src/HOL/Hahn_Banach/Function_Order.thy
author haftmann
Fri Nov 01 18:51:14 2013 +0100 (2013-11-01)
changeset 54230 b1d955791529
parent 44887 7ca82df6e951
child 58744 c434e37f290e
permissions -rw-r--r--
more simplification rules on unary and binary minus
     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 type_synonym 'a graph = "('a \<times> real) set"
    25 
    26 definition graph :: "'a set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a graph"
    27   where "graph F f = {(x, f x) | x. x \<in> F}"
    28 
    29 lemma graphI [intro]: "x \<in> F \<Longrightarrow> (x, f x) \<in> graph F f"
    30   unfolding graph_def by blast
    31 
    32 lemma graphI2 [intro?]: "x \<in> F \<Longrightarrow> \<exists>t \<in> graph F f. t = (x, f x)"
    33   unfolding graph_def by blast
    34 
    35 lemma graphE [elim?]:
    36   assumes "(x, y) \<in> graph F f"
    37   obtains "x \<in> F" and "y = f x"
    38   using assms 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?]: "graph H h \<subseteq> graph H' h' \<Longrightarrow> x \<in> H \<Longrightarrow> h x = h' x"
    54   unfolding graph_def by blast
    55 
    56 lemma graph_extD2 [dest?]: "graph H h \<subseteq> graph H' h' \<Longrightarrow> H \<subseteq> H'"
    57   unfolding graph_def by blast
    58 
    59 
    60 subsection {* Domain and function of a graph *}
    61 
    62 text {*
    63   The inverse functions to @{text graph} are @{text domain} and @{text
    64   funct}.
    65 *}
    66 
    67 definition domain :: "'a graph \<Rightarrow> 'a set"
    68   where "domain g = {x. \<exists>y. (x, y) \<in> g}"
    69 
    70 definition funct :: "'a graph \<Rightarrow> ('a \<Rightarrow> real)"
    71   where "funct g = (\<lambda>x. (SOME y. (x, y) \<in> g))"
    72 
    73 text {*
    74   The following lemma states that @{text g} is the graph of a function
    75   if the relation induced by @{text g} is unique.
    76 *}
    77 
    78 lemma graph_domain_funct:
    79   assumes uniq: "\<And>x y z. (x, y) \<in> g \<Longrightarrow> (x, z) \<in> g \<Longrightarrow> z = y"
    80   shows "graph (domain g) (funct g) = g"
    81   unfolding domain_def funct_def graph_def
    82 proof auto  (* FIXME !? *)
    83   fix a b assume g: "(a, b) \<in> g"
    84   from g show "(a, SOME y. (a, y) \<in> g) \<in> g" by (rule someI2)
    85   from g show "\<exists>y. (a, y) \<in> g" ..
    86   from g show "b = (SOME y. (a, y) \<in> g)"
    87   proof (rule some_equality [symmetric])
    88     fix y assume "(a, y) \<in> g"
    89     with g show "y = b" by (rule uniq)
    90   qed
    91 qed
    92 
    93 
    94 subsection {* Norm-preserving extensions of a function *}
    95 
    96 text {*
    97   Given a linear form @{text f} on the space @{text F} and a seminorm
    98   @{text p} on @{text E}. The set of all linear extensions of @{text
    99   f}, to superspaces @{text H} of @{text F}, which are bounded by
   100   @{text p}, is defined as follows.
   101 *}
   102 
   103 definition
   104   norm_pres_extensions ::
   105     "'a::{plus, minus, uminus, zero} set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> real)
   106       \<Rightarrow> 'a graph set"
   107 where
   108   "norm_pres_extensions E p F f
   109     = {g. \<exists>H h. g = graph H h
   110         \<and> linearform H h
   111         \<and> H \<unlhd> E
   112         \<and> F \<unlhd> H
   113         \<and> graph F f \<subseteq> graph H h
   114         \<and> (\<forall>x \<in> H. h x \<le> p x)}"
   115 
   116 lemma norm_pres_extensionE [elim]:
   117   assumes "g \<in> norm_pres_extensions E p F f"
   118   obtains H h
   119     where "g = graph H h"
   120     and "linearform H h"
   121     and "H \<unlhd> E"
   122     and "F \<unlhd> H"
   123     and "graph F f \<subseteq> graph H h"
   124     and "\<forall>x \<in> H. h x \<le> p x"
   125   using assms 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