Safe Haskell | Safe |
---|---|
Language | Haskell2010 |
Data.Prd.Property
Contents
Description
Synopsis
- symmetric :: Prd r => r -> r -> Bool
- coreflexive :: (Eq r, Prd r) => r -> r -> Bool
- reflexive_eq :: Prd r => r -> Bool
- transitive_eq :: Prd r => r -> r -> r -> Bool
- antisymmetric :: Prd r => r -> r -> Bool
- reflexive_le :: Prd r => r -> Bool
- transitive_le :: Prd r => r -> r -> r -> Bool
- connex :: Prd r => r -> r -> Bool
- asymmetric :: Eq r => Prd r => r -> r -> Bool
- transitive_lt :: Eq r => Prd r => r -> r -> r -> Bool
- irreflexive_lt :: Eq r => Prd r => r -> Bool
- semiconnex :: Eq r => Prd r => r -> r -> Bool
- trichotomous :: Eq r => Prd r => r -> r -> Bool
- chain_22 :: Eq r => Prd r => r -> r -> r -> r -> Bool
- chain_31 :: Eq r => Prd r => r -> r -> r -> r -> Bool
Equivalence relations
coreflexive :: (Eq r, Prd r) => r -> r -> Bool Source #
∀x,y:x\eqy⇔x==y
=~
is a coreflexive relation.
See https://en.wikipedia.org/wiki/Reflexive_relation#Related_terms.
This is a required property.
reflexive_eq :: Prd r => r -> Bool Source #
transitive_eq :: Prd r => r -> r -> r -> Bool Source #
Partial orders
Non-strict partial orders
antisymmetric :: Prd r => r -> r -> Bool Source #
reflexive_le :: Prd r => r -> Bool Source #
transitive_le :: Prd r => r -> r -> r -> Bool Source #
Connex non-strict partial orders
Strict partial orders
Semiconnex strict partial orders
semiconnex :: Eq r => Prd r => r -> r -> Bool Source #
∀a,b:¬(a\eqb)⇒((a<b)∨(b<a))
lt
is a semiconnex relation.
trichotomous :: Eq r => Prd r => r -> r -> Bool Source #
∀a,b,c:((a<b)∨(a\eqb)∨(b<a))∧¬((a<b)∧(a\eqb)∧(b<a))
In other words, exactly one of a<b, a\eqb, or b<a holds.
If lt
is a trichotomous relation then the set is totally ordered.