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src/HOL/Real/HahnBanach/Linearform.thy

author | wenzelm |

Tue, 21 Sep 1999 17:31:20 +0200 | |

changeset 7566 | c5a3f980a7af |

parent 7535 | 599d3414b51d |

child 7656 | 2f18c0ffc348 |

permissions | -rw-r--r-- |

accomodate refined facts handling;

(* Title: HOL/Real/HahnBanach/Linearform.thy ID: $Id$ Author: Gertrud Bauer, TU Munich *) theory Linearform = LinearSpace:; section {* linearforms *}; 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 (simp! add: vs_mult_minus_1); 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!!]: "[| 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;