1.1 --- a/src/HOL/Real/HahnBanach/Linearform.thy	Tue Dec 30 08:18:54 2008 +0100
     1.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.3 @@ -1,60 +0,0 @@
     1.4 -(*  Title:      HOL/Real/HahnBanach/Linearform.thy
     1.5 -    Author:     Gertrud Bauer, TU Munich
     1.6 -*)
     1.7 -
     1.8 -header {* Linearforms *}
     1.9 -
    1.10 -theory Linearform
    1.11 -imports VectorSpace
    1.12 -begin
    1.13 -
    1.14 -text {*
    1.15 -  A \emph{linear form} is a function on a vector space into the reals
    1.16 -  that is additive and multiplicative.
    1.17 -*}
    1.18 -
    1.19 -locale linearform =
    1.20 -  fixes V :: "'a\<Colon>{minus, plus, zero, uminus} set" and f
    1.21 -  assumes add [iff]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> f (x + y) = f x + f y"
    1.22 -    and mult [iff]: "x \<in> V \<Longrightarrow> f (a \<cdot> x) = a * f x"
    1.23 -
    1.24 -declare linearform.intro [intro?]
    1.25 -
    1.26 -lemma (in linearform) neg [iff]:
    1.27 -  assumes "vectorspace V"
    1.28 -  shows "x \<in> V \<Longrightarrow> f (- x) = - f x"
    1.29 -proof -
    1.30 -  interpret vectorspace V by fact
    1.31 -  assume x: "x \<in> V"
    1.32 -  then have "f (- x) = f ((- 1) \<cdot> x)" by (simp add: negate_eq1)
    1.33 -  also from x have "\<dots> = (- 1) * (f x)" by (rule mult)
    1.34 -  also from x have "\<dots> = - (f x)" by simp
    1.35 -  finally show ?thesis .
    1.36 -qed
    1.37 -
    1.38 -lemma (in linearform) diff [iff]:
    1.39 -  assumes "vectorspace V"
    1.40 -  shows "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> f (x - y) = f x - f y"
    1.41 -proof -
    1.42 -  interpret vectorspace V by fact
    1.43 -  assume x: "x \<in> V" and y: "y \<in> V"
    1.44 -  then have "x - y = x + - y" by (rule diff_eq1)
    1.45 -  also have "f \<dots> = f x + f (- y)" by (rule add) (simp_all add: x y)
    1.46 -  also have "f (- y) = - f y" using `vectorspace V` y by (rule neg)
    1.47 -  finally show ?thesis by simp
    1.48 -qed
    1.49 -
    1.50 -text {* Every linear form yields @{text 0} for the @{text 0} vector. *}
    1.51 -
    1.52 -lemma (in linearform) zero [iff]:
    1.53 -  assumes "vectorspace V"
    1.54 -  shows "f 0 = 0"
    1.55 -proof -
    1.56 -  interpret vectorspace V by fact
    1.57 -  have "f 0 = f (0 - 0)" by simp
    1.58 -  also have "\<dots> = f 0 - f 0" using `vectorspace V` by (rule diff) simp_all
    1.59 -  also have "\<dots> = 0" by simp
    1.60 -  finally show ?thesis .
    1.61 -qed
    1.62 -
    1.63 -end