--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal/activities/tphols09/IDW/Inductive_Examples.thy	Wed Mar 30 17:27:34 2016 +0100
@@ -0,0 +1,92 @@
+theory Inductive_Examples
+imports Simple_Inductive_Package
+begin
+
+section {* Transitive closure *}
+
+simple_inductive
+  trcl for r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
+where
+  base: "trcl r x x"
+| step: "trcl r x y \<Longrightarrow> r y z \<Longrightarrow> trcl r x z"
+
+thm trcl_def
+thm trcl.induct
+thm base
+thm step
+thm trcl.intros
+
+lemma trcl_strong_induct:
+  assumes trcl: "trcl r x y"
+  and I1: "\<And>x. P x x"
+  and I2: "\<And>x y z. P x y \<Longrightarrow> trcl r x y \<Longrightarrow> r y z \<Longrightarrow> P x z"
+  shows "P x y" 
+proof -
+  from trcl
+  have "P x y \<and> trcl r x y"
+  proof induct
+    case (base x)
+    from I1 and trcl.base show ?case ..
+  next
+    case (step x y z)
+    then have "P x y" and "trcl r x y" by simp_all
+    from `P x y` `trcl r x y` `r y z` have "P x z"
+      by (rule I2)
+    moreover from `trcl r x y` `r y z` have "trcl r x z"
+      by (rule trcl.step)
+    ultimately show ?case ..
+  qed
+  then show ?thesis ..
+qed
+
+
+section {* Even and odd numbers *}
+
+simple_inductive
+  even and odd
+where
+  even0: "even 0"
+| evenS: "odd n \<Longrightarrow> even (Suc n)"
+| oddS: "even n \<Longrightarrow> odd (Suc n)"
+
+thm even_def odd_def
+thm even.induct odd.induct
+thm even0
+thm evenS
+thm oddS
+thm even_odd.intros
+
+
+section {* Accessible part *}
+
+simple_inductive
+  accpart for r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
+where
+  accpartI: "(\<And>y. r y x \<Longrightarrow> accpart r y) \<Longrightarrow> accpart r x"
+
+thm accpart_def
+thm accpart.induct
+thm accpartI
+
+
+section {* Accessible part in locale *}
+
+locale rel =
+  fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
+
+simple_inductive (in rel) accpart'
+where
+  accpartI': "\<And>x. (\<And>y. r y x \<Longrightarrow> accpart' y) \<Longrightarrow> accpart' x"
+
+context rel
+begin
+
+thm accpartI'
+thm accpart'.induct
+
+end
+
+thm rel.accpartI'
+thm rel.accpart'.induct
+
+end