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diff --git a/theories7/Logic/Classical_Pred_Set.v b/theories7/Logic/Classical_Pred_Set.v new file mode 100755 index 00000000..b1c26e6d --- /dev/null +++ b/theories7/Logic/Classical_Pred_Set.v @@ -0,0 +1,64 @@ +(************************************************************************) +(* 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. |