author | huffman |
Wed, 18 Feb 2009 15:01:53 -0800 | |
changeset 29981 | 7d0ed261b712 |
parent 28952 | 15a4b2cf8c34 |
permissions | -rw-r--r-- |
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(* Title: HOL/Quadratic_Reciprocity/EvenOdd.thy |
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Authors: Jeremy Avigad, David Gray, and Adam Kramer |
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*) |
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|
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header {*Parity: Even and Odd Integers*} |
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|
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moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
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theory EvenOdd |
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moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
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imports Int2 |
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moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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begin |
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definition |
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more robust syntax for definition/abbreviation/notation;
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zOdd :: "int set" where |
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"zOdd = {x. \<exists>k. x = 2 * k + 1}" |
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|
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more robust syntax for definition/abbreviation/notation;
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parents:
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definition |
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zEven :: "int set" where |
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"zEven = {x. \<exists>k. x = 2 * k}" |
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subsection {* Some useful properties about even and odd *} |
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lemma zOddI [intro?]: "x = 2 * k + 1 \<Longrightarrow> x \<in> zOdd" |
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and zOddE [elim?]: "x \<in> zOdd \<Longrightarrow> (!!k. x = 2 * k + 1 \<Longrightarrow> C) \<Longrightarrow> C" |
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by (auto simp add: zOdd_def) |
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lemma zEvenI [intro?]: "x = 2 * k \<Longrightarrow> x \<in> zEven" |
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and zEvenE [elim?]: "x \<in> zEven \<Longrightarrow> (!!k. x = 2 * k \<Longrightarrow> C) \<Longrightarrow> C" |
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by (auto simp add: zEven_def) |
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||
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lemma one_not_even: "~(1 \<in> zEven)" |
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30 |
proof |
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assume "1 \<in> zEven" |
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then obtain k :: int where "1 = 2 * k" .. |
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then show False by arith |
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qed |
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lemma even_odd_conj: "~(x \<in> zOdd & x \<in> zEven)" |
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proof - |
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{ |
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fix a b |
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assume "2 * (a::int) = 2 * (b::int) + 1" |
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then have "2 * (a::int) - 2 * (b :: int) = 1" |
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by arith |
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then have "2 * (a - b) = 1" |
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by (auto simp add: zdiff_zmult_distrib) |
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moreover have "(2 * (a - b)):zEven" |
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by (auto simp only: zEven_def) |
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ultimately have False |
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by (auto simp add: one_not_even) |
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} |
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then show ?thesis |
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by (auto simp add: zOdd_def zEven_def) |
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qed |
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lemma even_odd_disj: "(x \<in> zOdd | x \<in> zEven)" |
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by (simp add: zOdd_def zEven_def) arith |
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lemma not_odd_impl_even: "~(x \<in> zOdd) ==> x \<in> zEven" |
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using even_odd_disj by auto |
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lemma odd_mult_odd_prop: "(x*y):zOdd ==> x \<in> zOdd" |
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proof (rule classical) |
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assume "\<not> ?thesis" |
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then have "x \<in> zEven" by (rule not_odd_impl_even) |
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then obtain a where a: "x = 2 * a" .. |
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assume "x * y : zOdd" |
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then obtain b where "x * y = 2 * b + 1" .. |
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with a have "2 * a * y = 2 * b + 1" by simp |
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then have "2 * a * y - 2 * b = 1" |
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by arith |
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then have "2 * (a * y - b) = 1" |
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by (auto simp add: zdiff_zmult_distrib) |
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moreover have "(2 * (a * y - b)):zEven" |
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by (auto simp only: zEven_def) |
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ultimately have False |
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by (auto simp add: one_not_even) |
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then show ?thesis .. |
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qed |
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||
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lemma odd_minus_one_even: "x \<in> zOdd ==> (x - 1):zEven" |
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by (auto simp add: zOdd_def zEven_def) |
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lemma even_div_2_prop1: "x \<in> zEven ==> (x mod 2) = 0" |
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by (auto simp add: zEven_def) |
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lemma even_div_2_prop2: "x \<in> zEven ==> (2 * (x div 2)) = x" |
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by (auto simp add: zEven_def) |
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lemma even_plus_even: "[| x \<in> zEven; y \<in> zEven |] ==> x + y \<in> zEven" |
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apply (auto simp add: zEven_def) |
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apply (auto simp only: zadd_zmult_distrib2 [symmetric]) |
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done |
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lemma even_times_either: "x \<in> zEven ==> x * y \<in> zEven" |
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by (auto simp add: zEven_def) |
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lemma even_minus_even: "[| x \<in> zEven; y \<in> zEven |] ==> x - y \<in> zEven" |
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apply (auto simp add: zEven_def) |
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apply (auto simp only: zdiff_zmult_distrib2 [symmetric]) |
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done |
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lemma odd_minus_odd: "[| x \<in> zOdd; y \<in> zOdd |] ==> x - y \<in> zEven" |
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apply (auto simp add: zOdd_def zEven_def) |
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apply (auto simp only: zdiff_zmult_distrib2 [symmetric]) |
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done |
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lemma even_minus_odd: "[| x \<in> zEven; y \<in> zOdd |] ==> x - y \<in> zOdd" |
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apply (auto simp add: zOdd_def zEven_def) |
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apply (rule_tac x = "k - ka - 1" in exI) |
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apply auto |
110 |
done |
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lemma odd_minus_even: "[| x \<in> zOdd; y \<in> zEven |] ==> x - y \<in> zOdd" |
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apply (auto simp add: zOdd_def zEven_def) |
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apply (auto simp only: zdiff_zmult_distrib2 [symmetric]) |
115 |
done |
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lemma odd_times_odd: "[| x \<in> zOdd; y \<in> zOdd |] ==> x * y \<in> zOdd" |
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apply (auto simp add: zOdd_def zadd_zmult_distrib zadd_zmult_distrib2) |
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apply (rule_tac x = "2 * ka * k + ka + k" in exI) |
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apply (auto simp add: zadd_zmult_distrib) |
121 |
done |
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lemma odd_iff_not_even: "(x \<in> zOdd) = (~ (x \<in> zEven))" |
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using even_odd_conj even_odd_disj by auto |
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||
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lemma even_product: "x * y \<in> zEven ==> x \<in> zEven | y \<in> zEven" |
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using odd_iff_not_even odd_times_odd by auto |
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lemma even_diff: "x - y \<in> zEven = ((x \<in> zEven) = (y \<in> zEven))" |
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proof |
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assume xy: "x - y \<in> zEven" |
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{ |
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assume x: "x \<in> zEven" |
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have "y \<in> zEven" |
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proof (rule classical) |
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assume "\<not> ?thesis" |
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then have "y \<in> zOdd" |
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by (simp add: odd_iff_not_even) |
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with x have "x - y \<in> zOdd" |
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by (simp add: even_minus_odd) |
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with xy have False |
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by (auto simp add: odd_iff_not_even) |
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then show ?thesis .. |
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qed |
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} moreover { |
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assume y: "y \<in> zEven" |
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have "x \<in> zEven" |
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proof (rule classical) |
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149 |
assume "\<not> ?thesis" |
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then have "x \<in> zOdd" |
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by (auto simp add: odd_iff_not_even) |
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with y have "x - y \<in> zOdd" |
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by (simp add: odd_minus_even) |
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with xy have False |
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by (auto simp add: odd_iff_not_even) |
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then show ?thesis .. |
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qed |
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} |
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ultimately show "(x \<in> zEven) = (y \<in> zEven)" |
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by (auto simp add: odd_iff_not_even even_minus_even odd_minus_odd |
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even_minus_odd odd_minus_even) |
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next |
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assume "(x \<in> zEven) = (y \<in> zEven)" |
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then show "x - y \<in> zEven" |
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by (auto simp add: odd_iff_not_even even_minus_even odd_minus_odd |
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even_minus_odd odd_minus_even) |
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qed |
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168 |
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lemma neg_one_even_power: "[| x \<in> zEven; 0 \<le> x |] ==> (-1::int)^(nat x) = 1" |
170 |
proof - |
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assume "x \<in> zEven" and "0 \<le> x" |
172 |
from `x \<in> zEven` obtain a where "x = 2 * a" .. |
|
173 |
with `0 \<le> x` have "0 \<le> a" by simp |
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174 |
from `0 \<le> x` and `x = 2 * a` have "nat x = nat (2 * a)" |
|
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by simp |
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also from `x = 2 * a` have "nat (2 * a) = 2 * nat a" |
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by (simp add: nat_mult_distrib) |
178 |
finally have "(-1::int)^nat x = (-1)^(2 * nat a)" |
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179 |
by simp |
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180 |
also have "... = ((-1::int)^2)^ (nat a)" |
|
181 |
by (simp add: zpower_zpower [symmetric]) |
|
182 |
also have "(-1::int)^2 = 1" |
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183 |
by simp |
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184 |
finally show ?thesis |
|
185 |
by simp |
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186 |
qed |
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187 |
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lemma neg_one_odd_power: "[| x \<in> zOdd; 0 \<le> x |] ==> (-1::int)^(nat x) = -1" |
189 |
proof - |
|
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assume "x \<in> zOdd" and "0 \<le> x" |
191 |
from `x \<in> zOdd` obtain a where "x = 2 * a + 1" .. |
|
192 |
with `0 \<le> x` have a: "0 \<le> a" by simp |
|
193 |
with `0 \<le> x` and `x = 2 * a + 1` have "nat x = nat (2 * a + 1)" |
|
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by simp |
195 |
also from a have "nat (2 * a + 1) = 2 * nat a + 1" |
|
196 |
by (auto simp add: nat_mult_distrib nat_add_distrib) |
|
197 |
finally have "(-1::int)^nat x = (-1)^(2 * nat a + 1)" |
|
198 |
by simp |
|
199 |
also have "... = ((-1::int)^2)^ (nat a) * (-1)^1" |
|
200 |
by (auto simp add: zpower_zpower [symmetric] zpower_zadd_distrib) |
|
201 |
also have "(-1::int)^2 = 1" |
|
202 |
by simp |
|
203 |
finally show ?thesis |
|
204 |
by simp |
|
205 |
qed |
|
206 |
||
207 |
lemma neg_one_power_parity: "[| 0 \<le> x; 0 \<le> y; (x \<in> zEven) = (y \<in> zEven) |] ==> |
|
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(-1::int)^(nat x) = (-1::int)^(nat y)" |
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using even_odd_disj [of x] even_odd_disj [of y] |
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210 |
by (auto simp add: neg_one_even_power neg_one_odd_power) |
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211 |
|
18369 | 212 |
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213 |
lemma one_not_neg_one_mod_m: "2 < m ==> ~([1 = -1] (mod m))" |
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214 |
by (auto simp add: zcong_def zdvd_not_zless) |
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215 |
|
18369 | 216 |
lemma even_div_2_l: "[| y \<in> zEven; x < y |] ==> x div 2 < y div 2" |
217 |
proof - |
|
20369 | 218 |
assume "y \<in> zEven" and "x < y" |
219 |
from `y \<in> zEven` obtain k where k: "y = 2 * k" .. |
|
220 |
with `x < y` have "x < 2 * k" by simp |
|
18369 | 221 |
then have "x div 2 < k" by (auto simp add: div_prop1) |
222 |
also have "k = (2 * k) div 2" by simp |
|
223 |
finally have "x div 2 < 2 * k div 2" by simp |
|
224 |
with k show ?thesis by simp |
|
225 |
qed |
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226 |
|
18369 | 227 |
lemma even_sum_div_2: "[| x \<in> zEven; y \<in> zEven |] ==> (x + y) div 2 = x div 2 + y div 2" |
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parents:
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228 |
by (auto simp add: zEven_def) |
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229 |
|
18369 | 230 |
lemma even_prod_div_2: "[| x \<in> zEven |] ==> (x * y) div 2 = (x div 2) * y" |
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by (auto simp add: zEven_def) |
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|
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233 |
(* An odd prime is greater than 2 *) |
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|
234 |
|
18369 | 235 |
lemma zprime_zOdd_eq_grt_2: "zprime p ==> (p \<in> zOdd) = (2 < p)" |
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apply (auto simp add: zOdd_def zprime_def) |
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apply (drule_tac x = 2 in allE) |
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using odd_iff_not_even [of p] |
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apply (auto simp add: zOdd_def zEven_def) |
|
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done |
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|
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(* Powers of -1 and parity *) |
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lemma neg_one_special: "finite A ==> |
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((-1 :: int) ^ card A) * (-1 ^ card A) = 1" |
|
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by (induct set: finite) auto |
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lemma neg_one_power: "(-1::int)^n = 1 | (-1::int)^n = -1" |
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by (induct n) auto |
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|
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lemma neg_one_power_eq_mod_m: "[| 2 < m; [(-1::int)^j = (-1::int)^k] (mod m) |] |
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==> ((-1::int)^j = (-1::int)^k)" |
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using neg_one_power [of j] and ListMem.insert neg_one_power [of k] |
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by (auto simp add: one_not_neg_one_mod_m zcong_sym) |
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end |