(*  Title:      HOL/Real/HahnBanach/HahnBanachSupLemmas.thy

     ID:         $Id$

     Author:     Gertrud Bauer, TU Munich

 *)

 header {* The supremum w.r.t.~the function order *}

 theory HahnBanachSupLemmas

 imports FunctionNorm ZornLemma

 begin

 text {*

   This section contains some lemmas that will be used in the proof of

   the Hahn-Banach Theorem.  In this section the following context is

   presumed.  Let @{text E} be a real vector space with a seminorm

   @{text p} on @{text E}.  @{text F} is a subspace of @{text E} and

   @{text f} a linear form on @{text F}. We consider a chain @{text c}

   of norm-preserving extensions of @{text f}, such that @{text "\<Union>c =

   graph H h"}.  We will show some properties about the limit function

   @{text h}, i.e.\ the supremum of the chain @{text c}.

   \medskip Let @{text c} be a chain of norm-preserving extensions of

   the function @{text f} and let @{text "graph H h"} be the supremum

   of @{text c}.  Every element in @{text H} is member of one of the

   elements of the chain.

 *}

 lemmas [dest?] = chainD

 lemmas chainE2 [elim?] = chainD2 [elim_format, standard]

 lemma some_H'h't:

   assumes M: "M = norm_pres_extensions E p F f"

     and cM: "c \<in> chain M"

     and u: "graph H h = \<Union>c"

     and x: "x \<in> H"

   shows "\<exists>H' h'. graph H' h' \<in> c

     \<and> (x, h x) \<in> graph H' h'

     \<and> linearform H' h' \<and> H' \<unlhd> E

     \<and> F \<unlhd> H' \<and> graph F f \<subseteq> graph H' h'

     \<and> (\<forall>x \<in> H'. h' x \<le> p x)"

 proof -

   from x have "(x, h x) \<in> graph H h" ..

   also from u have "\<dots> = \<Union>c" .

   finally obtain g where gc: "g \<in> c" and gh: "(x, h x) \<in> g" by blast

   from cM have "c \<subseteq> M" ..

   with gc have "g \<in> M" ..

   also from M have "\<dots> = norm_pres_extensions E p F f" .

   finally obtain H' and h' where g: "g = graph H' h'"

     and * : "linearform H' h'"  "H' \<unlhd> E"  "F \<unlhd> H'"

       "graph F f \<subseteq> graph H' h'"  "\<forall>x \<in> H'. h' x \<le> p x" ..

   from gc and g have "graph H' h' \<in> c" by (simp only:)

   moreover from gh and g have "(x, h x) \<in> graph H' h'" by (simp only:)

   ultimately show ?thesis using * by blast

 qed

 text {*

   \medskip Let @{text c} be a chain of norm-preserving extensions of

   the function @{text f} and let @{text "graph H h"} be the supremum

   of @{text c}.  Every element in the domain @{text H} of the supremum

   function is member of the domain @{text H'} of some function @{text

   h'}, such that @{text h} extends @{text h'}.

 *}

 lemma some_H'h':

   assumes M: "M = norm_pres_extensions E p F f"

     and cM: "c \<in> chain M"

     and u: "graph H h = \<Union>c"

     and x: "x \<in> H"

   shows "\<exists>H' h'. x \<in> H' \<and> graph H' h' \<subseteq> graph H h

     \<and> linearform H' h' \<and> H' \<unlhd> E \<and> F \<unlhd> H'

     \<and> graph F f \<subseteq> graph H' h' \<and> (\<forall>x \<in> H'. h' x \<le> p x)"

 proof -

   from M cM u x obtain H' h' where

       x_hx: "(x, h x) \<in> graph H' h'"

     and c: "graph H' h' \<in> c"

     and * : "linearform H' h'"  "H' \<unlhd> E"  "F \<unlhd> H'"

       "graph F f \<subseteq> graph H' h'"  "\<forall>x \<in> H'. h' x \<le> p x"

     by (rule some_H'h't [elim_format]) blast

   from x_hx have "x \<in> H'" ..

   moreover from cM u c have "graph H' h' \<subseteq> graph H h"

     by (simp only: chain_ball_Union_upper)

   ultimately show ?thesis using * by blast

 qed

 text {*

   \medskip Any two elements @{text x} and @{text y} in the domain

   @{text H} of the supremum function @{text h} are both in the domain

   @{text H'} of some function @{text h'}, such that @{text h} extends

   @{text h'}.

 *}

 lemma some_H'h'2:

   assumes M: "M = norm_pres_extensions E p F f"

     and cM: "c \<in> chain M"

     and u: "graph H h = \<Union>c"

     and x: "x \<in> H"

     and y: "y \<in> H"

   shows "\<exists>H' h'. x \<in> H' \<and> y \<in> H'

     \<and> graph H' h' \<subseteq> graph H h

     \<and> linearform H' h' \<and> H' \<unlhd> E \<and> F \<unlhd> H'

     \<and> graph F f \<subseteq> graph H' h' \<and> (\<forall>x \<in> H'. h' x \<le> p x)"

 proof -

   txt {* @{text y} is in the domain @{text H''} of some function @{text h''},

   such that @{text h} extends @{text h''}. *}

   from M cM u and y obtain H' h' where

       y_hy: "(y, h y) \<in> graph H' h'"

     and c': "graph H' h' \<in> c"

     and * :

       "linearform H' h'"  "H' \<unlhd> E"  "F \<unlhd> H'"

       "graph F f \<subseteq> graph H' h'"  "\<forall>x \<in> H'. h' x \<le> p x"

     by (rule some_H'h't [elim_format]) blast

   txt {* @{text x} is in the domain @{text H'} of some function @{text h'},

     such that @{text h} extends @{text h'}. *}

   from M cM u and x obtain H'' h'' where

       x_hx: "(x, h x) \<in> graph H'' h''"

     and c'': "graph H'' h'' \<in> c"

     and ** :

       "linearform H'' h''"  "H'' \<unlhd> E"  "F \<unlhd> H''"

       "graph F f \<subseteq> graph H'' h''"  "\<forall>x \<in> H''. h'' x \<le> p x"

     by (rule some_H'h't [elim_format]) blast

   txt {* Since both @{text h'} and @{text h''} are elements of the chain,

     @{text h''} is an extension of @{text h'} or vice versa. Thus both

     @{text x} and @{text y} are contained in the greater

     one. \label{cases1}*}

   from cM c'' c' have "graph H'' h'' \<subseteq> graph H' h' \<or> graph H' h' \<subseteq> graph H'' h''"

     (is "?case1 \<or> ?case2") ..

   then show ?thesis

   proof

     assume ?case1

     have "(x, h x) \<in> graph H'' h''" by fact

     also have "\<dots> \<subseteq> graph H' h'" by fact

     finally have xh:"(x, h x) \<in> graph H' h'" .

     then have "x \<in> H'" ..

     moreover from y_hy have "y \<in> H'" ..

     moreover from cM u and c' have "graph H' h' \<subseteq> graph H h"

       by (simp only: chain_ball_Union_upper)

     ultimately show ?thesis using * by blast

   next

     assume ?case2

     from x_hx have "x \<in> H''" ..

     moreover {

       have "(y, h y) \<in> graph H' h'" by (rule y_hy)

       also have "\<dots> \<subseteq> graph H'' h''" by fact

       finally have "(y, h y) \<in> graph H'' h''" .

     } then have "y \<in> H''" ..

     moreover from cM u and c'' have "graph H'' h'' \<subseteq> graph H h"

       by (simp only: chain_ball_Union_upper)

     ultimately show ?thesis using ** by blast

   qed

 qed

 text {*

   \medskip The relation induced by the graph of the supremum of a

   chain @{text c} is definite, i.~e.~t is the graph of a function.

 *}

 lemma sup_definite:

   assumes M_def: "M \<equiv> norm_pres_extensions E p F f"

     and cM: "c \<in> chain M"

     and xy: "(x, y) \<in> \<Union>c"

     and xz: "(x, z) \<in> \<Union>c"

   shows "z = y"

 proof -

   from cM have c: "c \<subseteq> M" ..

   from xy obtain G1 where xy': "(x, y) \<in> G1" and G1: "G1 \<in> c" ..

   from xz obtain G2 where xz': "(x, z) \<in> G2" and G2: "G2 \<in> c" ..

   from G1 c have "G1 \<in> M" ..

   then obtain H1 h1 where G1_rep: "G1 = graph H1 h1"

     unfolding M_def by blast

   from G2 c have "G2 \<in> M" ..

   then obtain H2 h2 where G2_rep: "G2 = graph H2 h2"

     unfolding M_def by blast

   txt {* @{text "G\<^sub>1"} is contained in @{text "G\<^sub>2"}

     or vice versa, since both @{text "G\<^sub>1"} and @{text

     "G\<^sub>2"} are members of @{text c}. \label{cases2}*}

   from cM G1 G2 have "G1 \<subseteq> G2 \<or> G2 \<subseteq> G1" (is "?case1 \<or> ?case2") ..

   then show ?thesis

   proof

     assume ?case1

     with xy' G2_rep have "(x, y) \<in> graph H2 h2" by blast

     then have "y = h2 x" ..

     also

     from xz' G2_rep have "(x, z) \<in> graph H2 h2" by (simp only:)

     then have "z = h2 x" ..

     finally show ?thesis .

   next

     assume ?case2

     with xz' G1_rep have "(x, z) \<in> graph H1 h1" by blast

     then have "z = h1 x" ..

     also

     from xy' G1_rep have "(x, y) \<in> graph H1 h1" by (simp only:)

     then have "y = h1 x" ..

     finally show ?thesis ..

   qed

 qed

 text {*

   \medskip The limit function @{text h} is linear. Every element

   @{text x} in the domain of @{text h} is in the domain of a function

   @{text h'} in the chain of norm preserving extensions.  Furthermore,

   @{text h} is an extension of @{text h'} so the function values of

   @{text x} are identical for @{text h'} and @{text h}.  Finally, the

   function @{text h'} is linear by construction of @{text M}.

 *}

 lemma sup_lf:

   assumes M: "M = norm_pres_extensions E p F f"

     and cM: "c \<in> chain M"

     and u: "graph H h = \<Union>c"

   shows "linearform H h"

 proof

   fix x y assume x: "x \<in> H" and y: "y \<in> H"

   with M cM u obtain H' h' where

         x': "x \<in> H'" and y': "y \<in> H'"

       and b: "graph H' h' \<subseteq> graph H h"

       and linearform: "linearform H' h'"

       and subspace: "H' \<unlhd> E"

     by (rule some_H'h'2 [elim_format]) blast

   show "h (x + y) = h x + h y"

   proof -

     from linearform x' y' have "h' (x + y) = h' x + h' y"

       by (rule linearform.add)

     also from b x' have "h' x = h x" ..

     also from b y' have "h' y = h y" ..

     also from subspace x' y' have "x + y \<in> H'"

       by (rule subspace.add_closed)

     with b have "h' (x + y) = h (x + y)" ..

     finally show ?thesis .

   qed

 next

   fix x a assume x: "x \<in> H"

   with M cM u obtain H' h' where

         x': "x \<in> H'"

       and b: "graph H' h' \<subseteq> graph H h"

       and linearform: "linearform H' h'"

       and subspace: "H' \<unlhd> E"

     by (rule some_H'h' [elim_format]) blast

   show "h (a \<cdot> x) = a * h x"

   proof -

     from linearform x' have "h' (a \<cdot> x) = a * h' x"

       by (rule linearform.mult)

     also from b x' have "h' x = h x" ..

     also from subspace x' have "a \<cdot> x \<in> H'"

       by (rule subspace.mult_closed)

     with b have "h' (a \<cdot> x) = h (a \<cdot> x)" ..

     finally show ?thesis .

   qed

 qed

 text {*

   \medskip The limit of a non-empty chain of norm preserving

   extensions of @{text f} is an extension of @{text f}, since every

   element of the chain is an extension of @{text f} and the supremum

   is an extension for every element of the chain.

 *}

 lemma sup_ext:

   assumes graph: "graph H h = \<Union>c"

     and M: "M = norm_pres_extensions E p F f"

     and cM: "c \<in> chain M"

     and ex: "\<exists>x. x \<in> c"

   shows "graph F f \<subseteq> graph H h"

 proof -

   from ex obtain x where xc: "x \<in> c" ..

   from cM have "c \<subseteq> M" ..

   with xc have "x \<in> M" ..

   with M have "x \<in> norm_pres_extensions E p F f"

     by (simp only:)

   then obtain G g where "x = graph G g" and "graph F f \<subseteq> graph G g" ..

   then have "graph F f \<subseteq> x" by (simp only:)

   also from xc have "\<dots> \<subseteq> \<Union>c" by blast

   also from graph have "\<dots> = graph H h" ..

   finally show ?thesis .

 qed

 text {*

   \medskip The domain @{text H} of the limit function is a superspace

   of @{text F}, since @{text F} is a subset of @{text H}. The

   existence of the @{text 0} element in @{text F} and the closure

   properties follow from the fact that @{text F} is a vector space.

 *}

 lemma sup_supF:

   assumes graph: "graph H h = \<Union>c"

     and M: "M = norm_pres_extensions E p F f"

     and cM: "c \<in> chain M"

     and ex: "\<exists>x. x \<in> c"

     and FE: "F \<unlhd> E"

   shows "F \<unlhd> H"

 proof

   from FE show "F \<noteq> {}" by (rule subspace.non_empty)

   from graph M cM ex have "graph F f \<subseteq> graph H h" by (rule sup_ext)

   then show "F \<subseteq> H" ..

   fix x y assume "x \<in> F" and "y \<in> F"

   with FE show "x + y \<in> F" by (rule subspace.add_closed)

 next

   fix x a assume "x \<in> F"

   with FE show "a \<cdot> x \<in> F" by (rule subspace.mult_closed)

 qed

 text {*

   \medskip The domain @{text H} of the limit function is a subspace of

   @{text E}.

 *}

 lemma sup_subE:

   assumes graph: "graph H h = \<Union>c"

     and M: "M = norm_pres_extensions E p F f"

     and cM: "c \<in> chain M"

     and ex: "\<exists>x. x \<in> c"

     and FE: "F \<unlhd> E"

     and E: "vectorspace E"

   shows "H \<unlhd> E"

 proof

   show "H \<noteq> {}"

   proof -

     from FE E have "0 \<in> F" by (rule subspace.zero)

     also from graph M cM ex FE have "F \<unlhd> H" by (rule sup_supF)

     then have "F \<subseteq> H" ..

     finally show ?thesis by blast

   qed

   show "H \<subseteq> E"

   proof

     fix x assume "x \<in> H"

     with M cM graph

     obtain H' h' where x: "x \<in> H'" and H'E: "H' \<unlhd> E"

       by (rule some_H'h' [elim_format]) blast

     from H'E have "H' \<subseteq> E" ..

     with x show "x \<in> E" ..

   qed

   fix x y assume x: "x \<in> H" and y: "y \<in> H"

   show "x + y \<in> H"

   proof -

     from M cM graph x y obtain H' h' where

           x': "x \<in> H'" and y': "y \<in> H'" and H'E: "H' \<unlhd> E"

         and graphs: "graph H' h' \<subseteq> graph H h"

       by (rule some_H'h'2 [elim_format]) blast

     from H'E x' y' have "x + y \<in> H'"

       by (rule subspace.add_closed)

     also from graphs have "H' \<subseteq> H" ..

     finally show ?thesis .

   qed

 next

   fix x a assume x: "x \<in> H"

   show "a \<cdot> x \<in> H"

   proof -

     from M cM graph x

     obtain H' h' where x': "x \<in> H'" and H'E: "H' \<unlhd> E"

         and graphs: "graph H' h' \<subseteq> graph H h"

       by (rule some_H'h' [elim_format]) blast

     from H'E x' have "a \<cdot> x \<in> H'" by (rule subspace.mult_closed)

     also from graphs have "H' \<subseteq> H" ..

     finally show ?thesis .

   qed

 qed

 text {*

   \medskip The limit function is bounded by the norm @{text p} as

   well, since all elements in the chain are bounded by @{text p}.

 *}

 lemma sup_norm_pres:

   assumes graph: "graph H h = \<Union>c"

     and M: "M = norm_pres_extensions E p F f"

     and cM: "c \<in> chain M"

   shows "\<forall>x \<in> H. h x \<le> p x"

 proof

   fix x assume "x \<in> H"

   with M cM graph obtain H' h' where x': "x \<in> H'"

       and graphs: "graph H' h' \<subseteq> graph H h"

       and a: "\<forall>x \<in> H'. h' x \<le> p x"

     by (rule some_H'h' [elim_format]) blast

   from graphs x' have [symmetric]: "h' x = h x" ..

   also from a x' have "h' x \<le> p x " ..

   finally show "h x \<le> p x" .

 qed

 text {*

   \medskip The following lemma is a property of linear forms on real

   vector spaces. It will be used for the lemma @{text abs_HahnBanach}

   (see page \pageref{abs-HahnBanach}). \label{abs-ineq-iff} For real

   vector spaces the following inequations are equivalent:

   \begin{center}

   \begin{tabular}{lll}

   @{text "\<forall>x \<in> H. \<bar>h x\<bar> \<le> p x"} & and &

   @{text "\<forall>x \<in> H. h x \<le> p x"} \\

   \end{tabular}

   \end{center}

 *}

 lemma abs_ineq_iff:

   assumes "subspace H E" and "vectorspace E" and "seminorm E p"

     and "linearform H h"

   shows "(\<forall>x \<in> H. \<bar>h x\<bar> \<le> p x) = (\<forall>x \<in> H. h x \<le> p x)" (is "?L = ?R")

 proof

   interpret subspace H E by fact

   interpret vectorspace E by fact

   interpret seminorm E p by fact

   interpret linearform H h by fact

   have H: "vectorspace H" using `vectorspace E` ..

   {

     assume l: ?L

     show ?R

     proof

       fix x assume x: "x \<in> H"

       have "h x \<le> \<bar>h x\<bar>" by arith

       also from l x have "\<dots> \<le> p x" ..

       finally show "h x \<le> p x" .

     qed

   next

     assume r: ?R

     show ?L

     proof

       fix x assume x: "x \<in> H"

       show "\<And>a b :: real. - a \<le> b \<Longrightarrow> b \<le> a \<Longrightarrow> \<bar>b\<bar> \<le> a"

         by arith

       from `linearform H h` and H x

       have "- h x = h (- x)" by (rule linearform.neg [symmetric])

       also

       from H x have "- x \<in> H" by (rule vectorspace.neg_closed)

       with r have "h (- x) \<le> p (- x)" ..

       also have "\<dots> = p x"

 	using `seminorm E p` `vectorspace E`

       proof (rule seminorm.minus)

         from x show "x \<in> E" ..

       qed

       finally have "- h x \<le> p x" .

       then show "- p x \<le> h x" by simp

       from r x show "h x \<le> p x" ..

     qed

   }

 qed

 end