src/HOL/Parity.thy
author haftmann
Sun Oct 19 18:05:26 2014 +0200 (2014-10-19)
changeset 58709 efdc6c533bd3
parent 58691 68f8d22a6867
child 58710 7216a10d69ba
permissions -rw-r--r--
prefer generic elimination rules for even/odd over specialized unfold rules for nat
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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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  [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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context semiring_div_parity
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begin
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lemma even_iff_mod_2_eq_zero [presburger]:
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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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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
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  shows "even (m - n) \<longleftrightarrow> m < n \<or> even (m + n)"
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  by (simp add: even_def two_dvd_diff_nat_iff)
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lemma even_int_iff:
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  "even (int n) \<longleftrightarrow> even n"
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  by (simp add: even_def dvd_int_iff)
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lemma even_nat_iff:
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  "0 \<le> k \<Longrightarrow> even (nat k) \<longleftrightarrow> even k"
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  by (simp add: even_int_iff [symmetric])
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subsubsection {* Parity and powers *}
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context comm_ring_1
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begin
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lemma power_minus_even [simp]:
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  "even n \<Longrightarrow> (- a) ^ n = a ^ n"
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  by (auto elim: evenE)
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lemma power_minus_odd [simp]:
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  "odd n \<Longrightarrow> (- a) ^ n = - (a ^ n)"
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  by (auto elim: oddE)
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lemma neg_power_if:
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  "(- a) ^ n = (if even n then a ^ n else - (a ^ n))"
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  by simp
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lemma neg_one_even_power [simp]:
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  "even n \<Longrightarrow> (- 1) ^ n = 1"
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  by simp
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lemma neg_one_odd_power [simp]:
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  "odd n \<Longrightarrow> (- 1) ^ n = - 1"
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  by simp
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end  
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   342
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   343
lemma zero_less_power_nat_eq_numeral [simp]: -- \<open>FIXME move\<close>
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   344
  "0 < (n :: nat) ^ numeral w \<longleftrightarrow> 0 < n \<or> numeral w = (0 :: nat)"
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   345
  by (fact nat_zero_less_power_iff)
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   346
haftmann@58689
   347
context linordered_idom
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   348
begin
haftmann@58689
   349
haftmann@58689
   350
lemma power_eq_0_iff' [simp]: -- \<open>FIXME move\<close>
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   351
  "a ^ n = 0 \<longleftrightarrow> a = 0 \<and> n > 0"
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   352
  by (induct n) auto
haftmann@58689
   353
haftmann@58689
   354
lemma power2_less_eq_zero_iff [simp]: -- \<open>FIXME move\<close>
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   355
  "a\<^sup>2 \<le> 0 \<longleftrightarrow> a = 0"
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   356
proof (cases "a = 0")
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   357
  case True then show ?thesis by simp
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   358
next
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   359
  case False then have "a < 0 \<or> a > 0" by auto
haftmann@58689
   360
  then have "a\<^sup>2 > 0" by auto
haftmann@58689
   361
  then have "\<not> a\<^sup>2 \<le> 0" by (simp add: not_le)
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   362
  with False show ?thesis by simp
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   363
qed
haftmann@58689
   364
haftmann@58689
   365
lemma zero_le_even_power:
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   366
  "even n \<Longrightarrow> 0 \<le> a ^ n"
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   367
  by (auto elim: evenE)
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   368
haftmann@58689
   369
lemma zero_le_odd_power:
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   370
  "odd n \<Longrightarrow> 0 \<le> a ^ n \<longleftrightarrow> 0 \<le> a"
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   371
  by (auto simp add: power_even_eq zero_le_mult_iff elim: oddE)
haftmann@58689
   372
haftmann@58689
   373
lemma zero_le_power_iff [presburger]:
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   374
  "0 \<le> a ^ n \<longleftrightarrow> 0 \<le> a \<or> even n"
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   375
proof (cases "even n")
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   376
  case True
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   377
  then obtain k where "n = 2 * k" ..
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   378
  then show ?thesis by simp
haftmann@58689
   379
next
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   380
  case False
haftmann@58689
   381
  then obtain k where "n = 2 * k + 1" ..
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   382
  moreover have "a ^ (2 * k) \<le> 0 \<Longrightarrow> a = 0"
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   383
    by (induct k) (auto simp add: zero_le_mult_iff mult_le_0_iff)
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   384
  ultimately show ?thesis
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   385
    by (auto simp add: zero_le_mult_iff zero_le_even_power)
haftmann@58689
   386
qed
haftmann@58689
   387
haftmann@58689
   388
lemma zero_le_power_eq [presburger]: -- \<open>FIXME weaker version of @{text zero_le_power_iff}\<close>
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   389
  "0 \<le> a ^ n \<longleftrightarrow> even n \<or> odd n \<and> 0 \<le> a"
haftmann@58689
   390
  using zero_le_power_iff [of a n] by auto
haftmann@58689
   391
haftmann@58689
   392
lemma zero_less_power_eq [presburger]:
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   393
  "0 < a ^ n \<longleftrightarrow> n = 0 \<or> even n \<and> a \<noteq> 0 \<or> odd n \<and> 0 < a"
haftmann@58689
   394
proof -
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   395
  have [simp]: "0 = a ^ n \<longleftrightarrow> a = 0 \<and> n > 0"
haftmann@58689
   396
    unfolding power_eq_0_iff' [of a n, symmetric] by blast
haftmann@58689
   397
  show ?thesis
haftmann@58689
   398
  unfolding less_le zero_le_power_iff by auto
haftmann@58689
   399
qed
haftmann@58689
   400
haftmann@58689
   401
lemma power_less_zero_eq [presburger]:
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   402
  "a ^ n < 0 \<longleftrightarrow> odd n \<and> a < 0"
haftmann@58689
   403
  unfolding not_le [symmetric] zero_le_power_eq by auto
haftmann@58689
   404
  
haftmann@58689
   405
lemma power_le_zero_eq [presburger]:
haftmann@58689
   406
  "a ^ n \<le> 0 \<longleftrightarrow> n > 0 \<and> (odd n \<and> a \<le> 0 \<or> even n \<and> a = 0)"
haftmann@58689
   407
  unfolding not_less [symmetric] zero_less_power_eq by auto 
haftmann@58689
   408
haftmann@58689
   409
lemma power_even_abs:
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   410
  "even n \<Longrightarrow> \<bar>a\<bar> ^ n = a ^ n"
haftmann@58689
   411
  using power_abs [of a n] by (simp add: zero_le_even_power)
haftmann@58689
   412
haftmann@58689
   413
lemma power_mono_even:
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   414
  assumes "even n" and "\<bar>a\<bar> \<le> \<bar>b\<bar>"
haftmann@58689
   415
  shows "a ^ n \<le> b ^ n"
haftmann@58689
   416
proof -
haftmann@58689
   417
  have "0 \<le> \<bar>a\<bar>" by auto
haftmann@58689
   418
  with `\<bar>a\<bar> \<le> \<bar>b\<bar>`
haftmann@58689
   419
  have "\<bar>a\<bar> ^ n \<le> \<bar>b\<bar> ^ n" by (rule power_mono)
haftmann@58689
   420
  with `even n` show ?thesis by (simp add: power_even_abs)  
haftmann@58689
   421
qed
haftmann@58689
   422
haftmann@58689
   423
lemma power_mono_odd:
haftmann@58689
   424
  assumes "odd n" and "a \<le> b"
haftmann@58689
   425
  shows "a ^ n \<le> b ^ n"
haftmann@58689
   426
proof (cases "b < 0")
haftmann@58689
   427
  case True with `a \<le> b` have "- b \<le> - a" and "0 \<le> - b" by auto
haftmann@58689
   428
  hence "(- b) ^ n \<le> (- a) ^ n" by (rule power_mono)
haftmann@58689
   429
  with `odd n` show ?thesis by simp
haftmann@58689
   430
next
haftmann@58689
   431
  case False then have "0 \<le> b" by auto
haftmann@58689
   432
  show ?thesis
haftmann@58689
   433
  proof (cases "a < 0")
haftmann@58689
   434
    case True then have "n \<noteq> 0" and "a \<le> 0" using `odd n` [THEN odd_pos] by auto
haftmann@58689
   435
    then have "a ^ n \<le> 0" unfolding power_le_zero_eq using `odd n` by auto
haftmann@58689
   436
    moreover
haftmann@58689
   437
    from `0 \<le> b` have "0 \<le> b ^ n" by auto
haftmann@58689
   438
    ultimately show ?thesis by auto
haftmann@58689
   439
  next
haftmann@58689
   440
    case False then have "0 \<le> a" by auto
haftmann@58689
   441
    with `a \<le> b` show ?thesis using power_mono by auto
haftmann@58689
   442
  qed
haftmann@58689
   443
qed
haftmann@58689
   444
 
haftmann@58689
   445
text {* Simplify, when the exponent is a numeral *}
haftmann@58689
   446
haftmann@58689
   447
lemma zero_le_power_eq_numeral [simp]:
haftmann@58689
   448
  "0 \<le> a ^ numeral w \<longleftrightarrow> even (numeral w :: nat) \<or> odd (numeral w :: nat) \<and> 0 \<le> a"
haftmann@58689
   449
  by (fact zero_le_power_eq)
haftmann@58689
   450
haftmann@58689
   451
lemma zero_less_power_eq_numeral [simp]:
haftmann@58689
   452
  "0 < a ^ numeral w \<longleftrightarrow> numeral w = (0 :: nat)
haftmann@58689
   453
    \<or> even (numeral w :: nat) \<and> a \<noteq> 0 \<or> odd (numeral w :: nat) \<and> 0 < a"
haftmann@58689
   454
  by (fact zero_less_power_eq)
haftmann@58689
   455
haftmann@58689
   456
lemma power_le_zero_eq_numeral [simp]:
haftmann@58689
   457
  "a ^ numeral w \<le> 0 \<longleftrightarrow> (0 :: nat) < numeral w
haftmann@58689
   458
    \<and> (odd (numeral w :: nat) \<and> a \<le> 0 \<or> even (numeral w :: nat) \<and> a = 0)"
haftmann@58689
   459
  by (fact power_le_zero_eq)
haftmann@58689
   460
haftmann@58689
   461
lemma power_less_zero_eq_numeral [simp]:
haftmann@58689
   462
  "a ^ numeral w < 0 \<longleftrightarrow> odd (numeral w :: nat) \<and> a < 0"
haftmann@58689
   463
  by (fact power_less_zero_eq)
haftmann@58689
   464
haftmann@58689
   465
lemma power_eq_0_iff_numeral [simp]:
haftmann@58689
   466
  "a ^ numeral w = (0 :: nat) \<longleftrightarrow> a = 0 \<and> numeral w \<noteq> (0 :: nat)"
haftmann@58689
   467
  by (fact power_eq_0_iff)
haftmann@58689
   468
haftmann@58689
   469
lemma power_even_abs_numeral [simp]:
haftmann@58689
   470
  "even (numeral w :: nat) \<Longrightarrow> \<bar>a\<bar> ^ numeral w = a ^ numeral w"
haftmann@58689
   471
  by (fact power_even_abs)
haftmann@58689
   472
haftmann@58689
   473
end
haftmann@58689
   474
haftmann@58689
   475
haftmann@58687
   476
subsubsection {* Tools setup *}
haftmann@58687
   477
haftmann@58679
   478
declare transfer_morphism_int_nat [transfer add return:
haftmann@58679
   479
  even_int_iff
haftmann@33318
   480
]
wenzelm@21256
   481
haftmann@58679
   482
lemma [presburger]:
haftmann@58679
   483
  "even n \<longleftrightarrow> even (int n)"
haftmann@58679
   484
  using even_int_iff [of n] by simp
haftmann@25600
   485
haftmann@58687
   486
lemma (in semiring_parity) [presburger]:
haftmann@58680
   487
  "even (a + b) \<longleftrightarrow> even a \<and> even b \<or> odd a \<and> odd b"
haftmann@58680
   488
  by auto
wenzelm@21256
   489
haftmann@58687
   490
lemma [presburger, algebra]:
haftmann@58687
   491
  fixes m n :: nat
haftmann@58687
   492
  shows "even (m - n) \<longleftrightarrow> m < n \<or> even m \<and> even n \<or> odd m \<and> odd n"
haftmann@58687
   493
  by auto
haftmann@58687
   494
haftmann@58687
   495
lemma [presburger, algebra]:
haftmann@58687
   496
  fixes m n :: nat
haftmann@58687
   497
  shows "even (m ^ n) \<longleftrightarrow> even m \<and> 0 < n"
haftmann@58687
   498
  by simp
haftmann@58687
   499
haftmann@58687
   500
lemma [presburger]:
haftmann@58687
   501
  fixes k :: int
haftmann@58687
   502
  shows "(k + 1) div 2 = k div 2 \<longleftrightarrow> even k"
haftmann@58687
   503
  by presburger
haftmann@58687
   504
haftmann@58687
   505
lemma [presburger]:
haftmann@58687
   506
  fixes k :: int
haftmann@58687
   507
  shows "(k + 1) div 2 = k div 2 + 1 \<longleftrightarrow> odd k"
haftmann@58687
   508
  by presburger
haftmann@58687
   509
  
haftmann@58687
   510
lemma [presburger]:
haftmann@58687
   511
  "Suc n div Suc (Suc 0) = n div Suc (Suc 0) \<longleftrightarrow> even n"
haftmann@58687
   512
  by presburger
haftmann@58687
   513
wenzelm@21256
   514
haftmann@58688
   515
subsubsection {* Miscellaneous *}
haftmann@58680
   516
haftmann@58688
   517
lemma even_nat_plus_one_div_two:
haftmann@58688
   518
  "even (x::nat) ==> (Suc x) div Suc (Suc 0) = x div Suc (Suc 0)"
haftmann@58688
   519
  by presburger
haftmann@58680
   520
haftmann@58688
   521
lemma odd_nat_plus_one_div_two:
haftmann@58688
   522
  "odd (x::nat) ==> (Suc x) div Suc (Suc 0) = Suc (x div Suc (Suc 0))"
haftmann@58688
   523
  by presburger
haftmann@58680
   524
haftmann@58687
   525
lemma even_nat_mod_two_eq_zero:
haftmann@58687
   526
  "even (x::nat) ==> x mod Suc (Suc 0) = 0"
haftmann@58680
   527
  by presburger
wenzelm@21256
   528
haftmann@58687
   529
lemma odd_nat_mod_two_eq_one:
haftmann@58687
   530
  "odd (x::nat) ==> x mod Suc (Suc 0) = Suc 0"
haftmann@58680
   531
  by presburger
wenzelm@21256
   532
haftmann@58687
   533
lemma even_nat_equiv_def:
haftmann@58687
   534
  "even (x::nat) = (x mod Suc (Suc 0) = 0)"
haftmann@58680
   535
  by presburger
wenzelm@21256
   536
haftmann@58687
   537
lemma odd_nat_equiv_def:
haftmann@58687
   538
  "odd (x::nat) = (x mod Suc (Suc 0) = Suc 0)"
haftmann@58687
   539
  by presburger
wenzelm@21256
   540
haftmann@58687
   541
lemma even_nat_div_two_times_two:
haftmann@58687
   542
  "even (x::nat) ==> Suc (Suc 0) * (x div Suc (Suc 0)) = x"
haftmann@58687
   543
  by presburger
wenzelm@21256
   544
haftmann@58687
   545
lemma odd_nat_div_two_times_two_plus_one:
haftmann@58687
   546
  "odd (x::nat) ==> Suc (Suc (Suc 0) * (x div Suc (Suc 0))) = x"
haftmann@58687
   547
  by presburger
wenzelm@21256
   548
haftmann@58691
   549
lemma lemma_even_div2 [simp]:
haftmann@58691
   550
  "even (n::nat) ==> (n + 1) div 2 = n div 2"
haftmann@58691
   551
  by presburger
haftmann@58688
   552
haftmann@58691
   553
lemma lemma_odd_div2 [simp]:
haftmann@58691
   554
  "odd n ==> (n + 1) div 2 = Suc (n div 2)"
haftmann@58688
   555
  by presburger
haftmann@58688
   556
haftmann@58691
   557
lemma even_num_iff:
haftmann@58691
   558
  "0 < n ==> even n = (odd (n - 1 :: nat))"
haftmann@58688
   559
  by presburger
haftmann@58688
   560
haftmann@58691
   561
lemma even_even_mod_4_iff:
haftmann@58691
   562
  "even (n::nat) = even (n mod 4)"
haftmann@58691
   563
  by presburger
haftmann@58688
   564
haftmann@58691
   565
lemma lemma_odd_mod_4_div_2:
haftmann@58691
   566
  "n mod 4 = (3::nat) ==> odd((n - 1) div 2)"
haftmann@58691
   567
  by presburger
haftmann@58688
   568
haftmann@58691
   569
lemma lemma_even_mod_4_div_2:
haftmann@58691
   570
  "n mod 4 = (1::nat) ==> even ((n - 1) div 2)"
haftmann@58688
   571
  by presburger
haftmann@58688
   572
wenzelm@21256
   573
end
haftmann@54227
   574