Section MAIN.
Variable A : Setoid.
(** - Constant functions and their compatibility %\label{constmap}% *)
Definition const_map : forall (X Y : Setoid) (y : Y), MAP X Y.
intros.
apply (Build_Map (Ap:=fun x : X => y)).
red in |- *.
intros.
apply Refl.
Defined.
Definition const_seq : forall (n : Nat) (a : A), seq n A.
intros.
apply (const_map (fin n) a); auto with algebra.
Defined.
simpl in |- *.
red in |- *.
intros.
simpl in |- *.
apply seq_S_O_contains_single_elt.
Qed.
intros.
intro i.
simpl in |- *.
induction n.
auto with algebra.
case i.
auto with algebra.
Qed.

Lemma cons_const_seq :
forall (n : Nat) (a a' a'' : A),
a =' a' in _ ->
a' =' a'' in _ -> a;; const_seq n a' =' const_seq (S n) a'' in _.