--- a/src/HOL/Real/HahnBanach/FunctionOrder.thy Sun Jul 16 21:00:32 2000 +0200
+++ b/src/HOL/Real/HahnBanach/FunctionOrder.thy Mon Jul 17 13:58:18 2000 +0200
@@ -12,7 +12,7 @@
text{* We define the \emph{graph} of a (real) function $f$ with
domain $F$ as the set
\[
-\{(x, f\ap x). \ap x:F\}
+\{(x, f\ap x). \ap x \in F\}
\]
So we are modeling partial functions by specifying the domain and
the mapping function. We use the term ``function'' also for its graph.
@@ -22,18 +22,18 @@
constdefs
graph :: "['a set, 'a => real] => 'a graph "
- "graph F f == {(x, f x) | x. x:F}"
+ "graph F f == {(x, f x) | x. x \<in> F}"
-lemma graphI [intro??]: "x:F ==> (x, f x) : graph F f"
+lemma graphI [intro??]: "x \<in> F ==> (x, f x) \<in> graph F f"
by (unfold graph_def, intro CollectI exI) force
-lemma graphI2 [intro??]: "x:F ==> EX t: (graph F f). t = (x, f x)"
+lemma graphI2 [intro??]: "x \<in> F ==> \<exists>t\<in> (graph F f). t = (x, f x)"
by (unfold graph_def, force)
-lemma graphD1 [intro??]: "(x, y): graph F f ==> x:F"
+lemma graphD1 [intro??]: "(x, y) \<in> graph F f ==> x \<in> F"
by (unfold graph_def, elim CollectE exE) force
-lemma graphD2 [intro??]: "(x, y): graph H h ==> y = h x"
+lemma graphD2 [intro??]: "(x, y) \<in> graph H h ==> y = h x"
by (unfold graph_def, elim CollectE exE) force
subsection {* Functions ordered by domain extension *}
@@ -42,12 +42,12 @@
$h$ is a subset of the graph of $h'$.*}
lemma graph_extI:
- "[| !! x. x: H ==> h x = h' x; H <= H'|]
+ "[| !! x. x \<in> H ==> h x = h' x; H <= H'|]
==> graph H h <= graph H' h'"
by (unfold graph_def, force)
lemma graph_extD1 [intro??]:
- "[| graph H h <= graph H' h'; x:H |] ==> h x = h' x"
+ "[| graph H h <= graph H' h'; x \<in> H |] ==> h x = h' x"
by (unfold graph_def, force)
lemma graph_extD2 [intro??]:
@@ -61,10 +61,10 @@
constdefs
domain :: "'a graph => 'a set"
- "domain g == {x. EX y. (x, y):g}"
+ "domain g == {x. \<exists>y. (x, y) \<in> g}"
funct :: "'a graph => ('a => real)"
- "funct g == \<lambda>x. (SOME y. (x, y):g)"
+ "funct g == \<lambda>x. (SOME y. (x, y) \<in> g)"
(*text{* The equations
\begin{matharray}
@@ -78,16 +78,16 @@
if the relation induced by $g$ is unique. *}
lemma graph_domain_funct:
- "(!!x y z. (x, y):g ==> (x, z):g ==> z = y)
+ "(!!x y z. (x, y) \<in> g ==> (x, z) \<in> g ==> z = y)
==> graph (domain g) (funct g) = g"
proof (unfold domain_def funct_def graph_def, auto)
- fix a b assume "(a, b) : g"
- show "(a, SOME y. (a, y) : g) : g" by (rule selectI2)
- show "EX y. (a, y) : g" ..
- assume uniq: "!!x y z. (x, y):g ==> (x, z):g ==> z = y"
- show "b = (SOME y. (a, y) : g)"
+ fix a b assume "(a, b) \<in> g"
+ show "(a, SOME y. (a, y) \<in> g) \<in> g" by (rule selectI2)
+ show "\<exists>y. (a, y) \<in> g" ..
+ assume uniq: "!!x y z. (x, y) \<in> g ==> (x, z) \<in> g ==> z = y"
+ show "b = (SOME y. (a, y) \<in> g)"
proof (rule select_equality [RS sym])
- fix y assume "(a, y):g" show "y = b" by (rule uniq)
+ fix y assume "(a, y) \<in> g" show "y = b" by (rule uniq)
qed
qed
@@ -102,40 +102,40 @@
constdefs
norm_pres_extensions ::
- "['a::{minus, plus} set, 'a => real, 'a set, 'a => real]
+ "['a::{plus, minus, zero} set, 'a => real, 'a set, 'a => real]
=> 'a graph set"
"norm_pres_extensions E p F f
- == {g. EX H h. graph H h = g
- & is_linearform H h
- & is_subspace H E
- & is_subspace F H
- & graph F f <= graph H h
- & (ALL x:H. h x <= p x)}"
+ == {g. \<exists>H h. graph H h = g
+ \<and> is_linearform H h
+ \<and> is_subspace H E
+ \<and> is_subspace F H
+ \<and> graph F f <= graph H h
+ \<and> (\<forall>x \<in> H. h x <= p x)}"
lemma norm_pres_extension_D:
- "g : norm_pres_extensions E p F f
- ==> EX H h. graph H h = g
- & is_linearform H h
- & is_subspace H E
- & is_subspace F H
- & graph F f <= graph H h
- & (ALL x:H. h x <= p x)"
+ "g \<in> norm_pres_extensions E p F f
+ ==> \<exists>H h. graph H h = g
+ \<and> is_linearform H h
+ \<and> is_subspace H E
+ \<and> is_subspace F H
+ \<and> graph F f <= graph H h
+ \<and> (\<forall>x \<in> H. h x <= p x)"
by (unfold norm_pres_extensions_def) force
lemma norm_pres_extensionI2 [intro]:
"[| is_linearform H h; is_subspace H E; is_subspace F H;
- graph F f <= graph H h; ALL x:H. h x <= p x |]
- ==> (graph H h : norm_pres_extensions E p F f)"
+ graph F f <= graph H h; \<forall>x \<in> H. h x <= p x |]
+ ==> (graph H h \<in> norm_pres_extensions E p F f)"
by (unfold norm_pres_extensions_def) force
lemma norm_pres_extensionI [intro]:
- "EX H h. graph H h = g
- & is_linearform H h
- & is_subspace H E
- & is_subspace F H
- & graph F f <= graph H h
- & (ALL x:H. h x <= p x)
- ==> g: norm_pres_extensions E p F f"
+ "\<exists>H h. graph H h = g
+ \<and> is_linearform H h
+ \<and> is_subspace H E
+ \<and> is_subspace F H
+ \<and> graph F f <= graph H h
+ \<and> (\<forall>x \<in> H. h x <= p x)
+ ==> g\<in> norm_pres_extensions E p F f"
by (unfold norm_pres_extensions_def) force
end
\ No newline at end of file