diff -r 8e562d56d404 -r 0b8a5fd339ab src/HOL/Nat_Numeral.thy
--- a/src/HOL/Nat_Numeral.thy	Sat Mar 06 16:02:22 2010 -0800
+++ b/src/HOL/Nat_Numeral.thy	Sat Mar 06 18:24:30 2010 -0800
@@ -113,7 +113,7 @@
 
 end
 
-context linordered_ring_strict
+context linordered_ring
 begin
 
 lemma sum_squares_ge_zero:
@@ -124,6 +124,11 @@
   "\<not> x * x + y * y < 0"
   by (simp add: not_less sum_squares_ge_zero)
 
+end
+
+context linordered_ring_strict
+begin
+
 lemma sum_squares_eq_zero_iff:
   "x * x + y * y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
   by (simp add: add_nonneg_eq_0_iff)
@@ -134,14 +139,7 @@
 
 lemma sum_squares_gt_zero_iff:
   "0 < x * x + y * y \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
-proof -
-  have "x * x + y * y \<noteq> 0 \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
-    by (simp add: sum_squares_eq_zero_iff)
-  then have "0 \<noteq> x * x + y * y \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
-    by auto
-  then show ?thesis
-    by (simp add: less_le sum_squares_ge_zero)
-qed
+  by (simp add: not_le [symmetric] sum_squares_le_zero_iff)
 
 end