src/HOL/Hahn_Banach/Linearform.thy

author | wenzelm |

Sat, 07 Apr 2012 16:41:59 +0200 | |

changeset 47389 | e8552cba702d |

parent 31795 | be3e1cc5005c |

child 58744 | c434e37f290e |

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

explicit checks stable_finished_theory/stable_command allow parallel asynchronous command transactions;
tuned;

(* Title: HOL/Hahn_Banach/Linearform.thy Author: Gertrud Bauer, TU Munich *) header {* Linearforms *} theory Linearform imports Vector_Space begin text {* A \emph{linear form} is a function on a vector space into the reals that is additive and multiplicative. *} locale linearform = fixes V :: "'a\<Colon>{minus, plus, zero, uminus} set" and f assumes add [iff]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> f (x + y) = f x + f y" and mult [iff]: "x \<in> V \<Longrightarrow> f (a \<cdot> x) = a * f x" declare linearform.intro [intro?] lemma (in linearform) neg [iff]: assumes "vectorspace V" shows "x \<in> V \<Longrightarrow> f (- x) = - f x" proof - interpret vectorspace V by fact assume x: "x \<in> V" then have "f (- x) = f ((- 1) \<cdot> x)" by (simp add: negate_eq1) also from x have "\<dots> = (- 1) * (f x)" by (rule mult) also from x have "\<dots> = - (f x)" by simp finally show ?thesis . qed lemma (in linearform) diff [iff]: assumes "vectorspace V" shows "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> f (x - y) = f x - f y" proof - interpret vectorspace V by fact assume x: "x \<in> V" and y: "y \<in> V" then have "x - y = x + - y" by (rule diff_eq1) also have "f \<dots> = f x + f (- y)" by (rule add) (simp_all add: x y) also have "f (- y) = - f y" using `vectorspace V` y by (rule neg) finally show ?thesis by simp qed text {* Every linear form yields @{text 0} for the @{text 0} vector. *} lemma (in linearform) zero [iff]: assumes "vectorspace V" shows "f 0 = 0" proof - interpret vectorspace V by fact have "f 0 = f (0 - 0)" by simp also have "\<dots> = f 0 - f 0" using `vectorspace V` by (rule diff) simp_all also have "\<dots> = 0" by simp finally show ?thesis . qed end