| 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
symmetric :: Prd r => r -> r -> Bool Source #
\( \forall a, b: (a \eq b) \Leftrightarrow (b \eq a) \)
=~ is a symmetric relation.
This is a required property.
coreflexive :: (Eq r, Prd r) => r -> r -> Bool Source #
\( \forall x, y: x \eq y \Leftrightarrow x == y \)
=~ 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 #
\( \forall a, b, c: ((a \eq b) \wedge (b \eq c)) \Rightarrow (a \eq c) \)
=~ is a transitive relation.
This is a required property.
Partial orders
Non-strict partial orders
antisymmetric :: Prd r => r -> r -> Bool Source #
\( \forall a, b: (a \leq b) \wedge (b \leq a) \Rightarrow a \eq b \)
<~ is an antisymmetric relation.
This is a required property.
reflexive_le :: Prd r => r -> Bool Source #
transitive_le :: Prd r => r -> r -> r -> Bool Source #
\( \forall a, b, c: ((a \leq b) \wedge (b \leq c)) \Rightarrow (a \leq c) \)
<~ is an transitive relation.
This is a required property.
Connex non-strict partial orders
connex :: Prd r => r -> r -> Bool Source #
\( \forall a, b: ((a \leq b) \vee (b \leq a)) \)
<~ is a connex relation.
Strict partial orders
asymmetric :: Eq r => Prd r => r -> r -> Bool Source #
\( \forall a, b: (a \lt b) \Rightarrow \neg (b \lt a) \)
lt is an asymmetric relation.
This is a required property.
transitive_lt :: Eq r => Prd r => r -> r -> r -> Bool Source #
\( \forall a, b, c: ((a \lt b) \wedge (b \lt c)) \Rightarrow (a \lt c) \)
lt is a transitive relation.
This is a required property.
Semiconnex strict partial orders
semiconnex :: Eq r => Prd r => r -> r -> Bool Source #
\( \forall a, b: \neg (a \eq b) \Rightarrow ((a \lt b) \vee (b \lt a)) \)
lt is a semiconnex relation.
trichotomous :: Eq r => Prd r => r -> r -> Bool Source #
\( \forall a, b, c: ((a \lt b) \vee (a \eq b) \vee (b \lt a)) \wedge \neg ((a \lt b) \wedge (a \eq b) \wedge (b \lt a)) \)
In other words, exactly one of \(a \lt b\), \(a \eq b\), or \(b \lt a\) holds.
If lt is a trichotomous relation then the set is totally ordered.