src/HOL/Real/HahnBanach/FunctionOrder.thy
 author wenzelm Sun Jul 23 12:01:05 2000 +0200 (2000-07-23) changeset 9408 d3d56e1d2ec1 parent 9379 21cfeae6659d child 9503 3324cbbecef8 permissions -rw-r--r--
classical atts now intro! / intro / intro?;
     1 (*  Title:      HOL/Real/HahnBanach/FunctionOrder.thy

     2     ID:         $Id$

     3     Author:     Gertrud Bauer, TU Munich

     4 *)

     5

     6 header {* An order on functions *}

     7

     8 theory FunctionOrder = Subspace + Linearform:

     9

    10 subsection {* The graph of a function *}

    11

    12 text{* We define the \emph{graph} of a (real) function $f$ with

    13 domain $F$ as the set

    14 $  15 \{(x, f\ap x). \ap x \in F\}   16$

    17 So we are modeling partial functions by specifying the domain and

    18 the mapping function. We use the term function'' also for its graph.

    19 *}

    20

    21 types 'a graph = "('a * real) set"

    22

    23 constdefs

    24   graph :: "['a set, 'a => real] => 'a graph "

    25   "graph F f == {(x, f x) | x. x \\<in> F}"

    26

    27 lemma graphI [intro?]: "x \\<in> F ==> (x, f x) \\<in> graph F f"

    28   by (unfold graph_def, intro CollectI exI) force

    29

    30 lemma graphI2 [intro?]: "x \\<in> F ==> \\<exists>t\\<in> (graph F f). t = (x, f x)"

    31   by (unfold graph_def, force)

    32

    33 lemma graphD1 [intro?]: "(x, y) \\<in> graph F f ==> x \\<in> F"

    34   by (unfold graph_def, elim CollectE exE) force

    35

    36 lemma graphD2 [intro?]: "(x, y) \\<in> graph H h ==> y = h x"

    37   by (unfold graph_def, elim CollectE exE) force

    38

    39 subsection {* Functions ordered by domain extension *}

    40

    41 text{* A function $h'$ is an extension of $h$, iff the graph of

    42 $h$ is a subset of the graph of $h'$.*}

    43

    44 lemma graph_extI:

    45   "[| !! x. x \\<in> H ==> h x = h' x; H <= H'|]

    46   ==> graph H h <= graph H' h'"

    47   by (unfold graph_def, force)

    48

    49 lemma graph_extD1 [intro?]:

    50   "[| graph H h <= graph H' h'; x \\<in> H |] ==> h x = h' x"

    51   by (unfold graph_def, force)

    52

    53 lemma graph_extD2 [intro?]:

    54   "[| graph H h <= graph H' h' |] ==> H <= H'"

    55   by (unfold graph_def, force)

    56

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

    58

    59 text{* The inverse functions to $\idt{graph}$ are $\idt{domain}$ and

    60 $\idt{funct}$.*}

    61

    62 constdefs

    63   domain :: "'a graph => 'a set"

    64   "domain g == {x. \\<exists>y. (x, y) \\<in> g}"

    65

    66   funct :: "'a graph => ('a => real)"

    67   "funct g == \\<lambda>x. (SOME y. (x, y) \\<in> g)"

    68

    69

    70 text {* The following lemma states that $g$ is the graph of a function

    71 if the relation induced by $g$ is unique. *}

    72

    73 lemma graph_domain_funct:

    74   "(!!x y z. (x, y) \\<in> g ==> (x, z) \\<in> g ==> z = y)

    75   ==> graph (domain g) (funct g) = g"

    76 proof (unfold domain_def funct_def graph_def, auto)

    77   fix a b assume "(a, b) \\<in> g"

    78   show "(a, SOME y. (a, y) \\<in> g) \\<in> g" by (rule selectI2)

    79   show "\\<exists>y. (a, y) \\<in> g" ..

    80   assume uniq: "!!x y z. (x, y) \\<in> g ==> (x, z) \\<in> g ==> z = y"

    81   show "b = (SOME y. (a, y) \\<in> g)"

    82   proof (rule select_equality [RS sym])

    83     fix y assume "(a, y) \\<in> g" show "y = b" by (rule uniq)

    84   qed

    85 qed

    86

    87

    88

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

    90

    91 text {* Given a linear form $f$ on the space $F$ and a seminorm $p$ on

    92 $E$. The set of all linear extensions of $f$, to superspaces $H$ of

    93 $F$, which are bounded by $p$, is defined as follows. *}

    94

    95

    96 constdefs

    97   norm_pres_extensions ::

    98     "['a::{plus, minus, zero} set, 'a  => real, 'a set, 'a => real]

    99     => 'a graph set"

   100     "norm_pres_extensions E p F f

   101     == {g. \\<exists>H h. graph H h = g

   102                 \\<and> is_linearform H h

   103                 \\<and> is_subspace H E

   104                 \\<and> is_subspace F H

   105                 \\<and> graph F f <= graph H h

   106                 \\<and> (\\<forall>x \\<in> H. h x <= p x)}"

   107

   108 lemma norm_pres_extension_D:

   109   "g \\<in> norm_pres_extensions E p F f

   110   ==> \\<exists>H h. graph H h = g

   111             \\<and> is_linearform H h

   112             \\<and> is_subspace H E

   113             \\<and> is_subspace F H

   114             \\<and> graph F f <= graph H h

   115             \\<and> (\\<forall>x \\<in> H. h x <= p x)"

   116   by (unfold norm_pres_extensions_def) force

   117

   118 lemma norm_pres_extensionI2 [intro]:

   119   "[| is_linearform H h; is_subspace H E; is_subspace F H;

   120   graph F f <= graph H h; \\<forall>x \\<in> H. h x <= p x |]

   121   ==> (graph H h \\<in> norm_pres_extensions E p F f)"

   122  by (unfold norm_pres_extensions_def) force

   123

   124 lemma norm_pres_extensionI [intro]:

   125   "\\<exists>H h. graph H h = g

   126          \\<and> is_linearform H h

   127          \\<and> is_subspace H E

   128          \\<and> is_subspace F H

   129          \\<and> graph F f <= graph H h

   130          \\<and> (\\<forall>x \\<in> H. h x <= p x)

   131   ==> g \\<in> norm_pres_extensions E p F f"

   132   by (unfold norm_pres_extensions_def) force

   133

   134 end