matita/matita/contribs/ng_assembly2/common/ascii.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_base.ma".
  24 include "common/comp.ma".
  25 include "num/bool_lemmas.ma".
  26
  27 (* ************************** *)
  28 (* DEFINIZIONE ASCII MINIMALE *)
  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 nlemma eq_to_eqascii : ∀n1,n2.n1 = n2 → eq_ascii n1 n2 = true.
  44  #n1; #n2; #H;
  45  nrewrite > H;
  46  nelim n2;
  47  nnormalize;
  48  napply refl_eq.
  49 nqed.
  50
  51 nlemma neqascii_to_neq : ∀n1,n2.eq_ascii n1 n2 = false → n1 ≠ n2.
  52  #n1; #n2; #H;
  53  napply (not_to_not (n1 = n2) (eq_ascii n1 n2 = true) …);
  54  ##[ ##1: napply (eq_to_eqascii n1 n2)
  55  ##| ##2: napply (eqfalse_to_neqtrue … H)
  56  ##]
  57 nqed.
  58
  59 (* !!! per brevita... *)
  60 naxiom eqascii_to_eq : ∀c1,c2.eq_ascii c1 c2 = true → c1 = c2.
  61
  62 nlemma neq_to_neqascii : ∀n1,n2.n1 ≠ n2 → eq_ascii n1 n2 = false.
  63  #n1; #n2; #H;
  64  napply (neqtrue_to_eqfalse (eq_ascii n1 n2));
  65  napply (not_to_not (eq_ascii n1 n2 = true) (n1 = n2) ? H);
  66  napply (eqascii_to_eq n1 n2).
  67 nqed.
  68
  69 nlemma decidable_ascii : ∀x,y:ascii.decidable (x = y).
  70  #x; #y; nnormalize;
  71  napply (or2_elim (eq_ascii x y = true) (eq_ascii x y = false) ? (decidable_bexpr ?));
  72  ##[ ##1: #H; napply (or2_intro1 (x = y) (x ≠ y) (eqascii_to_eq … H))
  73  ##| ##2: #H; napply (or2_intro2 (x = y) (x ≠ y) (neqascii_to_neq … H))
  74  ##]
  75 nqed.
  76
  77 nlemma symmetric_eqascii : symmetricT ascii bool eq_ascii.
  78  #n1; #n2;
  79  napply (or2_elim (n1 = n2) (n1 ≠ n2) ? (decidable_ascii n1 n2));
  80  ##[ ##1: #H; nrewrite > H; napply refl_eq
  81  ##| ##2: #H; nrewrite > (neq_to_neqascii n1 n2 H);
  82           napply (symmetric_eq ? (eq_ascii n2 n1) false);
  83           napply (neq_to_neqascii n2 n1 (symmetric_neq ? n1 n2 H))
  84  ##]
  85 nqed.
  86
  87 nlemma ascii_is_comparable : comparable.
  88  @ ascii
  89   ##[ napply ch_0
  90   ##| napply forall_ascii
  91   ##| napply eq_ascii
  92   ##| napply eqascii_to_eq
  93   ##| napply eq_to_eqascii
  94   ##| napply neqascii_to_neq
  95   ##| napply neq_to_neqascii
  96   ##| napply decidable_ascii
  97   ##| napply symmetric_eqascii
  98   ##]
  99 nqed.
 100
 101 unification hint 0 ≔ ⊢ carr ascii_is_comparable ≡ ascii.