(* Title: HOL/Parity.thy
Author: Jeremy Avigad
Author: Jacques D. Fleuriot
*)
header {* Even and Odd for int and nat *}
theory Parity
imports Nat_Transfer
begin
subsection {* Preliminaries about divisibility on @{typ nat} and @{typ int} *}
lemma two_dvd_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 two_dvd_Suc_iff:
"2 dvd Suc n \<longleftrightarrow> \<not> 2 dvd n"
by (induct n) auto
lemma two_dvd_diff_nat_iff:
fixes m n :: nat
shows "2 dvd m - n \<longleftrightarrow> m < n \<or> 2 dvd m + n"
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
lemma two_dvd_diff_iff:
fixes k l :: int
shows "2 dvd k - l \<longleftrightarrow> 2 dvd k + l"
using dvd_add_times_triv_right_iff [of 2 "k - l" l] by (simp add: mult_2_right)
lemma two_dvd_abs_add_iff:
fixes k l :: int
shows "2 dvd \<bar>k\<bar> + l \<longleftrightarrow> 2 dvd k + l"
by (cases "k \<ge> 0") (simp_all add: two_dvd_diff_iff ac_simps)
lemma two_dvd_add_abs_iff:
fixes k l :: int
shows "2 dvd k + \<bar>l\<bar> \<longleftrightarrow> 2 dvd k + l"
using two_dvd_abs_add_iff [of l k] by (simp add: ac_simps)
subsection {* Ring structures with parity *}
class semiring_parity = semiring_dvd + semiring_numeral +
assumes two_not_dvd_one [simp]: "\<not> 2 dvd 1"
assumes not_dvd_not_dvd_dvd_add: "\<not> 2 dvd a \<Longrightarrow> \<not> 2 dvd b \<Longrightarrow> 2 dvd a + b"
assumes two_is_prime: "2 dvd a * b \<Longrightarrow> 2 dvd a \<or> 2 dvd b"
assumes not_dvd_ex_decrement: "\<not> 2 dvd a \<Longrightarrow> \<exists>b. a = b + 1"
begin
lemma two_dvd_plus_one_iff [simp]:
"2 dvd a + 1 \<longleftrightarrow> \<not> 2 dvd a"
by (auto simp add: dvd_add_right_iff intro: not_dvd_not_dvd_dvd_add)
lemma not_two_dvdE [elim?]:
assumes "\<not> 2 dvd a"
obtains b where "a = 2 * b + 1"
proof -
from assms obtain b where *: "a = b + 1"
by (blast dest: not_dvd_ex_decrement)
with assms have "2 dvd b + 2" by simp
then have "2 dvd b" by simp
then obtain c where "b = 2 * c" ..
with * have "a = 2 * c + 1" by simp
with that show thesis .
qed
end
instance nat :: semiring_parity
proof
show "\<not> (2 :: nat) dvd 1"
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 add: two_dvd_Suc_iff)
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 add: two_dvd_Suc_iff)
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
class ring_parity = comm_ring_1 + semiring_parity
instance int :: ring_parity
proof
show "\<not> (2 :: int) dvd 1" by (simp add: dvd_int_unfold_dvd_nat)
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: not_dvd_not_dvd_dvd_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 add: two_dvd_abs_add_iff two_dvd_add_abs_iff)
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 two_is_prime nat_abs_mult_distrib)
next
fix k :: int
have "k = (k - 1) + 1" by simp
then show "\<exists>l. k = l + 1" ..
qed
subsection {* Dedicated @{text even}/@{text odd} predicate *}
subsubsection {* Properties *}
context semiring_parity
begin
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 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"
using assms by (rule not_two_dvdE)
lemma even_times_iff [simp]:
"even (a * b) \<longleftrightarrow> even a \<or> even b"
by (auto simp add: dest: two_is_prime)
lemma even_zero [simp]:
"even 0"
by simp
lemma odd_one [simp]:
"odd 1"
by simp
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)
with dvd_add_times_triv_left_iff [of 2 "numeral n" 1]
have "2 dvd 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 not_dvd_not_dvd_dvd_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
context ring_parity
begin
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
subsubsection {* Particularities for @{typ nat} and @{typ int} *}
lemma even_Suc [simp]:
"even (Suc n) = odd n"
by (fact two_dvd_Suc_iff)
lemma odd_pos:
"odd (n :: nat) \<Longrightarrow> 0 < n"
by (auto elim: oddE)
lemma even_diff_nat [simp]:
fixes m n :: nat
shows "even (m - n) \<longleftrightarrow> m < n \<or> even (m + n)"
by (fact two_dvd_diff_nat_iff)
lemma even_int_iff:
"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])
lemma even_num_iff:
"0 < n \<Longrightarrow> even n = odd (n - 1 :: nat)"
by simp
text {* Parity and powers *}
context comm_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_power_if:
"(- a) ^ n = (if even n then a ^ n else - (a ^ n))"
by simp
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
lemma zero_less_power_nat_eq_numeral [simp]: -- \<open>FIXME move\<close>
"0 < (n :: nat) ^ numeral w \<longleftrightarrow> 0 < n \<or> numeral w = (0 :: nat)"
by (fact nat_zero_less_power_iff)
context linordered_idom
begin
lemma power_eq_0_iff' [simp]: -- \<open>FIXME move\<close>
"a ^ n = 0 \<longleftrightarrow> a = 0 \<and> n > 0"
by (induct n) auto
lemma power2_less_eq_zero_iff [simp]: -- \<open>FIXME move\<close>
"a\<^sup>2 \<le> 0 \<longleftrightarrow> a = 0"
proof (cases "a = 0")
case True then show ?thesis by simp
next
case False then have "a < 0 \<or> a > 0" by auto
then have "a\<^sup>2 > 0" by auto
then have "\<not> a\<^sup>2 \<le> 0" by (simp add: not_le)
with False show ?thesis by simp
qed
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_iff: -- \<open>FIXME cf. @{text zero_le_power_eq}\<close>
"0 \<le> a ^ n \<longleftrightarrow> 0 \<le> a \<or> even n"
proof (cases "even n")
case True
then obtain k where "n = 2 * k" ..
then show ?thesis by simp
next
case False
then obtain k where "n = 2 * k + 1" ..
moreover have "a ^ (2 * k) \<le> 0 \<Longrightarrow> a = 0"
by (induct k) (auto simp add: zero_le_mult_iff mult_le_0_iff)
ultimately show ?thesis
by (auto simp add: zero_le_mult_iff zero_le_even_power)
qed
lemma zero_le_power_eq:
"0 \<le> a ^ n \<longleftrightarrow> even n \<or> odd n \<and> 0 \<le> a"
using zero_le_power_iff [of a n] by auto
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:
"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 `\<bar>a\<bar> \<le> \<bar>b\<bar>`
have "\<bar>a\<bar> ^ n \<le> \<bar>b\<bar> ^ n" by (rule power_mono)
with `even n` 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 `a \<le> b` have "- b \<le> - a" and "0 \<le> - b" by auto
hence "(- b) ^ n \<le> (- a) ^ n" by (rule power_mono)
with `odd n` 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 `odd n` [THEN odd_pos] by auto
then have "a ^ n \<le> 0" unfolding power_le_zero_eq using `odd n` by auto
moreover
from `0 \<le> b` have "0 \<le> b ^ n" by auto
ultimately show ?thesis by auto
next
case False then have "0 \<le> a" by auto
with `a \<le> b` show ?thesis using power_mono by auto
qed
qed
text {* Simplify, when the exponent is a numeral *}
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_eq_0_iff_numeral [simp]:
"a ^ numeral w = (0 :: nat) \<longleftrightarrow> a = 0 \<and> numeral w \<noteq> (0 :: nat)"
by (fact power_eq_0_iff)
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 {* Tools setup *}
declare transfer_morphism_int_nat [transfer add return:
even_int_iff
]
end