src/HOL/Real/HahnBanach/Linearform.thy
 changeset 9035 371f023d3dbd parent 9013 9dd0274f76af child 9374 153853af318b
     1.1 --- a/src/HOL/Real/HahnBanach/Linearform.thy	Sun Jun 04 00:09:04 2000 +0200
1.2 +++ b/src/HOL/Real/HahnBanach/Linearform.thy	Sun Jun 04 19:39:29 2000 +0200
1.3 @@ -3,67 +3,67 @@
1.4      Author:     Gertrud Bauer, TU Munich
1.5  *)
1.6
1.9
1.10 -theory Linearform = VectorSpace:;
1.11 +theory Linearform = VectorSpace:
1.12
1.13  text{* A \emph{linear form} is a function on a vector
1.14 -space into the reals that is additive and multiplicative. *};
1.15 +space into the reals that is additive and multiplicative. *}
1.16
1.17  constdefs
1.18    is_linearform :: "['a::{minus, plus} set, 'a => real] => bool"
1.19    "is_linearform V f ==
1.20        (ALL x: V. ALL y: V. f (x + y) = f x + f y) &
1.21 -      (ALL x: V. ALL a. f (a (*) x) = a * (f x))";
1.22 +      (ALL x: V. ALL a. f (a (*) x) = a * (f x))"
1.23
1.24  lemma is_linearformI [intro]:
1.25    "[| !! x y. [| x : V; y : V |] ==> f (x + y) = f x + f y;
1.26      !! x c. x : V ==> f (c (*) x) = c * f x |]
1.27 - ==> is_linearform V f";
1.28 - by (unfold is_linearform_def) force;
1.29 + ==> is_linearform V f"
1.30 + by (unfold is_linearform_def) force
1.31
1.33 -  "[| is_linearform V f; x:V; y:V |] ==> f (x + y) = f x + f y";
1.34 -  by (unfold is_linearform_def) fast;
1.35 +  "[| is_linearform V f; x:V; y:V |] ==> f (x + y) = f x + f y"
1.36 +  by (unfold is_linearform_def) fast
1.37
1.38  lemma linearform_mult [intro??]:
1.39 -  "[| is_linearform V f; x:V |] ==>  f (a (*) x) = a * (f x)";
1.40 -  by (unfold is_linearform_def) fast;
1.41 +  "[| is_linearform V f; x:V |] ==>  f (a (*) x) = a * (f x)"
1.42 +  by (unfold is_linearform_def) fast
1.43
1.44  lemma linearform_neg [intro??]:
1.45    "[|  is_vectorspace V; is_linearform V f; x:V|]
1.46 -  ==> f (- x) = - f x";
1.47 -proof -;
1.48 -  assume "is_linearform V f" "is_vectorspace V" "x:V";
1.49 -  have "f (- x) = f ((- (#1::real)) (*) x)"; by (simp! add: negate_eq1);
1.50 -  also; have "... = (- #1) * (f x)"; by (rule linearform_mult);
1.51 -  also; have "... = - (f x)"; by (simp!);
1.52 -  finally; show ?thesis; .;
1.53 -qed;
1.54 +  ==> f (- x) = - f x"
1.55 +proof -
1.56 +  assume "is_linearform V f" "is_vectorspace V" "x:V"
1.57 +  have "f (- x) = f ((- #1) (*) x)" by (simp! add: negate_eq1)
1.58 +  also have "... = (- #1) * (f x)" by (rule linearform_mult)
1.59 +  also have "... = - (f x)" by (simp!)
1.60 +  finally show ?thesis .
1.61 +qed
1.62
1.63  lemma linearform_diff [intro??]:
1.64    "[| is_vectorspace V; is_linearform V f; x:V; y:V |]
1.65 -  ==> f (x - y) = f x - f y";
1.66 -proof -;
1.67 -  assume "is_vectorspace V" "is_linearform V f" "x:V" "y:V";
1.68 -  have "f (x - y) = f (x + - y)"; by (simp! only: diff_eq1);
1.69 -  also; have "... = f x + f (- y)";
1.70 -    by (rule linearform_add) (simp!)+;
1.71 -  also; have "f (- y) = - f y"; by (rule linearform_neg);
1.72 -  finally; show "f (x - y) = f x - f y"; by (simp!);
1.73 -qed;
1.74 +  ==> f (x - y) = f x - f y"
1.75 +proof -
1.76 +  assume "is_vectorspace V" "is_linearform V f" "x:V" "y:V"
1.77 +  have "f (x - y) = f (x + - y)" by (simp! only: diff_eq1)
1.78 +  also have "... = f x + f (- y)"
1.79 +    by (rule linearform_add) (simp!)+
1.80 +  also have "f (- y) = - f y" by (rule linearform_neg)
1.81 +  finally show "f (x - y) = f x - f y" by (simp!)
1.82 +qed
1.83
1.84 -text{* Every linear form yields $0$ for the $\zero$ vector.*};
1.85 +text{* Every linear form yields $0$ for the $\zero$ vector.*}
1.86
1.87  lemma linearform_zero [intro??, simp]:
1.88 -  "[| is_vectorspace V; is_linearform V f |] ==> f 00 = (#0::real)";
1.89 -proof -;
1.90 -  assume "is_vectorspace V" "is_linearform V f";
1.91 -  have "f 00 = f (00 - 00)"; by (simp!);
1.92 -  also; have "... = f 00 - f 00";
1.93 -    by (rule linearform_diff) (simp!)+;
1.94 -  also; have "... = (#0::real)"; by simp;
1.95 -  finally; show "f 00 = (#0::real)"; .;
1.96 -qed;
1.97 +  "[| is_vectorspace V; is_linearform V f |] ==> f 00 = #0"
1.98 +proof -
1.99 +  assume "is_vectorspace V" "is_linearform V f"
1.100 +  have "f 00 = f (00 - 00)" by (simp!)
1.101 +  also have "... = f 00 - f 00"
1.102 +    by (rule linearform_diff) (simp!)+
1.103 +  also have "... = #0" by simp
1.104 +  finally show "f 00 = #0" .
1.105 +qed
1.106
1.107 -end;
1.108 \ No newline at end of file
1.109 +end
1.110 \ No newline at end of file