Section WeakTheoryForIDomainPseudoDivision.
Variable R : idomainType.
Implicit Type p q r d : {poly R}.
Hint Resolve lc_expn_scalp_neq0 : core.
Variant edivp_spec (m d : {poly R}) :
nat * {poly R} * {poly R} -> bool -> Type :=
|Redivp_spec k (q r: {poly R}) of
(lead_coef d ^+ k) *: m = q * d + r & lead_coef d \notin GRing.unit &
(d != 0 -> size r < size d) : edivp_spec m d (k, q, r) false
|Fedivp_spec (q r: {poly R}) of m = q * d + r & (lead_coef d \in GRing.unit) &
(d != 0 -> size r < size d) : edivp_spec m d (0%N, q, r) true.

Lemma dvdp_eq q p : (q %| p) = (lead_coef q ^+ scalp p q *: p == (p %/ q) * q).