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