(*  Title:      HOL/Hahn_Banach/Linearform.thy

    Author:     Gertrud Bauer, TU Munich

*)

header {* Linearforms *}

theory Linearform

imports Vector_Space

begin

text {*

  A \emph{linear form} is a function on a vector space into the reals

  that is additive and multiplicative.

*}

locale linearform =

  fixes V :: "'a\<Colon>{minus, plus, zero, uminus} set" and f

  assumes add [iff]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> f (x + y) = f x + f y"

    and mult [iff]: "x \<in> V \<Longrightarrow> f (a \<cdot> x) = a * f x"

declare linearform.intro [intro?]

lemma (in linearform) neg [iff]:

  assumes "vectorspace V"

  shows "x \<in> V \<Longrightarrow> f (- x) = - f x"

proof -

  interpret vectorspace V by fact

  assume x: "x \<in> V"

  then have "f (- x) = f ((- 1) \<cdot> x)" by (simp add: negate_eq1)

  also from x have "\<dots> = (- 1) * (f x)" by (rule mult)

  also from x have "\<dots> = - (f x)" by simp

  finally show ?thesis .

qed

lemma (in linearform) diff [iff]:

  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"

  then have "x - y = x + - y" by (rule diff_eq1)

  also have "f \<dots> = f x + f (- y)" by (rule add) (simp_all add: x y)

  also have "f (- y) = - f y" using `vectorspace V` y by (rule neg)

  finally show ?thesis by simp

qed

text {* Every linear form yields @{text 0} for the @{text 0} vector. *}

lemma (in linearform) zero [iff]:

  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

  finally show ?thesis .

qed

end