src/HOL/Real/HahnBanach/FunctionOrder.thy
 changeset 10687 c186279eecea parent 9969 4753185f1dd2 child 11472 d08d4e17a5f6
     1.1 --- a/src/HOL/Real/HahnBanach/FunctionOrder.thy	Sat Dec 16 21:41:14 2000 +0100
1.2 +++ b/src/HOL/Real/HahnBanach/FunctionOrder.thy	Sat Dec 16 21:41:51 2000 +0100
1.3 @@ -9,75 +9,82 @@
1.4
1.5  subsection {* The graph of a function *}
1.6
1.7 -text{* We define the \emph{graph} of a (real) function $f$ with
1.8 -domain $F$ as the set
1.9 -$1.10 -\{(x, f\ap x). \ap x \in F\} 1.11 -$
1.12 -So we are modeling partial functions by specifying the domain and
1.13 -the mapping function. We use the term function'' also for its graph.
1.14 +text {*
1.15 +  We define the \emph{graph} of a (real) function @{text f} with
1.16 +  domain @{text F} as the set
1.17 +  \begin{center}
1.18 +  @{text "{(x, f x). x \<in> F}"}
1.19 +  \end{center}
1.20 +  So we are modeling partial functions by specifying the domain and
1.21 +  the mapping function. We use the term function'' also for its
1.22 +  graph.
1.23  *}
1.24
1.25  types 'a graph = "('a * real) set"
1.26
1.27  constdefs
1.28 -  graph :: "['a set, 'a => real] => 'a graph "
1.29 -  "graph F f == {(x, f x) | x. x \<in> F}"
1.30 +  graph :: "'a set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a graph "
1.31 +  "graph F f \<equiv> {(x, f x) | x. x \<in> F}"
1.32
1.33 -lemma graphI [intro?]: "x \<in> F ==> (x, f x) \<in> graph F f"
1.34 -  by (unfold graph_def, intro CollectI exI) force
1.35 +lemma graphI [intro?]: "x \<in> F \<Longrightarrow> (x, f x) \<in> graph F f"
1.36 +  by (unfold graph_def, intro CollectI exI) blast
1.37
1.38 -lemma graphI2 [intro?]: "x \<in> F ==> \<exists>t\<in> (graph F f). t = (x, f x)"
1.39 -  by (unfold graph_def, force)
1.40 +lemma graphI2 [intro?]: "x \<in> F \<Longrightarrow> \<exists>t\<in> (graph F f). t = (x, f x)"
1.41 +  by (unfold graph_def) blast
1.42
1.43 -lemma graphD1 [intro?]: "(x, y) \<in> graph F f ==> x \<in> F"
1.44 -  by (unfold graph_def, elim CollectE exE) force
1.45 +lemma graphD1 [intro?]: "(x, y) \<in> graph F f \<Longrightarrow> x \<in> F"
1.46 +  by (unfold graph_def) blast
1.47
1.48 -lemma graphD2 [intro?]: "(x, y) \<in> graph H h ==> y = h x"
1.49 -  by (unfold graph_def, elim CollectE exE) force
1.50 +lemma graphD2 [intro?]: "(x, y) \<in> graph H h \<Longrightarrow> y = h x"
1.51 +  by (unfold graph_def) blast
1.52 +
1.53
1.54  subsection {* Functions ordered by domain extension *}
1.55
1.56 -text{* A function $h'$ is an extension of $h$, iff the graph of
1.57 -$h$ is a subset of the graph of $h'$.*}
1.58 +text {* A function @{text h'} is an extension of @{text h}, iff the
1.59 +  graph of @{text h} is a subset of the graph of @{text h'}. *}
1.60
1.61 -lemma graph_extI:
1.62 -  "[| !! x. x \<in> H ==> h x = h' x; H <= H'|]
1.63 -  ==> graph H h <= graph H' h'"
1.64 -  by (unfold graph_def, force)
1.65 +lemma graph_extI:
1.66 +  "(\<And>x. x \<in> H \<Longrightarrow> h x = h' x) \<Longrightarrow> H \<subseteq> H'
1.67 +  \<Longrightarrow> graph H h \<subseteq> graph H' h'"
1.68 +  by (unfold graph_def) blast
1.69
1.70 -lemma graph_extD1 [intro?]:
1.71 -  "[| graph H h <= graph H' h'; x \<in> H |] ==> h x = h' x"
1.72 -  by (unfold graph_def, force)
1.73 +lemma graph_extD1 [intro?]:
1.74 +  "graph H h \<subseteq> graph H' h' \<Longrightarrow> x \<in> H \<Longrightarrow> h x = h' x"
1.75 +  by (unfold graph_def) blast
1.76
1.77 -lemma graph_extD2 [intro?]:
1.78 -  "[| graph H h <= graph H' h' |] ==> H <= H'"
1.79 -  by (unfold graph_def, force)
1.80 +lemma graph_extD2 [intro?]:
1.81 +  "graph H h \<subseteq> graph H' h' \<Longrightarrow> H \<subseteq> H'"
1.82 +  by (unfold graph_def) blast
1.83
1.84  subsection {* Domain and function of a graph *}
1.85
1.86 -text{* The inverse functions to $\idt{graph}$ are $\idt{domain}$ and
1.87 -$\idt{funct}$.*}
1.88 +text {*
1.89 +  The inverse functions to @{text graph} are @{text domain} and
1.90 +  @{text funct}.
1.91 +*}
1.92
1.93  constdefs
1.94 -  domain :: "'a graph => 'a set"
1.95 -  "domain g == {x. \<exists>y. (x, y) \<in> g}"
1.96 +  domain :: "'a graph \<Rightarrow> 'a set"
1.97 +  "domain g \<equiv> {x. \<exists>y. (x, y) \<in> g}"
1.98
1.99 -  funct :: "'a graph => ('a => real)"
1.100 -  "funct g == \<lambda>x. (SOME y. (x, y) \<in> g)"
1.101 +  funct :: "'a graph \<Rightarrow> ('a \<Rightarrow> real)"
1.102 +  "funct g \<equiv> \<lambda>x. (SOME y. (x, y) \<in> g)"
1.103
1.104
1.105 -text {* The following lemma states that $g$ is the graph of a function
1.106 -if the relation induced by $g$ is unique. *}
1.107 +text {*
1.108 +  The following lemma states that @{text g} is the graph of a function
1.109 +  if the relation induced by @{text g} is unique.
1.110 +*}
1.111
1.112 -lemma graph_domain_funct:
1.113 -  "(!!x y z. (x, y) \<in> g ==> (x, z) \<in> g ==> z = y)
1.114 -  ==> graph (domain g) (funct g) = g"
1.115 +lemma graph_domain_funct:
1.116 +  "(\<And>x y z. (x, y) \<in> g \<Longrightarrow> (x, z) \<in> g \<Longrightarrow> z = y)
1.117 +  \<Longrightarrow> graph (domain g) (funct g) = g"
1.118  proof (unfold domain_def funct_def graph_def, auto)
1.119    fix a b assume "(a, b) \<in> g"
1.120    show "(a, SOME y. (a, y) \<in> g) \<in> g" by (rule someI2)
1.121    show "\<exists>y. (a, y) \<in> g" ..
1.122 -  assume uniq: "!!x y z. (x, y) \<in> g ==> (x, z) \<in> g ==> z = y"
1.123 +  assume uniq: "\<And>x y z. (x, y) \<in> g \<Longrightarrow> (x, z) \<in> g \<Longrightarrow> z = y"
1.124    show "b = (SOME y. (a, y) \<in> g)"
1.125    proof (rule some_equality [symmetric])
1.126      fix y assume "(a, y) \<in> g" show "y = b" by (rule uniq)
1.127 @@ -88,47 +95,49 @@
1.128
1.129  subsection {* Norm-preserving extensions of a function *}
1.130
1.131 -text {* Given a linear form $f$ on the space $F$ and a seminorm $p$ on
1.132 -$E$. The set of all linear extensions of $f$, to superspaces $H$ of
1.133 -$F$, which are bounded by $p$, is defined as follows. *}
1.134 -
1.135 +text {*
1.136 +  Given a linear form @{text f} on the space @{text F} and a seminorm
1.137 +  @{text p} on @{text E}. The set of all linear extensions of @{text
1.138 +  f}, to superspaces @{text H} of @{text F}, which are bounded by
1.139 +  @{text p}, is defined as follows.
1.140 +*}
1.141
1.142  constdefs
1.143 -  norm_pres_extensions ::
1.144 -    "['a::{plus, minus, zero} set, 'a  => real, 'a set, 'a => real]
1.145 -    => 'a graph set"
1.146 -    "norm_pres_extensions E p F f
1.147 -    == {g. \<exists>H h. graph H h = g
1.148 -                \<and> is_linearform H h
1.149 +  norm_pres_extensions ::
1.150 +    "'a::{plus, minus, zero} set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> real)
1.151 +    \<Rightarrow> 'a graph set"
1.152 +    "norm_pres_extensions E p F f
1.153 +    \<equiv> {g. \<exists>H h. graph H h = g
1.154 +                \<and> is_linearform H h
1.155                  \<and> is_subspace H E
1.156                  \<and> is_subspace F H
1.157 -                \<and> graph F f <= graph H h
1.158 -                \<and> (\<forall>x \<in> H. h x <= p x)}"
1.159 -
1.160 -lemma norm_pres_extension_D:
1.161 +                \<and> graph F f \<subseteq> graph H h
1.162 +                \<and> (\<forall>x \<in> H. h x \<le> p x)}"
1.163 +
1.164 +lemma norm_pres_extension_D:
1.165    "g \<in> norm_pres_extensions E p F f
1.166 -  ==> \<exists>H h. graph H h = g
1.167 -            \<and> is_linearform H h
1.168 +  \<Longrightarrow> \<exists>H h. graph H h = g
1.169 +            \<and> is_linearform H h
1.170              \<and> is_subspace H E
1.171              \<and> is_subspace F H
1.172 -            \<and> graph F f <= graph H h
1.173 -            \<and> (\<forall>x \<in> H. h x <= p x)"
1.174 -  by (unfold norm_pres_extensions_def) force
1.175 +            \<and> graph F f \<subseteq> graph H h
1.176 +            \<and> (\<forall>x \<in> H. h x \<le> p x)"
1.177 +  by (unfold norm_pres_extensions_def) blast
1.178
1.179 -lemma norm_pres_extensionI2 [intro]:
1.180 -  "[| is_linearform H h; is_subspace H E; is_subspace F H;
1.181 -  graph F f <= graph H h; \<forall>x \<in> H. h x <= p x |]
1.182 -  ==> (graph H h \<in> norm_pres_extensions E p F f)"
1.183 - by (unfold norm_pres_extensions_def) force
1.184 +lemma norm_pres_extensionI2 [intro]:
1.185 +  "is_linearform H h \<Longrightarrow> is_subspace H E \<Longrightarrow> is_subspace F H \<Longrightarrow>
1.186 +  graph F f \<subseteq> graph H h \<Longrightarrow> \<forall>x \<in> H. h x \<le> p x
1.187 +  \<Longrightarrow> (graph H h \<in> norm_pres_extensions E p F f)"
1.188 + by (unfold norm_pres_extensions_def) blast
1.189
1.190 -lemma norm_pres_extensionI [intro]:
1.191 -  "\<exists>H h. graph H h = g
1.192 -         \<and> is_linearform H h
1.193 +lemma norm_pres_extensionI [intro]:
1.194 +  "\<exists>H h. graph H h = g
1.195 +         \<and> is_linearform H h
1.196           \<and> is_subspace H E
1.197           \<and> is_subspace F H
1.198 -         \<and> graph F f <= graph H h
1.199 -         \<and> (\<forall>x \<in> H. h x <= p x)
1.200 -  ==> g \<in> norm_pres_extensions E p F f"
1.201 -  by (unfold norm_pres_extensions_def) force
1.202 +         \<and> graph F f \<subseteq> graph H h
1.203 +         \<and> (\<forall>x \<in> H. h x \<le> p x)
1.204 +  \<Longrightarrow> g \<in> norm_pres_extensions E p F f"
1.205 +  by (unfold norm_pres_extensions_def) blast
1.206
1.207 -end
1.208 \ No newline at end of file
1.209 +end