author | haftmann |
Sat, 18 Nov 2006 00:20:24 +0100 | |
changeset 21416 | f23e4e75dfd3 |
parent 20318 | 0e0ea63fe768 |
permissions | -rw-r--r-- |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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|
1 |
(* |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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|
2 |
Abstract class ring (commutative, with 1) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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|
3 |
$Id$ |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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4 |
Author: Clemens Ballarin, started 9 December 1996 |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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5 |
*) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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6 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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|
7 |
(* |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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8 |
val a_assoc = thm "semigroup_add.add_assoc"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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9 |
val l_zero = thm "comm_monoid_add.add_0"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
10 |
val a_comm = thm "ab_semigroup_add.add_commute"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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11 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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12 |
section "Rings"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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13 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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14 |
fun make_left_commute assoc commute s = |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
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15 |
[rtac (commute RS trans) 1, rtac (assoc RS trans) 1, |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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changeset
|
16 |
rtac (commute RS arg_cong) 1]; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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17 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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18 |
(* addition *) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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19 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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20 |
qed_goal "a_lcomm" Ring2.thy "!!a::'a::ring. a+(b+c) = b+(a+c)" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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21 |
(make_left_commute a_assoc a_comm); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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22 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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23 |
val a_ac = [a_assoc, a_comm, a_lcomm]; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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24 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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25 |
Goal "!!a::'a::ring. a + 0 = a"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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26 |
by (rtac (a_comm RS trans) 1); |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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27 |
by (rtac l_zero 1); |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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28 |
qed "r_zero"; |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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29 |
|
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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30 |
Goal "!!a::'a::ring. a + (-a) = 0"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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31 |
by (rtac (a_comm RS trans) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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32 |
by (rtac l_neg 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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33 |
qed "r_neg"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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34 |
|
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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35 |
Goal "!! a::'a::ring. a + b = a + c ==> b = c"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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36 |
by (rtac box_equals 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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37 |
by (rtac l_zero 2); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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|
38 |
by (rtac l_zero 2); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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39 |
by (res_inst_tac [("a1", "a")] (l_neg RS subst) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
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40 |
by (asm_simp_tac (simpset() addsimps [a_assoc]) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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41 |
qed "a_lcancel"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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42 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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43 |
Goal "!! a::'a::ring. b + a = c + a ==> b = c"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
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44 |
by (rtac a_lcancel 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
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45 |
by (asm_simp_tac (simpset() addsimps a_ac) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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46 |
qed "a_rcancel"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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47 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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48 |
Goal "!! a::'a::ring. (a + b = a + c) = (b = c)"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
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49 |
by (auto_tac (claset() addSDs [a_lcancel], simpset())); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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50 |
qed "a_lcancel_eq"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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51 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
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52 |
Goal "!! a::'a::ring. (b + a = c + a) = (b = c)"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
53 |
by (simp_tac (simpset() addsimps [a_lcancel_eq, a_comm]) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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54 |
qed "a_rcancel_eq"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
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55 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
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56 |
Addsimps [a_lcancel_eq, a_rcancel_eq]; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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57 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
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58 |
Goal "!!a::'a::ring. -(-a) = a"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
59 |
by (rtac a_lcancel 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
60 |
by (rtac (r_neg RS trans) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
61 |
by (rtac (l_neg RS sym) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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|
62 |
qed "minus_minus"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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63 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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64 |
Goal "- 0 = (0::'a::ring)"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
65 |
by (rtac a_lcancel 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
66 |
by (rtac (r_neg RS trans) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
67 |
by (rtac (l_zero RS sym) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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|
68 |
qed "minus0"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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changeset
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69 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
70 |
Goal "!!a::'a::ring. -(a + b) = (-a) + (-b)"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
71 |
by (res_inst_tac [("a", "a+b")] a_lcancel 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
72 |
by (simp_tac (simpset() addsimps ([r_neg, l_neg, l_zero]@a_ac)) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
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73 |
qed "minus_add"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
74 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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|
75 |
(* multiplication *) |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
76 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
77 |
qed_goal "m_lcomm" Ring2.thy "!!a::'a::ring. a*(b*c) = b*(a*c)" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
78 |
(make_left_commute m_assoc m_comm); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
79 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
80 |
val m_ac = [m_assoc, m_comm, m_lcomm]; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
81 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
82 |
Goal "!!a::'a::ring. a * 1 = a"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
83 |
by (rtac (m_comm RS trans) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
84 |
by (rtac l_one 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
85 |
qed "r_one"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
86 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
87 |
(* distributive and derived *) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
88 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
89 |
Goal "!!a::'a::ring. a * (b + c) = a * b + a * c"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
90 |
by (rtac (m_comm RS trans) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
91 |
by (rtac (l_distr RS trans) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
92 |
by (simp_tac (simpset() addsimps [m_comm]) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
93 |
qed "r_distr"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
94 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
95 |
val m_distr = m_ac @ [l_distr, r_distr]; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
96 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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|
97 |
(* the following two proofs can be found in |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
98 |
Jacobson, Basic Algebra I, pp. 88-89 *) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
99 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
100 |
Goal "!!a::'a::ring. 0 * a = 0"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
101 |
by (rtac a_lcancel 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
102 |
by (rtac (l_distr RS sym RS trans) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
103 |
by (simp_tac (simpset() addsimps [r_zero]) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
104 |
qed "l_null"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
105 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
106 |
Goal "!!a::'a::ring. a * 0 = 0"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
107 |
by (rtac (m_comm RS trans) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
108 |
by (rtac l_null 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
109 |
qed "r_null"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
110 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
111 |
Goal "!!a::'a::ring. (-a) * b = - (a * b)"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
112 |
by (rtac a_lcancel 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
113 |
by (rtac (r_neg RS sym RSN (2, trans)) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
114 |
by (rtac (l_distr RS sym RS trans) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
115 |
by (simp_tac (simpset() addsimps [l_null, r_neg]) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
116 |
qed "l_minus"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
117 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
118 |
Goal "!!a::'a::ring. a * (-b) = - (a * b)"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
119 |
by (rtac a_lcancel 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
120 |
by (rtac (r_neg RS sym RSN (2, trans)) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
121 |
by (rtac (r_distr RS sym RS trans) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
122 |
by (simp_tac (simpset() addsimps [r_null, r_neg]) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
123 |
qed "r_minus"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
124 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
125 |
val m_minus = [l_minus, r_minus]; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
126 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
127 |
Addsimps [l_zero, r_zero, l_neg, r_neg, minus_minus, minus0, |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
128 |
l_one, r_one, l_null, r_null]; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
129 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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|
130 |
(* further rules *) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
131 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
132 |
Goal "!!a::'a::ring. -a = 0 ==> a = 0"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
133 |
by (res_inst_tac [("t", "a")] (minus_minus RS subst) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
134 |
by (Asm_simp_tac 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
135 |
qed "uminus_monom"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
136 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
137 |
Goal "!!a::'a::ring. a ~= 0 ==> -a ~= 0"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
138 |
by (blast_tac (claset() addIs [uminus_monom]) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
139 |
qed "uminus_monom_neq"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
140 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
141 |
Goal "!!a::'a::ring. a * b ~= 0 ==> a ~= 0"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
142 |
by Auto_tac; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
143 |
qed "l_nullD"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
144 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
145 |
Goal "!!a::'a::ring. a * b ~= 0 ==> b ~= 0"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
146 |
by Auto_tac; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
147 |
qed "r_nullD"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
148 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
149 |
(* reflection between a = b and a -- b = 0 *) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
150 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
151 |
Goal "!!a::'a::ring. a = b ==> a + (-b) = 0"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
152 |
by (Asm_simp_tac 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
153 |
qed "eq_imp_diff_zero"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
154 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
155 |
Goal "!!a::'a::ring. a + (-b) = 0 ==> a = b"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
156 |
by (res_inst_tac [("a", "-b")] a_rcancel 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
157 |
by (Asm_simp_tac 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
158 |
qed "diff_zero_imp_eq"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
159 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
160 |
(* this could be a rewrite rule, but won't terminate |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
161 |
==> make it a simproc? |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
162 |
Goal "!!a::'a::ring. (a = b) = (a -- b = 0)"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
163 |
*) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
164 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
165 |
*) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
166 |
|
21416 | 167 |
val dvd_def = thm "dvd_def'"; |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
168 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
169 |
Goalw [dvd_def] |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
170 |
"!!a::'a::ring. [| a dvd 1; b dvd 1 |] ==> a * b dvd 1"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
171 |
by (Clarify_tac 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
172 |
by (res_inst_tac [("x", "k * ka")] exI 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
173 |
by (Asm_full_simp_tac 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
174 |
qed "unit_mult"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
175 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
176 |
Goal "!!a::'a::ring. a dvd 1 ==> a^n dvd 1"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
177 |
by (induct_tac "n" 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
178 |
by (Simp_tac 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
179 |
by (asm_simp_tac (simpset() addsimps [unit_mult]) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
180 |
qed "unit_power"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
181 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
182 |
Goalw [dvd_def] |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
183 |
"!! a::'a::ring. [| a dvd b; a dvd c |] ==> a dvd b + c"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
184 |
by (Clarify_tac 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
185 |
by (res_inst_tac [("x", "k + ka")] exI 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
186 |
by (simp_tac (simpset() addsimps [r_distr]) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
187 |
qed "dvd_add_right"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
188 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
189 |
Goalw [dvd_def] |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
190 |
"!! a::'a::ring. a dvd b ==> a dvd -b"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
191 |
by (Clarify_tac 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
192 |
by (res_inst_tac [("x", "-k")] exI 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
193 |
by (simp_tac (simpset() addsimps [r_minus]) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
194 |
qed "dvd_uminus_right"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
195 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
196 |
Goalw [dvd_def] |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
197 |
"!! a::'a::ring. a dvd b ==> a dvd c*b"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
198 |
by (Clarify_tac 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
199 |
by (res_inst_tac [("x", "c * k")] exI 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
200 |
by (Simp_tac 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
201 |
qed "dvd_l_mult_right"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
202 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
203 |
Goalw [dvd_def] |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
204 |
"!! a::'a::ring. a dvd b ==> a dvd b*c"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
205 |
by (Clarify_tac 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
206 |
by (res_inst_tac [("x", "k * c")] exI 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
207 |
by (Simp_tac 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
208 |
qed "dvd_r_mult_right"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
209 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
210 |
Addsimps [dvd_add_right, dvd_uminus_right, dvd_l_mult_right, dvd_r_mult_right]; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
211 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
212 |
(* Inverse of multiplication *) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
213 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
214 |
section "inverse"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
215 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
216 |
Goal "!! a::'a::ring. [| a * x = 1; a * y = 1 |] ==> x = y"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
217 |
by (res_inst_tac [("a", "(a*y)*x"), ("b", "y*(a*x)")] box_equals 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
218 |
by (Simp_tac 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
219 |
by Auto_tac; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
220 |
qed "inverse_unique"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
221 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
222 |
Goal "!! a::'a::ring. a dvd 1 ==> a * inverse a = 1"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
223 |
by (asm_full_simp_tac (simpset () addsimps [inverse_def, dvd_def] |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
224 |
delsimprocs [ring_simproc]) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
225 |
by (Clarify_tac 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
226 |
by (rtac theI 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
227 |
by (atac 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
228 |
by (rtac inverse_unique 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
229 |
by (atac 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
230 |
by (atac 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
231 |
qed "r_inverse_ring"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
232 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
233 |
Goal "!! a::'a::ring. a dvd 1 ==> inverse a * a= 1"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
234 |
by (asm_simp_tac (simpset() addsimps [r_inverse_ring]) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
235 |
qed "l_inverse_ring"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
236 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
237 |
(* Integral domain *) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
238 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
239 |
(* |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
240 |
section "Integral domains"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
241 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
242 |
Goal "0 ~= (1::'a::domain)"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
243 |
by (rtac not_sym 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
244 |
by (rtac one_not_zero 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
245 |
qed "zero_not_one"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
246 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
247 |
Goal "!! a::'a::domain. a dvd 1 ==> a ~= 0"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
248 |
by (auto_tac (claset() addDs [dvd_zero_left], |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
249 |
simpset() addsimps [one_not_zero] )); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
250 |
qed "unit_imp_nonzero"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
251 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
252 |
Goal "[| a * b = 0; a ~= 0 |] ==> (b::'a::domain) = 0"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
253 |
by (dtac integral 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
254 |
by (Fast_tac 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
255 |
qed "r_integral"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
256 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
257 |
Goal "[| a * b = 0; b ~= 0 |] ==> (a::'a::domain) = 0"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
258 |
by (dtac integral 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
259 |
by (Fast_tac 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
260 |
qed "l_integral"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
261 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
262 |
Goal "!! a::'a::domain. [| a ~= 0; b ~= 0 |] ==> a * b ~= 0"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
263 |
by (blast_tac (claset() addIs [l_integral]) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
264 |
qed "not_integral"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
265 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
266 |
Addsimps [not_integral, one_not_zero, zero_not_one]; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
267 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
268 |
Goal "!! a::'a::domain. [| a * x = x; x ~= 0 |] ==> a = 1"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
269 |
by (res_inst_tac [("a", "- 1")] a_lcancel 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
270 |
by (Simp_tac 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
271 |
by (rtac l_integral 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
272 |
by (assume_tac 2); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
273 |
by (asm_simp_tac (simpset() addsimps [l_distr, l_minus]) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
274 |
qed "l_one_integral"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
275 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
276 |
Goal "!! a::'a::domain. [| x * a = x; x ~= 0 |] ==> a = 1"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
277 |
by (res_inst_tac [("a", "- 1")] a_rcancel 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
278 |
by (Simp_tac 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
279 |
by (rtac r_integral 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
280 |
by (assume_tac 2); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
281 |
by (asm_simp_tac (simpset() addsimps [r_distr, r_minus]) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
282 |
qed "r_one_integral"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
283 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
284 |
(* cancellation laws for multiplication *) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
285 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
286 |
Goal "!! a::'a::domain. [| a ~= 0; a * b = a * c |] ==> b = c"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
287 |
by (rtac diff_zero_imp_eq 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
288 |
by (dtac eq_imp_diff_zero 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
289 |
by (full_simp_tac (simpset() addsimps [r_minus RS sym, r_distr RS sym]) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
290 |
by (fast_tac (claset() addIs [l_integral]) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
291 |
qed "m_lcancel"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
292 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
293 |
Goal "!! a::'a::domain. [| a ~= 0; b * a = c * a |] ==> b = c"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
294 |
by (rtac m_lcancel 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
295 |
by (assume_tac 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
296 |
by (Asm_full_simp_tac 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
297 |
qed "m_rcancel"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
298 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
299 |
Goal "!! a::'a::domain. a ~= 0 ==> (a * b = a * c) = (b = c)"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
300 |
by (auto_tac (claset() addDs [m_lcancel], simpset())); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
301 |
qed "m_lcancel_eq"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
302 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
303 |
Goal "!! a::'a::domain. a ~= 0 ==> (b * a = c * a) = (b = c)"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
304 |
by (asm_simp_tac (simpset() addsimps [m_lcancel_eq, m_comm]) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
305 |
qed "m_rcancel_eq"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
306 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
307 |
Addsimps [m_lcancel_eq, m_rcancel_eq]; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
308 |
*) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
309 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
310 |
(* Fields *) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
311 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
312 |
section "Fields"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
313 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
314 |
Goal "!! a::'a::field. (a dvd 1) = (a ~= 0)"; |
21416 | 315 |
by (auto_tac (claset() addDs [thm "field_ax", thm "dvd_zero_left"], |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
316 |
simpset() addsimps [thm "field_one_not_zero"])); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
317 |
qed "field_unit"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
318 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
319 |
(* required for instantiation of field < domain in file Field.thy *) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
320 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
321 |
Addsimps [field_unit]; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
322 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
323 |
val field_one_not_zero = thm "field_one_not_zero"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
324 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
325 |
Goal "!! a::'a::field. a ~= 0 ==> a * inverse a = 1"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
326 |
by (asm_full_simp_tac (simpset() addsimps [r_inverse_ring]) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
327 |
qed "r_inverse"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
328 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
329 |
Goal "!! a::'a::field. a ~= 0 ==> inverse a * a= 1"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
330 |
by (asm_full_simp_tac (simpset() addsimps [l_inverse_ring]) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
331 |
qed "l_inverse"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
332 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
333 |
Addsimps [l_inverse, r_inverse]; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
334 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
335 |
(* fields are integral domains *) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
336 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
337 |
Goal "!! a::'a::field. a * b = 0 ==> a = 0 | b = 0"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
338 |
by (Step_tac 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
339 |
by (res_inst_tac [("a", "(a*b)*inverse b")] box_equals 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
340 |
by (rtac refl 3); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
341 |
by (Simp_tac 2); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
342 |
by Auto_tac; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
343 |
qed "field_integral"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
344 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
345 |
(* fields are factorial domains *) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
346 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
347 |
Goalw [thm "prime_def", thm "irred_def"] |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
348 |
"!! a::'a::field. irred a ==> prime a"; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
349 |
by (blast_tac (claset() addIs [thm "field_ax"]) 1); |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
350 |
qed "field_fact_prime"; |