--- a/src/HOL/Library/Parity.thy Mon Jul 21 13:36:44 2008 +0200
+++ b/src/HOL/Library/Parity.thy Mon Jul 21 13:36:59 2008 +0200
@@ -41,14 +41,18 @@
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 (simp add: even_def zmod_zmult1_eq')
+ by algebra
-lemma anything_times_even: "even (y::int) ==> even (x * y)"
- by (simp add: even_def zmod_zmult1_eq)
+lemma anything_times_even: "even (y::int) ==> even (x * y)" by algebra
-lemma odd_times_odd: "odd (x::int) ==> odd y ==> odd (x * y)"
+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)"
@@ -71,7 +75,7 @@
lemma even_sum[presburger]: "even ((x::int) + y) = ((even x & even y) | (odd x & odd y))"
by presburger
-lemma even_neg[presburger]: "even (-(x::int)) = even x" 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
@@ -80,7 +84,8 @@
"even (x::int) ==> 0 < n ==> even (x^n)"
by (induct n) (auto simp add: even_product)
-lemma odd_pow_iff[presburger]: "odd ((x::int) ^ n) \<longleftrightarrow> (n = 0 \<or> odd x)"
+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)
@@ -120,19 +125,19 @@
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]: "even((x::nat) * y) = (even x | even y)"
+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]: "even ((x::nat) + y) =
+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]:
+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]: "even (Suc x) = odd x" by presburger
+lemma even_nat_Suc[presburger, algebra]: "even (Suc x) = odd x" by presburger
-lemma even_nat_power[presburger]: "even ((x::nat)^y) = (even x & 0 < y)"
+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
@@ -249,29 +254,11 @@
lemma zero_less_power_eq[presburger]: "(0 < (x::'a::{recpower,ordered_idom}) ^ n) =
(n = 0 | (even n & x ~= 0) | (odd n & 0 < x))"
- apply (rule iffI)
- apply clarsimp
- apply (rule conjI)
- apply clarsimp
- apply (rule ccontr)
- apply (subgoal_tac "~ (0 <= x^n)")
- apply simp
- apply (subst zero_le_odd_power)
- apply assumption
- apply simp
- apply (rule notI)
- apply (simp add: power_0_left)
- apply (rule notI)
- apply (simp add: power_0_left)
- apply auto
- apply (subgoal_tac "0 <= x^n")
- apply (frule order_le_imp_less_or_eq)
- apply simp
- apply (erule zero_le_even_power)
- done
+
+ 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)"
+ (odd n & x < 0)"
apply (subst linorder_not_le [symmetric])+
apply (subst zero_le_power_eq)
apply auto
@@ -324,6 +311,7 @@
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)
@@ -342,17 +330,11 @@
subsection {* More Even/Odd Results *}
-lemma even_mult_two_ex: "even(n) = (\<exists>m::nat. n = 2*m)"
-by (simp add: even_nat_equiv_def2 numeral_2_eq_2)
-
-lemma odd_Suc_mult_two_ex: "odd(n) = (\<exists>m. n = Suc (2*m))"
-by (simp add: odd_nat_equiv_def2 numeral_2_eq_2)
+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 even_add [simp]: "even(m + n::nat) = (even m = even n)"
-by auto
-
-lemma odd_add [simp]: "odd(m + n::nat) = (odd m \<noteq> odd n)"
-by auto
+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"
@@ -361,35 +343,18 @@
apply (rule div_add1_eq, simp)
done
-lemma lemma_even_div2 [simp]: "even (n::nat) ==> (n + 1) div 2 = n div 2"
-apply (simp add: numeral_2_eq_2)
-apply (subst div_Suc)
-apply (simp add: even_nat_mod_two_eq_zero)
-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)"
-apply (simp add: numeral_2_eq_2)
-apply (subst div_Suc)
-apply (simp add: odd_nat_mod_two_eq_one)
-done
-
-lemma even_num_iff: "0 < n ==> even n = (~ even(n - 1 :: nat))"
-by (case_tac "n", auto)
+by presburger
-lemma even_even_mod_4_iff: "even (n::nat) = even (n mod 4)"
-apply (induct n, simp)
-apply (subst mod_Suc, simp)
-done
+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)"
-apply (rule mod_div_equality [of n 4, THEN subst])
-apply (simp add: even_num_iff)
-done
+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)"
-apply (rule mod_div_equality [of n 4, THEN subst])
-apply simp
-done
+ by presburger
text {* Simplify, when the exponent is a numeral *}
@@ -441,8 +406,7 @@
subsection {* Miscellaneous *}
-lemma odd_pos: "odd (n::nat) \<Longrightarrow> 0 < n"
- by (cases n, simp_all)
+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