author wenzelm Fri, 12 Jan 2001 11:06:50 +0100 changeset 10875 1715cb147294 parent 10874 ad7113530c32 child 10876 e12892e4666a
 src/HOL/Induct/Sigma_Algebra.thy file | annotate | diff | comparison | revisions src/HOL/IsaMakefile file | annotate | diff | comparison | revisions
```--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Induct/Sigma_Algebra.thy	Fri Jan 12 11:06:50 2001 +0100
@@ -0,0 +1,50 @@
+(*  Title:      HOL/Induct/Sigma_Algebra.thy
+    ID:         \$Id\$
+    Author:     Markus Wenzel, TU Muenchen
+*)
+
+theory Sigma_Algebra = Main:
+
+text {*
+  This is just a tiny example demonstrating the use of inductive
+  definitions in classical mathematics.  We define the least @{text
+  \<sigma>}-algebra over a given set of sets.
+*}
+
+consts
+  sigma_algebra :: "'a set set => 'a set set"    ("\<sigma>'_algebra")
+
+inductive "\<sigma>_algebra A"
+  intros
+    basic: "a \<in> A ==> a \<in> \<sigma>_algebra A"
+    UNIV: "UNIV \<in> \<sigma>_algebra A"
+    complement: "a \<in> \<sigma>_algebra A ==> -a \<in> \<sigma>_algebra A"
+    Union: "(!!i::nat. a i \<in> \<sigma>_algebra A) ==> (\<Union>i. a i) \<in> \<sigma>_algebra A"
+
+text {*
+  The following basic facts are consequences of the closure properties
+  of any @{text \<sigma>}-algebra, merely using the introduction rules, but
+  no induction nor cases.
+*}
+
+theorem sigma_algebra_empty: "{} \<in> \<sigma>_algebra A"
+proof -
+  have "UNIV \<in> \<sigma>_algebra A" by (rule sigma_algebra.UNIV)
+  hence "-UNIV \<in> \<sigma>_algebra A" by (rule sigma_algebra.complement)
+  also have "-UNIV = {}" by simp
+  finally show ?thesis .
+qed
+
+theorem sigma_algebra_Inter:
+  "(!!i::nat. a i \<in> \<sigma>_algebra A) ==> (\<Inter>i. a i) \<in> \<sigma>_algebra A"
+proof -
+  assume "!!i::nat. a i \<in> \<sigma>_algebra A"
+  hence "!!i::nat. -(a i) \<in> \<sigma>_algebra A" by (rule sigma_algebra.complement)
+  hence "(\<Union>i. -(a i)) \<in> \<sigma>_algebra A" by (rule sigma_algebra.Union)
+  hence "-(\<Union>i. -(a i)) \<in> \<sigma>_algebra A" by (rule sigma_algebra.complement)
+  also have "-(\<Union>i. -(a i)) = (\<Inter>i. a i)" by simp
+  finally show ?thesis .
+qed
+
+end```
```--- a/src/HOL/IsaMakefile	Fri Jan 12 09:32:53 2001 +0100
+++ b/src/HOL/IsaMakefile	Fri Jan 12 11:06:50 2001 +0100
@@ -205,8 +205,8 @@
Induct/LList.ML Induct/LList.thy Induct/Mutil.ML Induct/Mutil.thy \
Induct/Perm.ML Induct/Perm.thy Induct/PropLog.ML \
Induct/PropLog.thy Induct/ROOT.ML Induct/Sexp.ML Induct/Sexp.thy \
-  Induct/SList.ML Induct/SList.thy Induct/ABexp.ML Induct/ABexp.thy \
-  Induct/Term.ML Induct/Term.thy
+  Induct/Sigma_Algebra.thy Induct/SList.ML Induct/SList.thy \
+  Induct/ABexp.ML Induct/ABexp.thy Induct/Term.ML Induct/Term.thy
@\$(ISATOOL) usedir \$(OUT)/HOL Induct

```