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author | Samuel Mimram <samuel.mimram@ens-lyon.org> | 2004-07-28 21:54:47 +0000 |
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committer | Samuel Mimram <samuel.mimram@ens-lyon.org> | 2004-07-28 21:54:47 +0000 |
commit | 6b649aba925b6f7462da07599fe67ebb12a3460e (patch) | |
tree | 43656bcaa51164548f3fa14e5b10de5ef1088574 /theories/Logic/Classical_Pred_Set.v |
Imported Upstream version 8.0pl1upstream/8.0pl1
Diffstat (limited to 'theories/Logic/Classical_Pred_Set.v')
-rwxr-xr-x | theories/Logic/Classical_Pred_Set.v | 70 |
1 files changed, 70 insertions, 0 deletions
diff --git a/theories/Logic/Classical_Pred_Set.v b/theories/Logic/Classical_Pred_Set.v new file mode 100755 index 00000000..c8f87fe8 --- /dev/null +++ b/theories/Logic/Classical_Pred_Set.v @@ -0,0 +1,70 @@ +(************************************************************************) +(* 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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