diff -r 71af1fd6a5e4 -r be3e1cc5005c src/HOL/Hahn_Banach/Function_Order.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Hahn_Banach/Function_Order.thy	Wed Jun 24 21:46:54 2009 +0200
@@ -0,0 +1,141 @@
+(*  Title:      HOL/Hahn_Banach/Function_Order.thy
+    Author:     Gertrud Bauer, TU Munich
+*)
+
+header {* An order on functions *}
+
+theory Function_Order
+imports Subspace Linearform
+begin
+
+subsection {* The graph of a function *}
+
+text {*
+  We define the \emph{graph} of a (real) function @{text f} with
+  domain @{text F} as the set
+  \begin{center}
+  @{text "{(x, f x). x \<in> F}"}
+  \end{center}
+  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 \<times> real) set"
+
+definition
+  graph :: "'a set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a graph" where
+  "graph F f = {(x, f x) | x. x \<in> F}"
+
+lemma graphI [intro]: "x \<in> F \<Longrightarrow> (x, f x) \<in> graph F f"
+  unfolding graph_def by blast
+
+lemma graphI2 [intro?]: "x \<in> F \<Longrightarrow> \<exists>t \<in> graph F f. t = (x, f x)"
+  unfolding graph_def by blast
+
+lemma graphE [elim?]:
+    "(x, y) \<in> graph F f \<Longrightarrow> (x \<in> F \<Longrightarrow> y = f x \<Longrightarrow> C) \<Longrightarrow> C"
+  unfolding graph_def by blast
+
+
+subsection {* Functions ordered by domain extension *}
+
+text {*
+  A function @{text h'} is an extension of @{text h}, iff the graph of
+  @{text h} is a subset of the graph of @{text h'}.
+*}
+
+lemma graph_extI:
+  "(\<And>x. x \<in> H \<Longrightarrow> h x = h' x) \<Longrightarrow> H \<subseteq> H'
+    \<Longrightarrow> graph H h \<subseteq> graph H' h'"
+  unfolding graph_def by blast
+
+lemma graph_extD1 [dest?]:
+  "graph H h \<subseteq> graph H' h' \<Longrightarrow> x \<in> H \<Longrightarrow> h x = h' x"
+  unfolding graph_def by blast
+
+lemma graph_extD2 [dest?]:
+  "graph H h \<subseteq> graph H' h' \<Longrightarrow> H \<subseteq> H'"
+  unfolding graph_def by blast
+
+
+subsection {* Domain and function of a graph *}
+
+text {*
+  The inverse functions to @{text graph} are @{text domain} and @{text
+  funct}.
+*}
+
+definition
+  "domain" :: "'a graph \<Rightarrow> 'a set" where
+  "domain g = {x. \<exists>y. (x, y) \<in> g}"
+
+definition
+  funct :: "'a graph \<Rightarrow> ('a \<Rightarrow> real)" where
+  "funct g = (\<lambda>x. (SOME y. (x, y) \<in> g))"
+
+text {*
+  The following lemma states that @{text g} is the graph of a function
+  if the relation induced by @{text g} is unique.
+*}
+
+lemma graph_domain_funct:
+  assumes uniq: "\<And>x y z. (x, y) \<in> g \<Longrightarrow> (x, z) \<in> g \<Longrightarrow> z = y"
+  shows "graph (domain g) (funct g) = g"
+  unfolding domain_def funct_def graph_def
+proof auto  (* FIXME !? *)
+  fix a b assume g: "(a, b) \<in> g"
+  from g show "(a, SOME y. (a, y) \<in> g) \<in> g" by (rule someI2)
+  from g show "\<exists>y. (a, y) \<in> g" ..
+  from g show "b = (SOME y. (a, y) \<in> g)"
+  proof (rule some_equality [symmetric])
+    fix y assume "(a, y) \<in> g"
+    with g show "y = b" by (rule uniq)
+  qed
+qed
+
+
+subsection {* Norm-preserving extensions of a function *}
+
+text {*
+  Given a linear form @{text f} on the space @{text F} and a seminorm
+  @{text p} on @{text E}. The set of all linear extensions of @{text
+  f}, to superspaces @{text H} of @{text F}, which are bounded by
+  @{text p}, is defined as follows.
+*}
+
+definition
+  norm_pres_extensions ::
+    "'a::{plus, minus, uminus, zero} set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> real)
+      \<Rightarrow> 'a graph set" where
+    "norm_pres_extensions E p F f
+      = {g. \<exists>H h. g = graph H h
+          \<and> linearform H h
+          \<and> H \<unlhd> E
+          \<and> F \<unlhd> H
+          \<and> graph F f \<subseteq> graph H h
+          \<and> (\<forall>x \<in> H. h x \<le> p x)}"
+
+lemma norm_pres_extensionE [elim]:
+  "g \<in> norm_pres_extensions E p F f
+  \<Longrightarrow> (\<And>H h. g = graph H h \<Longrightarrow> linearform H h
+        \<Longrightarrow> H \<unlhd> E \<Longrightarrow> F \<unlhd> H \<Longrightarrow> graph F f \<subseteq> graph H h
+        \<Longrightarrow> \<forall>x \<in> H. h x \<le> p x \<Longrightarrow> C) \<Longrightarrow> C"
+  unfolding norm_pres_extensions_def by blast
+
+lemma norm_pres_extensionI2 [intro]:
+  "linearform H h \<Longrightarrow> H \<unlhd> E \<Longrightarrow> F \<unlhd> H
+    \<Longrightarrow> graph F f \<subseteq> graph H h \<Longrightarrow> \<forall>x \<in> H. h x \<le> p x
+    \<Longrightarrow> graph H h \<in> norm_pres_extensions E p F f"
+  unfolding norm_pres_extensions_def by blast
+
+lemma norm_pres_extensionI:  (* FIXME ? *)
+  "\<exists>H h. g = graph H h
+    \<and> linearform H h
+    \<and> H \<unlhd> E
+    \<and> F \<unlhd> H
+    \<and> graph F f \<subseteq> graph H h
+    \<and> (\<forall>x \<in> H. h x \<le> p x) \<Longrightarrow> g \<in> norm_pres_extensions E p F f"
+  unfolding norm_pres_extensions_def by blast
+
+end