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src/HOL/Real/HahnBanach/Linearform.thy

author | wenzelm |

Thu Jun 14 00:22:45 2007 +0200 (2007-06-14) | |

changeset 23378 | 1d138d6bb461 |

parent 16417 | 9bc16273c2d4 |

child 25762 | c03e9d04b3e4 |

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

tuned proofs: avoid implicit prems;

1 (* Title: HOL/Real/HahnBanach/Linearform.thy

2 ID: $Id$

3 Author: Gertrud Bauer, TU Munich

4 *)

6 header {* Linearforms *}

8 theory Linearform imports VectorSpace begin

10 text {*

11 A \emph{linear form} is a function on a vector space into the reals

12 that is additive and multiplicative.

13 *}

15 locale linearform = var V + var f +

16 assumes add [iff]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> f (x + y) = f x + f y"

17 and mult [iff]: "x \<in> V \<Longrightarrow> f (a \<cdot> x) = a * f x"

19 declare linearform.intro [intro?]

21 lemma (in linearform) neg [iff]:

22 includes vectorspace

23 shows "x \<in> V \<Longrightarrow> f (- x) = - f x"

24 proof -

25 assume x: "x \<in> V"

26 hence "f (- x) = f ((- 1) \<cdot> x)" by (simp add: negate_eq1)

27 also from x have "... = (- 1) * (f x)" by (rule mult)

28 also from x have "... = - (f x)" by simp

29 finally show ?thesis .

30 qed

32 lemma (in linearform) diff [iff]:

33 includes vectorspace

34 shows "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> f (x - y) = f x - f y"

35 proof -

36 assume x: "x \<in> V" and y: "y \<in> V"

37 hence "x - y = x + - y" by (rule diff_eq1)

38 also have "f ... = f x + f (- y)" by (rule add) (simp_all add: x y)

39 also have "f (- y) = - f y" using `vectorspace V` y by (rule neg)

40 finally show ?thesis by simp

41 qed

43 text {* Every linear form yields @{text 0} for the @{text 0} vector. *}

45 lemma (in linearform) zero [iff]:

46 includes vectorspace

47 shows "f 0 = 0"

48 proof -

49 have "f 0 = f (0 - 0)" by simp

50 also have "\<dots> = f 0 - f 0" using `vectorspace V` by (rule diff) simp_all

51 also have "\<dots> = 0" by simp

52 finally show ?thesis .

53 qed

55 end