Fixpoint force_lengthn {A} n (xs: list A) (default: A) :=
  match n, xs with
  | O, _ => nil
  | S n0, nil => default :: force_lengthn n0 nil default
  | S n0, hd :: tl => hd :: force_lengthn n0 tl default
  end.

Lemma nth_force_lengthn_nil: forall {A} n i (default: A), nth i (force_lengthn n nil default) default = default.