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-(************************************************************************)
-(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
-(*   \VV/  **************************************************************)
-(*    //   *      This file is distributed under the terms of the       *)
-(*         *       GNU Lesser General Public License Version 2.1        *)
-(************************************************************************)
-
-(*i $Id: Classical_Pred_Set.v,v 1.1.2.1 2004/07/16 19:31:29 herbelin Exp $ i*)
-
-(** Classical Predicate Logic on Set*)
-
-Require Classical_Prop.
-
-Section Generic.
-Variable U: Set.
-
-(** de Morgan laws for quantifiers *)
-
-Lemma not_all_ex_not : (P:U->Prop)(~(n:U)(P n)) -> (EX n:U | ~(P n)).
-Proof.
-Unfold not; Intros P notall.
-Apply NNPP; Unfold not.
-Intro abs.
-Cut ((n:U)(P n)); Auto.
-Intro n; Apply NNPP.
-Unfold not; Intros.
-Apply abs; Exists n; Trivial.
-Qed.
-
-Lemma not_all_not_ex : (P:U->Prop)(~(n:U)~(P n)) -> (EX n:U |(P n)).
-Proof.
-Intros P H.
-Elim (not_all_ex_not [n:U]~(P n) H); Intros n Pn; Exists n.
-Apply NNPP; Trivial.
-Qed.
-
-Lemma not_ex_all_not : (P:U->Prop) (~(EX n:U |(P n))) -> (n:U)~(P n).
-Proof.
-Unfold not; Intros P notex n abs.
-Apply notex.
-Exists n; Trivial.
-Qed. 
-
-Lemma not_ex_not_all : (P:U->Prop)(~(EX n:U | ~(P n))) -> (n:U)(P n).
-Proof.
-Intros P H n.
-Apply NNPP.
-Red; Intro K; Apply H; Exists n; Trivial.
-Qed.
-
-Lemma ex_not_not_all : (P:U->Prop) (EX n:U | ~(P n)) -> ~(n:U)(P n).
-Proof.
-Unfold not; Intros P exnot allP.
-Elim exnot; Auto.
-Qed.
-
-Lemma all_not_not_ex : (P:U->Prop) ((n:U)~(P n)) -> ~(EX n:U |(P n)).
-Proof.
-Unfold not; Intros P allnot exP; Elim exP; Intros n p.
-Apply allnot with n; Auto.
-Qed.
-
-End Generic.