author | wenzelm |
Thu, 08 Dec 2005 12:50:04 +0100 | |
changeset 18369 | 694ea14ab4f2 |
parent 16663 | 13e9c402308b |
child 19670 | 2e4a143c73c5 |
permissions | -rw-r--r-- |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
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parents:
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1 |
(* Title: HOL/Quadratic_Reciprocity/Gauss.thy |
14981 | 2 |
ID: $Id$ |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
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parents:
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3 |
Authors: Jeremy Avigad, David Gray, and Adam Kramer |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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4 |
*) |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
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|
5 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
6 |
header {*Integers: Divisibility and Congruences*} |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
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parents:
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|
7 |
|
18369 | 8 |
theory Int2 imports Finite2 WilsonRuss begin |
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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|
9 |
|
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
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parents:
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|
10 |
text{*Note. This theory is being revised. See the web page |
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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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11 |
\url{http://www.andrew.cmu.edu/~avigad/isabelle}.*} |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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12 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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13 |
constdefs |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
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|
14 |
MultInv :: "int => int => int" |
18369 | 15 |
"MultInv p x == x ^ nat (p - 2)" |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
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parents:
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16 |
|
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
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|
17 |
(*****************************************************************) |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
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parents:
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|
18 |
(* *) |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
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parents:
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|
19 |
(* Useful lemmas about dvd and powers *) |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
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parents:
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|
20 |
(* *) |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
21 |
(*****************************************************************) |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
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|
22 |
|
18369 | 23 |
lemma zpower_zdvd_prop1: |
24 |
"0 < n \<Longrightarrow> p dvd y \<Longrightarrow> p dvd ((y::int) ^ n)" |
|
25 |
by (induct n) (auto simp add: zdvd_zmult zdvd_zmult2 [of p y]) |
|
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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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|
26 |
|
18369 | 27 |
lemma zdvd_bounds: "n dvd m ==> m \<le> (0::int) | n \<le> m" |
28 |
proof - |
|
29 |
assume "n dvd m" |
|
30 |
then have "~(0 < m & m < n)" |
|
31 |
using zdvd_not_zless [of m n] by auto |
|
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
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parents:
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32 |
then show ?thesis by auto |
18369 | 33 |
qed |
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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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34 |
|
16663 | 35 |
lemma zprime_zdvd_zmult_better: "[| zprime p; p dvd (m * n) |] ==> |
18369 | 36 |
(p dvd m) | (p dvd n)" |
37 |
apply (cases "0 \<le> m") |
|
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
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parents:
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|
38 |
apply (simp add: zprime_zdvd_zmult) |
18369 | 39 |
apply (insert zprime_zdvd_zmult [of "-m" p n]) |
40 |
apply auto |
|
41 |
done |
|
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
42 |
|
18369 | 43 |
lemma zpower_zdvd_prop2: |
44 |
"zprime p \<Longrightarrow> p dvd ((y::int) ^ n) \<Longrightarrow> 0 < n \<Longrightarrow> p dvd y" |
|
45 |
apply (induct n) |
|
46 |
apply simp |
|
47 |
apply (frule zprime_zdvd_zmult_better) |
|
48 |
apply simp |
|
49 |
apply force |
|
50 |
done |
|
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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|
51 |
|
18369 | 52 |
lemma div_prop1: "[| 0 < z; (x::int) < y * z |] ==> x div z < y" |
53 |
proof - |
|
54 |
assume "0 < z" |
|
55 |
then have "(x div z) * z \<le> (x div z) * z + x mod z" |
|
56 |
by arith |
|
57 |
also have "... = x" |
|
58 |
by (auto simp add: zmod_zdiv_equality [symmetric] zmult_ac) |
|
59 |
also assume "x < y * z" |
|
60 |
finally show ?thesis |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
13871
diff
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|
61 |
by (auto simp add: prems mult_less_cancel_right, insert prems, arith) |
18369 | 62 |
qed |
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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|
63 |
|
18369 | 64 |
lemma div_prop2: "[| 0 < z; (x::int) < (y * z) + z |] ==> x div z \<le> y" |
65 |
proof - |
|
66 |
assume "0 < z" and "x < (y * z) + z" |
|
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
67 |
then have "x < (y + 1) * z" by (auto simp add: int_distrib) |
18369 | 68 |
then have "x div z < y + 1" |
69 |
apply - |
|
70 |
apply (rule_tac y = "y + 1" in div_prop1) |
|
71 |
apply (auto simp add: prems) |
|
72 |
done |
|
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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|
73 |
then show ?thesis by auto |
18369 | 74 |
qed |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
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parents:
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75 |
|
18369 | 76 |
lemma zdiv_leq_prop: "[| 0 < y |] ==> y * (x div y) \<le> (x::int)" |
77 |
proof- |
|
78 |
assume "0 < y" |
|
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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|
79 |
from zmod_zdiv_equality have "x = y * (x div y) + x mod y" by auto |
18369 | 80 |
moreover have "0 \<le> x mod y" |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
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parents:
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|
81 |
by (auto simp add: prems pos_mod_sign) |
18369 | 82 |
ultimately show ?thesis |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
83 |
by arith |
18369 | 84 |
qed |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
85 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
86 |
(*****************************************************************) |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
87 |
(* *) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
88 |
(* Useful properties of congruences *) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
89 |
(* *) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
90 |
(*****************************************************************) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
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|
91 |
|
18369 | 92 |
lemma zcong_eq_zdvd_prop: "[x = 0](mod p) = (p dvd x)" |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
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parents:
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|
93 |
by (auto simp add: zcong_def) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
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parents:
diff
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|
94 |
|
18369 | 95 |
lemma zcong_id: "[m = 0] (mod m)" |
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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|
96 |
by (auto simp add: zcong_def zdvd_0_right) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
97 |
|
18369 | 98 |
lemma zcong_shift: "[a = b] (mod m) ==> [a + c = b + c] (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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|
99 |
by (auto simp add: zcong_refl zcong_zadd) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
100 |
|
18369 | 101 |
lemma zcong_zpower: "[x = y](mod m) ==> [x^z = y^z](mod m)" |
102 |
by (induct z) (auto simp add: zcong_zmult) |
|
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
103 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
104 |
lemma zcong_eq_trans: "[| [a = b](mod m); b = c; [c = d](mod m) |] ==> |
18369 | 105 |
[a = d](mod m)" |
106 |
apply (erule zcong_trans) |
|
107 |
apply simp |
|
108 |
done |
|
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
109 |
|
18369 | 110 |
lemma aux1: "a - b = (c::int) ==> a = c + b" |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
111 |
by auto |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
112 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
113 |
lemma zcong_zmult_prop1: "[a = b](mod m) ==> ([c = a * d](mod m) = |
18369 | 114 |
[c = b * d] (mod m))" |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
115 |
apply (auto simp add: zcong_def dvd_def) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
116 |
apply (rule_tac x = "ka + k * d" in exI) |
18369 | 117 |
apply (drule aux1)+ |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
118 |
apply (auto simp add: int_distrib) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
119 |
apply (rule_tac x = "ka - k * d" in exI) |
18369 | 120 |
apply (drule aux1)+ |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
121 |
apply (auto simp add: int_distrib) |
18369 | 122 |
done |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
123 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
124 |
lemma zcong_zmult_prop2: "[a = b](mod m) ==> |
18369 | 125 |
([c = d * a](mod m) = [c = d * b] (mod m))" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
126 |
by (auto simp add: zmult_ac zcong_zmult_prop1) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
127 |
|
16663 | 128 |
lemma zcong_zmult_prop3: "[| zprime p; ~[x = 0] (mod p); |
18369 | 129 |
~[y = 0] (mod p) |] ==> ~[x * y = 0] (mod p)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
130 |
apply (auto simp add: zcong_def) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
131 |
apply (drule zprime_zdvd_zmult_better, auto) |
18369 | 132 |
done |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
133 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
134 |
lemma zcong_less_eq: "[| 0 < x; 0 < y; 0 < m; [x = y] (mod m); |
18369 | 135 |
x < m; y < m |] ==> x = y" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
136 |
apply (simp add: zcong_zmod_eq) |
18369 | 137 |
apply (subgoal_tac "(x mod m) = x") |
138 |
apply (subgoal_tac "(y mod m) = y") |
|
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
139 |
apply simp |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
140 |
apply (rule_tac [1-2] mod_pos_pos_trivial) |
18369 | 141 |
apply auto |
142 |
done |
|
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
143 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
144 |
lemma zcong_neg_1_impl_ne_1: "[| 2 < p; [x = -1] (mod p) |] ==> |
18369 | 145 |
~([x = 1] (mod p))" |
146 |
proof |
|
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
147 |
assume "2 < p" and "[x = 1] (mod p)" and "[x = -1] (mod p)" |
18369 | 148 |
then have "[1 = -1] (mod p)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
149 |
apply (auto simp add: zcong_sym) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
150 |
apply (drule zcong_trans, auto) |
18369 | 151 |
done |
152 |
then have "[1 + 1 = -1 + 1] (mod p)" |
|
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
153 |
by (simp only: zcong_shift) |
18369 | 154 |
then have "[2 = 0] (mod p)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
155 |
by auto |
18369 | 156 |
then have "p dvd 2" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
157 |
by (auto simp add: dvd_def zcong_def) |
18369 | 158 |
with prems show False |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
159 |
by (auto simp add: zdvd_not_zless) |
18369 | 160 |
qed |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
161 |
|
18369 | 162 |
lemma zcong_zero_equiv_div: "[a = 0] (mod m) = (m dvd a)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
163 |
by (auto simp add: zcong_def) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
164 |
|
16663 | 165 |
lemma zcong_zprime_prod_zero: "[| zprime p; 0 < a |] ==> |
18369 | 166 |
[a * b = 0] (mod p) ==> [a = 0] (mod p) | [b = 0] (mod p)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
167 |
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
changeset
|
168 |
|
16663 | 169 |
lemma zcong_zprime_prod_zero_contra: "[| zprime p; 0 < a |] ==> |
18369 | 170 |
~[a = 0](mod p) & ~[b = 0](mod p) ==> ~[a * b = 0] (mod p)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
171 |
apply auto |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
172 |
apply (frule_tac a = a and b = b and p = p in zcong_zprime_prod_zero) |
18369 | 173 |
apply auto |
174 |
done |
|
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
175 |
|
18369 | 176 |
lemma zcong_not_zero: "[| 0 < x; x < m |] ==> ~[x = 0] (mod m)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
177 |
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
|
178 |
|
18369 | 179 |
lemma zcong_zero: "[| 0 \<le> x; x < m; [x = 0](mod m) |] ==> x = 0" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
180 |
apply (drule order_le_imp_less_or_eq, auto) |
18369 | 181 |
apply (frule_tac m = m in zcong_not_zero) |
182 |
apply auto |
|
183 |
done |
|
13871
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paulson
parents:
diff
changeset
|
184 |
|
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paulson
parents:
diff
changeset
|
185 |
lemma all_relprime_prod_relprime: "[| finite A; \<forall>x \<in> A. (zgcd(x,y) = 1) |] |
18369 | 186 |
==> zgcd (setprod id A,y) = 1" |
187 |
by (induct set: Finites) (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
changeset
|
188 |
|
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paulson
parents:
diff
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|
189 |
(*****************************************************************) |
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paulson
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|
190 |
(* *) |
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paulson
parents:
diff
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|
191 |
(* Some properties of MultInv *) |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
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parents:
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|
192 |
(* *) |
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paulson
parents:
diff
changeset
|
193 |
(*****************************************************************) |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
194 |
|
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
195 |
lemma MultInv_prop1: "[| 2 < p; [x = y] (mod p) |] ==> |
18369 | 196 |
[(MultInv p x) = (MultInv p y)] (mod p)" |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
197 |
by (auto simp add: MultInv_def zcong_zpower) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
198 |
|
16663 | 199 |
lemma MultInv_prop2: "[| 2 < p; zprime p; ~([x = 0](mod p)) |] ==> |
18369 | 200 |
[(x * (MultInv p x)) = 1] (mod p)" |
201 |
proof (simp add: MultInv_def zcong_eq_zdvd_prop) |
|
202 |
assume "2 < p" and "zprime p" and "~ p dvd x" |
|
203 |
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
|
204 |
by auto |
18369 | 205 |
also from prems have "nat (p - 2) + 1 = nat (p - 2 + 1)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
206 |
by (simp only: nat_add_distrib, auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
207 |
also have "p - 2 + 1 = p - 1" by arith |
18369 | 208 |
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
|
209 |
by (rule ssubst, auto) |
18369 | 210 |
also from prems have "[x ^ nat (p - 1) = 1] (mod p)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
211 |
by (auto simp add: Little_Fermat) |
18369 | 212 |
finally (zcong_trans) show "[x * x ^ nat (p - 2) = 1] (mod p)" . |
213 |
qed |
|
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
214 |
|
16663 | 215 |
lemma MultInv_prop2a: "[| 2 < p; zprime p; ~([x = 0](mod p)) |] ==> |
18369 | 216 |
[(MultInv p x) * x = 1] (mod p)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
217 |
by (auto simp add: MultInv_prop2 zmult_ac) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
218 |
|
18369 | 219 |
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
paulson
parents:
diff
changeset
|
220 |
by (simp add: nat_diff_distrib) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
221 |
|
18369 | 222 |
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
paulson
parents:
diff
changeset
|
223 |
by auto |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
224 |
|
16663 | 225 |
lemma MultInv_prop3: "[| 2 < p; zprime p; ~([x = 0](mod p)) |] ==> |
18369 | 226 |
~([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
|
227 |
apply (auto simp add: MultInv_def zcong_eq_zdvd_prop aux_1) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
228 |
apply (drule aux_2) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
229 |
apply (drule zpower_zdvd_prop2, auto) |
18369 | 230 |
done |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
231 |
|
16663 | 232 |
lemma aux__1: "[| 2 < p; zprime 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
|
233 |
[(MultInv p (MultInv p x)) = (x * (MultInv p x) * |
18369 | 234 |
(MultInv p (MultInv p x)))] (mod p)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
235 |
apply (drule MultInv_prop2, auto) |
18369 | 236 |
apply (drule_tac k = "MultInv p (MultInv p x)" in zcong_scalar, auto) |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
237 |
apply (auto simp add: zcong_sym) |
18369 | 238 |
done |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
239 |
|
16663 | 240 |
lemma aux__2: "[| 2 < p; zprime p; ~([x = 0](mod p))|] ==> |
18369 | 241 |
[(x * (MultInv p x) * (MultInv p (MultInv p x))) = x] (mod p)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
242 |
apply (frule MultInv_prop3, auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
243 |
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
|
244 |
apply (drule MultInv_prop2, auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
245 |
apply (drule_tac k = x in zcong_scalar2, auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
246 |
apply (auto simp add: zmult_ac) |
18369 | 247 |
done |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
248 |
|
16663 | 249 |
lemma MultInv_prop4: "[| 2 < p; zprime p; ~([x = 0](mod p)) |] ==> |
18369 | 250 |
[(MultInv p (MultInv p x)) = x] (mod p)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
251 |
apply (frule aux__1, auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
252 |
apply (drule aux__2, auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
253 |
apply (drule zcong_trans, auto) |
18369 | 254 |
done |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
255 |
|
16663 | 256 |
lemma MultInv_prop5: "[| 2 < p; zprime p; ~([x = 0](mod p)); |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
257 |
~([y = 0](mod p)); [(MultInv p x) = (MultInv p y)] (mod p) |] ==> |
18369 | 258 |
[x = y] (mod p)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
259 |
apply (drule_tac a = "MultInv p x" and b = "MultInv p y" and |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
260 |
m = p and k = x in zcong_scalar) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
261 |
apply (insert MultInv_prop2 [of p x], simp) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
262 |
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
|
263 |
apply (auto simp add: zmult_ac) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
264 |
apply (drule zcong_trans, auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
265 |
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
|
266 |
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
|
267 |
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
|
268 |
apply (auto simp add: zcong_sym) |
18369 | 269 |
done |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
270 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
271 |
lemma MultInv_zcong_prop1: "[| 2 < p; [j = k] (mod p) |] ==> |
18369 | 272 |
[a * MultInv p j = a * MultInv p k] (mod p)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
273 |
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
|
274 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
275 |
lemma aux___1: "[j = a * MultInv p k] (mod p) ==> |
18369 | 276 |
[j * k = a * MultInv p k * k] (mod p)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
277 |
by (auto simp add: zcong_scalar) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
278 |
|
16663 | 279 |
lemma aux___2: "[|2 < p; zprime p; ~([k = 0](mod p)); |
18369 | 280 |
[j * k = a * MultInv p k * k] (mod p) |] ==> [j * k = a] (mod p)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
281 |
apply (insert MultInv_prop2a [of p k] zcong_zmult_prop2 |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
282 |
[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
|
283 |
apply (auto simp add: zmult_ac) |
18369 | 284 |
done |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
285 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
286 |
lemma aux___3: "[j * k = a] (mod p) ==> [(MultInv p j) * j * k = |
18369 | 287 |
(MultInv p j) * a] (mod p)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
288 |
by (auto simp add: zmult_assoc zcong_scalar2) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
289 |
|
16663 | 290 |
lemma aux___4: "[|2 < p; zprime p; ~([j = 0](mod p)); |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
291 |
[(MultInv p j) * j * k = (MultInv p j) * a] (mod p) |] |
18369 | 292 |
==> [k = a * (MultInv p j)] (mod p)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
293 |
apply (insert MultInv_prop2a [of p j] zcong_zmult_prop1 |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
294 |
[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
|
295 |
apply (auto simp add: zmult_ac zcong_sym) |
18369 | 296 |
done |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
297 |
|
16663 | 298 |
lemma MultInv_zcong_prop2: "[| 2 < p; zprime p; ~([k = 0](mod p)); |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
299 |
~([j = 0](mod p)); [j = a * MultInv p k] (mod p) |] ==> |
18369 | 300 |
[k = a * MultInv p j] (mod p)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
301 |
apply (drule aux___1) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
302 |
apply (frule aux___2, auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
303 |
by (drule aux___3, drule aux___4, auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
304 |
|
16663 | 305 |
lemma MultInv_zcong_prop3: "[| 2 < p; zprime p; ~([a = 0](mod p)); |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
306 |
~([k = 0](mod p)); ~([j = 0](mod p)); |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
307 |
[a * MultInv p j = a * MultInv p k] (mod p) |] ==> |
18369 | 308 |
[j = k] (mod p)" |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
309 |
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
|
310 |
apply (frule zprime_imp_zrelprime, auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
311 |
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
|
312 |
apply (drule MultInv_prop5, auto) |
18369 | 313 |
done |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
314 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
315 |
end |