The Ackermann function is a classic example of a recursive function, notable especially because it is not a primitive recursive function. It grows very quickly in value, as does the size of its call tree.



The Ackermann function is usually defined as follows:







A
(
m
,
n
)
=


{



n
+
1



if 

m
=
0




A
(
m
−
1
,
1
)



if 

m
>
0

 and 

n
=
0




A
(
m
−
1
,
A
(
m
,
n
−
1
)
)



if 

m
>
0

 and 

n
>
0.








{\displaystyle A(m,n)={\begin{cases}n+1&{\mbox{if }}m=0\\A(m-1,1)&{\mbox{if }}m>0{\mbox{ and }}n=0\\A(m-1,A(m,n-1))&{\mbox{if }}m>0{\mbox{ and }}n>0.\end{cases}}}






Its arguments are never negative and it always terminates.



Task

Write a function which returns the value of 



A
(
m
,
n
)


{\displaystyle A(m,n)}

. Arbitrary precision is preferred (since the function grows so quickly), but not required.



See also

 Conway chained arrow notation for the Ackermann function.


:- func ack(integer, integer) = integer.
ack(M, N) = R :- ack(M, N, R).
 
:- pred ack(integer::in, integer::in, integer::out) is det.
ack(M, N, R) :-
	( ( M < integer(0)  
	  ; N < integer(0) ) -> throw(bounds_error)
	; M = integer(0)     -> R = N + integer(1)
	; N = integer(0)     -> ack(M - integer(1), integer(1), R)
	;                       ack(M - integer(1), ack(M, N - integer(1)), R) ).