--- a/src/HOL/Parity.thy Tue Apr 28 15:50:30 2009 +0200
+++ b/src/HOL/Parity.thy Tue Apr 28 15:50:30 2009 +0200
@@ -178,7 +178,7 @@
subsection {* Parity and powers *}
lemma minus_one_even_odd_power:
- "(even x --> (- 1::'a::{comm_ring_1,recpower})^x = 1) &
+ "(even x --> (- 1::'a::{comm_ring_1})^x = 1) &
(odd x --> (- 1::'a)^x = - 1)"
apply (induct x)
apply (rule conjI)
@@ -188,37 +188,37 @@
done
lemma minus_one_even_power [simp]:
- "even x ==> (- 1::'a::{comm_ring_1,recpower})^x = 1"
+ "even x ==> (- 1::'a::{comm_ring_1})^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"
+ "odd x ==> (- 1::'a::{comm_ring_1})^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) &
+ "(even x --> (-1::'a::{number_ring})^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"
+ "even x ==> (-1::'a::{number_ring})^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"
+ "odd x ==> (-1::'a::{number_ring})^x = -1"
using neg_one_even_odd_power by blast
lemma neg_power_if:
- "(-x::'a::{comm_ring_1,recpower}) ^ n =
+ "(-x::'a::{comm_ring_1}) ^ 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"
+ 0 <= (x::'a::{ordered_ring_strict,monoid_mult}) ^ n"
apply (simp add: even_nat_equiv_def2)
apply (erule exE)
apply (erule ssubst)
@@ -227,12 +227,12 @@
done
lemma zero_le_odd_power: "odd n ==>
- (0 <= (x::'a::{recpower,ordered_idom}) ^ n) = (0 <= x)"
+ (0 <= (x::'a::{ordered_idom}) ^ n) = (0 <= x)"
apply (auto simp: odd_nat_equiv_def2 power_Suc power_add zero_le_mult_iff)
apply (metis field_power_not_zero no_zero_divirors_neq0 order_antisym_conv zero_le_square)
done
-lemma zero_le_power_eq[presburger]: "(0 <= (x::'a::{recpower,ordered_idom}) ^ n) =
+lemma zero_le_power_eq[presburger]: "(0 <= (x::'a::{ordered_idom}) ^ n) =
(even n | (odd n & 0 <= x))"
apply auto
apply (subst zero_le_odd_power [symmetric])
@@ -240,19 +240,19 @@
apply (erule zero_le_even_power)
done
-lemma zero_less_power_eq[presburger]: "(0 < (x::'a::{recpower,ordered_idom}) ^ n) =
+lemma zero_less_power_eq[presburger]: "(0 < (x::'a::{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) =
+lemma power_less_zero_eq[presburger]: "((x::'a::{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) =
+lemma power_le_zero_eq[presburger]: "((x::'a::{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)
@@ -260,7 +260,7 @@
done
lemma power_even_abs: "even n ==>
- (abs (x::'a::{recpower,ordered_idom}))^n = x^n"
+ (abs (x::'a::{ordered_idom}))^n = x^n"
apply (subst power_abs [symmetric])
apply (simp add: zero_le_even_power)
done
@@ -269,18 +269,18 @@
by (induct n) auto
lemma power_minus_even [simp]: "even n ==>
- (- x)^n = (x^n::'a::{recpower,comm_ring_1})"
+ (- x)^n = (x^n::'a::{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})"
+ (- x)^n = - (x^n::'a::{comm_ring_1})"
apply (subst power_minus)
apply simp
done
-lemma power_mono_even: fixes x y :: "'a :: {recpower, ordered_idom}"
+lemma power_mono_even: fixes x y :: "'a :: {ordered_idom}"
assumes "even n" and "\<bar>x\<bar> \<le> \<bar>y\<bar>"
shows "x^n \<le> y^n"
proof -
@@ -292,7 +292,7 @@
lemma odd_pos: "odd (n::nat) \<Longrightarrow> 0 < n" by presburger
-lemma power_mono_odd: fixes x y :: "'a :: {recpower, ordered_idom}"
+lemma power_mono_odd: fixes x y :: "'a :: {ordered_idom}"
assumes "odd n" and "x \<le> y"
shows "x^n \<le> y^n"
proof (cases "y < 0")
@@ -406,11 +406,11 @@
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"
+ "a ^ (2*k) \<le> (0::'a::{ordered_idom}) ==> 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)"
+ "(0 \<le> a^n) = (0 \<le> (a::'a::{ordered_idom}) | even n)"
proof cases
assume even: "even n"
then obtain k where "n = 2*k"