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 (**************************************************************************)
15 set "baseuri" "cic:/matita/Z/plus".
18 include "nat/compare.ma".
19 include "nat/minus.ma".
21 definition Zplus :Z \to Z \to Z \def
28 | (pos n) \Rightarrow (pos (pred ((S m)+(S n))))
30 match nat_compare m n with
31 [ LT \Rightarrow (neg (pred (n-m)))
33 | GT \Rightarrow (pos (pred (m-n)))]]
38 match nat_compare m n with
39 [ LT \Rightarrow (pos (pred (n-m)))
41 | GT \Rightarrow (neg (pred (m-n)))]
42 | (neg n) \Rightarrow (neg (pred ((S m)+(S n))))]].
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).
47 theorem Zplus_z_OZ: \forall z:Z. z+OZ = z.
54 (* theorem symmetric_Zplus: symmetric Z Zplus. *)
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.
60 rewrite < plus_n_Sm. rewrite < plus_n_Sm.rewrite < sym_plus.reflexivity.
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.
74 theorem Zpred_Zplus_neg_O : \forall z:Z. Zpred z = (neg O)+z.
78 elim n.simplify.reflexivity.
82 theorem Zsucc_Zplus_pos_O : \forall z:Z. Zsucc z = (pos O)+z.
85 elim n.simplify.reflexivity.
90 theorem Zplus_pos_pos:
91 \forall n,m. (pos n)+(pos m) = (Zsucc (pos n))+(Zpred (pos m)).
97 simplify.rewrite < plus_n_Sm.
98 rewrite < plus_n_O.reflexivity.
99 simplify.rewrite < plus_n_Sm.
100 rewrite < plus_n_Sm.reflexivity.
103 theorem Zplus_pos_neg:
104 \forall n,m. (pos n)+(neg m) = (Zsucc (pos n))+(Zpred (neg m)).
108 theorem Zplus_neg_pos :
109 \forall n,m. (neg n)+(pos m) = (Zsucc (neg n))+(Zpred (pos m)).
112 simplify.reflexivity.
113 simplify.reflexivity.
115 simplify.reflexivity.
116 simplify.reflexivity.
119 theorem Zplus_neg_neg:
120 \forall n,m. (neg n)+(neg m) = (Zsucc (neg n))+(Zpred (neg m)).
123 simplify.reflexivity.
124 simplify.reflexivity.
126 simplify.rewrite > plus_n_Sm.reflexivity.
127 simplify.rewrite > plus_n_Sm.reflexivity.
130 theorem Zplus_Zsucc_Zpred:
131 \forall x,y. x+y = (Zsucc x)+(Zpred y).
134 simplify.reflexivity.
135 simplify.reflexivity.
136 rewrite < Zsucc_Zplus_pos_O.
137 rewrite > Zsucc_Zpred.reflexivity.
138 elim y.rewrite < sym_Zplus.rewrite < sym_Zplus (Zpred OZ).
139 rewrite < Zpred_Zplus_neg_O.
140 rewrite > Zpred_Zsucc.
141 simplify.reflexivity.
142 rewrite < Zplus_neg_neg.reflexivity.
144 elim y.simplify.reflexivity.
149 theorem Zplus_Zsucc_pos_pos :
150 \forall n,m. (Zsucc (pos n))+(pos m) = Zsucc ((pos n)+(pos m)).
154 theorem Zplus_Zsucc_pos_neg:
155 \forall n,m. (Zsucc (pos n))+(neg m) = (Zsucc ((pos n)+(neg m))).
158 (\lambda n,m. (Zsucc (pos n))+(neg m) = (Zsucc ((pos n)+(neg m)))).intro.
160 simplify. reflexivity.
161 elim n2.simplify. reflexivity.
162 simplify. reflexivity.
164 simplify. reflexivity.
165 simplify.reflexivity.
167 rewrite < (Zplus_pos_neg ? m1).
171 theorem Zplus_Zsucc_neg_neg :
172 \forall n,m. (Zsucc (neg n))+(neg m) = Zsucc ((neg n)+(neg m)).
175 (\lambda n,m. ((Zsucc (neg n))+(neg m)) = Zsucc ((neg n)+(neg m))).intro.
177 simplify. reflexivity.
178 elim n2.simplify. reflexivity.
179 simplify. reflexivity.
181 simplify. reflexivity.
182 simplify.reflexivity.
184 rewrite < (Zplus_neg_neg ? m1).
188 theorem Zplus_Zsucc_neg_pos:
189 \forall n,m. Zsucc (neg n)+(pos m) = Zsucc ((neg n)+(pos m)).
192 (\lambda n,m. (Zsucc (neg n))+(pos m) = Zsucc ((neg n)+(pos m))).
194 simplify. reflexivity.
195 elim n2.simplify. reflexivity.
196 simplify. reflexivity.
198 simplify. reflexivity.
199 simplify.reflexivity.
202 rewrite < (Zplus_neg_pos ? (S m1)).
206 theorem Zplus_Zsucc : \forall x,y:Z. (Zsucc x)+y = Zsucc (x+y).
207 intros.elim x.elim y.
208 simplify. reflexivity.
209 rewrite < Zsucc_Zplus_pos_O.reflexivity.
210 simplify.reflexivity.
211 elim y.rewrite < sym_Zplus.rewrite < sym_Zplus OZ.simplify.reflexivity.
212 apply Zplus_Zsucc_neg_neg.
213 apply Zplus_Zsucc_neg_pos.
215 rewrite < sym_Zplus OZ.reflexivity.
216 apply Zplus_Zsucc_pos_neg.
217 apply Zplus_Zsucc_pos_pos.
220 theorem Zplus_Zpred: \forall x,y:Z. (Zpred x)+y = Zpred (x+y).
222 cut Zpred (x+y) = Zpred ((Zsucc (Zpred x))+y).
224 rewrite > Zplus_Zsucc.
225 rewrite > Zpred_Zsucc.
227 rewrite > Zsucc_Zpred.
232 theorem associative_Zplus: associative Z Zplus.
233 change with \forall x,y,z:Z. (x + y) + z = x + (y + z).
235 intros.elim x.simplify.reflexivity.
236 elim n.rewrite < (Zpred_Zplus_neg_O (y+z)).
237 rewrite < (Zpred_Zplus_neg_O y).
238 rewrite < Zplus_Zpred.
240 rewrite > Zplus_Zpred (neg n1).
241 rewrite > Zplus_Zpred (neg n1).
242 rewrite > Zplus_Zpred ((neg n1)+y).
243 apply eq_f.assumption.
244 elim n.rewrite < Zsucc_Zplus_pos_O.
245 rewrite < Zsucc_Zplus_pos_O.
246 rewrite > Zplus_Zsucc.
248 rewrite > Zplus_Zsucc (pos n1).
249 rewrite > Zplus_Zsucc (pos n1).
250 rewrite > Zplus_Zsucc ((pos n1)+y).
251 apply eq_f.assumption.
254 variant assoc_Zplus : \forall x,y,z:Z. (x+y)+z = x+(y+z)
255 \def associative_Zplus.
258 definition Zopp : Z \to Z \def
259 \lambda x:Z. match x with
261 | (pos n) \Rightarrow (neg n)
262 | (neg n) \Rightarrow (pos n) ].
264 (*CSC: the URI must disappear: there is a bug now *)
265 interpretation "integer unary minus" 'uminus x = (cic:/matita/Z/plus/Zopp.con x).
267 theorem Zopp_Zplus: \forall x,y:Z. -(x+y) = -x + -y.
270 simplify. reflexivity.
271 simplify. reflexivity.
272 simplify. reflexivity.
274 simplify. reflexivity.
275 simplify. reflexivity.
276 simplify. apply nat_compare_elim.
277 intro.simplify.reflexivity.
278 intro.simplify.reflexivity.
279 intro.simplify.reflexivity.
281 simplify. reflexivity.
282 simplify. apply nat_compare_elim.
283 intro.simplify.reflexivity.
284 intro.simplify.reflexivity.
285 intro.simplify.reflexivity.
286 simplify.reflexivity.
289 (* --x non gli piace, ma lo stampa *)
290 theorem Zopp_Zopp: \forall x:Z. -(-x) = x.
292 reflexivity.reflexivity.reflexivity.
295 theorem Zplus_Zopp: \forall x:Z. x+ -x = OZ.
299 rewrite > nat_compare_n_n.
300 simplify.apply refl_eq.
302 rewrite > nat_compare_n_n.
303 simplify.apply refl_eq.