induction n. intros. simpl. auto.
intros. unfold nsum. fold nsum.
assert (HEADHEAD:head f = head f'). unfold head. apply H.
rewrite HEADHEAD.
assert (TLTL:nsum n (Seqtl f) =' nsum n (Seqtl f')). apply IHn.
change (Seqtl f =' Seqtl f' in _). apply Seqtl_comp. exact H.
rewrite TLTL. apply Refl. Qed.

Lemma nsum_increment4:
forall (n:Nat) (v:seq n Nat) (i:fin n) (a:Nat) (H:a =' (v i) + 1),
nsum n (modify_seq v i a) =' nsum n v + 1.