Section PS_incr.
Context {K : AbsRing} {V : NormedModule K}.
Definition PS_incr_1 (a : nat -> V) (n : nat) : V :=
match n with
| 0 => zero
| S n => a n
end.
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.