(************************)
simple induction l. simpl in |- *. intros. absurd (i < 0). apply lt_n_O.
exact H.
simple induction i.
intros. unfold elemlist in |- *. rewrite elemlist_inv. rewrite elemlist_inv.
generalize H1 H2. elim p. intros. absurd (0 <> 0). unfold not in |- *; auto with arith.
exact H4.
intros. simpl in |- *. rewrite elemlist_cons_O. rewrite elemlist_cons_O. trivial with arith.
simple induction p. intros. simpl in |- *.
rewrite elemlist_cons_S. rewrite elemlist_cons_S. trivial with arith.
intros. clear H1. simpl in |- *. do 2 rewrite elemlist_cons_S.
clear H0. apply H. apply lt_S_n. generalize H2; simpl in |- *; trivial with arith.
apply lt_S_n. generalize H3; simpl in |- *; trivial with arith.
generalize H4. red in |- *. auto with arith.
Qed.

(* Specifications: *)
Lemma length_replace :
forall (l : list A) (p : nat) (x : A),
p < length l -> length (replace l p x) = length l.