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author | herbelin <herbelin@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2003-11-29 16:15:58 +0000 |
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committer | herbelin <herbelin@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2003-11-29 16:15:58 +0000 |
commit | 9058fb97426307536f56c3e7447be2f70798e081 (patch) | |
tree | b9a5fcf2ace7ecec13ed264b93c33fc04b0f220f /theories7/Sets/Relations_1.v | |
parent | 95ad10e5eb2efc9b63382e0e6a2f9ada8da2ea2d (diff) |
Deplacement des fichiers ancienne syntaxe dans theories7, contrib7 et states7
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@5026 85f007b7-540e-0410-9357-904b9bb8a0f7
Diffstat (limited to 'theories7/Sets/Relations_1.v')
-rwxr-xr-x | theories7/Sets/Relations_1.v | 67 |
1 files changed, 67 insertions, 0 deletions
diff --git a/theories7/Sets/Relations_1.v b/theories7/Sets/Relations_1.v new file mode 100755 index 000000000..74c031726 --- /dev/null +++ b/theories7/Sets/Relations_1.v @@ -0,0 +1,67 @@ +(***********************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) +(* \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 $Id$ i*) + +Section Relations_1. + Variable U: Type. + + Definition Relation := U -> U -> Prop. + Variable R: Relation. + + Definition Reflexive : Prop := (x: U) (R x x). + + Definition Transitive : Prop := (x,y,z: U) (R x y) -> (R y z) -> (R x z). + + Definition Symmetric : Prop := (x,y: U) (R x y) -> (R y x). + + Definition Antisymmetric : Prop := + (x: U) (y: U) (R x y) -> (R y x) -> x == y. + + Definition contains : Relation -> Relation -> Prop := + [R,R': Relation] (x: U) (y: U) (R' x y) -> (R x y). + + Definition same_relation : Relation -> Relation -> Prop := + [R,R': Relation] (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. +Hints Unfold Reflexive Transitive Antisymmetric Symmetric contains + same_relation : sets v62. +Hints Resolve Definition_of_preorder Definition_of_order + Definition_of_equivalence Definition_of_PER : sets v62. |