// Ackermann function on unary natural numbers

// Originally the function is written as follows:
// fun ack(0,y) = y+1
//   | ack(x,0) = ack(x-1,1)        
//   | ack(x,y) = ack(x-1,ack(x,y-1));

// This is written one on unary natural nunbers:
// fun ack(Z   ,    y) = S(y)
//   | ack(S(x),    Z) = ack(x,S(Z))        
//   | ack(S(x), S(y)) = ack(x,ack(S(x),y));

// For easily translation to interaction nets,
// the first argument will be moved to the last:
// fun ack(y,       Z) = S(y)
//   | ack(Z,    S(x)) = ack(S(Z),x)        
//   | ack(S(y), S(x)) = ack(ack(y,S(x)),x);

// It is traslated directly as follows:
// ack(r,y)>< Z    => r~S(y);
// ack(r,y)>< S(x) => ack_Sx(r, x)~y;
// ack_Sx(r,x)>< Z    => ack(r, S(Z))~x;
// ack_Sx(r,x)>< S(y) => Dup(x1,x2)~x, ack(r,cnt)~x1, ack(cnt,y)~S(x2);

// The last connection can be optimised by applying the rule ack><S,
// so finally we get the following as the result of the translation:
ack(r,y)>< Z    => r~S(y);
ack(r,y)>< S(x) => ack_Sx(r, x)~y;
ack_Sx(r,x)>< Z    => ack(r, S(Z))~x;
ack_Sx(r,x)>< S(y) => Dup(x1,x2)~x, ack(r,cnt)~x1, ack_Sx(cnt,x2)~y;


// By the way, it is possible to write it by using abbreviation notation:
// ack(r,y)>< Z    => r~S(y);
// ack(r,y)>< S(x) => r<<ack_Sx(x,y);
// ack_Sx(r,x)>< Z    => r<<ack(S(Z),x);
// ack_Sx(r,x)>< S(y) => x1,x2<<Dup(x), r<<ack(cnt,x1), cnt<<ack_Sx(x2,y);


// Dup is defined as built-in:
// Dup(a1,a2)>< Z => a1~Z, a2~Z;
// Dup(a1,a2)>< S(x) => a1~S(w1), a2~S(w2), Dup(w1,w2)~x;

// ack(result, y)~x
// For computation `ack 3 5', it is written as follows:
ack(result, S(S(S(S(S(Z))))))~S(S(S(Z)));

prnat result;
exit;