src/HOL/Parity.thy
author eberlm
Mon, 02 Nov 2015 11:56:28 +0100
changeset 61531 ab2e862263e7
parent 60867 86e7560e07d0
child 61799 4cf66f21b764
permissions -rw-r--r--
Rounding function, uniform limits, cotangent, binomial identities
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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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section \<open>Parity in rings and semirings\<close>
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theory Parity
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imports Nat_Transfer
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begin
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subsection \<open>Ring structures with parity and @{text even}/@{text odd} predicates\<close>
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24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
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class semiring_parity = comm_semiring_1_cancel + numeral +
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  assumes odd_one [simp]: "\<not> 2 dvd 1"
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  assumes odd_even_add: "\<not> 2 dvd a \<Longrightarrow> \<not> 2 dvd b \<Longrightarrow> 2 dvd a + b"
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  assumes even_multD: "2 dvd a * b \<Longrightarrow> 2 dvd a \<or> 2 dvd b"
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  assumes odd_ex_decrement: "\<not> 2 dvd a \<Longrightarrow> \<exists>b. a = b + 1"
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begin
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subclass semiring_numeral ..
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abbreviation even :: "'a \<Rightarrow> bool"
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where
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  "even a \<equiv> 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> 2 dvd a"
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lemma even_zero [simp]:
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  "even 0"
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  by (fact dvd_0_right)
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lemma even_plus_one_iff [simp]:
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  "even (a + 1) \<longleftrightarrow> odd a"
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  by (auto simp add: dvd_add_right_iff intro: odd_even_add)
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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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  using assms by (rule dvdE)
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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 obtain b where *: "a = b + 1"
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    by (blast dest: odd_ex_decrement)
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  with assms have "even (b + 2)" by simp
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  then have "even 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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lemma even_times_iff [simp]:
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  "even (a * b) \<longleftrightarrow> even a \<or> even b"
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  by (auto dest: even_multD)
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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
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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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    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: dvd_add_right_iff dvd_add_left_iff odd_even_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]:
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  "even (a ^ n) \<longleftrightarrow> even a \<and> n > 0"
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  by (induct n) auto
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end
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class ring_parity = ring + semiring_parity
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begin
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subclass comm_ring_1 ..
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lemma even_minus [simp]:
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  "even (- a) \<longleftrightarrow> even a"
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  by (fact dvd_minus_iff)
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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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subsection \<open>Instances for @{typ nat} and @{typ int}\<close>
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lemma even_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 even_Suc [simp]:
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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 even_diff_nat [simp]:
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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 add: mult_2_right)
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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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instance nat :: semiring_parity
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proof
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  show "\<not> 2 dvd (1 :: nat)"
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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
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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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   163
    assume "\<not> 2 dvd n"
063698416239 correct sort constraints for abbreviations in type classes
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   164
    then have "2 dvd (Suc n)" by simp
58787
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   165
    then obtain r where "Suc n = 2 * r" ..
af9eb5e566dd eliminated redundancies;
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diff changeset
   166
    moreover from * obtain s where "m * n = 2 * s" ..
af9eb5e566dd eliminated redundancies;
haftmann
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   167
    then have "2 * s + m = m * Suc n" by simp
af9eb5e566dd eliminated redundancies;
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diff changeset
   168
    ultimately have " 2 * s + m = 2 * (m * r)" by (simp add: algebra_simps)
af9eb5e566dd eliminated redundancies;
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   169
    then have "m = 2 * (m * r - s)" by simp
60343
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   170
    then show "2 dvd m" ..
58787
af9eb5e566dd eliminated redundancies;
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   171
  qed
af9eb5e566dd eliminated redundancies;
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   172
next
af9eb5e566dd eliminated redundancies;
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   173
  fix n :: nat
60343
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   174
  assume "\<not> 2 dvd n"
58787
af9eb5e566dd eliminated redundancies;
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   175
  then show "\<exists>m. n = m + 1"
af9eb5e566dd eliminated redundancies;
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   176
    by (cases n) simp_all
af9eb5e566dd eliminated redundancies;
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   177
qed
58687
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   178
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   179
lemma odd_pos: 
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   180
  "odd (n :: nat) \<Longrightarrow> 0 < n"
58690
5c5c14844738 standard elimination rule for even
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   181
  by (auto elim: oddE)
60343
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   182
063698416239 correct sort constraints for abbreviations in type classes
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   183
lemma even_diff_iff [simp]:
063698416239 correct sort constraints for abbreviations in type classes
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   184
  fixes k l :: int
063698416239 correct sort constraints for abbreviations in type classes
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   185
  shows "2 dvd (k - l) \<longleftrightarrow> 2 dvd (k + l)"
063698416239 correct sort constraints for abbreviations in type classes
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   186
  using dvd_add_times_triv_right_iff [of 2 "k - l" l] by (simp add: mult_2_right)
063698416239 correct sort constraints for abbreviations in type classes
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diff changeset
   187
063698416239 correct sort constraints for abbreviations in type classes
haftmann
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diff changeset
   188
lemma even_abs_add_iff [simp]:
063698416239 correct sort constraints for abbreviations in type classes
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   189
  fixes k l :: int
063698416239 correct sort constraints for abbreviations in type classes
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diff changeset
   190
  shows "2 dvd (\<bar>k\<bar> + l) \<longleftrightarrow> 2 dvd (k + l)"
063698416239 correct sort constraints for abbreviations in type classes
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   191
  by (cases "k \<ge> 0") (simp_all add: ac_simps)
063698416239 correct sort constraints for abbreviations in type classes
haftmann
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diff changeset
   192
063698416239 correct sort constraints for abbreviations in type classes
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diff changeset
   193
lemma even_add_abs_iff [simp]:
063698416239 correct sort constraints for abbreviations in type classes
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   194
  fixes k l :: int
063698416239 correct sort constraints for abbreviations in type classes
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diff changeset
   195
  shows "2 dvd (k + \<bar>l\<bar>) \<longleftrightarrow> 2 dvd (k + l)"
063698416239 correct sort constraints for abbreviations in type classes
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   196
  using even_abs_add_iff [of l k] by (simp add: ac_simps)
063698416239 correct sort constraints for abbreviations in type classes
haftmann
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diff changeset
   197
60867
86e7560e07d0 slight cleanup of lemmas
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   198
lemma odd_Suc_minus_one [simp]:
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   199
  "odd n \<Longrightarrow> Suc (n - Suc 0) = n"
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   200
  by (auto elim: oddE)
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diff changeset
   201
58787
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   202
instance int :: ring_parity
af9eb5e566dd eliminated redundancies;
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diff changeset
   203
proof
60343
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diff changeset
   204
  show "\<not> 2 dvd (1 :: int)" by (simp add: dvd_int_unfold_dvd_nat)
58787
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diff changeset
   205
  fix k l :: int
60343
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haftmann
parents: 59816
diff changeset
   206
  assume "\<not> 2 dvd k"
063698416239 correct sort constraints for abbreviations in type classes
haftmann
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diff changeset
   207
  moreover assume "\<not> 2 dvd l"
063698416239 correct sort constraints for abbreviations in type classes
haftmann
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diff changeset
   208
  ultimately have "2 dvd (nat \<bar>k\<bar> + nat \<bar>l\<bar>)" 
58787
af9eb5e566dd eliminated redundancies;
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diff changeset
   209
    by (auto simp add: dvd_int_unfold_dvd_nat intro: odd_even_add)
60343
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haftmann
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diff changeset
   210
  then have "2 dvd (\<bar>k\<bar> + \<bar>l\<bar>)"
58787
af9eb5e566dd eliminated redundancies;
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diff changeset
   211
    by (simp add: dvd_int_unfold_dvd_nat nat_add_distrib)
60343
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diff changeset
   212
  then show "2 dvd (k + l)"
58787
af9eb5e566dd eliminated redundancies;
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diff changeset
   213
    by simp
af9eb5e566dd eliminated redundancies;
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   214
next
af9eb5e566dd eliminated redundancies;
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diff changeset
   215
  fix k l :: int
60343
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haftmann
parents: 59816
diff changeset
   216
  assume "2 dvd (k * l)"
063698416239 correct sort constraints for abbreviations in type classes
haftmann
parents: 59816
diff changeset
   217
  then show "2 dvd k \<or> 2 dvd l"
58787
af9eb5e566dd eliminated redundancies;
haftmann
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diff changeset
   218
    by (simp add: dvd_int_unfold_dvd_nat even_multD nat_abs_mult_distrib)
af9eb5e566dd eliminated redundancies;
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diff changeset
   219
next
af9eb5e566dd eliminated redundancies;
haftmann
parents: 58778
diff changeset
   220
  fix k :: int
af9eb5e566dd eliminated redundancies;
haftmann
parents: 58778
diff changeset
   221
  have "k = (k - 1) + 1" by simp
af9eb5e566dd eliminated redundancies;
haftmann
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diff changeset
   222
  then show "\<exists>l. k = l + 1" ..
af9eb5e566dd eliminated redundancies;
haftmann
parents: 58778
diff changeset
   223
qed
58680
6b2fa479945f more algebraic deductions for facts on even/odd
haftmann
parents: 58679
diff changeset
   224
58787
af9eb5e566dd eliminated redundancies;
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diff changeset
   225
lemma even_int_iff [simp]:
58679
33c90658448a more algebraic deductions for facts on even/odd
haftmann
parents: 58678
diff changeset
   226
  "even (int n) \<longleftrightarrow> even n"
58740
cb9d84d3e7f2 turn even into an abbreviation
haftmann
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diff changeset
   227
  by (simp add: dvd_int_iff)
33318
ddd97d9dfbfb moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 31718
diff changeset
   228
58687
5469874b0228 even more cleanup
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diff changeset
   229
lemma even_nat_iff:
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diff changeset
   230
  "0 \<le> k \<Longrightarrow> even (nat k) \<longleftrightarrow> even k"
5469874b0228 even more cleanup
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diff changeset
   231
  by (simp add: even_int_iff [symmetric])
5469874b0228 even more cleanup
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diff changeset
   232
5469874b0228 even more cleanup
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diff changeset
   233
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
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   234
subsection \<open>Parity and powers\<close>
58689
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   235
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
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   236
context ring_1
58689
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   237
begin
ee5bf401cfa7 tuned facts on even and power
haftmann
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diff changeset
   238
ee5bf401cfa7 tuned facts on even and power
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   239
lemma power_minus_even [simp]:
ee5bf401cfa7 tuned facts on even and power
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diff changeset
   240
  "even n \<Longrightarrow> (- a) ^ n = a ^ n"
58690
5c5c14844738 standard elimination rule for even
haftmann
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diff changeset
   241
  by (auto elim: evenE)
58689
ee5bf401cfa7 tuned facts on even and power
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diff changeset
   242
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   243
lemma power_minus_odd [simp]:
ee5bf401cfa7 tuned facts on even and power
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diff changeset
   244
  "odd n \<Longrightarrow> (- a) ^ n = - (a ^ n)"
58690
5c5c14844738 standard elimination rule for even
haftmann
parents: 58689
diff changeset
   245
  by (auto elim: oddE)
5c5c14844738 standard elimination rule for even
haftmann
parents: 58689
diff changeset
   246
58689
ee5bf401cfa7 tuned facts on even and power
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diff changeset
   247
lemma neg_one_even_power [simp]:
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   248
  "even n \<Longrightarrow> (- 1) ^ n = 1"
58690
5c5c14844738 standard elimination rule for even
haftmann
parents: 58689
diff changeset
   249
  by simp
58689
ee5bf401cfa7 tuned facts on even and power
haftmann
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diff changeset
   250
ee5bf401cfa7 tuned facts on even and power
haftmann
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diff changeset
   251
lemma neg_one_odd_power [simp]:
ee5bf401cfa7 tuned facts on even and power
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parents: 58688
diff changeset
   252
  "odd n \<Longrightarrow> (- 1) ^ n = - 1"
58690
5c5c14844738 standard elimination rule for even
haftmann
parents: 58689
diff changeset
   253
  by simp
58689
ee5bf401cfa7 tuned facts on even and power
haftmann
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diff changeset
   254
ee5bf401cfa7 tuned facts on even and power
haftmann
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diff changeset
   255
end  
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   256
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   257
context linordered_idom
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   258
begin
ee5bf401cfa7 tuned facts on even and power
haftmann
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diff changeset
   259
ee5bf401cfa7 tuned facts on even and power
haftmann
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diff changeset
   260
lemma zero_le_even_power:
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   261
  "even n \<Longrightarrow> 0 \<le> a ^ n"
58690
5c5c14844738 standard elimination rule for even
haftmann
parents: 58689
diff changeset
   262
  by (auto elim: evenE)
58689
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   263
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   264
lemma zero_le_odd_power:
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   265
  "odd n \<Longrightarrow> 0 \<le> a ^ n \<longleftrightarrow> 0 \<le> a"
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   266
  by (auto simp add: power_even_eq zero_le_mult_iff elim: oddE)
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   267
58770
ae5e9b4f8daf downshift of theory Parity in the hierarchy
haftmann
parents: 58769
diff changeset
   268
lemma zero_le_power_eq:
58689
ee5bf401cfa7 tuned facts on even and power
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diff changeset
   269
  "0 \<le> a ^ n \<longleftrightarrow> even n \<or> odd n \<and> 0 \<le> a"
58787
af9eb5e566dd eliminated redundancies;
haftmann
parents: 58778
diff changeset
   270
  by (auto simp add: zero_le_even_power zero_le_odd_power)
af9eb5e566dd eliminated redundancies;
haftmann
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diff changeset
   271
  
58770
ae5e9b4f8daf downshift of theory Parity in the hierarchy
haftmann
parents: 58769
diff changeset
   272
lemma zero_less_power_eq:
58689
ee5bf401cfa7 tuned facts on even and power
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parents: 58688
diff changeset
   273
  "0 < a ^ n \<longleftrightarrow> n = 0 \<or> even n \<and> a \<noteq> 0 \<or> odd n \<and> 0 < a"
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   274
proof -
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   275
  have [simp]: "0 = a ^ n \<longleftrightarrow> a = 0 \<and> n > 0"
58787
af9eb5e566dd eliminated redundancies;
haftmann
parents: 58778
diff changeset
   276
    unfolding power_eq_0_iff [of a n, symmetric] by blast
58689
ee5bf401cfa7 tuned facts on even and power
haftmann
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diff changeset
   277
  show ?thesis
58710
7216a10d69ba augmented and tuned facts on even/odd and division
haftmann
parents: 58709
diff changeset
   278
  unfolding less_le zero_le_power_eq by auto
58689
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   279
qed
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   280
58787
af9eb5e566dd eliminated redundancies;
haftmann
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diff changeset
   281
lemma power_less_zero_eq [simp]:
58689
ee5bf401cfa7 tuned facts on even and power
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parents: 58688
diff changeset
   282
  "a ^ n < 0 \<longleftrightarrow> odd n \<and> a < 0"
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   283
  unfolding not_le [symmetric] zero_le_power_eq by auto
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   284
  
58770
ae5e9b4f8daf downshift of theory Parity in the hierarchy
haftmann
parents: 58769
diff changeset
   285
lemma power_le_zero_eq:
58689
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   286
  "a ^ n \<le> 0 \<longleftrightarrow> n > 0 \<and> (odd n \<and> a \<le> 0 \<or> even n \<and> a = 0)"
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   287
  unfolding not_less [symmetric] zero_less_power_eq by auto 
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   288
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   289
lemma power_even_abs:
ee5bf401cfa7 tuned facts on even and power
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parents: 58688
diff changeset
   290
  "even n \<Longrightarrow> \<bar>a\<bar> ^ n = a ^ n"
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   291
  using power_abs [of a n] by (simp add: zero_le_even_power)
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   292
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   293
lemma power_mono_even:
ee5bf401cfa7 tuned facts on even and power
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parents: 58688
diff changeset
   294
  assumes "even n" and "\<bar>a\<bar> \<le> \<bar>b\<bar>"
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   295
  shows "a ^ n \<le> b ^ n"
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   296
proof -
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   297
  have "0 \<le> \<bar>a\<bar>" by auto
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60562
diff changeset
   298
  with \<open>\<bar>a\<bar> \<le> \<bar>b\<bar>\<close>
58689
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   299
  have "\<bar>a\<bar> ^ n \<le> \<bar>b\<bar> ^ n" by (rule power_mono)
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60562
diff changeset
   300
  with \<open>even n\<close> show ?thesis by (simp add: power_even_abs)  
58689
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   301
qed
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   302
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   303
lemma power_mono_odd:
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   304
  assumes "odd n" and "a \<le> b"
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   305
  shows "a ^ n \<le> b ^ n"
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   306
proof (cases "b < 0")
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60562
diff changeset
   307
  case True with \<open>a \<le> b\<close> have "- b \<le> - a" and "0 \<le> - b" by auto
58689
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   308
  hence "(- b) ^ n \<le> (- a) ^ n" by (rule power_mono)
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60562
diff changeset
   309
  with \<open>odd n\<close> show ?thesis by simp
58689
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   310
next
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   311
  case False then have "0 \<le> b" by auto
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   312
  show ?thesis
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   313
  proof (cases "a < 0")
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60562
diff changeset
   314
    case True then have "n \<noteq> 0" and "a \<le> 0" using \<open>odd n\<close> [THEN odd_pos] by auto
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60562
diff changeset
   315
    then have "a ^ n \<le> 0" unfolding power_le_zero_eq using \<open>odd n\<close> by auto
58689
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   316
    moreover
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60562
diff changeset
   317
    from \<open>0 \<le> b\<close> have "0 \<le> b ^ n" by auto
58689
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   318
    ultimately show ?thesis by auto
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   319
  next
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   320
    case False then have "0 \<le> a" by auto
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60562
diff changeset
   321
    with \<open>a \<le> b\<close> show ?thesis using power_mono by auto
58689
ee5bf401cfa7 tuned facts on even and power
haftmann
parents: 58688
diff changeset
   322
  qed
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qed
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text \<open>Simplify, when the exponent is a numeral\<close>
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lemma zero_le_power_eq_numeral [simp]:
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  "0 \<le> a ^ numeral w \<longleftrightarrow> even (numeral w :: nat) \<or> odd (numeral w :: nat) \<and> 0 \<le> a"
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  by (fact zero_le_power_eq)
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lemma zero_less_power_eq_numeral [simp]:
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  "0 < a ^ numeral w \<longleftrightarrow> numeral w = (0 :: nat)
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    \<or> even (numeral w :: nat) \<and> a \<noteq> 0 \<or> odd (numeral w :: nat) \<and> 0 < a"
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  by (fact zero_less_power_eq)
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lemma power_le_zero_eq_numeral [simp]:
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  "a ^ numeral w \<le> 0 \<longleftrightarrow> (0 :: nat) < numeral w
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    \<and> (odd (numeral w :: nat) \<and> a \<le> 0 \<or> even (numeral w :: nat) \<and> a = 0)"
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  by (fact power_le_zero_eq)
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lemma power_less_zero_eq_numeral [simp]:
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  "a ^ numeral w < 0 \<longleftrightarrow> odd (numeral w :: nat) \<and> a < 0"
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  by (fact power_less_zero_eq)
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lemma power_even_abs_numeral [simp]:
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  "even (numeral w :: nat) \<Longrightarrow> \<bar>a\<bar> ^ numeral w = a ^ numeral w"
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  by (fact power_even_abs)
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end
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subsubsection \<open>Tools setup\<close>
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declare transfer_morphism_int_nat [transfer add return:
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  even_int_iff
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]
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end