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src/HOL/Real/HahnBanach/Linearform.thy

changeset 7535 | 599d3414b51d |

child 7566 | c5a3f980a7af |

--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Real/HahnBanach/Linearform.thy Fri Sep 10 17:28:51 1999 +0200 @@ -0,0 +1,53 @@ + +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: "[| 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: "[| is_linearform V f; x:V |] ==> f (a [*] x) = a * (f x)"; + by (unfold is_linearform_def, auto); + +lemma linearform_neg_linear: + "[| 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 (asm_simp add: vs_mult_minus_1); + also; have "... = (- 1r) * (f x)"; by (rule linearform_mult_linear); + also; have "... = - (f x)"; by asm_simp; + finally; show ?thesis; .; +qed; + +lemma linearform_diff_linear: + "[| 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) (asm_simp+); + also; have "f ([-] y) = - f y"; by (rule linearform_neg_linear); + finally; show "f (x [-] y) = f x - f y"; by asm_simp; +qed; + +lemma linearform_zero: "[| 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 asm_simp; + also; have "... = f <0> - f <0>"; by (rule linearform_diff_linear) asm_simp+; + also; have "... = 0r"; by simp; + finally; show "f <0> = 0r"; .; +qed; + +end; +