author | ballarin |
Thu, 28 Nov 2002 10:50:42 +0100 | |
changeset 13735 | 7de9342aca7a |
parent 13550 | 5a176b8dda84 |
child 13783 | 3294f727e20d |
permissions | -rw-r--r-- |
7998 | 1 |
(* |
2 |
Abstract class ring (commutative, with 1) |
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3 |
$Id$ |
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4 |
Author: Clemens Ballarin, started 9 December 1996 |
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5 |
*) |
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6 |
||
13735 | 7 |
(* |
11778 | 8 |
val a_assoc = thm "plus_ac0.assoc"; |
9 |
val l_zero = thm "plus_ac0.zero"; |
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10 |
val a_comm = thm "plus_ac0.commute"; |
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11 |
||
7998 | 12 |
section "Rings"; |
13 |
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14 |
fun make_left_commute assoc commute s = |
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[rtac (commute RS trans) 1, rtac (assoc RS trans) 1, |
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rtac (commute RS arg_cong) 1]; |
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17 |
||
18 |
(* addition *) |
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19 |
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20 |
qed_goal "a_lcomm" Ring.thy "!!a::'a::ring. a+(b+c) = b+(a+c)" |
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(make_left_commute a_assoc a_comm); |
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22 |
||
23 |
val a_ac = [a_assoc, a_comm, a_lcomm]; |
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24 |
||
11093 | 25 |
Goal "!!a::'a::ring. a + 0 = a"; |
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paulson
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8707
diff
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26 |
by (rtac (a_comm RS trans) 1); |
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renamed // to / (which is what we want anyway) to avoid clash with the new
paulson
parents:
8707
diff
changeset
|
27 |
by (rtac l_zero 1); |
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8707
diff
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|
28 |
qed "r_zero"; |
7998 | 29 |
|
11093 | 30 |
Goal "!!a::'a::ring. a + (-a) = 0"; |
9390
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31 |
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
|
32 |
by (rtac l_neg 1); |
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renamed // to / (which is what we want anyway) to avoid clash with the new
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parents:
8707
diff
changeset
|
33 |
qed "r_neg"; |
7998 | 34 |
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35 |
Goal "!! a::'a::ring. a + b = a + c ==> b = c"; |
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36 |
by (rtac box_equals 1); |
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37 |
by (rtac l_zero 2); |
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38 |
by (rtac l_zero 2); |
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39 |
by (res_inst_tac [("a1", "a")] (l_neg RS subst) 1); |
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40 |
by (asm_simp_tac (simpset() addsimps [a_assoc]) 1); |
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qed "a_lcancel"; |
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42 |
||
43 |
Goal "!! a::'a::ring. b + a = c + a ==> b = c"; |
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44 |
by (rtac a_lcancel 1); |
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45 |
by (asm_simp_tac (simpset() addsimps a_ac) 1); |
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46 |
qed "a_rcancel"; |
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47 |
||
48 |
Goal "!! a::'a::ring. (a + b = a + c) = (b = c)"; |
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49 |
by (auto_tac (claset() addSDs [a_lcancel], simpset())); |
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50 |
qed "a_lcancel_eq"; |
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51 |
||
52 |
Goal "!! a::'a::ring. (b + a = c + a) = (b = c)"; |
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53 |
by (simp_tac (simpset() addsimps [a_lcancel_eq, a_comm]) 1); |
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qed "a_rcancel_eq"; |
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55 |
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56 |
Addsimps [a_lcancel_eq, a_rcancel_eq]; |
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57 |
||
58 |
Goal "!!a::'a::ring. -(-a) = a"; |
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59 |
by (rtac a_lcancel 1); |
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by (rtac (r_neg RS trans) 1); |
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61 |
by (rtac (l_neg RS sym) 1); |
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62 |
qed "minus_minus"; |
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63 |
||
11093 | 64 |
Goal "- 0 = (0::'a::ring)"; |
7998 | 65 |
by (rtac a_lcancel 1); |
66 |
by (rtac (r_neg RS trans) 1); |
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67 |
by (rtac (l_zero RS sym) 1); |
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68 |
qed "minus0"; |
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69 |
||
70 |
Goal "!!a::'a::ring. -(a + b) = (-a) + (-b)"; |
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by (res_inst_tac [("a", "a+b")] a_lcancel 1); |
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by (simp_tac (simpset() addsimps ([r_neg, l_neg, l_zero]@a_ac)) 1); |
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qed "minus_add"; |
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74 |
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75 |
(* multiplication *) |
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76 |
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77 |
qed_goal "m_lcomm" Ring.thy "!!a::'a::ring. a*(b*c) = b*(a*c)" |
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78 |
(make_left_commute m_assoc m_comm); |
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79 |
||
80 |
val m_ac = [m_assoc, m_comm, m_lcomm]; |
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81 |
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13735 | 82 |
Goal "!!a::'a::ring. a * 1 = a"; |
9390
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paulson
parents:
8707
diff
changeset
|
83 |
by (rtac (m_comm RS trans) 1); |
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paulson
parents:
8707
diff
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|
84 |
by (rtac l_one 1); |
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85 |
qed "r_one"; |
7998 | 86 |
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87 |
(* distributive and derived *) |
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88 |
||
89 |
Goal "!!a::'a::ring. a * (b + c) = a * b + a * c"; |
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90 |
by (rtac (m_comm RS trans) 1); |
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91 |
by (rtac (l_distr RS trans) 1); |
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92 |
by (simp_tac (simpset() addsimps [m_comm]) 1); |
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93 |
qed "r_distr"; |
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94 |
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95 |
val m_distr = m_ac @ [l_distr, r_distr]; |
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96 |
||
97 |
(* the following two proofs can be found in |
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98 |
Jacobson, Basic Algebra I, pp. 88-89 *) |
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99 |
||
11093 | 100 |
Goal "!!a::'a::ring. 0 * a = 0"; |
7998 | 101 |
by (rtac a_lcancel 1); |
102 |
by (rtac (l_distr RS sym RS trans) 1); |
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103 |
by (simp_tac (simpset() addsimps [r_zero]) 1); |
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104 |
qed "l_null"; |
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105 |
||
11093 | 106 |
Goal "!!a::'a::ring. a * 0 = 0"; |
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paulson
parents:
8707
diff
changeset
|
107 |
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
|
108 |
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
|
109 |
qed "r_null"; |
7998 | 110 |
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111 |
Goal "!!a::'a::ring. (-a) * b = - (a * b)"; |
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112 |
by (rtac a_lcancel 1); |
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113 |
by (rtac (r_neg RS sym RSN (2, trans)) 1); |
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by (rtac (l_distr RS sym RS trans) 1); |
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115 |
by (simp_tac (simpset() addsimps [l_null, r_neg]) 1); |
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116 |
qed "l_minus"; |
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117 |
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118 |
Goal "!!a::'a::ring. a * (-b) = - (a * b)"; |
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by (rtac a_lcancel 1); |
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by (rtac (r_neg RS sym RSN (2, trans)) 1); |
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121 |
by (rtac (r_distr RS sym RS trans) 1); |
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122 |
by (simp_tac (simpset() addsimps [r_null, r_neg]) 1); |
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123 |
qed "r_minus"; |
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124 |
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125 |
val m_minus = [l_minus, r_minus]; |
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126 |
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127 |
Addsimps [l_zero, r_zero, l_neg, r_neg, minus_minus, minus0, |
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11093 | 128 |
l_one, r_one, l_null, r_null]; |
7998 | 129 |
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130 |
(* further rules *) |
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131 |
||
11093 | 132 |
Goal "!!a::'a::ring. -a = 0 ==> a = 0"; |
7998 | 133 |
by (res_inst_tac [("t", "a")] (minus_minus RS subst) 1); |
134 |
by (Asm_simp_tac 1); |
|
135 |
qed "uminus_monom"; |
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136 |
||
11093 | 137 |
Goal "!!a::'a::ring. a ~= 0 ==> -a ~= 0"; |
10198 | 138 |
by (blast_tac (claset() addIs [uminus_monom]) 1); |
7998 | 139 |
qed "uminus_monom_neq"; |
140 |
||
11093 | 141 |
Goal "!!a::'a::ring. a * b ~= 0 ==> a ~= 0"; |
10198 | 142 |
by Auto_tac; |
7998 | 143 |
qed "l_nullD"; |
144 |
||
11093 | 145 |
Goal "!!a::'a::ring. a * b ~= 0 ==> b ~= 0"; |
10198 | 146 |
by Auto_tac; |
7998 | 147 |
qed "r_nullD"; |
148 |
||
11093 | 149 |
(* reflection between a = b and a -- b = 0 *) |
7998 | 150 |
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11093 | 151 |
Goal "!!a::'a::ring. a = b ==> a + (-b) = 0"; |
7998 | 152 |
by (Asm_simp_tac 1); |
153 |
qed "eq_imp_diff_zero"; |
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154 |
||
11093 | 155 |
Goal "!!a::'a::ring. a + (-b) = 0 ==> a = b"; |
7998 | 156 |
by (res_inst_tac [("a", "-b")] a_rcancel 1); |
157 |
by (Asm_simp_tac 1); |
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158 |
qed "diff_zero_imp_eq"; |
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159 |
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160 |
(* this could be a rewrite rule, but won't terminate |
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161 |
==> make it a simproc? |
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11093 | 162 |
Goal "!!a::'a::ring. (a = b) = (a -- b = 0)"; |
7998 | 163 |
*) |
164 |
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13735 | 165 |
*) |
166 |
||
7998 | 167 |
(* Power *) |
168 |
||
13735 | 169 |
Goal "!!a::'a::ring. a ^ 0 = 1"; |
170 |
by (simp_tac (simpset() addsimps [power_def]) 1); |
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7998 | 171 |
qed "power_0"; |
172 |
||
173 |
Goal "!!a::'a::ring. a ^ Suc n = a ^ n * a"; |
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13735 | 174 |
by (simp_tac (simpset() addsimps [power_def]) 1); |
7998 | 175 |
qed "power_Suc"; |
176 |
||
177 |
Addsimps [power_0, power_Suc]; |
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178 |
||
13735 | 179 |
Goal "1 ^ n = (1::'a::ring)"; |
8707 | 180 |
by (induct_tac "n" 1); |
181 |
by Auto_tac; |
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7998 | 182 |
qed "power_one"; |
183 |
||
11093 | 184 |
Goal "!!n. n ~= 0 ==> 0 ^ n = (0::'a::ring)"; |
7998 | 185 |
by (etac rev_mp 1); |
8707 | 186 |
by (induct_tac "n" 1); |
187 |
by Auto_tac; |
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7998 | 188 |
qed "power_zero"; |
189 |
||
190 |
Addsimps [power_zero, power_one]; |
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191 |
||
192 |
Goal "!! a::'a::ring. a ^ m * a ^ n = a ^ (m + n)"; |
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8707 | 193 |
by (induct_tac "m" 1); |
7998 | 194 |
by (Simp_tac 1); |
13735 | 195 |
by (Asm_simp_tac 1); |
7998 | 196 |
qed "power_mult"; |
197 |
||
198 |
(* Divisibility *) |
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199 |
section "Divisibility"; |
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200 |
||
11093 | 201 |
Goalw [dvd_def] "!! a::'a::ring. a dvd 0"; |
202 |
by (res_inst_tac [("x", "0")] exI 1); |
|
7998 | 203 |
by (Simp_tac 1); |
204 |
qed "dvd_zero_right"; |
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205 |
||
11093 | 206 |
Goalw [dvd_def] "!! a::'a::ring. 0 dvd a ==> a = 0"; |
7998 | 207 |
by Auto_tac; |
208 |
qed "dvd_zero_left"; |
|
209 |
||
210 |
Goalw [dvd_def] "!! a::'a::ring. a dvd a"; |
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13735 | 211 |
by (res_inst_tac [("x", "1")] exI 1); |
7998 | 212 |
by (Simp_tac 1); |
213 |
qed "dvd_refl_ring"; |
|
214 |
||
215 |
Goalw [dvd_def] "!! a::'a::ring. [| a dvd b; b dvd c |] ==> a dvd c"; |
|
216 |
by (Step_tac 1); |
|
217 |
by (res_inst_tac [("x", "k * ka")] exI 1); |
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13735 | 218 |
by (Asm_simp_tac 1); |
7998 | 219 |
qed "dvd_trans_ring"; |
220 |
||
221 |
Addsimps [dvd_zero_right, dvd_refl_ring]; |
|
222 |
||
223 |
Goalw [dvd_def] |
|
13735 | 224 |
"!!a::'a::ring. [| a dvd 1; b dvd 1 |] ==> a * b dvd 1"; |
7998 | 225 |
by (Clarify_tac 1); |
226 |
by (res_inst_tac [("x", "k * ka")] exI 1); |
|
13735 | 227 |
by (Asm_full_simp_tac 1); |
7998 | 228 |
qed "unit_mult"; |
229 |
||
13735 | 230 |
Goal "!!a::'a::ring. a dvd 1 ==> a^n dvd 1"; |
7998 | 231 |
by (induct_tac "n" 1); |
232 |
by (Simp_tac 1); |
|
233 |
by (asm_simp_tac (simpset() addsimps [unit_mult]) 1); |
|
234 |
qed "unit_power"; |
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235 |
||
236 |
Goalw [dvd_def] |
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10789
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nipkow
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10448
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changeset
|
237 |
"!! a::'a::ring. [| a dvd b; a dvd c |] ==> a dvd b + c"; |
7998 | 238 |
by (Clarify_tac 1); |
239 |
by (res_inst_tac [("x", "k + ka")] exI 1); |
|
240 |
by (simp_tac (simpset() addsimps [r_distr]) 1); |
|
241 |
qed "dvd_add_right"; |
|
242 |
||
243 |
Goalw [dvd_def] |
|
10789
260fa2c67e3e
Changed priority of dvd from 70 to 50 as befits a relation.
nipkow
parents:
10448
diff
changeset
|
244 |
"!! a::'a::ring. a dvd b ==> a dvd -b"; |
7998 | 245 |
by (Clarify_tac 1); |
246 |
by (res_inst_tac [("x", "-k")] exI 1); |
|
247 |
by (simp_tac (simpset() addsimps [r_minus]) 1); |
|
248 |
qed "dvd_uminus_right"; |
|
249 |
||
250 |
Goalw [dvd_def] |
|
10789
260fa2c67e3e
Changed priority of dvd from 70 to 50 as befits a relation.
nipkow
parents:
10448
diff
changeset
|
251 |
"!! a::'a::ring. a dvd b ==> a dvd c*b"; |
7998 | 252 |
by (Clarify_tac 1); |
253 |
by (res_inst_tac [("x", "c * k")] exI 1); |
|
13735 | 254 |
by (Simp_tac 1); |
7998 | 255 |
qed "dvd_l_mult_right"; |
256 |
||
257 |
Goalw [dvd_def] |
|
10789
260fa2c67e3e
Changed priority of dvd from 70 to 50 as befits a relation.
nipkow
parents:
10448
diff
changeset
|
258 |
"!! a::'a::ring. a dvd b ==> a dvd b*c"; |
7998 | 259 |
by (Clarify_tac 1); |
260 |
by (res_inst_tac [("x", "k * c")] exI 1); |
|
13735 | 261 |
by (Simp_tac 1); |
7998 | 262 |
qed "dvd_r_mult_right"; |
263 |
||
264 |
Addsimps [dvd_add_right, dvd_uminus_right, dvd_l_mult_right, dvd_r_mult_right]; |
|
265 |
||
266 |
(* Inverse of multiplication *) |
|
267 |
||
268 |
section "inverse"; |
|
269 |
||
13735 | 270 |
Goal "!! a::'a::ring. [| a * x = 1; a * y = 1 |] ==> x = y"; |
7998 | 271 |
by (res_inst_tac [("a", "(a*y)*x"), ("b", "y*(a*x)")] box_equals 1); |
13735 | 272 |
by (Simp_tac 1); |
7998 | 273 |
by Auto_tac; |
274 |
qed "inverse_unique"; |
|
275 |
||
13735 | 276 |
Goal "!! a::'a::ring. a dvd 1 ==> a * inverse a = 1"; |
277 |
by (asm_full_simp_tac (simpset () addsimps [inverse_def, dvd_def]) 1); |
|
7998 | 278 |
by (Clarify_tac 1); |
13735 | 279 |
by (rtac theI 1); |
280 |
by (atac 1); |
|
281 |
by (rtac inverse_unique 1); |
|
282 |
by (atac 1); |
|
283 |
by (atac 1); |
|
7998 | 284 |
qed "r_inverse_ring"; |
285 |
||
13735 | 286 |
Goal "!! a::'a::ring. a dvd 1 ==> inverse a * a= 1"; |
287 |
by (asm_simp_tac (simpset() addsimps [r_inverse_ring]) 1); |
|
7998 | 288 |
qed "l_inverse_ring"; |
289 |
||
290 |
(* Integral domain *) |
|
291 |
||
13735 | 292 |
(* |
7998 | 293 |
section "Integral domains"; |
294 |
||
13735 | 295 |
Goal "0 ~= (1::'a::domain)"; |
11093 | 296 |
by (rtac not_sym 1); |
297 |
by (rtac one_not_zero 1); |
|
298 |
qed "zero_not_one"; |
|
299 |
||
13735 | 300 |
Goal "!! a::'a::domain. a dvd 1 ==> a ~= 0"; |
11093 | 301 |
by (auto_tac (claset() addDs [dvd_zero_left], |
302 |
simpset() addsimps [one_not_zero] )); |
|
303 |
qed "unit_imp_nonzero"; |
|
304 |
||
305 |
Goal "[| a * b = 0; a ~= 0 |] ==> (b::'a::domain) = 0"; |
|
7998 | 306 |
by (dtac integral 1); |
307 |
by (Fast_tac 1); |
|
308 |
qed "r_integral"; |
|
309 |
||
11093 | 310 |
Goal "[| a * b = 0; b ~= 0 |] ==> (a::'a::domain) = 0"; |
7998 | 311 |
by (dtac integral 1); |
312 |
by (Fast_tac 1); |
|
313 |
qed "l_integral"; |
|
314 |
||
11093 | 315 |
Goal "!! a::'a::domain. [| a ~= 0; b ~= 0 |] ==> a * b ~= 0"; |
10230 | 316 |
by (blast_tac (claset() addIs [l_integral]) 1); |
7998 | 317 |
qed "not_integral"; |
318 |
||
11093 | 319 |
Addsimps [not_integral, one_not_zero, zero_not_one]; |
7998 | 320 |
|
13735 | 321 |
Goal "!! a::'a::domain. [| a * x = x; x ~= 0 |] ==> a = 1"; |
322 |
by (res_inst_tac [("a", "- 1")] a_lcancel 1); |
|
7998 | 323 |
by (Simp_tac 1); |
324 |
by (rtac l_integral 1); |
|
325 |
by (assume_tac 2); |
|
326 |
by (asm_simp_tac (simpset() addsimps [l_distr, l_minus]) 1); |
|
327 |
qed "l_one_integral"; |
|
328 |
||
13735 | 329 |
Goal "!! a::'a::domain. [| x * a = x; x ~= 0 |] ==> a = 1"; |
330 |
by (res_inst_tac [("a", "- 1")] a_rcancel 1); |
|
7998 | 331 |
by (Simp_tac 1); |
332 |
by (rtac r_integral 1); |
|
333 |
by (assume_tac 2); |
|
334 |
by (asm_simp_tac (simpset() addsimps [r_distr, r_minus]) 1); |
|
335 |
qed "r_one_integral"; |
|
336 |
||
337 |
(* cancellation laws for multiplication *) |
|
338 |
||
11093 | 339 |
Goal "!! a::'a::domain. [| a ~= 0; a * b = a * c |] ==> b = c"; |
7998 | 340 |
by (rtac diff_zero_imp_eq 1); |
341 |
by (dtac eq_imp_diff_zero 1); |
|
342 |
by (full_simp_tac (simpset() addsimps [r_minus RS sym, r_distr RS sym]) 1); |
|
343 |
by (fast_tac (claset() addIs [l_integral]) 1); |
|
344 |
qed "m_lcancel"; |
|
345 |
||
11093 | 346 |
Goal "!! a::'a::domain. [| a ~= 0; b * a = c * a |] ==> b = c"; |
7998 | 347 |
by (rtac m_lcancel 1); |
348 |
by (assume_tac 1); |
|
13735 | 349 |
by (Asm_full_simp_tac 1); |
7998 | 350 |
qed "m_rcancel"; |
351 |
||
11093 | 352 |
Goal "!! a::'a::domain. a ~= 0 ==> (a * b = a * c) = (b = c)"; |
7998 | 353 |
by (auto_tac (claset() addDs [m_lcancel], simpset())); |
354 |
qed "m_lcancel_eq"; |
|
355 |
||
11093 | 356 |
Goal "!! a::'a::domain. a ~= 0 ==> (b * a = c * a) = (b = c)"; |
7998 | 357 |
by (asm_simp_tac (simpset() addsimps [m_lcancel_eq, m_comm]) 1); |
358 |
qed "m_rcancel_eq"; |
|
359 |
||
360 |
Addsimps [m_lcancel_eq, m_rcancel_eq]; |
|
13735 | 361 |
*) |
7998 | 362 |
|
363 |
(* Fields *) |
|
364 |
||
365 |
section "Fields"; |
|
366 |
||
13735 | 367 |
Goal "!! a::'a::field. (a dvd 1) = (a ~= 0)"; |
368 |
by (auto_tac (claset() addDs [thm "field_ax", dvd_zero_left], |
|
369 |
simpset() addsimps [thm "field_one_not_zero"])); |
|
7998 | 370 |
qed "field_unit"; |
371 |
||
13735 | 372 |
(* required for instantiation of field < domain in file Field.thy *) |
373 |
||
7998 | 374 |
Addsimps [field_unit]; |
375 |
||
13735 | 376 |
val field_one_not_zero = thm "field_one_not_zero"; |
377 |
||
378 |
Goal "!! a::'a::field. a ~= 0 ==> a * inverse a = 1"; |
|
7998 | 379 |
by (asm_full_simp_tac (simpset() addsimps [r_inverse_ring]) 1); |
380 |
qed "r_inverse"; |
|
381 |
||
13735 | 382 |
Goal "!! a::'a::field. a ~= 0 ==> inverse a * a= 1"; |
7998 | 383 |
by (asm_full_simp_tac (simpset() addsimps [l_inverse_ring]) 1); |
384 |
qed "l_inverse"; |
|
385 |
||
386 |
Addsimps [l_inverse, r_inverse]; |
|
387 |
||
11093 | 388 |
(* fields are integral domains *) |
7998 | 389 |
|
11093 | 390 |
Goal "!! a::'a::field. a * b = 0 ==> a = 0 | b = 0"; |
7998 | 391 |
by (Step_tac 1); |
392 |
by (res_inst_tac [("a", "(a*b)*inverse b")] box_equals 1); |
|
393 |
by (rtac refl 3); |
|
13735 | 394 |
by (Simp_tac 2); |
7998 | 395 |
by Auto_tac; |
396 |
qed "field_integral"; |
|
397 |
||
11093 | 398 |
(* fields are factorial domains *) |
399 |
||
13735 | 400 |
Goalw [thm "prime_def", thm "irred_def"] |
401 |
"!! a::'a::field. irred a ==> prime a"; |
|
402 |
by (blast_tac (claset() addIs [thm "field_ax"]) 1); |
|
7998 | 403 |
qed "field_fact_prime"; |