author | ballarin |
Tue, 15 Jul 2008 16:50:09 +0200 | |
changeset 27611 | 2c01c0bdb385 |
parent 25762 | c03e9d04b3e4 |
child 27612 | d3eb431db035 |
permissions | -rw-r--r-- |
7566 | 1 |
(* Title: HOL/Real/HahnBanach/Linearform.thy |
2 |
ID: $Id$ |
|
3 |
Author: Gertrud Bauer, TU Munich |
|
4 |
*) |
|
7535
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
5 |
|
9035 | 6 |
header {* Linearforms *} |
7535
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
7 |
|
16417 | 8 |
theory Linearform imports VectorSpace begin |
7917 | 9 |
|
10687 | 10 |
text {* |
11 |
A \emph{linear form} is a function on a vector space into the reals |
|
12 |
that is additive and multiplicative. |
|
13 |
*} |
|
7535
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
14 |
|
13515 | 15 |
locale linearform = var V + var f + |
25762 | 16 |
constrains V :: "'a\<Colon>{minus, plus, zero, uminus} set" |
13515 | 17 |
assumes add [iff]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> f (x + y) = f x + f y" |
18 |
and mult [iff]: "x \<in> V \<Longrightarrow> f (a \<cdot> x) = a * f x" |
|
7535
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
19 |
|
14254
342634f38451
Isar/Locales: <loc>.intro and <loc>.axioms no longer intro? and elim? by
ballarin
parents:
13547
diff
changeset
|
20 |
declare linearform.intro [intro?] |
342634f38451
Isar/Locales: <loc>.intro and <loc>.axioms no longer intro? and elim? by
ballarin
parents:
13547
diff
changeset
|
21 |
|
13547 | 22 |
lemma (in linearform) neg [iff]: |
27611 | 23 |
assumes "vectorspace V" |
13547 | 24 |
shows "x \<in> V \<Longrightarrow> f (- x) = - f x" |
10687 | 25 |
proof - |
27611 | 26 |
interpret vectorspace [V] by fact |
13515 | 27 |
assume x: "x \<in> V" |
28 |
hence "f (- x) = f ((- 1) \<cdot> x)" by (simp add: negate_eq1) |
|
29 |
also from x have "... = (- 1) * (f x)" by (rule mult) |
|
30 |
also from x have "... = - (f x)" by simp |
|
9035 | 31 |
finally show ?thesis . |
32 |
qed |
|
7535
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
33 |
|
13547 | 34 |
lemma (in linearform) diff [iff]: |
27611 | 35 |
assumes "vectorspace V" |
13547 | 36 |
shows "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> f (x - y) = f x - f y" |
9035 | 37 |
proof - |
27611 | 38 |
interpret vectorspace [V] by fact |
13515 | 39 |
assume x: "x \<in> V" and y: "y \<in> V" |
40 |
hence "x - y = x + - y" by (rule diff_eq1) |
|
13547 | 41 |
also have "f ... = f x + f (- y)" by (rule add) (simp_all add: x y) |
23378 | 42 |
also have "f (- y) = - f y" using `vectorspace V` y by (rule neg) |
13515 | 43 |
finally show ?thesis by simp |
9035 | 44 |
qed |
7535
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
45 |
|
10687 | 46 |
text {* Every linear form yields @{text 0} for the @{text 0} vector. *} |
7917 | 47 |
|
13547 | 48 |
lemma (in linearform) zero [iff]: |
27611 | 49 |
assumes "vectorspace V" |
13547 | 50 |
shows "f 0 = 0" |
10687 | 51 |
proof - |
27611 | 52 |
interpret vectorspace [V] by fact |
13515 | 53 |
have "f 0 = f (0 - 0)" by simp |
23378 | 54 |
also have "\<dots> = f 0 - f 0" using `vectorspace V` by (rule diff) simp_all |
13515 | 55 |
also have "\<dots> = 0" by simp |
56 |
finally show ?thesis . |
|
10687 | 57 |
qed |
7535
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
58 |
|
10687 | 59 |
end |