Section Algebra.
(* Compatibility with evaluation *)
Hint Extern 5 =>
match goal with
| [ |- (Pos.max ?x ?y <= ?z)%positive ] =>
apply Pos.max_case_strong; intros; lia
| [ |- (?z <= Pos.max ?x ?y)%positive ] =>
apply Pos.max_case_strong; intros; lia
| [ |- (Pos.max ?x ?y < ?z)%positive ] =>
apply Pos.max_case_strong; intros; lia
| [ |- (?z < Pos.max ?x ?y)%positive ] =>
apply Pos.max_case_strong; intros; lia
| _ => zify; omega
end : core.
Hint Resolve Pos.le_max_r Pos.le_max_l : core.
Hint Constructors valid linear : core.
(* Compatibility of validity w.r.t algebraic operations *)

Lemma poly_mul_mon_valid_compat : forall k i p,
valid k p -> valid (Pos.max (Pos.succ i) k) (poly_mul_mon i p).