Section RingPseudoDivision.
Variable R : ringType.
Implicit Types d p q r : {poly R}.
(* Pseudo division, defined on an arbitrary ring *)
Definition redivp_rec (q : {poly R}) :=
let sq := size q in
let cq := lead_coef q in
fix loop (k : nat) (qq r : {poly R})(n : nat) {struct n} :=
if size r < sq then (k, qq, r) else
let m := (lead_coef r) *: 'X^(size r - sq) in
let qq1 := qq * cq%:P + m in
let r1 := r * cq%:P - m * q in
if n is n1.+1 then loop k.+1 qq1 r1 n1 else (k.+1, qq1, r1).
Definition redivp_expanded_def p q :=
if q == 0 then (0%N, 0, p) else redivp_rec q 0 0 p (size p).
Fact redivp_key : unit. Proof. by []. Qed.
Definition redivp : {poly R} -> {poly R} -> nat * {poly R} * {poly R} :=
locked_with redivp_key redivp_expanded_def.
Canonical redivp_unlockable := [unlockable fun redivp].
Definition rdivp p q := ((redivp p q).1).2.
Definition rmodp p q := (redivp p q).2.
Definition rscalp p q := ((redivp p q).1).1.
Definition rdvdp p q := rmodp q p == 0.
Variant comm_redivp_spec m d : nat * {poly R} * {poly R} -> Type :=
ComEdivnSpec k (q r : {poly R}) of
(GRing.comm d (lead_coef d)%:P -> m * (lead_coef d ^+ k)%:P = q * d + r) &
(d != 0 -> size r < size d) : comm_redivp_spec m d (k, q, r).

Lemma comm_redivpP m d : comm_redivp_spec m d (redivp m d).