(*  Title: 	ZF/nat.thy

     ID:         $Id$

     Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory

     Copyright   1992  University of Cambridge

 Natural numbers in Zermelo-Fraenkel Set Theory 

 *)

 Nat = Ord + Bool + 

 consts

     nat 	::      "i"

     nat_case    ::      "[i, i=>i, i]=>i"

     nat_rec     ::      "[i, i, [i,i]=>i]=>i"

 rules

     nat_def     "nat == lfp(Inf, %X. {0} Un {succ(i). i:X})"

     nat_case_def

 	"nat_case(a,b,k) == THE y. k=0 & y=a | (EX x. k=succ(x) & y=b(x))"

     nat_rec_def

 	"nat_rec(k,a,b) ==   \

 \   	  wfrec(Memrel(nat), k, %n f. nat_case(a, %m. b(m, f`m), n))"

 end