src/HOL/Real/HahnBanach/Subspace.thy
 author wenzelm Tue Sep 12 22:13:23 2000 +0200 (2000-09-12) changeset 9941 fe05af7ec816 parent 9906 5c027cca6262 child 9969 4753185f1dd2 permissions -rw-r--r--
renamed atts: rulify to rule_format, elimify to elim_format;
     1 (*  Title:      HOL/Real/HahnBanach/Subspace.thy

     2     ID:         $Id$

     3     Author:     Gertrud Bauer, TU Munich

     4 *)

     5

     6

     7 header {* Subspaces *}

     8

     9 theory Subspace = VectorSpace:

    10

    11

    12 subsection {* Definition *}

    13

    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. *}

    17

    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)"

    22

    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+)

    30

    31 lemma subspace_not_empty [intro?]: "is_subspace U V ==> U \<noteq> {}"

    32   by (unfold is_subspace_def) simp

    33

    34 lemma subspace_subset [intro?]: "is_subspace U V ==> U <= V"

    35   by (unfold is_subspace_def) simp

    36

    37 lemma subspace_subsetD [simp, intro?]:

    38   "[| is_subspace U V; x \<in> U |] ==> x \<in> V"

    39   by (unfold is_subspace_def) force

    40

    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

    44

    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

    48

    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)

    53

    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.*}

    57

    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

    73

    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)

    77

    78 text_raw {* \medskip *}

    79 text {* Further derived laws: every subspace is a vector space. *}

    80

    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)

    93   qed (simp! add: vs_add_mult_distrib1 vs_add_mult_distrib2)+

    94 qed

    95

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

    97

    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

   106

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

   108

   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" ..

   115

   116   have "U <= V" ..

   117   also have "V <= W" ..

   118   finally show "U <= W" .

   119

   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

   125

   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

   132

   133

   134

   135 subsection {* Linear closure *}

   136

   137 text {* The \emph{linear closure} of a vector $x$ is the set of all

   138 scalar multiples of $x$. *}

   139

   140 constdefs

   141   lin :: "('a::{minus,plus,zero}) => 'a set"

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

   143

   144 lemma linD: "x \<in> lin v = (\<exists>a::real. x = a \<cdot> v)"

   145   by (unfold lin_def) fast

   146

   147 lemma linI [intro?]: "a \<cdot> x0 \<in> lin x0"

   148   by (unfold lin_def) fast

   149

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

   151

   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

   157

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

   159

   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

   168

   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

   174

   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"

   183         by (simp! add: vs_add_mult_distrib2)

   184     qed simp

   185   qed

   186

   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

   198

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

   200

   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

   207

   208

   209

   210 subsection {* Sum of two vectorspaces *}

   211

   212 text {* The \emph{sum} of two vectorspaces $U$ and $V$ is the set of

   213 all sums of elements from $U$ and $V$. *}

   214

   215 instance set :: (plus) plus by intro_classes

   216

   217 defs vs_sum_def:

   218   "U + V == {u + v | u v. u \<in> U \<and> v \<in> V}" (***

   219

   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 ***)

   225

   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

   229

   230 lemmas vs_sumE = vs_sumD [THEN iffD1, elim_format, standard]

   231

   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

   235

   236 text{* $U$ is a subspace of $U + V$. *}

   237

   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

   259

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

   261

   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

   273

   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

   279

   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

   291

   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)"

   299         by (simp! add: vs_add_mult_distrib1)

   300     qed (simp!)+

   301   qed

   302 qed

   303

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

   305

   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

   313

   314

   315

   316 subsection {* Direct sums *}

   317

   318

   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.*}

   323

   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

   337

   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

   344

   345   show "v1 = v2"

   346   proof (rule vs_add_minus_eq [symmetric])

   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

   352

   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. *}

   358

   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'"

   367

   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 [THEN 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

   405

   406   from c show "y1 = y2" by simp

   407

   408   show  "a1 = a2"

   409   proof (rule vs_mult_right_cancel [THEN iffD1])

   410     from c show "a1 \<cdot> x' = a2 \<cdot> x'" by simp

   411   qed

   412 qed

   413

   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 $a = 0$.*}

   418

   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!)+

   432

   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. *}

   436

   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

   475

   476 end