author | paulson |
Tue, 29 Mar 2005 12:30:48 +0200 | |
changeset 15635 | 8408a06590a6 |
parent 14981 | e73f8140af78 |
child 16417 | 9bc16273c2d4 |
permissions | -rw-r--r-- |
13871
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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1 |
(* Title: HOL/Quadratic_Reciprocity/EvenOdd.thy |
14981 | 2 |
ID: $Id$ |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
3 |
Authors: Jeremy Avigad, David Gray, and Adam Kramer |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
4 |
*) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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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 {*Parity: Even and Odd Integers*} |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
7 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
8 |
theory EvenOdd = Int2:; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
9 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
10 |
text{*Note. This theory is being revised. See the web page |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
11 |
\url{http://www.andrew.cmu.edu/~avigad/isabelle}.*} |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
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:
diff
changeset
|
14 |
zOdd :: "int set" |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
15 |
"zOdd == {x. \<exists>k. x = 2*k + 1}" |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
16 |
zEven :: "int set" |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
17 |
"zEven == {x. \<exists>k. x = 2 * k}" |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
18 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
19 |
(***********************************************************) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
20 |
(* *) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
21 |
(* Some useful properties about even and odd *) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
22 |
(* *) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
23 |
(***********************************************************) |
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Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
24 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
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|
25 |
lemma one_not_even: "~(1 \<in> zEven)"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
26 |
apply (simp add: zEven_def) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
27 |
apply (rule allI, case_tac "k \<le> 0", auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
28 |
done |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
29 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
30 |
lemma even_odd_conj: "~(x \<in> zOdd & x \<in> zEven)"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
31 |
apply (auto simp add: zOdd_def zEven_def) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
32 |
proof -; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
33 |
fix a b; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
34 |
assume "2 * (a::int) = 2 * (b::int) + 1"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
35 |
then have "2 * (a::int) - 2 * (b :: int) = 1"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
36 |
by arith |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
37 |
then have "2 * (a - b) = 1"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
38 |
by (auto simp add: zdiff_zmult_distrib) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
39 |
moreover have "(2 * (a - b)):zEven"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
40 |
by (auto simp only: zEven_def) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
41 |
ultimately show "False"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
42 |
by (auto simp add: one_not_even) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
43 |
qed; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
44 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
45 |
lemma even_odd_disj: "(x \<in> zOdd | x \<in> zEven)"; |
14434 | 46 |
by (simp add: zOdd_def zEven_def, presburger) |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
47 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
48 |
lemma not_odd_impl_even: "~(x \<in> zOdd) ==> x \<in> zEven"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
49 |
by (insert even_odd_disj, auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
50 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
51 |
lemma odd_mult_odd_prop: "(x*y):zOdd ==> x \<in> zOdd"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
52 |
apply (case_tac "x \<in> zOdd", auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
53 |
apply (drule not_odd_impl_even) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
54 |
apply (auto simp add: zEven_def zOdd_def) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
55 |
proof -; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
56 |
fix a b; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
57 |
assume "2 * a * y = 2 * b + 1"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
58 |
then have "2 * a * y - 2 * b = 1"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
59 |
by arith |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
60 |
then have "2 * (a * y - b) = 1"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
61 |
by (auto simp add: zdiff_zmult_distrib) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
62 |
moreover have "(2 * (a * y - b)):zEven"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
63 |
by (auto simp only: zEven_def) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
64 |
ultimately show "False"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
65 |
by (auto simp add: one_not_even) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
66 |
qed; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
67 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
68 |
lemma odd_minus_one_even: "x \<in> zOdd ==> (x - 1):zEven"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
69 |
by (auto simp add: zOdd_def zEven_def) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
70 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
71 |
lemma even_div_2_prop1: "x \<in> zEven ==> (x mod 2) = 0"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
72 |
by (auto simp add: zEven_def) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
73 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
74 |
lemma even_div_2_prop2: "x \<in> zEven ==> (2 * (x div 2)) = x"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
75 |
by (auto simp add: zEven_def) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
76 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
77 |
lemma even_plus_even: "[| x \<in> zEven; y \<in> zEven |] ==> x + y \<in> zEven"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
78 |
apply (auto simp add: zEven_def) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
79 |
by (auto simp only: zadd_zmult_distrib2 [THEN sym]) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
80 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
81 |
lemma even_times_either: "x \<in> zEven ==> x * y \<in> zEven"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
82 |
by (auto simp add: zEven_def) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
83 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
84 |
lemma even_minus_even: "[| x \<in> zEven; y \<in> zEven |] ==> x - y \<in> zEven"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
85 |
apply (auto simp add: zEven_def) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
86 |
by (auto simp only: zdiff_zmult_distrib2 [THEN sym]) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
87 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
88 |
lemma odd_minus_odd: "[| x \<in> zOdd; y \<in> zOdd |] ==> x - y \<in> zEven"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
89 |
apply (auto simp add: zOdd_def zEven_def) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
90 |
by (auto simp only: zdiff_zmult_distrib2 [THEN sym]) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
91 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
92 |
lemma even_minus_odd: "[| x \<in> zEven; y \<in> zOdd |] ==> x - y \<in> zOdd"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
93 |
apply (auto simp add: zOdd_def zEven_def) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
94 |
apply (rule_tac x = "k - ka - 1" in exI) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
95 |
by auto |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
96 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
97 |
lemma odd_minus_even: "[| x \<in> zOdd; y \<in> zEven |] ==> x - y \<in> zOdd"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
98 |
apply (auto simp add: zOdd_def zEven_def) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
99 |
by (auto simp only: zdiff_zmult_distrib2 [THEN sym]) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
100 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
101 |
lemma odd_times_odd: "[| x \<in> zOdd; y \<in> zOdd |] ==> x * y \<in> zOdd"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
102 |
apply (auto simp add: zOdd_def zadd_zmult_distrib zadd_zmult_distrib2) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
103 |
apply (rule_tac x = "2 * ka * k + ka + k" in exI) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
104 |
by (auto simp add: zadd_zmult_distrib) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
105 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
106 |
lemma odd_iff_not_even: "(x \<in> zOdd) = (~ (x \<in> zEven))"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
107 |
by (insert even_odd_conj even_odd_disj, auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
108 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
109 |
lemma even_product: "x * y \<in> zEven ==> x \<in> zEven | y \<in> zEven"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
110 |
by (insert odd_iff_not_even odd_times_odd, auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
111 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
112 |
lemma even_diff: "x - y \<in> zEven = ((x \<in> zEven) = (y \<in> zEven))"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
113 |
apply (auto simp add: odd_iff_not_even even_minus_even odd_minus_odd |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
114 |
even_minus_odd odd_minus_even) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
115 |
proof -; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
116 |
assume "x - y \<in> zEven" and "x \<in> zEven"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
117 |
show "y \<in> zEven"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
118 |
proof (rule classical); |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
119 |
assume "~(y \<in> zEven)"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
120 |
then have "y \<in> zOdd" |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
121 |
by (auto simp add: odd_iff_not_even) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
122 |
with prems have "x - y \<in> zOdd"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
123 |
by (simp add: even_minus_odd) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
124 |
with prems have "False"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
125 |
by (auto simp add: odd_iff_not_even) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
126 |
thus ?thesis; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
127 |
by auto |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
128 |
qed; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
129 |
next assume "x - y \<in> zEven" and "y \<in> zEven"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
130 |
show "x \<in> zEven"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
131 |
proof (rule classical); |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
132 |
assume "~(x \<in> zEven)"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
133 |
then have "x \<in> zOdd" |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
134 |
by (auto simp add: odd_iff_not_even) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
135 |
with prems have "x - y \<in> zOdd"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
136 |
by (simp add: odd_minus_even) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
137 |
with prems have "False"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
138 |
by (auto simp add: odd_iff_not_even) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
139 |
thus ?thesis; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
140 |
by auto |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
141 |
qed; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
142 |
qed; |
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 neg_one_even_power: "[| x \<in> zEven; 0 \<le> x |] ==> (-1::int)^(nat x) = 1"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
145 |
proof -; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
146 |
assume "x \<in> zEven" and "0 \<le> x"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
147 |
then have "\<exists>k. x = 2 * k"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
148 |
by (auto simp only: zEven_def) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
149 |
then show ?thesis; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
150 |
proof; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
151 |
fix a; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
152 |
assume "x = 2 * a"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
153 |
from prems have a: "0 \<le> a"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
154 |
by arith |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
155 |
from prems have "nat x = nat(2 * a)"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
156 |
by auto |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
157 |
also from a have "nat (2 * a) = 2 * nat a"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
158 |
by (auto simp add: nat_mult_distrib) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
159 |
finally have "(-1::int)^nat x = (-1)^(2 * nat a)"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
160 |
by auto |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
161 |
also have "... = ((-1::int)^2)^ (nat a)"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
162 |
by (auto simp add: zpower_zpower [THEN sym]) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
163 |
also have "(-1::int)^2 = 1"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
164 |
by auto |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
165 |
finally; show ?thesis; |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
13871
diff
changeset
|
166 |
by auto |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
167 |
qed; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
168 |
qed; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
169 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
170 |
lemma neg_one_odd_power: "[| x \<in> zOdd; 0 \<le> x |] ==> (-1::int)^(nat x) = -1"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
171 |
proof -; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
172 |
assume "x \<in> zOdd" and "0 \<le> x"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
173 |
then have "\<exists>k. x = 2 * k + 1"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
174 |
by (auto simp only: zOdd_def) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
175 |
then show ?thesis; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
176 |
proof; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
177 |
fix a; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
178 |
assume "x = 2 * a + 1"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
179 |
from prems have a: "0 \<le> a"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
180 |
by arith |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
181 |
from prems have "nat x = nat(2 * a + 1)"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
182 |
by auto |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
183 |
also from a have "nat (2 * a + 1) = 2 * nat a + 1"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
184 |
by (auto simp add: nat_mult_distrib nat_add_distrib) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
185 |
finally have "(-1::int)^nat x = (-1)^(2 * nat a + 1)"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
186 |
by auto |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
187 |
also have "... = ((-1::int)^2)^ (nat a) * (-1)^1"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
188 |
by (auto simp add: zpower_zpower [THEN sym] zpower_zadd_distrib) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
189 |
also have "(-1::int)^2 = 1"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
190 |
by auto |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
191 |
finally; show ?thesis; |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
13871
diff
changeset
|
192 |
by auto |
13871
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
193 |
qed; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
194 |
qed; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
195 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
196 |
lemma neg_one_power_parity: "[| 0 \<le> x; 0 \<le> y; (x \<in> zEven) = (y \<in> zEven) |] ==> |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
197 |
(-1::int)^(nat x) = (-1::int)^(nat y)"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
198 |
apply (insert even_odd_disj [of x]) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
199 |
apply (insert even_odd_disj [of y]) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
200 |
by (auto simp add: neg_one_even_power neg_one_odd_power) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
201 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
202 |
lemma one_not_neg_one_mod_m: "2 < m ==> ~([1 = -1] (mod m))"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
203 |
by (auto simp add: zcong_def zdvd_not_zless) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
204 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
205 |
lemma even_div_2_l: "[| y \<in> zEven; x < y |] ==> x div 2 < y div 2"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
206 |
apply (auto simp only: zEven_def) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
207 |
proof -; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
208 |
fix k assume "x < 2 * k"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
209 |
then have "x div 2 < k" by (auto simp add: div_prop1) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
210 |
also have "k = (2 * k) div 2"; by auto |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
211 |
finally show "x div 2 < 2 * k div 2" by auto |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
212 |
qed; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
213 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
214 |
lemma even_sum_div_2: "[| x \<in> zEven; y \<in> zEven |] ==> (x + y) div 2 = x div 2 + y div 2"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
215 |
by (auto simp add: zEven_def, auto simp add: zdiv_zadd1_eq) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
216 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
217 |
lemma even_prod_div_2: "[| x \<in> zEven |] ==> (x * y) div 2 = (x div 2) * y"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
218 |
by (auto simp add: zEven_def) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
219 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
220 |
(* An odd prime is greater than 2 *) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
221 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
222 |
lemma zprime_zOdd_eq_grt_2: "p \<in> zprime ==> (p \<in> zOdd) = (2 < p)"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
223 |
apply (auto simp add: zOdd_def zprime_def) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
224 |
apply (drule_tac x = 2 in allE) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
225 |
apply (insert odd_iff_not_even [of p]) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
226 |
by (auto simp add: zOdd_def zEven_def) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
227 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
228 |
(* Powers of -1 and parity *) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
229 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
230 |
lemma neg_one_special: "finite A ==> |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
231 |
((-1 :: int) ^ card A) * (-1 ^ card A) = 1"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
232 |
by (induct set: Finites, auto) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
233 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
234 |
lemma neg_one_power: "(-1::int)^n = 1 | (-1::int)^n = -1"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
235 |
apply (induct_tac n) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
236 |
by auto |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
237 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
238 |
lemma neg_one_power_eq_mod_m: "[| 2 < m; [(-1::int)^j = (-1::int)^k] (mod m) |] |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
239 |
==> ((-1::int)^j = (-1::int)^k)"; |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
240 |
apply (insert neg_one_power [of j]) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
241 |
apply (insert neg_one_power [of k]) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
242 |
by (auto simp add: one_not_neg_one_mod_m zcong_sym) |
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
243 |
|
26e5f5e624f6
Gauss's law of quadratic reciprocity by Avigad, Gray and Kramer
paulson
parents:
diff
changeset
|
244 |
end; |