(*  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