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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 1.6.2.1 2004/07/16 19:31:06 herbelin Exp $ i*)
(** Classical Predicate Logic on Set*)
Require Import Classical_Prop.
Section Generic.
Variable U : Set.
(** de Morgan laws for quantifiers *)
Lemma not_all_ex_not :
forall P:U -> Prop, ~ (forall n:U, P n) -> exists n : U, ~ P n.
Proof.
unfold not in |- *; intros P notall.
apply NNPP; unfold not in |- *.
intro abs.
cut (forall n:U, P n); auto.
intro n; apply NNPP.
unfold not in |- *; intros.
apply abs; exists n; trivial.
Qed.
Lemma not_all_not_ex :
forall P:U -> Prop, ~ (forall n:U, ~ P n) -> exists n : U, P n.
Proof.
intros P H.
elim (not_all_ex_not (fun n:U => ~ P n) H); intros n Pn; exists n.
apply NNPP; trivial.
Qed.
Lemma not_ex_all_not :
forall P:U -> Prop, ~ (exists n : U, P n) -> forall n:U, ~ P n.
Proof.
unfold not in |- *; intros P notex n abs.
apply notex.
exists n; trivial.
Qed.
Lemma not_ex_not_all :
forall P:U -> Prop, ~ (exists n : U, ~ P n) -> forall n:U, P n.
Proof.
intros P H n.
apply NNPP.
red in |- *; intro K; apply H; exists n; trivial.
Qed.
Lemma ex_not_not_all :
forall P:U -> Prop, (exists n : U, ~ P n) -> ~ (forall n:U, P n).
Proof.
unfold not in |- *; intros P exnot allP.
elim exnot; auto.
Qed.
Lemma all_not_not_ex :
forall P:U -> Prop, (forall n:U, ~ P n) -> ~ (exists n : U, P n).
Proof.
unfold not in |- *; intros P allnot exP; elim exP; intros n p.
apply allnot with n; auto.
Qed.
End Generic.
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