(* Title: HOL/ex/Acc
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
Inductive definition of acc(r)
See Ch. Paulin-Mohring, Inductive Definitions in the System Coq.
Research Report 92-49, LIP, ENS Lyon. Dec 1992.
*)
open Acc;
(*The intended introduction rule*)
val prems = goal Acc.thy
"[| !!b. (b,a):r ==> b: acc(r) |] ==> a: acc(r)";
by (fast_tac (claset() addIs (prems @
map (rewrite_rule [pred_def]) acc.intrs)) 1);
qed "accI";
Goal "[| b: acc(r); (a,b): r |] ==> a: acc(r)";
by (etac acc.elim 1);
by (rewtac pred_def);
by (Fast_tac 1);
qed "acc_downward";
Goal "(b,a) : r^* ==> a : acc r --> b : acc r";
by (etac rtrancl_induct 1);
by (Blast_tac 1);
by (blast_tac (claset() addDs [acc_downward]) 1);
val lemma = result();
Goal "[| a : acc r; (b,a) : r^* |] ==> b : acc r";
by (blast_tac (claset() addDs [lemma]) 1);
qed "acc_downwards";
val [major,indhyp] = goal Acc.thy
"[| a : acc(r); \
\ !!x. [| x: acc(r); ALL y. (y,x):r --> P(y) |] ==> P(x) \
\ |] ==> P(a)";
by (rtac (major RS acc.induct) 1);
by (rtac indhyp 1);
by (resolve_tac acc.intrs 1);
by (rewtac pred_def);
by (Fast_tac 2);
by (etac (Int_lower1 RS Pow_mono RS subsetD) 1);
qed "acc_induct";
val [major] = goal Acc.thy "r <= (acc r) Times (acc r) ==> wf(r)";
by (rtac (major RS wfI) 1);
by (etac acc.induct 1);
by (rewtac pred_def);
by (Fast_tac 1);
qed "acc_wfI";
val [major] = goal Acc.thy "wf(r) ==> ALL x. (x,y): r | (y,x):r --> y: acc(r)";
by (rtac (major RS wf_induct) 1);
by (rtac (impI RS allI) 1);
by (rtac accI 1);
by (Fast_tac 1);
qed "acc_wfD_lemma";
val [major] = goal Acc.thy "wf(r) ==> r <= (acc r) Times (acc r)";
by (rtac subsetI 1);
by (res_inst_tac [("p", "x")] PairE 1);
by (fast_tac (claset() addSIs [SigmaI,
major RS acc_wfD_lemma RS spec RS mp]) 1);
qed "acc_wfD";
Goal "wf(r) = (r <= (acc r) Times (acc r))";
by (EVERY1 [rtac iffI, etac acc_wfD, etac acc_wfI]);
qed "wf_acc_iff";