diff options
Diffstat (limited to 'theories/Relations/Relation_Definitions.v')
| -rw-r--r-- | theories/Relations/Relation_Definitions.v | 28 |
1 files changed, 14 insertions, 14 deletions
diff --git a/theories/Relations/Relation_Definitions.v b/theories/Relations/Relation_Definitions.v index 762da1ff..c03c4b95 100644 --- a/theories/Relations/Relation_Definitions.v +++ b/theories/Relations/Relation_Definitions.v @@ -6,19 +6,19 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: Relation_Definitions.v 9245 2006-10-17 12:53:34Z notin $ i*) +(*i $Id$ i*) Section Relation_Definition. Variable A : Type. - + Definition relation := A -> A -> Prop. Variable R : relation. - + Section General_Properties_of_Relations. - + Definition reflexive : Prop := forall x:A, R x x. Definition transitive : Prop := forall x y z:A, R x y -> R y z -> R x z. Definition symmetric : Prop := forall x y:A, R x y -> R y x. @@ -32,33 +32,33 @@ Section Relation_Definition. Section Sets_of_Relations. - - Record preorder : Prop := + + Record preorder : Prop := { preord_refl : reflexive; preord_trans : transitive}. - - Record order : Prop := + + Record order : Prop := { ord_refl : reflexive; ord_trans : transitive; ord_antisym : antisymmetric}. - - Record equivalence : Prop := + + Record equivalence : Prop := { equiv_refl : reflexive; equiv_trans : transitive; equiv_sym : symmetric}. - + Record PER : Prop := {per_sym : symmetric; per_trans : transitive}. End Sets_of_Relations. Section Relations_of_Relations. - + Definition inclusion (R1 R2:relation) : Prop := forall x y:A, R1 x y -> R2 x y. - + Definition same_relation (R1 R2:relation) : Prop := inclusion R1 R2 /\ inclusion R2 R1. - + Definition commut (R1 R2:relation) : Prop := forall x y:A, R1 y x -> forall z:A, R2 z y -> exists2 y' : A, R2 y' x & R1 z y'. |