(* Title: HOL/Parity.thy
Author: Jeremy Avigad
Author: Jacques D. Fleuriot
*)
section \<open>Parity in rings and semirings\<close>
theory Parity
imports Nat_Transfer
begin
subsection \<open>Ring structures with parity and \<open>even\<close>/\<open>odd\<close> predicates\<close>
class semiring_parity = comm_semiring_1_cancel + numeral +
assumes odd_one [simp]: "\<not> 2 dvd 1"
assumes odd_even_add: "\<not> 2 dvd a \<Longrightarrow> \<not> 2 dvd b \<Longrightarrow> 2 dvd a + b"
assumes even_multD: "2 dvd a * b \<Longrightarrow> 2 dvd a \<or> 2 dvd b"
assumes odd_ex_decrement: "\<not> 2 dvd a \<Longrightarrow> \<exists>b. a = b + 1"
begin
subclass semiring_numeral ..
abbreviation even :: "'a \<Rightarrow> bool"
where "even a \<equiv> 2 dvd a"
abbreviation odd :: "'a \<Rightarrow> bool"
where "odd a \<equiv> \<not> 2 dvd a"
lemma even_zero [simp]: "even 0"
by (fact dvd_0_right)
lemma even_plus_one_iff [simp]: "even (a + 1) \<longleftrightarrow> odd a"
by (auto simp add: dvd_add_right_iff intro: odd_even_add)
lemma evenE [elim?]:
assumes "even a"
obtains b where "a = 2 * b"
using assms by (rule dvdE)
lemma oddE [elim?]:
assumes "odd a"
obtains b where "a = 2 * b + 1"
proof -
from assms obtain b where *: "a = b + 1"
by (blast dest: odd_ex_decrement)
with assms have "even (b + 2)" by simp
then have "even b" by simp
then obtain c where "b = 2 * c" ..
with * have "a = 2 * c + 1" by simp
with that show thesis .
qed
lemma even_times_iff [simp]: "even (a * b) \<longleftrightarrow> even a \<or> even b"
by (auto dest: even_multD)
lemma even_numeral [simp]: "even (numeral (Num.Bit0 n))"
proof -
have "even (2 * numeral n)"
unfolding even_times_iff by simp
then have "even (numeral n + numeral n)"
unfolding mult_2 .
then show ?thesis
unfolding numeral.simps .
qed
lemma odd_numeral [simp]: "odd (numeral (Num.Bit1 n))"
proof
assume "even (numeral (num.Bit1 n))"
then have "even (numeral n + numeral n + 1)"
unfolding numeral.simps .
then have "even (2 * numeral n + 1)"
unfolding mult_2 .
then have "2 dvd numeral n * 2 + 1"
by (simp add: ac_simps)
then have "2 dvd 1"
using dvd_add_times_triv_left_iff [of 2 "numeral n" 1] by simp
then show False by simp
qed
lemma even_add [simp]: "even (a + b) \<longleftrightarrow> (even a \<longleftrightarrow> even b)"
by (auto simp add: dvd_add_right_iff dvd_add_left_iff odd_even_add)
lemma odd_add [simp]: "odd (a + b) \<longleftrightarrow> (\<not> (odd a \<longleftrightarrow> odd b))"
by simp
lemma even_power [simp]: "even (a ^ n) \<longleftrightarrow> even a \<and> n > 0"
by (induct n) auto
end
class ring_parity = ring + semiring_parity
begin
subclass comm_ring_1 ..
lemma even_minus [simp]: "even (- a) \<longleftrightarrow> even a"
by (fact dvd_minus_iff)
lemma even_diff [simp]: "even (a - b) \<longleftrightarrow> even (a + b)"
using even_add [of a "- b"] by simp
end
subsection \<open>Instances for @{typ nat} and @{typ int}\<close>
lemma even_Suc_Suc_iff [simp]: "2 dvd Suc (Suc n) \<longleftrightarrow> 2 dvd n"
using dvd_add_triv_right_iff [of 2 n] by simp
lemma even_Suc [simp]: "2 dvd Suc n \<longleftrightarrow> \<not> 2 dvd n"
by (induct n) auto
lemma even_diff_nat [simp]: "2 dvd (m - n) \<longleftrightarrow> m < n \<or> 2 dvd (m + n)"
for m n :: nat
proof (cases "n \<le> m")
case True
then have "m - n + n * 2 = m + n" by (simp add: mult_2_right)
moreover have "2 dvd (m - n) \<longleftrightarrow> 2 dvd (m - n + n * 2)" by simp
ultimately have "2 dvd (m - n) \<longleftrightarrow> 2 dvd (m + n)" by (simp only:)
then show ?thesis by auto
next
case False
then show ?thesis by simp
qed
instance nat :: semiring_parity
proof
show "\<not> 2 dvd (1 :: nat)"
by (rule notI, erule dvdE) simp
next
fix m n :: nat
assume "\<not> 2 dvd m"
moreover assume "\<not> 2 dvd n"
ultimately have *: "2 dvd Suc m \<and> 2 dvd Suc n"
by simp
then have "2 dvd (Suc m + Suc n)"
by (blast intro: dvd_add)
also have "Suc m + Suc n = m + n + 2"
by simp
finally show "2 dvd (m + n)"
using dvd_add_triv_right_iff [of 2 "m + n"] by simp
next
fix m n :: nat
assume *: "2 dvd (m * n)"
show "2 dvd m \<or> 2 dvd n"
proof (rule disjCI)
assume "\<not> 2 dvd n"
then have "2 dvd (Suc n)" by simp
then obtain r where "Suc n = 2 * r" ..
moreover from * obtain s where "m * n = 2 * s" ..
then have "2 * s + m = m * Suc n" by simp
ultimately have " 2 * s + m = 2 * (m * r)"
by (simp add: algebra_simps)
then have "m = 2 * (m * r - s)" by simp
then show "2 dvd m" ..
qed
next
fix n :: nat
assume "\<not> 2 dvd n"
then show "\<exists>m. n = m + 1"
by (cases n) simp_all
qed
lemma odd_pos: "odd n \<Longrightarrow> 0 < n"
for n :: nat
by (auto elim: oddE)
lemma Suc_double_not_eq_double: "Suc (2 * m) \<noteq> 2 * n"
for m n :: nat
proof
assume "Suc (2 * m) = 2 * n"
moreover have "odd (Suc (2 * m))" and "even (2 * n)"
by simp_all
ultimately show False by simp
qed
lemma double_not_eq_Suc_double: "2 * m \<noteq> Suc (2 * n)"
for m n :: nat
using Suc_double_not_eq_double [of n m] by simp
lemma even_diff_iff [simp]: "2 dvd (k - l) \<longleftrightarrow> 2 dvd (k + l)"
for k l :: int
using dvd_add_times_triv_right_iff [of 2 "k - l" l] by (simp add: mult_2_right)
lemma even_abs_add_iff [simp]: "2 dvd (\<bar>k\<bar> + l) \<longleftrightarrow> 2 dvd (k + l)"
for k l :: int
by (cases "k \<ge> 0") (simp_all add: ac_simps)
lemma even_add_abs_iff [simp]: "2 dvd (k + \<bar>l\<bar>) \<longleftrightarrow> 2 dvd (k + l)"
for k l :: int
using even_abs_add_iff [of l k] by (simp add: ac_simps)
lemma odd_Suc_minus_one [simp]: "odd n \<Longrightarrow> Suc (n - Suc 0) = n"
by (auto elim: oddE)
instance int :: ring_parity
proof
show "\<not> 2 dvd (1 :: int)"
by (simp add: dvd_int_unfold_dvd_nat)
next
fix k l :: int
assume "\<not> 2 dvd k"
moreover assume "\<not> 2 dvd l"
ultimately have "2 dvd (nat \<bar>k\<bar> + nat \<bar>l\<bar>)"
by (auto simp add: dvd_int_unfold_dvd_nat intro: odd_even_add)
then have "2 dvd (\<bar>k\<bar> + \<bar>l\<bar>)"
by (simp add: dvd_int_unfold_dvd_nat nat_add_distrib)
then show "2 dvd (k + l)"
by simp
next
fix k l :: int
assume "2 dvd (k * l)"
then show "2 dvd k \<or> 2 dvd l"
by (simp add: dvd_int_unfold_dvd_nat even_multD nat_abs_mult_distrib)
next
fix k :: int
have "k = (k - 1) + 1" by simp
then show "\<exists>l. k = l + 1" ..
qed
lemma even_int_iff [simp]: "even (int n) \<longleftrightarrow> even n"
by (simp add: dvd_int_iff)
lemma even_nat_iff: "0 \<le> k \<Longrightarrow> even (nat k) \<longleftrightarrow> even k"
by (simp add: even_int_iff [symmetric])
subsection \<open>Parity and powers\<close>
context ring_1
begin
lemma power_minus_even [simp]: "even n \<Longrightarrow> (- a) ^ n = a ^ n"
by (auto elim: evenE)
lemma power_minus_odd [simp]: "odd n \<Longrightarrow> (- a) ^ n = - (a ^ n)"
by (auto elim: oddE)
lemma neg_one_even_power [simp]: "even n \<Longrightarrow> (- 1) ^ n = 1"
by simp
lemma neg_one_odd_power [simp]: "odd n \<Longrightarrow> (- 1) ^ n = - 1"
by simp
end
context linordered_idom
begin
lemma zero_le_even_power: "even n \<Longrightarrow> 0 \<le> a ^ n"
by (auto elim: evenE)
lemma zero_le_odd_power: "odd n \<Longrightarrow> 0 \<le> a ^ n \<longleftrightarrow> 0 \<le> a"
by (auto simp add: power_even_eq zero_le_mult_iff elim: oddE)
lemma zero_le_power_eq: "0 \<le> a ^ n \<longleftrightarrow> even n \<or> odd n \<and> 0 \<le> a"
by (auto simp add: zero_le_even_power zero_le_odd_power)
lemma zero_less_power_eq: "0 < a ^ n \<longleftrightarrow> n = 0 \<or> even n \<and> a \<noteq> 0 \<or> odd n \<and> 0 < a"
proof -
have [simp]: "0 = a ^ n \<longleftrightarrow> a = 0 \<and> n > 0"
unfolding power_eq_0_iff [of a n, symmetric] by blast
show ?thesis
unfolding less_le zero_le_power_eq by auto
qed
lemma power_less_zero_eq [simp]: "a ^ n < 0 \<longleftrightarrow> odd n \<and> a < 0"
unfolding not_le [symmetric] zero_le_power_eq by auto
lemma power_le_zero_eq: "a ^ n \<le> 0 \<longleftrightarrow> n > 0 \<and> (odd n \<and> a \<le> 0 \<or> even n \<and> a = 0)"
unfolding not_less [symmetric] zero_less_power_eq by auto
lemma power_even_abs: "even n \<Longrightarrow> \<bar>a\<bar> ^ n = a ^ n"
using power_abs [of a n] by (simp add: zero_le_even_power)
lemma power_mono_even:
assumes "even n" and "\<bar>a\<bar> \<le> \<bar>b\<bar>"
shows "a ^ n \<le> b ^ n"
proof -
have "0 \<le> \<bar>a\<bar>" by auto
with \<open>\<bar>a\<bar> \<le> \<bar>b\<bar>\<close> have "\<bar>a\<bar> ^ n \<le> \<bar>b\<bar> ^ n"
by (rule power_mono)
with \<open>even n\<close> show ?thesis
by (simp add: power_even_abs)
qed
lemma power_mono_odd:
assumes "odd n" and "a \<le> b"
shows "a ^ n \<le> b ^ n"
proof (cases "b < 0")
case True
with \<open>a \<le> b\<close> have "- b \<le> - a" and "0 \<le> - b" by auto
then have "(- b) ^ n \<le> (- a) ^ n" by (rule power_mono)
with \<open>odd n\<close> show ?thesis by simp
next
case False
then have "0 \<le> b" by auto
show ?thesis
proof (cases "a < 0")
case True
then have "n \<noteq> 0" and "a \<le> 0" using \<open>odd n\<close> [THEN odd_pos] by auto
then have "a ^ n \<le> 0" unfolding power_le_zero_eq using \<open>odd n\<close> by auto
moreover from \<open>0 \<le> b\<close> have "0 \<le> b ^ n" by auto
ultimately show ?thesis by auto
next
case False
then have "0 \<le> a" by auto
with \<open>a \<le> b\<close> show ?thesis
using power_mono by auto
qed
qed
lemma (in comm_ring_1) uminus_power_if: "(- x) ^ n = (if even n then x^n else - (x ^ n))"
by auto
text \<open>Simplify, when the exponent is a numeral\<close>
lemma zero_le_power_eq_numeral [simp]:
"0 \<le> a ^ numeral w \<longleftrightarrow> even (numeral w :: nat) \<or> odd (numeral w :: nat) \<and> 0 \<le> a"
by (fact zero_le_power_eq)
lemma zero_less_power_eq_numeral [simp]:
"0 < a ^ numeral w \<longleftrightarrow>
numeral w = (0 :: nat) \<or>
even (numeral w :: nat) \<and> a \<noteq> 0 \<or>
odd (numeral w :: nat) \<and> 0 < a"
by (fact zero_less_power_eq)
lemma power_le_zero_eq_numeral [simp]:
"a ^ numeral w \<le> 0 \<longleftrightarrow>
(0 :: nat) < numeral w \<and>
(odd (numeral w :: nat) \<and> a \<le> 0 \<or> even (numeral w :: nat) \<and> a = 0)"
by (fact power_le_zero_eq)
lemma power_less_zero_eq_numeral [simp]:
"a ^ numeral w < 0 \<longleftrightarrow> odd (numeral w :: nat) \<and> a < 0"
by (fact power_less_zero_eq)
lemma power_even_abs_numeral [simp]:
"even (numeral w :: nat) \<Longrightarrow> \<bar>a\<bar> ^ numeral w = a ^ numeral w"
by (fact power_even_abs)
end
subsubsection \<open>Tool setup\<close>
declare transfer_morphism_int_nat [transfer add return: even_int_iff]
end