--- a/src/HOL/Real/HahnBanach/HahnBanachExtLemmas.thy Tue Dec 30 08:18:54 2008 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,280 +0,0 @@
-(* Title: HOL/Real/HahnBanach/HahnBanachExtLemmas.thy
- Author: Gertrud Bauer, TU Munich
-*)
-
-header {* Extending non-maximal functions *}
-
-theory HahnBanachExtLemmas
-imports FunctionNorm
-begin
-
-text {*
- In this section the following context is presumed. Let @{text E} be
- a real vector space with a seminorm @{text q} on @{text E}. @{text
- F} is a subspace of @{text E} and @{text f} a linear function on
- @{text F}. We consider a subspace @{text H} of @{text E} that is a
- superspace of @{text F} and a linear form @{text h} on @{text
- H}. @{text H} is a not equal to @{text E} and @{text "x\<^sub>0"} is
- an element in @{text "E - H"}. @{text H} is extended to the direct
- sum @{text "H' = H + lin x\<^sub>0"}, so for any @{text "x \<in> H'"}
- the decomposition of @{text "x = y + a \<cdot> x"} with @{text "y \<in> H"} is
- unique. @{text h'} is defined on @{text H'} by @{text "h' x = h y +
- a \<cdot> \<xi>"} for a certain @{text \<xi>}.
-
- Subsequently we show some properties of this extension @{text h'} of
- @{text h}.
-
- \medskip This lemma will be used to show the existence of a linear
- extension of @{text f} (see page \pageref{ex-xi-use}). It is a
- consequence of the completeness of @{text \<real>}. To show
- \begin{center}
- \begin{tabular}{l}
- @{text "\<exists>\<xi>. \<forall>y \<in> F. a y \<le> \<xi> \<and> \<xi> \<le> b y"}
- \end{tabular}
- \end{center}
- \noindent it suffices to show that
- \begin{center}
- \begin{tabular}{l}
- @{text "\<forall>u \<in> F. \<forall>v \<in> F. a u \<le> b v"}
- \end{tabular}
- \end{center}
-*}
-
-lemma ex_xi:
- assumes "vectorspace F"
- assumes r: "\<And>u v. u \<in> F \<Longrightarrow> v \<in> F \<Longrightarrow> a u \<le> b v"
- shows "\<exists>xi::real. \<forall>y \<in> F. a y \<le> xi \<and> xi \<le> b y"
-proof -
- interpret vectorspace F by fact
- txt {* From the completeness of the reals follows:
- The set @{text "S = {a u. u \<in> F}"} has a supremum, if it is
- non-empty and has an upper bound. *}
-
- let ?S = "{a u | u. u \<in> F}"
- have "\<exists>xi. lub ?S xi"
- proof (rule real_complete)
- have "a 0 \<in> ?S" by blast
- then show "\<exists>X. X \<in> ?S" ..
- have "\<forall>y \<in> ?S. y \<le> b 0"
- proof
- fix y assume y: "y \<in> ?S"
- then obtain u where u: "u \<in> F" and y: "y = a u" by blast
- from u and zero have "a u \<le> b 0" by (rule r)
- with y show "y \<le> b 0" by (simp only:)
- qed
- then show "\<exists>u. \<forall>y \<in> ?S. y \<le> u" ..
- qed
- then obtain xi where xi: "lub ?S xi" ..
- {
- fix y assume "y \<in> F"
- then have "a y \<in> ?S" by blast
- with xi have "a y \<le> xi" by (rule lub.upper)
- } moreover {
- fix y assume y: "y \<in> F"
- from xi have "xi \<le> b y"
- proof (rule lub.least)
- fix au assume "au \<in> ?S"
- then obtain u where u: "u \<in> F" and au: "au = a u" by blast
- from u y have "a u \<le> b y" by (rule r)
- with au show "au \<le> b y" by (simp only:)
- qed
- } ultimately show "\<exists>xi. \<forall>y \<in> F. a y \<le> xi \<and> xi \<le> b y" by blast
-qed
-
-text {*
- \medskip The function @{text h'} is defined as a @{text "h' x = h y
- + a \<cdot> \<xi>"} where @{text "x = y + a \<cdot> \<xi>"} is a linear extension of
- @{text h} to @{text H'}.
-*}
-
-lemma h'_lf:
- assumes h'_def: "h' \<equiv> \<lambda>x. let (y, a) =
- SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H in h y + a * xi"
- and H'_def: "H' \<equiv> H + lin x0"
- and HE: "H \<unlhd> E"
- assumes "linearform H h"
- assumes x0: "x0 \<notin> H" "x0 \<in> E" "x0 \<noteq> 0"
- assumes E: "vectorspace E"
- shows "linearform H' h'"
-proof -
- interpret linearform H h by fact
- interpret vectorspace E by fact
- show ?thesis
- proof
- note E = `vectorspace E`
- have H': "vectorspace H'"
- proof (unfold H'_def)
- from `x0 \<in> E`
- have "lin x0 \<unlhd> E" ..
- with HE show "vectorspace (H + lin x0)" using E ..
- qed
- {
- fix x1 x2 assume x1: "x1 \<in> H'" and x2: "x2 \<in> H'"
- show "h' (x1 + x2) = h' x1 + h' x2"
- proof -
- from H' x1 x2 have "x1 + x2 \<in> H'"
- by (rule vectorspace.add_closed)
- with x1 x2 obtain y y1 y2 a a1 a2 where
- x1x2: "x1 + x2 = y + a \<cdot> x0" and y: "y \<in> H"
- and x1_rep: "x1 = y1 + a1 \<cdot> x0" and y1: "y1 \<in> H"
- and x2_rep: "x2 = y2 + a2 \<cdot> x0" and y2: "y2 \<in> H"
- unfolding H'_def sum_def lin_def by blast
-
- have ya: "y1 + y2 = y \<and> a1 + a2 = a" using E HE _ y x0
- proof (rule decomp_H') txt_raw {* \label{decomp-H-use} *}
- from HE y1 y2 show "y1 + y2 \<in> H"
- by (rule subspace.add_closed)
- from x0 and HE y y1 y2
- have "x0 \<in> E" "y \<in> E" "y1 \<in> E" "y2 \<in> E" by auto
- with x1_rep x2_rep have "(y1 + y2) + (a1 + a2) \<cdot> x0 = x1 + x2"
- by (simp add: add_ac add_mult_distrib2)
- also note x1x2
- finally show "(y1 + y2) + (a1 + a2) \<cdot> x0 = y + a \<cdot> x0" .
- qed
-
- from h'_def x1x2 E HE y x0
- have "h' (x1 + x2) = h y + a * xi"
- by (rule h'_definite)
- also have "\<dots> = h (y1 + y2) + (a1 + a2) * xi"
- by (simp only: ya)
- also from y1 y2 have "h (y1 + y2) = h y1 + h y2"
- by simp
- also have "\<dots> + (a1 + a2) * xi = (h y1 + a1 * xi) + (h y2 + a2 * xi)"
- by (simp add: left_distrib)
- also from h'_def x1_rep E HE y1 x0
- have "h y1 + a1 * xi = h' x1"
- by (rule h'_definite [symmetric])
- also from h'_def x2_rep E HE y2 x0
- have "h y2 + a2 * xi = h' x2"
- by (rule h'_definite [symmetric])
- finally show ?thesis .
- qed
- next
- fix x1 c assume x1: "x1 \<in> H'"
- show "h' (c \<cdot> x1) = c * (h' x1)"
- proof -
- from H' x1 have ax1: "c \<cdot> x1 \<in> H'"
- by (rule vectorspace.mult_closed)
- with x1 obtain y a y1 a1 where
- cx1_rep: "c \<cdot> x1 = y + a \<cdot> x0" and y: "y \<in> H"
- and x1_rep: "x1 = y1 + a1 \<cdot> x0" and y1: "y1 \<in> H"
- unfolding H'_def sum_def lin_def by blast
-
- have ya: "c \<cdot> y1 = y \<and> c * a1 = a" using E HE _ y x0
- proof (rule decomp_H')
- from HE y1 show "c \<cdot> y1 \<in> H"
- by (rule subspace.mult_closed)
- from x0 and HE y y1
- have "x0 \<in> E" "y \<in> E" "y1 \<in> E" by auto
- with x1_rep have "c \<cdot> y1 + (c * a1) \<cdot> x0 = c \<cdot> x1"
- by (simp add: mult_assoc add_mult_distrib1)
- also note cx1_rep
- finally show "c \<cdot> y1 + (c * a1) \<cdot> x0 = y + a \<cdot> x0" .
- qed
-
- from h'_def cx1_rep E HE y x0 have "h' (c \<cdot> x1) = h y + a * xi"
- by (rule h'_definite)
- also have "\<dots> = h (c \<cdot> y1) + (c * a1) * xi"
- by (simp only: ya)
- also from y1 have "h (c \<cdot> y1) = c * h y1"
- by simp
- also have "\<dots> + (c * a1) * xi = c * (h y1 + a1 * xi)"
- by (simp only: right_distrib)
- also from h'_def x1_rep E HE y1 x0 have "h y1 + a1 * xi = h' x1"
- by (rule h'_definite [symmetric])
- finally show ?thesis .
- qed
- }
- qed
-qed
-
-text {* \medskip The linear extension @{text h'} of @{text h}
- is bounded by the seminorm @{text p}. *}
-
-lemma h'_norm_pres:
- assumes h'_def: "h' \<equiv> \<lambda>x. let (y, a) =
- SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H in h y + a * xi"
- and H'_def: "H' \<equiv> H + lin x0"
- and x0: "x0 \<notin> H" "x0 \<in> E" "x0 \<noteq> 0"
- assumes E: "vectorspace E" and HE: "subspace H E"
- and "seminorm E p" and "linearform H h"
- assumes a: "\<forall>y \<in> H. h y \<le> p y"
- and a': "\<forall>y \<in> H. - p (y + x0) - h y \<le> xi \<and> xi \<le> p (y + x0) - h y"
- shows "\<forall>x \<in> H'. h' x \<le> p x"
-proof -
- interpret vectorspace E by fact
- interpret subspace H E by fact
- interpret seminorm E p by fact
- interpret linearform H h by fact
- show ?thesis
- proof
- fix x assume x': "x \<in> H'"
- show "h' x \<le> p x"
- proof -
- from a' have a1: "\<forall>ya \<in> H. - p (ya + x0) - h ya \<le> xi"
- and a2: "\<forall>ya \<in> H. xi \<le> p (ya + x0) - h ya" by auto
- from x' obtain y a where
- x_rep: "x = y + a \<cdot> x0" and y: "y \<in> H"
- unfolding H'_def sum_def lin_def by blast
- from y have y': "y \<in> E" ..
- from y have ay: "inverse a \<cdot> y \<in> H" by simp
-
- from h'_def x_rep E HE y x0 have "h' x = h y + a * xi"
- by (rule h'_definite)
- also have "\<dots> \<le> p (y + a \<cdot> x0)"
- proof (rule linorder_cases)
- assume z: "a = 0"
- then have "h y + a * xi = h y" by simp
- also from a y have "\<dots> \<le> p y" ..
- also from x0 y' z have "p y = p (y + a \<cdot> x0)" by simp
- finally show ?thesis .
- next
- txt {* In the case @{text "a < 0"}, we use @{text "a\<^sub>1"}
- with @{text ya} taken as @{text "y / a"}: *}
- assume lz: "a < 0" then have nz: "a \<noteq> 0" by simp
- from a1 ay
- have "- p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y) \<le> xi" ..
- with lz have "a * xi \<le>
- a * (- p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y))"
- by (simp add: mult_left_mono_neg order_less_imp_le)
-
- also have "\<dots> =
- - a * (p (inverse a \<cdot> y + x0)) - a * (h (inverse a \<cdot> y))"
- by (simp add: right_diff_distrib)
- also from lz x0 y' have "- a * (p (inverse a \<cdot> y + x0)) =
- p (a \<cdot> (inverse a \<cdot> y + x0))"
- by (simp add: abs_homogenous)
- also from nz x0 y' have "\<dots> = p (y + a \<cdot> x0)"
- by (simp add: add_mult_distrib1 mult_assoc [symmetric])
- also from nz y have "a * (h (inverse a \<cdot> y)) = h y"
- by simp
- finally have "a * xi \<le> p (y + a \<cdot> x0) - h y" .
- then show ?thesis by simp
- next
- txt {* In the case @{text "a > 0"}, we use @{text "a\<^sub>2"}
- with @{text ya} taken as @{text "y / a"}: *}
- assume gz: "0 < a" then have nz: "a \<noteq> 0" by simp
- from a2 ay
- have "xi \<le> p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y)" ..
- with gz have "a * xi \<le>
- a * (p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y))"
- by simp
- also have "\<dots> = a * p (inverse a \<cdot> y + x0) - a * h (inverse a \<cdot> y)"
- by (simp add: right_diff_distrib)
- also from gz x0 y'
- have "a * p (inverse a \<cdot> y + x0) = p (a \<cdot> (inverse a \<cdot> y + x0))"
- by (simp add: abs_homogenous)
- also from nz x0 y' have "\<dots> = p (y + a \<cdot> x0)"
- by (simp add: add_mult_distrib1 mult_assoc [symmetric])
- also from nz y have "a * h (inverse a \<cdot> y) = h y"
- by simp
- finally have "a * xi \<le> p (y + a \<cdot> x0) - h y" .
- then show ?thesis by simp
- qed
- also from x_rep have "\<dots> = p x" by (simp only:)
- finally show ?thesis .
- qed
- qed
-qed
-
-end