author  wenzelm 
Sat, 07 Apr 2012 16:41:59 +0200  
changeset 47389  e8552cba702d 
parent 47225  650318981557 
child 54227  63b441f49645 
permissions  rwrr 
41959  1 
(* Title: HOL/Parity.thy 
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Author: Jeremy Avigad 

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Author: Jacques D. Fleuriot 

21256  4 
*) 
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header {* Even and Odd for int and nat *} 

7 

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theory Parity 

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imports Main 
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begin 
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29608  12 
class even_odd = 
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fixes even :: "'a \<Rightarrow> bool" 
21256  14 

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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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26259  19 
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" 
22390  24 

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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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lemma transfer_int_nat_relations: 
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"even (int x) \<longleftrightarrow> even x" 
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by (simp add: even_nat_def) 
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declare transfer_morphism_int_nat[transfer add return: 
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transfer_int_nat_relations 
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] 
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lemma even_zero_int[simp]: "even (0::int)" by presburger 
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lemma odd_one_int[simp]: "odd (1::int)" by presburger 

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lemma even_zero_nat[simp]: "even (0::nat)" by presburger 

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lemma odd_1_nat [simp]: "odd (1::nat)" by presburger 
31148  47 

47224  48 
lemma even_numeral_int [simp]: "even (numeral (Num.Bit0 k) :: int)" 
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unfolding even_def by simp 

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lemma odd_numeral_int [simp]: "odd (numeral (Num.Bit1 k) :: int)" 

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unfolding even_def by simp 

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(* TODO: proper simp rules for Num.Bit0, Num.Bit1 *) 
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declare even_def[of "neg_numeral v", simp] for v 
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lemma even_numeral_nat [simp]: "even (numeral (Num.Bit0 k) :: nat)" 
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unfolding even_nat_def by simp 

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lemma odd_numeral_nat [simp]: "odd (numeral (Num.Bit1 k) :: nat)" 

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unfolding even_nat_def by simp 

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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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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" 

36840  76 
by presburger 
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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 mod_mult_right_eq) 
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lemma even_product[simp,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[simp,presburger]: 
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"even ((x::int) + y) = ((even x & even y)  (odd x & odd y))" 

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by presburger 

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31148  109 
lemma even_neg[simp,presburger,algebra]: "even ((x::int)) = even x" 
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by presburger 

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lemma even_difference[simp]: 
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"even ((x::int)  y) = ((even x & even y)  (odd x & odd y))" by presburger 
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lemma even_power[simp,presburger]: "even ((x::int)^n) = (even x & n \<noteq> 0)" 
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by (induct n) auto 

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lemma odd_pow: "odd x ==> odd((x::int)^n)" by simp 
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subsection {* Equivalent definitions *} 

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23522  123 
lemma two_times_even_div_two: "even (x::int) ==> 2 * (x div 2) = x" 
31148  124 
by presburger 
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lemma two_times_odd_div_two_plus_one: 
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"odd (x::int) ==> 2 * (x div 2) + 1 = x" 

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by presburger 

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23522  130 
lemma even_equiv_def: "even (x::int) = (EX y. x = 2 * y)" by presburger 
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23522  132 
lemma odd_equiv_def: "odd (x::int) = (EX y. x = 2 * y + 1)" by presburger 
21256  133 

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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_product_nat[simp,presburger,algebra]: 
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"even((x::nat) * y) = (even x  even y)" 

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by (simp add: even_nat_def int_mult) 

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31148  143 
lemma even_sum_nat[simp,presburger,algebra]: 
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"even ((x::nat) + y) = ((even x & even y)  (odd x & odd y))" 

23522  145 
by presburger 
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31148  147 
lemma even_difference_nat[simp,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_Suc[simp,presburger,algebra]: "even (Suc x) = odd x" 
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by presburger 

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lemma even_power_nat[simp,presburger,algebra]: 
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"even ((x::nat)^y) = (even x & 0 < y)" 

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by (simp add: even_nat_def int_power) 

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subsection {* Equivalent definitions *} 

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31148  161 
lemma nat_lt_two_imp_zero_or_one: 
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"(x::nat) < Suc (Suc 0) ==> x = 0  x = Suc 0" 

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by presburger 

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lemma even_nat_mod_two_eq_zero: "even (x::nat) ==> x mod (Suc (Suc 0)) = 0" 

31148  166 
by presburger 
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lemma odd_nat_mod_two_eq_one: "odd (x::nat) ==> x mod (Suc (Suc 0)) = Suc 0" 

23522  169 
by presburger 
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21263  171 
lemma even_nat_equiv_def: "even (x::nat) = (x mod Suc (Suc 0) = 0)" 
31148  172 
by presburger 
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lemma odd_nat_equiv_def: "odd (x::nat) = (x mod Suc (Suc 0) = Suc 0)" 

31148  175 
by presburger 
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21263  177 
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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21263  180 
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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25600  189 

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subsection {* Parity and powers *} 
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21263  192 
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_zero_nat, blast) 
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apply simp 
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done 
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lemma minus_one_even_power [simp]: 

31017  203 
"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::{comm_ring_1})^x = 1) & 
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(odd x > (1::'a)^x = 1)" 
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apply (induct x) 

35216  214 
apply (simp, simp) 
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done 
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lemma neg_one_even_power [simp]: 

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"even x ==> (1::'a::{comm_ring_1})^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::{comm_ring_1})^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 
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done 
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lemma zero_le_even_power: "even n ==> 
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0 <= (x::'a::{linordered_ring,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::{linordered_idom}) ^ n) = (0 <= x)" 
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apply (auto simp: odd_nat_equiv_def2 power_add zero_le_mult_iff) 
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apply (metis field_power_not_zero divisors_zero order_antisym_conv zero_le_square) 
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done 
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lemma zero_le_power_eq[presburger]: "(0 <= (x::'a::{linordered_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) 

21263  253 
done 
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lemma zero_less_power_eq[presburger]: "(0 < (x::'a::{linordered_idom}) ^ n) = 
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(n = 0  (even n & x ~= 0)  (odd n & 0 < x))" 
27668  257 

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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::{linordered_idom}) ^ n < 0) = 
27668  261 
(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::{linordered_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::{linordered_idom}))^n = x^n" 
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apply (subst power_abs [symmetric]) 
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apply (simp add: zero_le_even_power) 
21263  278 
done 
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23522  280 
lemma zero_less_power_nat_eq[presburger]: "(0 < (x::nat) ^ n) = (n = 0  0 < x)" 
21263  281 
by (induct n) auto 
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21263  283 
lemma power_minus_even [simp]: "even n ==> 
31017  284 
( x)^n = (x^n::'a::{comm_ring_1})" 
21256  285 
apply (subst power_minus) 
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apply simp 

21263  287 
done 
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21263  289 
lemma power_minus_odd [simp]: "odd n ==> 
31017  290 
( x)^n =  (x^n::'a::{comm_ring_1})" 
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apply (subst power_minus) 
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apply simp 

21263  293 
done 
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lemma power_mono_even: fixes x y :: "'a :: {linordered_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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thus ?thesis unfolding power_even_abs[OF `even n`] . 
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qed 
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lemma odd_pos: "odd (n::nat) \<Longrightarrow> 0 < n" by presburger 
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lemma power_mono_odd: fixes x y :: "'a :: {linordered_idom}" 
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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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hence "(y)^n \<le> (x)^n" by (rule power_mono) 
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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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show ?thesis 
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proof (cases "x < 0") 
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case True hence "n \<noteq> 0" and "x \<le> 0" using `odd n`[THEN odd_pos] by auto 
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hence "x^n \<le> 0" unfolding power_le_zero_eq using `odd n` by auto 
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moreover 
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from `\<not> y < 0` have "0 \<le> y" by auto 
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hence "0 \<le> y^n" by auto 
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ultimately show ?thesis by auto 
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next 
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case False hence "0 \<le> x" by auto 
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with `x \<le> y` show ?thesis using power_mono by auto 
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qed 
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qed 
21263  329 

25600  330 

331 
subsection {* More Even/Odd Results *} 

332 

27668  333 
lemma even_mult_two_ex: "even(n) = (\<exists>m::nat. n = 2*m)" by presburger 
334 
lemma odd_Suc_mult_two_ex: "odd(n) = (\<exists>m. n = Suc (2*m))" by presburger 

335 
lemma even_add [simp]: "even(m + n::nat) = (even m = even n)" by presburger 

25600  336 

27668  337 
lemma odd_add [simp]: "odd(m + n::nat) = (odd m \<noteq> odd n)" by presburger 
25600  338 

339 
lemma div_Suc: "Suc a div c = a div c + Suc 0 div c + 

340 
(a mod c + Suc 0 mod c) div c" 

341 
apply (subgoal_tac "Suc a = a + Suc 0") 

342 
apply (erule ssubst) 

343 
apply (rule div_add1_eq, simp) 

344 
done 

345 

27668  346 
lemma lemma_even_div2 [simp]: "even (n::nat) ==> (n + 1) div 2 = n div 2" by presburger 
25600  347 

348 
lemma lemma_not_even_div2 [simp]: "~even n ==> (n + 1) div 2 = Suc (n div 2)" 

27668  349 
by presburger 
25600  350 

27668  351 
lemma even_num_iff: "0 < n ==> even n = (~ even(n  1 :: nat))" by presburger 
352 
lemma even_even_mod_4_iff: "even (n::nat) = even (n mod 4)" by presburger 

25600  353 

27668  354 
lemma lemma_odd_mod_4_div_2: "n mod 4 = (3::nat) ==> odd((n  1) div 2)" by presburger 
25600  355 

356 
lemma lemma_even_mod_4_div_2: "n mod 4 = (1::nat) ==> even ((n  1) div 2)" 

27668  357 
by presburger 
25600  358 

21263  359 
text {* Simplify, when the exponent is a numeral *} 
21256  360 

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lemmas zero_le_power_eq_numeral [simp] = 
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zero_le_power_eq [of _ "numeral w"] for w 
21256  363 

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lemmas zero_less_power_eq_numeral [simp] = 
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zero_less_power_eq [of _ "numeral w"] for w 
21256  366 

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lemmas power_le_zero_eq_numeral [simp] = 
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power_le_zero_eq [of _ "numeral w"] for w 
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lemmas power_less_zero_eq_numeral [simp] = 
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power_less_zero_eq [of _ "numeral w"] for w 
21256  372 

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lemmas zero_less_power_nat_eq_numeral [simp] = 
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zero_less_power_nat_eq [of _ "numeral w"] for w 
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lemmas power_eq_0_iff_numeral [simp] = power_eq_0_iff [of _ "numeral w"] for w 
21256  377 

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lemmas power_even_abs_numeral [simp] = power_even_abs [of "numeral w" _] for w 
21256  379 

380 

381 
subsection {* An Equivalence for @{term [source] "0 \<le> a^n"} *} 

382 

383 
lemma even_power_le_0_imp_0: 

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"a ^ (2*k) \<le> (0::'a::{linordered_idom}) ==> a=0" 
35216  385 
by (induct k) (auto simp add: zero_le_mult_iff mult_le_0_iff) 
21256  386 

23522  387 
lemma zero_le_power_iff[presburger]: 
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"(0 \<le> a^n) = (0 \<le> (a::'a::{linordered_idom})  even n)" 
21256  389 
proof cases 
390 
assume even: "even n" 

391 
then obtain k where "n = 2*k" 

392 
by (auto simp add: even_nat_equiv_def2 numeral_2_eq_2) 

21263  393 
thus ?thesis by (simp add: zero_le_even_power even) 
21256  394 
next 
395 
assume odd: "odd n" 

396 
then obtain k where "n = Suc(2*k)" 

397 
by (auto simp add: odd_nat_equiv_def2 numeral_2_eq_2) 

398 
thus ?thesis 

35216  399 
by (auto simp add: zero_le_mult_iff zero_le_even_power 
21263  400 
dest!: even_power_le_0_imp_0) 
401 
qed 

402 

21256  403 

404 
subsection {* Miscellaneous *} 

405 

23522  406 
lemma [presburger]:"(x + 1) div 2 = x div 2 \<longleftrightarrow> even (x::int)" by presburger 
407 
lemma [presburger]: "(x + 1) div 2 = x div 2 + 1 \<longleftrightarrow> odd (x::int)" by presburger 

408 
lemma even_plus_one_div_two: "even (x::int) ==> (x + 1) div 2 = x div 2" by presburger 

409 
lemma odd_plus_one_div_two: "odd (x::int) ==> (x + 1) div 2 = x div 2 + 1" by presburger 

21256  410 

23522  411 
lemma [presburger]: "(Suc x) div Suc (Suc 0) = x div Suc (Suc 0) \<longleftrightarrow> even x" by presburger 
412 
lemma [presburger]: "(Suc x) div Suc (Suc 0) = x div Suc (Suc 0) \<longleftrightarrow> even x" by presburger 

21263  413 
lemma even_nat_plus_one_div_two: "even (x::nat) ==> 
23522  414 
(Suc x) div Suc (Suc 0) = x div Suc (Suc 0)" by presburger 
21256  415 

21263  416 
lemma odd_nat_plus_one_div_two: "odd (x::nat) ==> 
23522  417 
(Suc x) div Suc (Suc 0) = Suc (x div Suc (Suc 0))" by presburger 
21256  418 

419 
end 