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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_Type.v,v 1.1.2.1 2004/07/16 19:31:29 herbelin Exp $ i*)
+
+(** Classical Predicate Logic on Type *)
+
+Require Classical_Prop.
+
+Section Generic.
+Variable U: Type.
+
+(** de Morgan laws for quantifiers *)
+
+Lemma not_all_ex_not : (P:U->Prop)(~(n:U)(P n)) -> (EXT 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)) -> (EXT 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)(~(EXT 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)(~(EXT 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) (EXT 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)) -> ~(EXT n:U | (P n)).
+Proof.
+Unfold not; Intros P allnot exP; Elim exP; Intros n p.
+Apply allnot with n; Auto.
+Qed.
+
+End Generic.