theories/Sets

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+(****************************************************************************)
+(*                                                                          *)
+(*                         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.  *)
+(****************************************************************************)
+
+(* $Id$ *)
+
+Require Ensembles.
+
+Section Ensembles_finis.
+Variable U: Type.
+
+Inductive Finite : (Ensemble U) -> Prop :=
+      Empty_is_finite: (Finite (Empty_set U))
+   |  Union_is_finite:
+      (A: (Ensemble U)) (Finite A) -> 
+      (x: U) ~ (In U A x) -> (Finite (Add U A x)).
+
+Inductive cardinal : (Ensemble U) -> nat -> Prop :=
+      card_empty: (cardinal (Empty_set U) O)
+   |  card_add:
+       (A: (Ensemble U)) (n: nat) (cardinal A n) ->
+        (x: U) ~ (In U A x) -> (cardinal (Add U A x) (S n)).
+
+End Ensembles_finis.
+
+Hints Resolve Empty_is_finite Union_is_finite : sets v62.
+Hints Resolve card_empty card_add : sets v62.
+
+Require Constructive_sets.
+
+Section Ensembles_finis_facts.
+Variable U: Type.
+
+Lemma cardinal_invert :
+ (X: (Ensemble U)) (p:nat)(cardinal U X p) -> Case p of
+           X == (Empty_set U)
+   [n:nat] (EXT A | (EXT x | 
+           X == (Add U A x) /\ ~ (In U A x) /\ (cardinal U A n))) end.
+Proof.
+Induction 1; Simpl; Auto.
+Intros; Exists A; Exists x; Auto.
+Qed.
+
+Lemma cardinal_elim :
+ (X: (Ensemble U)) (p:nat)(cardinal U X p) -> Case p of
+                              X == (Empty_set U)
+                             [n:nat](Inhabited U X) end.
+Proof.
+Intros X p C; Elim C; Simpl; Trivial with sets.
+Qed.
+
+End Ensembles_finis_facts.