src/HOL/Hahn_Banach/Subspace.thy
author hoelzl
Thu Sep 02 10:14:32 2010 +0200 (2010-09-02)
changeset 39072 1030b1a166ef
parent 32960 69916a850301
child 44190 fe5504984937
permissions -rw-r--r--
Add lessThan_Suc_eq_insert_0
     1 (*  Title:      HOL/Hahn_Banach/Subspace.thy
     2     Author:     Gertrud Bauer, TU Munich
     3 *)
     4 
     5 header {* Subspaces *}
     6 
     7 theory Subspace
     8 imports Vector_Space
     9 begin
    10 
    11 subsection {* Definition *}
    12 
    13 text {*
    14   A non-empty subset @{text U} of a vector space @{text V} is a
    15   \emph{subspace} of @{text V}, iff @{text U} is closed under addition
    16   and scalar multiplication.
    17 *}
    18 
    19 locale subspace =
    20   fixes U :: "'a\<Colon>{minus, plus, zero, uminus} set" and V
    21   assumes non_empty [iff, intro]: "U \<noteq> {}"
    22     and subset [iff]: "U \<subseteq> V"
    23     and add_closed [iff]: "x \<in> U \<Longrightarrow> y \<in> U \<Longrightarrow> x + y \<in> U"
    24     and mult_closed [iff]: "x \<in> U \<Longrightarrow> a \<cdot> x \<in> U"
    25 
    26 notation (symbols)
    27   subspace  (infix "\<unlhd>" 50)
    28 
    29 declare vectorspace.intro [intro?] subspace.intro [intro?]
    30 
    31 lemma subspace_subset [elim]: "U \<unlhd> V \<Longrightarrow> U \<subseteq> V"
    32   by (rule subspace.subset)
    33 
    34 lemma (in subspace) subsetD [iff]: "x \<in> U \<Longrightarrow> x \<in> V"
    35   using subset by blast
    36 
    37 lemma subspaceD [elim]: "U \<unlhd> V \<Longrightarrow> x \<in> U \<Longrightarrow> x \<in> V"
    38   by (rule subspace.subsetD)
    39 
    40 lemma rev_subspaceD [elim?]: "x \<in> U \<Longrightarrow> U \<unlhd> V \<Longrightarrow> x \<in> V"
    41   by (rule subspace.subsetD)
    42 
    43 lemma (in subspace) diff_closed [iff]:
    44   assumes "vectorspace V"
    45   assumes x: "x \<in> U" and y: "y \<in> U"
    46   shows "x - y \<in> U"
    47 proof -
    48   interpret vectorspace V by fact
    49   from x y show ?thesis by (simp add: diff_eq1 negate_eq1)
    50 qed
    51 
    52 text {*
    53   \medskip Similar as for linear spaces, the existence of the zero
    54   element in every subspace follows from the non-emptiness of the
    55   carrier set and by vector space laws.
    56 *}
    57 
    58 lemma (in subspace) zero [intro]:
    59   assumes "vectorspace V"
    60   shows "0 \<in> U"
    61 proof -
    62   interpret V: vectorspace V by fact
    63   have "U \<noteq> {}" by (rule non_empty)
    64   then obtain x where x: "x \<in> U" by blast
    65   then have "x \<in> V" .. then have "0 = x - x" by simp
    66   also from `vectorspace V` x x have "\<dots> \<in> U" by (rule diff_closed)
    67   finally show ?thesis .
    68 qed
    69 
    70 lemma (in subspace) neg_closed [iff]:
    71   assumes "vectorspace V"
    72   assumes x: "x \<in> U"
    73   shows "- x \<in> U"
    74 proof -
    75   interpret vectorspace V by fact
    76   from x show ?thesis by (simp add: negate_eq1)
    77 qed
    78 
    79 text {* \medskip Further derived laws: every subspace is a vector space. *}
    80 
    81 lemma (in subspace) vectorspace [iff]:
    82   assumes "vectorspace V"
    83   shows "vectorspace U"
    84 proof -
    85   interpret vectorspace V by fact
    86   show ?thesis
    87   proof
    88     show "U \<noteq> {}" ..
    89     fix x y z assume x: "x \<in> U" and y: "y \<in> U" and z: "z \<in> U"
    90     fix a b :: real
    91     from x y show "x + y \<in> U" by simp
    92     from x show "a \<cdot> x \<in> U" by simp
    93     from x y z show "(x + y) + z = x + (y + z)" by (simp add: add_ac)
    94     from x y show "x + y = y + x" by (simp add: add_ac)
    95     from x show "x - x = 0" by simp
    96     from x show "0 + x = x" by simp
    97     from x y show "a \<cdot> (x + y) = a \<cdot> x + a \<cdot> y" by (simp add: distrib)
    98     from x show "(a + b) \<cdot> x = a \<cdot> x + b \<cdot> x" by (simp add: distrib)
    99     from x show "(a * b) \<cdot> x = a \<cdot> b \<cdot> x" by (simp add: mult_assoc)
   100     from x show "1 \<cdot> x = x" by simp
   101     from x show "- x = - 1 \<cdot> x" by (simp add: negate_eq1)
   102     from x y show "x - y = x + - y" by (simp add: diff_eq1)
   103   qed
   104 qed
   105 
   106 
   107 text {* The subspace relation is reflexive. *}
   108 
   109 lemma (in vectorspace) subspace_refl [intro]: "V \<unlhd> V"
   110 proof
   111   show "V \<noteq> {}" ..
   112   show "V \<subseteq> V" ..
   113   fix x y assume x: "x \<in> V" and y: "y \<in> V"
   114   fix a :: real
   115   from x y show "x + y \<in> V" by simp
   116   from x show "a \<cdot> x \<in> V" by simp
   117 qed
   118 
   119 text {* The subspace relation is transitive. *}
   120 
   121 lemma (in vectorspace) subspace_trans [trans]:
   122   "U \<unlhd> V \<Longrightarrow> V \<unlhd> W \<Longrightarrow> U \<unlhd> W"
   123 proof
   124   assume uv: "U \<unlhd> V" and vw: "V \<unlhd> W"
   125   from uv show "U \<noteq> {}" by (rule subspace.non_empty)
   126   show "U \<subseteq> W"
   127   proof -
   128     from uv have "U \<subseteq> V" by (rule subspace.subset)
   129     also from vw have "V \<subseteq> W" by (rule subspace.subset)
   130     finally show ?thesis .
   131   qed
   132   fix x y assume x: "x \<in> U" and y: "y \<in> U"
   133   from uv and x y show "x + y \<in> U" by (rule subspace.add_closed)
   134   from uv and x show "\<And>a. a \<cdot> x \<in> U" by (rule subspace.mult_closed)
   135 qed
   136 
   137 
   138 subsection {* Linear closure *}
   139 
   140 text {*
   141   The \emph{linear closure} of a vector @{text x} is the set of all
   142   scalar multiples of @{text x}.
   143 *}
   144 
   145 definition
   146   lin :: "('a::{minus, plus, zero}) \<Rightarrow> 'a set" where
   147   "lin x = {a \<cdot> x | a. True}"
   148 
   149 lemma linI [intro]: "y = a \<cdot> x \<Longrightarrow> y \<in> lin x"
   150   unfolding lin_def by blast
   151 
   152 lemma linI' [iff]: "a \<cdot> x \<in> lin x"
   153   unfolding lin_def by blast
   154 
   155 lemma linE [elim]: "x \<in> lin v \<Longrightarrow> (\<And>a::real. x = a \<cdot> v \<Longrightarrow> C) \<Longrightarrow> C"
   156   unfolding lin_def by blast
   157 
   158 
   159 text {* Every vector is contained in its linear closure. *}
   160 
   161 lemma (in vectorspace) x_lin_x [iff]: "x \<in> V \<Longrightarrow> x \<in> lin x"
   162 proof -
   163   assume "x \<in> V"
   164   then have "x = 1 \<cdot> x" by simp
   165   also have "\<dots> \<in> lin x" ..
   166   finally show ?thesis .
   167 qed
   168 
   169 lemma (in vectorspace) "0_lin_x" [iff]: "x \<in> V \<Longrightarrow> 0 \<in> lin x"
   170 proof
   171   assume "x \<in> V"
   172   then show "0 = 0 \<cdot> x" by simp
   173 qed
   174 
   175 text {* Any linear closure is a subspace. *}
   176 
   177 lemma (in vectorspace) lin_subspace [intro]:
   178   "x \<in> V \<Longrightarrow> lin x \<unlhd> V"
   179 proof
   180   assume x: "x \<in> V"
   181   then show "lin x \<noteq> {}" by (auto simp add: x_lin_x)
   182   show "lin x \<subseteq> V"
   183   proof
   184     fix x' assume "x' \<in> lin x"
   185     then obtain a where "x' = a \<cdot> x" ..
   186     with x show "x' \<in> V" by simp
   187   qed
   188   fix x' x'' assume x': "x' \<in> lin x" and x'': "x'' \<in> lin x"
   189   show "x' + x'' \<in> lin x"
   190   proof -
   191     from x' obtain a' where "x' = a' \<cdot> x" ..
   192     moreover from x'' obtain a'' where "x'' = a'' \<cdot> x" ..
   193     ultimately have "x' + x'' = (a' + a'') \<cdot> x"
   194       using x by (simp add: distrib)
   195     also have "\<dots> \<in> lin x" ..
   196     finally show ?thesis .
   197   qed
   198   fix a :: real
   199   show "a \<cdot> x' \<in> lin x"
   200   proof -
   201     from x' obtain a' where "x' = a' \<cdot> x" ..
   202     with x have "a \<cdot> x' = (a * a') \<cdot> x" by (simp add: mult_assoc)
   203     also have "\<dots> \<in> lin x" ..
   204     finally show ?thesis .
   205   qed
   206 qed
   207 
   208 
   209 text {* Any linear closure is a vector space. *}
   210 
   211 lemma (in vectorspace) lin_vectorspace [intro]:
   212   assumes "x \<in> V"
   213   shows "vectorspace (lin x)"
   214 proof -
   215   from `x \<in> V` have "subspace (lin x) V"
   216     by (rule lin_subspace)
   217   from this and vectorspace_axioms show ?thesis
   218     by (rule subspace.vectorspace)
   219 qed
   220 
   221 
   222 subsection {* Sum of two vectorspaces *}
   223 
   224 text {*
   225   The \emph{sum} of two vectorspaces @{text U} and @{text V} is the
   226   set of all sums of elements from @{text U} and @{text V}.
   227 *}
   228 
   229 instantiation "fun" :: (type, type) plus
   230 begin
   231 
   232 definition
   233   sum_def: "plus_fun U V = {u + v | u v. u \<in> U \<and> v \<in> V}"  (* FIXME not fully general!? *)
   234 
   235 instance ..
   236 
   237 end
   238 
   239 lemma sumE [elim]:
   240     "x \<in> U + V \<Longrightarrow> (\<And>u v. x = u + v \<Longrightarrow> u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> C) \<Longrightarrow> C"
   241   unfolding sum_def by blast
   242 
   243 lemma sumI [intro]:
   244     "u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> x = u + v \<Longrightarrow> x \<in> U + V"
   245   unfolding sum_def by blast
   246 
   247 lemma sumI' [intro]:
   248     "u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> u + v \<in> U + V"
   249   unfolding sum_def by blast
   250 
   251 text {* @{text U} is a subspace of @{text "U + V"}. *}
   252 
   253 lemma subspace_sum1 [iff]:
   254   assumes "vectorspace U" "vectorspace V"
   255   shows "U \<unlhd> U + V"
   256 proof -
   257   interpret vectorspace U by fact
   258   interpret vectorspace V by fact
   259   show ?thesis
   260   proof
   261     show "U \<noteq> {}" ..
   262     show "U \<subseteq> U + V"
   263     proof
   264       fix x assume x: "x \<in> U"
   265       moreover have "0 \<in> V" ..
   266       ultimately have "x + 0 \<in> U + V" ..
   267       with x show "x \<in> U + V" by simp
   268     qed
   269     fix x y assume x: "x \<in> U" and "y \<in> U"
   270     then show "x + y \<in> U" by simp
   271     from x show "\<And>a. a \<cdot> x \<in> U" by simp
   272   qed
   273 qed
   274 
   275 text {* The sum of two subspaces is again a subspace. *}
   276 
   277 lemma sum_subspace [intro?]:
   278   assumes "subspace U E" "vectorspace E" "subspace V E"
   279   shows "U + V \<unlhd> E"
   280 proof -
   281   interpret subspace U E by fact
   282   interpret vectorspace E by fact
   283   interpret subspace V E by fact
   284   show ?thesis
   285   proof
   286     have "0 \<in> U + V"
   287     proof
   288       show "0 \<in> U" using `vectorspace E` ..
   289       show "0 \<in> V" using `vectorspace E` ..
   290       show "(0::'a) = 0 + 0" by simp
   291     qed
   292     then show "U + V \<noteq> {}" by blast
   293     show "U + V \<subseteq> E"
   294     proof
   295       fix x assume "x \<in> U + V"
   296       then obtain u v where "x = u + v" and
   297         "u \<in> U" and "v \<in> V" ..
   298       then show "x \<in> E" by simp
   299     qed
   300     fix x y assume x: "x \<in> U + V" and y: "y \<in> U + V"
   301     show "x + y \<in> U + V"
   302     proof -
   303       from x obtain ux vx where "x = ux + vx" and "ux \<in> U" and "vx \<in> V" ..
   304       moreover
   305       from y obtain uy vy where "y = uy + vy" and "uy \<in> U" and "vy \<in> V" ..
   306       ultimately
   307       have "ux + uy \<in> U"
   308         and "vx + vy \<in> V"
   309         and "x + y = (ux + uy) + (vx + vy)"
   310         using x y by (simp_all add: add_ac)
   311       then show ?thesis ..
   312     qed
   313     fix a show "a \<cdot> x \<in> U + V"
   314     proof -
   315       from x obtain u v where "x = u + v" and "u \<in> U" and "v \<in> V" ..
   316       then have "a \<cdot> u \<in> U" and "a \<cdot> v \<in> V"
   317         and "a \<cdot> x = (a \<cdot> u) + (a \<cdot> v)" by (simp_all add: distrib)
   318       then show ?thesis ..
   319     qed
   320   qed
   321 qed
   322 
   323 text{* The sum of two subspaces is a vectorspace. *}
   324 
   325 lemma sum_vs [intro?]:
   326     "U \<unlhd> E \<Longrightarrow> V \<unlhd> E \<Longrightarrow> vectorspace E \<Longrightarrow> vectorspace (U + V)"
   327   by (rule subspace.vectorspace) (rule sum_subspace)
   328 
   329 
   330 subsection {* Direct sums *}
   331 
   332 text {*
   333   The sum of @{text U} and @{text V} is called \emph{direct}, iff the
   334   zero element is the only common element of @{text U} and @{text
   335   V}. For every element @{text x} of the direct sum of @{text U} and
   336   @{text V} the decomposition in @{text "x = u + v"} with
   337   @{text "u \<in> U"} and @{text "v \<in> V"} is unique.
   338 *}
   339 
   340 lemma decomp:
   341   assumes "vectorspace E" "subspace U E" "subspace V E"
   342   assumes direct: "U \<inter> V = {0}"
   343     and u1: "u1 \<in> U" and u2: "u2 \<in> U"
   344     and v1: "v1 \<in> V" and v2: "v2 \<in> V"
   345     and sum: "u1 + v1 = u2 + v2"
   346   shows "u1 = u2 \<and> v1 = v2"
   347 proof -
   348   interpret vectorspace E by fact
   349   interpret subspace U E by fact
   350   interpret subspace V E by fact
   351   show ?thesis
   352   proof
   353     have U: "vectorspace U"  (* FIXME: use interpret *)
   354       using `subspace U E` `vectorspace E` by (rule subspace.vectorspace)
   355     have V: "vectorspace V"
   356       using `subspace V E` `vectorspace E` by (rule subspace.vectorspace)
   357     from u1 u2 v1 v2 and sum have eq: "u1 - u2 = v2 - v1"
   358       by (simp add: add_diff_swap)
   359     from u1 u2 have u: "u1 - u2 \<in> U"
   360       by (rule vectorspace.diff_closed [OF U])
   361     with eq have v': "v2 - v1 \<in> U" by (simp only:)
   362     from v2 v1 have v: "v2 - v1 \<in> V"
   363       by (rule vectorspace.diff_closed [OF V])
   364     with eq have u': " u1 - u2 \<in> V" by (simp only:)
   365     
   366     show "u1 = u2"
   367     proof (rule add_minus_eq)
   368       from u1 show "u1 \<in> E" ..
   369       from u2 show "u2 \<in> E" ..
   370       from u u' and direct show "u1 - u2 = 0" by blast
   371     qed
   372     show "v1 = v2"
   373     proof (rule add_minus_eq [symmetric])
   374       from v1 show "v1 \<in> E" ..
   375       from v2 show "v2 \<in> E" ..
   376       from v v' and direct show "v2 - v1 = 0" by blast
   377     qed
   378   qed
   379 qed
   380 
   381 text {*
   382   An application of the previous lemma will be used in the proof of
   383   the Hahn-Banach Theorem (see page \pageref{decomp-H-use}): for any
   384   element @{text "y + a \<cdot> x\<^sub>0"} of the direct sum of a
   385   vectorspace @{text H} and the linear closure of @{text "x\<^sub>0"}
   386   the components @{text "y \<in> H"} and @{text a} are uniquely
   387   determined.
   388 *}
   389 
   390 lemma decomp_H':
   391   assumes "vectorspace E" "subspace H E"
   392   assumes y1: "y1 \<in> H" and y2: "y2 \<in> H"
   393     and x': "x' \<notin> H"  "x' \<in> E"  "x' \<noteq> 0"
   394     and eq: "y1 + a1 \<cdot> x' = y2 + a2 \<cdot> x'"
   395   shows "y1 = y2 \<and> a1 = a2"
   396 proof -
   397   interpret vectorspace E by fact
   398   interpret subspace H E by fact
   399   show ?thesis
   400   proof
   401     have c: "y1 = y2 \<and> a1 \<cdot> x' = a2 \<cdot> x'"
   402     proof (rule decomp)
   403       show "a1 \<cdot> x' \<in> lin x'" ..
   404       show "a2 \<cdot> x' \<in> lin x'" ..
   405       show "H \<inter> lin x' = {0}"
   406       proof
   407         show "H \<inter> lin x' \<subseteq> {0}"
   408         proof
   409           fix x assume x: "x \<in> H \<inter> lin x'"
   410           then obtain a where xx': "x = a \<cdot> x'"
   411             by blast
   412           have "x = 0"
   413           proof cases
   414             assume "a = 0"
   415             with xx' and x' show ?thesis by simp
   416           next
   417             assume a: "a \<noteq> 0"
   418             from x have "x \<in> H" ..
   419             with xx' have "inverse a \<cdot> a \<cdot> x' \<in> H" by simp
   420             with a and x' have "x' \<in> H" by (simp add: mult_assoc2)
   421             with `x' \<notin> H` show ?thesis by contradiction
   422           qed
   423           then show "x \<in> {0}" ..
   424         qed
   425         show "{0} \<subseteq> H \<inter> lin x'"
   426         proof -
   427           have "0 \<in> H" using `vectorspace E` ..
   428           moreover have "0 \<in> lin x'" using `x' \<in> E` ..
   429           ultimately show ?thesis by blast
   430         qed
   431       qed
   432       show "lin x' \<unlhd> E" using `x' \<in> E` ..
   433     qed (rule `vectorspace E`, rule `subspace H E`, rule y1, rule y2, rule eq)
   434     then show "y1 = y2" ..
   435     from c have "a1 \<cdot> x' = a2 \<cdot> x'" ..
   436     with x' show "a1 = a2" by (simp add: mult_right_cancel)
   437   qed
   438 qed
   439 
   440 text {*
   441   Since for any element @{text "y + a \<cdot> x'"} of the direct sum of a
   442   vectorspace @{text H} and the linear closure of @{text x'} the
   443   components @{text "y \<in> H"} and @{text a} are unique, it follows from
   444   @{text "y \<in> H"} that @{text "a = 0"}.
   445 *}
   446 
   447 lemma decomp_H'_H:
   448   assumes "vectorspace E" "subspace H E"
   449   assumes t: "t \<in> H"
   450     and x': "x' \<notin> H"  "x' \<in> E"  "x' \<noteq> 0"
   451   shows "(SOME (y, a). t = y + a \<cdot> x' \<and> y \<in> H) = (t, 0)"
   452 proof -
   453   interpret vectorspace E by fact
   454   interpret subspace H E by fact
   455   show ?thesis
   456   proof (rule, simp_all only: split_paired_all split_conv)
   457     from t x' show "t = t + 0 \<cdot> x' \<and> t \<in> H" by simp
   458     fix y and a assume ya: "t = y + a \<cdot> x' \<and> y \<in> H"
   459     have "y = t \<and> a = 0"
   460     proof (rule decomp_H')
   461       from ya x' show "y + a \<cdot> x' = t + 0 \<cdot> x'" by simp
   462       from ya show "y \<in> H" ..
   463     qed (rule `vectorspace E`, rule `subspace H E`, rule t, (rule x')+)
   464     with t x' show "(y, a) = (y + a \<cdot> x', 0)" by simp
   465   qed
   466 qed
   467 
   468 text {*
   469   The components @{text "y \<in> H"} and @{text a} in @{text "y + a \<cdot> x'"}
   470   are unique, so the function @{text h'} defined by
   471   @{text "h' (y + a \<cdot> x') = h y + a \<cdot> \<xi>"} is definite.
   472 *}
   473 
   474 lemma h'_definite:
   475   fixes H
   476   assumes h'_def:
   477     "h' \<equiv> (\<lambda>x. let (y, a) = SOME (y, a). (x = y + a \<cdot> x' \<and> y \<in> H)
   478                 in (h y) + a * xi)"
   479     and x: "x = y + a \<cdot> x'"
   480   assumes "vectorspace E" "subspace H E"
   481   assumes y: "y \<in> H"
   482     and x': "x' \<notin> H"  "x' \<in> E"  "x' \<noteq> 0"
   483   shows "h' x = h y + a * xi"
   484 proof -
   485   interpret vectorspace E by fact
   486   interpret subspace H E by fact
   487   from x y x' have "x \<in> H + lin x'" by auto
   488   have "\<exists>!p. (\<lambda>(y, a). x = y + a \<cdot> x' \<and> y \<in> H) p" (is "\<exists>!p. ?P p")
   489   proof (rule ex_ex1I)
   490     from x y show "\<exists>p. ?P p" by blast
   491     fix p q assume p: "?P p" and q: "?P q"
   492     show "p = q"
   493     proof -
   494       from p have xp: "x = fst p + snd p \<cdot> x' \<and> fst p \<in> H"
   495         by (cases p) simp
   496       from q have xq: "x = fst q + snd q \<cdot> x' \<and> fst q \<in> H"
   497         by (cases q) simp
   498       have "fst p = fst q \<and> snd p = snd q"
   499       proof (rule decomp_H')
   500         from xp show "fst p \<in> H" ..
   501         from xq show "fst q \<in> H" ..
   502         from xp and xq show "fst p + snd p \<cdot> x' = fst q + snd q \<cdot> x'"
   503           by simp
   504       qed (rule `vectorspace E`, rule `subspace H E`, (rule x')+)
   505       then show ?thesis by (cases p, cases q) simp
   506     qed
   507   qed
   508   then have eq: "(SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H) = (y, a)"
   509     by (rule some1_equality) (simp add: x y)
   510   with h'_def show "h' x = h y + a * xi" by (simp add: Let_def)
   511 qed
   512 
   513 end