author | haftmann |
Tue, 28 Apr 2009 15:50:30 +0200 | |
changeset 31017 | 2c227493ea56 |
parent 30738 | 0842e906300c |
child 31148 | 7ba7c1f8bc22 |
permissions | -rw-r--r-- |
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(* Title: HOL/Library/Parity.thy |
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Author: Jeremy Avigad, 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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class even_odd = |
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fixes even :: "'a \<Rightarrow> bool" |
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abbreviation |
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odd :: "'a\<Colon>even_odd \<Rightarrow> bool" where |
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"odd x \<equiv> \<not> even x" |
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instantiation nat and int :: even_odd |
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begin |
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definition |
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even_def [presburger]: "even x \<longleftrightarrow> (x\<Colon>int) mod 2 = 0" |
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definition |
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even_nat_def [presburger]: "even x \<longleftrightarrow> even (int x)" |
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instance .. |
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end |
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subsection {* Even and odd are mutually exclusive *} |
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lemma int_pos_lt_two_imp_zero_or_one: |
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"0 <= x ==> (x::int) < 2 ==> x = 0 | x = 1" |
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by presburger |
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|
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lemma neq_one_mod_two [simp, presburger]: |
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"((x::int) mod 2 ~= 0) = (x mod 2 = 1)" by presburger |
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subsection {* Behavior under integer arithmetic operations *} |
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declare dvd_def[algebra] |
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lemma nat_even_iff_2_dvd[algebra]: "even (x::nat) \<longleftrightarrow> 2 dvd x" |
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by (presburger add: even_nat_def even_def) |
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lemma int_even_iff_2_dvd[algebra]: "even (x::int) \<longleftrightarrow> 2 dvd x" |
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by presburger |
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lemma even_times_anything: "even (x::int) ==> even (x * y)" |
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by algebra |
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lemma anything_times_even: "even (y::int) ==> even (x * y)" by algebra |
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lemma odd_times_odd: "odd (x::int) ==> odd y ==> odd (x * y)" |
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by (simp add: even_def zmod_zmult1_eq) |
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lemma even_product[presburger]: "even((x::int) * y) = (even x | even y)" |
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apply (auto simp add: even_times_anything anything_times_even) |
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apply (rule ccontr) |
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apply (auto simp add: odd_times_odd) |
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done |
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lemma even_plus_even: "even (x::int) ==> even y ==> even (x + y)" |
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by presburger |
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lemma even_plus_odd: "even (x::int) ==> odd y ==> odd (x + y)" |
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by presburger |
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lemma odd_plus_even: "odd (x::int) ==> even y ==> odd (x + y)" |
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by presburger |
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lemma odd_plus_odd: "odd (x::int) ==> odd y ==> even (x + y)" by presburger |
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lemma even_sum[presburger]: "even ((x::int) + y) = ((even x & even y) | (odd x & odd y))" |
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by presburger |
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lemma even_neg[presburger, algebra]: "even (-(x::int)) = even x" by presburger |
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lemma even_difference: |
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"even ((x::int) - y) = ((even x & even y) | (odd x & odd y))" by presburger |
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lemma even_pow_gt_zero: |
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"even (x::int) ==> 0 < n ==> even (x^n)" |
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by (induct n) (auto simp add: even_product) |
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lemma odd_pow_iff[presburger, algebra]: |
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"odd ((x::int) ^ n) \<longleftrightarrow> (n = 0 \<or> odd x)" |
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apply (induct n, simp_all) |
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apply presburger |
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apply (case_tac n, auto) |
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apply (simp_all add: even_product) |
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done |
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lemma odd_pow: "odd x ==> odd((x::int)^n)" by (simp add: odd_pow_iff) |
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lemma even_power[presburger]: "even ((x::int)^n) = (even x & 0 < n)" |
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apply (auto simp add: even_pow_gt_zero) |
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apply (erule contrapos_pp, erule odd_pow) |
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apply (erule contrapos_pp, simp add: even_def) |
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done |
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lemma even_zero[presburger]: "even (0::int)" by presburger |
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lemma odd_one[presburger]: "odd (1::int)" by presburger |
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lemmas even_odd_simps [simp] = even_def[of "number_of v",standard] even_zero |
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odd_one even_product even_sum even_neg even_difference even_power |
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subsection {* Equivalent definitions *} |
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lemma two_times_even_div_two: "even (x::int) ==> 2 * (x div 2) = x" |
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by presburger |
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lemma two_times_odd_div_two_plus_one: "odd (x::int) ==> |
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2 * (x div 2) + 1 = x" by presburger |
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lemma even_equiv_def: "even (x::int) = (EX y. x = 2 * y)" by presburger |
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lemma odd_equiv_def: "odd (x::int) = (EX y. x = 2 * y + 1)" by presburger |
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subsection {* even and odd for nats *} |
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lemma pos_int_even_equiv_nat_even: "0 \<le> x ==> even x = even (nat x)" |
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by (simp add: even_nat_def) |
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lemma even_nat_product[presburger, algebra]: "even((x::nat) * y) = (even x | even y)" |
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by (simp add: even_nat_def int_mult) |
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lemma even_nat_sum[presburger, algebra]: "even ((x::nat) + y) = |
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((even x & even y) | (odd x & odd y))" by presburger |
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lemma even_nat_difference[presburger, algebra]: |
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"even ((x::nat) - y) = (x < y | (even x & even y) | (odd x & odd y))" |
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by presburger |
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lemma even_nat_Suc[presburger, algebra]: "even (Suc x) = odd x" by presburger |
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lemma even_nat_power[presburger, algebra]: "even ((x::nat)^y) = (even x & 0 < y)" |
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by (simp add: even_nat_def int_power) |
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lemma even_nat_zero[presburger]: "even (0::nat)" by presburger |
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lemmas even_odd_nat_simps [simp] = even_nat_def[of "number_of v",standard] |
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even_nat_zero even_nat_Suc even_nat_product even_nat_sum even_nat_power |
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subsection {* Equivalent definitions *} |
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lemma nat_lt_two_imp_zero_or_one: "(x::nat) < Suc (Suc 0) ==> |
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x = 0 | x = Suc 0" by presburger |
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lemma even_nat_mod_two_eq_zero: "even (x::nat) ==> x mod (Suc (Suc 0)) = 0" |
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by presburger |
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lemma odd_nat_mod_two_eq_one: "odd (x::nat) ==> x mod (Suc (Suc 0)) = Suc 0" |
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by presburger |
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lemma even_nat_equiv_def: "even (x::nat) = (x mod Suc (Suc 0) = 0)" |
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by presburger |
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lemma odd_nat_equiv_def: "odd (x::nat) = (x mod Suc (Suc 0) = Suc 0)" |
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by presburger |
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lemma even_nat_div_two_times_two: "even (x::nat) ==> |
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Suc (Suc 0) * (x div Suc (Suc 0)) = x" by presburger |
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lemma odd_nat_div_two_times_two_plus_one: "odd (x::nat) ==> |
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Suc( Suc (Suc 0) * (x div Suc (Suc 0))) = x" by presburger |
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lemma even_nat_equiv_def2: "even (x::nat) = (EX y. x = Suc (Suc 0) * y)" |
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by presburger |
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lemma odd_nat_equiv_def2: "odd (x::nat) = (EX y. x = Suc(Suc (Suc 0) * y))" |
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by presburger |
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subsection {* Parity and powers *} |
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lemma minus_one_even_odd_power: |
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"(even x --> (- 1::'a::{comm_ring_1})^x = 1) & |
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(odd x --> (- 1::'a)^x = - 1)" |
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apply (induct x) |
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apply (rule conjI) |
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apply simp |
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apply (insert even_nat_zero, blast) |
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apply (simp add: power_Suc) |
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done |
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lemma minus_one_even_power [simp]: |
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"even x ==> (- 1::'a::{comm_ring_1})^x = 1" |
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using minus_one_even_odd_power by blast |
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lemma minus_one_odd_power [simp]: |
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"odd x ==> (- 1::'a::{comm_ring_1})^x = - 1" |
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using minus_one_even_odd_power by blast |
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lemma neg_one_even_odd_power: |
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"(even x --> (-1::'a::{number_ring})^x = 1) & |
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(odd x --> (-1::'a)^x = -1)" |
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apply (induct x) |
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apply (simp, simp add: power_Suc) |
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done |
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lemma neg_one_even_power [simp]: |
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"even x ==> (-1::'a::{number_ring})^x = 1" |
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using neg_one_even_odd_power by blast |
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lemma neg_one_odd_power [simp]: |
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"odd x ==> (-1::'a::{number_ring})^x = -1" |
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using neg_one_even_odd_power by blast |
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lemma neg_power_if: |
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"(-x::'a::{comm_ring_1}) ^ n = |
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(if even n then (x ^ n) else -(x ^ n))" |
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apply (induct n) |
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apply (simp_all split: split_if_asm add: power_Suc) |
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done |
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lemma zero_le_even_power: "even n ==> |
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0 <= (x::'a::{ordered_ring_strict,monoid_mult}) ^ n" |
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apply (simp add: even_nat_equiv_def2) |
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apply (erule exE) |
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apply (erule ssubst) |
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apply (subst power_add) |
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apply (rule zero_le_square) |
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done |
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lemma zero_le_odd_power: "odd n ==> |
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(0 <= (x::'a::{ordered_idom}) ^ n) = (0 <= x)" |
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apply (auto simp: odd_nat_equiv_def2 power_Suc power_add zero_le_mult_iff) |
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apply (metis field_power_not_zero no_zero_divirors_neq0 order_antisym_conv zero_le_square) |
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done |
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lemma zero_le_power_eq[presburger]: "(0 <= (x::'a::{ordered_idom}) ^ n) = |
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(even n | (odd n & 0 <= x))" |
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apply auto |
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apply (subst zero_le_odd_power [symmetric]) |
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apply assumption+ |
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apply (erule zero_le_even_power) |
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done |
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lemma zero_less_power_eq[presburger]: "(0 < (x::'a::{ordered_idom}) ^ n) = |
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(n = 0 | (even n & x ~= 0) | (odd n & 0 < x))" |
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unfolding order_less_le zero_le_power_eq by auto |
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lemma power_less_zero_eq[presburger]: "((x::'a::{ordered_idom}) ^ n < 0) = |
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(odd n & x < 0)" |
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apply (subst linorder_not_le [symmetric])+ |
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apply (subst zero_le_power_eq) |
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apply auto |
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done |
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lemma power_le_zero_eq[presburger]: "((x::'a::{ordered_idom}) ^ n <= 0) = |
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(n ~= 0 & ((odd n & x <= 0) | (even n & x = 0)))" |
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apply (subst linorder_not_less [symmetric])+ |
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apply (subst zero_less_power_eq) |
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apply auto |
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done |
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lemma power_even_abs: "even n ==> |
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(abs (x::'a::{ordered_idom}))^n = x^n" |
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apply (subst power_abs [symmetric]) |
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apply (simp add: zero_le_even_power) |
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done |
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lemma zero_less_power_nat_eq[presburger]: "(0 < (x::nat) ^ n) = (n = 0 | 0 < x)" |
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by (induct n) auto |
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lemma power_minus_even [simp]: "even n ==> |
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(- x)^n = (x^n::'a::{comm_ring_1})" |
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apply (subst power_minus) |
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apply simp |
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done |
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lemma power_minus_odd [simp]: "odd n ==> |
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(- x)^n = - (x^n::'a::{comm_ring_1})" |
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apply (subst power_minus) |
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apply simp |
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done |
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lemma power_mono_even: fixes x y :: "'a :: {ordered_idom}" |
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assumes "even n" and "\<bar>x\<bar> \<le> \<bar>y\<bar>" |
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shows "x^n \<le> y^n" |
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proof - |
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have "0 \<le> \<bar>x\<bar>" by auto |
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with `\<bar>x\<bar> \<le> \<bar>y\<bar>` |
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have "\<bar>x\<bar>^n \<le> \<bar>y\<bar>^n" by (rule power_mono) |
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290 |
thus ?thesis unfolding power_even_abs[OF `even n`] . |
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291 |
qed |
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292 |
|
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Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
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293 |
lemma odd_pos: "odd (n::nat) \<Longrightarrow> 0 < n" by presburger |
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294 |
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31017 | 295 |
lemma power_mono_odd: fixes x y :: "'a :: {ordered_idom}" |
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296 |
assumes "odd n" and "x \<le> y" |
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shows "x^n \<le> y^n" |
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proof (cases "y < 0") |
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case True with `x \<le> y` have "-y \<le> -x" and "0 \<le> -y" by auto |
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Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
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hence "(-y)^n \<le> (-x)^n" by (rule power_mono) |
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Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
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thus ?thesis unfolding power_minus_odd[OF `odd n`] by auto |
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next |
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case False |
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304 |
show ?thesis |
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305 |
proof (cases "x < 0") |
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306 |
case True hence "n \<noteq> 0" and "x \<le> 0" using `odd n`[THEN odd_pos] by auto |
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307 |
hence "x^n \<le> 0" unfolding power_le_zero_eq using `odd n` by auto |
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308 |
moreover |
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309 |
from `\<not> y < 0` have "0 \<le> y" by auto |
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310 |
hence "0 \<le> y^n" by auto |
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311 |
ultimately show ?thesis by auto |
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312 |
next |
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313 |
case False hence "0 \<le> x" by auto |
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314 |
with `x \<le> y` show ?thesis using power_mono by auto |
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315 |
qed |
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316 |
qed |
21263 | 317 |
|
25600 | 318 |
subsection {* General Lemmas About Division *} |
319 |
||
320 |
lemma Suc_times_mod_eq: "1<k ==> Suc (k * m) mod k = 1" |
|
321 |
apply (induct "m") |
|
322 |
apply (simp_all add: mod_Suc) |
|
323 |
done |
|
324 |
||
325 |
declare Suc_times_mod_eq [of "number_of w", standard, simp] |
|
326 |
||
327 |
lemma [simp]: "n div k \<le> (Suc n) div k" |
|
328 |
by (simp add: div_le_mono) |
|
329 |
||
330 |
lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2" |
|
331 |
by arith |
|
332 |
||
333 |
lemma div_2_gt_zero [simp]: "(1::nat) < n ==> 0 < n div 2" |
|
334 |
by arith |
|
335 |
||
27668 | 336 |
(* Potential use of algebra : Equality modulo n*) |
25600 | 337 |
lemma mod_mult_self3 [simp]: "(k*n + m) mod n = m mod (n::nat)" |
338 |
by (simp add: mult_ac add_ac) |
|
339 |
||
340 |
lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n" |
|
341 |
proof - |
|
342 |
have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp |
|
343 |
also have "... = Suc m mod n" by (rule mod_mult_self3) |
|
344 |
finally show ?thesis . |
|
345 |
qed |
|
346 |
||
347 |
lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n" |
|
348 |
apply (subst mod_Suc [of m]) |
|
349 |
apply (subst mod_Suc [of "m mod n"], simp) |
|
350 |
done |
|
351 |
||
352 |
||
353 |
subsection {* More Even/Odd Results *} |
|
354 |
||
27668 | 355 |
lemma even_mult_two_ex: "even(n) = (\<exists>m::nat. n = 2*m)" by presburger |
356 |
lemma odd_Suc_mult_two_ex: "odd(n) = (\<exists>m. n = Suc (2*m))" by presburger |
|
357 |
lemma even_add [simp]: "even(m + n::nat) = (even m = even n)" by presburger |
|
25600 | 358 |
|
27668 | 359 |
lemma odd_add [simp]: "odd(m + n::nat) = (odd m \<noteq> odd n)" by presburger |
25600 | 360 |
|
361 |
lemma div_Suc: "Suc a div c = a div c + Suc 0 div c + |
|
362 |
(a mod c + Suc 0 mod c) div c" |
|
363 |
apply (subgoal_tac "Suc a = a + Suc 0") |
|
364 |
apply (erule ssubst) |
|
365 |
apply (rule div_add1_eq, simp) |
|
366 |
done |
|
367 |
||
27668 | 368 |
lemma lemma_even_div2 [simp]: "even (n::nat) ==> (n + 1) div 2 = n div 2" by presburger |
25600 | 369 |
|
370 |
lemma lemma_not_even_div2 [simp]: "~even n ==> (n + 1) div 2 = Suc (n div 2)" |
|
27668 | 371 |
by presburger |
25600 | 372 |
|
27668 | 373 |
lemma even_num_iff: "0 < n ==> even n = (~ even(n - 1 :: nat))" by presburger |
374 |
lemma even_even_mod_4_iff: "even (n::nat) = even (n mod 4)" by presburger |
|
25600 | 375 |
|
27668 | 376 |
lemma lemma_odd_mod_4_div_2: "n mod 4 = (3::nat) ==> odd((n - 1) div 2)" by presburger |
25600 | 377 |
|
378 |
lemma lemma_even_mod_4_div_2: "n mod 4 = (1::nat) ==> even ((n - 1) div 2)" |
|
27668 | 379 |
by presburger |
25600 | 380 |
|
21263 | 381 |
text {* Simplify, when the exponent is a numeral *} |
21256 | 382 |
|
383 |
lemmas power_0_left_number_of = power_0_left [of "number_of w", standard] |
|
384 |
declare power_0_left_number_of [simp] |
|
385 |
||
21263 | 386 |
lemmas zero_le_power_eq_number_of [simp] = |
21256 | 387 |
zero_le_power_eq [of _ "number_of w", standard] |
388 |
||
21263 | 389 |
lemmas zero_less_power_eq_number_of [simp] = |
21256 | 390 |
zero_less_power_eq [of _ "number_of w", standard] |
391 |
||
21263 | 392 |
lemmas power_le_zero_eq_number_of [simp] = |
21256 | 393 |
power_le_zero_eq [of _ "number_of w", standard] |
394 |
||
21263 | 395 |
lemmas power_less_zero_eq_number_of [simp] = |
21256 | 396 |
power_less_zero_eq [of _ "number_of w", standard] |
397 |
||
21263 | 398 |
lemmas zero_less_power_nat_eq_number_of [simp] = |
21256 | 399 |
zero_less_power_nat_eq [of _ "number_of w", standard] |
400 |
||
21263 | 401 |
lemmas power_eq_0_iff_number_of [simp] = power_eq_0_iff [of _ "number_of w", standard] |
21256 | 402 |
|
21263 | 403 |
lemmas power_even_abs_number_of [simp] = power_even_abs [of "number_of w" _, standard] |
21256 | 404 |
|
405 |
||
406 |
subsection {* An Equivalence for @{term [source] "0 \<le> a^n"} *} |
|
407 |
||
408 |
lemma even_power_le_0_imp_0: |
|
31017 | 409 |
"a ^ (2*k) \<le> (0::'a::{ordered_idom}) ==> a=0" |
21263 | 410 |
by (induct k) (auto simp add: zero_le_mult_iff mult_le_0_iff power_Suc) |
21256 | 411 |
|
23522 | 412 |
lemma zero_le_power_iff[presburger]: |
31017 | 413 |
"(0 \<le> a^n) = (0 \<le> (a::'a::{ordered_idom}) | even n)" |
21256 | 414 |
proof cases |
415 |
assume even: "even n" |
|
416 |
then obtain k where "n = 2*k" |
|
417 |
by (auto simp add: even_nat_equiv_def2 numeral_2_eq_2) |
|
21263 | 418 |
thus ?thesis by (simp add: zero_le_even_power even) |
21256 | 419 |
next |
420 |
assume odd: "odd n" |
|
421 |
then obtain k where "n = Suc(2*k)" |
|
422 |
by (auto simp add: odd_nat_equiv_def2 numeral_2_eq_2) |
|
423 |
thus ?thesis |
|
21263 | 424 |
by (auto simp add: power_Suc zero_le_mult_iff zero_le_even_power |
425 |
dest!: even_power_le_0_imp_0) |
|
426 |
qed |
|
427 |
||
21256 | 428 |
|
429 |
subsection {* Miscellaneous *} |
|
430 |
||
23522 | 431 |
lemma [presburger]:"(x + 1) div 2 = x div 2 \<longleftrightarrow> even (x::int)" by presburger |
432 |
lemma [presburger]: "(x + 1) div 2 = x div 2 + 1 \<longleftrightarrow> odd (x::int)" by presburger |
|
433 |
lemma even_plus_one_div_two: "even (x::int) ==> (x + 1) div 2 = x div 2" by presburger |
|
434 |
lemma odd_plus_one_div_two: "odd (x::int) ==> (x + 1) div 2 = x div 2 + 1" by presburger |
|
21256 | 435 |
|
23522 | 436 |
lemma [presburger]: "(Suc x) div Suc (Suc 0) = x div Suc (Suc 0) \<longleftrightarrow> even x" by presburger |
437 |
lemma [presburger]: "(Suc x) div Suc (Suc 0) = x div Suc (Suc 0) \<longleftrightarrow> even x" by presburger |
|
21263 | 438 |
lemma even_nat_plus_one_div_two: "even (x::nat) ==> |
23522 | 439 |
(Suc x) div Suc (Suc 0) = x div Suc (Suc 0)" by presburger |
21256 | 440 |
|
21263 | 441 |
lemma odd_nat_plus_one_div_two: "odd (x::nat) ==> |
23522 | 442 |
(Suc x) div Suc (Suc 0) = Suc (x div Suc (Suc 0))" by presburger |
21256 | 443 |
|
444 |
end |