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Diffstat (limited to 'theories/Sets/Relations_1.v')
| -rwxr-xr-x | theories/Sets/Relations_1.v | 67 |
1 files changed, 67 insertions, 0 deletions
diff --git a/theories/Sets/Relations_1.v b/theories/Sets/Relations_1.v new file mode 100755 index 00000000..e33746a9 --- /dev/null +++ b/theories/Sets/Relations_1.v @@ -0,0 +1,67 @@ +(************************************************************************) +(* 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 *) +(************************************************************************) +(****************************************************************************) +(* *) +(* Naive Set Theory in Coq *) +(* *) +(* INRIA INRIA *) +(* Rocquencourt Sophia-Antipolis *) +(* *) +(* Coq V6.1 *) +(* *) +(* Gilles Kahn *) +(* Gerard Huet *) +(* *) +(* *) +(* *) +(* Acknowledgments: This work was started in July 1993 by F. Prost. Thanks *) +(* to the Newton Institute for providing an exceptional work environment *) +(* in Summer 1995. Several developments by E. Ledinot were an inspiration. *) +(****************************************************************************) + +(*i $Id: Relations_1.v,v 1.4.2.1 2004/07/16 19:31:18 herbelin Exp $ i*) + +Section Relations_1. + Variable U : Type. + + Definition Relation := U -> U -> Prop. + Variable R : Relation. + + Definition Reflexive : Prop := forall x:U, R x x. + + Definition Transitive : Prop := forall x y z:U, R x y -> R y z -> R x z. + + Definition Symmetric : Prop := forall x y:U, R x y -> R y x. + + Definition Antisymmetric : Prop := forall x y:U, R x y -> R y x -> x = y. + + Definition contains (R R':Relation) : Prop := + forall x y:U, R' x y -> R x y. + + Definition same_relation (R R':Relation) : Prop := + contains R R' /\ contains R' R. + + Inductive Preorder : Prop := + Definition_of_preorder : Reflexive -> Transitive -> Preorder. + + Inductive Order : Prop := + Definition_of_order : + Reflexive -> Transitive -> Antisymmetric -> Order. + + Inductive Equivalence : Prop := + Definition_of_equivalence : + Reflexive -> Transitive -> Symmetric -> Equivalence. + + Inductive PER : Prop := + Definition_of_PER : Symmetric -> Transitive -> PER. + +End Relations_1. +Hint Unfold Reflexive Transitive Antisymmetric Symmetric contains + same_relation: sets v62. +Hint Resolve Definition_of_preorder Definition_of_order + Definition_of_equivalence Definition_of_PER: sets v62.
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