite > (H n4 H1);
 112                    napply (refl_eq ??)
 113           ##]
 114  ##]
 115 nqed.