src/HOL/Hahn_Banach/Hahn_Banach_Sup_Lemmas.thy
author hoelzl
Thu Sep 02 10:14:32 2010 +0200 (2010-09-02)
changeset 39072 1030b1a166ef
parent 32960 69916a850301
child 44887 7ca82df6e951
permissions -rw-r--r--
Add lessThan_Suc_eq_insert_0
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(*  Title:      HOL/Hahn_Banach/Hahn_Banach_Sup_Lemmas.thy
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    Author:     Gertrud Bauer, TU Munich
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*)
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header {* The supremum w.r.t.~the function order *}
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theory Hahn_Banach_Sup_Lemmas
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imports Function_Norm Zorn_Lemma
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begin
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text {*
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  This section contains some lemmas that will be used in the proof of
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  the Hahn-Banach Theorem.  In this section the following context is
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  presumed.  Let @{text E} be a real vector space with a seminorm
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  @{text p} on @{text E}.  @{text F} is a subspace of @{text E} and
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  @{text f} a linear form on @{text F}. We consider a chain @{text c}
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  of norm-preserving extensions of @{text f}, such that @{text "\<Union>c =
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  graph H h"}.  We will show some properties about the limit function
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  @{text h}, i.e.\ the supremum of the chain @{text c}.
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  \medskip Let @{text c} be a chain of norm-preserving extensions of
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  the function @{text f} and let @{text "graph H h"} be the supremum
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  of @{text c}.  Every element in @{text H} is member of one of the
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  elements of the chain.
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*}
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lemmas [dest?] = chainD
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lemmas chainE2 [elim?] = chainD2 [elim_format, standard]
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lemma some_H'h't:
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  assumes M: "M = norm_pres_extensions E p F f"
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    and cM: "c \<in> chain M"
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    and u: "graph H h = \<Union>c"
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    and x: "x \<in> H"
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  shows "\<exists>H' h'. graph H' h' \<in> c
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    \<and> (x, h x) \<in> graph H' h'
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    \<and> linearform H' h' \<and> H' \<unlhd> E
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    \<and> F \<unlhd> H' \<and> graph F f \<subseteq> graph H' h'
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    \<and> (\<forall>x \<in> H'. h' x \<le> p x)"
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proof -
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  from x have "(x, h x) \<in> graph H h" ..
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  also from u have "\<dots> = \<Union>c" .
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  finally obtain g where gc: "g \<in> c" and gh: "(x, h x) \<in> g" by blast
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  from cM have "c \<subseteq> M" ..
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  with gc have "g \<in> M" ..
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  also from M have "\<dots> = norm_pres_extensions E p F f" .
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  finally obtain H' and h' where g: "g = graph H' h'"
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    and * : "linearform H' h'"  "H' \<unlhd> E"  "F \<unlhd> H'"
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      "graph F f \<subseteq> graph H' h'"  "\<forall>x \<in> H'. h' x \<le> p x" ..
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  from gc and g have "graph H' h' \<in> c" by (simp only:)
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  moreover from gh and g have "(x, h x) \<in> graph H' h'" by (simp only:)
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  ultimately show ?thesis using * by blast
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qed
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text {*
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  \medskip Let @{text c} be a chain of norm-preserving extensions of
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  the function @{text f} and let @{text "graph H h"} be the supremum
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  of @{text c}.  Every element in the domain @{text H} of the supremum
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  function is member of the domain @{text H'} of some function @{text
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  h'}, such that @{text h} extends @{text h'}.
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*}
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lemma some_H'h':
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  assumes M: "M = norm_pres_extensions E p F f"
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    and cM: "c \<in> chain M"
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    and u: "graph H h = \<Union>c"
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    and x: "x \<in> H"
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  shows "\<exists>H' h'. x \<in> H' \<and> graph H' h' \<subseteq> graph H h
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    \<and> linearform H' h' \<and> H' \<unlhd> E \<and> F \<unlhd> H'
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    \<and> graph F f \<subseteq> graph H' h' \<and> (\<forall>x \<in> H'. h' x \<le> p x)"
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proof -
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  from M cM u x obtain H' h' where
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      x_hx: "(x, h x) \<in> graph H' h'"
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    and c: "graph H' h' \<in> c"
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    and * : "linearform H' h'"  "H' \<unlhd> E"  "F \<unlhd> H'"
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      "graph F f \<subseteq> graph H' h'"  "\<forall>x \<in> H'. h' x \<le> p x"
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    by (rule some_H'h't [elim_format]) blast
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  from x_hx have "x \<in> H'" ..
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  moreover from cM u c have "graph H' h' \<subseteq> graph H h"
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    by (simp only: chain_ball_Union_upper)
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  ultimately show ?thesis using * by blast
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qed
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text {*
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  \medskip Any two elements @{text x} and @{text y} in the domain
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  @{text H} of the supremum function @{text h} are both in the domain
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  @{text H'} of some function @{text h'}, such that @{text h} extends
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  @{text h'}.
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*}
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lemma some_H'h'2:
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  assumes M: "M = norm_pres_extensions E p F f"
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    and cM: "c \<in> chain M"
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    and u: "graph H h = \<Union>c"
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    and x: "x \<in> H"
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    and y: "y \<in> H"
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  shows "\<exists>H' h'. x \<in> H' \<and> y \<in> H'
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    \<and> graph H' h' \<subseteq> graph H h
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    \<and> linearform H' h' \<and> H' \<unlhd> E \<and> F \<unlhd> H'
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    \<and> graph F f \<subseteq> graph H' h' \<and> (\<forall>x \<in> H'. h' x \<le> p x)"
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proof -
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  txt {* @{text y} is in the domain @{text H''} of some function @{text h''},
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  such that @{text h} extends @{text h''}. *}
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  from M cM u and y obtain H' h' where
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      y_hy: "(y, h y) \<in> graph H' h'"
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    and c': "graph H' h' \<in> c"
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    and * :
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      "linearform H' h'"  "H' \<unlhd> E"  "F \<unlhd> H'"
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      "graph F f \<subseteq> graph H' h'"  "\<forall>x \<in> H'. h' x \<le> p x"
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    by (rule some_H'h't [elim_format]) blast
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  txt {* @{text x} is in the domain @{text H'} of some function @{text h'},
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    such that @{text h} extends @{text h'}. *}
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  from M cM u and x obtain H'' h'' where
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      x_hx: "(x, h x) \<in> graph H'' h''"
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    and c'': "graph H'' h'' \<in> c"
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    and ** :
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      "linearform H'' h''"  "H'' \<unlhd> E"  "F \<unlhd> H''"
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      "graph F f \<subseteq> graph H'' h''"  "\<forall>x \<in> H''. h'' x \<le> p x"
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    by (rule some_H'h't [elim_format]) blast
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  txt {* Since both @{text h'} and @{text h''} are elements of the chain,
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    @{text h''} is an extension of @{text h'} or vice versa. Thus both
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    @{text x} and @{text y} are contained in the greater
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    one. \label{cases1}*}
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  from cM c'' c' have "graph H'' h'' \<subseteq> graph H' h' \<or> graph H' h' \<subseteq> graph H'' h''"
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    (is "?case1 \<or> ?case2") ..
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  then show ?thesis
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  proof
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    assume ?case1
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    have "(x, h x) \<in> graph H'' h''" by fact
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    also have "\<dots> \<subseteq> graph H' h'" by fact
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    finally have xh:"(x, h x) \<in> graph H' h'" .
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    then have "x \<in> H'" ..
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    moreover from y_hy have "y \<in> H'" ..
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    moreover from cM u and c' have "graph H' h' \<subseteq> graph H h"
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      by (simp only: chain_ball_Union_upper)
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    ultimately show ?thesis using * by blast
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  next
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    assume ?case2
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    from x_hx have "x \<in> H''" ..
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    moreover {
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      have "(y, h y) \<in> graph H' h'" by (rule y_hy)
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      also have "\<dots> \<subseteq> graph H'' h''" by fact
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      finally have "(y, h y) \<in> graph H'' h''" .
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    } then have "y \<in> H''" ..
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    moreover from cM u and c'' have "graph H'' h'' \<subseteq> graph H h"
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      by (simp only: chain_ball_Union_upper)
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    ultimately show ?thesis using ** by blast
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  qed
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qed
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text {*
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  \medskip The relation induced by the graph of the supremum of a
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  chain @{text c} is definite, i.~e.~t is the graph of a function.
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*}
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lemma sup_definite:
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  assumes M_def: "M \<equiv> norm_pres_extensions E p F f"
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    and cM: "c \<in> chain M"
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    and xy: "(x, y) \<in> \<Union>c"
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    and xz: "(x, z) \<in> \<Union>c"
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  shows "z = y"
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proof -
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  from cM have c: "c \<subseteq> M" ..
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  from xy obtain G1 where xy': "(x, y) \<in> G1" and G1: "G1 \<in> c" ..
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  from xz obtain G2 where xz': "(x, z) \<in> G2" and G2: "G2 \<in> c" ..
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  from G1 c have "G1 \<in> M" ..
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  then obtain H1 h1 where G1_rep: "G1 = graph H1 h1"
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    unfolding M_def by blast
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  from G2 c have "G2 \<in> M" ..
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  then obtain H2 h2 where G2_rep: "G2 = graph H2 h2"
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    unfolding M_def by blast
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  txt {* @{text "G\<^sub>1"} is contained in @{text "G\<^sub>2"}
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    or vice versa, since both @{text "G\<^sub>1"} and @{text
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    "G\<^sub>2"} are members of @{text c}. \label{cases2}*}
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  from cM G1 G2 have "G1 \<subseteq> G2 \<or> G2 \<subseteq> G1" (is "?case1 \<or> ?case2") ..
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  then show ?thesis
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  proof
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    assume ?case1
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    with xy' G2_rep have "(x, y) \<in> graph H2 h2" by blast
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    then have "y = h2 x" ..
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    also
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    from xz' G2_rep have "(x, z) \<in> graph H2 h2" by (simp only:)
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    then have "z = h2 x" ..
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    finally show ?thesis .
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  next
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    assume ?case2
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    with xz' G1_rep have "(x, z) \<in> graph H1 h1" by blast
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    then have "z = h1 x" ..
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    also
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    from xy' G1_rep have "(x, y) \<in> graph H1 h1" by (simp only:)
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    then have "y = h1 x" ..
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    finally show ?thesis ..
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  qed
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qed
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text {*
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  \medskip The limit function @{text h} is linear. Every element
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  @{text x} in the domain of @{text h} is in the domain of a function
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  @{text h'} in the chain of norm preserving extensions.  Furthermore,
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  @{text h} is an extension of @{text h'} so the function values of
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  @{text x} are identical for @{text h'} and @{text h}.  Finally, the
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  function @{text h'} is linear by construction of @{text M}.
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*}
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lemma sup_lf:
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  assumes M: "M = norm_pres_extensions E p F f"
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    and cM: "c \<in> chain M"
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    and u: "graph H h = \<Union>c"
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  shows "linearform H h"
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proof
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  fix x y assume x: "x \<in> H" and y: "y \<in> H"
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  with M cM u obtain H' h' where
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        x': "x \<in> H'" and y': "y \<in> H'"
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      and b: "graph H' h' \<subseteq> graph H h"
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      and linearform: "linearform H' h'"
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      and subspace: "H' \<unlhd> E"
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    by (rule some_H'h'2 [elim_format]) blast
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  show "h (x + y) = h x + h y"
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  proof -
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    from linearform x' y' have "h' (x + y) = h' x + h' y"
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      by (rule linearform.add)
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    also from b x' have "h' x = h x" ..
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    also from b y' have "h' y = h y" ..
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    also from subspace x' y' have "x + y \<in> H'"
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      by (rule subspace.add_closed)
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    with b have "h' (x + y) = h (x + y)" ..
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    finally show ?thesis .
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  qed
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next
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  fix x a assume x: "x \<in> H"
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  with M cM u obtain H' h' where
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        x': "x \<in> H'"
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      and b: "graph H' h' \<subseteq> graph H h"
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      and linearform: "linearform H' h'"
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      and subspace: "H' \<unlhd> E"
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    by (rule some_H'h' [elim_format]) blast
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  show "h (a \<cdot> x) = a * h x"
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  proof -
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    from linearform x' have "h' (a \<cdot> x) = a * h' x"
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      by (rule linearform.mult)
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    also from b x' have "h' x = h x" ..
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    also from subspace x' have "a \<cdot> x \<in> H'"
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      by (rule subspace.mult_closed)
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    with b have "h' (a \<cdot> x) = h (a \<cdot> x)" ..
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    finally show ?thesis .
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  qed
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qed
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text {*
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  \medskip The limit of a non-empty chain of norm preserving
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  extensions of @{text f} is an extension of @{text f}, since every
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  element of the chain is an extension of @{text f} and the supremum
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  is an extension for every element of the chain.
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*}
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lemma sup_ext:
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  assumes graph: "graph H h = \<Union>c"
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    and M: "M = norm_pres_extensions E p F f"
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    and cM: "c \<in> chain M"
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    and ex: "\<exists>x. x \<in> c"
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  shows "graph F f \<subseteq> graph H h"
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proof -
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  from ex obtain x where xc: "x \<in> c" ..
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  from cM have "c \<subseteq> M" ..
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  with xc have "x \<in> M" ..
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  with M have "x \<in> norm_pres_extensions E p F f"
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    by (simp only:)
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  then obtain G g where "x = graph G g" and "graph F f \<subseteq> graph G g" ..
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  then have "graph F f \<subseteq> x" by (simp only:)
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  also from xc have "\<dots> \<subseteq> \<Union>c" by blast
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  also from graph have "\<dots> = graph H h" ..
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  finally show ?thesis .
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qed
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text {*
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  \medskip The domain @{text H} of the limit function is a superspace
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  of @{text F}, since @{text F} is a subset of @{text H}. The
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  existence of the @{text 0} element in @{text F} and the closure
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  properties follow from the fact that @{text F} is a vector space.
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*}
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lemma sup_supF:
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  assumes graph: "graph H h = \<Union>c"
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    and M: "M = norm_pres_extensions E p F f"
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    and cM: "c \<in> chain M"
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    and ex: "\<exists>x. x \<in> c"
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    and FE: "F \<unlhd> E"
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  shows "F \<unlhd> H"
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proof
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   302
  from FE show "F \<noteq> {}" by (rule subspace.non_empty)
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   303
  from graph M cM ex have "graph F f \<subseteq> graph H h" by (rule sup_ext)
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  then show "F \<subseteq> H" ..
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   305
  fix x y assume "x \<in> F" and "y \<in> F"
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   306
  with FE show "x + y \<in> F" by (rule subspace.add_closed)
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   307
next
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   308
  fix x a assume "x \<in> F"
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  with FE show "a \<cdot> x \<in> F" by (rule subspace.mult_closed)
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qed
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text {*
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  \medskip The domain @{text H} of the limit function is a subspace of
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  @{text E}.
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*}
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   317
lemma sup_subE:
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  assumes graph: "graph H h = \<Union>c"
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    and M: "M = norm_pres_extensions E p F f"
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    and cM: "c \<in> chain M"
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    and ex: "\<exists>x. x \<in> c"
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    and FE: "F \<unlhd> E"
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    and E: "vectorspace E"
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  shows "H \<unlhd> E"
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   325
proof
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  show "H \<noteq> {}"
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   327
  proof -
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    from FE E have "0 \<in> F" by (rule subspace.zero)
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    also from graph M cM ex FE have "F \<unlhd> H" by (rule sup_supF)
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   330
    then have "F \<subseteq> H" ..
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   331
    finally show ?thesis by blast
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   332
  qed
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   333
  show "H \<subseteq> E"
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   334
  proof
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   335
    fix x assume "x \<in> H"
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   336
    with M cM graph
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   337
    obtain H' h' where x: "x \<in> H'" and H'E: "H' \<unlhd> E"
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   338
      by (rule some_H'h' [elim_format]) blast
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   339
    from H'E have "H' \<subseteq> E" ..
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   340
    with x show "x \<in> E" ..
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   341
  qed
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   342
  fix x y assume x: "x \<in> H" and y: "y \<in> H"
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   343
  show "x + y \<in> H"
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   344
  proof -
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   345
    from M cM graph x y obtain H' h' where
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   346
          x': "x \<in> H'" and y': "y \<in> H'" and H'E: "H' \<unlhd> E"
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   347
        and graphs: "graph H' h' \<subseteq> graph H h"
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   348
      by (rule some_H'h'2 [elim_format]) blast
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   349
    from H'E x' y' have "x + y \<in> H'"
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   350
      by (rule subspace.add_closed)
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   351
    also from graphs have "H' \<subseteq> H" ..
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   352
    finally show ?thesis .
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   353
  qed
wenzelm@13515
   354
next
wenzelm@13515
   355
  fix x a assume x: "x \<in> H"
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   356
  show "a \<cdot> x \<in> H"
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   357
  proof -
wenzelm@13515
   358
    from M cM graph x
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   359
    obtain H' h' where x': "x \<in> H'" and H'E: "H' \<unlhd> E"
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   360
        and graphs: "graph H' h' \<subseteq> graph H h"
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   361
      by (rule some_H'h' [elim_format]) blast
wenzelm@13515
   362
    from H'E x' have "a \<cdot> x \<in> H'" by (rule subspace.mult_closed)
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   363
    also from graphs have "H' \<subseteq> H" ..
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   364
    finally show ?thesis .
bauerg@9261
   365
  qed
bauerg@9261
   366
qed
wenzelm@7917
   367
wenzelm@10687
   368
text {*
wenzelm@10687
   369
  \medskip The limit function is bounded by the norm @{text p} as
wenzelm@10687
   370
  well, since all elements in the chain are bounded by @{text p}.
bauerg@9261
   371
*}
wenzelm@7917
   372
bauerg@9374
   373
lemma sup_norm_pres:
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   374
  assumes graph: "graph H h = \<Union>c"
wenzelm@13515
   375
    and M: "M = norm_pres_extensions E p F f"
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   376
    and cM: "c \<in> chain M"
wenzelm@13515
   377
  shows "\<forall>x \<in> H. h x \<le> p x"
bauerg@9261
   378
proof
wenzelm@9503
   379
  fix x assume "x \<in> H"
wenzelm@13515
   380
  with M cM graph obtain H' h' where x': "x \<in> H'"
wenzelm@13515
   381
      and graphs: "graph H' h' \<subseteq> graph H h"
wenzelm@10687
   382
      and a: "\<forall>x \<in> H'. h' x \<le> p x"
wenzelm@13515
   383
    by (rule some_H'h' [elim_format]) blast
wenzelm@13515
   384
  from graphs x' have [symmetric]: "h' x = h x" ..
wenzelm@13515
   385
  also from a x' have "h' x \<le> p x " ..
wenzelm@13515
   386
  finally show "h x \<le> p x" .
bauerg@9261
   387
qed
wenzelm@7917
   388
wenzelm@10687
   389
text {*
wenzelm@10687
   390
  \medskip The following lemma is a property of linear forms on real
wenzelm@31795
   391
  vector spaces. It will be used for the lemma @{text abs_Hahn_Banach}
wenzelm@31795
   392
  (see page \pageref{abs-Hahn_Banach}). \label{abs-ineq-iff} For real
wenzelm@10687
   393
  vector spaces the following inequations are equivalent:
wenzelm@10687
   394
  \begin{center}
wenzelm@10687
   395
  \begin{tabular}{lll}
wenzelm@10687
   396
  @{text "\<forall>x \<in> H. \<bar>h x\<bar> \<le> p x"} & and &
wenzelm@10687
   397
  @{text "\<forall>x \<in> H. h x \<le> p x"} \\
wenzelm@10687
   398
  \end{tabular}
wenzelm@10687
   399
  \end{center}
bauerg@9261
   400
*}
wenzelm@7917
   401
wenzelm@10687
   402
lemma abs_ineq_iff:
ballarin@27611
   403
  assumes "subspace H E" and "vectorspace E" and "seminorm E p"
ballarin@27611
   404
    and "linearform H h"
wenzelm@13515
   405
  shows "(\<forall>x \<in> H. \<bar>h x\<bar> \<le> p x) = (\<forall>x \<in> H. h x \<le> p x)" (is "?L = ?R")
wenzelm@13515
   406
proof
ballarin@29234
   407
  interpret subspace H E by fact
ballarin@29234
   408
  interpret vectorspace E by fact
ballarin@29234
   409
  interpret seminorm E p by fact
ballarin@29234
   410
  interpret linearform H h by fact
wenzelm@23378
   411
  have H: "vectorspace H" using `vectorspace E` ..
wenzelm@13515
   412
  {
bauerg@9261
   413
    assume l: ?L
bauerg@9261
   414
    show ?R
bauerg@9261
   415
    proof
wenzelm@9503
   416
      fix x assume x: "x \<in> H"
wenzelm@13515
   417
      have "h x \<le> \<bar>h x\<bar>" by arith
wenzelm@13515
   418
      also from l x have "\<dots> \<le> p x" ..
wenzelm@10687
   419
      finally show "h x \<le> p x" .
bauerg@9261
   420
    qed
bauerg@9261
   421
  next
bauerg@9261
   422
    assume r: ?R
bauerg@9261
   423
    show ?L
wenzelm@10687
   424
    proof
wenzelm@13515
   425
      fix x assume x: "x \<in> H"
wenzelm@10687
   426
      show "\<And>a b :: real. - a \<le> b \<Longrightarrow> b \<le> a \<Longrightarrow> \<bar>b\<bar> \<le> a"
bauerg@9261
   427
        by arith
wenzelm@23378
   428
      from `linearform H h` and H x
wenzelm@23378
   429
      have "- h x = h (- x)" by (rule linearform.neg [symmetric])
bauerg@14710
   430
      also
bauerg@14710
   431
      from H x have "- x \<in> H" by (rule vectorspace.neg_closed)
bauerg@14710
   432
      with r have "h (- x) \<le> p (- x)" ..
bauerg@14710
   433
      also have "\<dots> = p x"
wenzelm@32960
   434
        using `seminorm E p` `vectorspace E`
bauerg@14710
   435
      proof (rule seminorm.minus)
bauerg@14710
   436
        from x show "x \<in> E" ..
bauerg@9261
   437
      qed
bauerg@14710
   438
      finally have "- h x \<le> p x" .
bauerg@14710
   439
      then show "- p x \<le> h x" by simp
wenzelm@13515
   440
      from r x show "h x \<le> p x" ..
bauerg@9261
   441
    qed
wenzelm@13515
   442
  }
wenzelm@10687
   443
qed
wenzelm@7917
   444
wenzelm@10687
   445
end