num(z).
num(s(A)) :- num(A).

decode(z, 0).
decode(s(A), B) :- decode(A, C), B is C + 1.

encode(A, B) :- decode(B, A), !.

add(z, A, A).
add(s(A), B, s(C)) :- add(A, B, C).

sub(A, B, C) :- add(B, C, A).

mul(z, A, z).
mul(s(A), B, C) :- mul(A, B, D), add(D, B, C).

div(A, B, C) :- mul(B, C, A).