author | wenzelm |
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changeset 10332 | b4f7f8693f8e |
parent 10248 | d99e5eeb16f4 |
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permissions | -rw-r--r-- |
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The accessible part of a relation (from HOL/Induct/Acc);
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(* Title: HOL/Library/Accessible_Part.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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*) |
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|
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header {* |
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\title{The accessible part of a relation} |
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\author{Lawrence C Paulson} |
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*} |
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|
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theory Accessible_Part = Main: |
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|
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subsection {* Inductive definition *} |
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text {* |
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Inductive definition of the accessible part @{term "acc r"} of a |
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relation; see also \cite{paulin-tlca}. |
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*} |
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|
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consts |
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acc :: "('a \<times> 'a) set => 'a set" |
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inductive "acc r" |
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intros |
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accI [rule_format]: "\<forall>y. (y, x) \<in> r --> y \<in> acc r ==> x \<in> acc r" |
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|
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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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subsection {* Induction rules *} |
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theorem acc_induct [induct set: acc]: |
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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) ==> 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 |