--- a/src/HOL/Library/Inner_Product.thy Tue Aug 09 10:30:00 2011 -0700
+++ b/src/HOL/Library/Inner_Product.thy Tue Aug 09 10:42:07 2011 -0700
@@ -123,8 +123,7 @@
unfolding power2_sum power2_norm_eq_inner
by (simp add: inner_add inner_commute)
show "0 \<le> norm x + norm y"
- unfolding norm_eq_sqrt_inner
- by (simp add: add_nonneg_nonneg)
+ unfolding norm_eq_sqrt_inner by simp
qed
have "sqrt (a\<twosuperior> * inner x x) = \<bar>a\<bar> * sqrt (inner x x)"
by (simp add: real_sqrt_mult_distrib)
@@ -217,7 +216,7 @@
show "inner (scaleR r x) y = r * inner x y"
unfolding inner_complex_def by (simp add: right_distrib)
show "0 \<le> inner x x"
- unfolding inner_complex_def by (simp add: add_nonneg_nonneg)
+ unfolding inner_complex_def by simp
show "inner x x = 0 \<longleftrightarrow> x = 0"
unfolding inner_complex_def
by (simp add: add_nonneg_eq_0_iff complex_Re_Im_cancel_iff)
--- a/src/HOL/Library/Product_Vector.thy Tue Aug 09 10:30:00 2011 -0700
+++ b/src/HOL/Library/Product_Vector.thy Tue Aug 09 10:42:07 2011 -0700
@@ -453,9 +453,9 @@
assumes x: "0 \<le> x" and y: "0 \<le> y"
shows "sqrt (x + y) \<le> sqrt x + sqrt y"
apply (rule power2_le_imp_le)
-apply (simp add: real_sum_squared_expand add_nonneg_nonneg x y)
+apply (simp add: real_sum_squared_expand x y)
apply (simp add: mult_nonneg_nonneg x y)
-apply (simp add: add_nonneg_nonneg x y)
+apply (simp add: x y)
done
lemma bounded_linear_Pair: