author | wenzelm |
Sat, 13 Jun 2015 20:07:54 +0200 | |
changeset 60458 | 0d10ae17e3ee |
parent 58999 | ed09ae4ea2d8 |
child 61486 | 3590367b0ce9 |
permissions | -rw-r--r-- |
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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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||
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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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|
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text \<open> |
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In this section the following context is presumed. Let @{text E} be |
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a real vector space with a seminorm @{text q} on @{text E}. @{text |
|
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F} is a subspace of @{text E} and @{text f} a linear function on |
|
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@{text F}. We consider a subspace @{text H} of @{text E} that is a |
|
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superspace of @{text F} and a linear form @{text h} on @{text |
|
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H}. @{text H} is a not equal to @{text E} and @{text "x\<^sub>0"} is |
|
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an element in @{text "E - H"}. @{text H} is extended to the direct |
|
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sum @{text "H' = H + lin x\<^sub>0"}, so for any @{text "x \<in> H'"} |
|
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the decomposition of @{text "x = y + a \<cdot> x"} with @{text "y \<in> H"} is |
|
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unique. @{text h'} is defined on @{text H'} by @{text "h' x = h y + |
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a \<cdot> \<xi>"} for a certain @{text \<xi>}. |
|
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|
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Subsequently we show some properties of this extension @{text h'} of |
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@{text h}. |
|
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|
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\medskip This lemma will be used to show the existence of a linear |
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extension of @{text f} (see page \pageref{ex-xi-use}). It is a |
|
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consequence of the completeness of @{text \<real>}. To show |
|
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\begin{center} |
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\begin{tabular}{l} |
|
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@{text "\<exists>\<xi>. \<forall>y \<in> F. a y \<le> \<xi> \<and> \<xi> \<le> b y"} |
|
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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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@{text "\<forall>u \<in> F. \<forall>v \<in> F. a u \<le> b v"} |
|
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\end{tabular} |
|
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\end{center} |
|
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\<close> |
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|
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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 @{text "S = {a u. u \<in> F}"} has a supremum, if it is |
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non-empty and has an upper bound.\<close> |
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|
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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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|
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text \<open> |
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\medskip The function @{text h'} is defined as a @{text "h' x = h y |
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+ a \<cdot> \<xi>"} where @{text "x = y + a \<cdot> \<xi>"} is a linear extension of |
|
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@{text h} to @{text H'}. |
|
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\<close> |
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|
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lemma h'_lf: |
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assumes h'_def: "h' \<equiv> \<lambda>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' \<equiv> 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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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 |
105 |
proof |
|
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note E = \<open>vectorspace E\<close> |
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have H': "vectorspace H'" |
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proof (unfold H'_def) |
|
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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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{ |
|
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fix x1 x2 assume x1: "x1 \<in> H'" and x2: "x2 \<in> H'" |
|
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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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unfolding H'_def sum_def lin_def by blast |
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124 |
|
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have ya: "y1 + y2 = y \<and> a1 + a2 = a" using E HE _ y x0 |
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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" |
128 |
by (rule subspace.add_closed) |
|
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from x0 and HE y y1 y2 |
|
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have "x0 \<in> E" "y \<in> E" "y1 \<in> E" "y2 \<in> E" by auto |
|
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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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qed |
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from h'_def x1x2 E HE y x0 |
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have "h' (x1 + x2) = h y + a * xi" |
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by (rule h'_definite) |
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also have "\<dots> = h (y1 + y2) + (a1 + a2) * xi" |
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by (simp only: ya) |
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also from y1 y2 have "h (y1 + y2) = h y1 + h y2" |
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by simp |
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also have "\<dots> + (a1 + a2) * xi = (h y1 + a1 * xi) + (h y2 + a2 * xi)" |
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Renamed {left,right}_distrib to distrib_{right,left}.
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by (simp add: distrib_right) |
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also from h'_def x1_rep E HE y1 x0 |
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have "h y1 + a1 * xi = h' x1" |
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by (rule h'_definite [symmetric]) |
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also from h'_def x2_rep E HE y2 x0 |
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have "h y2 + a2 * xi = h' x2" |
27611 | 151 |
by (rule h'_definite [symmetric]) |
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finally show ?thesis . |
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qed |
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next |
155 |
fix x1 c assume x1: "x1 \<in> H'" |
|
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show "h' (c \<cdot> x1) = c * (h' x1)" |
|
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proof - |
|
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from H' x1 have ax1: "c \<cdot> x1 \<in> H'" |
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by (rule vectorspace.mult_closed) |
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with x1 obtain y a y1 a1 where |
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cx1_rep: "c \<cdot> x1 = 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" |
27612 | 163 |
unfolding H'_def sum_def lin_def by blast |
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164 |
|
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165 |
have ya: "c \<cdot> y1 = y \<and> c * a1 = a" using E HE _ y x0 |
69916a850301
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166 |
proof (rule decomp_H') |
27611 | 167 |
from HE y1 show "c \<cdot> y1 \<in> H" |
168 |
by (rule subspace.mult_closed) |
|
169 |
from x0 and HE y y1 |
|
170 |
have "x0 \<in> E" "y \<in> E" "y1 \<in> E" by auto |
|
171 |
with x1_rep have "c \<cdot> y1 + (c * a1) \<cdot> x0 = c \<cdot> x1" |
|
172 |
by (simp add: mult_assoc add_mult_distrib1) |
|
173 |
also note cx1_rep |
|
174 |
finally show "c \<cdot> y1 + (c * a1) \<cdot> x0 = y + a \<cdot> x0" . |
|
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175 |
qed |
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176 |
|
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177 |
from h'_def cx1_rep E HE y x0 have "h' (c \<cdot> x1) = h y + a * xi" |
27611 | 178 |
by (rule h'_definite) |
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179 |
also have "\<dots> = h (c \<cdot> y1) + (c * a1) * xi" |
27611 | 180 |
by (simp only: ya) |
32960
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181 |
also from y1 have "h (c \<cdot> y1) = c * h y1" |
27611 | 182 |
by simp |
32960
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183 |
also have "\<dots> + (c * a1) * xi = c * (h y1 + a1 * xi)" |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
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184 |
by (simp only: distrib_left) |
32960
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185 |
also from h'_def x1_rep E HE y1 x0 have "h y1 + a1 * xi = h' x1" |
27611 | 186 |
by (rule h'_definite [symmetric]) |
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187 |
finally show ?thesis . |
10007 | 188 |
qed |
27611 | 189 |
} |
190 |
qed |
|
10007 | 191 |
qed |
7917 | 192 |
|
58744 | 193 |
text \<open>\medskip The linear extension @{text h'} of @{text h} |
194 |
is bounded by the seminorm @{text p}.\<close> |
|
7917 | 195 |
|
9374 | 196 |
lemma h'_norm_pres: |
13515 | 197 |
assumes h'_def: "h' \<equiv> \<lambda>x. let (y, a) = |
198 |
SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H in h y + a * xi" |
|
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199 |
and H'_def: "H' \<equiv> H + lin x0" |
13515 | 200 |
and x0: "x0 \<notin> H" "x0 \<in> E" "x0 \<noteq> 0" |
27611 | 201 |
assumes E: "vectorspace E" and HE: "subspace H E" |
202 |
and "seminorm E p" and "linearform H h" |
|
13515 | 203 |
assumes a: "\<forall>y \<in> H. h y \<le> p y" |
204 |
and a': "\<forall>y \<in> H. - p (y + x0) - h y \<le> xi \<and> xi \<le> p (y + x0) - h y" |
|
205 |
shows "\<forall>x \<in> H'. h' x \<le> p x" |
|
27611 | 206 |
proof - |
29234 | 207 |
interpret vectorspace E by fact |
208 |
interpret subspace H E by fact |
|
209 |
interpret seminorm E p by fact |
|
210 |
interpret linearform H h by fact |
|
27612 | 211 |
show ?thesis |
212 |
proof |
|
27611 | 213 |
fix x assume x': "x \<in> H'" |
214 |
show "h' x \<le> p x" |
|
215 |
proof - |
|
216 |
from a' have a1: "\<forall>ya \<in> H. - p (ya + x0) - h ya \<le> xi" |
|
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217 |
and a2: "\<forall>ya \<in> H. xi \<le> p (ya + x0) - h ya" by auto |
27611 | 218 |
from x' obtain y a where |
27612 | 219 |
x_rep: "x = y + a \<cdot> x0" and y: "y \<in> H" |
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220 |
unfolding H'_def sum_def lin_def by blast |
27611 | 221 |
from y have y': "y \<in> E" .. |
222 |
from y have ay: "inverse a \<cdot> y \<in> H" by simp |
|
223 |
||
224 |
from h'_def x_rep E HE y x0 have "h' x = h y + a * xi" |
|
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225 |
by (rule h'_definite) |
27611 | 226 |
also have "\<dots> \<le> p (y + a \<cdot> x0)" |
227 |
proof (rule linorder_cases) |
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228 |
assume z: "a = 0" |
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229 |
then have "h y + a * xi = h y" by simp |
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230 |
also from a y have "\<dots> \<le> p y" .. |
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231 |
also from x0 y' z have "p y = p (y + a \<cdot> x0)" by simp |
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232 |
finally show ?thesis . |
27611 | 233 |
next |
58744 | 234 |
txt \<open>In the case @{text "a < 0"}, we use @{text "a\<^sub>1"} |
235 |
with @{text ya} taken as @{text "y / a"}:\<close> |
|
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236 |
assume lz: "a < 0" then have nz: "a \<noteq> 0" by simp |
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237 |
from a1 ay |
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238 |
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
|
239 |
with lz have "a * xi \<le> |
13515 | 240 |
a * (- p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y))" |
27611 | 241 |
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
|
242 |
|
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31795
diff
changeset
|
243 |
also have "\<dots> = |
13515 | 244 |
- 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
|
245 |
by (simp add: right_diff_distrib) |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31795
diff
changeset
|
246 |
also from lz x0 y' have "- a * (p (inverse a \<cdot> y + x0)) = |
13515 | 247 |
p (a \<cdot> (inverse a \<cdot> y + x0))" |
27611 | 248 |
by (simp add: abs_homogenous) |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31795
diff
changeset
|
249 |
also from nz x0 y' have "\<dots> = p (y + a \<cdot> x0)" |
27611 | 250 |
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
|
251 |
also from nz y have "a * (h (inverse a \<cdot> y)) = h y" |
27611 | 252 |
by simp |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31795
diff
changeset
|
253 |
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
|
254 |
then show ?thesis by simp |
27611 | 255 |
next |
58744 | 256 |
txt \<open>In the case @{text "a > 0"}, we use @{text "a\<^sub>2"} |
257 |
with @{text ya} taken as @{text "y / a"}:\<close> |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31795
diff
changeset
|
258 |
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
|
259 |
from a2 ay |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31795
diff
changeset
|
260 |
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
|
261 |
with gz have "a * xi \<le> |
13515 | 262 |
a * (p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y))" |
27611 | 263 |
by simp |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31795
diff
changeset
|
264 |
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
|
265 |
by (simp add: right_diff_distrib) |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31795
diff
changeset
|
266 |
also from gz x0 y' |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31795
diff
changeset
|
267 |
have "a * p (inverse a \<cdot> y + x0) = p (a \<cdot> (inverse a \<cdot> y + x0))" |
27611 | 268 |
by (simp add: abs_homogenous) |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31795
diff
changeset
|
269 |
also from nz x0 y' have "\<dots> = p (y + a \<cdot> x0)" |
27611 | 270 |
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
|
271 |
also from nz y have "a * h (inverse a \<cdot> y) = h y" |
27611 | 272 |
by simp |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31795
diff
changeset
|
273 |
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
|
274 |
then show ?thesis by simp |
27611 | 275 |
qed |
276 |
also from x_rep have "\<dots> = p x" by (simp only:) |
|
277 |
finally show ?thesis . |
|
10007 | 278 |
qed |
279 |
qed |
|
13515 | 280 |
qed |
7917 | 281 |
|
10007 | 282 |
end |