author | wenzelm |
Fri, 06 Aug 2010 12:37:00 +0200 | |
changeset 38159 | e9b4835a54ee |
parent 32479 | 521cc9bf2958 |
child 41541 | 1fa4725c4656 |
permissions | -rw-r--r-- |
38159 | 1 |
(* Title: HOL/Old_Number_Theory/Int2.thy |
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2 |
Authors: Jeremy Avigad, David Gray, and Adam Kramer |
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paulson
parents:
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3 |
*) |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
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parents:
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4 |
|
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
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parents:
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5 |
header {*Integers: Divisibility and Congruences*} |
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parents:
diff
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6 |
|
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theory Int2 |
8 |
imports Finite2 WilsonRuss |
|
9 |
begin |
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38159 | 11 |
definition MultInv :: "int => int => int" |
12 |
where "MultInv p x = x ^ nat (p - 2)" |
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13 |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
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subsection {* Useful lemmas about dvd and powers *} |
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|
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lemma zpower_zdvd_prop1: |
18 |
"0 < n \<Longrightarrow> p dvd y \<Longrightarrow> p dvd ((y::int) ^ n)" |
|
30042 | 19 |
by (induct n) (auto simp add: dvd_mult2 [of p y]) |
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diff
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20 |
|
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lemma zdvd_bounds: "n dvd m ==> m \<le> (0::int) | n \<le> m" |
22 |
proof - |
|
23 |
assume "n dvd m" |
|
24 |
then have "~(0 < m & m < n)" |
|
25 |
using zdvd_not_zless [of m n] by auto |
|
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parents:
diff
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then show ?thesis by auto |
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qed |
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parents:
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28 |
|
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lemma zprime_zdvd_zmult_better: "[| zprime p; p dvd (m * n) |] ==> |
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(p dvd m) | (p dvd n)" |
31 |
apply (cases "0 \<le> m") |
|
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32 |
apply (simp add: zprime_zdvd_zmult) |
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apply (insert zprime_zdvd_zmult [of "-m" p n]) |
34 |
apply auto |
|
35 |
done |
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parents:
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36 |
|
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lemma zpower_zdvd_prop2: |
38 |
"zprime p \<Longrightarrow> p dvd ((y::int) ^ n) \<Longrightarrow> 0 < n \<Longrightarrow> p dvd y" |
|
39 |
apply (induct n) |
|
40 |
apply simp |
|
41 |
apply (frule zprime_zdvd_zmult_better) |
|
42 |
apply simp |
|
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apply (force simp del:dvd_mult) |
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done |
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parents:
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45 |
|
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lemma div_prop1: "[| 0 < z; (x::int) < y * z |] ==> x div z < y" |
47 |
proof - |
|
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assume "0 < z" then have modth: "x mod z \<ge> 0" by simp |
49 |
have "(x div z) * z \<le> (x div z) * z" by simp |
|
50 |
then have "(x div z) * z \<le> (x div z) * z + x mod z" using modth by arith |
|
51 |
also have "\<dots> = x" |
|
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by (auto simp add: zmod_zdiv_equality [symmetric] zmult_ac) |
53 |
also assume "x < y * z" |
|
54 |
finally show ?thesis |
|
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Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
13871
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55 |
by (auto simp add: prems mult_less_cancel_right, insert prems, arith) |
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qed |
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parents:
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57 |
|
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lemma div_prop2: "[| 0 < z; (x::int) < (y * z) + z |] ==> x div z \<le> y" |
59 |
proof - |
|
60 |
assume "0 < z" and "x < (y * z) + z" |
|
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61 |
then have "x < (y + 1) * z" by (auto simp add: int_distrib) |
18369 | 62 |
then have "x div z < y + 1" |
63 |
apply - |
|
64 |
apply (rule_tac y = "y + 1" in div_prop1) |
|
65 |
apply (auto simp add: prems) |
|
66 |
done |
|
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parents:
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then show ?thesis by auto |
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qed |
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parents:
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69 |
|
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lemma zdiv_leq_prop: "[| 0 < y |] ==> y * (x div y) \<le> (x::int)" |
71 |
proof- |
|
72 |
assume "0 < y" |
|
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from zmod_zdiv_equality have "x = y * (x div y) + x mod y" by auto |
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moreover have "0 \<le> x mod y" |
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75 |
by (auto simp add: prems pos_mod_sign) |
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ultimately show ?thesis |
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by arith |
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qed |
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79 |
|
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|
81 |
subsection {* Useful properties of congruences *} |
|
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82 |
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lemma zcong_eq_zdvd_prop: "[x = 0](mod p) = (p dvd x)" |
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parents:
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84 |
by (auto simp add: zcong_def) |
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parents:
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85 |
|
18369 | 86 |
lemma zcong_id: "[m = 0] (mod m)" |
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by (auto simp add: zcong_def) |
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88 |
|
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lemma zcong_shift: "[a = b] (mod m) ==> [a + c = b + c] (mod m)" |
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parents:
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90 |
by (auto simp add: zcong_refl zcong_zadd) |
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parents:
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|
91 |
|
18369 | 92 |
lemma zcong_zpower: "[x = y](mod m) ==> [x^z = y^z](mod m)" |
93 |
by (induct z) (auto simp add: zcong_zmult) |
|
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94 |
|
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lemma zcong_eq_trans: "[| [a = b](mod m); b = c; [c = d](mod m) |] ==> |
18369 | 96 |
[a = d](mod m)" |
97 |
apply (erule zcong_trans) |
|
98 |
apply simp |
|
99 |
done |
|
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100 |
|
18369 | 101 |
lemma aux1: "a - b = (c::int) ==> a = c + b" |
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102 |
by auto |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
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parents:
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|
103 |
|
19670 | 104 |
lemma zcong_zmult_prop1: "[a = b](mod m) ==> ([c = a * d](mod m) = |
18369 | 105 |
[c = b * d] (mod m))" |
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parents:
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|
106 |
apply (auto simp add: zcong_def dvd_def) |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
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parents:
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107 |
apply (rule_tac x = "ka + k * d" in exI) |
18369 | 108 |
apply (drule aux1)+ |
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parents:
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|
109 |
apply (auto simp add: int_distrib) |
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parents:
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|
110 |
apply (rule_tac x = "ka - k * d" in exI) |
18369 | 111 |
apply (drule aux1)+ |
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parents:
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112 |
apply (auto simp add: int_distrib) |
18369 | 113 |
done |
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parents:
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|
114 |
|
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lemma zcong_zmult_prop2: "[a = b](mod m) ==> |
18369 | 116 |
([c = d * a](mod m) = [c = d * b] (mod m))" |
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parents:
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117 |
by (auto simp add: zmult_ac zcong_zmult_prop1) |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
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parents:
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|
118 |
|
19670 | 119 |
lemma zcong_zmult_prop3: "[| zprime p; ~[x = 0] (mod p); |
18369 | 120 |
~[y = 0] (mod p) |] ==> ~[x * y = 0] (mod p)" |
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parents:
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121 |
apply (auto simp add: zcong_def) |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
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parents:
diff
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122 |
apply (drule zprime_zdvd_zmult_better, auto) |
18369 | 123 |
done |
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parents:
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|
124 |
|
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lemma zcong_less_eq: "[| 0 < x; 0 < y; 0 < m; [x = y] (mod m); |
18369 | 126 |
x < m; y < m |] ==> x = y" |
25675 | 127 |
by (metis zcong_not zcong_sym zless_linear) |
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parents:
diff
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|
128 |
|
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lemma zcong_neg_1_impl_ne_1: "[| 2 < p; [x = -1] (mod p) |] ==> |
18369 | 130 |
~([x = 1] (mod p))" |
131 |
proof |
|
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
132 |
assume "2 < p" and "[x = 1] (mod p)" and "[x = -1] (mod p)" |
18369 | 133 |
then have "[1 = -1] (mod p)" |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
134 |
apply (auto simp add: zcong_sym) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
135 |
apply (drule zcong_trans, auto) |
18369 | 136 |
done |
137 |
then have "[1 + 1 = -1 + 1] (mod p)" |
|
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
138 |
by (simp only: zcong_shift) |
18369 | 139 |
then have "[2 = 0] (mod p)" |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
140 |
by auto |
18369 | 141 |
then have "p dvd 2" |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
142 |
by (auto simp add: dvd_def zcong_def) |
18369 | 143 |
with prems show False |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
144 |
by (auto simp add: zdvd_not_zless) |
18369 | 145 |
qed |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
146 |
|
18369 | 147 |
lemma zcong_zero_equiv_div: "[a = 0] (mod m) = (m dvd a)" |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
148 |
by (auto simp add: zcong_def) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
149 |
|
19670 | 150 |
lemma zcong_zprime_prod_zero: "[| zprime p; 0 < a |] ==> |
151 |
[a * b = 0] (mod p) ==> [a = 0] (mod p) | [b = 0] (mod p)" |
|
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
152 |
by (auto simp add: zcong_zero_equiv_div zprime_zdvd_zmult) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
153 |
|
16663 | 154 |
lemma zcong_zprime_prod_zero_contra: "[| zprime p; 0 < a |] ==> |
18369 | 155 |
~[a = 0](mod p) & ~[b = 0](mod p) ==> ~[a * b = 0] (mod p)" |
19670 | 156 |
apply auto |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
157 |
apply (frule_tac a = a and b = b and p = p in zcong_zprime_prod_zero) |
18369 | 158 |
apply auto |
159 |
done |
|
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
160 |
|
19670 | 161 |
lemma zcong_not_zero: "[| 0 < x; x < m |] ==> ~[x = 0] (mod m)" |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
162 |
by (auto simp add: zcong_zero_equiv_div zdvd_not_zless) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
163 |
|
18369 | 164 |
lemma zcong_zero: "[| 0 \<le> x; x < m; [x = 0](mod m) |] ==> x = 0" |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
165 |
apply (drule order_le_imp_less_or_eq, auto) |
18369 | 166 |
apply (frule_tac m = m in zcong_not_zero) |
167 |
apply auto |
|
168 |
done |
|
13871
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paulson
parents:
diff
changeset
|
169 |
|
27556 | 170 |
lemma all_relprime_prod_relprime: "[| finite A; \<forall>x \<in> A. zgcd x y = 1 |] |
171 |
==> zgcd (setprod id A) y = 1" |
|
22274 | 172 |
by (induct set: finite) (auto simp add: zgcd_zgcd_zmult) |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
173 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
174 |
|
19670 | 175 |
subsection {* Some properties of MultInv *} |
176 |
||
177 |
lemma MultInv_prop1: "[| 2 < p; [x = y] (mod p) |] ==> |
|
18369 | 178 |
[(MultInv p x) = (MultInv p y)] (mod p)" |
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paulson
parents:
diff
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|
179 |
by (auto simp add: MultInv_def zcong_zpower) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
180 |
|
19670 | 181 |
lemma MultInv_prop2: "[| 2 < p; zprime p; ~([x = 0](mod p)) |] ==> |
18369 | 182 |
[(x * (MultInv p x)) = 1] (mod p)" |
183 |
proof (simp add: MultInv_def zcong_eq_zdvd_prop) |
|
184 |
assume "2 < p" and "zprime p" and "~ p dvd x" |
|
185 |
have "x * x ^ nat (p - 2) = x ^ (nat (p - 2) + 1)" |
|
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
186 |
by auto |
18369 | 187 |
also from prems have "nat (p - 2) + 1 = nat (p - 2 + 1)" |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19670
diff
changeset
|
188 |
by (simp only: nat_add_distrib) |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
189 |
also have "p - 2 + 1 = p - 1" by arith |
18369 | 190 |
finally have "[x * x ^ nat (p - 2) = x ^ nat (p - 1)] (mod p)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
191 |
by (rule ssubst, auto) |
18369 | 192 |
also from prems have "[x ^ nat (p - 1) = 1] (mod p)" |
19670 | 193 |
by (auto simp add: Little_Fermat) |
18369 | 194 |
finally (zcong_trans) show "[x * x ^ nat (p - 2) = 1] (mod p)" . |
195 |
qed |
|
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
196 |
|
19670 | 197 |
lemma MultInv_prop2a: "[| 2 < p; zprime p; ~([x = 0](mod p)) |] ==> |
18369 | 198 |
[(MultInv p x) * x = 1] (mod p)" |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
199 |
by (auto simp add: MultInv_prop2 zmult_ac) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
200 |
|
18369 | 201 |
lemma aux_1: "2 < p ==> ((nat p) - 2) = (nat (p - 2))" |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
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diff
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|
202 |
by (simp add: nat_diff_distrib) |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
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parents:
diff
changeset
|
203 |
|
18369 | 204 |
lemma aux_2: "2 < p ==> 0 < nat (p - 2)" |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
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parents:
diff
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|
205 |
by auto |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
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parents:
diff
changeset
|
206 |
|
19670 | 207 |
lemma MultInv_prop3: "[| 2 < p; zprime p; ~([x = 0](mod p)) |] ==> |
18369 | 208 |
~([MultInv p x = 0](mod p))" |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
209 |
apply (auto simp add: MultInv_def zcong_eq_zdvd_prop aux_1) |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
210 |
apply (drule aux_2) |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
211 |
apply (drule zpower_zdvd_prop2, auto) |
18369 | 212 |
done |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
213 |
|
19670 | 214 |
lemma aux__1: "[| 2 < p; zprime p; ~([x = 0](mod p))|] ==> |
215 |
[(MultInv p (MultInv p x)) = (x * (MultInv p x) * |
|
18369 | 216 |
(MultInv p (MultInv p x)))] (mod p)" |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
217 |
apply (drule MultInv_prop2, auto) |
18369 | 218 |
apply (drule_tac k = "MultInv p (MultInv p x)" in zcong_scalar, auto) |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
219 |
apply (auto simp add: zcong_sym) |
18369 | 220 |
done |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
221 |
|
16663 | 222 |
lemma aux__2: "[| 2 < p; zprime p; ~([x = 0](mod p))|] ==> |
18369 | 223 |
[(x * (MultInv p x) * (MultInv p (MultInv p x))) = x] (mod p)" |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
224 |
apply (frule MultInv_prop3, auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
225 |
apply (insert MultInv_prop2 [of p "MultInv p x"], auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
226 |
apply (drule MultInv_prop2, auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
227 |
apply (drule_tac k = x in zcong_scalar2, auto) |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
228 |
apply (auto simp add: zmult_ac) |
18369 | 229 |
done |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
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parents:
diff
changeset
|
230 |
|
19670 | 231 |
lemma MultInv_prop4: "[| 2 < p; zprime p; ~([x = 0](mod p)) |] ==> |
18369 | 232 |
[(MultInv p (MultInv p x)) = x] (mod p)" |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
233 |
apply (frule aux__1, auto) |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
234 |
apply (drule aux__2, auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
235 |
apply (drule zcong_trans, auto) |
18369 | 236 |
done |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
237 |
|
19670 | 238 |
lemma MultInv_prop5: "[| 2 < p; zprime p; ~([x = 0](mod p)); |
239 |
~([y = 0](mod p)); [(MultInv p x) = (MultInv p y)] (mod p) |] ==> |
|
18369 | 240 |
[x = y] (mod p)" |
19670 | 241 |
apply (drule_tac a = "MultInv p x" and b = "MultInv p y" and |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
242 |
m = p and k = x in zcong_scalar) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
243 |
apply (insert MultInv_prop2 [of p x], simp) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
244 |
apply (auto simp only: zcong_sym [of "MultInv p x * x"]) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
245 |
apply (auto simp add: zmult_ac) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
246 |
apply (drule zcong_trans, auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
247 |
apply (drule_tac a = "x * MultInv p y" and k = y in zcong_scalar, auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
248 |
apply (insert MultInv_prop2a [of p y], auto simp add: zmult_ac) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
249 |
apply (insert zcong_zmult_prop2 [of "y * MultInv p y" 1 p y x]) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
250 |
apply (auto simp add: zcong_sym) |
18369 | 251 |
done |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
252 |
|
19670 | 253 |
lemma MultInv_zcong_prop1: "[| 2 < p; [j = k] (mod p) |] ==> |
18369 | 254 |
[a * MultInv p j = a * MultInv p k] (mod p)" |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
255 |
by (drule MultInv_prop1, auto simp add: zcong_scalar2) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
256 |
|
19670 | 257 |
lemma aux___1: "[j = a * MultInv p k] (mod p) ==> |
18369 | 258 |
[j * k = a * MultInv p k * k] (mod p)" |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
259 |
by (auto simp add: zcong_scalar) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
260 |
|
19670 | 261 |
lemma aux___2: "[|2 < p; zprime p; ~([k = 0](mod p)); |
18369 | 262 |
[j * k = a * MultInv p k * k] (mod p) |] ==> [j * k = a] (mod p)" |
19670 | 263 |
apply (insert MultInv_prop2a [of p k] zcong_zmult_prop2 |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
264 |
[of "MultInv p k * k" 1 p "j * k" a]) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
265 |
apply (auto simp add: zmult_ac) |
18369 | 266 |
done |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
267 |
|
19670 | 268 |
lemma aux___3: "[j * k = a] (mod p) ==> [(MultInv p j) * j * k = |
18369 | 269 |
(MultInv p j) * a] (mod p)" |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
270 |
by (auto simp add: zmult_assoc zcong_scalar2) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
271 |
|
19670 | 272 |
lemma aux___4: "[|2 < p; zprime p; ~([j = 0](mod p)); |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
273 |
[(MultInv p j) * j * k = (MultInv p j) * a] (mod p) |] |
18369 | 274 |
==> [k = a * (MultInv p j)] (mod p)" |
19670 | 275 |
apply (insert MultInv_prop2a [of p j] zcong_zmult_prop1 |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
276 |
[of "MultInv p j * j" 1 p "MultInv p j * a" k]) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
277 |
apply (auto simp add: zmult_ac zcong_sym) |
18369 | 278 |
done |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
279 |
|
19670 | 280 |
lemma MultInv_zcong_prop2: "[| 2 < p; zprime p; ~([k = 0](mod p)); |
281 |
~([j = 0](mod p)); [j = a * MultInv p k] (mod p) |] ==> |
|
18369 | 282 |
[k = a * MultInv p j] (mod p)" |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
283 |
apply (drule aux___1) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
284 |
apply (frule aux___2, auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
285 |
by (drule aux___3, drule aux___4, auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
286 |
|
19670 | 287 |
lemma MultInv_zcong_prop3: "[| 2 < p; zprime p; ~([a = 0](mod p)); |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
288 |
~([k = 0](mod p)); ~([j = 0](mod p)); |
19670 | 289 |
[a * MultInv p j = a * MultInv p k] (mod p) |] ==> |
18369 | 290 |
[j = k] (mod p)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
291 |
apply (auto simp add: zcong_eq_zdvd_prop [of a p]) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
292 |
apply (frule zprime_imp_zrelprime, auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
293 |
apply (insert zcong_cancel2 [of p a "MultInv p j" "MultInv p k"], auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
294 |
apply (drule MultInv_prop5, auto) |
18369 | 295 |
done |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
296 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
297 |
end |