src/HOL/Real/HahnBanach/Subspace.thy
 author wenzelm Sun Jul 23 12:01:05 2000 +0200 (2000-07-23) changeset 9408 d3d56e1d2ec1 parent 9374 153853af318b child 9623 3ade112482af permissions -rw-r--r--
classical atts now intro! / intro / intro?;
1 (*  Title:      HOL/Real/HahnBanach/Subspace.thy
2     ID:         $Id$
3     Author:     Gertrud Bauer, TU Munich
4 *)
7 header {* Subspaces *}
9 theory Subspace = VectorSpace:
12 subsection {* Definition *}
14 text {* A non-empty subset $U$ of a vector space $V$ is a
15 \emph{subspace} of $V$, iff $U$ is closed under addition and
16 scalar multiplication. *}
18 constdefs
19   is_subspace ::  "['a::{plus, minus, zero} set, 'a set] => bool"
20   "is_subspace U V == U \<noteq> {} \<and> U <= V
21      \<and> (\<forall>x \<in> U. \<forall>y \<in> U. \<forall>a. x + y \<in> U \<and> a \<cdot> x\<in> U)"
23 lemma subspaceI [intro]:
24   "[| 0 \<in> U; U <= V; \<forall>x \<in> U. \<forall>y \<in> U. (x + y \<in> U);
25   \<forall>x \<in> U. \<forall>a. a \<cdot> x \<in> U |]
26   ==> is_subspace U V"
27 proof (unfold is_subspace_def, intro conjI)
28   assume "0 \<in> U" thus "U \<noteq> {}" by fast
29 qed (simp+)
31 lemma subspace_not_empty [intro?]: "is_subspace U V ==> U \<noteq> {}"
32   by (unfold is_subspace_def) simp
34 lemma subspace_subset [intro?]: "is_subspace U V ==> U <= V"
35   by (unfold is_subspace_def) simp
37 lemma subspace_subsetD [simp, intro?]:
38   "[| is_subspace U V; x \<in> U |] ==> x \<in> V"
39   by (unfold is_subspace_def) force
41 lemma subspace_add_closed [simp, intro?]:
42   "[| is_subspace U V; x \<in> U; y \<in> U |] ==> x + y \<in> U"
43   by (unfold is_subspace_def) simp
45 lemma subspace_mult_closed [simp, intro?]:
46   "[| is_subspace U V; x \<in> U |] ==> a \<cdot> x \<in> U"
47   by (unfold is_subspace_def) simp
49 lemma subspace_diff_closed [simp, intro?]:
50   "[| is_subspace U V; is_vectorspace V; x \<in> U; y \<in> U |]
51   ==> x - y \<in> U"
52   by (simp! add: diff_eq1 negate_eq1)
54 text {* Similar as for linear spaces, the existence of the
55 zero element in every subspace follows from the non-emptiness
56 of the carrier set and by vector space laws.*}
58 lemma zero_in_subspace [intro?]:
59   "[| is_subspace U V; is_vectorspace V |] ==> 0 \<in> U"
60 proof -
61   assume "is_subspace U V" and v: "is_vectorspace V"
62   have "U \<noteq> {}" ..
63   hence "\<exists>x. x \<in> U" by force
64   thus ?thesis
65   proof
66     fix x assume u: "x \<in> U"
67     hence "x \<in> V" by (simp!)
68     with v have "0 = x - x" by (simp!)
69     also have "... \<in> U" by (rule subspace_diff_closed)
70     finally show ?thesis .
71   qed
72 qed
74 lemma subspace_neg_closed [simp, intro?]:
75   "[| is_subspace U V; is_vectorspace V; x \<in> U |] ==> - x \<in> U"
76   by (simp add: negate_eq1)
78 text_raw {* \medskip *}
79 text {* Further derived laws: every subspace is a vector space. *}
81 lemma subspace_vs [intro?]:
82   "[| is_subspace U V; is_vectorspace V |] ==> is_vectorspace U"
83 proof -
84   assume "is_subspace U V" "is_vectorspace V"
85   show ?thesis
86   proof
87     show "0 \<in> U" ..
88     show "\<forall>x \<in> U. \<forall>a. a \<cdot> x \<in> U" by (simp!)
89     show "\<forall>x \<in> U. \<forall>y \<in> U. x + y \<in> U" by (simp!)
90     show "\<forall>x \<in> U. - x = -#1 \<cdot> x" by (simp! add: negate_eq1)
91     show "\<forall>x \<in> U. \<forall>y \<in> U. x - y =  x + - y"
92       by (simp! add: diff_eq1)
94 qed
96 text {* The subspace relation is reflexive. *}
98 lemma subspace_refl [intro]: "is_vectorspace V ==> is_subspace V V"
99 proof
100   assume "is_vectorspace V"
101   show "0 \<in> V" ..
102   show "V <= V" ..
103   show "\<forall>x \<in> V. \<forall>y \<in> V. x + y \<in> V" by (simp!)
104   show "\<forall>x \<in> V. \<forall>a. a \<cdot> x \<in> V" by (simp!)
105 qed
107 text {* The subspace relation is transitive. *}
109 lemma subspace_trans:
110   "[| is_subspace U V; is_vectorspace V; is_subspace V W |]
111   ==> is_subspace U W"
112 proof
113   assume "is_subspace U V" "is_subspace V W" "is_vectorspace V"
114   show "0 \<in> U" ..
116   have "U <= V" ..
117   also have "V <= W" ..
118   finally show "U <= W" .
120   show "\<forall>x \<in> U. \<forall>y \<in> U. x + y \<in> U"
121   proof (intro ballI)
122     fix x y assume "x \<in> U" "y \<in> U"
123     show "x + y \<in> U" by (simp!)
124   qed
126   show "\<forall>x \<in> U. \<forall>a. a \<cdot> x \<in> U"
127   proof (intro ballI allI)
128     fix x a assume "x \<in> U"
129     show "a \<cdot> x \<in> U" by (simp!)
130   qed
131 qed
135 subsection {* Linear closure *}
137 text {* The \emph{linear closure} of a vector $x$ is the set of all
138 scalar multiples of $x$. *}
140 constdefs
141   lin :: "('a::{minus,plus,zero}) => 'a set"
142   "lin x == {a \<cdot> x | a. True}"
144 lemma linD: "x \<in> lin v = (\<exists>a::real. x = a \<cdot> v)"
145   by (unfold lin_def) fast
147 lemma linI [intro?]: "a \<cdot> x0 \<in> lin x0"
148   by (unfold lin_def) fast
150 text {* Every vector is contained in its linear closure. *}
152 lemma x_lin_x: "[| is_vectorspace V; x \<in> V |] ==> x \<in> lin x"
153 proof (unfold lin_def, intro CollectI exI conjI)
154   assume "is_vectorspace V" "x \<in> V"
155   show "x = #1 \<cdot> x" by (simp!)
156 qed simp
158 text {* Any linear closure is a subspace. *}
160 lemma lin_subspace [intro?]:
161   "[| is_vectorspace V; x \<in> V |] ==> is_subspace (lin x) V"
162 proof
163   assume "is_vectorspace V" "x \<in> V"
164   show "0 \<in> lin x"
165   proof (unfold lin_def, intro CollectI exI conjI)
166     show "0 = (#0::real) \<cdot> x" by (simp!)
167   qed simp
169   show "lin x <= V"
170   proof (unfold lin_def, intro subsetI, elim CollectE exE conjE)
171     fix xa a assume "xa = a \<cdot> x"
172     show "xa \<in> V" by (simp!)
173   qed
175   show "\<forall>x1 \<in> lin x. \<forall>x2 \<in> lin x. x1 + x2 \<in> lin x"
176   proof (intro ballI)
177     fix x1 x2 assume "x1 \<in> lin x" "x2 \<in> lin x"
178     thus "x1 + x2 \<in> lin x"
179     proof (unfold lin_def, elim CollectE exE conjE,
180       intro CollectI exI conjI)
181       fix a1 a2 assume "x1 = a1 \<cdot> x" "x2 = a2 \<cdot> x"
182       show "x1 + x2 = (a1 + a2) \<cdot> x"
184     qed simp
185   qed
187   show "\<forall>xa \<in> lin x. \<forall>a. a \<cdot> xa \<in> lin x"
188   proof (intro ballI allI)
189     fix x1 a assume "x1 \<in> lin x"
190     thus "a \<cdot> x1 \<in> lin x"
191     proof (unfold lin_def, elim CollectE exE conjE,
192       intro CollectI exI conjI)
193       fix a1 assume "x1 = a1 \<cdot> x"
194       show "a \<cdot> x1 = (a * a1) \<cdot> x" by (simp!)
195     qed simp
196   qed
197 qed
199 text {* Any linear closure is a vector space. *}
201 lemma lin_vs [intro?]:
202   "[| is_vectorspace V; x \<in> V |] ==> is_vectorspace (lin x)"
203 proof (rule subspace_vs)
204   assume "is_vectorspace V" "x \<in> V"
205   show "is_subspace (lin x) V" ..
206 qed
210 subsection {* Sum of two vectorspaces *}
212 text {* The \emph{sum} of two vectorspaces $U$ and $V$ is the set of
213 all sums of elements from $U$ and $V$. *}
215 instance set :: (plus) plus by intro_classes
217 defs vs_sum_def:
218   "U + V == {u + v | u v. u \<in> U \<and> v \<in> V}" (***
220 constdefs
221   vs_sum ::
222   "['a::{plus, minus, zero} set, 'a set] => 'a set"         (infixl "+" 65)
223   "vs_sum U V == {x. \<exists>u \<in> U. \<exists>v \<in> V. x = u + v}";
224 ***)
226 lemma vs_sumD:
227   "x \<in> U + V = (\<exists>u \<in> U. \<exists>v \<in> V. x = u + v)"
228     by (unfold vs_sum_def) fast
230 lemmas vs_sumE = vs_sumD [RS iffD1, elimify]
232 lemma vs_sumI [intro?]:
233   "[| x \<in> U; y \<in> V; t= x + y |] ==> t \<in> U + V"
234   by (unfold vs_sum_def) fast
236 text{* $U$ is a subspace of $U + V$. *}
238 lemma subspace_vs_sum1 [intro?]:
239   "[| is_vectorspace U; is_vectorspace V |]
240   ==> is_subspace U (U + V)"
241 proof
242   assume "is_vectorspace U" "is_vectorspace V"
243   show "0 \<in> U" ..
244   show "U <= U + V"
245   proof (intro subsetI vs_sumI)
246   fix x assume "x \<in> U"
247     show "x = x + 0" by (simp!)
248     show "0 \<in> V" by (simp!)
249   qed
250   show "\<forall>x \<in> U. \<forall>y \<in> U. x + y \<in> U"
251   proof (intro ballI)
252     fix x y assume "x \<in> U" "y \<in> U" show "x + y \<in> U" by (simp!)
253   qed
254   show "\<forall>x \<in> U. \<forall>a. a \<cdot> x \<in> U"
255   proof (intro ballI allI)
256     fix x a assume "x \<in> U" show "a \<cdot> x \<in> U" by (simp!)
257   qed
258 qed
260 text{* The sum of two subspaces is again a subspace.*}
262 lemma vs_sum_subspace [intro?]:
263   "[| is_subspace U E; is_subspace V E; is_vectorspace E |]
264   ==> is_subspace (U + V) E"
265 proof
266   assume "is_subspace U E" "is_subspace V E" "is_vectorspace E"
267   show "0 \<in> U + V"
268   proof (intro vs_sumI)
269     show "0 \<in> U" ..
270     show "0 \<in> V" ..
271     show "(0::'a) = 0 + 0" by (simp!)
272   qed
274   show "U + V <= E"
275   proof (intro subsetI, elim vs_sumE bexE)
276     fix x u v assume "u \<in> U" "v \<in> V" "x = u + v"
277     show "x \<in> E" by (simp!)
278   qed
280   show "\<forall>x \<in> U + V. \<forall>y \<in> U + V. x + y \<in> U + V"
281   proof (intro ballI)
282     fix x y assume "x \<in> U + V" "y \<in> U + V"
283     thus "x + y \<in> U + V"
284     proof (elim vs_sumE bexE, intro vs_sumI)
285       fix ux vx uy vy
286       assume "ux \<in> U" "vx \<in> V" "x = ux + vx"
287 	and "uy \<in> U" "vy \<in> V" "y = uy + vy"
288       show "x + y = (ux + uy) + (vx + vy)" by (simp!)
289     qed (simp!)+
290   qed
292   show "\<forall>x \<in> U + V. \<forall>a. a \<cdot> x \<in> U + V"
293   proof (intro ballI allI)
294     fix x a assume "x \<in> U + V"
295     thus "a \<cdot> x \<in> U + V"
296     proof (elim vs_sumE bexE, intro vs_sumI)
297       fix a x u v assume "u \<in> U" "v \<in> V" "x = u + v"
298       show "a \<cdot> x = (a \<cdot> u) + (a \<cdot> v)"
300     qed (simp!)+
301   qed
302 qed
304 text{* The sum of two subspaces is a vectorspace. *}
306 lemma vs_sum_vs [intro?]:
307   "[| is_subspace U E; is_subspace V E; is_vectorspace E |]
308   ==> is_vectorspace (U + V)"
309 proof (rule subspace_vs)
310   assume "is_subspace U E" "is_subspace V E" "is_vectorspace E"
311   show "is_subspace (U + V) E" ..
312 qed
316 subsection {* Direct sums *}
319 text {* The sum of $U$ and $V$ is called \emph{direct}, iff the zero
320 element is the only common element of $U$ and $V$. For every element
321 $x$ of the direct sum of $U$ and $V$ the decomposition in
322 $x = u + v$ with $u \in U$ and $v \in V$ is unique.*}
324 lemma decomp:
325   "[| is_vectorspace E; is_subspace U E; is_subspace V E;
326   U \<inter> V = {0}; u1 \<in> U; u2 \<in> U; v1 \<in> V; v2 \<in> V; u1 + v1 = u2 + v2 |]
327   ==> u1 = u2 \<and> v1 = v2"
328 proof
329   assume "is_vectorspace E" "is_subspace U E" "is_subspace V E"
330     "U \<inter> V = {0}" "u1 \<in> U" "u2 \<in> U" "v1 \<in> V" "v2 \<in> V"
331     "u1 + v1 = u2 + v2"
332   have eq: "u1 - u2 = v2 - v1" by (simp! add: vs_add_diff_swap)
333   have u: "u1 - u2 \<in> U" by (simp!)
334   with eq have v': "v2 - v1 \<in> U" by simp
335   have v: "v2 - v1 \<in> V" by (simp!)
336   with eq have u': "u1 - u2 \<in> V" by simp
338   show "u1 = u2"
339   proof (rule vs_add_minus_eq)
340     show "u1 - u2 = 0" by (rule Int_singletonD [OF _ u u'])
341     show "u1 \<in> E" ..
342     show "u2 \<in> E" ..
343   qed
345   show "v1 = v2"
346   proof (rule vs_add_minus_eq [RS sym])
347     show "v2 - v1 = 0" by (rule Int_singletonD [OF _ v' v])
348     show "v1 \<in> E" ..
349     show "v2 \<in> E" ..
350   qed
351 qed
353 text {* An application of the previous lemma will be used in the proof
354 of the Hahn-Banach Theorem (see page \pageref{decomp-H-use}): for any
355 element $y + a \mult x_0$ of the direct sum of a vectorspace $H$ and
356 the linear closure of $x_0$ the components $y \in H$ and $a$ are
357 uniquely determined. *}
359 lemma decomp_H':
360   "[| is_vectorspace E; is_subspace H E; y1 \<in> H; y2 \<in> H;
361   x' \<notin> H; x' \<in> E; x' \<noteq> 0; y1 + a1 \<cdot> x' = y2 + a2 \<cdot> x' |]
362   ==> y1 = y2 \<and> a1 = a2"
363 proof
364   assume "is_vectorspace E" and h: "is_subspace H E"
365      and "y1 \<in> H" "y2 \<in> H" "x' \<notin> H" "x' \<in> E" "x' \<noteq> 0"
366          "y1 + a1 \<cdot> x' = y2 + a2 \<cdot> x'"
368   have c: "y1 = y2 \<and> a1 \<cdot> x' = a2 \<cdot> x'"
369   proof (rule decomp)
370     show "a1 \<cdot> x' \<in> lin x'" ..
371     show "a2 \<cdot> x' \<in> lin x'" ..
372     show "H \<inter> (lin x') = {0}"
373     proof
374       show "H \<inter> lin x' <= {0}"
375       proof (intro subsetI, elim IntE, rule singleton_iff[RS iffD2])
376         fix x assume "x \<in> H" "x \<in> lin x'"
377         thus "x = 0"
378         proof (unfold lin_def, elim CollectE exE conjE)
379           fix a assume "x = a \<cdot> x'"
380           show ?thesis
381           proof cases
382             assume "a = (#0::real)" show ?thesis by (simp!)
383           next
384             assume "a \<noteq> (#0::real)"
385             from h have "rinv a \<cdot> a \<cdot> x' \<in> H"
386               by (rule subspace_mult_closed) (simp!)
387             also have "rinv a \<cdot> a \<cdot> x' = x'" by (simp!)
388             finally have "x' \<in> H" .
389             thus ?thesis by contradiction
390           qed
391        qed
392       qed
393       show "{0} <= H \<inter> lin x'"
394       proof -
395 	have "0 \<in> H \<inter> lin x'"
396 	proof (rule IntI)
397 	  show "0 \<in> H" ..
398 	  from lin_vs show "0 \<in> lin x'" ..
399 	qed
400 	thus ?thesis by simp
401       qed
402     qed
403     show "is_subspace (lin x') E" ..
404   qed
406   from c show "y1 = y2" by simp
408   show  "a1 = a2"
409   proof (rule vs_mult_right_cancel [RS iffD1])
410     from c show "a1 \<cdot> x' = a2 \<cdot> x'" by simp
411   qed
412 qed
414 text {* Since for any element $y + a \mult x'$ of the direct sum
415 of a vectorspace $H$ and the linear closure of $x'$ the components
416 $y\in H$ and $a$ are unique, it follows from $y\in H$ that
417 $a = 0$.*}
419 lemma decomp_H'_H:
420   "[| is_vectorspace E; is_subspace H E; t \<in> H; x' \<notin> H; x' \<in> E;
421   x' \<noteq> 0 |]
422   ==> (SOME (y, a). t = y + a \<cdot> x' \<and> y \<in> H) = (t, (#0::real))"
423 proof (rule, unfold split_tupled_all)
424   assume "is_vectorspace E" "is_subspace H E" "t \<in> H" "x' \<notin> H" "x' \<in> E"
425     "x' \<noteq> 0"
426   have h: "is_vectorspace H" ..
427   fix y a presume t1: "t = y + a \<cdot> x'" and "y \<in> H"
428   have "y = t \<and> a = (#0::real)"
429     by (rule decomp_H') (assumption | (simp!))+
430   thus "(y, a) = (t, (#0::real))" by (simp!)
431 qed (simp!)+
433 text {* The components $y\in H$ and $a$ in $y \plus a \mult x'$
434 are unique, so the function $h'$ defined by
435 $h' (y \plus a \mult x') = h y + a \cdot \xi$ is definite. *}
437 lemma h'_definite:
438   "[| h' == (\<lambda>x. let (y, a) = SOME (y, a). (x = y + a \<cdot> x' \<and> y \<in> H)
439                 in (h y) + a * xi);
440   x = y + a \<cdot> x'; is_vectorspace E; is_subspace H E;
441   y \<in> H; x' \<notin> H; x' \<in> E; x' \<noteq> 0 |]
442   ==> h' x = h y + a * xi"
443 proof -
444   assume
445     "h' == (\<lambda>x. let (y, a) = SOME (y, a). (x = y + a \<cdot> x' \<and> y \<in> H)
446                in (h y) + a * xi)"
447     "x = y + a \<cdot> x'" "is_vectorspace E" "is_subspace H E"
448     "y \<in> H" "x' \<notin> H" "x' \<in> E" "x' \<noteq> 0"
449   have "x \<in> H + (lin x')"
450     by (simp! add: vs_sum_def lin_def) force+
451   have "\<exists>! xa. ((\<lambda>(y, a). x = y + a \<cdot> x' \<and> y \<in> H) xa)"
452   proof
453     show "\<exists>xa. ((\<lambda>(y, a). x = y + a \<cdot> x' \<and> y \<in> H) xa)"
454       by (force!)
455   next
456     fix xa ya
457     assume "(\<lambda>(y,a). x = y + a \<cdot> x' \<and> y \<in> H) xa"
458            "(\<lambda>(y,a). x = y + a \<cdot> x' \<and> y \<in> H) ya"
459     show "xa = ya"
460     proof -
461       show "fst xa = fst ya \<and> snd xa = snd ya ==> xa = ya"
462         by (simp add: Pair_fst_snd_eq)
463       have x: "x = fst xa + snd xa \<cdot> x' \<and> fst xa \<in> H"
464         by (force!)
465       have y: "x = fst ya + snd ya \<cdot> x' \<and> fst ya \<in> H"
466         by (force!)
467       from x y show "fst xa = fst ya \<and> snd xa = snd ya"
468         by (elim conjE) (rule decomp_H', (simp!)+)
469     qed
470   qed
471   hence eq: "(SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H) = (y, a)"
472     by (rule select1_equality) (force!)
473   thus "h' x = h y + a * xi" by (simp! add: Let_def)
474 qed
476 end