(* Title: CCL/ex/nat.thy
ID: $Id$
Author: Martin Coen, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
Programs defined over the natural numbers
*)
Nat = Wfd +
consts
not :: "i=>i"
"#+","#*","#-",
"##","#<","#<=" :: "[i,i]=>i" (infixr 60)
ackermann :: "[i,i]=>i"
rules
not_def "not(b) == if b then false else true"
add_def "a #+ b == nrec(a,b,%x g.succ(g))"
mult_def "a #* b == nrec(a,zero,%x g.b #+ g)"
sub_def "a #- b == letrec sub x y be ncase(y,x,%yy.ncase(x,zero,%xx.sub(xx,yy)))
in sub(a,b)"
le_def "a #<= b == letrec le x y be ncase(x,true,%xx.ncase(y,false,%yy.le(xx,yy)))
in le(a,b)"
lt_def "a #< b == not(b #<= a)"
div_def "a ## b == letrec div x y be if x #< y then zero else succ(div(x#-y,y))
in div(a,b)"
ack_def
"ackermann(a,b) == letrec ack n m be ncase(n,succ(m),%x.
ncase(m,ack(x,succ(zero)),%y.ack(x,ack(succ(x),y))))
in ack(a,b)"
end