src/HOL/Parity.thy

   apply (induct n)
   apply (simp_all split: split_if_asm add: power_Suc)
   done
 
 lemma zero_le_even_power: "even n ==>
-    0 <= (x::'a::{ordered_ring_strict,monoid_mult}) ^ n"
+    0 <= (x::'a::{linlinordered_ring_strict,monoid_mult}) ^ 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::{ordered_idom}) ^ n) = (0 <= x)"
+    (0 <= (x::'a::{linordered_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::{ordered_idom}) ^ n) =
+lemma zero_le_power_eq[presburger]: "(0 <= (x::'a::{linordered_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::{ordered_idom}) ^ n) =
+lemma zero_less_power_eq[presburger]: "(0 < (x::'a::{linordered_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::{ordered_idom}) ^ n < 0) =
+lemma power_less_zero_eq[presburger]: "((x::'a::{linordered_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::{ordered_idom}) ^ n <= 0) =
+lemma power_le_zero_eq[presburger]: "((x::'a::{linordered_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::{ordered_idom}))^n = x^n"
+    (abs (x::'a::{linordered_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)"

     (- x)^n = - (x^n::'a::{comm_ring_1})"
   apply (subst power_minus)
   apply simp
   done
 
-lemma power_mono_even: fixes x y :: "'a :: {ordered_idom}"
+lemma power_mono_even: fixes x y :: "'a :: {linordered_idom}"
   assumes "even n" and "\<bar>x\<bar> \<le> \<bar>y\<bar>"
   shows "x^n \<le> y^n"
 proof -
   have "0 \<le> \<bar>x\<bar>" by auto
   with `\<bar>x\<bar> \<le> \<bar>y\<bar>`

   thus ?thesis unfolding power_even_abs[OF `even n`] .
 qed
 
 lemma odd_pos: "odd (n::nat) \<Longrightarrow> 0 < n" by presburger
 
-lemma power_mono_odd: fixes x y :: "'a :: {ordered_idom}"
+lemma power_mono_odd: fixes x y :: "'a :: {linordered_idom}"
   assumes "odd n" and "x \<le> y"
   shows "x^n \<le> y^n"
 proof (cases "y < 0")
   case True with `x \<le> y` have "-y \<le> -x" and "0 \<le> -y" by auto
   hence "(-y)^n \<le> (-x)^n" by (rule power_mono)

 
 
 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}) ==> a=0"
+    "a ^ (2*k) \<le> (0::'a::{linordered_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}) | even n)"
+  "(0 \<le> a^n) = (0 \<le> (a::'a::{linordered_idom}) | 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)