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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.
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