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