src/HOL/Induct/Acc.ML
author berghofe
Fri, 24 Jul 1998 13:39:47 +0200
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Renamed '$' to 'Scons' because of clashes with constants of the same name in theories using datatypes.
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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 (claset() 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 "[| 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 1);
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qed "acc_downward";
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Goal "(b,a) : r^* ==> a : acc r --> b : acc r";
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by (etac rtrancl_induct 1);
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by (Blast_tac 1);
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 by (blast_tac (claset() addDs [acc_downward]) 1);
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val lemma = result();
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Goal "[| a : acc r; (b,a) : r^* |] ==> b : acc r";
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by (blast_tac (claset() addDs [lemma]) 1);
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qed "acc_downwards";
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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 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 <= (acc r) Times (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 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 1);
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qed "acc_wfD_lemma";
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val [major] = goal Acc.thy "wf(r) ==> r <= (acc r) Times (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 (claset() 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 "wf(r)  =  (r <= (acc r) Times (acc r))";
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by (EVERY1 [rtac iffI, etac acc_wfD, etac acc_wfI]);
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qed "wf_acc_iff";