author | haftmann |
Fri, 07 Dec 2007 15:07:59 +0100 | |
changeset 25571 | c9e39eafc7a0 |
parent 25502 | 9200b36280c0 |
child 25594 | 43c718438f9f |
permissions | -rw-r--r-- |
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(* Title: HOL/Library/Parity.thy |
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ID: $Id$ |
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Author: Jeremy Avigad |
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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 = type + |
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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 int and nat :: 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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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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lemma even_times_anything: "even (x::int) ==> even (x * y)" |
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by (simp add: even_def zmod_zmult1_eq') |
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lemma anything_times_even: "even (y::int) ==> even (x * y)" |
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by (simp add: even_def zmod_zmult1_eq) |
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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]: "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]: "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]: "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]: "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]: |
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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]: "even (Suc x) = odd x" by presburger |
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lemma even_nat_power[presburger]: "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,recpower})^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,recpower})^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,recpower})^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,recpower})^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,recpower})^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,recpower})^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,recpower}) ^ 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::{recpower,ordered_ring_strict}) ^ 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::{recpower,ordered_idom}) ^ n) = (0 <= x)" |
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apply (simp add: odd_nat_equiv_def2) |
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apply (erule exE) |
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apply (erule ssubst) |
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apply (subst power_Suc) |
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apply (subst power_add) |
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apply (subst zero_le_mult_iff) |
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apply auto |
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apply (subgoal_tac "x = 0 & y > 0") |
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apply (erule conjE, assumption) |
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apply (subst power_eq_0_iff [symmetric]) |
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apply (subgoal_tac "0 <= x^y * x^y") |
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apply simp |
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apply (rule zero_le_square)+ |
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done |
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lemma zero_le_power_eq[presburger]: "(0 <= (x::'a::{recpower,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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apply (subst zero_le_odd_power) |
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apply assumption+ |
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done |
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lemma zero_less_power_eq[presburger]: "(0 < (x::'a::{recpower,ordered_idom}) ^ n) = |
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(n = 0 | (even n & x ~= 0) | (odd n & 0 < x))" |
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apply (rule iffI) |
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apply clarsimp |
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apply (rule conjI) |
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apply clarsimp |
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apply (rule ccontr) |
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apply (subgoal_tac "~ (0 <= x^n)") |
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apply simp |
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apply (subst zero_le_odd_power) |
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apply assumption |
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apply simp |
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apply (rule notI) |
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apply (simp add: power_0_left) |
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apply (rule notI) |
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apply (simp add: power_0_left) |
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apply auto |
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apply (subgoal_tac "0 <= x^n") |
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apply (frule order_le_imp_less_or_eq) |
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apply simp |
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apply (erule zero_le_even_power) |
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apply (subgoal_tac "0 <= x^n") |
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apply (frule order_le_imp_less_or_eq) |
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apply auto |
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apply (subst zero_le_odd_power) |
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apply assumption |
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apply (erule order_less_imp_le) |
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done |
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lemma power_less_zero_eq[presburger]: "((x::'a::{recpower,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::{recpower,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::{recpower,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::{recpower,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::{recpower,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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text {* Simplify, when the exponent is a numeral *} |
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lemmas power_0_left_number_of = power_0_left [of "number_of w", standard] |
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declare power_0_left_number_of [simp] |
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lemmas zero_le_power_eq_number_of [simp] = |
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zero_le_power_eq [of _ "number_of w", standard] |
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lemmas zero_less_power_eq_number_of [simp] = |
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zero_less_power_eq [of _ "number_of w", standard] |
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lemmas power_le_zero_eq_number_of [simp] = |
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power_le_zero_eq [of _ "number_of w", standard] |
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lemmas power_less_zero_eq_number_of [simp] = |
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power_less_zero_eq [of _ "number_of w", standard] |
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lemmas zero_less_power_nat_eq_number_of [simp] = |
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zero_less_power_nat_eq [of _ "number_of w", standard] |
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lemmas power_eq_0_iff_number_of [simp] = power_eq_0_iff [of _ "number_of w", standard] |
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lemmas power_even_abs_number_of [simp] = power_even_abs [of "number_of w" _, standard] |
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subsection {* An Equivalence for @{term [source] "0 \<le> a^n"} *} |
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lemma even_power_le_0_imp_0: |
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"a ^ (2*k) \<le> (0::'a::{ordered_idom,recpower}) ==> a=0" |
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by (induct k) (auto simp add: zero_le_mult_iff mult_le_0_iff power_Suc) |
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lemma zero_le_power_iff[presburger]: |
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"(0 \<le> a^n) = (0 \<le> (a::'a::{ordered_idom,recpower}) | even n)" |
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proof cases |
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assume even: "even n" |
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then obtain k where "n = 2*k" |
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by (auto simp add: even_nat_equiv_def2 numeral_2_eq_2) |
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thus ?thesis by (simp add: zero_le_even_power even) |
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next |
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assume odd: "odd n" |
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then obtain k where "n = Suc(2*k)" |
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by (auto simp add: odd_nat_equiv_def2 numeral_2_eq_2) |
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thus ?thesis |
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by (auto simp add: power_Suc zero_le_mult_iff zero_le_even_power |
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dest!: even_power_le_0_imp_0) |
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qed |
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subsection {* Miscellaneous *} |
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lemma [presburger]:"(x + 1) div 2 = x div 2 \<longleftrightarrow> even (x::int)" by presburger |
366 |
lemma [presburger]: "(x + 1) div 2 = x div 2 + 1 \<longleftrightarrow> odd (x::int)" by presburger |
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367 |
lemma even_plus_one_div_two: "even (x::int) ==> (x + 1) div 2 = x div 2" by presburger |
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368 |
lemma odd_plus_one_div_two: "odd (x::int) ==> (x + 1) div 2 = x div 2 + 1" by presburger |
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21256 | 369 |
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21263 | 370 |
lemma div_Suc: "Suc a div c = a div c + Suc 0 div c + |
23522 | 371 |
(a mod c + Suc 0 mod c) div c" |
21256 | 372 |
apply (subgoal_tac "Suc a = a + Suc 0") |
373 |
apply (erule ssubst) |
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374 |
apply (rule div_add1_eq, simp) |
|
375 |
done |
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376 |
||
23522 | 377 |
lemma [presburger]: "(Suc x) div Suc (Suc 0) = x div Suc (Suc 0) \<longleftrightarrow> even x" by presburger |
378 |
lemma [presburger]: "(Suc x) div Suc (Suc 0) = x div Suc (Suc 0) \<longleftrightarrow> even x" by presburger |
|
21263 | 379 |
lemma even_nat_plus_one_div_two: "even (x::nat) ==> |
23522 | 380 |
(Suc x) div Suc (Suc 0) = x div Suc (Suc 0)" by presburger |
21256 | 381 |
|
21263 | 382 |
lemma odd_nat_plus_one_div_two: "odd (x::nat) ==> |
23522 | 383 |
(Suc x) div Suc (Suc 0) = Suc (x div Suc (Suc 0))" by presburger |
21256 | 384 |
|
385 |
end |