Definition make_last {A} n (x def : A) :=
Pos.peano_rect (fun _ => list A)
(cons x nil)
(fun _ F => cons def F) n.
(* Replace the nth element of a list *)
Fixpoint list_replace l n b :=
match l with
| nil => make_last n b false
| cons a l =>
Pos.peano_rect _
(cons b l) (fun n _ => cons a (list_replace l n b)) n
end.
(** Extract a non-null witness from a polynomial *)
Existing Instance Decidable_null.
Fixpoint boolean_witness p :=
match p with
| Cst c => nil
| Poly p i q =>
if decide (null p) then
let var := boolean_witness q in
list_replace var i true
else
let var := boolean_witness p in
list_replace var i false
end.

Lemma make_last_nth_1 : forall A n i x def, i <> n ->
list_nth i (@make_last A n x def) def = def.