diff -r 3cb310f40a3a -r 5e5b9813cce7 src/HOL/Real/HahnBanach/Linearform.thy
--- 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;