clasohm@0: (*  Title: 	ZF/nat.thy
clasohm@0:     ID:         $Id$
clasohm@0:     Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
clasohm@0:     Copyright   1992  University of Cambridge
clasohm@0: 
clasohm@0: Natural numbers in Zermelo-Fraenkel Set Theory 
clasohm@0: *)
clasohm@0: 
clasohm@0: Nat = Ord + Bool + 
clasohm@0: consts
clasohm@0:     nat 	::      "i"
clasohm@0:     nat_case    ::      "[i, i, i=>i]=>i"
clasohm@0:     nat_rec     ::      "[i, i, [i,i]=>i]=>i"
clasohm@0: 
clasohm@0: rules
clasohm@0: 
clasohm@0:     nat_def     "nat == lfp(Inf, %X. {0} Un {succ(i). i:X})"
clasohm@0: 
clasohm@0:     nat_case_def
clasohm@0: 	"nat_case(k,a,b) == THE y. k=0 & y=a | (EX x. k=succ(x) & y=b(x))"
clasohm@0: 
clasohm@0:     nat_rec_def
clasohm@0: 	"nat_rec(k,a,b) ==   \
clasohm@0: \   	  wfrec(Memrel(nat), k, %n f. nat_case(n, a, %m. b(m, f`m)))"
clasohm@0: 
clasohm@0: end