Theory Function_Order

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theory Function_Order
imports Subspace Linearform

(*  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 ∈ 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 × real) set"

definition
graph :: "'a set => ('a => real) => 'a graph" where
"graph F f = {(x, f x) | x. x ∈ F}"


lemma graphI [intro]: "x ∈ F ==> (x, f x) ∈ graph F f"
unfolding graph_def by blast

lemma graphI2 [intro?]: "x ∈ F ==> ∃t ∈ graph F f. t = (x, f x)"
unfolding graph_def by blast

lemma graphE [elim?]:
"(x, y) ∈ graph F f ==> (x ∈ F ==> y = f x ==> C) ==> 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:
"(!!x. x ∈ H ==> h x = h' x) ==> H ⊆ H'
==> graph H h ⊆ graph H' h'"

unfolding graph_def by blast

lemma graph_extD1 [dest?]:
"graph H h ⊆ graph H' h' ==> x ∈ H ==> h x = h' x"

unfolding graph_def by blast

lemma graph_extD2 [dest?]:
"graph H h ⊆ graph H' h' ==> H ⊆ 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 => 'a set" where
"domain g = {x. ∃y. (x, y) ∈ g}"


definition
funct :: "'a graph => ('a => real)" where
"funct g = (λx. (SOME y. (x, y) ∈ 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: "!!x y z. (x, y) ∈ g ==> (x, z) ∈ g ==> 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) ∈ g"
from g show "(a, SOME y. (a, y) ∈ g) ∈ g" by (rule someI2)
from g show "∃y. (a, y) ∈ g" ..
from g show "b = (SOME y. (a, y) ∈ g)"
proof (rule some_equality [symmetric])
fix y assume "(a, y) ∈ 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 => ('a => real) => 'a set => ('a => real)
=> 'a graph set"
where
"norm_pres_extensions E p F f
= {g. ∃H h. g = graph H h
∧ linearform H h
∧ H \<unlhd> E
∧ F \<unlhd> H
∧ graph F f ⊆ graph H h
∧ (∀x ∈ H. h x ≤ p x)}"


lemma norm_pres_extensionE [elim]:
"g ∈ norm_pres_extensions E p F f
==> (!!H h. g = graph H h ==> linearform H h
==> H \<unlhd> E ==> F \<unlhd> H ==> graph F f ⊆ graph H h
==> ∀x ∈ H. h x ≤ p x ==> C) ==> C"

unfolding norm_pres_extensions_def by blast

lemma norm_pres_extensionI2 [intro]:
"linearform H h ==> H \<unlhd> E ==> F \<unlhd> H
==> graph F f ⊆ graph H h ==> ∀x ∈ H. h x ≤ p x
==> graph H h ∈ norm_pres_extensions E p F f"

unfolding norm_pres_extensions_def by blast

lemma norm_pres_extensionI: (* FIXME ? *)
"∃H h. g = graph H h
∧ linearform H h
∧ H \<unlhd> E
∧ F \<unlhd> H
∧ graph F f ⊆ graph H h
∧ (∀x ∈ H. h x ≤ p x) ==> g ∈ norm_pres_extensions E p F f"

unfolding norm_pres_extensions_def by blast

end