matitaB/matita/contribs/ng_assembly/common/ascii_lemmas.ma

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  14
  15 (* ********************************************************************** *)
  16 (*                          Progetto FreeScale                            *)
  17 (*                                                                        *)
  18 (*   Sviluppato da: Ing. Cosimo Oliboni, oliboni@cs.unibo.it              *)
  19 (*   Sviluppo: 2008-2010                                                  *)
  20 (*                                                                        *)
  21 (* ********************************************************************** *)
  22
  23 include "common/ascii.ma".
  24 include "num/bool_lemmas.ma".
  25
  26 (* ************************** *)
  27 (* DEFINIZIONE ASCII MINIMALE *)
  28 (* ************************** *)
  29
  30 (*
  31 ndefinition ascii_destruct_aux ≝
  32 Πc1,c2.ΠP:Prop.c1 = c2 →
  33  match eq_ascii c1 c2 with [ true ⇒ P → P | false ⇒ P ].
  34
  35 nlemma ascii_destruct : ascii_destruct_aux.
  36  #c1; #c2; #P; #H;
  37  nrewrite < H;
  38  nelim c1;
  39  nnormalize;
  40  napply (λx.x).
  41 nqed.
  42 *)
  43
  44 nlemma eq_to_eqascii : ∀n1,n2.n1 = n2 → eq_ascii n1 n2 = true.
  45  #n1; #n2; #H;
  46  nrewrite > H;
  47  nelim n2;
  48  nnormalize;
  49  napply refl_eq.
  50 nqed.
  51
  52 nlemma neqascii_to_neq : ∀n1,n2.eq_ascii n1 n2 = false → n1 ≠ n2.
  53  #n1; #n2; #H;
  54  napply (not_to_not (n1 = n2) (eq_ascii n1 n2 = true) …);
  55  ##[ ##1: napply (eq_to_eqascii n1 n2)
  56  ##| ##2: napply (eqfalse_to_neqtrue … H)
  57  ##]
  58 nqed.
  59
  60 (* !!! per brevita... *)
  61 naxiom eqascii_to_eq : ∀c1,c2.eq_ascii c1 c2 = true → c1 = c2.
  62
  63 nlemma neq_to_neqascii : ∀n1,n2.n1 ≠ n2 → eq_ascii n1 n2 = false.
  64  #n1; #n2; #H;
  65  napply (neqtrue_to_eqfalse (eq_ascii n1 n2));
  66  napply (not_to_not (eq_ascii n1 n2 = true) (n1 = n2) ? H);
  67  napply (eqascii_to_eq n1 n2).
  68 nqed.
  69
  70 nlemma decidable_ascii : ∀x,y:ascii.decidable (x = y).
  71  #x; #y; nnormalize;
  72  napply (or2_elim (eq_ascii x y = true) (eq_ascii x y = false) ? (decidable_bexpr ?));
  73  ##[ ##1: #H; napply (or2_intro1 (x = y) (x ≠ y) (eqascii_to_eq … H))
  74  ##| ##2: #H; napply (or2_intro2 (x = y) (x ≠ y) (neqascii_to_neq … H))
  75  ##]
  76 nqed.
  77
  78 nlemma symmetric_eqascii : symmetricT ascii bool eq_ascii.
  79  #n1; #n2;
  80  napply (or2_elim (n1 = n2) (n1 ≠ n2) ? (decidable_ascii n1 n2));
  81  ##[ ##1: #H; nrewrite > H; napply refl_eq
  82  ##| ##2: #H; nrewrite > (neq_to_neqascii n1 n2 H);
  83           napply (symmetric_eq ? (eq_ascii n2 n1) false);
  84           napply (neq_to_neqascii n2 n1 (symmetric_neq ? n1 n2 H))
  85  ##]
  86 nqed.