Section Lists_replace.
Variable A : Set.
(* Remplace l'element de l a la position par new *)
Fixpoint replace (l : list A) : nat -> A -> list A :=
fun (position : nat) (new : A) =>
match l return (list A) with
| nil =>
(* nil *)  nil | x :: l' => (* (cons x l') *)
match position return (list A) with
| O =>
(* O *)  new :: l' (* (S n') *)
| S n' => x :: replace l' n' new
end
end.
(***************)
simple induction l. simpl in |- *. intros. absurd (i < 0). apply lt_n_O.
exact H.
simple induction i. intros. simpl in |- *.
unfold elemlist in |- *. rewrite strip_O. simpl in |- *. rewrite trunc_O; trivial with arith.
intros. simpl in |- *. clear H0. unfold elemlist in |- *. rewrite strip_cons_S.
rewrite elemlist_inv.
apply H. apply lt_S_n. generalize H1. simpl in |- *. trivial with arith.
Qed.
(************************)
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.
(*******************)
simple induction l. simpl in |- *. try trivial with arith.
simple induction p. intros. simpl in |- *. apply eq_S. try trivial with arith.
intros. clear H0. simpl in |- *. rewrite H. try trivial with arith.
auto with arith.
Qed.

Lemma replace_sym :
forall (l : list A) (p p' : nat) (x x' : A),
p < length l ->
p' < length l ->
p <> p' -> replace (replace l p' x') p x = replace (replace l p x) p' x'.