#[local]
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
| _ => lia
end : core.

Lemma linear_reduce : forall k p, valid k p -> linear k (reduce p).