helm/software/matita/contribs/ng_assembly/common/string_lemmas.ma

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   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: Ing. Cosimo Oliboni, oliboni@cs.unibo.it              *)
  19 (*   Ultima modifica: 05/08/2009                                          *)
  20 (*                                                                        *)
  21 (* ********************************************************************** *)
  22
  23 include "common/string.ma".
  24 include "common/ascii_lemmas2.ma".
  25 include "common/list_utility_lemmas.ma".
  26
  27 (* ************************ *)
  28 (* MANIPOLAZIONE DI STRINGA *)
  29 (* ************************ *)
  30
  31 nlemma symmetric_eqstr : symmetricT (list ascii) bool eq_str.
  32  #s1; #s2;
  33  napply (symmetric_bfoldrightlist2 ascii eq_ascii s1 s2 symmetric_eqascii).
  34 nqed.
  35
  36 nlemma eqstr_to_eq : ∀s,s'.eq_str s s' = true → s = s'.
  37  #s1; #s2;
  38  napply (bfoldrightlist2_to_eq ascii eq_ascii s1 s2 eqascii_to_eq).
  39 nqed.
  40
  41 nlemma eq_to_eqstr : ∀s,s'.s = s' → eq_str s s' = true.
  42  #s1; #s2;
  43  napply (eq_to_bfoldrightlist2 ascii eq_ascii s1 s2 eq_to_eqascii).
  44 nqed.
  45
  46 (* ************ *)
  47 (* STRINGA + ID *)
  48 (* ************ *)
  49
  50 nlemma strid_destruct_1 : ∀x1,x2,y1,y2.mk_strId x1 y1 = mk_strId x2 y2 → x1 = x2.
  51  #x1; #x2; #y1; #y2; #H;
  52  nchange with (match mk_strId x2 y2 with [ mk_strId a _ ⇒ x1 = a ]);
  53  nrewrite < H;
  54  nnormalize;
  55  napply refl_eq.
  56 nqed.
  57
  58 nlemma strid_destruct_2 : ∀x1,x2,y1,y2.mk_strId x1 y1 = mk_strId x2 y2 → y1 = y2.
  59  #x1; #x2; #y1; #y2; #H;
  60  nchange with (match mk_strId x2 y2 with [ mk_strId _ b ⇒ y1 = b ]);
  61  nrewrite < H;
  62  nnormalize;
  63  napply refl_eq.
  64 nqed.
  65
  66 nlemma symmetric_eqstrid : symmetricT strId bool eq_strId.
  67  #si1; #si2;
  68  nchange with (
  69   ((eq_str (str_elem si1) (str_elem si2))⊗(eq_nat (id_elem si1) (id_elem si2))) =
  70   ((eq_str (str_elem si2) (str_elem si1))⊗(eq_nat (id_elem si2) (id_elem si1))));
  71  nrewrite > (symmetric_eqstr (str_elem si1) (str_elem si2));
  72  nrewrite > (symmetric_eqnat (id_elem si1) (id_elem si2));
  73  napply refl_eq.
  74 nqed.
  75
  76 nlemma eqstrid_to_eq : ∀s,s'.eq_strId s s' = true → s = s'.
  77  #si1; #si2;
  78  nelim si1;
  79  #l1; #n1;
  80  nelim si2;
  81  #l2; #n2; #H;
  82  nchange in H:(%) with (((eq_str l1 l2)⊗(eq_nat n1 n2)) = true);
  83  nrewrite > (eqstr_to_eq l1 l2 (andb_true_true_l … H));
  84  nrewrite > (eqnat_to_eq n1 n2 (andb_true_true_r … H));
  85  napply refl_eq.
  86 nqed.
  87
  88 nlemma eq_to_eqstrid : ∀s,s'.s = s' → eq_strId s s' = true.
  89  #si1; #si2;
  90  nelim si1;
  91  #l1; #n1;
  92  nelim si2;
  93  #l2; #n2; #H;
  94  nchange with (((eq_str l1 l2)⊗(eq_nat n1 n2)) = true);
  95  nrewrite > (strid_destruct_1 … H);
  96  nrewrite > (strid_destruct_2 … H);
  97  nrewrite > (eq_to_eqstr l2 l2 (refl_eq …));
  98  nrewrite > (eq_to_eqnat n2 n2 (refl_eq …));
  99  nnormalize;
 100  napply refl_eq.
 101 nqed.