--- a/src/HOL/Library/Parity.thy Wed Dec 03 09:53:58 2008 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,424 +0,0 @@
-(* Title: HOL/Library/Parity.thy
- ID: $Id$
- Author: Jeremy Avigad, Jacques D. Fleuriot
-*)
-
-header {* Even and Odd for int and nat *}
-
-theory Parity
-imports Plain "~~/src/HOL/Presburger"
-begin
-
-class even_odd = type +
- fixes even :: "'a \<Rightarrow> bool"
-
-abbreviation
- odd :: "'a\<Colon>even_odd \<Rightarrow> bool" where
- "odd x \<equiv> \<not> even x"
-
-instantiation nat and int :: even_odd
-begin
-
-definition
- even_def [presburger]: "even x \<longleftrightarrow> (x\<Colon>int) mod 2 = 0"
-
-definition
- even_nat_def [presburger]: "even x \<longleftrightarrow> even (int x)"
-
-instance ..
-
-end
-
-
-subsection {* Even and odd are mutually exclusive *}
-
-lemma int_pos_lt_two_imp_zero_or_one:
- "0 <= x ==> (x::int) < 2 ==> x = 0 | x = 1"
- by presburger
-
-lemma neq_one_mod_two [simp, presburger]:
- "((x::int) mod 2 ~= 0) = (x mod 2 = 1)" by presburger
-
-
-subsection {* Behavior under integer arithmetic operations *}
-declare dvd_def[algebra]
-lemma nat_even_iff_2_dvd[algebra]: "even (x::nat) \<longleftrightarrow> 2 dvd x"
- by (presburger add: even_nat_def even_def)
-lemma int_even_iff_2_dvd[algebra]: "even (x::int) \<longleftrightarrow> 2 dvd x"
- by presburger
-
-lemma even_times_anything: "even (x::int) ==> even (x * y)"
- by algebra
-
-lemma anything_times_even: "even (y::int) ==> even (x * y)" by algebra
-
-lemma odd_times_odd: "odd (x::int) ==> odd y ==> odd (x * y)"
- by (simp add: even_def zmod_zmult1_eq)
-
-lemma even_product[presburger]: "even((x::int) * y) = (even x | even y)"
- apply (auto simp add: even_times_anything anything_times_even)
- apply (rule ccontr)
- apply (auto simp add: odd_times_odd)
- done
-
-lemma even_plus_even: "even (x::int) ==> even y ==> even (x + y)"
- by presburger
-
-lemma even_plus_odd: "even (x::int) ==> odd y ==> odd (x + y)"
- by presburger
-
-lemma odd_plus_even: "odd (x::int) ==> even y ==> odd (x + y)"
- by presburger
-
-lemma odd_plus_odd: "odd (x::int) ==> odd y ==> even (x + y)" by presburger
-
-lemma even_sum[presburger]: "even ((x::int) + y) = ((even x & even y) | (odd x & odd y))"
- by presburger
-
-lemma even_neg[presburger, algebra]: "even (-(x::int)) = even x" by presburger
-
-lemma even_difference:
- "even ((x::int) - y) = ((even x & even y) | (odd x & odd y))" by presburger
-
-lemma even_pow_gt_zero:
- "even (x::int) ==> 0 < n ==> even (x^n)"
- by (induct n) (auto simp add: even_product)
-
-lemma odd_pow_iff[presburger, algebra]:
- "odd ((x::int) ^ n) \<longleftrightarrow> (n = 0 \<or> odd x)"
- apply (induct n, simp_all)
- apply presburger
- apply (case_tac n, auto)
- apply (simp_all add: even_product)
- done
-
-lemma odd_pow: "odd x ==> odd((x::int)^n)" by (simp add: odd_pow_iff)
-
-lemma even_power[presburger]: "even ((x::int)^n) = (even x & 0 < n)"
- apply (auto simp add: even_pow_gt_zero)
- apply (erule contrapos_pp, erule odd_pow)
- apply (erule contrapos_pp, simp add: even_def)
- done
-
-lemma even_zero[presburger]: "even (0::int)" by presburger
-
-lemma odd_one[presburger]: "odd (1::int)" by presburger
-
-lemmas even_odd_simps [simp] = even_def[of "number_of v",standard] even_zero
- odd_one even_product even_sum even_neg even_difference even_power
-
-
-subsection {* Equivalent definitions *}
-
-lemma two_times_even_div_two: "even (x::int) ==> 2 * (x div 2) = x"
- by presburger
-
-lemma two_times_odd_div_two_plus_one: "odd (x::int) ==>
- 2 * (x div 2) + 1 = x" by presburger
-
-lemma even_equiv_def: "even (x::int) = (EX y. x = 2 * y)" by presburger
-
-lemma odd_equiv_def: "odd (x::int) = (EX y. x = 2 * y + 1)" by presburger
-
-subsection {* even and odd for nats *}
-
-lemma pos_int_even_equiv_nat_even: "0 \<le> x ==> even x = even (nat x)"
- by (simp add: even_nat_def)
-
-lemma even_nat_product[presburger, algebra]: "even((x::nat) * y) = (even x | even y)"
- by (simp add: even_nat_def int_mult)
-
-lemma even_nat_sum[presburger, algebra]: "even ((x::nat) + y) =
- ((even x & even y) | (odd x & odd y))" by presburger
-
-lemma even_nat_difference[presburger, algebra]:
- "even ((x::nat) - y) = (x < y | (even x & even y) | (odd x & odd y))"
-by presburger
-
-lemma even_nat_Suc[presburger, algebra]: "even (Suc x) = odd x" by presburger
-
-lemma even_nat_power[presburger, algebra]: "even ((x::nat)^y) = (even x & 0 < y)"
- by (simp add: even_nat_def int_power)
-
-lemma even_nat_zero[presburger]: "even (0::nat)" by presburger
-
-lemmas even_odd_nat_simps [simp] = even_nat_def[of "number_of v",standard]
- even_nat_zero even_nat_Suc even_nat_product even_nat_sum even_nat_power
-
-
-subsection {* Equivalent definitions *}
-
-lemma nat_lt_two_imp_zero_or_one: "(x::nat) < Suc (Suc 0) ==>
- x = 0 | x = Suc 0" by presburger
-
-lemma even_nat_mod_two_eq_zero: "even (x::nat) ==> x mod (Suc (Suc 0)) = 0"
- by presburger
-
-lemma odd_nat_mod_two_eq_one: "odd (x::nat) ==> x mod (Suc (Suc 0)) = Suc 0"
-by presburger
-
-lemma even_nat_equiv_def: "even (x::nat) = (x mod Suc (Suc 0) = 0)"
- by presburger
-
-lemma odd_nat_equiv_def: "odd (x::nat) = (x mod Suc (Suc 0) = Suc 0)"
- by presburger
-
-lemma even_nat_div_two_times_two: "even (x::nat) ==>
- Suc (Suc 0) * (x div Suc (Suc 0)) = x" by presburger
-
-lemma odd_nat_div_two_times_two_plus_one: "odd (x::nat) ==>
- Suc( Suc (Suc 0) * (x div Suc (Suc 0))) = x" by presburger
-
-lemma even_nat_equiv_def2: "even (x::nat) = (EX y. x = Suc (Suc 0) * y)"
- by presburger
-
-lemma odd_nat_equiv_def2: "odd (x::nat) = (EX y. x = Suc(Suc (Suc 0) * y))"
- by presburger
-
-
-subsection {* Parity and powers *}
-
-lemma minus_one_even_odd_power:
- "(even x --> (- 1::'a::{comm_ring_1,recpower})^x = 1) &
- (odd x --> (- 1::'a)^x = - 1)"
- apply (induct x)
- apply (rule conjI)
- apply simp
- apply (insert even_nat_zero, blast)
- apply (simp add: power_Suc)
- done
-
-lemma minus_one_even_power [simp]:
- "even x ==> (- 1::'a::{comm_ring_1,recpower})^x = 1"
- using minus_one_even_odd_power by blast
-
-lemma minus_one_odd_power [simp]:
- "odd x ==> (- 1::'a::{comm_ring_1,recpower})^x = - 1"
- using minus_one_even_odd_power by blast
-
-lemma neg_one_even_odd_power:
- "(even x --> (-1::'a::{number_ring,recpower})^x = 1) &
- (odd x --> (-1::'a)^x = -1)"
- apply (induct x)
- apply (simp, simp add: power_Suc)
- done
-
-lemma neg_one_even_power [simp]:
- "even x ==> (-1::'a::{number_ring,recpower})^x = 1"
- using neg_one_even_odd_power by blast
-
-lemma neg_one_odd_power [simp]:
- "odd x ==> (-1::'a::{number_ring,recpower})^x = -1"
- using neg_one_even_odd_power by blast
-
-lemma neg_power_if:
- "(-x::'a::{comm_ring_1,recpower}) ^ n =
- (if even n then (x ^ n) else -(x ^ n))"
- apply (induct n)
- apply (simp_all split: split_if_asm add: power_Suc)
- done
-
-lemma zero_le_even_power: "even n ==>
- 0 <= (x::'a::{recpower,ordered_ring_strict}) ^ n"
- apply (simp add: even_nat_equiv_def2)
- apply (erule exE)
- apply (erule ssubst)
- apply (subst power_add)
- apply (rule zero_le_square)
- done
-
-lemma zero_le_odd_power: "odd n ==>
- (0 <= (x::'a::{recpower,ordered_idom}) ^ n) = (0 <= x)"
- apply (simp add: odd_nat_equiv_def2)
- apply (erule exE)
- apply (erule ssubst)
- apply (subst power_Suc)
- apply (subst power_add)
- apply (subst zero_le_mult_iff)
- apply auto
- apply (subgoal_tac "x = 0 & y > 0")
- apply (erule conjE, assumption)
- apply (subst power_eq_0_iff [symmetric])
- apply (subgoal_tac "0 <= x^y * x^y")
- apply simp
- apply (rule zero_le_square)+
- done
-
-lemma zero_le_power_eq[presburger]: "(0 <= (x::'a::{recpower,ordered_idom}) ^ n) =
- (even n | (odd n & 0 <= x))"
- apply auto
- apply (subst zero_le_odd_power [symmetric])
- apply assumption+
- apply (erule zero_le_even_power)
- done
-
-lemma zero_less_power_eq[presburger]: "(0 < (x::'a::{recpower,ordered_idom}) ^ n) =
- (n = 0 | (even n & x ~= 0) | (odd n & 0 < x))"
-
- unfolding order_less_le zero_le_power_eq by auto
-
-lemma power_less_zero_eq[presburger]: "((x::'a::{recpower,ordered_idom}) ^ n < 0) =
- (odd n & x < 0)"
- apply (subst linorder_not_le [symmetric])+
- apply (subst zero_le_power_eq)
- apply auto
- done
-
-lemma power_le_zero_eq[presburger]: "((x::'a::{recpower,ordered_idom}) ^ n <= 0) =
- (n ~= 0 & ((odd n & x <= 0) | (even n & x = 0)))"
- apply (subst linorder_not_less [symmetric])+
- apply (subst zero_less_power_eq)
- apply auto
- done
-
-lemma power_even_abs: "even n ==>
- (abs (x::'a::{recpower,ordered_idom}))^n = x^n"
- apply (subst power_abs [symmetric])
- apply (simp add: zero_le_even_power)
- done
-
-lemma zero_less_power_nat_eq[presburger]: "(0 < (x::nat) ^ n) = (n = 0 | 0 < x)"
- by (induct n) auto
-
-lemma power_minus_even [simp]: "even n ==>
- (- x)^n = (x^n::'a::{recpower,comm_ring_1})"
- apply (subst power_minus)
- apply simp
- done
-
-lemma power_minus_odd [simp]: "odd n ==>
- (- x)^n = - (x^n::'a::{recpower,comm_ring_1})"
- apply (subst power_minus)
- apply simp
- done
-
-
-subsection {* General Lemmas About Division *}
-
-lemma Suc_times_mod_eq: "1<k ==> Suc (k * m) mod k = 1"
-apply (induct "m")
-apply (simp_all add: mod_Suc)
-done
-
-declare Suc_times_mod_eq [of "number_of w", standard, simp]
-
-lemma [simp]: "n div k \<le> (Suc n) div k"
-by (simp add: div_le_mono)
-
-lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"
-by arith
-
-lemma div_2_gt_zero [simp]: "(1::nat) < n ==> 0 < n div 2"
-by arith
-
- (* Potential use of algebra : Equality modulo n*)
-lemma mod_mult_self3 [simp]: "(k*n + m) mod n = m mod (n::nat)"
-by (simp add: mult_ac add_ac)
-
-lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n"
-proof -
- have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp
- also have "... = Suc m mod n" by (rule mod_mult_self3)
- finally show ?thesis .
-qed
-
-lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"
-apply (subst mod_Suc [of m])
-apply (subst mod_Suc [of "m mod n"], simp)
-done
-
-
-subsection {* More Even/Odd Results *}
-
-lemma even_mult_two_ex: "even(n) = (\<exists>m::nat. n = 2*m)" by presburger
-lemma odd_Suc_mult_two_ex: "odd(n) = (\<exists>m. n = Suc (2*m))" by presburger
-lemma even_add [simp]: "even(m + n::nat) = (even m = even n)" by presburger
-
-lemma odd_add [simp]: "odd(m + n::nat) = (odd m \<noteq> odd n)" by presburger
-
-lemma div_Suc: "Suc a div c = a div c + Suc 0 div c +
- (a mod c + Suc 0 mod c) div c"
- apply (subgoal_tac "Suc a = a + Suc 0")
- apply (erule ssubst)
- apply (rule div_add1_eq, simp)
- done
-
-lemma lemma_even_div2 [simp]: "even (n::nat) ==> (n + 1) div 2 = n div 2" by presburger
-
-lemma lemma_not_even_div2 [simp]: "~even n ==> (n + 1) div 2 = Suc (n div 2)"
-by presburger
-
-lemma even_num_iff: "0 < n ==> even n = (~ even(n - 1 :: nat))" by presburger
-lemma even_even_mod_4_iff: "even (n::nat) = even (n mod 4)" by presburger
-
-lemma lemma_odd_mod_4_div_2: "n mod 4 = (3::nat) ==> odd((n - 1) div 2)" by presburger
-
-lemma lemma_even_mod_4_div_2: "n mod 4 = (1::nat) ==> even ((n - 1) div 2)"
- by presburger
-
-text {* Simplify, when the exponent is a numeral *}
-
-lemmas power_0_left_number_of = power_0_left [of "number_of w", standard]
-declare power_0_left_number_of [simp]
-
-lemmas zero_le_power_eq_number_of [simp] =
- zero_le_power_eq [of _ "number_of w", standard]
-
-lemmas zero_less_power_eq_number_of [simp] =
- zero_less_power_eq [of _ "number_of w", standard]
-
-lemmas power_le_zero_eq_number_of [simp] =
- power_le_zero_eq [of _ "number_of w", standard]
-
-lemmas power_less_zero_eq_number_of [simp] =
- power_less_zero_eq [of _ "number_of w", standard]
-
-lemmas zero_less_power_nat_eq_number_of [simp] =
- zero_less_power_nat_eq [of _ "number_of w", standard]
-
-lemmas power_eq_0_iff_number_of [simp] = power_eq_0_iff [of _ "number_of w", standard]
-
-lemmas power_even_abs_number_of [simp] = power_even_abs [of "number_of w" _, standard]
-
-
-subsection {* An Equivalence for @{term [source] "0 \<le> a^n"} *}
-
-lemma even_power_le_0_imp_0:
- "a ^ (2*k) \<le> (0::'a::{ordered_idom,recpower}) ==> a=0"
- by (induct k) (auto simp add: zero_le_mult_iff mult_le_0_iff power_Suc)
-
-lemma zero_le_power_iff[presburger]:
- "(0 \<le> a^n) = (0 \<le> (a::'a::{ordered_idom,recpower}) | even n)"
-proof cases
- assume even: "even n"
- then obtain k where "n = 2*k"
- by (auto simp add: even_nat_equiv_def2 numeral_2_eq_2)
- thus ?thesis by (simp add: zero_le_even_power even)
-next
- assume odd: "odd n"
- then obtain k where "n = Suc(2*k)"
- by (auto simp add: odd_nat_equiv_def2 numeral_2_eq_2)
- thus ?thesis
- by (auto simp add: power_Suc zero_le_mult_iff zero_le_even_power
- dest!: even_power_le_0_imp_0)
-qed
-
-
-subsection {* Miscellaneous *}
-
-lemma odd_pos: "odd (n::nat) \<Longrightarrow> 0 < n" by presburger
-
-lemma [presburger]:"(x + 1) div 2 = x div 2 \<longleftrightarrow> even (x::int)" by presburger
-lemma [presburger]: "(x + 1) div 2 = x div 2 + 1 \<longleftrightarrow> odd (x::int)" by presburger
-lemma even_plus_one_div_two: "even (x::int) ==> (x + 1) div 2 = x div 2" by presburger
-lemma odd_plus_one_div_two: "odd (x::int) ==> (x + 1) div 2 = x div 2 + 1" by presburger
-
-lemma [presburger]: "(Suc x) div Suc (Suc 0) = x div Suc (Suc 0) \<longleftrightarrow> even x" by presburger
-lemma [presburger]: "(Suc x) div Suc (Suc 0) = x div Suc (Suc 0) \<longleftrightarrow> even x" by presburger
-lemma even_nat_plus_one_div_two: "even (x::nat) ==>
- (Suc x) div Suc (Suc 0) = x div Suc (Suc 0)" by presburger
-
-lemma odd_nat_plus_one_div_two: "odd (x::nat) ==>
- (Suc x) div Suc (Suc 0) = Suc (x div Suc (Suc 0))" by presburger
-
-end