Section Ensembles.
Variable U : Type.
Definition Ensemble := U -> Prop.
Definition In (B : Ensemble) (x : U) : Prop := B x.
Definition Included (B C : Ensemble) : Prop :=
forall x : U, In B x -> In C x.
Inductive Empty_set : Ensemble :=.
Inductive Singleton (x : U) : Ensemble :=
In_singleton : In (Singleton x) x.
Inductive Union (B C : Ensemble) : Ensemble :=
| Union_introl : forall x : U, In B x -> In (Union B C) x
| Union_intror : forall x : U, In C x -> In (Union B C) x.
Inductive Intersection (B C : Ensemble) : Ensemble :=
Intersection_intro :
forall x : U, In B x -> In C x -> In (Intersection B C) x.
Inductive Couple (x y : U) : Ensemble :=
| Couple_l : In (Couple x y) x
| Couple_r : In (Couple x y) y.
Inductive Setminus (B C : Ensemble) : Ensemble :=
Setminus_intro :
forall x : U, In B x -> ~ In C x -> In (Setminus B C) x.
Inductive Disjoint (B C : Ensemble) : Prop :=
Disjoint_intro :
(forall x : U, ~ In (Intersection B C) x) -> Disjoint B C.
Inductive Non_empty (B : Ensemble) : Prop :=
Non_empty_intro : forall x : U, In B x -> Non_empty B.
Inductive Finite : Ensemble -> Prop :=
| Empty_is_finite : Finite Empty_set
| Union_is_finite :
forall A : Ensemble,
Finite A -> forall x : U, ~ In A x -> Finite (Union (Singleton x) A).
(*************************************************************************)
(*** Initialement, la definition de Same_set etait :  Inductive Same_set [B,C: Ensemble]: Prop := Same_set_intro: (Included B C) -> (Included C B) -> (Same_set B C).  *** mais dans le lemme Same_set_equivalence de ps.v (par exemple) (Same_set U) n'est plus unifie avec [x,y:(Ensemble U)](and (Included x y) (Included y x))  *** on redefinit donc Same_set et Same_Set_intro de la maniere suivante : ***)
(*************************************************************************)
Definition Same_set (B C : Ensemble) : Prop :=
Included B C /\ Included C B.

Lemma Same_set_intro :
forall B C : Ensemble, Included B C -> Included C B -> Same_set B C.