1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| A.Asperti, C.Sacerdoti Coen, *)
8 (* ||A|| E.Tassi, S.Zacchiroli *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU Lesser General Public License Version 2.1 *)
13 (**************************************************************************)
16 include "nat/minus.ma".
18 definition Zplus :Z \to Z \to Z \def
25 | (pos n) \Rightarrow (pos (pred ((S m)+(S n))))
27 match nat_compare m n with
28 [ LT \Rightarrow (neg (pred (n-m)))
30 | GT \Rightarrow (pos (pred (m-n)))] ]
35 match nat_compare m n with
36 [ LT \Rightarrow (pos (pred (n-m)))
38 | GT \Rightarrow (neg (pred (m-n)))]
39 | (neg n) \Rightarrow (neg (pred ((S m)+(S n))))] ].
41 interpretation "integer plus" 'plus x y = (Zplus x y).
43 theorem Zplus_z_OZ: \forall z:Z. z+OZ = z.
50 (* theorem symmetric_Zplus: symmetric Z Zplus. *)
52 theorem sym_Zplus : \forall x,y:Z. x+y = y+x.
53 intros.elim x.rewrite > Zplus_z_OZ.reflexivity.
54 elim y.simplify.reflexivity.
56 rewrite < plus_n_Sm. rewrite < plus_n_Sm.rewrite < sym_plus.reflexivity.
58 rewrite > nat_compare_n_m_m_n.
59 simplify.elim nat_compare.simplify.reflexivity.
60 simplify. reflexivity.
61 simplify. reflexivity.
62 elim y.simplify.reflexivity.
63 simplify.rewrite > nat_compare_n_m_m_n.
64 simplify.elim nat_compare.simplify.reflexivity.
65 simplify. reflexivity.
66 simplify. reflexivity.
67 simplify.rewrite < plus_n_Sm. rewrite < plus_n_Sm.rewrite < sym_plus.reflexivity.
70 theorem Zpred_Zplus_neg_O : \forall z:Z. Zpred z = (neg O)+z.
79 theorem Zsucc_Zplus_pos_O : \forall z:Z. Zsucc z = (pos O)+z.
88 theorem Zplus_pos_pos:
89 \forall n,m. (pos n)+(pos m) = (Zsucc (pos n))+(Zpred (pos m)).
95 simplify.rewrite < plus_n_Sm.
96 rewrite < plus_n_O.reflexivity.
97 simplify.rewrite < plus_n_Sm.
98 rewrite < plus_n_Sm.reflexivity.
101 theorem Zplus_pos_neg:
102 \forall n,m. (pos n)+(neg m) = (Zsucc (pos n))+(Zpred (neg m)).
106 theorem Zplus_neg_pos :
107 \forall n,m. (neg n)+(pos m) = (Zsucc (neg n))+(Zpred (pos m)).
110 simplify.reflexivity.
111 simplify.reflexivity.
113 simplify.reflexivity.
114 simplify.reflexivity.
117 theorem Zplus_neg_neg:
118 \forall n,m. (neg n)+(neg m) = (Zsucc (neg n))+(Zpred (neg m)).
121 simplify.reflexivity.
122 simplify.reflexivity.
124 simplify.rewrite > plus_n_Sm.reflexivity.
125 simplify.rewrite > plus_n_Sm.reflexivity.
128 theorem Zplus_Zsucc_Zpred:
129 \forall x,y. x+y = (Zsucc x)+(Zpred y).
132 simplify.reflexivity.
133 rewrite < Zsucc_Zplus_pos_O.rewrite > Zsucc_Zpred.reflexivity.
134 simplify.reflexivity.
136 simplify.reflexivity.
140 rewrite < sym_Zplus.rewrite < (sym_Zplus (Zpred OZ)).
141 rewrite < Zpred_Zplus_neg_O.rewrite > Zpred_Zsucc.simplify.reflexivity.
143 rewrite < Zplus_neg_neg.reflexivity.
146 theorem Zplus_Zsucc_pos_pos :
147 \forall n,m. (Zsucc (pos n))+(pos m) = Zsucc ((pos n)+(pos m)).
151 theorem Zplus_Zsucc_pos_neg:
152 \forall n,m. (Zsucc (pos n))+(neg m) = (Zsucc ((pos n)+(neg m))).
155 (\lambda n,m. (Zsucc (pos n))+(neg m) = (Zsucc ((pos n)+(neg m))))).intro.
157 simplify. reflexivity.
158 elim n2.simplify. reflexivity.
159 simplify. reflexivity.
161 simplify. reflexivity.
162 simplify.reflexivity.
164 rewrite < (Zplus_pos_neg ? m1).
168 theorem Zplus_Zsucc_neg_neg :
169 \forall n,m. Zsucc (neg n) + neg m = Zsucc (neg n + neg m).
172 (\lambda n,m. Zsucc (neg n) + neg m = Zsucc (neg n + neg m))).intro.
174 simplify. reflexivity.
175 elim n2.simplify. reflexivity.
176 simplify. reflexivity.
178 simplify. reflexivity.
179 simplify.reflexivity.
181 rewrite < (Zplus_neg_neg ? m1).
185 theorem Zplus_Zsucc_neg_pos:
186 \forall n,m. Zsucc (neg n)+(pos m) = Zsucc ((neg n)+(pos m)).
189 (\lambda n,m. Zsucc (neg n) + (pos m) = Zsucc (neg n + pos m))).
191 simplify. reflexivity.
192 elim n2.simplify. reflexivity.
193 simplify. reflexivity.
195 simplify. reflexivity.
196 simplify.reflexivity.
199 rewrite < (Zplus_neg_pos ? (S m1)).
203 theorem Zplus_Zsucc : \forall x,y:Z. (Zsucc x)+y = Zsucc (x+y).
206 simplify. reflexivity.
207 simplify.reflexivity.
208 rewrite < Zsucc_Zplus_pos_O.reflexivity.
210 rewrite < (sym_Zplus OZ).reflexivity.
211 apply Zplus_Zsucc_pos_pos.
212 apply Zplus_Zsucc_pos_neg.
214 rewrite < sym_Zplus.rewrite < (sym_Zplus OZ).simplify.reflexivity.
215 apply Zplus_Zsucc_neg_pos.
216 apply Zplus_Zsucc_neg_neg.
219 theorem Zplus_Zpred: \forall x,y:Z. (Zpred x)+y = Zpred (x+y).
221 cut (Zpred (x+y) = Zpred ((Zsucc (Zpred x))+y)).
223 rewrite > Zplus_Zsucc.
224 rewrite > Zpred_Zsucc.
226 rewrite > Zsucc_Zpred.
231 theorem associative_Zplus: associative Z Zplus.
232 change with (\forall x,y,z:Z. (x + y) + z = x + (y + z)).
235 simplify.reflexivity.
237 rewrite < Zsucc_Zplus_pos_O.rewrite < Zsucc_Zplus_pos_O.
238 rewrite > Zplus_Zsucc.reflexivity.
239 rewrite > (Zplus_Zsucc (pos n1)).rewrite > (Zplus_Zsucc (pos n1)).
240 rewrite > (Zplus_Zsucc ((pos n1)+y)).apply eq_f.assumption.
242 rewrite < (Zpred_Zplus_neg_O (y+z)).rewrite < (Zpred_Zplus_neg_O y).
243 rewrite < Zplus_Zpred.reflexivity.
244 rewrite > (Zplus_Zpred (neg n1)).rewrite > (Zplus_Zpred (neg n1)).
245 rewrite > (Zplus_Zpred ((neg n1)+y)).apply eq_f.assumption.
248 variant assoc_Zplus : \forall x,y,z:Z. (x+y)+z = x+(y+z)
249 \def associative_Zplus.
252 definition Zopp : Z \to Z \def
253 \lambda x:Z. match x with
255 | (pos n) \Rightarrow (neg n)
256 | (neg n) \Rightarrow (pos n) ].
258 interpretation "integer unary minus" 'uminus x = (Zopp x).
260 theorem eq_OZ_Zopp_OZ : OZ = (- OZ).
264 theorem Zopp_Zplus: \forall x,y:Z. -(x+y) = -x + -y.
267 simplify. reflexivity.
268 simplify. reflexivity.
269 simplify. reflexivity.
271 simplify. reflexivity.
272 simplify. reflexivity.
273 simplify. apply nat_compare_elim.
274 intro.simplify.reflexivity.
275 intro.simplify.reflexivity.
276 intro.simplify.reflexivity.
278 simplify. reflexivity.
279 simplify. apply nat_compare_elim.
280 intro.simplify.reflexivity.
281 intro.simplify.reflexivity.
282 intro.simplify.reflexivity.
283 simplify.reflexivity.
286 theorem Zopp_Zopp: \forall x:Z. --x = x.
288 reflexivity.reflexivity.reflexivity.
291 theorem Zplus_Zopp: \forall x:Z. x+ -x = OZ.
295 rewrite > nat_compare_n_n.
296 simplify.apply refl_eq.
298 rewrite > nat_compare_n_n.
299 simplify.apply refl_eq.
302 theorem injective_Zplus_l: \forall x:Z.injective Z Z (\lambda y.y+x).
303 intro.simplify.intros (z y).
304 rewrite < Zplus_z_OZ.
305 rewrite < (Zplus_z_OZ y).
306 rewrite < (Zplus_Zopp x).
307 rewrite < assoc_Zplus.
308 rewrite < assoc_Zplus.
310 [assumption|reflexivity]
313 theorem injective_Zplus_r: \forall x:Z.injective Z Z (\lambda y.x+y).
314 intro.simplify.intros (z y).
315 apply (injective_Zplus_l x).
322 definition Zminus : Z \to Z \to Z \def \lambda x,y:Z. x + (-y).
324 interpretation "integer minus" 'minus x y = (Zminus x y).