Inductive Finite : Ensemble U -> Prop :=
    | Empty_is_finite : Finite (Empty_set U)
    | Union_is_finite :
      forall A:Ensemble U,
        Finite A -> forall x:U, ~ In U A x -> Finite (Add U A x).Inductive Finite : Ensemble U -> Prop :=
    | Empty_is_finite : Finite (Empty_set U)
    | Union_is_finite :
      forall A:Ensemble U,
        Finite A -> forall x:U, ~ In U A x -> Finite (Add U A x).Inductive Finite : Ensemble U -> Prop :=
    | Empty_is_finite : Finite (Empty_set U)
    | Union_is_finite :
      forall A:Ensemble U,
        Finite A -> forall x:U, ~ In U A x -> Finite (Add U A x).Inductive cardinal : Ensemble U -> nat -> Prop :=
    | card_empty : cardinal (Empty_set U) 0
    | card_add :
      forall (A:Ensemble U) (n:nat),
        cardinal A n -> forall x:U, ~ In U A x -> cardinal (Add U A x) (S n).Inductive cardinal : Ensemble U -> nat -> Prop :=
    | card_empty : cardinal (Empty_set U) 0
    | card_add :
      forall (A:Ensemble U) (n:nat),
        cardinal A n -> forall x:U, ~ In U A x -> cardinal (Add U A x) (S n).Inductive cardinal : Ensemble U -> nat -> Prop :=
    | card_empty : cardinal (Empty_set U) 0
    | card_add :
      forall (A:Ensemble U) (n:nat),
        cardinal A n -> forall x:U, ~ In U A x -> cardinal (Add U A x) (S n).

(* Specifications: *)
Lemma cardinal_invert : forall (X:Ensemble U) (p:nat), cardinal U X p -> match p with | O => X = Empty_set U | S n => exists A : _, (exists x : _, X = Add U A x /\ ~ In U A x /\ cardinal U A n) end.
Lemma cardinal_elim : forall (X:Ensemble U) (p:nat), cardinal U X p -> match p with | O => X = Empty_set U | S n => Inhabited U X end.