# HG changeset patch
# User paulson
# Date 972377331 -7200
# Node ID ec30a7d15f76c2d8e611ef1c98d914f6f1357196
# Parent  5b36035e4dff598b2ff51e2d13bea9afeaa55b7c
Acc example

diff -r 5b36035e4dff -r ec30a7d15f76 doc-src/TutorialI/Inductive/Acc.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/TutorialI/Inductive/Acc.thy	Tue Oct 24 10:48:51 2000 +0200
@@ -0,0 +1,78 @@
+(*  Title:      HOL/ex/Acc.thy
+    ID:         $Id$
+    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
+    Copyright   1994  University of Cambridge
+
+Inductive definition of acc(r)
+
+See Ch. Paulin-Mohring, Inductive Definitions in the System Coq.
+Research Report 92-49, LIP, ENS Lyon.  Dec 1992.
+*)
+
+header {* The accessible part of a relation *}
+
+theory Acc = Main:
+
+consts
+  acc  :: "('a \<times> 'a) set => 'a set"  -- {* accessible part *}
+
+inductive "acc r"
+  intros
+    accI [rule_format]:
+      "\<forall>y. (y, x) \<in> r --> y \<in> acc r ==> x \<in> acc r"
+
+syntax
+  termi :: "('a \<times> 'a) set => 'a set"
+translations
+  "termi r" == "acc (r^-1)"
+
+
+theorem acc_induct:
+  "[| a \<in> acc r;
+      !!x. [| x \<in> acc r;  \<forall>y. (y, x) \<in> r --> P y |] ==> P x
+  |] ==> P a"
+proof -
+  assume major: "a \<in> acc r"
+  assume hyp: "!!x. [| x \<in> acc r;  \<forall>y. (y, x) \<in> r --> P y |] ==> P x"
+  show ?thesis
+    apply (rule major [THEN acc.induct])
+    apply (rule hyp)
+     apply (rule accI)
+     apply fast
+    apply fast
+    done
+qed
+
+theorem acc_downward: "[| b \<in> acc r; (a, b) \<in> r |] ==> a \<in> acc r"
+  apply (erule acc.elims)
+  apply fast
+  done
+
+lemma acc_downwards_aux: "(b, a) \<in> r^* ==> a \<in> acc r --> b \<in> acc r"
+  apply (erule rtrancl_induct)
+   apply blast
+  apply (blast dest: acc_downward)
+  done
+
+theorem acc_downwards: "[| a \<in> acc r; (b, a) \<in> r^* |] ==> b \<in> acc r"
+  apply (blast dest: acc_downwards_aux)
+  done
+
+theorem acc_wfI: "\<forall>x. x \<in> acc r ==> wf r"
+  apply (rule wfUNIVI)
+  apply (induct_tac P x rule: acc_induct)
+   apply blast
+  apply blast
+  done
+
+theorem acc_wfD: "wf r ==> x \<in> acc r"
+  apply (erule wf_induct)
+  apply (rule accI)
+  apply blast
+  done
+
+theorem wf_acc_iff: "wf r = (\<forall>x. x \<in> acc r)"
+  apply (blast intro: acc_wfI dest: acc_wfD)
+  done
+
+end