(* Title: HOL/Hahn_Banach/Function_Order.thy
Author: Gertrud Bauer, TU Munich
*)
header {* An order on functions *}
theory Function_Order
imports Subspace Linearform
begin
subsection {* The graph of a function *}
text {*
We define the \emph{graph} of a (real) function @{text f} with
domain @{text F} as the set
\begin{center}
@{text "{(x, f x). x \<in> F}"}
\end{center}
So we are modeling partial functions by specifying the domain and
the mapping function. We use the term ``function'' also for its
graph.
*}
types 'a graph = "('a \<times> real) set"
definition
graph :: "'a set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a graph" where
"graph F f = {(x, f x) | x. x \<in> F}"
lemma graphI [intro]: "x \<in> F \<Longrightarrow> (x, f x) \<in> graph F f"
unfolding graph_def by blast
lemma graphI2 [intro?]: "x \<in> F \<Longrightarrow> \<exists>t \<in> graph F f. t = (x, f x)"
unfolding graph_def by blast
lemma graphE [elim?]:
"(x, y) \<in> graph F f \<Longrightarrow> (x \<in> F \<Longrightarrow> y = f x \<Longrightarrow> C) \<Longrightarrow> C"
unfolding graph_def by blast
subsection {* Functions ordered by domain extension *}
text {*
A function @{text h'} is an extension of @{text h}, iff the graph of
@{text h} is a subset of the graph of @{text h'}.
*}
lemma graph_extI:
"(\<And>x. x \<in> H \<Longrightarrow> h x = h' x) \<Longrightarrow> H \<subseteq> H'
\<Longrightarrow> graph H h \<subseteq> graph H' h'"
unfolding graph_def by blast
lemma graph_extD1 [dest?]:
"graph H h \<subseteq> graph H' h' \<Longrightarrow> x \<in> H \<Longrightarrow> h x = h' x"
unfolding graph_def by blast
lemma graph_extD2 [dest?]:
"graph H h \<subseteq> graph H' h' \<Longrightarrow> H \<subseteq> H'"
unfolding graph_def by blast
subsection {* Domain and function of a graph *}
text {*
The inverse functions to @{text graph} are @{text domain} and @{text
funct}.
*}
definition
"domain" :: "'a graph \<Rightarrow> 'a set" where
"domain g = {x. \<exists>y. (x, y) \<in> g}"
definition
funct :: "'a graph \<Rightarrow> ('a \<Rightarrow> real)" where
"funct g = (\<lambda>x. (SOME y. (x, y) \<in> g))"
text {*
The following lemma states that @{text g} is the graph of a function
if the relation induced by @{text g} is unique.
*}
lemma graph_domain_funct:
assumes uniq: "\<And>x y z. (x, y) \<in> g \<Longrightarrow> (x, z) \<in> g \<Longrightarrow> z = y"
shows "graph (domain g) (funct g) = g"
unfolding domain_def funct_def graph_def
proof auto (* FIXME !? *)
fix a b assume g: "(a, b) \<in> g"
from g show "(a, SOME y. (a, y) \<in> g) \<in> g" by (rule someI2)
from g show "\<exists>y. (a, y) \<in> g" ..
from g show "b = (SOME y. (a, y) \<in> g)"
proof (rule some_equality [symmetric])
fix y assume "(a, y) \<in> g"
with g show "y = b" by (rule uniq)
qed
qed
subsection {* Norm-preserving extensions of a function *}
text {*
Given a linear form @{text f} on the space @{text F} and a seminorm
@{text p} on @{text E}. The set of all linear extensions of @{text
f}, to superspaces @{text H} of @{text F}, which are bounded by
@{text p}, is defined as follows.
*}
definition
norm_pres_extensions ::
"'a::{plus, minus, uminus, zero} set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> real)
\<Rightarrow> 'a graph set" where
"norm_pres_extensions E p F f
= {g. \<exists>H h. g = 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)}"
lemma norm_pres_extensionE [elim]:
"g \<in> norm_pres_extensions E p F f
\<Longrightarrow> (\<And>H h. g = graph H h \<Longrightarrow> linearform H h
\<Longrightarrow> H \<unlhd> E \<Longrightarrow> F \<unlhd> H \<Longrightarrow> graph F f \<subseteq> graph H h
\<Longrightarrow> \<forall>x \<in> H. h x \<le> p x \<Longrightarrow> C) \<Longrightarrow> C"
unfolding norm_pres_extensions_def by blast
lemma norm_pres_extensionI2 [intro]:
"linearform H h \<Longrightarrow> H \<unlhd> E \<Longrightarrow> F \<unlhd> H
\<Longrightarrow> graph F f \<subseteq> graph H h \<Longrightarrow> \<forall>x \<in> H. h x \<le> p x
\<Longrightarrow> graph H h \<in> norm_pres_extensions E p F f"
unfolding norm_pres_extensions_def by blast
lemma norm_pres_extensionI: (* FIXME ? *)
"\<exists>H h. g = 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) \<Longrightarrow> g \<in> norm_pres_extensions E p F f"
unfolding norm_pres_extensions_def by blast
end