src/HOL/Real/HahnBanach/FunctionOrder.thy
 author wenzelm Tue Jul 15 19:39:37 2008 +0200 (2008-07-15) changeset 27612 d3eb431db035 parent 25762 c03e9d04b3e4 permissions -rw-r--r--
modernized specifications and proofs;
tuned document;
     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

     9 imports Subspace Linearform

    10 begin

    11

    12 subsection {* The graph of a function *}

    13

    14 text {*

    15   We define the \emph{graph} of a (real) function @{text f} with

    16   domain @{text F} as the set

    17   \begin{center}

    18   @{text "{(x, f x). x \<in> F}"}

    19   \end{center}

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

    21   the mapping function. We use the term function'' also for its

    22   graph.

    23 *}

    24

    25 types 'a graph = "('a \<times> real) set"

    26

    27 definition

    28   graph :: "'a set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a graph" where

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

    30

    31 lemma graphI [intro]: "x \<in> F \<Longrightarrow> (x, f x) \<in> graph F f"

    32   unfolding graph_def by blast

    33

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

    35   unfolding graph_def by blast

    36

    37 lemma graphE [elim?]:

    38     "(x, y) \<in> graph F f \<Longrightarrow> (x \<in> F \<Longrightarrow> y = f x \<Longrightarrow> C) \<Longrightarrow> C"

    39   unfolding graph_def by blast

    40

    41

    42 subsection {* Functions ordered by domain extension *}

    43

    44 text {*

    45   A function @{text h'} is an extension of @{text h}, iff the graph of

    46   @{text h} is a subset of the graph of @{text h'}.

    47 *}

    48

    49 lemma graph_extI:

    50   "(\<And>x. x \<in> H \<Longrightarrow> h x = h' x) \<Longrightarrow> H \<subseteq> H'

    51     \<Longrightarrow> graph H h \<subseteq> graph H' h'"

    52   unfolding graph_def by blast

    53

    54 lemma graph_extD1 [dest?]:

    55   "graph H h \<subseteq> graph H' h' \<Longrightarrow> x \<in> H \<Longrightarrow> h x = h' x"

    56   unfolding graph_def by blast

    57

    58 lemma graph_extD2 [dest?]:

    59   "graph H h \<subseteq> graph H' h' \<Longrightarrow> H \<subseteq> H'"

    60   unfolding graph_def by blast

    61

    62

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

    64

    65 text {*

    66   The inverse functions to @{text graph} are @{text domain} and @{text

    67   funct}.

    68 *}

    69

    70 definition

    71   "domain" :: "'a graph \<Rightarrow> 'a set" where

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

    73

    74 definition

    75   funct :: "'a graph \<Rightarrow> ('a \<Rightarrow> real)" where

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

    77

    78 text {*

    79   The following lemma states that @{text g} is the graph of a function

    80   if the relation induced by @{text g} is unique.

    81 *}

    82

    83 lemma graph_domain_funct:

    84   assumes uniq: "\<And>x y z. (x, y) \<in> g \<Longrightarrow> (x, z) \<in> g \<Longrightarrow> z = y"

    85   shows "graph (domain g) (funct g) = g"

    86   unfolding domain_def funct_def graph_def

    87 proof auto  (* FIXME !? *)

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

    89   from g show "(a, SOME y. (a, y) \<in> g) \<in> g" by (rule someI2)

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

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

    92   proof (rule some_equality [symmetric])

    93     fix y assume "(a, y) \<in> g"

    94     with g show "y = b" by (rule uniq)

    95   qed

    96 qed

    97

    98

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

   100

   101 text {*

   102   Given a linear form @{text f} on the space @{text F} and a seminorm

   103   @{text p} on @{text E}. The set of all linear extensions of @{text

   104   f}, to superspaces @{text H} of @{text F}, which are bounded by

   105   @{text p}, is defined as follows.

   106 *}

   107

   108 definition

   109   norm_pres_extensions ::

   110     "'a::{plus, minus, uminus, zero} set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> real)

   111       \<Rightarrow> 'a graph set" where

   112     "norm_pres_extensions E p F f

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

   114           \<and> linearform H h

   115           \<and> H \<unlhd> E

   116           \<and> F \<unlhd> H

   117           \<and> graph F f \<subseteq> graph H h

   118           \<and> (\<forall>x \<in> H. h x \<le> p x)}"

   119

   120 lemma norm_pres_extensionE [elim]:

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

   122   \<Longrightarrow> (\<And>H h. g = graph H h \<Longrightarrow> linearform H h

   123         \<Longrightarrow> H \<unlhd> E \<Longrightarrow> F \<unlhd> H \<Longrightarrow> graph F f \<subseteq> graph H h

   124         \<Longrightarrow> \<forall>x \<in> H. h x \<le> p x \<Longrightarrow> C) \<Longrightarrow> C"

   125   unfolding norm_pres_extensions_def by blast

   126

   127 lemma norm_pres_extensionI2 [intro]:

   128   "linearform H h \<Longrightarrow> H \<unlhd> E \<Longrightarrow> F \<unlhd> H

   129     \<Longrightarrow> graph F f \<subseteq> graph H h \<Longrightarrow> \<forall>x \<in> H. h x \<le> p x

   130     \<Longrightarrow> graph H h \<in> norm_pres_extensions E p F f"

   131   unfolding norm_pres_extensions_def by blast

   132

   133 lemma norm_pres_extensionI:  (* FIXME ? *)

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

   135     \<and> linearform H h

   136     \<and> H \<unlhd> E

   137     \<and> F \<unlhd> H

   138     \<and> graph F f \<subseteq> graph H h

   139     \<and> (\<forall>x \<in> H. h x \<le> p x) \<Longrightarrow> g \<in> norm_pres_extensions E p F f"

   140   unfolding norm_pres_extensions_def by blast

   141

   142 end