mul(zz, zz, zz).
mul(qq, qq, qq).
mul(zz, qq, qq).

mul(poly(A), poly(A), poly(A)) :- ring(A).
mul(A, vec(B), vec(C)) :- mul(A, B, C).
mul(vec(A), B, vec(C)) :- mul(A, B, C).
mul(A, poly(B), poly(C)) :- mul(A, vec(B), vec(C)).
mul(poly(A), B, poly(C)) :- mul(vec(A), B, vec(C)).
mul(mat(A), mat(B), mat(C)) :- mul(A, B, C), add(C, C, C).
mul(A, mat(B), mat(C)) :- mul(A, vec(B), vec(C)).

mul(A, B, B) :- coerce(A, B), mul(B, B, B).

coerce(A, poly(A)).
coerce(poly(A), poly(B)) :- coerce(A, B).
coerce(zz, qq).

add(zz, zz, zz).
add(qq, qq, qq).

add(poly(A), poly(B), poly(C)) :- add(A, B, C), zero(A), zero(B).
add(A, poly(B), poly(B)) :- add(A, B, B), zero(B).

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

add(A, B, B) :- coerce(A, B), add(B, B, B).
add(vec(A), vec(B), vec(C)) :- add(A, B, C).


zero(zz).
zero(qq).
zero(poly(_)).
zero(vec(A)) :- zero(A).
zero(mat(A)) :- zero(vec(A)).

ring(A) :- mul(A, A, A), add(A, A, A), zero(A).
module(A, M) :- ring(A), mul(A, M, M), add(M, M, M), zero(M).