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