src/HOL/Hahn_Banach/Hahn_Banach_Sup_Lemmas.thy
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(*  Title:      HOL/Hahn_Banach/Hahn_Banach_Sup_Lemmas.thy
Author:     Gertrud Bauer, TU Munich
*)

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

theory Hahn_Banach_Sup_Lemmas
imports Function_Norm Zorn_Lemma
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]

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"
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'"
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' 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'"
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_Hahn_Banach}
(see page \pageref{abs-Hahn_Banach}). \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