src/HOL/Hahn_Banach/Vector_Space.thy
 author wenzelm Mon Apr 25 16:09:26 2016 +0200 (2016-04-25) changeset 63040 eb4ddd18d635 parent 61879 e4f9d8f094fe child 69597 ff784d5a5bfb permissions -rw-r--r--
eliminated old 'def';
1 (*  Title:      HOL/Hahn_Banach/Vector_Space.thy
2     Author:     Gertrud Bauer, TU Munich
3 *)
5 section \<open>Vector spaces\<close>
7 theory Vector_Space
8 imports Complex_Main Bounds
9 begin
11 subsection \<open>Signature\<close>
13 text \<open>
14   For the definition of real vector spaces a type @{typ 'a} of the sort
15   \<open>{plus, minus, zero}\<close> is considered, on which a real scalar multiplication
16   \<open>\<cdot>\<close> is declared.
17 \<close>
19 consts
20   prod :: "real \<Rightarrow> 'a::{plus,minus,zero} \<Rightarrow> 'a"  (infixr "\<cdot>" 70)
23 subsection \<open>Vector space laws\<close>
25 text \<open>
26   A \<^emph>\<open>vector space\<close> is a non-empty set \<open>V\<close> of elements from @{typ 'a} with the
27   following vector space laws: The set \<open>V\<close> is closed under addition and scalar
28   multiplication, addition is associative and commutative; \<open>- x\<close> is the
29   inverse of \<open>x\<close> wrt.\ addition and \<open>0\<close> is the neutral element of addition.
30   Addition and multiplication are distributive; scalar multiplication is
31   associative and the real number \<open>1\<close> is the neutral element of scalar
32   multiplication.
33 \<close>
35 locale vectorspace =
36   fixes V
37   assumes non_empty [iff, intro?]: "V \<noteq> {}"
38     and add_closed [iff]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + y \<in> V"
39     and mult_closed [iff]: "x \<in> V \<Longrightarrow> a \<cdot> x \<in> V"
40     and add_assoc: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> (x + y) + z = x + (y + z)"
41     and add_commute: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + y = y + x"
42     and diff_self [simp]: "x \<in> V \<Longrightarrow> x - x = 0"
43     and add_zero_left [simp]: "x \<in> V \<Longrightarrow> 0 + x = x"
44     and add_mult_distrib1: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> a \<cdot> (x + y) = a \<cdot> x + a \<cdot> y"
45     and add_mult_distrib2: "x \<in> V \<Longrightarrow> (a + b) \<cdot> x = a \<cdot> x + b \<cdot> x"
46     and mult_assoc: "x \<in> V \<Longrightarrow> (a * b) \<cdot> x = a \<cdot> (b \<cdot> x)"
47     and mult_1 [simp]: "x \<in> V \<Longrightarrow> 1 \<cdot> x = x"
48     and negate_eq1: "x \<in> V \<Longrightarrow> - x = (- 1) \<cdot> x"
49     and diff_eq1: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x - y = x + - y"
50 begin
52 lemma negate_eq2: "x \<in> V \<Longrightarrow> (- 1) \<cdot> x = - x"
53   by (rule negate_eq1 [symmetric])
55 lemma negate_eq2a: "x \<in> V \<Longrightarrow> -1 \<cdot> x = - x"
56   by (simp add: negate_eq1)
58 lemma diff_eq2: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + - y = x - y"
59   by (rule diff_eq1 [symmetric])
61 lemma diff_closed [iff]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x - y \<in> V"
62   by (simp add: diff_eq1 negate_eq1)
64 lemma neg_closed [iff]: "x \<in> V \<Longrightarrow> - x \<in> V"
65   by (simp add: negate_eq1)
68   "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> x + (y + z) = y + (x + z)"
69 proof -
70   assume xyz: "x \<in> V"  "y \<in> V"  "z \<in> V"
71   then have "x + (y + z) = (x + y) + z"
72     by (simp only: add_assoc)
73   also from xyz have "\<dots> = (y + x) + z" by (simp only: add_commute)
74   also from xyz have "\<dots> = y + (x + z)" by (simp only: add_assoc)
75   finally show ?thesis .
76 qed
81 text \<open>
82   The existence of the zero element of a vector space follows from the
83   non-emptiness of carrier set.
84 \<close>
86 lemma zero [iff]: "0 \<in> V"
87 proof -
88   from non_empty obtain x where x: "x \<in> V" by blast
89   then have "0 = x - x" by (rule diff_self [symmetric])
90   also from x x have "\<dots> \<in> V" by (rule diff_closed)
91   finally show ?thesis .
92 qed
94 lemma add_zero_right [simp]: "x \<in> V \<Longrightarrow>  x + 0 = x"
95 proof -
96   assume x: "x \<in> V"
97   from this and zero have "x + 0 = 0 + x" by (rule add_commute)
98   also from x have "\<dots> = x" by (rule add_zero_left)
99   finally show ?thesis .
100 qed
102 lemma mult_assoc2: "x \<in> V \<Longrightarrow> a \<cdot> b \<cdot> x = (a * b) \<cdot> x"
103   by (simp only: mult_assoc)
105 lemma diff_mult_distrib1: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> a \<cdot> (x - y) = a \<cdot> x - a \<cdot> y"
106   by (simp add: diff_eq1 negate_eq1 add_mult_distrib1 mult_assoc2)
108 lemma diff_mult_distrib2: "x \<in> V \<Longrightarrow> (a - b) \<cdot> x = a \<cdot> x - (b \<cdot> x)"
109 proof -
110   assume x: "x \<in> V"
111   have " (a - b) \<cdot> x = (a + - b) \<cdot> x"
112     by simp
113   also from x have "\<dots> = a \<cdot> x + (- b) \<cdot> x"
114     by (rule add_mult_distrib2)
115   also from x have "\<dots> = a \<cdot> x + - (b \<cdot> x)"
116     by (simp add: negate_eq1 mult_assoc2)
117   also from x have "\<dots> = a \<cdot> x - (b \<cdot> x)"
118     by (simp add: diff_eq1)
119   finally show ?thesis .
120 qed
122 lemmas distrib =
124   diff_mult_distrib1 diff_mult_distrib2
127 text \<open>\<^medskip> Further derived laws:\<close>
129 lemma mult_zero_left [simp]: "x \<in> V \<Longrightarrow> 0 \<cdot> x = 0"
130 proof -
131   assume x: "x \<in> V"
132   have "0 \<cdot> x = (1 - 1) \<cdot> x" by simp
133   also have "\<dots> = (1 + - 1) \<cdot> x" by simp
134   also from x have "\<dots> =  1 \<cdot> x + (- 1) \<cdot> x"
135     by (rule add_mult_distrib2)
136   also from x have "\<dots> = x + (- 1) \<cdot> x" by simp
137   also from x have "\<dots> = x + - x" by (simp add: negate_eq2a)
138   also from x have "\<dots> = x - x" by (simp add: diff_eq2)
139   also from x have "\<dots> = 0" by simp
140   finally show ?thesis .
141 qed
143 lemma mult_zero_right [simp]: "a \<cdot> 0 = (0::'a)"
144 proof -
145   have "a \<cdot> 0 = a \<cdot> (0 - (0::'a))" by simp
146   also have "\<dots> =  a \<cdot> 0 - a \<cdot> 0"
147     by (rule diff_mult_distrib1) simp_all
148   also have "\<dots> = 0" by simp
149   finally show ?thesis .
150 qed
152 lemma minus_mult_cancel [simp]: "x \<in> V \<Longrightarrow> (- a) \<cdot> - x = a \<cdot> x"
153   by (simp add: negate_eq1 mult_assoc2)
155 lemma add_minus_left_eq_diff: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> - x + y = y - x"
156 proof -
157   assume xy: "x \<in> V"  "y \<in> V"
158   then have "- x + y = y + - x" by (simp add: add_commute)
159   also from xy have "\<dots> = y - x" by (simp add: diff_eq1)
160   finally show ?thesis .
161 qed
163 lemma add_minus [simp]: "x \<in> V \<Longrightarrow> x + - x = 0"
164   by (simp add: diff_eq2)
166 lemma add_minus_left [simp]: "x \<in> V \<Longrightarrow> - x + x = 0"
169 lemma minus_minus [simp]: "x \<in> V \<Longrightarrow> - (- x) = x"
170   by (simp add: negate_eq1 mult_assoc2)
172 lemma minus_zero [simp]: "- (0::'a) = 0"
173   by (simp add: negate_eq1)
175 lemma minus_zero_iff [simp]:
176   assumes x: "x \<in> V"
177   shows "(- x = 0) = (x = 0)"
178 proof
179   from x have "x = - (- x)" by simp
180   also assume "- x = 0"
181   also have "- \<dots> = 0" by (rule minus_zero)
182   finally show "x = 0" .
183 next
184   assume "x = 0"
185   then show "- x = 0" by simp
186 qed
188 lemma add_minus_cancel [simp]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + (- x + y) = y"
191 lemma minus_add_cancel [simp]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> - x + (x + y) = y"
194 lemma minus_add_distrib [simp]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> - (x + y) = - x + - y"
197 lemma diff_zero [simp]: "x \<in> V \<Longrightarrow> x - 0 = x"
198   by (simp add: diff_eq1)
200 lemma diff_zero_right [simp]: "x \<in> V \<Longrightarrow> 0 - x = - x"
201   by (simp add: diff_eq1)
204   assumes x: "x \<in> V" and y: "y \<in> V" and z: "z \<in> V"
205   shows "(x + y = x + z) = (y = z)"
206 proof
207   from y have "y = 0 + y" by simp
208   also from x y have "\<dots> = (- x + x) + y" by simp
209   also from x y have "\<dots> = - x + (x + y)" by (simp add: add.assoc)
210   also assume "x + y = x + z"
211   also from x z have "- x + (x + z) = - x + x + z" by (simp add: add.assoc)
212   also from x z have "\<dots> = z" by simp
213   finally show "y = z" .
214 next
215   assume "y = z"
216   then show "x + y = x + z" by (simp only:)
217 qed
220     "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> (y + x = z + x) = (y = z)"
224   "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x' \<in> V \<Longrightarrow> y' \<in> V \<Longrightarrow> z \<in> V
225     \<Longrightarrow> x + y = x' + y' \<Longrightarrow> x + (y + z) = x' + (y' + z)"
226   by (simp only: add_assoc [symmetric])
228 lemma mult_left_commute: "x \<in> V \<Longrightarrow> a \<cdot> b \<cdot> x = b \<cdot> a \<cdot> x"
229   by (simp add: mult.commute mult_assoc2)
231 lemma mult_zero_uniq:
232   assumes x: "x \<in> V"  "x \<noteq> 0" and ax: "a \<cdot> x = 0"
233   shows "a = 0"
234 proof (rule classical)
235   assume a: "a \<noteq> 0"
236   from x a have "x = (inverse a * a) \<cdot> x" by simp
237   also from \<open>x \<in> V\<close> have "\<dots> = inverse a \<cdot> (a \<cdot> x)" by (rule mult_assoc)
238   also from ax have "\<dots> = inverse a \<cdot> 0" by simp
239   also have "\<dots> = 0" by simp
240   finally have "x = 0" .
241   with \<open>x \<noteq> 0\<close> show "a = 0" by contradiction
242 qed
244 lemma mult_left_cancel:
245   assumes x: "x \<in> V" and y: "y \<in> V" and a: "a \<noteq> 0"
246   shows "(a \<cdot> x = a \<cdot> y) = (x = y)"
247 proof
248   from x have "x = 1 \<cdot> x" by simp
249   also from a have "\<dots> = (inverse a * a) \<cdot> x" by simp
250   also from x have "\<dots> = inverse a \<cdot> (a \<cdot> x)"
251     by (simp only: mult_assoc)
252   also assume "a \<cdot> x = a \<cdot> y"
253   also from a y have "inverse a \<cdot> \<dots> = y"
254     by (simp add: mult_assoc2)
255   finally show "x = y" .
256 next
257   assume "x = y"
258   then show "a \<cdot> x = a \<cdot> y" by (simp only:)
259 qed
261 lemma mult_right_cancel:
262   assumes x: "x \<in> V" and neq: "x \<noteq> 0"
263   shows "(a \<cdot> x = b \<cdot> x) = (a = b)"
264 proof
265   from x have "(a - b) \<cdot> x = a \<cdot> x - b \<cdot> x"
266     by (simp add: diff_mult_distrib2)
267   also assume "a \<cdot> x = b \<cdot> x"
268   with x have "a \<cdot> x - b \<cdot> x = 0" by simp
269   finally have "(a - b) \<cdot> x = 0" .
270   with x neq have "a - b = 0" by (rule mult_zero_uniq)
271   then show "a = b" by simp
272 next
273   assume "a = b"
274   then show "a \<cdot> x = b \<cdot> x" by (simp only:)
275 qed
277 lemma eq_diff_eq:
278   assumes x: "x \<in> V" and y: "y \<in> V" and z: "z \<in> V"
279   shows "(x = z - y) = (x + y = z)"
280 proof
281   assume "x = z - y"
282   then have "x + y = z - y + y" by simp
283   also from y z have "\<dots> = z + - y + y"
284     by (simp add: diff_eq1)
285   also have "\<dots> = z + (- y + y)"
286     by (rule add_assoc) (simp_all add: y z)
287   also from y z have "\<dots> = z + 0"
288     by (simp only: add_minus_left)
289   also from z have "\<dots> = z"
290     by (simp only: add_zero_right)
291   finally show "x + y = z" .
292 next
293   assume "x + y = z"
294   then have "z - y = (x + y) - y" by simp
295   also from x y have "\<dots> = x + y + - y"
296     by (simp add: diff_eq1)
297   also have "\<dots> = x + (y + - y)"
298     by (rule add_assoc) (simp_all add: x y)
299   also from x y have "\<dots> = x" by simp
300   finally show "x = z - y" ..
301 qed
304   assumes x: "x \<in> V" and y: "y \<in> V" and xy: "x + y = 0"
305   shows "x = - y"
306 proof -
307   from x y have "x = (- y + y) + x" by simp
308   also from x y have "\<dots> = - y + (x + y)" by (simp add: add_ac)
309   also note xy
310   also from y have "- y + 0 = - y" by simp
311   finally show "x = - y" .
312 qed
315   assumes x: "x \<in> V" and y: "y \<in> V" and xy: "x - y = 0"
316   shows "x = y"
317 proof -
318   from x y xy have eq: "x + - y = 0" by (simp add: diff_eq1)
319   with _ _ have "x = - (- y)"
320     by (rule add_minus_eq_minus) (simp_all add: x y)
321   with x y show "x = y" by simp
322 qed
325   assumes vs: "a \<in> V"  "b \<in> V"  "c \<in> V"  "d \<in> V"
326     and eq: "a + b = c + d"
327   shows "a - c = d - b"
328 proof -
329   from assms have "- c + (a + b) = - c + (c + d)"
331   also have "\<dots> = d" using \<open>c \<in> V\<close> \<open>d \<in> V\<close> by (rule minus_add_cancel)
332   finally have eq: "- c + (a + b) = d" .
333   from vs have "a - c = (- c + (a + b)) + - b"
335   also from vs eq have "\<dots>  = d + - b"
337   also from vs have "\<dots> = d - b" by (simp add: diff_eq2)
338   finally show "a - c = d - b" .
339 qed
342   assumes vs: "x \<in> V"  "y \<in> V"  "z \<in> V"  "u \<in> V"
343   shows "(x + (y + z) = y + u) = (x + z = u)"
344 proof
345   from vs have "x + z = - y + y + (x + z)" by simp
346   also have "\<dots> = - y + (y + (x + z))"
347     by (rule add_assoc) (simp_all add: vs)
348   also from vs have "y + (x + z) = x + (y + z)"
350   also assume "x + (y + z) = y + u"
351   also from vs have "- y + (y + u) = u" by simp
352   finally show "x + z = u" .
353 next
354   assume "x + z = u"
355   with vs show "x + (y + z) = y + u"
356     by (simp only: add_left_commute [of x])
357 qed
360   assumes vs: "x \<in> V"  "y \<in> V"  "z \<in> V"
361   shows "(x + (y + z) = y) = (x = - z)"
362 proof
363   assume "x + (y + z) = y"
364   with vs have "(x + z) + y = 0 + y" by (simp add: add_ac)
365   with vs have "x + z = 0" by (simp only: add_right_cancel add_closed zero)
366   with vs show "x = - z" by (simp add: add_minus_eq_minus)
367 next
368   assume eq: "x = - z"
369   then have "x + (y + z) = - z + (y + z)" by simp
370   also have "\<dots> = y + (- z + z)" by (rule add_left_commute) (simp_all add: vs)
371   also from vs have "\<dots> = y"  by simp
372   finally show "x + (y + z) = y" .
373 qed
375 end
377 end