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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(****************************************************************************)
(* *)
(* Naive Set Theory in Coq *)
(* *)
(* INRIA INRIA *)
(* Rocquencourt Sophia-Antipolis *)
(* *)
(* Coq V6.1 *)
(* *)
(* Gilles Kahn *)
(* Gerard Huet *)
(* *)
(* *)
(* *)
(* Acknowledgments: This work was started in July 1993 by F. Prost. Thanks *)
(* to the Newton Institute for providing an exceptional work environment *)
(* in Summer 1995. Several developments by E. Ledinot were an inspiration. *)
(****************************************************************************)
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.
Hint Resolve card_empty card_add: sets.
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.
Proof.
induction 1; simpl; auto.
exists A; exists x; auto.
Qed.
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.
Proof.
intros X p C; elim C; simpl; trivial with sets.
Qed.
End Ensembles_finis_facts.
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