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src/HOL/Hahn_Banach/Linearform.thy

author | hoelzl |

Thu Sep 02 10:14:32 2010 +0200 (2010-09-02) | |

changeset 39072 | 1030b1a166ef |

parent 31795 | be3e1cc5005c |

child 58744 | c434e37f290e |

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

Add lessThan_Suc_eq_insert_0

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