src/HOL/Hahn_Banach/Hahn_Banach_Ext_Lemmas.thy
 author blanchet Thu Sep 11 18:54:36 2014 +0200 (2014-09-11) changeset 58306 117ba6cbe414 parent 49962 a8cc904a6820 child 58744 c434e37f290e permissions -rw-r--r--
renamed 'rep_datatype' to 'old_rep_datatype' (HOL)
     1 (*  Title:      HOL/Hahn_Banach/Hahn_Banach_Ext_Lemmas.thy

     2     Author:     Gertrud Bauer, TU Munich

     3 *)

     4

     5 header {* Extending non-maximal functions *}

     6

     7 theory Hahn_Banach_Ext_Lemmas

     8 imports Function_Norm

     9 begin

    10

    11 text {*

    12   In this section the following context is presumed.  Let @{text E} be

    13   a real vector space with a seminorm @{text q} on @{text E}. @{text

    14   F} is a subspace of @{text E} and @{text f} a linear function on

    15   @{text F}. We consider a subspace @{text H} of @{text E} that is a

    16   superspace of @{text F} and a linear form @{text h} on @{text

    17   H}. @{text H} is a not equal to @{text E} and @{text "x\<^sub>0"} is

    18   an element in @{text "E - H"}.  @{text H} is extended to the direct

    19   sum @{text "H' = H + lin x\<^sub>0"}, so for any @{text "x \<in> H'"}

    20   the decomposition of @{text "x = y + a \<cdot> x"} with @{text "y \<in> H"} is

    21   unique. @{text h'} is defined on @{text H'} by @{text "h' x = h y +

    22   a \<cdot> \<xi>"} for a certain @{text \<xi>}.

    23

    24   Subsequently we show some properties of this extension @{text h'} of

    25   @{text h}.

    26

    27   \medskip This lemma will be used to show the existence of a linear

    28   extension of @{text f} (see page \pageref{ex-xi-use}). It is a

    29   consequence of the completeness of @{text \<real>}. To show

    30   \begin{center}

    31   \begin{tabular}{l}

    32   @{text "\<exists>\<xi>. \<forall>y \<in> F. a y \<le> \<xi> \<and> \<xi> \<le> b y"}

    33   \end{tabular}

    34   \end{center}

    35   \noindent it suffices to show that

    36   \begin{center}

    37   \begin{tabular}{l}

    38   @{text "\<forall>u \<in> F. \<forall>v \<in> F. a u \<le> b v"}

    39   \end{tabular}

    40   \end{center}

    41 *}

    42

    43 lemma ex_xi:

    44   assumes "vectorspace F"

    45   assumes r: "\<And>u v. u \<in> F \<Longrightarrow> v \<in> F \<Longrightarrow> a u \<le> b v"

    46   shows "\<exists>xi::real. \<forall>y \<in> F. a y \<le> xi \<and> xi \<le> b y"

    47 proof -

    48   interpret vectorspace F by fact

    49   txt {* From the completeness of the reals follows:

    50     The set @{text "S = {a u. u \<in> F}"} has a supremum, if it is

    51     non-empty and has an upper bound. *}

    52

    53   let ?S = "{a u | u. u \<in> F}"

    54   have "\<exists>xi. lub ?S xi"

    55   proof (rule real_complete)

    56     have "a 0 \<in> ?S" by blast

    57     then show "\<exists>X. X \<in> ?S" ..

    58     have "\<forall>y \<in> ?S. y \<le> b 0"

    59     proof

    60       fix y assume y: "y \<in> ?S"

    61       then obtain u where u: "u \<in> F" and y: "y = a u" by blast

    62       from u and zero have "a u \<le> b 0" by (rule r)

    63       with y show "y \<le> b 0" by (simp only:)

    64     qed

    65     then show "\<exists>u. \<forall>y \<in> ?S. y \<le> u" ..

    66   qed

    67   then obtain xi where xi: "lub ?S xi" ..

    68   {

    69     fix y assume "y \<in> F"

    70     then have "a y \<in> ?S" by blast

    71     with xi have "a y \<le> xi" by (rule lub.upper)

    72   } moreover {

    73     fix y assume y: "y \<in> F"

    74     from xi have "xi \<le> b y"

    75     proof (rule lub.least)

    76       fix au assume "au \<in> ?S"

    77       then obtain u where u: "u \<in> F" and au: "au = a u" by blast

    78       from u y have "a u \<le> b y" by (rule r)

    79       with au show "au \<le> b y" by (simp only:)

    80     qed

    81   } ultimately show "\<exists>xi. \<forall>y \<in> F. a y \<le> xi \<and> xi \<le> b y" by blast

    82 qed

    83

    84 text {*

    85   \medskip The function @{text h'} is defined as a @{text "h' x = h y

    86   + a \<cdot> \<xi>"} where @{text "x = y + a \<cdot> \<xi>"} is a linear extension of

    87   @{text h} to @{text H'}.

    88 *}

    89

    90 lemma h'_lf:

    91   assumes h'_def: "h' \<equiv> \<lambda>x. let (y, a) =

    92       SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H in h y + a * xi"

    93     and H'_def: "H' \<equiv> H + lin x0"

    94     and HE: "H \<unlhd> E"

    95   assumes "linearform H h"

    96   assumes x0: "x0 \<notin> H"  "x0 \<in> E"  "x0 \<noteq> 0"

    97   assumes E: "vectorspace E"

    98   shows "linearform H' h'"

    99 proof -

   100   interpret linearform H h by fact

   101   interpret vectorspace E by fact

   102   show ?thesis

   103   proof

   104     note E = vectorspace E

   105     have H': "vectorspace H'"

   106     proof (unfold H'_def)

   107       from x0 \<in> E

   108       have "lin x0 \<unlhd> E" ..

   109       with HE show "vectorspace (H + lin x0)" using E ..

   110     qed

   111     {

   112       fix x1 x2 assume x1: "x1 \<in> H'" and x2: "x2 \<in> H'"

   113       show "h' (x1 + x2) = h' x1 + h' x2"

   114       proof -

   115         from H' x1 x2 have "x1 + x2 \<in> H'"

   116           by (rule vectorspace.add_closed)

   117         with x1 x2 obtain y y1 y2 a a1 a2 where

   118           x1x2: "x1 + x2 = y + a \<cdot> x0" and y: "y \<in> H"

   119           and x1_rep: "x1 = y1 + a1 \<cdot> x0" and y1: "y1 \<in> H"

   120           and x2_rep: "x2 = y2 + a2 \<cdot> x0" and y2: "y2 \<in> H"

   121           unfolding H'_def sum_def lin_def by blast

   122

   123         have ya: "y1 + y2 = y \<and> a1 + a2 = a" using E HE _ y x0

   124         proof (rule decomp_H') txt_raw {* \label{decomp-H-use} *}

   125           from HE y1 y2 show "y1 + y2 \<in> H"

   126             by (rule subspace.add_closed)

   127           from x0 and HE y y1 y2

   128           have "x0 \<in> E"  "y \<in> E"  "y1 \<in> E"  "y2 \<in> E" by auto

   129           with x1_rep x2_rep have "(y1 + y2) + (a1 + a2) \<cdot> x0 = x1 + x2"

   130             by (simp add: add_ac add_mult_distrib2)

   131           also note x1x2

   132           finally show "(y1 + y2) + (a1 + a2) \<cdot> x0 = y + a \<cdot> x0" .

   133         qed

   134

   135         from h'_def x1x2 E HE y x0

   136         have "h' (x1 + x2) = h y + a * xi"

   137           by (rule h'_definite)

   138         also have "\<dots> = h (y1 + y2) + (a1 + a2) * xi"

   139           by (simp only: ya)

   140         also from y1 y2 have "h (y1 + y2) = h y1 + h y2"

   141           by simp

   142         also have "\<dots> + (a1 + a2) * xi = (h y1 + a1 * xi) + (h y2 + a2 * xi)"

   143           by (simp add: distrib_right)

   144         also from h'_def x1_rep E HE y1 x0

   145         have "h y1 + a1 * xi = h' x1"

   146           by (rule h'_definite [symmetric])

   147         also from h'_def x2_rep E HE y2 x0

   148         have "h y2 + a2 * xi = h' x2"

   149           by (rule h'_definite [symmetric])

   150         finally show ?thesis .

   151       qed

   152     next

   153       fix x1 c assume x1: "x1 \<in> H'"

   154       show "h' (c \<cdot> x1) = c * (h' x1)"

   155       proof -

   156         from H' x1 have ax1: "c \<cdot> x1 \<in> H'"

   157           by (rule vectorspace.mult_closed)

   158         with x1 obtain y a y1 a1 where

   159             cx1_rep: "c \<cdot> x1 = y + a \<cdot> x0" and y: "y \<in> H"

   160           and x1_rep: "x1 = y1 + a1 \<cdot> x0" and y1: "y1 \<in> H"

   161           unfolding H'_def sum_def lin_def by blast

   162

   163         have ya: "c \<cdot> y1 = y \<and> c * a1 = a" using E HE _ y x0

   164         proof (rule decomp_H')

   165           from HE y1 show "c \<cdot> y1 \<in> H"

   166             by (rule subspace.mult_closed)

   167           from x0 and HE y y1

   168           have "x0 \<in> E"  "y \<in> E"  "y1 \<in> E" by auto

   169           with x1_rep have "c \<cdot> y1 + (c * a1) \<cdot> x0 = c \<cdot> x1"

   170             by (simp add: mult_assoc add_mult_distrib1)

   171           also note cx1_rep

   172           finally show "c \<cdot> y1 + (c * a1) \<cdot> x0 = y + a \<cdot> x0" .

   173         qed

   174

   175         from h'_def cx1_rep E HE y x0 have "h' (c \<cdot> x1) = h y + a * xi"

   176           by (rule h'_definite)

   177         also have "\<dots> = h (c \<cdot> y1) + (c * a1) * xi"

   178           by (simp only: ya)

   179         also from y1 have "h (c \<cdot> y1) = c * h y1"

   180           by simp

   181         also have "\<dots> + (c * a1) * xi = c * (h y1 + a1 * xi)"

   182           by (simp only: distrib_left)

   183         also from h'_def x1_rep E HE y1 x0 have "h y1 + a1 * xi = h' x1"

   184           by (rule h'_definite [symmetric])

   185         finally show ?thesis .

   186       qed

   187     }

   188   qed

   189 qed

   190

   191 text {* \medskip The linear extension @{text h'} of @{text h}

   192   is bounded by the seminorm @{text p}. *}

   193

   194 lemma h'_norm_pres:

   195   assumes h'_def: "h' \<equiv> \<lambda>x. let (y, a) =

   196       SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H in h y + a * xi"

   197     and H'_def: "H' \<equiv> H + lin x0"

   198     and x0: "x0 \<notin> H"  "x0 \<in> E"  "x0 \<noteq> 0"

   199   assumes E: "vectorspace E" and HE: "subspace H E"

   200     and "seminorm E p" and "linearform H h"

   201   assumes a: "\<forall>y \<in> H. h y \<le> p y"

   202     and a': "\<forall>y \<in> H. - p (y + x0) - h y \<le> xi \<and> xi \<le> p (y + x0) - h y"

   203   shows "\<forall>x \<in> H'. h' x \<le> p x"

   204 proof -

   205   interpret vectorspace E by fact

   206   interpret subspace H E by fact

   207   interpret seminorm E p by fact

   208   interpret linearform H h by fact

   209   show ?thesis

   210   proof

   211     fix x assume x': "x \<in> H'"

   212     show "h' x \<le> p x"

   213     proof -

   214       from a' have a1: "\<forall>ya \<in> H. - p (ya + x0) - h ya \<le> xi"

   215         and a2: "\<forall>ya \<in> H. xi \<le> p (ya + x0) - h ya" by auto

   216       from x' obtain y a where

   217           x_rep: "x = y + a \<cdot> x0" and y: "y \<in> H"

   218         unfolding H'_def sum_def lin_def by blast

   219       from y have y': "y \<in> E" ..

   220       from y have ay: "inverse a \<cdot> y \<in> H" by simp

   221

   222       from h'_def x_rep E HE y x0 have "h' x = h y + a * xi"

   223         by (rule h'_definite)

   224       also have "\<dots> \<le> p (y + a \<cdot> x0)"

   225       proof (rule linorder_cases)

   226         assume z: "a = 0"

   227         then have "h y + a * xi = h y" by simp

   228         also from a y have "\<dots> \<le> p y" ..

   229         also from x0 y' z have "p y = p (y + a \<cdot> x0)" by simp

   230         finally show ?thesis .

   231       next

   232         txt {* In the case @{text "a < 0"}, we use @{text "a\<^sub>1"}

   233           with @{text ya} taken as @{text "y / a"}: *}

   234         assume lz: "a < 0" then have nz: "a \<noteq> 0" by simp

   235         from a1 ay

   236         have "- p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y) \<le> xi" ..

   237         with lz have "a * xi \<le>

   238           a * (- p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y))"

   239           by (simp add: mult_left_mono_neg order_less_imp_le)

   240

   241         also have "\<dots> =

   242           - a * (p (inverse a \<cdot> y + x0)) - a * (h (inverse a \<cdot> y))"

   243           by (simp add: right_diff_distrib)

   244         also from lz x0 y' have "- a * (p (inverse a \<cdot> y + x0)) =

   245           p (a \<cdot> (inverse a \<cdot> y + x0))"

   246           by (simp add: abs_homogenous)

   247         also from nz x0 y' have "\<dots> = p (y + a \<cdot> x0)"

   248           by (simp add: add_mult_distrib1 mult_assoc [symmetric])

   249         also from nz y have "a * (h (inverse a \<cdot> y)) =  h y"

   250           by simp

   251         finally have "a * xi \<le> p (y + a \<cdot> x0) - h y" .

   252         then show ?thesis by simp

   253       next

   254         txt {* In the case @{text "a > 0"}, we use @{text "a\<^sub>2"}

   255           with @{text ya} taken as @{text "y / a"}: *}

   256         assume gz: "0 < a" then have nz: "a \<noteq> 0" by simp

   257         from a2 ay

   258         have "xi \<le> p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y)" ..

   259         with gz have "a * xi \<le>

   260           a * (p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y))"

   261           by simp

   262         also have "\<dots> = a * p (inverse a \<cdot> y + x0) - a * h (inverse a \<cdot> y)"

   263           by (simp add: right_diff_distrib)

   264         also from gz x0 y'

   265         have "a * p (inverse a \<cdot> y + x0) = p (a \<cdot> (inverse a \<cdot> y + x0))"

   266           by (simp add: abs_homogenous)

   267         also from nz x0 y' have "\<dots> = p (y + a \<cdot> x0)"

   268           by (simp add: add_mult_distrib1 mult_assoc [symmetric])

   269         also from nz y have "a * h (inverse a \<cdot> y) = h y"

   270           by simp

   271         finally have "a * xi \<le> p (y + a \<cdot> x0) - h y" .

   272         then show ?thesis by simp

   273       qed

   274       also from x_rep have "\<dots> = p x" by (simp only:)

   275       finally show ?thesis .

   276     qed

   277   qed

   278 qed

   279

   280 end