Section PolyXY_ComRing.
Variable R : comRingType.
Implicit Types (u : {poly {poly R}}) (p : {poly R}) (x y : R).

Lemma div_annihilant_in_ideal p q :
1 < size p -> 1 < size q ->
{uv : {poly {poly R}} * {poly {poly R}}
| size uv.1 < size q /\ size uv.2 < size p
& forall x y,
(div_annihilant p q).[y] = uv.1.[x, y] * p.[x * y] + uv.2.[x, y] * q.[x]}.