src/HOL/Real/HahnBanach/FunctionOrder.thy

final update by Gertrud Bauer;

(*  Title:      HOL/Real/HahnBanach/FunctionOrder.thy
    ID:         $Id$
    Author:     Gertrud Bauer, TU Munich
*)

header {* An order on functions *};

theory FunctionOrder = Subspace + Linearform:;

subsection {* The graph of a function *};

text{* We define the \emph{graph} of a (real) function $f$ with
domain $F$ as the set 
\[
\{(x, f\ap x). \ap x:F\}
\]
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";

constdefs
  graph :: "['a set, 'a => real] => 'a graph "
  "graph F f == {(x, f x) | x. x:F}"; 

lemma graphI [intro!!]: "x:F ==> (x, f x) : graph F f";
  by (unfold graph_def, intro CollectI exI) force;

lemma graphI2 [intro!!]: "x:F ==> EX t: (graph F f). t = (x, f x)";
  by (unfold graph_def, force);

lemma graphD1 [intro!!]: "(x, y): graph F f ==> x:F";
  by (unfold graph_def, elim CollectE exE) force;

lemma graphD2 [intro!!]: "(x, y): graph H h ==> y = h x";
  by (unfold graph_def, elim CollectE exE) force; 

subsection {* Functions ordered by domain extension *};

text{* A function $h'$ is an extension of $h$, iff the graph of 
$h$ is a subset of the graph of $h'$.*};

lemma graph_extI: 
  "[| !! x. x: H ==> h x = h' x; H <= H'|]
  ==> graph H h <= graph H' h'";
  by (unfold graph_def, force);

lemma graph_extD1 [intro!!]: 
  "[| graph H h <= graph H' h'; x:H |] ==> h x = h' x";
  by (unfold graph_def, force);

lemma graph_extD2 [intro!!]: 
  "[| graph H h <= graph H' h' |] ==> H <= H'";
  by (unfold graph_def, force);

subsection {* Domain and function of a graph *};

text{* The inverse functions to $\idt{graph}$ are $\idt{domain}$ and 
$\idt{funct}$.*};

constdefs
  domain :: "'a graph => 'a set" 
  "domain g == {x. EX y. (x, y):g}"

  funct :: "'a graph => ('a => real)"
  "funct g == \<lambda>x. (SOME y. (x, y):g)";

(*text{*  The equations 
\begin{matharray}
\idt{domain} graph F f = F {\rm and}\\ 
\idt{funct} graph F f = f
\end{matharray}
hold, but are not proved here.
*};*)

text {* The following lemma states that $g$ is the graph of a function
if the relation induced by $g$ is unique. *};

lemma graph_domain_funct: 
  "(!!x y z. (x, y):g ==> (x, z):g ==> z = y) 
  ==> graph (domain g) (funct g) = g";
proof (unfold domain_def funct_def graph_def, auto);
  fix a b; assume "(a, b) : g";
  show "(a, SOME y. (a, y) : g) : g"; by (rule selectI2);
  show "EX y. (a, y) : g"; ..;
  assume uniq: "!!x y z. (x, y):g ==> (x, z):g ==> z = y";
  show "b = (SOME y. (a, y) : g)";
  proof (rule select_equality [RS sym]);
    fix y; assume "(a, y):g"; show "y = b"; by (rule uniq);
  qed;
qed;



subsection {* Norm-preserving extensions of a function *};

text {* Given a linear form $f$ on the space $F$ and a seminorm $p$ on 
$E$. The set of all linear extensions of $f$, to superspaces $H$ of 
$F$, which are bounded by $p$, is defined as follows. *};


constdefs
  norm_pres_extensions :: 
    "['a::{minus, plus} set, 'a  => real, 'a set, 'a => real] 
    => 'a graph set"
    "norm_pres_extensions E p F f 
    == {g. EX H h. graph H h = g 
                & is_linearform H h 
                & is_subspace H E
                & is_subspace F H
                & graph F f <= graph H h
                & (ALL x:H. h x <= p x)}";
                       
lemma norm_pres_extension_D:  
  "g : norm_pres_extensions E p F f
  ==> EX H h. graph H h = g 
            & is_linearform H h 
            & is_subspace H E
            & is_subspace F H
            & graph F f <= graph H h
            & (ALL x:H. h x <= p x)";
  by (unfold norm_pres_extensions_def) force;

lemma norm_pres_extensionI2 [intro]:  
  "[| is_linearform H h; is_subspace H E; is_subspace F H;
  graph F f <= graph H h; ALL x:H. h x <= p x |]
  ==> (graph H h : norm_pres_extensions E p F f)";
 by (unfold norm_pres_extensions_def) force;

lemma norm_pres_extensionI [intro]:  
  "EX H h. graph H h = g 
         & is_linearform H h    
         & is_subspace H E
         & is_subspace F H
         & graph F f <= graph H h
         & (ALL x:H. h x <= p x)
  ==> g: norm_pres_extensions E p F f";
  by (unfold norm_pres_extensions_def) force;

end;