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