--- a/src/HOL/Real/HahnBanach/Linearform.thy Fri Oct 22 18:41:00 1999 +0200
+++ b/src/HOL/Real/HahnBanach/Linearform.thy Fri Oct 22 20:14:31 1999 +0200
@@ -5,33 +5,38 @@
header {* Linearforms *};
-theory Linearform = LinearSpace:;
+theory Linearform = VectorSpace:;
+
+text{* A \emph{linearform} is a function on a vector
+space into the reals that is linear w.~r.~t.~addition and skalar
+multiplikation. *};
constdefs
- is_linearform :: "['a set, 'a => real] => bool"
+ is_linearform :: "['a::{minus, plus} 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))";
+ (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 |]
+ "[| !! 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;
+ "[| is_linearform V f; x:V; y:V |] ==> f (x + y) = f x + f y";
+ by (unfold is_linearform_def) fast;
lemma linearform_mult_linear [intro!!]:
- "[| is_linearform V f; x:V |] ==> f (a [*] x) = a * (f x)";
- by (unfold is_linearform_def) auto;
+ "[| is_linearform V f; x:V |] ==> f (a <*> x) = a * (f x)";
+ by (unfold is_linearform_def) fast;
lemma linearform_neg_linear [intro!!]:
- "[| is_vectorspace V; is_linearform V f; x:V|] ==> f ([-] x) = - f x";
+ "[| 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;
+ have "f (- x) = f ((- 1r) <*> x)"; by (simp! add: negate_eq1);
also; have "... = (- 1r) * (f x)"; by (rule linearform_mult_linear);
also; have "... = - (f x)"; by (simp!);
finally; show ?thesis; .;
@@ -39,21 +44,23 @@
lemma linearform_diff_linear [intro!!]:
"[| is_vectorspace V; is_linearform V f; x:V; y:V |]
- ==> f (x [-] y) = f x - f y";
+ ==> 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)";
+ have "f (x - y) = f (x + - y)"; by (simp! only: diff_eq1);
+ 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!);
+ also; have "f (- y) = - f y"; by (rule linearform_neg_linear);
+ finally; show "f (x - y) = f x - f y"; by (simp!);
qed;
+text{* Every linearform yields $0$ for the $\zero$ vector.*};
+
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!);
+ 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;