src/HOL/Real/HahnBanach/Linearform.thy
changeset 7535 599d3414b51d
child 7566 c5a3f980a7af
     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.2 +++ b/src/HOL/Real/HahnBanach/Linearform.thy	Fri Sep 10 17:28:51 1999 +0200
     1.3 @@ -0,0 +1,53 @@
     1.4 +
     1.5 +theory Linearform = LinearSpace:;
     1.6 +
     1.7 +section {* linearforms *};
     1.8 +
     1.9 +constdefs
    1.10 +  is_linearform :: "['a set, 'a => real] => bool" 
    1.11 +  "is_linearform V f == 
    1.12 +      (ALL x: V. ALL y: V. f (x [+] y) = f x + f y) &
    1.13 +      (ALL x: V. ALL a. f (a [*] x) = a * (f x))"; 
    1.14 +
    1.15 +lemma is_linearformI [intro]: "[| !! x y. [| x : V; y : V |] ==> f (x [+] y) = f x + f y;
    1.16 +    !! x c. x : V ==> f (c [*] x) = c * f x |]
    1.17 + ==> is_linearform V f";
    1.18 + by (unfold is_linearform_def, force);
    1.19 +
    1.20 +lemma linearform_add_linear: "[| is_linearform V f; x:V; y:V |] ==> f (x [+] y) = f x + f y";
    1.21 + by (unfold is_linearform_def, auto);
    1.22 +
    1.23 +lemma linearform_mult_linear: "[| is_linearform V f; x:V |] ==>  f (a [*] x) = a * (f x)"; 
    1.24 + by (unfold is_linearform_def, auto);
    1.25 +
    1.26 +lemma linearform_neg_linear:
    1.27 +  "[|  is_vectorspace V; is_linearform V f; x:V|] ==> f ([-] x) = - f x";
    1.28 +proof -; 
    1.29 +  assume "is_linearform V f" "is_vectorspace V" "x:V"; 
    1.30 +  have "f ([-] x) = f ((- 1r) [*] x)"; by (asm_simp add: vs_mult_minus_1);
    1.31 +  also; have "... = (- 1r) * (f x)"; by (rule linearform_mult_linear);
    1.32 +  also; have "... = - (f x)"; by asm_simp;
    1.33 +  finally; show ?thesis; .;
    1.34 +qed;
    1.35 +
    1.36 +lemma linearform_diff_linear: 
    1.37 +  "[| is_vectorspace V; is_linearform V f; x:V; y:V |] ==> f (x [-] y) = f x - f y";  
    1.38 +proof -;
    1.39 +  assume "is_vectorspace V" "is_linearform V f" "x:V" "y:V";
    1.40 +  have "f (x [-] y) = f (x [+] [-] y)"; by (simp only: diff_def);
    1.41 +  also; have "... = f x + f ([-] y)"; by (rule linearform_add_linear) (asm_simp+);
    1.42 +  also; have "f ([-] y) = - f y"; by (rule linearform_neg_linear);
    1.43 +  finally; show "f (x [-] y) = f x - f y"; by asm_simp;
    1.44 +qed;
    1.45 +
    1.46 +lemma linearform_zero: "[| is_vectorspace V; is_linearform V f |] ==> f <0> = 0r"; 
    1.47 +proof -; 
    1.48 +  assume "is_vectorspace V" "is_linearform V f";
    1.49 +  have "f <0> = f (<0> [-] <0>)"; by asm_simp;
    1.50 +  also; have "... = f <0> - f <0>"; by (rule linearform_diff_linear) asm_simp+;
    1.51 +  also; have "... = 0r"; by simp;
    1.52 +  finally; show "f <0> = 0r"; .;
    1.53 +qed; 
    1.54 +
    1.55 +end;
    1.56 +