# Library Coq.Sets.Finite_sets

Require Import Ensembles.
Section Ensembles_finis.
Variable U : Type.
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).
End Ensembles_finis.
Hint Resolve Empty_is_finite Union_is_finite: sets v62.
Hint Resolve card_empty card_add: sets v62.
Require Import Constructive_sets.
Section Ensembles_finis_facts.
Variable U : Type.
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.
End Ensembles_finis_facts.