Fixpoint PS_incr_n (a : nat -> V) (n k : nat) : V :=
  match n with
    | O => a k
    | S n => PS_incr_1 (PS_incr_n a n) k
  end.

Lemma PS_incr_n_simplify (a : nat -> V) (n k : nat) : PS_incr_n a n k = match (le_lt_dec n k) with | left _ => a (k-n)%nat | right _ => zero end.