src/HOL/Hahn_Banach/Function_Order.thy
author hoelzl
Thu Sep 02 10:14:32 2010 +0200 (2010-09-02)
changeset 39072 1030b1a166ef
parent 31795 be3e1cc5005c
child 41818 6d4c3ee8219d
permissions -rw-r--r--
Add lessThan_Suc_eq_insert_0
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(*  Title:      HOL/Hahn_Banach/Function_Order.thy
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    Author:     Gertrud Bauer, TU Munich
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*)
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header {* An order on functions *}
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theory Function_Order
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imports Subspace Linearform
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begin
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subsection {* The graph of a function *}
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text {*
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  We define the \emph{graph} of a (real) function @{text f} with
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  domain @{text F} as the set
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  \begin{center}
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  @{text "{(x, f x). x \<in> F}"}
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  \end{center}
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  So we are modeling partial functions by specifying the domain and
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  the mapping function. We use the term ``function'' also for its
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  graph.
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*}
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types 'a graph = "('a \<times> real) set"
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definition
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  graph :: "'a set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a graph" where
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  "graph F f = {(x, f x) | x. x \<in> F}"
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lemma graphI [intro]: "x \<in> F \<Longrightarrow> (x, f x) \<in> graph F f"
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  unfolding graph_def by blast
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lemma graphI2 [intro?]: "x \<in> F \<Longrightarrow> \<exists>t \<in> graph F f. t = (x, f x)"
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  unfolding graph_def by blast
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lemma graphE [elim?]:
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    "(x, y) \<in> graph F f \<Longrightarrow> (x \<in> F \<Longrightarrow> y = f x \<Longrightarrow> C) \<Longrightarrow> C"
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  unfolding graph_def by blast
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subsection {* Functions ordered by domain extension *}
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text {*
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  A function @{text h'} is an extension of @{text h}, iff the graph of
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  @{text h} is a subset of the graph of @{text h'}.
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*}
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lemma graph_extI:
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  "(\<And>x. x \<in> H \<Longrightarrow> h x = h' x) \<Longrightarrow> H \<subseteq> H'
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    \<Longrightarrow> graph H h \<subseteq> graph H' h'"
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  unfolding graph_def by blast
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lemma graph_extD1 [dest?]:
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  "graph H h \<subseteq> graph H' h' \<Longrightarrow> x \<in> H \<Longrightarrow> h x = h' x"
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  unfolding graph_def by blast
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lemma graph_extD2 [dest?]:
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  "graph H h \<subseteq> graph H' h' \<Longrightarrow> H \<subseteq> H'"
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  unfolding graph_def by blast
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subsection {* Domain and function of a graph *}
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text {*
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  The inverse functions to @{text graph} are @{text domain} and @{text
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  funct}.
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*}
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definition
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  "domain" :: "'a graph \<Rightarrow> 'a set" where
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  "domain g = {x. \<exists>y. (x, y) \<in> g}"
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definition
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  funct :: "'a graph \<Rightarrow> ('a \<Rightarrow> real)" where
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  "funct g = (\<lambda>x. (SOME y. (x, y) \<in> g))"
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text {*
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  The following lemma states that @{text g} is the graph of a function
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  if the relation induced by @{text g} is unique.
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*}
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lemma graph_domain_funct:
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  assumes uniq: "\<And>x y z. (x, y) \<in> g \<Longrightarrow> (x, z) \<in> g \<Longrightarrow> z = y"
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  shows "graph (domain g) (funct g) = g"
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  unfolding domain_def funct_def graph_def
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proof auto  (* FIXME !? *)
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  fix a b assume g: "(a, b) \<in> g"
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  from g show "(a, SOME y. (a, y) \<in> g) \<in> g" by (rule someI2)
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  from g show "\<exists>y. (a, y) \<in> g" ..
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  from g show "b = (SOME y. (a, y) \<in> g)"
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  proof (rule some_equality [symmetric])
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    fix y assume "(a, y) \<in> g"
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    with g show "y = b" by (rule uniq)
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  qed
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qed
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subsection {* Norm-preserving extensions of a function *}
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text {*
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  Given a linear form @{text f} on the space @{text F} and a seminorm
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  @{text p} on @{text E}. The set of all linear extensions of @{text
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  f}, to superspaces @{text H} of @{text F}, which are bounded by
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  @{text p}, is defined as follows.
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*}
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definition
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  norm_pres_extensions ::
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    "'a::{plus, minus, uminus, zero} set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> real)
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      \<Rightarrow> 'a graph set" where
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    "norm_pres_extensions E p F f
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      = {g. \<exists>H h. g = graph H h
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          \<and> linearform H h
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          \<and> H \<unlhd> E
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          \<and> F \<unlhd> H
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          \<and> graph F f \<subseteq> graph H h
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          \<and> (\<forall>x \<in> H. h x \<le> p x)}"
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lemma norm_pres_extensionE [elim]:
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  "g \<in> norm_pres_extensions E p F f
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  \<Longrightarrow> (\<And>H h. g = graph H h \<Longrightarrow> linearform H h
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        \<Longrightarrow> H \<unlhd> E \<Longrightarrow> F \<unlhd> H \<Longrightarrow> graph F f \<subseteq> graph H h
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        \<Longrightarrow> \<forall>x \<in> H. h x \<le> p x \<Longrightarrow> C) \<Longrightarrow> C"
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  unfolding norm_pres_extensions_def by blast
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lemma norm_pres_extensionI2 [intro]:
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  "linearform H h \<Longrightarrow> H \<unlhd> E \<Longrightarrow> F \<unlhd> H
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    \<Longrightarrow> graph F f \<subseteq> graph H h \<Longrightarrow> \<forall>x \<in> H. h x \<le> p x
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    \<Longrightarrow> graph H h \<in> norm_pres_extensions E p F f"
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  unfolding norm_pres_extensions_def by blast
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lemma norm_pres_extensionI:  (* FIXME ? *)
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  "\<exists>H h. g = graph H h
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    \<and> linearform H h
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    \<and> H \<unlhd> E
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    \<and> F \<unlhd> H
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    \<and> graph F f \<subseteq> graph H h
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    \<and> (\<forall>x \<in> H. h x \<le> p x) \<Longrightarrow> g \<in> norm_pres_extensions E p F f"
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  unfolding norm_pres_extensions_def by blast
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end