src/HOL/Library/Accessible_Part.thy
 author nipkow Wed, 18 Aug 2004 11:09:40 +0200 changeset 15140 322485b816ac parent 15131 c69542757a4d child 18241 afdba6b3e383 permissions -rw-r--r--
import -> imports
```
(*  Title:      HOL/Library/Accessible_Part.thy
ID:         \$Id\$
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
*)

header {* The accessible part of a relation *}

theory Accessible_Part
imports Main
begin

subsection {* Inductive definition *}

text {*
Inductive definition of the accessible part @{term "acc r"} of a
*}

consts
acc :: "('a \<times> 'a) set => 'a set"
inductive "acc r"
intros
accI: "(!!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\<inverse>)"

subsection {* Induction rules *}

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

theorems acc_induct_rule = acc_induct [rule_format, induct set: acc]

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\<^sup>* ==> 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\<^sup>* ==> 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
```