--- a/src/HOL/Real/HahnBanach/FunctionNorm.thy Tue Sep 30 15:10:26 2003 +0200
+++ b/src/HOL/Real/HahnBanach/FunctionNorm.thy Tue Sep 30 15:10:59 2003 +0200
@@ -151,7 +151,7 @@
shows "b \<le> \<parallel>f\<parallel>\<hyphen>V"
proof -
have "lub (B V f) (\<parallel>f\<parallel>\<hyphen>V)"
- by (unfold B_def fn_norm_def) (rule fn_norm_works)
+ by (unfold B_def fn_norm_def) (rule fn_norm_works [OF _ continuous.intro])
from this and b show ?thesis ..
qed
@@ -161,7 +161,7 @@
shows "\<parallel>f\<parallel>\<hyphen>V \<le> y"
proof -
have "lub (B V f) (\<parallel>f\<parallel>\<hyphen>V)"
- by (unfold B_def fn_norm_def) (rule fn_norm_works)
+ by (unfold B_def fn_norm_def) (rule fn_norm_works [OF _ continuous.intro])
from this and b show ?thesis ..
qed
@@ -175,7 +175,7 @@
So it is @{text "\<ge> 0"} if all elements in @{text B} are @{text "\<ge>
0"}, provided the supremum exists and @{text B} is not empty. *}
have "lub (B V f) (\<parallel>f\<parallel>\<hyphen>V)"
- by (unfold B_def fn_norm_def) (rule fn_norm_works)
+ by (unfold B_def fn_norm_def) (rule fn_norm_works [OF _ continuous.intro])
moreover have "0 \<in> B V f" ..
ultimately show ?thesis ..
qed
@@ -198,7 +198,9 @@
also have "\<bar>\<dots>\<bar> = 0" by simp
also have "\<dots> \<le> \<parallel>f\<parallel>\<hyphen>V * \<parallel>x\<parallel>"
proof (rule real_le_mult_order1a)
- show "0 \<le> \<parallel>f\<parallel>\<hyphen>V" by (unfold B_def fn_norm_def) rule
+ show "0 \<le> \<parallel>f\<parallel>\<hyphen>V"
+ by (unfold B_def fn_norm_def)
+ (rule fn_norm_ge_zero [OF _ continuous.intro])
show "0 \<le> norm x" ..
qed
finally show "\<bar>f x\<bar> \<le> \<parallel>f\<parallel>\<hyphen>V * \<parallel>x\<parallel>" .
@@ -212,7 +214,7 @@
from x and neq have "\<bar>f x\<bar> * inverse \<parallel>x\<parallel> \<in> B V f"
by (auto simp add: B_def real_divide_def)
then show "\<bar>f x\<bar> * inverse \<parallel>x\<parallel> \<le> \<parallel>f\<parallel>\<hyphen>V"
- by (unfold B_def fn_norm_def) (rule fn_norm_ub)
+ by (unfold B_def fn_norm_def) (rule fn_norm_ub [OF _ continuous.intro])
qed
finally show ?thesis .
qed
@@ -254,6 +256,6 @@
qed
finally show ?thesis .
qed
-qed
+qed (simp_all! add: continuous_def)
end