diff -r 8f84a608883d -r ea97aa6aeba2 src/HOL/HahnBanach/HahnBanachExtLemmas.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/HahnBanach/HahnBanachExtLemmas.thy	Tue Dec 30 11:10:01 2008 +0100
@@ -0,0 +1,280 @@
+(*  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