author | huffman |
Wed, 20 Jun 2007 17:28:55 +0200 | |
changeset 23438 | dd824e86fa8a |
parent 23431 | 25ca91279a9b |
child 23522 | 7e8255828502 |
permissions | -rw-r--r-- |
21263 | 1 |
(* 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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instance int :: even_odd |
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even_def: "even x \<equiv> x mod 2 = 0" .. |
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instance nat :: even_odd |
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change simp rules for of_nat to work like int did previously (reorient of_nat_Suc, remove of_nat_mult [simp]); preserve original variable names in legacy int theorems
huffman
parents:
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even_nat_def: "even x \<equiv> even (int x)" .. |
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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 auto |
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||
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lemma neq_one_mod_two [simp]: "((x::int) mod 2 ~= 0) = (x mod 2 = 1)" |
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proof - |
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have "x mod 2 = 0 | x mod 2 = 1" |
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by (rule int_pos_lt_two_imp_zero_or_one) auto |
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then show ?thesis by force |
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qed |
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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: "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 (simp add: even_def zmod_zadd1_eq) |
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lemma even_plus_odd: "even (x::int) ==> odd y ==> odd (x + y)" |
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by (simp add: even_def zmod_zadd1_eq) |
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lemma odd_plus_even: "odd (x::int) ==> even y ==> odd (x + y)" |
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by (simp add: even_def zmod_zadd1_eq) |
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lemma odd_plus_odd: "odd (x::int) ==> odd y ==> even (x + y)" |
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by (simp add: even_def zmod_zadd1_eq) |
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lemma even_sum: "even ((x::int) + y) = ((even x & even y) | (odd x & odd y))" |
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apply (auto intro: even_plus_even odd_plus_odd) |
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apply (rule ccontr, simp add: even_plus_odd) |
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apply (rule ccontr, simp add: odd_plus_even) |
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done |
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lemma even_neg: "even (-(x::int)) = even x" |
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by (auto simp add: even_def zmod_zminus1_eq_if) |
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||
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lemma even_difference: |
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"even ((x::int) - y) = ((even x & even y) | (odd x & odd y))" |
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by (simp only: diff_minus even_sum even_neg) |
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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: "odd x ==> odd((x::int)^n)" |
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apply (induct n) |
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apply (simp add: even_def) |
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apply (simp add: even_product) |
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done |
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lemma even_power: "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: "even (0::int)" |
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by (simp add: even_def) |
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lemma odd_one: "odd (1::int)" |
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by (simp add: even_def) |
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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 (auto simp add: even_def) |
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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" |
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apply (insert zmod_zdiv_equality [of x 2, symmetric]) |
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apply (simp add: even_def) |
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done |
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lemma even_equiv_def: "even (x::int) = (EX y. x = 2 * y)" |
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apply auto |
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apply (rule exI) |
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apply (erule two_times_even_div_two [symmetric]) |
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done |
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lemma odd_equiv_def: "odd (x::int) = (EX y. x = 2 * y + 1)" |
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apply auto |
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apply (rule exI) |
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apply (erule two_times_odd_div_two_plus_one [symmetric]) |
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done |
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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: "even((x::nat) * y) = (even x | even y)" |
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23431
25ca91279a9b
change simp rules for of_nat to work like int did previously (reorient of_nat_Suc, remove of_nat_mult [simp]); preserve original variable names in legacy int theorems
huffman
parents:
23309
diff
changeset
|
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by (simp add: even_nat_def int_mult) |
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lemma even_nat_sum: "even ((x::nat) + y) = |
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((even x & even y) | (odd x & odd y))" |
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by (unfold even_nat_def, simp) |
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||
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lemma even_nat_difference: |
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"even ((x::nat) - y) = (x < y | (even x & even y) | (odd x & odd y))" |
23438
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parents:
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diff
changeset
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apply (auto simp add: even_nat_def zdiff_int [symmetric]) |
dd824e86fa8a
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huffman
parents:
23431
diff
changeset
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147 |
apply (case_tac "x < y", auto simp add: zdiff_int [symmetric]) |
dd824e86fa8a
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parents:
23431
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changeset
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apply (case_tac "x < y", auto simp add: zdiff_int [symmetric]) |
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done |
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lemma even_nat_Suc: "even (Suc x) = odd x" |
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by (simp add: even_nat_def) |
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lemma even_nat_power: "even ((x::nat)^y) = (even x & 0 < y)" |
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23431
25ca91279a9b
change simp rules for of_nat to work like int did previously (reorient of_nat_Suc, remove of_nat_mult [simp]); preserve original variable names in legacy int theorems
huffman
parents:
23309
diff
changeset
|
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by (simp add: even_nat_def int_power) |
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lemma even_nat_zero: "even (0::nat)" |
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by (simp add: even_nat_def) |
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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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163 |
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subsection {* Equivalent definitions *} |
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||
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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" |
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by auto |
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lemma even_nat_mod_two_eq_zero: "even (x::nat) ==> x mod (Suc (Suc 0)) = 0" |
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apply (insert mod_div_equality [of x "Suc (Suc 0)", symmetric]) |
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apply (drule subst, assumption) |
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apply (subgoal_tac "x mod Suc (Suc 0) = 0 | x mod Suc (Suc 0) = Suc 0") |
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apply force |
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apply (subgoal_tac "0 < Suc (Suc 0)") |
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apply (frule mod_less_divisor [of "Suc (Suc 0)" x]) |
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apply (erule nat_lt_two_imp_zero_or_one, auto) |
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done |
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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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apply (insert mod_div_equality [of x "Suc (Suc 0)", symmetric]) |
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apply (drule subst, assumption) |
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apply (subgoal_tac "x mod Suc (Suc 0) = 0 | x mod Suc (Suc 0) = Suc 0") |
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21263 | 184 |
apply force |
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apply (subgoal_tac "0 < Suc (Suc 0)") |
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apply (frule mod_less_divisor [of "Suc (Suc 0)" x]) |
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apply (erule nat_lt_two_imp_zero_or_one, auto) |
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done |
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lemma even_nat_equiv_def: "even (x::nat) = (x mod Suc (Suc 0) = 0)" |
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apply (rule iffI) |
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apply (erule even_nat_mod_two_eq_zero) |
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apply (insert odd_nat_mod_two_eq_one [of x], auto) |
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done |
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lemma odd_nat_equiv_def: "odd (x::nat) = (x mod Suc (Suc 0) = Suc 0)" |
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apply (auto simp add: even_nat_equiv_def) |
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apply (subgoal_tac "x mod (Suc (Suc 0)) < Suc (Suc 0)") |
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apply (frule nat_lt_two_imp_zero_or_one, auto) |
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done |
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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" |
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apply (insert mod_div_equality [of x "Suc (Suc 0)", symmetric]) |
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apply (drule even_nat_mod_two_eq_zero, simp) |
206 |
done |
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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" |
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apply (insert mod_div_equality [of x "Suc (Suc 0)", symmetric]) |
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apply (drule odd_nat_mod_two_eq_one, simp) |
212 |
done |
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213 |
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214 |
lemma even_nat_equiv_def2: "even (x::nat) = (EX y. x = Suc (Suc 0) * y)" |
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215 |
apply (rule iffI, rule exI) |
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apply (erule even_nat_div_two_times_two [symmetric], auto) |
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done |
218 |
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lemma odd_nat_equiv_def2: "odd (x::nat) = (EX y. x = Suc(Suc (Suc 0) * y))" |
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apply (rule iffI, rule exI) |
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apply (erule odd_nat_div_two_times_two_plus_one [symmetric], auto) |
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done |
223 |
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subsection {* Parity and powers *} |
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225 |
||
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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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21263 | 234 |
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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21256 | 239 |
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lemma minus_one_odd_power [simp]: |
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21263 | 241 |
"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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21256 | 243 |
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lemma neg_one_even_odd_power: |
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21263 | 245 |
"(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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249 |
done |
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250 |
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lemma neg_one_even_power [simp]: |
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21263 | 252 |
"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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21256 | 254 |
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lemma neg_one_odd_power [simp]: |
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21263 | 256 |
"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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21256 | 258 |
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lemma neg_power_if: |
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21263 | 260 |
"(-x::'a::{comm_ring_1,recpower}) ^ n = |
21256 | 261 |
(if even n then (x ^ n) else -(x ^ n))" |
21263 | 262 |
apply (induct n) |
263 |
apply (simp_all split: split_if_asm add: power_Suc) |
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264 |
done |
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21256 | 265 |
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21263 | 266 |
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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270 |
apply (erule ssubst) |
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271 |
apply (subst power_add) |
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apply (rule zero_le_square) |
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273 |
done |
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274 |
||
21263 | 275 |
lemma zero_le_odd_power: "odd n ==> |
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(0 <= (x::'a::{recpower,ordered_idom}) ^ n) = (0 <= x)" |
277 |
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 & 0 < y") |
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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") |
288 |
apply simp |
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289 |
apply (rule zero_le_square)+ |
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21263 | 290 |
done |
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lemma zero_le_power_eq: "(0 <= (x::'a::{recpower,ordered_idom}) ^ n) = |
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(even n | (odd n & 0 <= x))" |
294 |
apply auto |
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21263 | 295 |
apply (subst zero_le_odd_power [symmetric]) |
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apply assumption+ |
297 |
apply (erule zero_le_even_power) |
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21263 | 298 |
apply (subst zero_le_odd_power) |
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apply assumption+ |
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done |
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|
21263 | 302 |
lemma zero_less_power_eq: "(0 < (x::'a::{recpower,ordered_idom}) ^ n) = |
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(n = 0 | (even n & x ~= 0) | (odd n & 0 < x))" |
304 |
apply (rule iffI) |
|
305 |
apply clarsimp |
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306 |
apply (rule conjI) |
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307 |
apply clarsimp |
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308 |
apply (rule ccontr) |
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309 |
apply (subgoal_tac "~ (0 <= x^n)") |
|
310 |
apply simp |
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311 |
apply (subst zero_le_odd_power) |
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21263 | 312 |
apply assumption |
21256 | 313 |
apply simp |
314 |
apply (rule notI) |
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315 |
apply (simp add: power_0_left) |
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316 |
apply (rule notI) |
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317 |
apply (simp add: power_0_left) |
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318 |
apply auto |
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319 |
apply (subgoal_tac "0 <= x^n") |
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320 |
apply (frule order_le_imp_less_or_eq) |
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321 |
apply simp |
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322 |
apply (erule zero_le_even_power) |
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323 |
apply (subgoal_tac "0 <= x^n") |
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324 |
apply (frule order_le_imp_less_or_eq) |
|
325 |
apply auto |
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326 |
apply (subst zero_le_odd_power) |
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327 |
apply assumption |
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328 |
apply (erule order_less_imp_le) |
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21263 | 329 |
done |
21256 | 330 |
|
331 |
lemma power_less_zero_eq: "((x::'a::{recpower,ordered_idom}) ^ n < 0) = |
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21263 | 332 |
(odd n & x < 0)" |
333 |
apply (subst linorder_not_le [symmetric])+ |
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21256 | 334 |
apply (subst zero_le_power_eq) |
335 |
apply auto |
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21263 | 336 |
done |
21256 | 337 |
|
338 |
lemma power_le_zero_eq: "((x::'a::{recpower,ordered_idom}) ^ n <= 0) = |
|
339 |
(n ~= 0 & ((odd n & x <= 0) | (even n & x = 0)))" |
|
21263 | 340 |
apply (subst linorder_not_less [symmetric])+ |
21256 | 341 |
apply (subst zero_less_power_eq) |
342 |
apply auto |
|
21263 | 343 |
done |
21256 | 344 |
|
21263 | 345 |
lemma power_even_abs: "even n ==> |
21256 | 346 |
(abs (x::'a::{recpower,ordered_idom}))^n = x^n" |
21263 | 347 |
apply (subst power_abs [symmetric]) |
21256 | 348 |
apply (simp add: zero_le_even_power) |
21263 | 349 |
done |
21256 | 350 |
|
351 |
lemma zero_less_power_nat_eq: "(0 < (x::nat) ^ n) = (n = 0 | 0 < x)" |
|
21263 | 352 |
by (induct n) auto |
21256 | 353 |
|
21263 | 354 |
lemma power_minus_even [simp]: "even n ==> |
21256 | 355 |
(- x)^n = (x^n::'a::{recpower,comm_ring_1})" |
356 |
apply (subst power_minus) |
|
357 |
apply simp |
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21263 | 358 |
done |
21256 | 359 |
|
21263 | 360 |
lemma power_minus_odd [simp]: "odd n ==> |
21256 | 361 |
(- x)^n = - (x^n::'a::{recpower,comm_ring_1})" |
362 |
apply (subst power_minus) |
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363 |
apply simp |
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21263 | 364 |
done |
21256 | 365 |
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21263 | 366 |
|
367 |
text {* Simplify, when the exponent is a numeral *} |
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21256 | 368 |
|
369 |
lemmas power_0_left_number_of = power_0_left [of "number_of w", standard] |
|
370 |
declare power_0_left_number_of [simp] |
|
371 |
||
21263 | 372 |
lemmas zero_le_power_eq_number_of [simp] = |
21256 | 373 |
zero_le_power_eq [of _ "number_of w", standard] |
374 |
||
21263 | 375 |
lemmas zero_less_power_eq_number_of [simp] = |
21256 | 376 |
zero_less_power_eq [of _ "number_of w", standard] |
377 |
||
21263 | 378 |
lemmas power_le_zero_eq_number_of [simp] = |
21256 | 379 |
power_le_zero_eq [of _ "number_of w", standard] |
380 |
||
21263 | 381 |
lemmas power_less_zero_eq_number_of [simp] = |
21256 | 382 |
power_less_zero_eq [of _ "number_of w", standard] |
383 |
||
21263 | 384 |
lemmas zero_less_power_nat_eq_number_of [simp] = |
21256 | 385 |
zero_less_power_nat_eq [of _ "number_of w", standard] |
386 |
||
21263 | 387 |
lemmas power_eq_0_iff_number_of [simp] = power_eq_0_iff [of _ "number_of w", standard] |
21256 | 388 |
|
21263 | 389 |
lemmas power_even_abs_number_of [simp] = power_even_abs [of "number_of w" _, standard] |
21256 | 390 |
|
391 |
||
392 |
subsection {* An Equivalence for @{term [source] "0 \<le> a^n"} *} |
|
393 |
||
394 |
lemma even_power_le_0_imp_0: |
|
21263 | 395 |
"a ^ (2*k) \<le> (0::'a::{ordered_idom,recpower}) ==> a=0" |
396 |
by (induct k) (auto simp add: zero_le_mult_iff mult_le_0_iff power_Suc) |
|
21256 | 397 |
|
398 |
lemma zero_le_power_iff: |
|
21263 | 399 |
"(0 \<le> a^n) = (0 \<le> (a::'a::{ordered_idom,recpower}) | even n)" |
21256 | 400 |
proof cases |
401 |
assume even: "even n" |
|
402 |
then obtain k where "n = 2*k" |
|
403 |
by (auto simp add: even_nat_equiv_def2 numeral_2_eq_2) |
|
21263 | 404 |
thus ?thesis by (simp add: zero_le_even_power even) |
21256 | 405 |
next |
406 |
assume odd: "odd n" |
|
407 |
then obtain k where "n = Suc(2*k)" |
|
408 |
by (auto simp add: odd_nat_equiv_def2 numeral_2_eq_2) |
|
409 |
thus ?thesis |
|
21263 | 410 |
by (auto simp add: power_Suc zero_le_mult_iff zero_le_even_power |
411 |
dest!: even_power_le_0_imp_0) |
|
412 |
qed |
|
413 |
||
21256 | 414 |
|
415 |
subsection {* Miscellaneous *} |
|
416 |
||
417 |
lemma even_plus_one_div_two: "even (x::int) ==> (x + 1) div 2 = x div 2" |
|
418 |
apply (subst zdiv_zadd1_eq) |
|
419 |
apply (simp add: even_def) |
|
420 |
done |
|
421 |
||
422 |
lemma odd_plus_one_div_two: "odd (x::int) ==> (x + 1) div 2 = x div 2 + 1" |
|
423 |
apply (subst zdiv_zadd1_eq) |
|
424 |
apply (simp add: even_def) |
|
425 |
done |
|
426 |
||
21263 | 427 |
lemma div_Suc: "Suc a div c = a div c + Suc 0 div c + |
21256 | 428 |
(a mod c + Suc 0 mod c) div c" |
429 |
apply (subgoal_tac "Suc a = a + Suc 0") |
|
430 |
apply (erule ssubst) |
|
431 |
apply (rule div_add1_eq, simp) |
|
432 |
done |
|
433 |
||
21263 | 434 |
lemma even_nat_plus_one_div_two: "even (x::nat) ==> |
435 |
(Suc x) div Suc (Suc 0) = x div Suc (Suc 0)" |
|
21256 | 436 |
apply (subst div_Suc) |
437 |
apply (simp add: even_nat_equiv_def) |
|
438 |
done |
|
439 |
||
21263 | 440 |
lemma odd_nat_plus_one_div_two: "odd (x::nat) ==> |
21256 | 441 |
(Suc x) div Suc (Suc 0) = Suc (x div Suc (Suc 0))" |
442 |
apply (subst div_Suc) |
|
443 |
apply (simp add: odd_nat_equiv_def) |
|
444 |
done |
|
445 |
||
446 |
end |