12088
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(* Title: ZF/ex/acc
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ID: $Id$
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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Copyright 1993 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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(*The introduction rule must require a \\<in> field(r)
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Otherwise acc(r) would be a proper class! *)
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(*The intended introduction rule*)
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val prems = goal Acc.thy
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"[| !!b. <b,a>:r ==> b \\<in> acc(r); a \\<in> field(r) |] ==> a \\<in> acc(r)";
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by (fast_tac (claset() addIs prems@acc.intrs) 1);
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qed "accI";
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Goal "[| b \\<in> acc(r); <a,b>: r |] ==> a \\<in> acc(r)";
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by (etac acc.elim 1);
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by (Fast_tac 1);
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qed "acc_downward";
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val [major,indhyp] = goal Acc.thy
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"[| a \\<in> acc(r); \
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\ !!x. [| x \\<in> acc(r); \\<forall>y. <y,x>:r --> P(y) |] ==> P(x) \
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\ |] ==> P(a)";
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by (rtac (major RS acc.induct) 1);
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by (rtac indhyp 1);
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by (Fast_tac 2);
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by (resolve_tac acc.intrs 1);
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by (assume_tac 2);
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by (etac (Collect_subset RS Pow_mono RS subsetD) 1);
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qed "acc_induct";
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Goal "wf[acc(r)](r)";
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by (rtac wf_onI2 1);
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by (etac acc_induct 1);
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by (Fast_tac 1);
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qed "wf_on_acc";
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(* field(r) \\<subseteq> acc(r) ==> wf(r) *)
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val acc_wfI = wf_on_acc RS wf_on_subset_A RS wf_on_field_imp_wf;
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val [major] = goal Acc.thy "wf(r) ==> field(r) \\<subseteq> acc(r)";
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by (rtac subsetI 1);
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by (etac (major RS wf_induct2) 1);
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by (rtac subset_refl 1);
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by (rtac accI 1);
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by (assume_tac 2);
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by (Fast_tac 1);
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qed "acc_wfD";
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Goal "wf(r) <-> field(r) \\<subseteq> acc(r)";
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by (EVERY1 [rtac iffI, etac acc_wfD, etac acc_wfI]);
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qed "wf_acc_iff";
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