src/HOL/HahnBanach/Subspace.thy
 author haftmann Mon Dec 29 14:08:08 2008 +0100 (2008-12-29) changeset 29197 6d4cb27ed19c parent 27612 src/HOL/Real/HahnBanach/Subspace.thy@d3eb431db035 child 29252 ea97aa6aeba2 permissions -rw-r--r--
adapted HOL source structure to distribution layout
     1 (*  Title:      HOL/Real/HahnBanach/Subspace.thy

     2     ID:         $Id$

     3     Author:     Gertrud Bauer, TU Munich

     4 *)

     5

     6 header {* Subspaces *}

     7

     8 theory Subspace

     9 imports VectorSpace

    10 begin

    11

    12 subsection {* Definition *}

    13

    14 text {*

    15   A non-empty subset @{text U} of a vector space @{text V} is a

    16   \emph{subspace} of @{text V}, iff @{text U} is closed under addition

    17   and scalar multiplication.

    18 *}

    19

    20 locale subspace = var U + var V +

    21   constrains U :: "'a\<Colon>{minus, plus, zero, uminus} set"

    22   assumes non_empty [iff, intro]: "U \<noteq> {}"

    23     and subset [iff]: "U \<subseteq> V"

    24     and add_closed [iff]: "x \<in> U \<Longrightarrow> y \<in> U \<Longrightarrow> x + y \<in> U"

    25     and mult_closed [iff]: "x \<in> U \<Longrightarrow> a \<cdot> x \<in> U"

    26

    27 notation (symbols)

    28   subspace  (infix "\<unlhd>" 50)

    29

    30 declare vectorspace.intro [intro?] subspace.intro [intro?]

    31

    32 lemma subspace_subset [elim]: "U \<unlhd> V \<Longrightarrow> U \<subseteq> V"

    33   by (rule subspace.subset)

    34

    35 lemma (in subspace) subsetD [iff]: "x \<in> U \<Longrightarrow> x \<in> V"

    36   using subset by blast

    37

    38 lemma subspaceD [elim]: "U \<unlhd> V \<Longrightarrow> x \<in> U \<Longrightarrow> x \<in> V"

    39   by (rule subspace.subsetD)

    40

    41 lemma rev_subspaceD [elim?]: "x \<in> U \<Longrightarrow> U \<unlhd> V \<Longrightarrow> x \<in> V"

    42   by (rule subspace.subsetD)

    43

    44 lemma (in subspace) diff_closed [iff]:

    45   assumes "vectorspace V"

    46   assumes x: "x \<in> U" and y: "y \<in> U"

    47   shows "x - y \<in> U"

    48 proof -

    49   interpret vectorspace [V] by fact

    50   from x y show ?thesis by (simp add: diff_eq1 negate_eq1)

    51 qed

    52

    53 text {*

    54   \medskip Similar as for linear spaces, the existence of the zero

    55   element in every subspace follows from the non-emptiness of the

    56   carrier set and by vector space laws.

    57 *}

    58

    59 lemma (in subspace) zero [intro]:

    60   assumes "vectorspace V"

    61   shows "0 \<in> U"

    62 proof -

    63   interpret vectorspace [V] by fact

    64   have "U \<noteq> {}" by (rule U_V.non_empty)

    65   then obtain x where x: "x \<in> U" by blast

    66   then have "x \<in> V" .. then have "0 = x - x" by simp

    67   also from vectorspace V x x have "\<dots> \<in> U" by (rule U_V.diff_closed)

    68   finally show ?thesis .

    69 qed

    70

    71 lemma (in subspace) neg_closed [iff]:

    72   assumes "vectorspace V"

    73   assumes x: "x \<in> U"

    74   shows "- x \<in> U"

    75 proof -

    76   interpret vectorspace [V] by fact

    77   from x show ?thesis by (simp add: negate_eq1)

    78 qed

    79

    80 text {* \medskip Further derived laws: every subspace is a vector space. *}

    81

    82 lemma (in subspace) vectorspace [iff]:

    83   assumes "vectorspace V"

    84   shows "vectorspace U"

    85 proof -

    86   interpret vectorspace [V] by fact

    87   show ?thesis

    88   proof

    89     show "U \<noteq> {}" ..

    90     fix x y z assume x: "x \<in> U" and y: "y \<in> U" and z: "z \<in> U"

    91     fix a b :: real

    92     from x y show "x + y \<in> U" by simp

    93     from x show "a \<cdot> x \<in> U" by simp

    94     from x y z show "(x + y) + z = x + (y + z)" by (simp add: add_ac)

    95     from x y show "x + y = y + x" by (simp add: add_ac)

    96     from x show "x - x = 0" by simp

    97     from x show "0 + x = x" by simp

    98     from x y show "a \<cdot> (x + y) = a \<cdot> x + a \<cdot> y" by (simp add: distrib)

    99     from x show "(a + b) \<cdot> x = a \<cdot> x + b \<cdot> x" by (simp add: distrib)

   100     from x show "(a * b) \<cdot> x = a \<cdot> b \<cdot> x" by (simp add: mult_assoc)

   101     from x show "1 \<cdot> x = x" by simp

   102     from x show "- x = - 1 \<cdot> x" by (simp add: negate_eq1)

   103     from x y show "x - y = x + - y" by (simp add: diff_eq1)

   104   qed

   105 qed

   106

   107

   108 text {* The subspace relation is reflexive. *}

   109

   110 lemma (in vectorspace) subspace_refl [intro]: "V \<unlhd> V"

   111 proof

   112   show "V \<noteq> {}" ..

   113   show "V \<subseteq> V" ..

   114   fix x y assume x: "x \<in> V" and y: "y \<in> V"

   115   fix a :: real

   116   from x y show "x + y \<in> V" by simp

   117   from x show "a \<cdot> x \<in> V" by simp

   118 qed

   119

   120 text {* The subspace relation is transitive. *}

   121

   122 lemma (in vectorspace) subspace_trans [trans]:

   123   "U \<unlhd> V \<Longrightarrow> V \<unlhd> W \<Longrightarrow> U \<unlhd> W"

   124 proof

   125   assume uv: "U \<unlhd> V" and vw: "V \<unlhd> W"

   126   from uv show "U \<noteq> {}" by (rule subspace.non_empty)

   127   show "U \<subseteq> W"

   128   proof -

   129     from uv have "U \<subseteq> V" by (rule subspace.subset)

   130     also from vw have "V \<subseteq> W" by (rule subspace.subset)

   131     finally show ?thesis .

   132   qed

   133   fix x y assume x: "x \<in> U" and y: "y \<in> U"

   134   from uv and x y show "x + y \<in> U" by (rule subspace.add_closed)

   135   from uv and x show "\<And>a. a \<cdot> x \<in> U" by (rule subspace.mult_closed)

   136 qed

   137

   138

   139 subsection {* Linear closure *}

   140

   141 text {*

   142   The \emph{linear closure} of a vector @{text x} is the set of all

   143   scalar multiples of @{text x}.

   144 *}

   145

   146 definition

   147   lin :: "('a::{minus, plus, zero}) \<Rightarrow> 'a set" where

   148   "lin x = {a \<cdot> x | a. True}"

   149

   150 lemma linI [intro]: "y = a \<cdot> x \<Longrightarrow> y \<in> lin x"

   151   unfolding lin_def by blast

   152

   153 lemma linI' [iff]: "a \<cdot> x \<in> lin x"

   154   unfolding lin_def by blast

   155

   156 lemma linE [elim]: "x \<in> lin v \<Longrightarrow> (\<And>a::real. x = a \<cdot> v \<Longrightarrow> C) \<Longrightarrow> C"

   157   unfolding lin_def by blast

   158

   159

   160 text {* Every vector is contained in its linear closure. *}

   161

   162 lemma (in vectorspace) x_lin_x [iff]: "x \<in> V \<Longrightarrow> x \<in> lin x"

   163 proof -

   164   assume "x \<in> V"

   165   then have "x = 1 \<cdot> x" by simp

   166   also have "\<dots> \<in> lin x" ..

   167   finally show ?thesis .

   168 qed

   169

   170 lemma (in vectorspace) "0_lin_x" [iff]: "x \<in> V \<Longrightarrow> 0 \<in> lin x"

   171 proof

   172   assume "x \<in> V"

   173   then show "0 = 0 \<cdot> x" by simp

   174 qed

   175

   176 text {* Any linear closure is a subspace. *}

   177

   178 lemma (in vectorspace) lin_subspace [intro]:

   179   "x \<in> V \<Longrightarrow> lin x \<unlhd> V"

   180 proof

   181   assume x: "x \<in> V"

   182   then show "lin x \<noteq> {}" by (auto simp add: x_lin_x)

   183   show "lin x \<subseteq> V"

   184   proof

   185     fix x' assume "x' \<in> lin x"

   186     then obtain a where "x' = a \<cdot> x" ..

   187     with x show "x' \<in> V" by simp

   188   qed

   189   fix x' x'' assume x': "x' \<in> lin x" and x'': "x'' \<in> lin x"

   190   show "x' + x'' \<in> lin x"

   191   proof -

   192     from x' obtain a' where "x' = a' \<cdot> x" ..

   193     moreover from x'' obtain a'' where "x'' = a'' \<cdot> x" ..

   194     ultimately have "x' + x'' = (a' + a'') \<cdot> x"

   195       using x by (simp add: distrib)

   196     also have "\<dots> \<in> lin x" ..

   197     finally show ?thesis .

   198   qed

   199   fix a :: real

   200   show "a \<cdot> x' \<in> lin x"

   201   proof -

   202     from x' obtain a' where "x' = a' \<cdot> x" ..

   203     with x have "a \<cdot> x' = (a * a') \<cdot> x" by (simp add: mult_assoc)

   204     also have "\<dots> \<in> lin x" ..

   205     finally show ?thesis .

   206   qed

   207 qed

   208

   209

   210 text {* Any linear closure is a vector space. *}

   211

   212 lemma (in vectorspace) lin_vectorspace [intro]:

   213   assumes "x \<in> V"

   214   shows "vectorspace (lin x)"

   215 proof -

   216   from x \<in> V have "subspace (lin x) V"

   217     by (rule lin_subspace)

   218   from this and vectorspace_axioms show ?thesis

   219     by (rule subspace.vectorspace)

   220 qed

   221

   222

   223 subsection {* Sum of two vectorspaces *}

   224

   225 text {*

   226   The \emph{sum} of two vectorspaces @{text U} and @{text V} is the

   227   set of all sums of elements from @{text U} and @{text V}.

   228 *}

   229

   230 instantiation "fun" :: (type, type) plus

   231 begin

   232

   233 definition

   234   sum_def: "plus_fun U V = {u + v | u v. u \<in> U \<and> v \<in> V}"  (* FIXME not fully general!? *)

   235

   236 instance ..

   237

   238 end

   239

   240 lemma sumE [elim]:

   241     "x \<in> U + V \<Longrightarrow> (\<And>u v. x = u + v \<Longrightarrow> u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> C) \<Longrightarrow> C"

   242   unfolding sum_def by blast

   243

   244 lemma sumI [intro]:

   245     "u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> x = u + v \<Longrightarrow> x \<in> U + V"

   246   unfolding sum_def by blast

   247

   248 lemma sumI' [intro]:

   249     "u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> u + v \<in> U + V"

   250   unfolding sum_def by blast

   251

   252 text {* @{text U} is a subspace of @{text "U + V"}. *}

   253

   254 lemma subspace_sum1 [iff]:

   255   assumes "vectorspace U" "vectorspace V"

   256   shows "U \<unlhd> U + V"

   257 proof -

   258   interpret vectorspace [U] by fact

   259   interpret vectorspace [V] by fact

   260   show ?thesis

   261   proof

   262     show "U \<noteq> {}" ..

   263     show "U \<subseteq> U + V"

   264     proof

   265       fix x assume x: "x \<in> U"

   266       moreover have "0 \<in> V" ..

   267       ultimately have "x + 0 \<in> U + V" ..

   268       with x show "x \<in> U + V" by simp

   269     qed

   270     fix x y assume x: "x \<in> U" and "y \<in> U"

   271     then show "x + y \<in> U" by simp

   272     from x show "\<And>a. a \<cdot> x \<in> U" by simp

   273   qed

   274 qed

   275

   276 text {* The sum of two subspaces is again a subspace. *}

   277

   278 lemma sum_subspace [intro?]:

   279   assumes "subspace U E" "vectorspace E" "subspace V E"

   280   shows "U + V \<unlhd> E"

   281 proof -

   282   interpret subspace [U E] by fact

   283   interpret vectorspace [E] by fact

   284   interpret subspace [V E] by fact

   285   show ?thesis

   286   proof

   287     have "0 \<in> U + V"

   288     proof

   289       show "0 \<in> U" using vectorspace E ..

   290       show "0 \<in> V" using vectorspace E ..

   291       show "(0::'a) = 0 + 0" by simp

   292     qed

   293     then show "U + V \<noteq> {}" by blast

   294     show "U + V \<subseteq> E"

   295     proof

   296       fix x assume "x \<in> U + V"

   297       then obtain u v where "x = u + v" and

   298 	"u \<in> U" and "v \<in> V" ..

   299       then show "x \<in> E" by simp

   300     qed

   301     fix x y assume x: "x \<in> U + V" and y: "y \<in> U + V"

   302     show "x + y \<in> U + V"

   303     proof -

   304       from x obtain ux vx where "x = ux + vx" and "ux \<in> U" and "vx \<in> V" ..

   305       moreover

   306       from y obtain uy vy where "y = uy + vy" and "uy \<in> U" and "vy \<in> V" ..

   307       ultimately

   308       have "ux + uy \<in> U"

   309 	and "vx + vy \<in> V"

   310 	and "x + y = (ux + uy) + (vx + vy)"

   311 	using x y by (simp_all add: add_ac)

   312       then show ?thesis ..

   313     qed

   314     fix a show "a \<cdot> x \<in> U + V"

   315     proof -

   316       from x obtain u v where "x = u + v" and "u \<in> U" and "v \<in> V" ..

   317       then have "a \<cdot> u \<in> U" and "a \<cdot> v \<in> V"

   318 	and "a \<cdot> x = (a \<cdot> u) + (a \<cdot> v)" by (simp_all add: distrib)

   319       then show ?thesis ..

   320     qed

   321   qed

   322 qed

   323

   324 text{* The sum of two subspaces is a vectorspace. *}

   325

   326 lemma sum_vs [intro?]:

   327     "U \<unlhd> E \<Longrightarrow> V \<unlhd> E \<Longrightarrow> vectorspace E \<Longrightarrow> vectorspace (U + V)"

   328   by (rule subspace.vectorspace) (rule sum_subspace)

   329

   330

   331 subsection {* Direct sums *}

   332

   333 text {*

   334   The sum of @{text U} and @{text V} is called \emph{direct}, iff the

   335   zero element is the only common element of @{text U} and @{text

   336   V}. For every element @{text x} of the direct sum of @{text U} and

   337   @{text V} the decomposition in @{text "x = u + v"} with

   338   @{text "u \<in> U"} and @{text "v \<in> V"} is unique.

   339 *}

   340

   341 lemma decomp:

   342   assumes "vectorspace E" "subspace U E" "subspace V E"

   343   assumes direct: "U \<inter> V = {0}"

   344     and u1: "u1 \<in> U" and u2: "u2 \<in> U"

   345     and v1: "v1 \<in> V" and v2: "v2 \<in> V"

   346     and sum: "u1 + v1 = u2 + v2"

   347   shows "u1 = u2 \<and> v1 = v2"

   348 proof -

   349   interpret vectorspace [E] by fact

   350   interpret subspace [U E] by fact

   351   interpret subspace [V E] by fact

   352   show ?thesis

   353   proof

   354     have U: "vectorspace U"  (* FIXME: use interpret *)

   355       using subspace U E vectorspace E by (rule subspace.vectorspace)

   356     have V: "vectorspace V"

   357       using subspace V E vectorspace E by (rule subspace.vectorspace)

   358     from u1 u2 v1 v2 and sum have eq: "u1 - u2 = v2 - v1"

   359       by (simp add: add_diff_swap)

   360     from u1 u2 have u: "u1 - u2 \<in> U"

   361       by (rule vectorspace.diff_closed [OF U])

   362     with eq have v': "v2 - v1 \<in> U" by (simp only:)

   363     from v2 v1 have v: "v2 - v1 \<in> V"

   364       by (rule vectorspace.diff_closed [OF V])

   365     with eq have u': " u1 - u2 \<in> V" by (simp only:)

   366

   367     show "u1 = u2"

   368     proof (rule add_minus_eq)

   369       from u1 show "u1 \<in> E" ..

   370       from u2 show "u2 \<in> E" ..

   371       from u u' and direct show "u1 - u2 = 0" by blast

   372     qed

   373     show "v1 = v2"

   374     proof (rule add_minus_eq [symmetric])

   375       from v1 show "v1 \<in> E" ..

   376       from v2 show "v2 \<in> E" ..

   377       from v v' and direct show "v2 - v1 = 0" by blast

   378     qed

   379   qed

   380 qed

   381

   382 text {*

   383   An application of the previous lemma will be used in the proof of

   384   the Hahn-Banach Theorem (see page \pageref{decomp-H-use}): for any

   385   element @{text "y + a \<cdot> x\<^sub>0"} of the direct sum of a

   386   vectorspace @{text H} and the linear closure of @{text "x\<^sub>0"}

   387   the components @{text "y \<in> H"} and @{text a} are uniquely

   388   determined.

   389 *}

   390

   391 lemma decomp_H':

   392   assumes "vectorspace E" "subspace H E"

   393   assumes y1: "y1 \<in> H" and y2: "y2 \<in> H"

   394     and x': "x' \<notin> H"  "x' \<in> E"  "x' \<noteq> 0"

   395     and eq: "y1 + a1 \<cdot> x' = y2 + a2 \<cdot> x'"

   396   shows "y1 = y2 \<and> a1 = a2"

   397 proof -

   398   interpret vectorspace [E] by fact

   399   interpret subspace [H E] by fact

   400   show ?thesis

   401   proof

   402     have c: "y1 = y2 \<and> a1 \<cdot> x' = a2 \<cdot> x'"

   403     proof (rule decomp)

   404       show "a1 \<cdot> x' \<in> lin x'" ..

   405       show "a2 \<cdot> x' \<in> lin x'" ..

   406       show "H \<inter> lin x' = {0}"

   407       proof

   408 	show "H \<inter> lin x' \<subseteq> {0}"

   409 	proof

   410           fix x assume x: "x \<in> H \<inter> lin x'"

   411           then obtain a where xx': "x = a \<cdot> x'"

   412             by blast

   413           have "x = 0"

   414           proof cases

   415             assume "a = 0"

   416             with xx' and x' show ?thesis by simp

   417           next

   418             assume a: "a \<noteq> 0"

   419             from x have "x \<in> H" ..

   420             with xx' have "inverse a \<cdot> a \<cdot> x' \<in> H" by simp

   421             with a and x' have "x' \<in> H" by (simp add: mult_assoc2)

   422             with x' \<notin> H show ?thesis by contradiction

   423           qed

   424           then show "x \<in> {0}" ..

   425 	qed

   426 	show "{0} \<subseteq> H \<inter> lin x'"

   427 	proof -

   428           have "0 \<in> H" using vectorspace E ..

   429           moreover have "0 \<in> lin x'" using x' \<in> E ..

   430           ultimately show ?thesis by blast

   431 	qed

   432       qed

   433       show "lin x' \<unlhd> E" using x' \<in> E ..

   434     qed (rule vectorspace E, rule subspace H E, rule y1, rule y2, rule eq)

   435     then show "y1 = y2" ..

   436     from c have "a1 \<cdot> x' = a2 \<cdot> x'" ..

   437     with x' show "a1 = a2" by (simp add: mult_right_cancel)

   438   qed

   439 qed

   440

   441 text {*

   442   Since for any element @{text "y + a \<cdot> x'"} of the direct sum of a

   443   vectorspace @{text H} and the linear closure of @{text x'} the

   444   components @{text "y \<in> H"} and @{text a} are unique, it follows from

   445   @{text "y \<in> H"} that @{text "a = 0"}.

   446 *}

   447

   448 lemma decomp_H'_H:

   449   assumes "vectorspace E" "subspace H E"

   450   assumes t: "t \<in> H"

   451     and x': "x' \<notin> H"  "x' \<in> E"  "x' \<noteq> 0"

   452   shows "(SOME (y, a). t = y + a \<cdot> x' \<and> y \<in> H) = (t, 0)"

   453 proof -

   454   interpret vectorspace [E] by fact

   455   interpret subspace [H E] by fact

   456   show ?thesis

   457   proof (rule, simp_all only: split_paired_all split_conv)

   458     from t x' show "t = t + 0 \<cdot> x' \<and> t \<in> H" by simp

   459     fix y and a assume ya: "t = y + a \<cdot> x' \<and> y \<in> H"

   460     have "y = t \<and> a = 0"

   461     proof (rule decomp_H')

   462       from ya x' show "y + a \<cdot> x' = t + 0 \<cdot> x'" by simp

   463       from ya show "y \<in> H" ..

   464     qed (rule vectorspace E, rule subspace H E, rule t, (rule x')+)

   465     with t x' show "(y, a) = (y + a \<cdot> x', 0)" by simp

   466   qed

   467 qed

   468

   469 text {*

   470   The components @{text "y \<in> H"} and @{text a} in @{text "y + a \<cdot> x'"}

   471   are unique, so the function @{text h'} defined by

   472   @{text "h' (y + a \<cdot> x') = h y + a \<cdot> \<xi>"} is definite.

   473 *}

   474

   475 lemma h'_definite:

   476   fixes H

   477   assumes h'_def:

   478     "h' \<equiv> (\<lambda>x. let (y, a) = SOME (y, a). (x = y + a \<cdot> x' \<and> y \<in> H)

   479                 in (h y) + a * xi)"

   480     and x: "x = y + a \<cdot> x'"

   481   assumes "vectorspace E" "subspace H E"

   482   assumes y: "y \<in> H"

   483     and x': "x' \<notin> H"  "x' \<in> E"  "x' \<noteq> 0"

   484   shows "h' x = h y + a * xi"

   485 proof -

   486   interpret vectorspace [E] by fact

   487   interpret subspace [H E] by fact

   488   from x y x' have "x \<in> H + lin x'" by auto

   489   have "\<exists>!p. (\<lambda>(y, a). x = y + a \<cdot> x' \<and> y \<in> H) p" (is "\<exists>!p. ?P p")

   490   proof (rule ex_ex1I)

   491     from x y show "\<exists>p. ?P p" by blast

   492     fix p q assume p: "?P p" and q: "?P q"

   493     show "p = q"

   494     proof -

   495       from p have xp: "x = fst p + snd p \<cdot> x' \<and> fst p \<in> H"

   496         by (cases p) simp

   497       from q have xq: "x = fst q + snd q \<cdot> x' \<and> fst q \<in> H"

   498         by (cases q) simp

   499       have "fst p = fst q \<and> snd p = snd q"

   500       proof (rule decomp_H')

   501         from xp show "fst p \<in> H" ..

   502         from xq show "fst q \<in> H" ..

   503         from xp and xq show "fst p + snd p \<cdot> x' = fst q + snd q \<cdot> x'"

   504           by simp

   505       qed (rule vectorspace E, rule subspace H E, (rule x')+)

   506       then show ?thesis by (cases p, cases q) simp

   507     qed

   508   qed

   509   then have eq: "(SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H) = (y, a)"

   510     by (rule some1_equality) (simp add: x y)

   511   with h'_def show "h' x = h y + a * xi" by (simp add: Let_def)

   512 qed

   513

   514 end