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/times".
19 definition Ztimes :Z \to Z \to Z \def
26 | (pos n) \Rightarrow (pos (pred (times (S m) (S n))))
27 | (neg n) \Rightarrow (neg (pred (times (S m) (S n))))]
31 | (pos n) \Rightarrow (neg (pred (times (S m) (S n))))
32 | (neg n) \Rightarrow (pos (pred (times (S m) (S n))))]].
34 (*CSC: the URI must disappear: there is a bug now *)
35 interpretation "integer times" 'times x y = (cic:/matita/Z/times/Ztimes.con x y).
37 theorem Ztimes_z_OZ: \forall z:Z. z*OZ = OZ.
44 (*CSC: da qui in avanti niente notazione *)
46 theorem symmetric_Ztimes : symmetric Z Ztimes.
47 change with \forall x,y:Z. eq Z (Ztimes x y) (Ztimes y x).
48 intros.elim x.rewrite > Ztimes_z_OZ.reflexivity.
49 elim y.simplify.reflexivity.
50 change with eq Z (pos (pred (times (S e1) (S e)))) (pos (pred (times (S e) (S e1)))).
51 rewrite < sym_times.reflexivity.
52 change with eq Z (neg (pred (times (S e1) (S e2)))) (neg (pred (times (S e2) (S e1)))).
53 rewrite < sym_times.reflexivity.
54 elim y.simplify.reflexivity.
55 change with eq Z (neg (pred (times (S e2) (S e1)))) (neg (pred (times (S e1) (S e2)))).
56 rewrite < sym_times.reflexivity.
57 change with eq Z (pos (pred (times (S e2) (S e)))) (pos (pred (times (S e) (S e2)))).
58 rewrite < sym_times.reflexivity.
61 variant sym_Ztimes : \forall x,y:Z. eq Z (Ztimes x y) (Ztimes y x)
62 \def symmetric_Ztimes.
64 theorem associative_Ztimes: associative Z Ztimes.
65 change with \forall x,y,z:Z.eq Z (Ztimes (Ztimes x y) z) (Ztimes x (Ztimes y z)).
67 elim x.simplify.reflexivity.
68 elim y.simplify.reflexivity.
69 elim z.simplify.reflexivity.
71 eq Z (neg (pred (times (S (pred (times (S e1) (S e)))) (S e2))))
72 (neg (pred (times (S e1) (S (pred (times (S e) (S e2))))))).
75 theorem Zpred_Zplus_neg_O : \forall z:Z. eq Z (Zpred z) (Zplus (neg O) z).
79 elim e2.simplify.reflexivity.
83 theorem Zsucc_Zplus_pos_O : \forall z:Z. eq Z (Zsucc z) (Zplus (pos O) z).
86 elim e1.simplify.reflexivity.
91 theorem Zplus_pos_pos:
92 \forall n,m. eq Z (Zplus (pos n) (pos m)) (Zplus (Zsucc (pos n)) (Zpred (pos m))).
99 rewrite < plus_n_O.reflexivity.
101 rewrite < plus_n_Sm.reflexivity.
104 theorem Zplus_pos_neg:
105 \forall n,m. eq Z (Zplus (pos n) (neg m)) (Zplus (Zsucc (pos n)) (Zpred (neg m))).
109 theorem Zplus_neg_pos :
110 \forall n,m. eq Z (Zplus (neg n) (pos m)) (Zplus (Zsucc (neg n)) (Zpred (pos m))).
113 simplify.reflexivity.
114 simplify.reflexivity.
116 simplify.reflexivity.
117 simplify.reflexivity.
120 theorem Zplus_neg_neg:
121 \forall n,m. eq Z (Zplus (neg n) (neg m)) (Zplus (Zsucc (neg n)) (Zpred (neg m))).
124 simplify.reflexivity.
125 simplify.reflexivity.
127 simplify.rewrite < plus_n_Sm.reflexivity.
128 simplify.rewrite > plus_n_Sm.reflexivity.
131 theorem Zplus_Zsucc_Zpred:
132 \forall x,y. eq Z (Zplus x y) (Zplus (Zsucc x) (Zpred y)).
135 simplify.reflexivity.
136 simplify.reflexivity.
137 rewrite < Zsucc_Zplus_pos_O.
138 rewrite > Zsucc_Zpred.reflexivity.
139 elim y.rewrite < sym_Zplus.rewrite < sym_Zplus (Zpred OZ).
140 rewrite < Zpred_Zplus_neg_O.
141 rewrite > Zpred_Zsucc.
142 simplify.reflexivity.
143 rewrite < Zplus_neg_neg.reflexivity.
145 elim y.simplify.reflexivity.
150 theorem Zplus_Zsucc_pos_pos :
151 \forall n,m. eq Z (Zplus (Zsucc (pos n)) (pos m)) (Zsucc (Zplus (pos n) (pos m))).
155 theorem Zplus_Zsucc_pos_neg:
156 \forall n,m. eq Z (Zplus (Zsucc (pos n)) (neg m)) (Zsucc (Zplus (pos n) (neg m))).
159 (\lambda n,m. eq Z (Zplus (Zsucc (pos n)) (neg m)) (Zsucc (Zplus (pos n) (neg m)))).intro.
161 simplify. reflexivity.
162 elim e1.simplify. reflexivity.
163 simplify. reflexivity.
165 simplify. reflexivity.
166 simplify.reflexivity.
168 rewrite < (Zplus_pos_neg ? m1).
172 theorem Zplus_Zsucc_neg_neg :
173 \forall n,m. eq Z (Zplus (Zsucc (neg n)) (neg m)) (Zsucc (Zplus (neg n) (neg m))).
176 (\lambda n,m. eq Z (Zplus (Zsucc (neg n)) (neg m)) (Zsucc (Zplus (neg n) (neg m)))).intro.
178 simplify. reflexivity.
179 elim e1.simplify. reflexivity.
180 simplify. reflexivity.
182 simplify. reflexivity.
183 simplify.reflexivity.
185 rewrite < (Zplus_neg_neg ? m1).
189 theorem Zplus_Zsucc_neg_pos:
190 \forall n,m. eq Z (Zplus (Zsucc (neg n)) (pos m)) (Zsucc (Zplus (neg n) (pos m))).
193 (\lambda n,m. eq Z (Zplus (Zsucc (neg n)) (pos m)) (Zsucc (Zplus (neg n) (pos m)))).
195 simplify. reflexivity.
196 elim e1.simplify. reflexivity.
197 simplify. reflexivity.
199 simplify. reflexivity.
200 simplify.reflexivity.
203 rewrite < (Zplus_neg_pos ? (S m1)).
207 theorem Zplus_Zsucc : \forall x,y:Z. eq Z (Zplus (Zsucc x) y) (Zsucc (Zplus x y)).
208 intros.elim x.elim y.
209 simplify. reflexivity.
210 rewrite < Zsucc_Zplus_pos_O.reflexivity.
211 simplify.reflexivity.
212 elim y.rewrite < sym_Zplus.rewrite < sym_Zplus OZ.simplify.reflexivity.
213 apply Zplus_Zsucc_neg_neg.
214 apply Zplus_Zsucc_neg_pos.
216 rewrite < sym_Zplus OZ.reflexivity.
217 apply Zplus_Zsucc_pos_neg.
218 apply Zplus_Zsucc_pos_pos.
221 theorem Zplus_Zpred: \forall x,y:Z. eq Z (Zplus (Zpred x) y) (Zpred (Zplus x y)).
223 cut eq Z (Zpred (Zplus x y)) (Zpred (Zplus (Zsucc (Zpred x)) y)).
225 rewrite > Zplus_Zsucc.
226 rewrite > Zpred_Zsucc.
228 rewrite > Zsucc_Zpred.
233 theorem associative_Zplus: associative Z Zplus.
234 change with \forall x,y,z:Z. eq Z (Zplus (Zplus x y) z) (Zplus x (Zplus y z)).
236 intros.elim x.simplify.reflexivity.
237 elim e1.rewrite < (Zpred_Zplus_neg_O (Zplus y z)).
238 rewrite < (Zpred_Zplus_neg_O y).
239 rewrite < Zplus_Zpred.
241 rewrite > Zplus_Zpred (neg e).
242 rewrite > Zplus_Zpred (neg e).
243 rewrite > Zplus_Zpred (Zplus (neg e) y).
244 apply eq_f.assumption.
245 elim e2.rewrite < Zsucc_Zplus_pos_O.
246 rewrite < Zsucc_Zplus_pos_O.
247 rewrite > Zplus_Zsucc.
249 rewrite > Zplus_Zsucc (pos e1).
250 rewrite > Zplus_Zsucc (pos e1).
251 rewrite > Zplus_Zsucc (Zplus (pos e1) y).
252 apply eq_f.assumption.
255 variant assoc_Zplus : \forall x,y,z:Z. eq Z (Zplus (Zplus x y) z) (Zplus x (Zplus y z))
256 \def associative_Zplus.