Copyright	(c) 2015-2020 Rudy Matela
License	3-Clause BSD (see the file LICENSE)
Maintainer	Rudy Matela <rudy@matela.com.br>
Safe Haskell	None
Language	Haskell2010

Test.LeanCheck.Utils.Operators

Contents

Combining properties
Properties of unary functions
Properties of operators (binary functions)
Properties of relations (binary functions returning truth values)
Properties of order relations
Ternary comparison operators
Properties for typeclass instances
Deprecated functions

Description

This module is part of LeanCheck, a simple enumerative property-based testing library.

Some operators for property-based testing.

Synopsis

(==>) :: Bool -> Bool -> Bool
(===) :: Eq b => (a -> b) -> (a -> b) -> a -> Bool
(====) :: Eq c => (a -> b -> c) -> (a -> b -> c) -> a -> b -> Bool
(&&&) :: (a -> Bool) -> (a -> Bool) -> a -> Bool
(&&&&) :: (a -> b -> Bool) -> (a -> b -> Bool) -> a -> b -> Bool
(&&&&&) :: (a -> b -> c -> Bool) -> (a -> b -> c -> Bool) -> a -> b -> c -> Bool
(|||) :: (a -> Bool) -> (a -> Bool) -> a -> Bool
(||||) :: (a -> b -> Bool) -> (a -> b -> Bool) -> a -> b -> Bool
isIdempotent :: Eq a => (a -> a) -> a -> Bool
isIdentity :: Eq a => (a -> a) -> a -> Bool
isNeverIdentity :: Eq a => (a -> a) -> a -> Bool
isCommutative :: Eq b => (a -> a -> b) -> a -> a -> Bool
isAssociative :: Eq a => (a -> a -> a) -> a -> a -> a -> Bool
isDistributiveOver :: Eq a => (a -> a -> a) -> (a -> a -> a) -> a -> a -> a -> Bool
isLeftDistributiveOver :: Eq a => (a -> a -> a) -> (a -> a -> a) -> a -> a -> a -> Bool
isRightDistributiveOver :: Eq a => (a -> a -> a) -> (a -> a -> a) -> a -> a -> a -> Bool
isFlipped :: Eq c => (a -> b -> c) -> (b -> a -> c) -> a -> b -> Bool
isTransitive :: (a -> a -> Bool) -> a -> a -> a -> Bool
isReflexive :: (a -> a -> Bool) -> a -> Bool
isIrreflexive :: (a -> a -> Bool) -> a -> Bool
isSymmetric :: (a -> a -> Bool) -> a -> a -> Bool
isAsymmetric :: (a -> a -> Bool) -> a -> a -> Bool
isAntisymmetric :: Eq a => (a -> a -> Bool) -> a -> a -> Bool
isEquivalence :: (a -> a -> Bool) -> a -> a -> a -> Bool
isPartialOrder :: Eq a => (a -> a -> Bool) -> a -> a -> a -> Bool
isStrictPartialOrder :: (a -> a -> Bool) -> a -> a -> a -> Bool
isTotalOrder :: Eq a => (a -> a -> Bool) -> a -> a -> a -> Bool
isStrictTotalOrder :: Eq a => (a -> a -> Bool) -> a -> a -> a -> Bool
isComparison :: (a -> a -> Ordering) -> a -> a -> a -> Bool
(=$) :: Eq b => a -> (a -> b) -> a -> Bool
($=) :: (a -> Bool) -> a -> Bool
(=|) :: Eq a => [a] -> Int -> [a] -> Bool
(|=) :: (a -> Bool) -> a -> Bool
okEq :: Eq a => a -> a -> a -> Bool
okOrd :: Ord a => a -> a -> a -> Bool
okEqOrd :: (Eq a, Ord a) => a -> a -> a -> Bool
okNum :: (Eq a, Num a) => a -> a -> a -> Bool
okNumNonNegative :: (Eq a, Num a) => a -> a -> a -> Bool
idempotent :: Eq a => (a -> a) -> a -> Bool
identity :: Eq a => (a -> a) -> a -> Bool
neverIdentity :: Eq a => (a -> a) -> a -> Bool
commutative :: Eq b => (a -> a -> b) -> a -> a -> Bool
associative :: Eq a => (a -> a -> a) -> a -> a -> a -> Bool
distributive :: Eq a => (a -> a -> a) -> (a -> a -> a) -> a -> a -> a -> Bool
symmetric2 :: Eq c => (a -> b -> c) -> (b -> a -> c) -> a -> b -> Bool
transitive :: (a -> a -> Bool) -> a -> a -> a -> Bool
reflexive :: (a -> a -> Bool) -> a -> Bool
irreflexive :: (a -> a -> Bool) -> a -> Bool
symmetric :: (a -> a -> Bool) -> a -> a -> Bool
asymmetric :: (a -> a -> Bool) -> a -> a -> Bool
antisymmetric :: Eq a => (a -> a -> Bool) -> a -> a -> Bool
equivalence :: (a -> a -> Bool) -> a -> a -> a -> Bool
partialOrder :: Eq a => (a -> a -> Bool) -> a -> a -> a -> Bool
strictPartialOrder :: (a -> a -> Bool) -> a -> a -> a -> Bool
totalOrder :: Eq a => (a -> a -> Bool) -> a -> a -> a -> Bool
strictTotalOrder :: Eq a => (a -> a -> Bool) -> a -> a -> a -> Bool
comparison :: (a -> a -> Ordering) -> a -> a -> a -> Bool

Combining properties

(==>) :: Bool -> Bool -> Bool infixr 0 Source #

Boolean implication operator. Useful for defining conditional properties:

prop_something x y  =  condition x y ==> something x y

Examples:

> prop_addMonotonic x y  =  y > 0 ==> x + y > x
> check prop_addMonotonic
+++ OK, passed 200 tests.

(===) :: Eq b => (a -> b) -> (a -> b) -> a -> Bool infix 4 Source #

Allows building equality properties between functions.

prop_id_idempotent  =  id === id . id

> check $ id === (+0)
+++ OK, passed 200 tests.

> check $ id === id . id
+++ OK, passed 1 tests (exhausted).

> check $ id === (+1)
*** Failed! Falsifiable (after 1 tests):
0

(====) :: Eq c => (a -> b -> c) -> (a -> b -> c) -> a -> b -> Bool infix 4 Source #

Allows building equality properties between two-argument functions.

> holds 100 $ const ==== asTypeOf
True

> holds 100 $ (+) ==== flip (+)
True

> holds 100 $ (+) ==== (*)
False

(&&&) :: (a -> Bool) -> (a -> Bool) -> a -> Bool infixr 3 Source #

And (&&) operator over one-argument properties.

Allows building conjuntions between one-argument properties:

> holds 100 $ id === (+0) &&& id === (id . id)
True

(&&&&) :: (a -> b -> Bool) -> (a -> b -> Bool) -> a -> b -> Bool infixr 3 Source #

And (&&) operator over two-argument properties.

Allows building conjuntions between two-argument properties:

> holds 100 $ (+) ==== flip (+) &&&& (+) ==== (*)
False

(&&&&&) :: (a -> b -> c -> Bool) -> (a -> b -> c -> Bool) -> a -> b -> c -> Bool infixr 3 Source #

And operator over three-argument properties.

(|||) :: (a -> Bool) -> (a -> Bool) -> a -> Bool infixr 2 Source #

Or (||) operator over one-argument properties.

Allows building disjunctions between one-argument properties:

> holds 100 $ id === (+0) ||| id === (id . id)
True

(||||) :: (a -> b -> Bool) -> (a -> b -> Bool) -> a -> b -> Bool infixr 2 Source #

Or (||) operator over two-argument properties.

Allows building conjuntions between two-argument properties:

> holds 100 $ (+) ==== flip (+) |||| (+) ==== (*)
True

Properties of unary functions

isIdempotent :: Eq a => (a -> a) -> a -> Bool Source #

Is the given function idempotent? f (f x) == x

> check $ isIdempotent abs
+++ OK, passed 200 tests.

> check $ isIdempotent sort
+++ OK, passed 200 tests.

> check $ isIdempotent negate
*** Failed! Falsifiable (after 2 tests):
1

isIdentity :: Eq a => (a -> a) -> a -> Bool Source #

Is the given function an identity? f x == x

> check $ isIdentity (+0)
+++ OK, passed 200 tests.

> check $ isIdentity (sort :: [()]->[()])
+++ OK, passed 200 tests.

> check $ isIdentity (not . not)
+++ OK, passed 2 tests (exhausted).

isNeverIdentity :: Eq a => (a -> a) -> a -> Bool Source #

Is the given function never an identity? f x /= x

> check $ neverIdentity not
+++ OK, passed 2 tests (exhausted).

> check $ neverIdentity negate
*** Failed! Falsifiable (after 1 tests):
0

Note: this is not the same as not being an identity.

Properties of operators (binary functions)

isCommutative :: Eq b => (a -> a -> b) -> a -> a -> Bool Source #

Is a given operator commutative? x + y = y + x

> check $ isCommutative (+)
+++ OK, passed 200 tests.

> import Data.List
> check $ isCommutative (union :: [Int]->[Int]->[Int])
*** Failed! Falsifiable (after 4 tests):
[] [0,0]

isAssociative :: Eq a => (a -> a -> a) -> a -> a -> a -> Bool Source #

Is a given operator associative? x + (y + z) = (x + y) + z

> check $ isAssociative (+)
+++ OK, passed 200 tests.

> check $ isAssociative (-)
*** Failed! Falsifiable (after 2 tests):
0 0 1

isDistributiveOver :: Eq a => (a -> a -> a) -> (a -> a -> a) -> a -> a -> a -> Bool Source #

Does the first operator, left-distributes over the second?

This is an alias to isLeftDistributiveOver.

isLeftDistributiveOver :: Eq a => (a -> a -> a) -> (a -> a -> a) -> a -> a -> a -> Bool Source #

Does the first operator, left-distributes over the second? x * (y + z) = (x * y) + (x * z)

> check $ (*) `isLeftDistributiveOver` (+)
+++ OK, passed 200 tests.

> check $ (+) `isLeftDistributiveOver` (*)
*** Failed! Falsifiable (after 8 tests):
1 0 1

isRightDistributiveOver :: Eq a => (a -> a -> a) -> (a -> a -> a) -> a -> a -> a -> Bool Source #

Does the first operator, right-distributes over the second? (y + z) * x = (y * x) + (z * x)

> check $ (*) `isRightDistributiveOver` (+)
+++ OK, passed 200 tests.

> check $ (+) `isRightDistributiveOver` (*)
*** Failed! Falsifiable (after 8 tests):
1 0 1

isFlipped :: Eq c => (a -> b -> c) -> (b -> a -> c) -> a -> b -> Bool Source #

Are two operators flipped versions of each other?

> check $ ((<) `isFlipped` (>) :: Int -> Int -> Bool)
+++ OK, passed 200 tests.

> check $ ((<=) `isFlipped` (>=) :: Int -> Int -> Bool)
+++ OK, passed 200 tests.

> check $ ((<) `isFlipped` (>=) :: Int -> Int -> Bool)
*** Failed! Falsifiable (after 1 tests):
0 0

> check $ ((<=) `isFlipped` (>) :: Int -> Int -> Bool)
*** Failed! Falsifiable (after 1 tests):
0 0

Properties of relations (binary functions returning truth values)

isTransitive :: (a -> a -> Bool) -> a -> a -> a -> Bool Source #

Is a given relation transitive?

A relation is transitive when if a is related to b then b is related to c.

> check $ isTransitive ((==) :: Int->Int->Bool)
+++ OK, passed 200 tests.

> check $ isTransitive ((/=) :: Int->Int->Bool)
*** Failed! Falsifiable (after 3 tests):
0 1 0

isReflexive :: (a -> a -> Bool) -> a -> Bool Source #

Is a given relation reflexive?

A relation is reflexive when an element is always related to itself.

> check $ isReflexive ((==) :: Int->Int->Bool)
+++ OK, passed 200 tests.

> check $ isReflexive ((/=) :: Int->Int->Bool)
*** Failed! Falsifiable (after 1 tests):
0

isIrreflexive :: (a -> a -> Bool) -> a -> Bool Source #

Is a given relation irreflexive?

A given relation is irreflexive or anti-reflexive when an element is _never_ related to itself.

This is not the negation of isReflexive.

> check $ isIrreflexive ((==) :: Int->Int->Bool)
*** Failed! Falsifiable (after 1 tests):
0

> check $ isIrreflexive ((/=) :: Int->Int->Bool)
+++ OK, passed 200 tests.

isSymmetric :: (a -> a -> Bool) -> a -> a -> Bool Source #

Is a given relation symmetric?

A relation is symmetric when if a is related to b, then b is related to a.

> check $ isSymmetric (&&)
+++ OK, passed 4 tests (exhausted).

> check $ isSymmetric (==>)
*** Failed! Falsifiable (after 2 tests):
False True

This is a type-restricted version of isCommutative.

isAsymmetric :: (a -> a -> Bool) -> a -> a -> Bool Source #

Is a given relation asymmetric?

Not to be confused with not isSymmetric and isAntisymmetric.

> check $ isAsymmetric ((<=) :: Int->Int->Bool)
*** Failed! Falsifiable (after 1 tests):
0 0

> check $ isAsymmetric ((<) :: Int->Int->Bool)
+++ OK, passed 200 tests.

isAntisymmetric :: Eq a => (a -> a -> Bool) -> a -> a -> Bool Source #

Is a given relation antisymmetric?

Not to be confused with isAsymmetric. Not to be confused with the negation of isSymmetric.

> check $ isAntisymmetric ((<=) :: Int->Int->Bool)
+++ OK, passed 200 tests.

> check $ isAntisymmetric ((/=) :: Int->Int->Bool)
*** Failed! Falsifiable (after 2 tests):
0 1

Properties of order relations

isEquivalence :: (a -> a -> Bool) -> a -> a -> a -> Bool Source #

Is the given binary relation an equivalence?

In other words, is the given relation reflexive, symmetric and transitive?

> check (isEquivalence (==) :: Int -> Int -> Int -> Bool)
+++ OK, passed 200 tests.

> check (isEquivalence (<=) :: Int -> Int -> Int -> Bool)
*** Failed! Falsifiable (after 3 tests):
0 1 0

Or, using Test.LeanCheck.Utils.TypeBinding:

> check $ isEquivalence (<=) -:> int
*** Failed! Falsifiable (after 3 tests):
0 1 0

isPartialOrder :: Eq a => (a -> a -> Bool) -> a -> a -> a -> Bool Source #

Is the given binary relation a partial order?

In other words, is the given relation reflexive, antisymmetric and transitive?

> check $ isPartialOrder ((<) :: Int->Int->Bool)
*** Failed! Falsifiable (after 1 tests):
0 0 0

> check $ isPartialOrder ((<=) :: Int->Int->Bool)
+++ OK, passed 200 tests.

> check $ isPartialOrder isSubsetOf
+++ OK, passed 200 tests.

isStrictPartialOrder :: (a -> a -> Bool) -> a -> a -> a -> Bool Source #

Is the given binary relation a strict partial order?

In other words, is the given relation irreflexive, asymmetric and transitive?

> check $ isStrictPartialOrder ((<) :: Int->Int->Bool)
+++ OK, passed 200 tests.

> check $ isStrictPartialOrder ((<=) :: Int->Int->Bool)
*** Failed! Falsifiable (after 1 tests):
0 0 0

isTotalOrder :: Eq a => (a -> a -> Bool) -> a -> a -> a -> Bool Source #

Is the given binary relation a total order?

> check $ isTotalOrder ((<) :: Int->Int->Bool)
*** Failed! Falsifiable (after 1 tests):
0 0 0
> check $ isTotalOrder ((<=) :: Int->Int->Bool)
+++ OK, passed 200 tests.

isStrictTotalOrder :: Eq a => (a -> a -> Bool) -> a -> a -> a -> Bool Source #

Is the given binary relation a strict total order?

> check $ isStrictTotalOrder ((<=) :: Int->Int->Bool)
*** Failed! Falsifiable (after 1 tests):
0 0 0

> check $ isStrictTotalOrder ((<) :: Int->Int->Bool)
+++ OK, passed 200 tests.

isComparison :: (a -> a -> Ordering) -> a -> a -> a -> Bool Source #

Does the given compare function follow the required properties?

This is useful for testing custom Ord instances.

> check $ isComparison (compare :: Int->Int->Ordering)
+++ OK, passed 200 tests.

Ternary comparison operators

(=$) :: Eq b => a -> (a -> b) -> a -> Bool infixl 4 Source #

Equal under, a ternary operator with the same fixity as ==.

x =$ f $= y  =  f x == f y

> [1,2,3,4,5] =$ take 2 $= [1,2,4,8,16]
True

> [1,2,3,4,5] =$ take 3 $= [1,2,4,8,16]
False

> [1,2,3] =$ sort $= [3,2,1]
True

> 42 =$ (`mod` 10) $= 16842
True

> 42 =$ (`mod`  9) $= 16842
False

> 'a' =$ isLetter $= 'b'
True

> 'a' =$ isLetter $= '1'
False

($=) :: (a -> Bool) -> a -> Bool infixl 4 Source #

See =$

(=|) :: Eq a => [a] -> Int -> [a] -> Bool infixl 4 Source #

Check if two lists are equal for n values. This operator has the same fixity of ==.

xs =| n |= ys  =  take n xs == take n ys

[1,2,3,4,5] =| 2 |= [1,2,4,8,16] -- > True
[1,2,3,4,5] =| 3 |= [1,2,4,8,16] -- > False

(|=) :: (a -> Bool) -> a -> Bool infixl 4 Source #

See =|

Properties for typeclass instances

okEq :: Eq a => a -> a -> a -> Bool Source #

Is this Eq instance valid?

This is useful for testing your custom Eq instances against required properties.

In particular, this function tests that == is an equivalence and that /= is the negation of ==.

> check $ (okEq :: Int -> Int -> Int -> Bool)
+++ OK, passed 200 tests.

> check $ (okEq :: Bool -> Bool -> Bool -> Bool)
+++ OK, passed 8 tests (exhausted).

okOrd :: Ord a => a -> a -> a -> Bool Source #

Is this Ord instance valid?

This is useful for testing your custom Ord instances against required properties.

> check $ (okOrd :: Int -> Int -> Int -> Bool)
+++ OK, passed 200 tests.

> check $ (okOrd :: Bool -> Bool -> Bool -> Bool)
+++ OK, passed 8 tests (exhausted).

okEqOrd :: (Eq a, Ord a) => a -> a -> a -> Bool Source #

Is this Eq and Ord instance valid and consistent?

This is useful for testing your custom Eq and Ord instances against required properties.

> check $ (okEqOrd :: Int -> Int -> Int -> Bool)
+++ OK, passed 200 tests.

> check $ (okEqOrd :: Bool -> Bool -> Bool -> Bool)
+++ OK, passed 8 tests (exhausted).

okNum :: (Eq a, Num a) => a -> a -> a -> Bool Source #

Is this Num instance valid?

This is useful for testing your custom Num instances against required properties.

> check (okNum :: Int -> Int -> Int -> Bool)
+++ OK, passed 200 tests.

Double is mostly valid, but not entirely valid:

> check (okNum :: Double -> Double -> Double -> Bool)
*** Failed! Falsifiable (after 6 tests):

0 0.0 Infinity

okNumNonNegative :: (Eq a, Num a) => a -> a -> a -> Bool Source #

Like okNum but restricted to zero and positives.

> check (okNumNonNegative :: Natural -> Natural -> Natural -> Bool)
+++ OK, passed 200 tests.

Deprecated functions

idempotent :: Eq a => (a -> a) -> a -> Bool Source #

Deprecated: Use isIdempotent.

Deprecated: use isIdempotent.

identity :: Eq a => (a -> a) -> a -> Bool Source #

Deprecated: Use isIdentity.

Deprecated: use isIdentity.

neverIdentity :: Eq a => (a -> a) -> a -> Bool Source #

Deprecated: Use isNeverIdentity.

Deprecated: use isNeverIdentity.

commutative :: Eq b => (a -> a -> b) -> a -> a -> Bool Source #

Deprecated: Use isCommutative.

Deprecated: use isCommutative.

associative :: Eq a => (a -> a -> a) -> a -> a -> a -> Bool Source #

Deprecated: Use isAssociative.

Deprecated: use isAssociative.

distributive :: Eq a => (a -> a -> a) -> (a -> a -> a) -> a -> a -> a -> Bool Source #

Deprecated: Use isDistributiveOver.

Deprecated: use isDistributiveOver.

symmetric2 :: Eq c => (a -> b -> c) -> (b -> a -> c) -> a -> b -> Bool Source #

Deprecated: Use isFlipped.

Deprecated: use isFlipped.

transitive :: (a -> a -> Bool) -> a -> a -> a -> Bool Source #

Deprecated: Use isTransitive.

Deprecated: use isTransitive.

reflexive :: (a -> a -> Bool) -> a -> Bool Source #

Deprecated: Use isReflexive.

Deprecated: use isReflexive.

irreflexive :: (a -> a -> Bool) -> a -> Bool Source #

Deprecated: Use isIrreflexive.

Deprecated: use isIrreflexive.

symmetric :: (a -> a -> Bool) -> a -> a -> Bool Source #

Deprecated: Use isSymmetric.

Deprecated: use isSymmetric.

asymmetric :: (a -> a -> Bool) -> a -> a -> Bool Source #

Deprecated: Use isAsymmetric.

Deprecated: use isAsymmetric.

antisymmetric :: Eq a => (a -> a -> Bool) -> a -> a -> Bool Source #

Deprecated: Use isAntisymmetric.

Deprecated: use isAntisymmetric.

equivalence :: (a -> a -> Bool) -> a -> a -> a -> Bool Source #

Deprecated: Use isEquivalence.

Deprecated: use isEquivalence.

partialOrder :: Eq a => (a -> a -> Bool) -> a -> a -> a -> Bool Source #

Deprecated: Use isPartialOrder.

Deprecated: use isPartialOrder.

strictPartialOrder :: (a -> a -> Bool) -> a -> a -> a -> Bool Source #

Deprecated: Use isStrictPartialOrder.

Deprecated: use isStrictPartialOrder.

totalOrder :: Eq a => (a -> a -> Bool) -> a -> a -> a -> Bool Source #

Deprecated: Use isTotalOrder.

Deprecated: use isTotalOrder.

strictTotalOrder :: Eq a => (a -> a -> Bool) -> a -> a -> a -> Bool Source #

Deprecated: Use isStrictTotalOrder.

Deprecated: use isStrictTotalOrder.

comparison :: (a -> a -> Ordering) -> a -> a -> a -> Bool Source #

Deprecated: Use isComparison.

Deprecated: use isComparison.