(* Title: HOL/Hahn_Banach/Linearform.thy
Author: Gertrud Bauer, TU Munich
*)
section \<open>Linearforms\<close>
theory Linearform
imports Vector_Space
begin
text \<open>
A \emph{linear form} is a function on a vector space into the reals
that is additive and multiplicative.
\<close>
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 \<open>vectorspace V\<close> y by (rule neg)
finally show ?thesis by simp
qed
text \<open>Every linear form yields @{text 0} for the @{text 0} vector.\<close>
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 \<open>vectorspace V\<close> by (rule diff) simp_all
also have "\<dots> = 0" by simp
finally show ?thesis .
qed
end