(*  Title:      HOL/Hahn_Banach/Hahn_Banach_Ext_Lemmas.thy

     Author:     Gertrud Bauer, TU Munich

 *)

 header {* Extending non-maximal functions *}

 theory Hahn_Banach_Ext_Lemmas

 imports Function_Norm

 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