src/HOL/Real/HahnBanach/FunctionOrder.thy
 author wenzelm Sun Jul 23 12:01:05 2000 +0200 (2000-07-23) changeset 9408 d3d56e1d2ec1 parent 9379 21cfeae6659d child 9503 3324cbbecef8 permissions -rw-r--r--
classical atts now intro! / intro / intro?;
1 (*  Title:      HOL/Real/HahnBanach/FunctionOrder.thy
2     ID:         $Id$
3     Author:     Gertrud Bauer, TU Munich
4 *)
6 header {* An order on functions *}
8 theory FunctionOrder = Subspace + Linearform:
10 subsection {* The graph of a function *}
12 text{* We define the \emph{graph} of a (real) function $f$ with
13 domain $F$ as the set
14 $15 \{(x, f\ap x). \ap x \in F\} 16$
17 So we are modeling partial functions by specifying the domain and
18 the mapping function. We use the term function'' also for its graph.
19 *}
21 types 'a graph = "('a * real) set"
23 constdefs
24   graph :: "['a set, 'a => real] => 'a graph "
25   "graph F f == {(x, f x) | x. x \\<in> F}"
27 lemma graphI [intro?]: "x \\<in> F ==> (x, f x) \\<in> graph F f"
28   by (unfold graph_def, intro CollectI exI) force
30 lemma graphI2 [intro?]: "x \\<in> F ==> \\<exists>t\\<in> (graph F f). t = (x, f x)"
31   by (unfold graph_def, force)
33 lemma graphD1 [intro?]: "(x, y) \\<in> graph F f ==> x \\<in> F"
34   by (unfold graph_def, elim CollectE exE) force
36 lemma graphD2 [intro?]: "(x, y) \\<in> graph H h ==> y = h x"
37   by (unfold graph_def, elim CollectE exE) force
39 subsection {* Functions ordered by domain extension *}
41 text{* A function $h'$ is an extension of $h$, iff the graph of
42 $h$ is a subset of the graph of $h'$.*}
44 lemma graph_extI:
45   "[| !! x. x \\<in> H ==> h x = h' x; H <= H'|]
46   ==> graph H h <= graph H' h'"
47   by (unfold graph_def, force)
49 lemma graph_extD1 [intro?]:
50   "[| graph H h <= graph H' h'; x \\<in> H |] ==> h x = h' x"
51   by (unfold graph_def, force)
53 lemma graph_extD2 [intro?]:
54   "[| graph H h <= graph H' h' |] ==> H <= H'"
55   by (unfold graph_def, force)
57 subsection {* Domain and function of a graph *}
59 text{* The inverse functions to $\idt{graph}$ are $\idt{domain}$ and
60 $\idt{funct}$.*}
62 constdefs
63   domain :: "'a graph => 'a set"
64   "domain g == {x. \\<exists>y. (x, y) \\<in> g}"
66   funct :: "'a graph => ('a => real)"
67   "funct g == \\<lambda>x. (SOME y. (x, y) \\<in> g)"
70 text {* The following lemma states that $g$ is the graph of a function
71 if the relation induced by $g$ is unique. *}
73 lemma graph_domain_funct:
74   "(!!x y z. (x, y) \\<in> g ==> (x, z) \\<in> g ==> z = y)
75   ==> graph (domain g) (funct g) = g"
76 proof (unfold domain_def funct_def graph_def, auto)
77   fix a b assume "(a, b) \\<in> g"
78   show "(a, SOME y. (a, y) \\<in> g) \\<in> g" by (rule selectI2)
79   show "\\<exists>y. (a, y) \\<in> g" ..
80   assume uniq: "!!x y z. (x, y) \\<in> g ==> (x, z) \\<in> g ==> z = y"
81   show "b = (SOME y. (a, y) \\<in> g)"
82   proof (rule select_equality [RS sym])
83     fix y assume "(a, y) \\<in> g" show "y = b" by (rule uniq)
84   qed
85 qed
89 subsection {* Norm-preserving extensions of a function *}
91 text {* Given a linear form $f$ on the space $F$ and a seminorm $p$ on
92 $E$. The set of all linear extensions of $f$, to superspaces $H$ of
93 $F$, which are bounded by $p$, is defined as follows. *}
96 constdefs
97   norm_pres_extensions ::
98     "['a::{plus, minus, zero} set, 'a  => real, 'a set, 'a => real]
99     => 'a graph set"
100     "norm_pres_extensions E p F f
101     == {g. \\<exists>H h. graph H h = g
102                 \\<and> is_linearform H h
103                 \\<and> is_subspace H E
104                 \\<and> is_subspace F H
105                 \\<and> graph F f <= graph H h
106                 \\<and> (\\<forall>x \\<in> H. h x <= p x)}"
108 lemma norm_pres_extension_D:
109   "g \\<in> norm_pres_extensions E p F f
110   ==> \\<exists>H h. graph H h = g
111             \\<and> is_linearform H h
112             \\<and> is_subspace H E
113             \\<and> is_subspace F H
114             \\<and> graph F f <= graph H h
115             \\<and> (\\<forall>x \\<in> H. h x <= p x)"
116   by (unfold norm_pres_extensions_def) force
118 lemma norm_pres_extensionI2 [intro]:
119   "[| is_linearform H h; is_subspace H E; is_subspace F H;
120   graph F f <= graph H h; \\<forall>x \\<in> H. h x <= p x |]
121   ==> (graph H h \\<in> norm_pres_extensions E p F f)"
122  by (unfold norm_pres_extensions_def) force
124 lemma norm_pres_extensionI [intro]:
125   "\\<exists>H h. graph H h = g
126          \\<and> is_linearform H h
127          \\<and> is_subspace H E
128          \\<and> is_subspace F H
129          \\<and> graph F f <= graph H h
130          \\<and> (\\<forall>x \\<in> H. h x <= p x)
131   ==> g \\<in> norm_pres_extensions E p F f"
132   by (unfold norm_pres_extensions_def) force
134 end