--- a/src/HOL/Real/HahnBanach/Linearform.thy Tue Jul 15 16:02:10 2008 +0200
+++ b/src/HOL/Real/HahnBanach/Linearform.thy Tue Jul 15 16:50:09 2008 +0200
@@ -20,9 +20,10 @@
declare linearform.intro [intro?]
lemma (in linearform) neg [iff]:
- includes vectorspace
+ assumes "vectorspace V"
shows "x \<in> V \<Longrightarrow> f (- x) = - f x"
proof -
+ interpret vectorspace [V] by fact
assume x: "x \<in> V"
hence "f (- x) = f ((- 1) \<cdot> x)" by (simp add: negate_eq1)
also from x have "... = (- 1) * (f x)" by (rule mult)
@@ -31,9 +32,10 @@
qed
lemma (in linearform) diff [iff]:
- includes vectorspace
+ assumes "vectorspace V"
shows "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> f (x - y) = f x - f y"
proof -
+ interpret vectorspace [V] by fact
assume x: "x \<in> V" and y: "y \<in> V"
hence "x - y = x + - y" by (rule diff_eq1)
also have "f ... = f x + f (- y)" by (rule add) (simp_all add: x y)
@@ -44,9 +46,10 @@
text {* Every linear form yields @{text 0} for the @{text 0} vector. *}
lemma (in linearform) zero [iff]:
- includes vectorspace
+ assumes "vectorspace V"
shows "f 0 = 0"
proof -
+ interpret vectorspace [V] by fact
have "f 0 = f (0 - 0)" by simp
also have "\<dots> = f 0 - f 0" using `vectorspace V` by (rule diff) simp_all
also have "\<dots> = 0" by simp