src/HOL/Hahn_Banach/Hahn_Banach_Ext_Lemmas.thy
author wenzelm
Mon, 25 Apr 2016 16:09:26 +0200
changeset 63040 eb4ddd18d635
parent 61879 e4f9d8f094fe
permissions -rw-r--r--
eliminated old 'def'; tuned comments;
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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(*  Title:      HOL/Hahn_Banach/Hahn_Banach_Ext_Lemmas.thy
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    Author:     Gertrud Bauer, TU Munich
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*)
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section \<open>Extending non-maximal functions\<close>
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theory Hahn_Banach_Ext_Lemmas
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imports Function_Norm
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begin
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text \<open>
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  In this section the following context is presumed. Let \<open>E\<close> be a real vector
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  space with a seminorm \<open>q\<close> on \<open>E\<close>. \<open>F\<close> is a subspace of \<open>E\<close> and \<open>f\<close> a linear
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  function on \<open>F\<close>. We consider a subspace \<open>H\<close> of \<open>E\<close> that is a superspace of
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  \<open>F\<close> and a linear form \<open>h\<close> on \<open>H\<close>. \<open>H\<close> is a not equal to \<open>E\<close> and \<open>x\<^sub>0\<close> is an
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  element in \<open>E - H\<close>. \<open>H\<close> is extended to the direct sum \<open>H' = H + lin x\<^sub>0\<close>, so
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  for any \<open>x \<in> H'\<close> the decomposition of \<open>x = y + a \<cdot> x\<close> with \<open>y \<in> H\<close> is
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  unique. \<open>h'\<close> is defined on \<open>H'\<close> by \<open>h' x = h y + a \<cdot> \<xi>\<close> for a certain \<open>\<xi>\<close>.
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  Subsequently we show some properties of this extension \<open>h'\<close> of \<open>h\<close>.
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  \<^medskip>
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  This lemma will be used to show the existence of a linear extension of \<open>f\<close>
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  (see page \pageref{ex-xi-use}). It is a consequence of the completeness of
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  \<open>\<real>\<close>. To show
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  \begin{center}
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  \begin{tabular}{l}
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  \<open>\<exists>\<xi>. \<forall>y \<in> F. a y \<le> \<xi> \<and> \<xi> \<le> b y\<close>
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  \end{tabular}
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  \end{center}
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  \<^noindent> it suffices to show that
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  \begin{center}
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  \begin{tabular}{l}
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  \<open>\<forall>u \<in> F. \<forall>v \<in> F. a u \<le> b v\<close>
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  \end{tabular}
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  \end{center}
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\<close>
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lemma ex_xi:
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  assumes "vectorspace F"
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  assumes r: "\<And>u v. u \<in> F \<Longrightarrow> v \<in> F \<Longrightarrow> a u \<le> b v"
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  shows "\<exists>xi::real. \<forall>y \<in> F. a y \<le> xi \<and> xi \<le> b y"
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proof -
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  interpret vectorspace F by fact
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  txt \<open>From the completeness of the reals follows:
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    The set \<open>S = {a u. u \<in> F}\<close> has a supremum, if it is
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    non-empty and has an upper bound.\<close>
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  let ?S = "{a u | u. u \<in> F}"
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  have "\<exists>xi. lub ?S xi"
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  proof (rule real_complete)
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    have "a 0 \<in> ?S" by blast
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    then show "\<exists>X. X \<in> ?S" ..
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    have "\<forall>y \<in> ?S. y \<le> b 0"
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    proof
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      fix y assume y: "y \<in> ?S"
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      then obtain u where u: "u \<in> F" and y: "y = a u" by blast
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      from u and zero have "a u \<le> b 0" by (rule r)
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      with y show "y \<le> b 0" by (simp only:)
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    qed
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    then show "\<exists>u. \<forall>y \<in> ?S. y \<le> u" ..
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  qed
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  then obtain xi where xi: "lub ?S xi" ..
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  {
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    fix y assume "y \<in> F"
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    then have "a y \<in> ?S" by blast
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    with xi have "a y \<le> xi" by (rule lub.upper)
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  }
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  moreover {
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    fix y assume y: "y \<in> F"
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    from xi have "xi \<le> b y"
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    proof (rule lub.least)
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      fix au assume "au \<in> ?S"
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      then obtain u where u: "u \<in> F" and au: "au = a u" by blast
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      from u y have "a u \<le> b y" by (rule r)
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      with au show "au \<le> b y" by (simp only:)
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    qed
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  }
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  ultimately show "\<exists>xi. \<forall>y \<in> F. a y \<le> xi \<and> xi \<le> b y" by blast
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qed
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text \<open>
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  \<^medskip>
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  The function \<open>h'\<close> is defined as a \<open>h' x = h y + a \<cdot> \<xi>\<close> where \<open>x = y + a \<cdot> \<xi>\<close>
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  is a linear extension of \<open>h\<close> to \<open>H'\<close>.
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\<close>
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lemma h'_lf:
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  assumes h'_def: "\<And>x. h' x = (let (y, a) =
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      SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H in h y + a * xi)"
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    and H'_def: "H' = H + lin x0"
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    and HE: "H \<unlhd> E"
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  assumes "linearform H h"
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  assumes x0: "x0 \<notin> H"  "x0 \<in> E"  "x0 \<noteq> 0"
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  assumes E: "vectorspace E"
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  shows "linearform H' h'"
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    97
proof -
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  interpret linearform H h by fact
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  interpret vectorspace E by fact
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  show ?thesis
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  proof
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   102
    note E = \<open>vectorspace E\<close>
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   103
    have H': "vectorspace H'"
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   104
    proof (unfold H'_def)
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   105
      from \<open>x0 \<in> E\<close>
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      have "lin x0 \<unlhd> E" ..
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      with HE show "vectorspace (H + lin x0)" using E ..
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    qed
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   109
    {
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      fix x1 x2 assume x1: "x1 \<in> H'" and x2: "x2 \<in> H'"
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   111
      show "h' (x1 + x2) = h' x1 + h' x2"
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      proof -
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        from H' x1 x2 have "x1 + x2 \<in> H'"
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          by (rule vectorspace.add_closed)
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        with x1 x2 obtain y y1 y2 a a1 a2 where
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          x1x2: "x1 + x2 = y + a \<cdot> x0" and y: "y \<in> H"
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          and x1_rep: "x1 = y1 + a1 \<cdot> x0" and y1: "y1 \<in> H"
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          and x2_rep: "x2 = y2 + a2 \<cdot> x0" and y2: "y2 \<in> H"
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   119
          unfolding H'_def sum_def lin_def by blast
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   120
        
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        have ya: "y1 + y2 = y \<and> a1 + a2 = a" using E HE _ y x0
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   122
        proof (rule decomp_H') text_raw \<open>\label{decomp-H-use}\<close>
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          from HE y1 y2 show "y1 + y2 \<in> H"
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            by (rule subspace.add_closed)
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   125
          from x0 and HE y y1 y2
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   126
          have "x0 \<in> E"  "y \<in> E"  "y1 \<in> E"  "y2 \<in> E" by auto
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   127
          with x1_rep x2_rep have "(y1 + y2) + (a1 + a2) \<cdot> x0 = x1 + x2"
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            by (simp add: add_ac add_mult_distrib2)
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          also note x1x2
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          finally show "(y1 + y2) + (a1 + a2) \<cdot> x0 = y + a \<cdot> x0" .
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   131
        qed
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   133
        from h'_def x1x2 E HE y x0
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   134
        have "h' (x1 + x2) = h y + a * xi"
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   135
          by (rule h'_definite)
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   136
        also have "\<dots> = h (y1 + y2) + (a1 + a2) * xi"
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   137
          by (simp only: ya)
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   138
        also from y1 y2 have "h (y1 + y2) = h y1 + h y2"
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   139
          by simp
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diff changeset
   140
        also have "\<dots> + (a1 + a2) * xi = (h y1 + a1 * xi) + (h y2 + a2 * xi)"
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 47445
diff changeset
   141
          by (simp add: distrib_right)
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   142
        also from h'_def x1_rep E HE y1 x0
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   143
        have "h y1 + a1 * xi = h' x1"
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   144
          by (rule h'_definite [symmetric])
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   145
        also from h'_def x2_rep E HE y2 x0
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   146
        have "h y2 + a2 * xi = h' x2"
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   147
          by (rule h'_definite [symmetric])
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   148
        finally show ?thesis .
10007
64bf7da1994a isar-strip-terminators;
wenzelm
parents: 9906
diff changeset
   149
      qed
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   150
    next
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   151
      fix x1 c assume x1: "x1 \<in> H'"
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   152
      show "h' (c \<cdot> x1) = c * (h' x1)"
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   153
      proof -
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   154
        from H' x1 have ax1: "c \<cdot> x1 \<in> H'"
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   155
          by (rule vectorspace.mult_closed)
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   156
        with x1 obtain y a y1 a1 where
27612
d3eb431db035 modernized specifications and proofs;
wenzelm
parents: 27611
diff changeset
   157
            cx1_rep: "c \<cdot> x1 = y + a \<cdot> x0" and y: "y \<in> H"
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   158
          and x1_rep: "x1 = y1 + a1 \<cdot> x0" and y1: "y1 \<in> H"
27612
d3eb431db035 modernized specifications and proofs;
wenzelm
parents: 27611
diff changeset
   159
          unfolding H'_def sum_def lin_def by blast
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   160
        
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   161
        have ya: "c \<cdot> y1 = y \<and> c * a1 = a" using E HE _ y x0
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   162
        proof (rule decomp_H')
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   163
          from HE y1 show "c \<cdot> y1 \<in> H"
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   164
            by (rule subspace.mult_closed)
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   165
          from x0 and HE y y1
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   166
          have "x0 \<in> E"  "y \<in> E"  "y1 \<in> E" by auto
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   167
          with x1_rep have "c \<cdot> y1 + (c * a1) \<cdot> x0 = c \<cdot> x1"
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   168
            by (simp add: mult_assoc add_mult_distrib1)
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   169
          also note cx1_rep
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   170
          finally show "c \<cdot> y1 + (c * a1) \<cdot> x0 = y + a \<cdot> x0" .
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   171
        qed
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   172
        
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   173
        from h'_def cx1_rep E HE y x0 have "h' (c \<cdot> x1) = h y + a * xi"
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   174
          by (rule h'_definite)
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   175
        also have "\<dots> = h (c \<cdot> y1) + (c * a1) * xi"
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   176
          by (simp only: ya)
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   177
        also from y1 have "h (c \<cdot> y1) = c * h y1"
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   178
          by simp
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   179
        also have "\<dots> + (c * a1) * xi = c * (h y1 + a1 * xi)"
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 47445
diff changeset
   180
          by (simp only: distrib_left)
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   181
        also from h'_def x1_rep E HE y1 x0 have "h y1 + a1 * xi = h' x1"
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   182
          by (rule h'_definite [symmetric])
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   183
        finally show ?thesis .
10007
64bf7da1994a isar-strip-terminators;
wenzelm
parents: 9906
diff changeset
   184
      qed
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   185
    }
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   186
  qed
10007
64bf7da1994a isar-strip-terminators;
wenzelm
parents: 9906
diff changeset
   187
qed
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents:
diff changeset
   188
61486
3590367b0ce9 tuned document;
wenzelm
parents: 60458
diff changeset
   189
text \<open>
3590367b0ce9 tuned document;
wenzelm
parents: 60458
diff changeset
   190
  \<^medskip>
61540
f92bf6674699 tuned document;
wenzelm
parents: 61539
diff changeset
   191
  The linear extension \<open>h'\<close> of \<open>h\<close> is bounded by the seminorm \<open>p\<close>.
f92bf6674699 tuned document;
wenzelm
parents: 61539
diff changeset
   192
\<close>
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents:
diff changeset
   193
9374
153853af318b - xsymbols for
bauerg
parents: 9256
diff changeset
   194
lemma h'_norm_pres:
63040
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 61879
diff changeset
   195
  assumes h'_def: "\<And>x. h' x = (let (y, a) =
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 61879
diff changeset
   196
      SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H in h y + a * xi)"
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 61879
diff changeset
   197
    and H'_def: "H' = H + lin x0"
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   198
    and x0: "x0 \<notin> H"  "x0 \<in> E"  "x0 \<noteq> 0"
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   199
  assumes E: "vectorspace E" and HE: "subspace H E"
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   200
    and "seminorm E p" and "linearform H h"
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   201
  assumes a: "\<forall>y \<in> H. h y \<le> p y"
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   202
    and a': "\<forall>y \<in> H. - p (y + x0) - h y \<le> xi \<and> xi \<le> p (y + x0) - h y"
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   203
  shows "\<forall>x \<in> H'. h' x \<le> p x"
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   204
proof -
29234
60f7fb56f8cd More porting to new locales.
ballarin
parents: 27612
diff changeset
   205
  interpret vectorspace E by fact
60f7fb56f8cd More porting to new locales.
ballarin
parents: 27612
diff changeset
   206
  interpret subspace H E by fact
60f7fb56f8cd More porting to new locales.
ballarin
parents: 27612
diff changeset
   207
  interpret seminorm E p by fact
60f7fb56f8cd More porting to new locales.
ballarin
parents: 27612
diff changeset
   208
  interpret linearform H h by fact
27612
d3eb431db035 modernized specifications and proofs;
wenzelm
parents: 27611
diff changeset
   209
  show ?thesis
d3eb431db035 modernized specifications and proofs;
wenzelm
parents: 27611
diff changeset
   210
  proof
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   211
    fix x assume x': "x \<in> H'"
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   212
    show "h' x \<le> p x"
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   213
    proof -
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   214
      from a' have a1: "\<forall>ya \<in> H. - p (ya + x0) - h ya \<le> xi"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   215
        and a2: "\<forall>ya \<in> H. xi \<le> p (ya + x0) - h ya" by auto
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   216
      from x' obtain y a where
27612
d3eb431db035 modernized specifications and proofs;
wenzelm
parents: 27611
diff changeset
   217
          x_rep: "x = y + a \<cdot> x0" and y: "y \<in> H"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   218
        unfolding H'_def sum_def lin_def by blast
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   219
      from y have y': "y \<in> E" ..
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   220
      from y have ay: "inverse a \<cdot> y \<in> H" by simp
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   221
      
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   222
      from h'_def x_rep E HE y x0 have "h' x = h y + a * xi"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   223
        by (rule h'_definite)
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   224
      also have "\<dots> \<le> p (y + a \<cdot> x0)"
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   225
      proof (rule linorder_cases)
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   226
        assume z: "a = 0"
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   227
        then have "h y + a * xi = h y" by simp
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   228
        also from a y have "\<dots> \<le> p y" ..
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   229
        also from x0 y' z have "p y = p (y + a \<cdot> x0)" by simp
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   230
        finally show ?thesis .
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   231
      next
61539
a29295dac1ca isabelle update_cartouches -t;
wenzelm
parents: 61486
diff changeset
   232
        txt \<open>In the case \<open>a < 0\<close>, we use \<open>a\<^sub>1\<close>
a29295dac1ca isabelle update_cartouches -t;
wenzelm
parents: 61486
diff changeset
   233
          with \<open>ya\<close> taken as \<open>y / a\<close>:\<close>
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   234
        assume lz: "a < 0" then have nz: "a \<noteq> 0" by simp
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   235
        from a1 ay
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   236
        have "- p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y) \<le> xi" ..
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   237
        with lz have "a * xi \<le>
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   238
          a * (- p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y))"
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   239
          by (simp add: mult_left_mono_neg order_less_imp_le)
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   240
        
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   241
        also have "\<dots> =
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   242
          - a * (p (inverse a \<cdot> y + x0)) - a * (h (inverse a \<cdot> y))"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   243
          by (simp add: right_diff_distrib)
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   244
        also from lz x0 y' have "- a * (p (inverse a \<cdot> y + x0)) =
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   245
          p (a \<cdot> (inverse a \<cdot> y + x0))"
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   246
          by (simp add: abs_homogenous)
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   247
        also from nz x0 y' have "\<dots> = p (y + a \<cdot> x0)"
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   248
          by (simp add: add_mult_distrib1 mult_assoc [symmetric])
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   249
        also from nz y have "a * (h (inverse a \<cdot> y)) =  h y"
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   250
          by simp
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   251
        finally have "a * xi \<le> p (y + a \<cdot> x0) - h y" .
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   252
        then show ?thesis by simp
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   253
      next
61539
a29295dac1ca isabelle update_cartouches -t;
wenzelm
parents: 61486
diff changeset
   254
        txt \<open>In the case \<open>a > 0\<close>, we use \<open>a\<^sub>2\<close>
a29295dac1ca isabelle update_cartouches -t;
wenzelm
parents: 61486
diff changeset
   255
          with \<open>ya\<close> taken as \<open>y / a\<close>:\<close>
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   256
        assume gz: "0 < a" then have nz: "a \<noteq> 0" by simp
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   257
        from a2 ay
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   258
        have "xi \<le> p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y)" ..
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   259
        with gz have "a * xi \<le>
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   260
          a * (p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y))"
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   261
          by simp
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   262
        also have "\<dots> = a * p (inverse a \<cdot> y + x0) - a * h (inverse a \<cdot> y)"
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   263
          by (simp add: right_diff_distrib)
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   264
        also from gz x0 y'
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   265
        have "a * p (inverse a \<cdot> y + x0) = p (a \<cdot> (inverse a \<cdot> y + x0))"
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   266
          by (simp add: abs_homogenous)
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   267
        also from nz x0 y' have "\<dots> = p (y + a \<cdot> x0)"
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   268
          by (simp add: add_mult_distrib1 mult_assoc [symmetric])
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   269
        also from nz y have "a * h (inverse a \<cdot> y) = h y"
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   270
          by simp
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   271
        finally have "a * xi \<le> p (y + a \<cdot> x0) - h y" .
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   272
        then show ?thesis by simp
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   273
      qed
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   274
      also from x_rep have "\<dots> = p x" by (simp only:)
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   275
      finally show ?thesis .
10007
64bf7da1994a isar-strip-terminators;
wenzelm
parents: 9906
diff changeset
   276
    qed
64bf7da1994a isar-strip-terminators;
wenzelm
parents: 9906
diff changeset
   277
  qed
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   278
qed
7917
5e5b9813cce7 HahnBanach update by Gertrud Bauer;
wenzelm
parents:
diff changeset
   279
10007
64bf7da1994a isar-strip-terminators;
wenzelm
parents: 9906
diff changeset
   280
end