src/HOL/Parity.thy
author wenzelm
Mon Oct 20 17:00:13 2014 +0200 (2014-10-20)
changeset 58718 48395c763059
parent 58711 3f7886cd75b9
child 58740 cb9d84d3e7f2
permissions -rw-r--r--
repared document;
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(*  Title:      HOL/Parity.thy
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    Author:     Jeremy Avigad
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    Author:     Jacques D. Fleuriot
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*)
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header {* Even and Odd for int and nat *}
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theory Parity
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imports Main
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begin
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subsection {* Preliminaries about divisibility on @{typ nat} and @{typ int} *}
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lemma two_dvd_Suc_Suc_iff [simp]:
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  "2 dvd Suc (Suc n) \<longleftrightarrow> 2 dvd n"
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  using dvd_add_triv_right_iff [of 2 n] by simp
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lemma two_dvd_Suc_iff:
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  "2 dvd Suc n \<longleftrightarrow> \<not> 2 dvd n"
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  by (induct n) auto
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lemma two_dvd_diff_nat_iff:
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  fixes m n :: nat
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  shows "2 dvd m - n \<longleftrightarrow> m < n \<or> 2 dvd m + n"
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proof (cases "n \<le> m")
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  case True
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  then have "m - n + n * 2 = m + n" by simp
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  moreover have "2 dvd m - n \<longleftrightarrow> 2 dvd m - n + n * 2" by simp
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  ultimately have "2 dvd m - n \<longleftrightarrow> 2 dvd m + n" by (simp only:)
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  then show ?thesis by auto
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next
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  case False
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  then show ?thesis by simp
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qed 
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lemma two_dvd_diff_iff:
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  fixes k l :: int
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  shows "2 dvd k - l \<longleftrightarrow> 2 dvd k + l"
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  using dvd_add_times_triv_right_iff [of 2 "k - l" l] by (simp add: ac_simps)
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lemma two_dvd_abs_add_iff:
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  fixes k l :: int
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  shows "2 dvd \<bar>k\<bar> + l \<longleftrightarrow> 2 dvd k + l"
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  by (cases "k \<ge> 0") (simp_all add: two_dvd_diff_iff ac_simps)
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lemma two_dvd_add_abs_iff:
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  fixes k l :: int
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  shows "2 dvd k + \<bar>l\<bar> \<longleftrightarrow> 2 dvd k + l"
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  using two_dvd_abs_add_iff [of l k] by (simp add: ac_simps)
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subsection {* Ring structures with parity *}
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class semiring_parity = semiring_dvd + semiring_numeral +
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  assumes two_not_dvd_one [simp]: "\<not> 2 dvd 1"
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  assumes not_dvd_not_dvd_dvd_add: "\<not> 2 dvd a \<Longrightarrow> \<not> 2 dvd b \<Longrightarrow> 2 dvd a + b"
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  assumes two_is_prime: "2 dvd a * b \<Longrightarrow> 2 dvd a \<or> 2 dvd b"
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  assumes not_dvd_ex_decrement: "\<not> 2 dvd a \<Longrightarrow> \<exists>b. a = b + 1"
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begin
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lemma two_dvd_plus_one_iff [simp]:
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  "2 dvd a + 1 \<longleftrightarrow> \<not> 2 dvd a"
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  by (auto simp add: dvd_add_right_iff intro: not_dvd_not_dvd_dvd_add)
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lemma not_two_dvdE [elim?]:
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  assumes "\<not> 2 dvd a"
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  obtains b where "a = 2 * b + 1"
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proof -
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  from assms obtain b where *: "a = b + 1"
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    by (blast dest: not_dvd_ex_decrement)
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  with assms have "2 dvd b + 2" by simp
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  then have "2 dvd b" by simp
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  then obtain c where "b = 2 * c" ..
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  with * have "a = 2 * c + 1" by simp
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  with that show thesis .
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qed
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end
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instance nat :: semiring_parity
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proof
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  show "\<not> (2 :: nat) dvd 1"
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    by (rule notI, erule dvdE) simp
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next
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  fix m n :: nat
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  assume "\<not> 2 dvd m"
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  moreover assume "\<not> 2 dvd n"
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  ultimately have *: "2 dvd Suc m \<and> 2 dvd Suc n"
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    by (simp add: two_dvd_Suc_iff)
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  then have "2 dvd Suc m + Suc n"
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    by (blast intro: dvd_add)
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  also have "Suc m + Suc n = m + n + 2"
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    by simp
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  finally show "2 dvd m + n"
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    using dvd_add_triv_right_iff [of 2 "m + n"] by simp
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next
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  fix m n :: nat
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  assume *: "2 dvd m * n"
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  show "2 dvd m \<or> 2 dvd n"
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  proof (rule disjCI)
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    assume "\<not> 2 dvd n"
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    then have "2 dvd Suc n" by (simp add: two_dvd_Suc_iff)
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    then obtain r where "Suc n = 2 * r" ..
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    moreover from * obtain s where "m * n = 2 * s" ..
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    then have "2 * s + m = m * Suc n" by simp
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    ultimately have " 2 * s + m = 2 * (m * r)" by simp
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    then have "m = 2 * (m * r - s)" by simp
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    then show "2 dvd m" ..
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  qed
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next
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  fix n :: nat
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  assume "\<not> 2 dvd n"
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  then show "\<exists>m. n = m + 1"
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    by (cases n) simp_all
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qed
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class ring_parity = comm_ring_1 + semiring_parity
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instance int :: ring_parity
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proof
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  show "\<not> (2 :: int) dvd 1" by (simp add: dvd_int_unfold_dvd_nat)
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  fix k l :: int
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  assume "\<not> 2 dvd k"
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  moreover assume "\<not> 2 dvd l"
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  ultimately have "2 dvd nat \<bar>k\<bar> + nat \<bar>l\<bar>" 
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    by (auto simp add: dvd_int_unfold_dvd_nat intro: not_dvd_not_dvd_dvd_add)
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  then have "2 dvd \<bar>k\<bar> + \<bar>l\<bar>"
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    by (simp add: dvd_int_unfold_dvd_nat nat_add_distrib)
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  then show "2 dvd k + l"
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    by (simp add: two_dvd_abs_add_iff two_dvd_add_abs_iff)
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next
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  fix k l :: int
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  assume "2 dvd k * l"
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  then show "2 dvd k \<or> 2 dvd l"
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    by (simp add: dvd_int_unfold_dvd_nat two_is_prime nat_abs_mult_distrib)
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next
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  fix k :: int
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  have "k = (k - 1) + 1" by simp
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  then show "\<exists>l. k = l + 1" ..
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qed
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context semiring_div_parity
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begin
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subclass semiring_parity
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proof (unfold_locales, unfold dvd_eq_mod_eq_0 not_mod_2_eq_0_eq_1)
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  fix a b c
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  show "(c * a + b) mod a = 0 \<longleftrightarrow> b mod a = 0"
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    by simp
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next
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  fix a b c
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  assume "(b + c) mod a = 0"
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  with mod_add_eq [of b c a]
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  have "(b mod a + c mod a) mod a = 0"
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    by simp
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  moreover assume "b mod a = 0"
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  ultimately show "c mod a = 0"
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    by simp
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next
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  show "1 mod 2 = 1"
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    by (fact one_mod_two_eq_one)
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next
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  fix a b
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  assume "a mod 2 = 1"
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  moreover assume "b mod 2 = 1"
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  ultimately show "(a + b) mod 2 = 0"
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    using mod_add_eq [of a b 2] by simp
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next
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  fix a b
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  assume "(a * b) mod 2 = 0"
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  then have "(a mod 2) * (b mod 2) = 0"
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    by (cases "a mod 2 = 0") (simp_all add: mod_mult_eq [of a b 2])
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  then show "a mod 2 = 0 \<or> b mod 2 = 0"
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    by (rule divisors_zero)
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next
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  fix a
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  assume "a mod 2 = 1"
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  then have "a = a div 2 * 2 + 1" using mod_div_equality [of a 2] by simp
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  then show "\<exists>b. a = b + 1" ..
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qed
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end
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subsection {* Dedicated @{text even}/@{text odd} predicate *}
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subsubsection {* Properties *}
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context semiring_parity
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begin
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definition even :: "'a \<Rightarrow> bool"
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where
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  [presburger, algebra]: "even a \<longleftrightarrow> 2 dvd a"
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abbreviation odd :: "'a \<Rightarrow> bool"
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where
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  "odd a \<equiv> \<not> even a"
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lemma evenE [elim?]:
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  assumes "even a"
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  obtains b where "a = 2 * b"
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proof -
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  from assms have "2 dvd a" by (simp add: even_def)
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  then show thesis using that ..
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qed
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lemma oddE [elim?]:
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  assumes "odd a"
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  obtains b where "a = 2 * b + 1"
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proof -
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  from assms have "\<not> 2 dvd a" by (simp add: even_def)
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  then show thesis using that by (rule not_two_dvdE)
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qed
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lemma even_times_iff [simp, presburger, algebra]:
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  "even (a * b) \<longleftrightarrow> even a \<or> even b"
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  by (auto simp add: even_def dest: two_is_prime)
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lemma even_zero [simp]:
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  "even 0"
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  by (simp add: even_def)
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lemma odd_one [simp]:
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  "odd 1"
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  by (simp add: even_def)
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lemma even_numeral [simp]:
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  "even (numeral (Num.Bit0 n))"
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proof -
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  have "even (2 * numeral n)"
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    unfolding even_times_iff by (simp add: even_def)
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  then have "even (numeral n + numeral n)"
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    unfolding mult_2 .
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  then show ?thesis
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    unfolding numeral.simps .
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qed
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lemma odd_numeral [simp]:
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  "odd (numeral (Num.Bit1 n))"
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proof
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  assume "even (numeral (num.Bit1 n))"
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  then have "even (numeral n + numeral n + 1)"
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    unfolding numeral.simps .
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  then have "even (2 * numeral n + 1)"
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    unfolding mult_2 .
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  then have "2 dvd numeral n * 2 + 1"
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    unfolding even_def by (simp add: ac_simps)
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  with dvd_add_times_triv_left_iff [of 2 "numeral n" 1]
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    have "2 dvd 1"
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    by simp
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  then show False by simp
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qed
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lemma even_add [simp]:
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  "even (a + b) \<longleftrightarrow> (even a \<longleftrightarrow> even b)"
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  by (auto simp add: even_def dvd_add_right_iff dvd_add_left_iff not_dvd_not_dvd_dvd_add)
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lemma odd_add [simp]:
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  "odd (a + b) \<longleftrightarrow> (\<not> (odd a \<longleftrightarrow> odd b))"
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  by simp
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lemma even_power [simp, presburger]:
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  "even (a ^ n) \<longleftrightarrow> even a \<and> n \<noteq> 0"
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  by (induct n) auto
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end
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context ring_parity
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begin
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lemma even_minus [simp, presburger, algebra]:
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  "even (- a) \<longleftrightarrow> even a"
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  by (simp add: even_def)
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lemma even_diff [simp]:
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  "even (a - b) \<longleftrightarrow> even (a + b)"
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  using even_add [of a "- b"] by simp
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end
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subsubsection {* Parity and division *}
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context semiring_div_parity
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begin
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lemma one_div_two_eq_zero [simp]: -- \<open>FIXME move\<close>
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  "1 div 2 = 0"
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proof (cases "2 = 0")
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  case True then show ?thesis by simp
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next
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  case False
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  from mod_div_equality have "1 div 2 * 2 + 1 mod 2 = 1" .
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  with one_mod_two_eq_one have "1 div 2 * 2 + 1 = 1" by simp
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  then have "1 div 2 * 2 = 0" by (simp add: ac_simps add_left_imp_eq)
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  then have "1 div 2 = 0 \<or> 2 = 0" by (rule divisors_zero)
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  with False show ?thesis by auto
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qed
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lemma even_iff_mod_2_eq_zero:
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  "even a \<longleftrightarrow> a mod 2 = 0"
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  by (simp add: even_def dvd_eq_mod_eq_0)
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lemma even_succ_div_two [simp]:
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  "even a \<Longrightarrow> (a + 1) div 2 = a div 2"
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  by (cases "a = 0") (auto elim!: evenE dest: mult_not_zero)
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lemma odd_succ_div_two [simp]:
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  "odd a \<Longrightarrow> (a + 1) div 2 = a div 2 + 1"
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  by (auto elim!: oddE simp add: zero_not_eq_two [symmetric] add.assoc)
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lemma even_two_times_div_two:
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  "even a \<Longrightarrow> 2 * (a div 2) = a"
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  by (rule dvd_mult_div_cancel) (simp add: even_def)
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lemma odd_two_times_div_two_succ:
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  "odd a \<Longrightarrow> 2 * (a div 2) + 1 = a"
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  using mod_div_equality2 [of 2 a] by (simp add: even_iff_mod_2_eq_zero)
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end
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subsubsection {* Particularities for @{typ nat} and @{typ int} *}
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lemma even_Suc [simp, presburger, algebra]:
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  "even (Suc n) = odd n"
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  by (simp add: even_def two_dvd_Suc_iff)
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lemma odd_pos: 
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  "odd (n :: nat) \<Longrightarrow> 0 < n"
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  by (auto elim: oddE)
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lemma even_diff_nat [simp]:
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  fixes m n :: nat
haftmann@58687
   336
  shows "even (m - n) \<longleftrightarrow> m < n \<or> even (m + n)"
haftmann@58687
   337
  by (simp add: even_def two_dvd_diff_nat_iff)
haftmann@58680
   338
haftmann@58679
   339
lemma even_int_iff:
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   340
  "even (int n) \<longleftrightarrow> even n"
haftmann@58679
   341
  by (simp add: even_def dvd_int_iff)
haftmann@33318
   342
haftmann@58687
   343
lemma even_nat_iff:
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   344
  "0 \<le> k \<Longrightarrow> even (nat k) \<longleftrightarrow> even k"
haftmann@58687
   345
  by (simp add: even_int_iff [symmetric])
haftmann@58687
   346
haftmann@58710
   347
lemma even_num_iff:
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   348
  "0 < n \<Longrightarrow> even n = odd (n - 1 :: nat)"
haftmann@58710
   349
  by simp
haftmann@58687
   350
haftmann@58710
   351
lemma even_Suc_div_two [simp]:
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   352
  "even n \<Longrightarrow> Suc n div 2 = n div 2"
haftmann@58710
   353
  using even_succ_div_two [of n] by simp
haftmann@58710
   354
  
haftmann@58710
   355
lemma odd_Suc_div_two [simp]:
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   356
  "odd n \<Longrightarrow> Suc n div 2 = Suc (n div 2)"
haftmann@58710
   357
  using odd_succ_div_two [of n] by simp
haftmann@58710
   358
haftmann@58710
   359
lemma odd_two_times_div_two_Suc:
haftmann@58710
   360
  "odd n \<Longrightarrow> Suc (2 * (n div 2)) = n"
haftmann@58710
   361
  using odd_two_times_div_two_succ [of n] by simp
haftmann@58710
   362
  
haftmann@58710
   363
text {* Nice facts about division by @{term 4} *}  
haftmann@58710
   364
haftmann@58710
   365
lemma even_even_mod_4_iff:
haftmann@58710
   366
  "even (n::nat) \<longleftrightarrow> even (n mod 4)"
haftmann@58710
   367
  by presburger
haftmann@58710
   368
haftmann@58710
   369
lemma odd_mod_4_div_2:
haftmann@58710
   370
  "n mod 4 = (3::nat) \<Longrightarrow> odd ((n - 1) div 2)"
haftmann@58710
   371
  by presburger
haftmann@58710
   372
haftmann@58710
   373
lemma even_mod_4_div_2:
haftmann@58710
   374
  "n mod 4 = (1::nat) \<Longrightarrow> even ((n - 1) div 2)"
haftmann@58710
   375
  by presburger
haftmann@58710
   376
  
haftmann@58710
   377
text {* Parity and powers *}
haftmann@58689
   378
haftmann@58689
   379
context comm_ring_1
haftmann@58689
   380
begin
haftmann@58689
   381
haftmann@58689
   382
lemma power_minus_even [simp]:
haftmann@58689
   383
  "even n \<Longrightarrow> (- a) ^ n = a ^ n"
haftmann@58690
   384
  by (auto elim: evenE)
haftmann@58689
   385
haftmann@58689
   386
lemma power_minus_odd [simp]:
haftmann@58689
   387
  "odd n \<Longrightarrow> (- a) ^ n = - (a ^ n)"
haftmann@58690
   388
  by (auto elim: oddE)
haftmann@58690
   389
haftmann@58690
   390
lemma neg_power_if:
haftmann@58690
   391
  "(- a) ^ n = (if even n then a ^ n else - (a ^ n))"
haftmann@58690
   392
  by simp
haftmann@58689
   393
haftmann@58689
   394
lemma neg_one_even_power [simp]:
haftmann@58689
   395
  "even n \<Longrightarrow> (- 1) ^ n = 1"
haftmann@58690
   396
  by simp
haftmann@58689
   397
haftmann@58689
   398
lemma neg_one_odd_power [simp]:
haftmann@58689
   399
  "odd n \<Longrightarrow> (- 1) ^ n = - 1"
haftmann@58690
   400
  by simp
haftmann@58689
   401
haftmann@58689
   402
end  
haftmann@58689
   403
haftmann@58689
   404
lemma zero_less_power_nat_eq_numeral [simp]: -- \<open>FIXME move\<close>
haftmann@58689
   405
  "0 < (n :: nat) ^ numeral w \<longleftrightarrow> 0 < n \<or> numeral w = (0 :: nat)"
haftmann@58689
   406
  by (fact nat_zero_less_power_iff)
haftmann@58689
   407
haftmann@58689
   408
context linordered_idom
haftmann@58689
   409
begin
haftmann@58689
   410
haftmann@58689
   411
lemma power_eq_0_iff' [simp]: -- \<open>FIXME move\<close>
haftmann@58689
   412
  "a ^ n = 0 \<longleftrightarrow> a = 0 \<and> n > 0"
haftmann@58689
   413
  by (induct n) auto
haftmann@58689
   414
haftmann@58689
   415
lemma power2_less_eq_zero_iff [simp]: -- \<open>FIXME move\<close>
haftmann@58689
   416
  "a\<^sup>2 \<le> 0 \<longleftrightarrow> a = 0"
haftmann@58689
   417
proof (cases "a = 0")
haftmann@58689
   418
  case True then show ?thesis by simp
haftmann@58689
   419
next
haftmann@58689
   420
  case False then have "a < 0 \<or> a > 0" by auto
haftmann@58689
   421
  then have "a\<^sup>2 > 0" by auto
haftmann@58689
   422
  then have "\<not> a\<^sup>2 \<le> 0" by (simp add: not_le)
haftmann@58689
   423
  with False show ?thesis by simp
haftmann@58689
   424
qed
haftmann@58689
   425
haftmann@58689
   426
lemma zero_le_even_power:
haftmann@58689
   427
  "even n \<Longrightarrow> 0 \<le> a ^ n"
haftmann@58690
   428
  by (auto elim: evenE)
haftmann@58689
   429
haftmann@58689
   430
lemma zero_le_odd_power:
haftmann@58689
   431
  "odd n \<Longrightarrow> 0 \<le> a ^ n \<longleftrightarrow> 0 \<le> a"
haftmann@58689
   432
  by (auto simp add: power_even_eq zero_le_mult_iff elim: oddE)
haftmann@58689
   433
wenzelm@58718
   434
lemma zero_le_power_iff [presburger]: -- \<open>FIXME cf. @{text zero_le_power_eq}\<close>
haftmann@58689
   435
  "0 \<le> a ^ n \<longleftrightarrow> 0 \<le> a \<or> even n"
haftmann@58689
   436
proof (cases "even n")
haftmann@58689
   437
  case True
haftmann@58689
   438
  then obtain k where "n = 2 * k" ..
haftmann@58690
   439
  then show ?thesis by simp
haftmann@58689
   440
next
haftmann@58689
   441
  case False
haftmann@58689
   442
  then obtain k where "n = 2 * k + 1" ..
haftmann@58689
   443
  moreover have "a ^ (2 * k) \<le> 0 \<Longrightarrow> a = 0"
haftmann@58689
   444
    by (induct k) (auto simp add: zero_le_mult_iff mult_le_0_iff)
haftmann@58689
   445
  ultimately show ?thesis
haftmann@58689
   446
    by (auto simp add: zero_le_mult_iff zero_le_even_power)
haftmann@58689
   447
qed
haftmann@58689
   448
haftmann@58710
   449
lemma zero_le_power_eq [presburger]:
haftmann@58689
   450
  "0 \<le> a ^ n \<longleftrightarrow> even n \<or> odd n \<and> 0 \<le> a"
haftmann@58689
   451
  using zero_le_power_iff [of a n] by auto
haftmann@58689
   452
haftmann@58689
   453
lemma zero_less_power_eq [presburger]:
haftmann@58689
   454
  "0 < a ^ n \<longleftrightarrow> n = 0 \<or> even n \<and> a \<noteq> 0 \<or> odd n \<and> 0 < a"
haftmann@58689
   455
proof -
haftmann@58689
   456
  have [simp]: "0 = a ^ n \<longleftrightarrow> a = 0 \<and> n > 0"
haftmann@58689
   457
    unfolding power_eq_0_iff' [of a n, symmetric] by blast
haftmann@58689
   458
  show ?thesis
haftmann@58710
   459
  unfolding less_le zero_le_power_eq by auto
haftmann@58689
   460
qed
haftmann@58689
   461
haftmann@58689
   462
lemma power_less_zero_eq [presburger]:
haftmann@58689
   463
  "a ^ n < 0 \<longleftrightarrow> odd n \<and> a < 0"
haftmann@58689
   464
  unfolding not_le [symmetric] zero_le_power_eq by auto
haftmann@58689
   465
  
haftmann@58689
   466
lemma power_le_zero_eq [presburger]:
haftmann@58689
   467
  "a ^ n \<le> 0 \<longleftrightarrow> n > 0 \<and> (odd n \<and> a \<le> 0 \<or> even n \<and> a = 0)"
haftmann@58689
   468
  unfolding not_less [symmetric] zero_less_power_eq by auto 
haftmann@58689
   469
haftmann@58689
   470
lemma power_even_abs:
haftmann@58689
   471
  "even n \<Longrightarrow> \<bar>a\<bar> ^ n = a ^ n"
haftmann@58689
   472
  using power_abs [of a n] by (simp add: zero_le_even_power)
haftmann@58689
   473
haftmann@58689
   474
lemma power_mono_even:
haftmann@58689
   475
  assumes "even n" and "\<bar>a\<bar> \<le> \<bar>b\<bar>"
haftmann@58689
   476
  shows "a ^ n \<le> b ^ n"
haftmann@58689
   477
proof -
haftmann@58689
   478
  have "0 \<le> \<bar>a\<bar>" by auto
haftmann@58689
   479
  with `\<bar>a\<bar> \<le> \<bar>b\<bar>`
haftmann@58689
   480
  have "\<bar>a\<bar> ^ n \<le> \<bar>b\<bar> ^ n" by (rule power_mono)
haftmann@58689
   481
  with `even n` show ?thesis by (simp add: power_even_abs)  
haftmann@58689
   482
qed
haftmann@58689
   483
haftmann@58689
   484
lemma power_mono_odd:
haftmann@58689
   485
  assumes "odd n" and "a \<le> b"
haftmann@58689
   486
  shows "a ^ n \<le> b ^ n"
haftmann@58689
   487
proof (cases "b < 0")
haftmann@58689
   488
  case True with `a \<le> b` have "- b \<le> - a" and "0 \<le> - b" by auto
haftmann@58689
   489
  hence "(- b) ^ n \<le> (- a) ^ n" by (rule power_mono)
haftmann@58689
   490
  with `odd n` show ?thesis by simp
haftmann@58689
   491
next
haftmann@58689
   492
  case False then have "0 \<le> b" by auto
haftmann@58689
   493
  show ?thesis
haftmann@58689
   494
  proof (cases "a < 0")
haftmann@58689
   495
    case True then have "n \<noteq> 0" and "a \<le> 0" using `odd n` [THEN odd_pos] by auto
haftmann@58689
   496
    then have "a ^ n \<le> 0" unfolding power_le_zero_eq using `odd n` by auto
haftmann@58689
   497
    moreover
haftmann@58689
   498
    from `0 \<le> b` have "0 \<le> b ^ n" by auto
haftmann@58689
   499
    ultimately show ?thesis by auto
haftmann@58689
   500
  next
haftmann@58689
   501
    case False then have "0 \<le> a" by auto
haftmann@58689
   502
    with `a \<le> b` show ?thesis using power_mono by auto
haftmann@58689
   503
  qed
haftmann@58689
   504
qed
haftmann@58689
   505
 
haftmann@58689
   506
text {* Simplify, when the exponent is a numeral *}
haftmann@58689
   507
haftmann@58689
   508
lemma zero_le_power_eq_numeral [simp]:
haftmann@58689
   509
  "0 \<le> a ^ numeral w \<longleftrightarrow> even (numeral w :: nat) \<or> odd (numeral w :: nat) \<and> 0 \<le> a"
haftmann@58689
   510
  by (fact zero_le_power_eq)
haftmann@58689
   511
haftmann@58689
   512
lemma zero_less_power_eq_numeral [simp]:
haftmann@58689
   513
  "0 < a ^ numeral w \<longleftrightarrow> numeral w = (0 :: nat)
haftmann@58689
   514
    \<or> even (numeral w :: nat) \<and> a \<noteq> 0 \<or> odd (numeral w :: nat) \<and> 0 < a"
haftmann@58689
   515
  by (fact zero_less_power_eq)
haftmann@58689
   516
haftmann@58689
   517
lemma power_le_zero_eq_numeral [simp]:
haftmann@58689
   518
  "a ^ numeral w \<le> 0 \<longleftrightarrow> (0 :: nat) < numeral w
haftmann@58689
   519
    \<and> (odd (numeral w :: nat) \<and> a \<le> 0 \<or> even (numeral w :: nat) \<and> a = 0)"
haftmann@58689
   520
  by (fact power_le_zero_eq)
haftmann@58689
   521
haftmann@58689
   522
lemma power_less_zero_eq_numeral [simp]:
haftmann@58689
   523
  "a ^ numeral w < 0 \<longleftrightarrow> odd (numeral w :: nat) \<and> a < 0"
haftmann@58689
   524
  by (fact power_less_zero_eq)
haftmann@58689
   525
haftmann@58689
   526
lemma power_eq_0_iff_numeral [simp]:
haftmann@58689
   527
  "a ^ numeral w = (0 :: nat) \<longleftrightarrow> a = 0 \<and> numeral w \<noteq> (0 :: nat)"
haftmann@58689
   528
  by (fact power_eq_0_iff)
haftmann@58689
   529
haftmann@58689
   530
lemma power_even_abs_numeral [simp]:
haftmann@58689
   531
  "even (numeral w :: nat) \<Longrightarrow> \<bar>a\<bar> ^ numeral w = a ^ numeral w"
haftmann@58689
   532
  by (fact power_even_abs)
haftmann@58689
   533
haftmann@58689
   534
end
haftmann@58689
   535
haftmann@58689
   536
haftmann@58687
   537
subsubsection {* Tools setup *}
haftmann@58687
   538
haftmann@58679
   539
declare transfer_morphism_int_nat [transfer add return:
haftmann@58679
   540
  even_int_iff
haftmann@33318
   541
]
wenzelm@21256
   542
haftmann@58679
   543
lemma [presburger]:
haftmann@58679
   544
  "even n \<longleftrightarrow> even (int n)"
haftmann@58679
   545
  using even_int_iff [of n] by simp
haftmann@25600
   546
haftmann@58687
   547
lemma (in semiring_parity) [presburger]:
haftmann@58680
   548
  "even (a + b) \<longleftrightarrow> even a \<and> even b \<or> odd a \<and> odd b"
haftmann@58680
   549
  by auto
wenzelm@21256
   550
haftmann@58687
   551
lemma [presburger, algebra]:
haftmann@58687
   552
  fixes m n :: nat
haftmann@58687
   553
  shows "even (m - n) \<longleftrightarrow> m < n \<or> even m \<and> even n \<or> odd m \<and> odd n"
haftmann@58687
   554
  by auto
haftmann@58687
   555
haftmann@58687
   556
lemma [presburger, algebra]:
haftmann@58687
   557
  fixes m n :: nat
haftmann@58687
   558
  shows "even (m ^ n) \<longleftrightarrow> even m \<and> 0 < n"
haftmann@58687
   559
  by simp
haftmann@58687
   560
haftmann@58687
   561
lemma [presburger]:
haftmann@58687
   562
  fixes k :: int
haftmann@58687
   563
  shows "(k + 1) div 2 = k div 2 \<longleftrightarrow> even k"
haftmann@58687
   564
  by presburger
haftmann@58687
   565
haftmann@58687
   566
lemma [presburger]:
haftmann@58687
   567
  fixes k :: int
haftmann@58687
   568
  shows "(k + 1) div 2 = k div 2 + 1 \<longleftrightarrow> odd k"
haftmann@58687
   569
  by presburger
haftmann@58687
   570
  
haftmann@58687
   571
lemma [presburger]:
haftmann@58687
   572
  "Suc n div Suc (Suc 0) = n div Suc (Suc 0) \<longleftrightarrow> even n"
haftmann@58687
   573
  by presburger
haftmann@58687
   574
wenzelm@21256
   575
end
haftmann@54227
   576