(* Title: HOL/Real/HahnBanach/Linearform.thy
ID: $Id$
Author: Gertrud Bauer, TU Munich
*)
header {* Linearforms *};
theory Linearform = LinearSpace:;
constdefs
is_linearform :: "['a set, 'a => real] => bool"
"is_linearform V f ==
(ALL x: V. ALL y: V. f (x [+] y) = f x + f y) &
(ALL x: V. ALL a. f (a [*] x) = a * (f x))";
lemma is_linearformI [intro]:
"[| !! x y. [| x : V; y : V |] ==> f (x [+] y) = f x + f y;
!! x c. x : V ==> f (c [*] x) = c * f x |]
==> is_linearform V f";
by (unfold is_linearform_def) force;
lemma linearform_add_linear [intro!!]:
"[| is_linearform V f; x:V; y:V |] ==> f (x [+] y) = f x + f y";
by (unfold is_linearform_def) auto;
lemma linearform_mult_linear [intro!!]:
"[| is_linearform V f; x:V |] ==> f (a [*] x) = a * (f x)";
by (unfold is_linearform_def) auto;
lemma linearform_neg_linear [intro!!]:
"[| is_vectorspace V; is_linearform V f; x:V|] ==> f ([-] x) = - f x";
proof -;
assume "is_linearform V f" "is_vectorspace V" "x:V";
have "f ([-] x) = f ((- 1r) [*] x)"; by (unfold negate_def) simp;
also; have "... = (- 1r) * (f x)"; by (rule linearform_mult_linear);
also; have "... = - (f x)"; by (simp!);
finally; show ?thesis; .;
qed;
lemma linearform_diff_linear [intro!!]:
"[| is_vectorspace V; is_linearform V f; x:V; y:V |]
==> f (x [-] y) = f x - f y";
proof -;
assume "is_vectorspace V" "is_linearform V f" "x:V" "y:V";
have "f (x [-] y) = f (x [+] [-] y)"; by (simp only: diff_def);
also; have "... = f x + f ([-] y)";
by (rule linearform_add_linear) (simp!)+;
also; have "f ([-] y) = - f y"; by (rule linearform_neg_linear);
finally; show "f (x [-] y) = f x - f y"; by (simp!);
qed;
lemma linearform_zero [intro!!, simp]:
"[| is_vectorspace V; is_linearform V f |] ==> f <0> = 0r";
proof -;
assume "is_vectorspace V" "is_linearform V f";
have "f <0> = f (<0> [-] <0>)"; by (simp!);
also; have "... = f <0> - f <0>";
by (rule linearform_diff_linear) (simp!)+;
also; have "... = 0r"; by simp;
finally; show "f <0> = 0r"; .;
qed;
end;