1 (* Title: HOL/Real/HahnBanach/Linearform.thy |
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2 ID: $Id$ |
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3 Author: Gertrud Bauer, TU Munich |
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4 *) |
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5 |
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6 header {* Linearforms *} |
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7 |
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8 theory Linearform |
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9 imports VectorSpace |
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10 begin |
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11 |
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12 text {* |
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13 A \emph{linear form} is a function on a vector space into the reals |
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14 that is additive and multiplicative. |
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15 *} |
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16 |
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17 locale linearform = var V + var f + |
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18 constrains V :: "'a\<Colon>{minus, plus, zero, uminus} set" |
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19 assumes add [iff]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> f (x + y) = f x + f y" |
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20 and mult [iff]: "x \<in> V \<Longrightarrow> f (a \<cdot> x) = a * f x" |
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21 |
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22 declare linearform.intro [intro?] |
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23 |
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24 lemma (in linearform) neg [iff]: |
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25 assumes "vectorspace V" |
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26 shows "x \<in> V \<Longrightarrow> f (- x) = - f x" |
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27 proof - |
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28 interpret vectorspace [V] by fact |
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29 assume x: "x \<in> V" |
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30 then have "f (- x) = f ((- 1) \<cdot> x)" by (simp add: negate_eq1) |
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31 also from x have "\<dots> = (- 1) * (f x)" by (rule mult) |
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32 also from x have "\<dots> = - (f x)" by simp |
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33 finally show ?thesis . |
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34 qed |
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35 |
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36 lemma (in linearform) diff [iff]: |
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37 assumes "vectorspace V" |
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38 shows "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> f (x - y) = f x - f y" |
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39 proof - |
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40 interpret vectorspace [V] by fact |
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41 assume x: "x \<in> V" and y: "y \<in> V" |
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42 then have "x - y = x + - y" by (rule diff_eq1) |
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43 also have "f \<dots> = f x + f (- y)" by (rule add) (simp_all add: x y) |
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44 also have "f (- y) = - f y" using `vectorspace V` y by (rule neg) |
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45 finally show ?thesis by simp |
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46 qed |
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47 |
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48 text {* Every linear form yields @{text 0} for the @{text 0} vector. *} |
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49 |
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50 lemma (in linearform) zero [iff]: |
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51 assumes "vectorspace V" |
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52 shows "f 0 = 0" |
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53 proof - |
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54 interpret vectorspace [V] by fact |
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55 have "f 0 = f (0 - 0)" by simp |
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56 also have "\<dots> = f 0 - f 0" using `vectorspace V` by (rule diff) simp_all |
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57 also have "\<dots> = 0" by simp |
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58 finally show ?thesis . |
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59 qed |
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60 |
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61 end |
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