Section UniqueList.
Variable A : Type.
Variable eqA_dec : forall a b : A, {a = b} + {a <> b}.
(* A list is unique if there is not twice the same element in the list *)
Inductive ulist : list A -> Prop :=
| ulist_nil : ulist nil
| ulist_cons : forall a l, ~ In a l -> ulist l -> ulist (a :: l).
Hint Constructors ulist : core.
Proof using.
intros a l H; inversion H; auto.
Qed.
Proof using.
intros L1; elim L1; simpl in |- *; auto.
intros a l H l2 H0 H1 H2; apply ulist_cons; simpl in |- *; auto.
red in |- *; intros H3; case in_app_or with (1 := H3); auto; intros H4.
inversion H0; auto.
apply H2 with a; auto.
apply H; auto.
apply ulist_inv with (1 := H0); auto.
intros a0 H3 H4; apply (H2 a0); auto.
Qed.
Proof using.
intros l1; elim l1; simpl in |- *; auto.
intros a l H l2 a0 H0 [H1| H1] H2.
inversion H0; auto.
case H5; rewrite H1; auto with datatypes.
apply (H l2 a0); auto.
apply ulist_inv with (1 := H0); auto.
Qed.
Proof using.
intros l1; elim l1; simpl in |- *; auto.
intros a l H l2 H0; inversion H0; apply ulist_cons; auto.
Contradict H3; auto with datatypes.
apply H with l2; auto.
Qed.

(* Inversion theorem the appended list *)
Theorem ulist_app_inv_r : forall l1 l2 : list A, ulist (l1 ++ l2) -> ulist l2.