Does a property hold up to a number of test values?

> holds 1000 $ \xs -> length (sort xs) == length xs
True

> holds 1000 $ \x -> x == x + 1
False

The suggested number of test values are 500, 1 000 or 10 000. With more than that you may or may not run out of memory depending on the types being tested. This also applies to fails, exists, etc.

(cf. fails, counterExample)

Does a property fail for a number of test values?

> fails 1000 $ \xs -> xs ++ ys == ys ++ xs
True

> holds 1000 $ \xs -> length (sort xs) == length xs
False

This is the negation of holds.

There exists an assignment of values that satisfies a property up to a number of test values?

> exists 1000 $ \x -> x > 10
True

Up to a number of tests to a property, returns Just the first counter-example or Nothing if there is none.

> counterExample 100 $ \xs -> [] `union` xs == (xs::[Int])
Just ["[0,0]"]

Lists all counter-examples for a number of tests to a property,

> counterExamples 12 $ \xs -> xs == nub (xs :: [Int])
[["[0,0]"],["[0,0,0]"],["[0,0,0,0]"],["[0,0,1]"],["[0,1,0]"]]

Up to a number of tests to a property, returns Just the first witness or Nothing if there is none.

> witness 1000 (\x -> x > 1 && x < 77 && 77 `rem` x == 0)
Just ["7"]

Lists all witnesses up to a number of tests to a property.

> witnesses 1000 (\x -> x > 1 && x < 77 && 77 `rem` x == 0)
[["7"],["11"]]

Testable values are functions of Listable arguments that return boolean values.

•  Bool

•  Listable a => a -> Bool

•  (Listable a, Listable b) => a -> b -> Bool

•  (Listable a, Listable b, Listable c) => a -> b -> c -> Bool

•  (Listable a, Listable b, Listable c, ...) => a -> b -> c -> ... -> Bool

For example:

•  Int -> Bool

•  String -> [Int] -> Bool

(cf. results)

List all results of a Testable property. Each result is a pair of a list of strings and a boolean. The list of strings is a printable representation of one possible choice of argument values for the property. Each boolean paired with such a list indicates whether the property holds for this choice. The outer list is potentially infinite and lazily evaluated.

> results (<)
[ (["0","0"],    False)
, (["0","1"],    True)
, (["1","0"],    False)
, (["0","(-1)"], False)
, (["1","1"],    False)
, (["(-1)","0"], True)
, (["0","2"],    True)
, (["1","(-1)"], False)
, ...
]

> take 10 $ results (\xs -> xs == nub (xs :: [Int]))
[ (["[]"],      True)
, (["[0]"],     True)
, (["[0,0]"],   False)
, (["[1]"],     True)
, (["[0,0,0]"], False)
, ...
]

A type is Listable when there exists a function that is able to list (ideally all of) its values.

Ideally, instances should be defined by a tiers function that returns a (potentially infinite) list of finite sub-lists (tiers): the first sub-list contains elements of size 0, the second sub-list contains elements of size 1 and so on. Size here is defined by the implementor of the type-class instance.

For algebraic data types, the general form for tiers is

tiers  =  cons<N> ConstructorA
       \/ cons<N> ConstructorB
       \/ ...
       \/ cons<N> ConstructorZ

where N is the number of arguments of each constructor A...Z.

Here is a datatype with 4 constructors and its listable instance:

data MyType  =  MyConsA
             |  MyConsB Int
             |  MyConsC Int Char
             |  MyConsD String

instance Listable MyType where
  tiers =  cons0 MyConsA
        \/ cons1 MyConsB
        \/ cons2 MyConsC
        \/ cons1 MyConsD

The instance for Hutton's Razor is given by:

data Expr  =  Val Int
           |  Add Expr Expr

instance Listable Expr where
  tiers  =  cons1 Val
         \/ cons2 Add

Instances can be alternatively defined by list. In this case, each sub-list in tiers is a singleton list (each succeeding element of list has +1 size).

The function deriveListable from Test.LeanCheck.Derive can automatically derive instances of this typeclass.

A Listable instance for functions is also available but is not exported by default. Import Test.LeanCheck.Function if you need to test higher-order properties.

Given a constructor with no arguments, returns tiers of all possible applications of this constructor.

Since in this case there is only one possible application (to no arguments), only a single value, of size/weight 0, will be present in the resulting list of tiers.

To be used in the declaration of tiers in Listable instances.

instance Listable <Type> where
  tiers  =  ...
         \/ cons0 <Constructor>
         \/ ...

Given a constructor with one Listable argument, return tiers of applications of this constructor.

By default, returned values will have size/weight of 1.

To be used in the declaration of tiers in Listable instances.

instance Listable <Type> where
  tiers  =  ...
         \/ cons1 <Constructor>
         \/ ...

Given a constructor with two Listable arguments, return tiers of applications of this constructor.

By default, returned values will have size/weight of 1.

To be used in the declaration of tiers in Listable instances.

instance Listable <Type> where
  tiers  =  ...
         \/ cons2 <Constructor>
         \/ ...

Returns tiers of applications of a 3-argument constructor.

To be used in the declaration of tiers in Listable instances.

instance Listable <Type> where
  tiers  =  ...
         \/ cons3 <Constructor>
         \/ ...

Returns tiers of applications of a 4-argument constructor.

To be used in the declaration of tiers in Listable instances.

Returns tiers of applications of a 5-argument constructor.

To be used in the declaration of tiers in Listable instances.

Delays the enumeration of tiers. Conceptually this function adds to the weight of a constructor.

delay [xs, ys, zs, ... ]  =  [[], xs, ys, zs, ...]

delay [[x,...], [y,...], ...]  =  [[], [x,...], [y,...], ...]

Typically used when defining Listable instances:

instance Listable <Type> where
  tiers  =  ...
         \/ delay (cons<N> <Constructor>)
         \/ ...

Resets any delays in a list-of tiers. Conceptually this function makes a constructor "weightless", assuring the first tier is non-empty.

reset [[], [], ..., xs, ys, zs, ...]  =  [xs, ys, zs, ...]

reset [[], xs, ys, zs, ...]  =  [xs, ys, zs, ...]

reset [[], [], ..., [x], [y], [z], ...]  =  [[x], [y], [z], ...]

Typically used when defining Listable instances:

instance Listable <Type> where
  tiers  =  ...
         \/ reset (cons<N> <Constructor>)
         \/ ...

Be careful: do not apply reset to recursive data structure constructors. In general this will make the list of size 0 infinite, breaking the tiers invariant (each tier must be finite).

Tiers of values that follow a property.

Typically used in the definition of Listable tiers:

instance Listable <Type> where
  tiers  =  ...
         \/ cons<N> `suchThat` <condition>
         \/ ...

Examples:

> tiers `suchThat` odd
[[], [1], [-1], [], [], [3], [-3], [], [], [5], ...]

> tiers `suchThat` even
[[0], [], [], [2], [-2], [], [], [4], [-4], [], ...]

This function is just a flipped version of filterT.

Append tiers --- sum of two tiers enumerations.

[xs,ys,zs,...] \/ [as,bs,cs,...]  =  [xs++as, ys++bs, zs++cs, ...]

Interleave tiers --- sum of two tiers enumerations. When in doubt, use \/ instead.

[xs,ys,zs,...] \/ [as,bs,cs,...]  =  [xs+|as, ys+|bs, zs+|cs, ...]

Take a tiered product of lists of tiers.

[t0,t1,t2,...] >< [u0,u1,u2,...]  =
  [ t0**u0
  , t0**u1 ++ t1**u0
  , t0**u2 ++ t1**u1 ++ t2**u0
  , ...       ...       ...       ...
  ]
  where
  xs ** ys  =  [(x,y) | x <- xs, y <- ys]

Example:

[[0],[1],[2],...] >< [[0],[1],[2],...]  =
  [ [(0,0)]
  , [(1,0),(0,1)]
  , [(2,0),(1,1),(0,2)]
  , [(3,0),(2,1),(1,2),(0,3)]
  , ...
  ]

(cf. productWith)

Take a tiered product of lists of tiers. productWith can be defined by ><, as:

productWith f xss yss  =  map (uncurry f) $ xss >< yss

(cf. ><)

map over tiers

mapT f [[x], [y,z], [w,...], ...]  =  [[f x], [f y, f z], [f w, ...], ...]

mapT f [xs, ys, zs, ...]  =  [map f xs, map f ys, map f zs]

filter tiers

filterT p [xs, yz, zs, ...]  =  [filter p xs, filter p ys, filter p zs]

filterT odd tiers  =  [[], [1], [-1], [], [], [3], [-3], [], [], [5], ...]

concat tiers of tiers

concatT [ [xss0, yss0, zss0, ...]
        , [xss1, yss1, zss1, ...]
        , [xss2, yss2, zss2, ...]
        , ...
        ]
  =  xss0 \/ yss0 \/ zss0 \/ ...
          \/ delay (xss1 \/ yss1 \/ zss1 \/ ...
                         \/ delay (xss2 \/ yss2 \/ zss2 \/ ...
                                        \/ (delay ...)))

(cf. concatMapT)

concatMap over tiers

concatMapT f [ [x0, y0, z0]
             , [x1, y1, z1]
             , [x2, y2, z2]
             , ...
             ]
  =  f x0 \/ f y0 \/ f z0 \/ ...
          \/ delay (f x1 \/ f y1 \/ f z1 \/ ...
                         \/ delay (f x2 \/ f y2 \/ f z2 \/ ...
                                        \/ (delay ...)))

(cf. concatT)

Takes a list of values xs and transform it into tiers on which each tier is occupied by a single element from xs.

toTiers [x, y, z, ...]  =  [[x], [y], [z], ...]

To convert back to a list, just concat.

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.

Lazily interleaves two lists, switching between elements of the two. Union/sum of the elements in the lists.

[x,y,z,...] +| [a,b,c,...]  =  [x,a,y,b,z,c,...]

Tiers of Integral values. Can be used as a default implementation of list for Integral types.

For types with negative values, like Int, the list starts with 0 then intercalates between positives and negatives.

listIntegral  =  [0, 1, -1, 2, -2, 3, -3, 4, -4, ...]

For types without negative values, like Word, the list starts with 0 followed by positives of increasing magnitude.

listIntegral  =  [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...]

This function will not work for types that throw errors when the result of an arithmetic operation is negative such as Natural. For these, use [0..] as the list implementation.

Listing of Fractional values. This can be used as the implementation of list for Fractional types.

listFractional :: [[Rational]]  =
  [0 % 1, 1 % 1, (-1) % 1, 1 % 2, (-1) % 2, 2 % 1, (-2) % 1, 1 % 3, ...]

All rationals are included without repetition in their most simple form. This is the Calkin-Wilf sequence computed with the help of the fusc function (EWD 570).

This also works for unsigned types that wrap around zero, yielding:

listFractional :: [Ratio Word]  =
  [0 % 1, 1 % 1, 1 % 2, 2 % 1, 1 % 3, 3 % 2, 2 % 3, 3 % 1, 1 % 4, ...]

Listing of Floating values. This can be used as the implementation of list for Floating types.

listFloating :: [Double]  =
  [0.0, 1.0, -1.0, 0.5, -0.5, 2.0, Infinity, -Infinity, -2.0, 0.333, ...]

This follow the same Calkin-Wilf sequence of listFractional but positive and negative infinities are artificially included after two.

NaN and -0 are excluded from this enumeration.