helm/matita/library/Z/plus.ma

   1 (**************************************************************************)
   2 (*       ___                                                                *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||       A.Asperti, C.Sacerdoti Coen,                          *)
   8 (*      ||A||       E.Tassi, S.Zacchiroli                                 *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU Lesser General Public License Version 2.1         *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 set "baseuri" "cic:/matita/Z/plus".
  16
  17 include "Z/z.ma".
  18 include "nat/compare.ma".
  19 include "nat/minus.ma".
  20
  21 definition Zplus :Z \to Z \to Z \def
  22 \lambda x,y.
  23   match x with
  24     [ OZ \Rightarrow y
  25     | (pos m) \Rightarrow
  26         match y with
  27          [ OZ \Rightarrow x
  28          | (pos n) \Rightarrow (pos (pred ((S m)+(S n))))
  29          | (neg n) \Rightarrow
  30               match nat_compare m n with
  31                 [ LT \Rightarrow (neg (pred (n-m)))
  32                 | EQ \Rightarrow OZ
  33                 | GT \Rightarrow (pos (pred (m-n)))]]
  34     | (neg m) \Rightarrow
  35         match y with
  36          [ OZ \Rightarrow x
  37          | (pos n) \Rightarrow
  38               match nat_compare m n with
  39                 [ LT \Rightarrow (pos (pred (n-m)))
  40                 | EQ \Rightarrow OZ
  41                 | GT \Rightarrow (neg (pred (m-n)))]
  42          | (neg n) \Rightarrow (neg (pred ((S m)+(S n))))]].
  43
  44 (*CSC: the URI must disappear: there is a bug now *)
  45 interpretation "integer plus" 'plus x y = (cic:/matita/Z/plus/Zplus.con x y).
  46
  47 theorem Zplus_z_OZ:  \forall z:Z. z+OZ = z.
  48 intro.elim z.
  49 simplify.reflexivity.
  50 simplify.reflexivity.
  51 simplify.reflexivity.
  52 qed.
  53
  54 (* theorem symmetric_Zplus: symmetric Z Zplus. *)
  55
  56 theorem sym_Zplus : \forall x,y:Z. x+y = y+x.
  57 intros.elim x.rewrite > Zplus_z_OZ.reflexivity.
  58 elim y.simplify.reflexivity.
  59 simplify.
  60 rewrite < plus_n_Sm. rewrite < plus_n_Sm.rewrite < sym_plus.reflexivity.
  61 simplify.
  62 rewrite > nat_compare_n_m_m_n.
  63 simplify.elim nat_compare ? ?.simplify.reflexivity.
  64 simplify. reflexivity.
  65 simplify. reflexivity.
  66 elim y.simplify.reflexivity.
  67 simplify.rewrite > nat_compare_n_m_m_n.
  68 simplify.elim nat_compare ? ?.simplify.reflexivity.
  69 simplify. reflexivity.
  70 simplify. reflexivity.
  71 simplify.rewrite < plus_n_Sm. rewrite < plus_n_Sm.rewrite < sym_plus.reflexivity.
  72 qed.
  73
  74 theorem Zpred_Zplus_neg_O : \forall z:Z. Zpred z = (neg O)+z.
  75 intros.elim z.
  76   simplify.reflexivity.
  77   elim n.
  78     simplify.reflexivity.
  79     simplify.reflexivity.
  80   simplify.reflexivity.
  81 qed.
  82
  83 theorem Zsucc_Zplus_pos_O : \forall z:Z. Zsucc z = (pos O)+z.
  84 intros.elim z.
  85   simplify.reflexivity.
  86   simplify.reflexivity.
  87   elim n.
  88     simplify.reflexivity.
  89     simplify.reflexivity.
  90 qed.
  91
  92 theorem Zplus_pos_pos:
  93 \forall n,m. (pos n)+(pos m) = (Zsucc (pos n))+(Zpred (pos m)).
  94 intros.
  95 elim n.elim m.
  96 simplify.reflexivity.
  97 simplify.reflexivity.
  98 elim m.
  99 simplify.rewrite < plus_n_Sm.
 100 rewrite < plus_n_O.reflexivity.
 101 simplify.rewrite < plus_n_Sm.
 102 rewrite < plus_n_Sm.reflexivity.
 103 qed.
 104
 105 theorem Zplus_pos_neg:
 106 \forall n,m. (pos n)+(neg m) = (Zsucc (pos n))+(Zpred (neg m)).
 107 intros.reflexivity.
 108 qed.
 109
 110 theorem Zplus_neg_pos :
 111 \forall n,m. (neg n)+(pos m) = (Zsucc (neg n))+(Zpred (pos m)).
 112 intros.
 113 elim n.elim m.
 114 simplify.reflexivity.
 115 simplify.reflexivity.
 116 elim m.
 117 simplify.reflexivity.
 118 simplify.reflexivity.
 119 qed.
 120
 121 theorem Zplus_neg_neg:
 122 \forall n,m. (neg n)+(neg m) = (Zsucc (neg n))+(Zpred (neg m)).
 123 intros.
 124 elim n.elim m.
 125 simplify.reflexivity.
 126 simplify.reflexivity.
 127 elim m.
 128 simplify.rewrite > plus_n_Sm.reflexivity.
 129 simplify.rewrite > plus_n_Sm.reflexivity.
 130 qed.
 131
 132 theorem Zplus_Zsucc_Zpred:
 133 \forall x,y. x+y = (Zsucc x)+(Zpred y).
 134 intros.elim x.
 135   elim y.
 136     simplify.reflexivity.
 137     rewrite < Zsucc_Zplus_pos_O.rewrite > Zsucc_Zpred.reflexivity.
 138     simplify.reflexivity.
 139   elim y.
 140     simplify.reflexivity.
 141     apply Zplus_pos_pos.
 142     apply Zplus_pos_neg.
 143   elim y.
 144     rewrite < sym_Zplus.rewrite < sym_Zplus (Zpred OZ).
 145      rewrite < Zpred_Zplus_neg_O.rewrite > Zpred_Zsucc.simplify.reflexivity.
 146     apply Zplus_neg_pos.
 147     rewrite < Zplus_neg_neg.reflexivity.
 148 qed.
 149
 150 theorem Zplus_Zsucc_pos_pos :
 151 \forall n,m. (Zsucc (pos n))+(pos m) = Zsucc ((pos n)+(pos m)).
 152 intros.reflexivity.
 153 qed.
 154
 155 theorem Zplus_Zsucc_pos_neg:
 156 \forall n,m. (Zsucc (pos n))+(neg m) = (Zsucc ((pos n)+(neg m))).
 157 intros.
 158 apply nat_elim2
 159 (\lambda n,m. (Zsucc (pos n))+(neg m) = (Zsucc ((pos n)+(neg m)))).intro.
 160 intros.elim n1.
 161 simplify. reflexivity.
 162 elim n2.simplify. reflexivity.
 163 simplify. reflexivity.
 164 intros. elim n1.
 165 simplify. reflexivity.
 166 simplify.reflexivity.
 167 intros.
 168 rewrite < (Zplus_pos_neg ? m1).
 169 elim H.reflexivity.
 170 qed.
 171
 172 theorem Zplus_Zsucc_neg_neg :
 173 \forall n,m. (Zsucc (neg n))+(neg m) = Zsucc ((neg n)+(neg m)).
 174 intros.
 175 apply nat_elim2
 176 (\lambda n,m. ((Zsucc (neg n))+(neg m)) = Zsucc ((neg n)+(neg m))).intro.
 177 intros.elim n1.
 178 simplify. reflexivity.
 179 elim n2.simplify. reflexivity.
 180 simplify. reflexivity.
 181 intros. elim n1.
 182 simplify. reflexivity.
 183 simplify.reflexivity.
 184 intros.
 185 rewrite < (Zplus_neg_neg ? m1).
 186 reflexivity.
 187 qed.
 188
 189 theorem Zplus_Zsucc_neg_pos:
 190 \forall n,m. Zsucc (neg n)+(pos m) = Zsucc ((neg n)+(pos m)).
 191 intros.
 192 apply nat_elim2
 193 (\lambda n,m. (Zsucc (neg n))+(pos m) = Zsucc ((neg n)+(pos m))).
 194 intros.elim n1.
 195 simplify. reflexivity.
 196 elim n2.simplify. reflexivity.
 197 simplify. reflexivity.
 198 intros. elim n1.
 199 simplify. reflexivity.
 200 simplify.reflexivity.
 201 intros.
 202 rewrite < H.
 203 rewrite < (Zplus_neg_pos ? (S m1)).
 204 reflexivity.
 205 qed.
 206
 207 theorem Zplus_Zsucc : \forall x,y:Z. (Zsucc x)+y = Zsucc (x+y).
 208 intros.elim x.
 209   elim y.
 210     simplify. reflexivity.
 211     simplify.reflexivity.
 212     rewrite < Zsucc_Zplus_pos_O.reflexivity.
 213   elim y.
 214     rewrite < sym_Zplus OZ.reflexivity.
 215     apply Zplus_Zsucc_pos_pos.
 216     apply Zplus_Zsucc_pos_neg.
 217   elim y.
 218     rewrite < sym_Zplus.rewrite < sym_Zplus OZ.simplify.reflexivity.
 219     apply Zplus_Zsucc_neg_pos.
 220     apply Zplus_Zsucc_neg_neg.
 221 qed.
 222
 223 theorem Zplus_Zpred: \forall x,y:Z. (Zpred x)+y = Zpred (x+y).
 224 intros.
 225 cut Zpred (x+y) = Zpred ((Zsucc (Zpred x))+y).
 226 rewrite > Hcut.
 227 rewrite > Zplus_Zsucc.
 228 rewrite > Zpred_Zsucc.
 229 reflexivity.
 230 rewrite > Zsucc_Zpred.
 231 reflexivity.
 232 qed.
 233
 234
 235 theorem associative_Zplus: associative Z Zplus.
 236 change with \forall x,y,z:Z. (x + y) + z = x + (y + z).
 237 (* simplify. *)
 238 intros.elim x.
 239   simplify.reflexivity.
 240   elim n.
 241     rewrite < Zsucc_Zplus_pos_O.rewrite < Zsucc_Zplus_pos_O.
 242      rewrite > Zplus_Zsucc.reflexivity.
 243     rewrite > Zplus_Zsucc (pos n1).rewrite > Zplus_Zsucc (pos n1).
 244      rewrite > Zplus_Zsucc ((pos n1)+y).apply eq_f.assumption.
 245   elim n.
 246     rewrite < (Zpred_Zplus_neg_O (y+z)).rewrite < (Zpred_Zplus_neg_O y).
 247      rewrite < Zplus_Zpred.reflexivity.
 248     rewrite > Zplus_Zpred (neg n1).rewrite > Zplus_Zpred (neg n1).
 249      rewrite > Zplus_Zpred ((neg n1)+y).apply eq_f.assumption.
 250 qed.
 251
 252 variant assoc_Zplus : \forall x,y,z:Z.  (x+y)+z = x+(y+z)
 253 \def associative_Zplus.
 254
 255 (* Zopp *)
 256 definition Zopp : Z \to Z \def
 257 \lambda x:Z. match x with
 258 [ OZ \Rightarrow OZ
 259 | (pos n) \Rightarrow (neg n)
 260 | (neg n) \Rightarrow (pos n) ].
 261
 262 (*CSC: the URI must disappear: there is a bug now *)
 263 interpretation "integer unary minus" 'uminus x = (cic:/matita/Z/plus/Zopp.con x).
 264
 265 theorem Zopp_Zplus: \forall x,y:Z. -(x+y) = -x + -y.
 266 intros.
 267 elim x.elim y.
 268 simplify. reflexivity.
 269 simplify. reflexivity.
 270 simplify. reflexivity.
 271 elim y.
 272 simplify. reflexivity.
 273 simplify. reflexivity.
 274 simplify. apply nat_compare_elim.
 275 intro.simplify.reflexivity.
 276 intro.simplify.reflexivity.
 277 intro.simplify.reflexivity.
 278 elim y.
 279 simplify. reflexivity.
 280 simplify. apply nat_compare_elim.
 281 intro.simplify.reflexivity.
 282 intro.simplify.reflexivity.
 283 intro.simplify.reflexivity.
 284 simplify.reflexivity.
 285 qed.
 286
 287 theorem Zopp_Zopp: \forall x:Z. --x = x.
 288 intro. elim x.
 289 reflexivity.reflexivity.reflexivity.
 290 qed.
 291
 292 theorem Zplus_Zopp: \forall x:Z. x+ -x = OZ.
 293 intro.elim x.
 294 apply refl_eq.
 295 simplify.
 296 rewrite > nat_compare_n_n.
 297 simplify.apply refl_eq.
 298 simplify.
 299 rewrite > nat_compare_n_n.
 300 simplify.apply refl_eq.
 301 qed.
 302