diff options
Diffstat (limited to 'theories/Sets/Relations_1.v')
| -rw-r--r-- | theories/Sets/Relations_1.v | 24 |
1 files changed, 12 insertions, 12 deletions
diff --git a/theories/Sets/Relations_1.v b/theories/Sets/Relations_1.v index f15bf19e6..85d0cffcc 100644 --- a/theories/Sets/Relations_1.v +++ b/theories/Sets/Relations_1.v @@ -28,38 +28,38 @@ 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. |