src/HOL/Hahn_Banach/Function_Order.thy
 author hoelzl Tue Jan 18 21:37:23 2011 +0100 (2011-01-18) changeset 41654 32fe42892983 parent 31795 be3e1cc5005c child 41818 6d4c3ee8219d permissions -rw-r--r--
Gauge measure removed
     1 (*  Title:      HOL/Hahn_Banach/Function_Order.thy

     2     Author:     Gertrud Bauer, TU Munich

     3 *)

     4

     5 header {* An order on functions *}

     6

     7 theory Function_Order

     8 imports Subspace Linearform

     9 begin

    10

    11 subsection {* The graph of a function *}

    12

    13 text {*

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

    15   domain @{text F} as the set

    16   \begin{center}

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

    18   \end{center}

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

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

    21   graph.

    22 *}

    23

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

    25

    26 definition

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

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

    29

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

    31   unfolding graph_def by blast

    32

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

    34   unfolding graph_def by blast

    35

    36 lemma graphE [elim?]:

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

    38   unfolding graph_def by blast

    39

    40

    41 subsection {* Functions ordered by domain extension *}

    42

    43 text {*

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

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

    46 *}

    47

    48 lemma graph_extI:

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

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

    51   unfolding graph_def by blast

    52

    53 lemma graph_extD1 [dest?]:

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

    55   unfolding graph_def by blast

    56

    57 lemma graph_extD2 [dest?]:

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

    59   unfolding graph_def by blast

    60

    61

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

    63

    64 text {*

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

    66   funct}.

    67 *}

    68

    69 definition

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

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

    72

    73 definition

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

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

    76

    77 text {*

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

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

    80 *}

    81

    82 lemma graph_domain_funct:

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

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

    85   unfolding domain_def funct_def graph_def

    86 proof auto  (* FIXME !? *)

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

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

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

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

    91   proof (rule some_equality [symmetric])

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

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

    94   qed

    95 qed

    96

    97

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

    99

   100 text {*

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

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

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

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

   105 *}

   106

   107 definition

   108   norm_pres_extensions ::

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

   110       \<Rightarrow> 'a graph set" where

   111     "norm_pres_extensions E p F f

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

   113           \<and> linearform H h

   114           \<and> H \<unlhd> E

   115           \<and> F \<unlhd> H

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

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

   118

   119 lemma norm_pres_extensionE [elim]:

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

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

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

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

   124   unfolding norm_pres_extensions_def by blast

   125

   126 lemma norm_pres_extensionI2 [intro]:

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

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

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

   130   unfolding norm_pres_extensions_def by blast

   131

   132 lemma norm_pres_extensionI:  (* FIXME ? *)

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

   134     \<and> linearform H h

   135     \<and> H \<unlhd> E

   136     \<and> F \<unlhd> H

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

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

   139   unfolding norm_pres_extensions_def by blast

   140

   141 end