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(* Title: HOL/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 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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open Acc;
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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: acc(r) |] ==> a: acc(r)";
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by (fast_tac (set_cs addIs (prems @
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map (rewrite_rule [pred_def]) acc.intrs)) 1);
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qed "accI";
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goal Acc.thy "!!a b r. [| b: acc(r); <a,b>: r |] ==> a: acc(r)";
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by (etac acc.elim 1);
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by (rewtac pred_def);
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by (fast_tac set_cs 1);
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qed "acc_downward";
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val [major,indhyp] = goal Acc.thy
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"[| a : acc(r); \
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\ !!x. [| x: acc(r); ALL 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 (resolve_tac acc.intrs 1);
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by (rewtac pred_def);
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by (fast_tac set_cs 2);
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by (etac (Int_lower1 RS Pow_mono RS subsetD) 1);
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qed "acc_induct";
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val [major] = goal Acc.thy "r <= Sigma(acc(r), %u. acc(r)) ==> wf(r)";
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by (rtac (major RS wfI) 1);
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by (etac acc.induct 1);
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by (rewtac pred_def);
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by (fast_tac set_cs 1);
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qed "acc_wfI";
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val [major] = goal Acc.thy "wf(r) ==> ALL x. <x,y>: r | <y,x>:r --> y: acc(r)";
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by (rtac (major RS wf_induct) 1);
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by (rtac (impI RS allI) 1);
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by (rtac accI 1);
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by (fast_tac set_cs 1);
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qed "acc_wfD_lemma";
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val [major] = goal Acc.thy "wf(r) ==> r <= Sigma(acc(r), %u. acc(r))";
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by (rtac subsetI 1);
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by (res_inst_tac [("p", "x")] PairE 1);
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by (fast_tac (set_cs addSIs [SigmaI,
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major RS acc_wfD_lemma RS spec RS mp]) 1);
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qed "acc_wfD";
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goal Acc.thy "wf(r) = (r <= Sigma(acc(r), %u. 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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