author | haftmann |
Thu, 31 Oct 2013 11:44:20 +0100 | |
changeset 54227 | 63b441f49645 |
parent 47225 | 650318981557 |
child 54228 | 229282d53781 |
permissions | -rw-r--r-- |
41959 | 1 |
(* 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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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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begin |
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abbreviation odd :: "'a \<Rightarrow> bool" |
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where |
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"odd x \<equiv> \<not> even x" |
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end |
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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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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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45 |
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 |
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47224 | 51 |
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 |
31148 | 59 |
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47224 | 60 |
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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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 |
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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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21256 | 75 |
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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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|
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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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31148 | 94 |
by presburger |
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lemma odd_plus_even: "odd (x::int) ==> even y ==> odd (x + y)" |
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by presburger |
21256 | 98 |
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lemma odd_plus_odd: "odd (x::int) ==> odd y ==> even (x + y)" by presburger |
21256 | 100 |
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31148 | 101 |
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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21256 | 104 |
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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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||
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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: |
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"odd (x::int) ==> 2 * (x div 2) + 1 = x" |
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by presburger |
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21256 | 125 |
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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_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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lemma even_sum_nat[simp,presburger,algebra]: |
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"even ((x::nat) + y) = ((even x & even y) | (odd x & odd y))" |
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by presburger |
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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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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_zero_nat, blast) |
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apply simp |
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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::{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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35216 | 206 |
apply (simp, simp) |
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done |
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209 |
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 |
21256 | 212 |
|
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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 |
21256 | 216 |
|
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lemma neg_power_if: |
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31017 | 218 |
"(-x::'a::{comm_ring_1}) ^ n = |
21256 | 219 |
(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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|
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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)" |
35216 | 235 |
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) = |
21256 | 240 |
(even n | (odd n & 0 <= x))" |
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apply auto |
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21263 | 242 |
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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21263 | 245 |
done |
21256 | 246 |
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lemma zero_less_power_eq[presburger]: "(0 < (x::'a::{linordered_idom}) ^ n) = |
21256 | 248 |
(n = 0 | (even n & x ~= 0) | (odd n & 0 < x))" |
27668 | 249 |
|
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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 | 253 |
(odd n & x < 0)" |
21263 | 254 |
apply (subst linorder_not_le [symmetric])+ |
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apply (subst zero_le_power_eq) |
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apply auto |
|
21263 | 257 |
done |
21256 | 258 |
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lemma power_le_zero_eq[presburger]: "((x::'a::{linordered_idom}) ^ n <= 0) = |
21256 | 260 |
(n ~= 0 & ((odd n & x <= 0) | (even n & x = 0)))" |
21263 | 261 |
apply (subst linorder_not_less [symmetric])+ |
21256 | 262 |
apply (subst zero_less_power_eq) |
263 |
apply auto |
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21263 | 264 |
done |
21256 | 265 |
|
21263 | 266 |
lemma power_even_abs: "even n ==> |
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(abs (x::'a::{linordered_idom}))^n = x^n" |
21263 | 268 |
apply (subst power_abs [symmetric]) |
21256 | 269 |
apply (simp add: zero_le_even_power) |
21263 | 270 |
done |
21256 | 271 |
|
21263 | 272 |
lemma power_minus_even [simp]: "even n ==> |
31017 | 273 |
(- x)^n = (x^n::'a::{comm_ring_1})" |
21256 | 274 |
apply (subst power_minus) |
275 |
apply simp |
|
21263 | 276 |
done |
21256 | 277 |
|
21263 | 278 |
lemma power_minus_odd [simp]: "odd n ==> |
31017 | 279 |
(- x)^n = - (x^n::'a::{comm_ring_1})" |
21256 | 280 |
apply (subst power_minus) |
281 |
apply simp |
|
21263 | 282 |
done |
21256 | 283 |
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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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286 |
shows "x^n \<le> y^n" |
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287 |
proof - |
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288 |
have "0 \<le> \<bar>x\<bar>" by auto |
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with `\<bar>x\<bar> \<le> \<bar>y\<bar>` |
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290 |
have "\<bar>x\<bar>^n \<le> \<bar>y\<bar>^n" by (rule power_mono) |
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291 |
thus ?thesis unfolding power_even_abs[OF `even n`] . |
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292 |
qed |
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293 |
|
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294 |
lemma odd_pos: "odd (n::nat) \<Longrightarrow> 0 < n" by presburger |
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295 |
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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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303 |
next |
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304 |
case False |
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show ?thesis |
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306 |
proof (cases "x < 0") |
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307 |
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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309 |
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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312 |
ultimately show ?thesis by auto |
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313 |
next |
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314 |
case False hence "0 \<le> x" by auto |
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315 |
with `x \<le> y` show ?thesis using power_mono by auto |
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316 |
qed |
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qed |
21263 | 318 |
|
25600 | 319 |
|
320 |
subsection {* More Even/Odd Results *} |
|
321 |
||
27668 | 322 |
lemma even_mult_two_ex: "even(n) = (\<exists>m::nat. n = 2*m)" by presburger |
323 |
lemma odd_Suc_mult_two_ex: "odd(n) = (\<exists>m. n = Suc (2*m))" by presburger |
|
324 |
lemma even_add [simp]: "even(m + n::nat) = (even m = even n)" by presburger |
|
25600 | 325 |
|
27668 | 326 |
lemma odd_add [simp]: "odd(m + n::nat) = (odd m \<noteq> odd n)" by presburger |
25600 | 327 |
|
27668 | 328 |
lemma lemma_even_div2 [simp]: "even (n::nat) ==> (n + 1) div 2 = n div 2" by presburger |
25600 | 329 |
|
330 |
lemma lemma_not_even_div2 [simp]: "~even n ==> (n + 1) div 2 = Suc (n div 2)" |
|
27668 | 331 |
by presburger |
25600 | 332 |
|
27668 | 333 |
lemma even_num_iff: "0 < n ==> even n = (~ even(n - 1 :: nat))" by presburger |
334 |
lemma even_even_mod_4_iff: "even (n::nat) = even (n mod 4)" by presburger |
|
25600 | 335 |
|
27668 | 336 |
lemma lemma_odd_mod_4_div_2: "n mod 4 = (3::nat) ==> odd((n - 1) div 2)" by presburger |
25600 | 337 |
|
338 |
lemma lemma_even_mod_4_div_2: "n mod 4 = (1::nat) ==> even ((n - 1) div 2)" |
|
27668 | 339 |
by presburger |
25600 | 340 |
|
21263 | 341 |
text {* Simplify, when the exponent is a numeral *} |
21256 | 342 |
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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 | 345 |
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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 | 348 |
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lemmas power_le_zero_eq_numeral [simp] = |
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power_le_zero_eq [of _ "numeral w"] for w |
21256 | 351 |
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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 | 354 |
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355 |
lemmas zero_less_power_nat_eq_numeral [simp] = |
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nat_zero_less_power_iff [of _ "numeral w"] for w |
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lemmas power_eq_0_iff_numeral [simp] = |
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359 |
power_eq_0_iff [of _ "numeral w"] for w |
21256 | 360 |
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361 |
lemmas power_even_abs_numeral [simp] = |
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362 |
power_even_abs [of "numeral w" _] for w |
21256 | 363 |
|
364 |
||
365 |
subsection {* An Equivalence for @{term [source] "0 \<le> a^n"} *} |
|
366 |
||
23522 | 367 |
lemma zero_le_power_iff[presburger]: |
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368 |
"(0 \<le> a^n) = (0 \<le> (a::'a::{linordered_idom}) | even n)" |
21256 | 369 |
proof cases |
370 |
assume even: "even n" |
|
371 |
then obtain k where "n = 2*k" |
|
372 |
by (auto simp add: even_nat_equiv_def2 numeral_2_eq_2) |
|
21263 | 373 |
thus ?thesis by (simp add: zero_le_even_power even) |
21256 | 374 |
next |
375 |
assume odd: "odd n" |
|
376 |
then obtain k where "n = Suc(2*k)" |
|
377 |
by (auto simp add: odd_nat_equiv_def2 numeral_2_eq_2) |
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378 |
moreover have "a ^ (2 * k) \<le> 0 \<Longrightarrow> a = 0" |
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379 |
by (induct k) (auto simp add: zero_le_mult_iff mult_le_0_iff) |
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380 |
ultimately show ?thesis |
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381 |
by (auto simp add: zero_le_mult_iff zero_le_even_power) |
21263 | 382 |
qed |
383 |
||
21256 | 384 |
|
385 |
subsection {* Miscellaneous *} |
|
386 |
||
23522 | 387 |
lemma [presburger]:"(x + 1) div 2 = x div 2 \<longleftrightarrow> even (x::int)" by presburger |
388 |
lemma [presburger]: "(x + 1) div 2 = x div 2 + 1 \<longleftrightarrow> odd (x::int)" by presburger |
|
389 |
lemma even_plus_one_div_two: "even (x::int) ==> (x + 1) div 2 = x div 2" by presburger |
|
390 |
lemma odd_plus_one_div_two: "odd (x::int) ==> (x + 1) div 2 = x div 2 + 1" by presburger |
|
21256 | 391 |
|
23522 | 392 |
lemma [presburger]: "(Suc x) div Suc (Suc 0) = x div Suc (Suc 0) \<longleftrightarrow> even x" by presburger |
21263 | 393 |
lemma even_nat_plus_one_div_two: "even (x::nat) ==> |
23522 | 394 |
(Suc x) div Suc (Suc 0) = x div Suc (Suc 0)" by presburger |
21256 | 395 |
|
21263 | 396 |
lemma odd_nat_plus_one_div_two: "odd (x::nat) ==> |
23522 | 397 |
(Suc x) div Suc (Suc 0) = Suc (x div Suc (Suc 0))" by presburger |
21256 | 398 |
|
399 |
end |
|
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400 |