Set Implicit Arguments.
Unset Strict Implicit.
Require Export finite.
Require Export Parts.

Definition distinct (A : Setoid) (n : Nat) (v : seq n A) :=
  ∀ i j : fin n, ¬ i =' j in _ → ¬ v i =' v j in _.

Lemma distinct_comp :
 ∀ (A : Setoid) (n : Nat) (v v' : seq n A),
 distinct v → v =' v' in _ → distinct v'.
unfold distinct in |- ×.
intros.
simpl in H0; red in H0.
red in H; red in |- *; intro.
apply H with i j; auto with algebra.
apply Trans with (v' i); auto with algebra.
apply Trans with (v' j); auto with algebra.
Qed.

Hint Resolve distinct_comp: algebra.

Definition distinct_pred (A : Setoid) (n : Nat) : Predicate (seq n A).
intros.
apply (Build_Predicate (Pred_fun:=distinct (A:=A) (n:=n))); auto with algebra.
red in |- ×.
intros.
apply distinct_comp with x; auto with algebra.
Defined.