author | paulson |
Thu, 12 Oct 2000 12:16:58 +0200 | |
changeset 10198 | 2b255b772585 |
parent 9970 | dfe4747c8318 |
child 10230 | 5eb935d6cc69 |
permissions | -rw-r--r-- |
7998 | 1 |
(* |
2 |
Abstract class ring (commutative, with 1) |
|
3 |
$Id$ |
|
4 |
Author: Clemens Ballarin, started 9 December 1996 |
|
5 |
*) |
|
6 |
||
7 |
Blast.overloaded ("Divides.op dvd", domain_type); |
|
8 |
||
9 |
section "Rings"; |
|
10 |
||
11 |
fun make_left_commute assoc commute s = |
|
12 |
[rtac (commute RS trans) 1, rtac (assoc RS trans) 1, |
|
13 |
rtac (commute RS arg_cong) 1]; |
|
14 |
||
15 |
(* addition *) |
|
16 |
||
17 |
qed_goal "a_lcomm" Ring.thy "!!a::'a::ring. a+(b+c) = b+(a+c)" |
|
18 |
(make_left_commute a_assoc a_comm); |
|
19 |
||
20 |
val a_ac = [a_assoc, a_comm, a_lcomm]; |
|
21 |
||
9390
e6b96d953965
renamed // to / (which is what we want anyway) to avoid clash with the new
paulson
parents:
8707
diff
changeset
|
22 |
Goal "!!a::'a::ring. a + <0> = a"; |
e6b96d953965
renamed // to / (which is what we want anyway) to avoid clash with the new
paulson
parents:
8707
diff
changeset
|
23 |
by (rtac (a_comm RS trans) 1); |
e6b96d953965
renamed // to / (which is what we want anyway) to avoid clash with the new
paulson
parents:
8707
diff
changeset
|
24 |
by (rtac l_zero 1); |
e6b96d953965
renamed // to / (which is what we want anyway) to avoid clash with the new
paulson
parents:
8707
diff
changeset
|
25 |
qed "r_zero"; |
7998 | 26 |
|
9390
e6b96d953965
renamed // to / (which is what we want anyway) to avoid clash with the new
paulson
parents:
8707
diff
changeset
|
27 |
Goal "!!a::'a::ring. a + (-a) = <0>"; |
e6b96d953965
renamed // to / (which is what we want anyway) to avoid clash with the new
paulson
parents:
8707
diff
changeset
|
28 |
by (rtac (a_comm RS trans) 1); |
e6b96d953965
renamed // to / (which is what we want anyway) to avoid clash with the new
paulson
parents:
8707
diff
changeset
|
29 |
by (rtac l_neg 1); |
e6b96d953965
renamed // to / (which is what we want anyway) to avoid clash with the new
paulson
parents:
8707
diff
changeset
|
30 |
qed "r_neg"; |
7998 | 31 |
|
32 |
Goal "!! a::'a::ring. a + b = a + c ==> b = c"; |
|
33 |
by (rtac box_equals 1); |
|
34 |
by (rtac l_zero 2); |
|
35 |
by (rtac l_zero 2); |
|
36 |
by (res_inst_tac [("a1", "a")] (l_neg RS subst) 1); |
|
37 |
by (asm_simp_tac (simpset() addsimps [a_assoc]) 1); |
|
38 |
qed "a_lcancel"; |
|
39 |
||
40 |
Goal "!! a::'a::ring. b + a = c + a ==> b = c"; |
|
41 |
by (rtac a_lcancel 1); |
|
42 |
by (asm_simp_tac (simpset() addsimps a_ac) 1); |
|
43 |
qed "a_rcancel"; |
|
44 |
||
45 |
Goal "!! a::'a::ring. (a + b = a + c) = (b = c)"; |
|
46 |
by (auto_tac (claset() addSDs [a_lcancel], simpset())); |
|
47 |
qed "a_lcancel_eq"; |
|
48 |
||
49 |
Goal "!! a::'a::ring. (b + a = c + a) = (b = c)"; |
|
50 |
by (simp_tac (simpset() addsimps [a_lcancel_eq, a_comm]) 1); |
|
51 |
qed "a_rcancel_eq"; |
|
52 |
||
53 |
Addsimps [a_lcancel_eq, a_rcancel_eq]; |
|
54 |
||
55 |
Goal "!!a::'a::ring. -(-a) = a"; |
|
56 |
by (rtac a_lcancel 1); |
|
57 |
by (rtac (r_neg RS trans) 1); |
|
58 |
by (rtac (l_neg RS sym) 1); |
|
59 |
qed "minus_minus"; |
|
60 |
||
61 |
Goal "- <0> = (<0>::'a::ring)"; |
|
62 |
by (rtac a_lcancel 1); |
|
63 |
by (rtac (r_neg RS trans) 1); |
|
64 |
by (rtac (l_zero RS sym) 1); |
|
65 |
qed "minus0"; |
|
66 |
||
67 |
Goal "!!a::'a::ring. -(a + b) = (-a) + (-b)"; |
|
68 |
by (res_inst_tac [("a", "a+b")] a_lcancel 1); |
|
69 |
by (simp_tac (simpset() addsimps ([r_neg, l_neg, l_zero]@a_ac)) 1); |
|
70 |
qed "minus_add"; |
|
71 |
||
72 |
(* multiplication *) |
|
73 |
||
74 |
qed_goal "m_lcomm" Ring.thy "!!a::'a::ring. a*(b*c) = b*(a*c)" |
|
75 |
(make_left_commute m_assoc m_comm); |
|
76 |
||
77 |
val m_ac = [m_assoc, m_comm, m_lcomm]; |
|
78 |
||
9390
e6b96d953965
renamed // to / (which is what we want anyway) to avoid clash with the new
paulson
parents:
8707
diff
changeset
|
79 |
Goal "!!a::'a::ring. a * <1> = a"; |
e6b96d953965
renamed // to / (which is what we want anyway) to avoid clash with the new
paulson
parents:
8707
diff
changeset
|
80 |
by (rtac (m_comm RS trans) 1); |
e6b96d953965
renamed // to / (which is what we want anyway) to avoid clash with the new
paulson
parents:
8707
diff
changeset
|
81 |
by (rtac l_one 1); |
e6b96d953965
renamed // to / (which is what we want anyway) to avoid clash with the new
paulson
parents:
8707
diff
changeset
|
82 |
qed "r_one"; |
7998 | 83 |
|
84 |
(* distributive and derived *) |
|
85 |
||
86 |
Goal "!!a::'a::ring. a * (b + c) = a * b + a * c"; |
|
87 |
by (rtac (m_comm RS trans) 1); |
|
88 |
by (rtac (l_distr RS trans) 1); |
|
89 |
by (simp_tac (simpset() addsimps [m_comm]) 1); |
|
90 |
qed "r_distr"; |
|
91 |
||
92 |
val m_distr = m_ac @ [l_distr, r_distr]; |
|
93 |
||
94 |
(* the following two proofs can be found in |
|
95 |
Jacobson, Basic Algebra I, pp. 88-89 *) |
|
96 |
||
97 |
Goal "!!a::'a::ring. <0> * a = <0>"; |
|
98 |
by (rtac a_lcancel 1); |
|
99 |
by (rtac (l_distr RS sym RS trans) 1); |
|
100 |
by (simp_tac (simpset() addsimps [r_zero]) 1); |
|
101 |
qed "l_null"; |
|
102 |
||
9390
e6b96d953965
renamed // to / (which is what we want anyway) to avoid clash with the new
paulson
parents:
8707
diff
changeset
|
103 |
Goal "!!a::'a::ring. a * <0> = <0>"; |
e6b96d953965
renamed // to / (which is what we want anyway) to avoid clash with the new
paulson
parents:
8707
diff
changeset
|
104 |
by (rtac (m_comm RS trans) 1); |
e6b96d953965
renamed // to / (which is what we want anyway) to avoid clash with the new
paulson
parents:
8707
diff
changeset
|
105 |
by (rtac l_null 1); |
e6b96d953965
renamed // to / (which is what we want anyway) to avoid clash with the new
paulson
parents:
8707
diff
changeset
|
106 |
qed "r_null"; |
7998 | 107 |
|
108 |
Goal "!!a::'a::ring. (-a) * b = - (a * b)"; |
|
109 |
by (rtac a_lcancel 1); |
|
110 |
by (rtac (r_neg RS sym RSN (2, trans)) 1); |
|
111 |
by (rtac (l_distr RS sym RS trans) 1); |
|
112 |
by (simp_tac (simpset() addsimps [l_null, r_neg]) 1); |
|
113 |
qed "l_minus"; |
|
114 |
||
115 |
Goal "!!a::'a::ring. a * (-b) = - (a * b)"; |
|
116 |
by (rtac a_lcancel 1); |
|
117 |
by (rtac (r_neg RS sym RSN (2, trans)) 1); |
|
118 |
by (rtac (r_distr RS sym RS trans) 1); |
|
119 |
by (simp_tac (simpset() addsimps [r_null, r_neg]) 1); |
|
120 |
qed "r_minus"; |
|
121 |
||
122 |
val m_minus = [l_minus, r_minus]; |
|
123 |
||
124 |
(* one and zero are distinct *) |
|
125 |
||
9390
e6b96d953965
renamed // to / (which is what we want anyway) to avoid clash with the new
paulson
parents:
8707
diff
changeset
|
126 |
Goal "<0> ~= (<1>::'a::ring)"; |
e6b96d953965
renamed // to / (which is what we want anyway) to avoid clash with the new
paulson
parents:
8707
diff
changeset
|
127 |
by (rtac not_sym 1); |
e6b96d953965
renamed // to / (which is what we want anyway) to avoid clash with the new
paulson
parents:
8707
diff
changeset
|
128 |
by (rtac one_not_zero 1); |
e6b96d953965
renamed // to / (which is what we want anyway) to avoid clash with the new
paulson
parents:
8707
diff
changeset
|
129 |
qed "zero_not_one"; |
7998 | 130 |
|
131 |
Addsimps [l_zero, r_zero, l_neg, r_neg, minus_minus, minus0, |
|
9390
e6b96d953965
renamed // to / (which is what we want anyway) to avoid clash with the new
paulson
parents:
8707
diff
changeset
|
132 |
l_one, r_one, l_null, r_null, |
e6b96d953965
renamed // to / (which is what we want anyway) to avoid clash with the new
paulson
parents:
8707
diff
changeset
|
133 |
one_not_zero, zero_not_one]; |
7998 | 134 |
|
135 |
(* further rules *) |
|
136 |
||
137 |
Goal "!!a::'a::ring. -a = <0> ==> a = <0>"; |
|
138 |
by (res_inst_tac [("t", "a")] (minus_minus RS subst) 1); |
|
139 |
by (Asm_simp_tac 1); |
|
140 |
qed "uminus_monom"; |
|
141 |
||
142 |
Goal "!!a::'a::ring. a ~= <0> ==> -a ~= <0>"; |
|
10198 | 143 |
by (blast_tac (claset() addIs [uminus_monom]) 1); |
7998 | 144 |
qed "uminus_monom_neq"; |
145 |
||
146 |
Goal "!!a::'a::ring. a * b ~= <0> ==> a ~= <0>"; |
|
10198 | 147 |
by Auto_tac; |
7998 | 148 |
qed "l_nullD"; |
149 |
||
150 |
Goal "!!a::'a::ring. a * b ~= <0> ==> b ~= <0>"; |
|
10198 | 151 |
by Auto_tac; |
7998 | 152 |
qed "r_nullD"; |
153 |
||
154 |
(* reflection between a = b and a -- b = <0> *) |
|
155 |
||
156 |
Goal "!!a::'a::ring. a = b ==> a + (-b) = <0>"; |
|
157 |
by (Asm_simp_tac 1); |
|
158 |
qed "eq_imp_diff_zero"; |
|
159 |
||
160 |
Goal "!!a::'a::ring. a + (-b) = <0> ==> a = b"; |
|
161 |
by (res_inst_tac [("a", "-b")] a_rcancel 1); |
|
162 |
by (Asm_simp_tac 1); |
|
163 |
qed "diff_zero_imp_eq"; |
|
164 |
||
165 |
(* this could be a rewrite rule, but won't terminate |
|
166 |
==> make it a simproc? |
|
167 |
Goal "!!a::'a::ring. (a = b) = (a -- b = <0>)"; |
|
168 |
*) |
|
169 |
||
170 |
(* Power *) |
|
171 |
||
172 |
Goal "!!a::'a::ring. a ^ 0 = <1>"; |
|
173 |
by (simp_tac (simpset() addsimps [power_ax]) 1); |
|
174 |
qed "power_0"; |
|
175 |
||
176 |
Goal "!!a::'a::ring. a ^ Suc n = a ^ n * a"; |
|
177 |
by (simp_tac (simpset() addsimps [power_ax]) 1); |
|
178 |
qed "power_Suc"; |
|
179 |
||
180 |
Addsimps [power_0, power_Suc]; |
|
181 |
||
182 |
Goal "<1> ^ n = (<1>::'a::ring)"; |
|
8707 | 183 |
by (induct_tac "n" 1); |
184 |
by Auto_tac; |
|
7998 | 185 |
qed "power_one"; |
186 |
||
187 |
Goal "!!n. n ~= 0 ==> <0> ^ n = (<0>::'a::ring)"; |
|
188 |
by (etac rev_mp 1); |
|
8707 | 189 |
by (induct_tac "n" 1); |
190 |
by Auto_tac; |
|
7998 | 191 |
qed "power_zero"; |
192 |
||
193 |
Addsimps [power_zero, power_one]; |
|
194 |
||
195 |
Goal "!! a::'a::ring. a ^ m * a ^ n = a ^ (m + n)"; |
|
8707 | 196 |
by (induct_tac "m" 1); |
7998 | 197 |
by (Simp_tac 1); |
198 |
by (asm_simp_tac (simpset() addsimps m_ac) 1); |
|
199 |
qed "power_mult"; |
|
200 |
||
201 |
(* Divisibility *) |
|
202 |
section "Divisibility"; |
|
203 |
||
204 |
Goalw [dvd_def] "!! a::'a::ring. a dvd <0>"; |
|
205 |
by (res_inst_tac [("x", "<0>")] exI 1); |
|
206 |
by (Simp_tac 1); |
|
207 |
qed "dvd_zero_right"; |
|
208 |
||
209 |
Goalw [dvd_def] "!! a::'a::ring. <0> dvd a ==> a = <0>"; |
|
210 |
by Auto_tac; |
|
211 |
qed "dvd_zero_left"; |
|
212 |
||
213 |
Goalw [dvd_def] "!! a::'a::ring. a dvd a"; |
|
214 |
by (res_inst_tac [("x", "<1>")] exI 1); |
|
215 |
by (Simp_tac 1); |
|
216 |
qed "dvd_refl_ring"; |
|
217 |
||
218 |
Goalw [dvd_def] "!! a::'a::ring. [| a dvd b; b dvd c |] ==> a dvd c"; |
|
219 |
by (Step_tac 1); |
|
220 |
by (res_inst_tac [("x", "k * ka")] exI 1); |
|
221 |
by (simp_tac (simpset() addsimps m_ac) 1); |
|
222 |
qed "dvd_trans_ring"; |
|
223 |
||
224 |
Addsimps [dvd_zero_right, dvd_refl_ring]; |
|
225 |
||
226 |
Goal "!! a::'a::ring. a dvd <1> ==> a ~= <0>"; |
|
227 |
by (auto_tac (claset() addDs [dvd_zero_left], simpset())); |
|
228 |
qed "unit_imp_nonzero"; |
|
229 |
||
230 |
Goalw [dvd_def] |
|
231 |
"!!a::'a::ring. [| a dvd <1>; b dvd <1> |] ==> a * b dvd <1>"; |
|
232 |
by (Clarify_tac 1); |
|
233 |
by (res_inst_tac [("x", "k * ka")] exI 1); |
|
234 |
by (asm_full_simp_tac (simpset() addsimps m_ac) 1); |
|
235 |
qed "unit_mult"; |
|
236 |
||
237 |
Goal "!!a::'a::ring. a dvd <1> ==> a^n dvd <1>"; |
|
238 |
by (induct_tac "n" 1); |
|
239 |
by (Simp_tac 1); |
|
240 |
by (asm_simp_tac (simpset() addsimps [unit_mult]) 1); |
|
241 |
qed "unit_power"; |
|
242 |
||
243 |
Goalw [dvd_def] |
|
244 |
"!! a::'a::ring. [| a dvd b; a dvd c |] ==> a dvd (b + c)"; |
|
245 |
by (Clarify_tac 1); |
|
246 |
by (res_inst_tac [("x", "k + ka")] exI 1); |
|
247 |
by (simp_tac (simpset() addsimps [r_distr]) 1); |
|
248 |
qed "dvd_add_right"; |
|
249 |
||
250 |
Goalw [dvd_def] |
|
251 |
"!! a::'a::ring. a dvd b ==> a dvd (-b)"; |
|
252 |
by (Clarify_tac 1); |
|
253 |
by (res_inst_tac [("x", "-k")] exI 1); |
|
254 |
by (simp_tac (simpset() addsimps [r_minus]) 1); |
|
255 |
qed "dvd_uminus_right"; |
|
256 |
||
257 |
Goalw [dvd_def] |
|
258 |
"!! a::'a::ring. a dvd b ==> a dvd (c * b)"; |
|
259 |
by (Clarify_tac 1); |
|
260 |
by (res_inst_tac [("x", "c * k")] exI 1); |
|
261 |
by (simp_tac (simpset() addsimps m_ac) 1); |
|
262 |
qed "dvd_l_mult_right"; |
|
263 |
||
264 |
Goalw [dvd_def] |
|
265 |
"!! a::'a::ring. a dvd b ==> a dvd (b * c)"; |
|
266 |
by (Clarify_tac 1); |
|
267 |
by (res_inst_tac [("x", "k * c")] exI 1); |
|
268 |
by (simp_tac (simpset() addsimps m_ac) 1); |
|
269 |
qed "dvd_r_mult_right"; |
|
270 |
||
271 |
Addsimps [dvd_add_right, dvd_uminus_right, dvd_l_mult_right, dvd_r_mult_right]; |
|
272 |
||
273 |
(* Inverse of multiplication *) |
|
274 |
||
275 |
section "inverse"; |
|
276 |
||
277 |
Goal "!! a::'a::ring. [| a * x = <1>; a * y = <1> |] ==> x = y"; |
|
278 |
by (res_inst_tac [("a", "(a*y)*x"), ("b", "y*(a*x)")] box_equals 1); |
|
279 |
by (simp_tac (simpset() addsimps m_ac) 1); |
|
280 |
by Auto_tac; |
|
281 |
qed "inverse_unique"; |
|
282 |
||
283 |
Goalw [inverse_def, dvd_def] |
|
284 |
"!! a::'a::ring. a dvd <1> ==> a * inverse a = <1>"; |
|
285 |
by (Asm_simp_tac 1); |
|
286 |
by (Clarify_tac 1); |
|
9970 | 287 |
by (rtac someI 1); |
7998 | 288 |
by (rtac sym 1); |
289 |
by (assume_tac 1); |
|
290 |
qed "r_inverse_ring"; |
|
291 |
||
292 |
Goal "!! a::'a::ring. a dvd <1> ==> inverse a * a= <1>"; |
|
293 |
by (asm_simp_tac (simpset() addsimps r_inverse_ring::m_ac) 1); |
|
294 |
qed "l_inverse_ring"; |
|
295 |
||
296 |
(* Integral domain *) |
|
297 |
||
298 |
section "Integral domains"; |
|
299 |
||
9390
e6b96d953965
renamed // to / (which is what we want anyway) to avoid clash with the new
paulson
parents:
8707
diff
changeset
|
300 |
Goal "[| a * b = <0>; a ~= <0> |] ==> (b::'a::domain) = <0>"; |
7998 | 301 |
by (dtac integral 1); |
302 |
by (Fast_tac 1); |
|
303 |
qed "r_integral"; |
|
304 |
||
9390
e6b96d953965
renamed // to / (which is what we want anyway) to avoid clash with the new
paulson
parents:
8707
diff
changeset
|
305 |
Goal "[| a * b = <0>; b ~= <0> |] ==> (a::'a::domain) = <0>"; |
7998 | 306 |
by (dtac integral 1); |
307 |
by (Fast_tac 1); |
|
308 |
qed "l_integral"; |
|
309 |
||
9390
e6b96d953965
renamed // to / (which is what we want anyway) to avoid clash with the new
paulson
parents:
8707
diff
changeset
|
310 |
Goal "!! a::'a::domain. [| a ~= <0>; b ~= <0> |] ==> a * b ~= <0>"; |
7998 | 311 |
by (etac contrapos 1); |
312 |
by (rtac l_integral 1); |
|
313 |
by (assume_tac 1); |
|
314 |
by (assume_tac 1); |
|
315 |
qed "not_integral"; |
|
316 |
||
317 |
Addsimps [not_integral]; |
|
318 |
||
319 |
Goal "!! a::'a::domain. [| a * x = x; x ~= <0> |] ==> a = <1>"; |
|
320 |
by (res_inst_tac [("a", "- <1>")] a_lcancel 1); |
|
321 |
by (Simp_tac 1); |
|
322 |
by (rtac l_integral 1); |
|
323 |
by (assume_tac 2); |
|
324 |
by (asm_simp_tac (simpset() addsimps [l_distr, l_minus]) 1); |
|
325 |
qed "l_one_integral"; |
|
326 |
||
327 |
Goal "!! a::'a::domain. [| x * a = x; x ~= <0> |] ==> a = <1>"; |
|
328 |
by (res_inst_tac [("a", "- <1>")] a_rcancel 1); |
|
329 |
by (Simp_tac 1); |
|
330 |
by (rtac r_integral 1); |
|
331 |
by (assume_tac 2); |
|
332 |
by (asm_simp_tac (simpset() addsimps [r_distr, r_minus]) 1); |
|
333 |
qed "r_one_integral"; |
|
334 |
||
335 |
(* cancellation laws for multiplication *) |
|
336 |
||
337 |
Goal "!! a::'a::domain. [| a ~= <0>; a * b = a * c |] ==> b = c"; |
|
338 |
by (rtac diff_zero_imp_eq 1); |
|
339 |
by (dtac eq_imp_diff_zero 1); |
|
340 |
by (full_simp_tac (simpset() addsimps [r_minus RS sym, r_distr RS sym]) 1); |
|
341 |
by (fast_tac (claset() addIs [l_integral]) 1); |
|
342 |
qed "m_lcancel"; |
|
343 |
||
344 |
Goal "!! a::'a::domain. [| a ~= <0>; b * a = c * a |] ==> b = c"; |
|
345 |
by (rtac m_lcancel 1); |
|
346 |
by (assume_tac 1); |
|
347 |
by (asm_full_simp_tac (simpset() addsimps m_ac) 1); |
|
348 |
qed "m_rcancel"; |
|
349 |
||
350 |
Goal "!! a::'a::domain. a ~= <0> ==> (a * b = a * c) = (b = c)"; |
|
351 |
by (auto_tac (claset() addDs [m_lcancel], simpset())); |
|
352 |
qed "m_lcancel_eq"; |
|
353 |
||
354 |
Goal "!! a::'a::domain. a ~= <0> ==> (b * a = c * a) = (b = c)"; |
|
355 |
by (asm_simp_tac (simpset() addsimps [m_lcancel_eq, m_comm]) 1); |
|
356 |
qed "m_rcancel_eq"; |
|
357 |
||
358 |
Addsimps [m_lcancel_eq, m_rcancel_eq]; |
|
359 |
||
360 |
(* Fields *) |
|
361 |
||
362 |
section "Fields"; |
|
363 |
||
364 |
Goal "!! a::'a::field. a dvd <1> = (a ~= <0>)"; |
|
365 |
by (blast_tac (claset() addDs [field_ax, unit_imp_nonzero]) 1); |
|
366 |
qed "field_unit"; |
|
367 |
||
368 |
Addsimps [field_unit]; |
|
369 |
||
370 |
Goal "!! a::'a::field. a ~= <0> ==> a * inverse a = <1>"; |
|
371 |
by (asm_full_simp_tac (simpset() addsimps [r_inverse_ring]) 1); |
|
372 |
qed "r_inverse"; |
|
373 |
||
374 |
Goal "!! a::'a::field. a ~= <0> ==> inverse a * a= <1>"; |
|
375 |
by (asm_full_simp_tac (simpset() addsimps [l_inverse_ring]) 1); |
|
376 |
qed "l_inverse"; |
|
377 |
||
378 |
Addsimps [l_inverse, r_inverse]; |
|
379 |
||
380 |
(* fields are factorial domains *) |
|
381 |
||
382 |
Goal "!! a::'a::field. a * b = <0> ==> a = <0> | b = <0>"; |
|
383 |
by (Step_tac 1); |
|
384 |
by (res_inst_tac [("a", "(a*b)*inverse b")] box_equals 1); |
|
385 |
by (rtac refl 3); |
|
386 |
by (simp_tac (simpset() addsimps m_ac) 2); |
|
387 |
by Auto_tac; |
|
388 |
qed "field_integral"; |
|
389 |
||
390 |
Goalw [prime_def, irred_def] "!! a::'a::field. irred a ==> prime a"; |
|
391 |
by (blast_tac (claset() addIs [field_ax]) 1); |
|
392 |
qed "field_fact_prime"; |