author ballarin Tue, 30 Sep 2003 15:10:59 +0200 changeset 14214 5369d671f100 parent 14213 7bf882b0a51e child 14215 ebf291f3b449
Improvements to Isar/Locales: premises generated by "includes" elements changed.
```--- 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

end```
```--- a/src/HOL/Real/HahnBanach/HahnBanach.thy	Tue Sep 30 15:10:26 2003 +0200
+++ b/src/HOL/Real/HahnBanach/HahnBanach.thy	Tue Sep 30 15:10:59 2003 +0200
@@ -339,10 +339,11 @@
have FE: "F \<unlhd> E" .
have F: "vectorspace F" ..
have linearform: "linearform F f" .
-  have F_norm: "normed_vectorspace F norm" ..
+  have F_norm: "normed_vectorspace F norm"
+    by (rule subspace_normed_vs [OF _ _ _ norm.intro])
have ge_zero: "0 \<le> \<parallel>f\<parallel>\<hyphen>F"
by (rule normed_vectorspace.fn_norm_ge_zero
-      [OF F_norm, folded B_def fn_norm_def])
+      [OF F_norm _ continuous.intro, folded B_def fn_norm_def])

txt {* We define a function @{text p} on @{text E} as follows:
@{text "p x = \<parallel>f\<parallel> \<cdot> \<parallel>x\<parallel>"} *}
@@ -394,7 +395,7 @@
fix x assume "x \<in> F"
show "\<bar>f x\<bar> \<le> p x"
by (unfold p_def) (rule normed_vectorspace.fn_norm_le_cong
-        [OF F_norm, folded B_def fn_norm_def])
+        [OF F_norm _ continuous.intro, folded B_def fn_norm_def])
qed

txt {* Using the fact that @{text p} is a seminorm and @{text f} is bounded
@@ -443,7 +444,7 @@
with b show "\<bar>g x\<bar> \<le> \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>"
by (simp only: p_def)
qed
-    from continuous.axioms [OF g_cont] this ge_zero
+    from linearformE g_cont this ge_zero
show "\<parallel>g\<parallel>\<hyphen>E \<le> \<parallel>f\<parallel>\<hyphen>F"
by (rule fn_norm_least [of g, folded B_def fn_norm_def])

@@ -456,7 +457,7 @@
fix x assume x: "x \<in> F"
from a have "g x = f x" ..
hence "\<bar>f x\<bar> = \<bar>g x\<bar>" by (simp only:)
-	also from continuous.axioms [OF g_cont]
+	also from linearformE g_cont
have "\<dots> \<le> \<parallel>g\<parallel>\<hyphen>E * \<parallel>x\<parallel>"
proof (rule fn_norm_le_cong [of g, folded B_def fn_norm_def])
from FE x show "x \<in> E" ..
@@ -464,8 +465,10 @@
finally show "\<bar>f x\<bar> \<le> \<parallel>g\<parallel>\<hyphen>E * \<parallel>x\<parallel>" .
qed
show "0 \<le> \<parallel>g\<parallel>\<hyphen>E"
-	using continuous.axioms [OF g_cont]
+	using linearformE g_cont
by (rule fn_norm_ge_zero [of g, folded B_def fn_norm_def])
+    next
+      show "continuous F norm f" by (rule continuous.intro)
qed
qed
with linearformE a g_cont show ?thesis by blast```