src/HOL/Hahn_Banach/Linearform.thy
 author blanchet Thu Sep 11 18:54:36 2014 +0200 (2014-09-11) changeset 58306 117ba6cbe414 parent 31795 be3e1cc5005c child 58744 c434e37f290e permissions -rw-r--r--
renamed 'rep_datatype' to 'old_rep_datatype' (HOL)
1 (*  Title:      HOL/Hahn_Banach/Linearform.thy
2     Author:     Gertrud Bauer, TU Munich
3 *)
5 header {* Linearforms *}
7 theory Linearform
8 imports Vector_Space
9 begin
11 text {*
12   A \emph{linear form} is a function on a vector space into the reals
13   that is additive and multiplicative.
14 *}
16 locale linearform =
17   fixes V :: "'a\<Colon>{minus, plus, zero, uminus} set" and f
18   assumes add [iff]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> f (x + y) = f x + f y"
19     and mult [iff]: "x \<in> V \<Longrightarrow> f (a \<cdot> x) = a * f x"
21 declare linearform.intro [intro?]
23 lemma (in linearform) neg [iff]:
24   assumes "vectorspace V"
25   shows "x \<in> V \<Longrightarrow> f (- x) = - f x"
26 proof -
27   interpret vectorspace V by fact
28   assume x: "x \<in> V"
29   then have "f (- x) = f ((- 1) \<cdot> x)" by (simp add: negate_eq1)
30   also from x have "\<dots> = (- 1) * (f x)" by (rule mult)
31   also from x have "\<dots> = - (f x)" by simp
32   finally show ?thesis .
33 qed
35 lemma (in linearform) diff [iff]:
36   assumes "vectorspace V"
37   shows "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> f (x - y) = f x - f y"
38 proof -
39   interpret vectorspace V by fact
40   assume x: "x \<in> V" and y: "y \<in> V"
41   then have "x - y = x + - y" by (rule diff_eq1)
42   also have "f \<dots> = f x + f (- y)" by (rule add) (simp_all add: x y)
43   also have "f (- y) = - f y" using vectorspace V y by (rule neg)
44   finally show ?thesis by simp
45 qed
47 text {* Every linear form yields @{text 0} for the @{text 0} vector. *}
49 lemma (in linearform) zero [iff]:
50   assumes "vectorspace V"
51   shows "f 0 = 0"
52 proof -
53   interpret vectorspace V by fact
54   have "f 0 = f (0 - 0)" by simp
55   also have "\<dots> = f 0 - f 0" using vectorspace V by (rule diff) simp_all
56   also have "\<dots> = 0" by simp
57   finally show ?thesis .
58 qed
60 end