End InfinitePrimitiveElementTheorem.
Section Separable.
Variables (F : fieldType) (L : fieldExtType F).
Implicit Types (U V W : {vspace L}) (E K M : {subfield L}) (D : 'End(L)).
Section Derivation.
Variables (K : {vspace L}) (D : 'End(L)).
(* A deriviation only needs to be additive and satify Lebniz's law, but all   *)
(* the deriviations used here are going to be linear, so we only define       *)
(* the Derivation predicate for linear endomorphisms.                         *)
Definition Derivation (s := vbasis K) : bool :=
all (fun u => all (fun v => D (u * v) == D u * v + u * D v) s) s.
Hypothesis derD : Derivation.

(* Specifications: *)
Lemma Derivation_mul : {in K &, forall u v, D (u * v) = D u * v + u * D v}.
Lemma Derivation_mul_poly (Dp := map_poly D) :
{in polyOver K &, forall p q, Dp (p * q) = Dp p * q + p * Dp q}.