Checks a property printing results on stdout

> check $ \xs -> sort (sort xs) == sort (xs::[Int])
+++ OK, passed 360 tests.

> check $ \xs ys -> xs `union` ys == ys `union` (xs::[Int])
*** Failed! Falsifiable (after 4 tests):
[] [0,0]

Generalization:
[] (x:x:_)

Check a property printing results on stdout and returning True on success.

There is no option to silence this function: for silence, you should use holds.

Use for to configure the number of tests performed by check.

> check `for` 10080 $ \xs -> sort (sort xs) == sort (xs :: [Int])
+++ OK, passed 10080 tests.

Don't forget the dollar ($)!

Use withBackground to provide additional functions to appear in side-conditions.

check `withBackground` [value "isSpace" isSpace] $ \xs -> unwords (words xs) == xs
*** Failed! Falsifiable (after 4 tests):
" "

Generalization:
' ':_

Conditional Generalization:
c:_  when  isSpace c

Use withConditionSize to configure the maximum condition size allowed.

Extrapolate can generalize counter-examples of any types that are Generalizable.

Generalizable types must first be instances of

• Listable, so Extrapolate knows how to enumerate values;
• Express, so Extrapolate can represent then manipulate values;
• Name, so Extrapolate knows how to name variables.

There are no required functions, so we can define instances with:

instance Generalizable OurType

However, it is desirable to define both background and subInstances.

The following example shows a datatype and its instance:

data Stack a = Stack a (Stack a) | Empty

instance Generalizable a => Generalizable (Stack a) where
  subInstances s  =  instances (argTy1of1 s)

To declare instances it may be useful to use type binding operators such as: argTy1of1, argTy1of2 and argTy2of2.

Instances can be automatically derived using deriveGeneralizable which will also automatically derivate Listable, Express and Name when needed.

Used in the definition of subInstances in Generalizable typeclass instances.

Values of type Expr represent objects or applications between objects. Each object is encapsulated together with its type and string representation. Values encoded in Exprs are always monomorphic.

An Expr can be constructed using:

• val, for values that are Show instances;
• value, for values that are not Show instances, like functions;
• :$, for applications between Exprs.

> val False
False :: Bool

> value "not" not :$ val False
not False :: Bool

An Expr can be evaluated using evaluate, eval or evl.

> evl $ val (1 :: Int) :: Int
1

> evaluate $ val (1 :: Int) :: Maybe Bool
Nothing

> eval 'a' (val 'b')
'b'

Showing a value of type Expr will return a pretty-printed representation of the expression together with its type.

> show (value "not" not :$ val False)
"not False :: Bool"

Expr is like Dynamic but has support for applications and variables (:$, var).

The var underscore convention: Functions that manipulate Exprs usually follow the convention where a value whose String representation starts with '_' represents a variable.

O(1). It takes a string representation of a value and a value, returning an Expr with that terminal value. For instances of Show, it is preferable to use val.

> value "0" (0 :: Integer)
0 :: Integer

> value "'a'" 'a'
'a' :: Char

> value "True" True
True :: Bool

> value "id" (id :: Int -> Int)
id :: Int -> Int

> value "(+)" ((+) :: Int -> Int -> Int)
(+) :: Int -> Int -> Int

> value "sort" (sort :: [Bool] -> [Bool])
sort :: [Bool] -> [Bool]

O(1). A shorthand for value for values that are Show instances.

> val (0 :: Int)
0 :: Int

> val 'a'
'a' :: Char

> val True
True :: Bool

Example equivalences to value:

val 0     =  value "0" 0
val 'a'   =  value "'a'" 'a'
val True  =  value "True" True

O(1). Reifies an Eq instance into a list of Exprs. The list will contain == and /= for the given type. (cf. mkEq, mkEquation)

> reifyEq (undefined :: Int)
[ (==) :: Int -> Int -> Bool
, (/=) :: Int -> Int -> Bool ]

> reifyEq (undefined :: Bool)
[ (==) :: Bool -> Bool -> Bool
, (/=) :: Bool -> Bool -> Bool ]

> reifyEq (undefined :: String)
[ (==) :: [Char] -> [Char] -> Bool
, (/=) :: [Char] -> [Char] -> Bool ]

O(1). Reifies an Ord instance into a list of Exprs. The list will contain compare, <= and < for the given type. (cf. mkOrd, mkOrdLessEqual, mkComparisonLE, mkComparisonLT)

> reifyOrd (undefined :: Int)
[ (<=) :: Int -> Int -> Bool
, (<) :: Int -> Int -> Bool ]

> reifyOrd (undefined :: Bool)
[ (<=) :: Bool -> Bool -> Bool
, (<) :: Bool -> Bool -> Bool ]

> reifyOrd (undefined :: [Bool])
[ (<=) :: [Bool] -> [Bool] -> Bool
, (<) :: [Bool] -> [Bool] -> Bool ]

O(1). Reifies Eq and Ord instances into a list of Expr.

Derives a Generalizable instance for a given type Name.

If needed, this function also automatically derivates Listable, Express and Name instances using respectively deriveListable, deriveExpress and deriveName.

Consider the following Stack datatype:

data Stack a = Stack a (Stack a) | Empty

Writing

deriveGeneralizable ''Stack

will automatically derive the following Generalizable instance:

instance Generalizable a => Generalizable (Stack a) where
  instances s = this "s" s
              $ let Stack x y = Stack undefined undefined `asTypeOf` s
                in instances x
                 . instances y

This function needs the TemplateHaskell extension.

Same as deriveGeneralizable but does not warn when instance already exists (deriveGeneralizable is preferable).

Derives a Generalizable instance for a given type Name cascading derivation of type arguments as well.

If we were to come up with a variable name for the given type what name would it be?

An instance for a given type Ty is simply given by:

instance Name Ty where name _ = "x"

Examples:

> name (undefined :: Int)
"x"

> name (undefined :: Bool)
"p"

> name (undefined :: [Int])
"xs"

This is then used to generate an infinite list of variable names:

> names (undefined :: Int)
["x", "y", "z", "x'", "y'", "z'", "x''", "y''", "z''", ...]

> names (undefined :: Bool)
["p", "q", "r", "p'", "q'", "r'", "p''", "q''", "r''", ...]

> names (undefined :: [Int])
["xs", "ys", "zs", "xs'", "ys'", "zs'", "xs''", "ys''", ...]

Express typeclass instances provide an expr function that allows values to be deeply encoded as applications of Exprs.

expr False  =  val False
expr (Just True)  =  value "Just" (Just :: Bool -> Maybe Bool) :$ val True

The function expr can be contrasted with the function val:

• val always encodes values as atomic Value Exprs -- shallow encoding.
• expr ideally encodes expressions as applications (:$) between Value Exprs -- deep encoding.

Depending on the situation, one or the other may be desirable.

Instances can be automatically derived using the TH function deriveExpress.

The following example shows a datatype and its instance:

data Stack a = Stack a (Stack a) | Empty

instance Express a => Express (Stack a) where
  expr s@(Stack x y) = value "Stack" (Stack ->>: s) :$ expr x :$ expr y
  expr s@Empty       = value "Empty" (Empty   -: s)

To declare expr it may be useful to use auxiliary type binding operators: -:, ->:, ->>:, ->>>:, ->>>>:, ->>>>>:, ...

For types with atomic values, just declare expr = val

Takes as argument an integer length and tiers of element values; returns tiers of lists of element values of the given length.

listsOfLength 3 [[0],[1],[2],[3],[4]...] =
  [ [[0,0,0]]
  , [[0,0,1],[0,1,0],[1,0,0]]
  , [[0,0,2],[0,1,1],[0,2,0],[1,0,1],[1,1,0],[2,0,0]]
  , ...
  ]

Takes as argument tiers of element values; returns tiers of size-ordered lists of elements without repetition.

setsOf [[0],[1],[2],...] =
  [ [[]]
  , [[0]]
  , [[1]]
  , [[0,1],[2]]
  , [[0,2],[3]]
  , [[0,3],[1,2],[4]]
  , [[0,1,2],[0,4],[1,3],[5]]
  , ...
  ]

Can be used in the constructor of specialized Listable instances. For Set (from Data.Set), we would have:

instance Listable a => Listable (Set a) where
  tiers  =  mapT fromList $ setsOf tiers

Takes as argument tiers of element values; returns tiers of size-ordered lists of elements possibly with repetition.

bagsOf [[0],[1],[2],...] =
  [ [[]]
  , [[0]]
  , [[0,0],[1]]
  , [[0,0,0],[0,1],[2]]
  , [[0,0,0,0],[0,0,1],[0,2],[1,1],[3]]
  , [[0,0,0,0,0],[0,0,0,1],[0,0,2],[0,1,1],[0,3],[1,2],[4]]
  , ...
  ]

Takes as argument tiers of element values; returns tiers of lists with no repeated elements.

noDupListsOf [[0],[1],[2],...] ==
  [ [[]]
  , [[0]]
  , [[1]]
  , [[0,1],[1,0],[2]]
  , [[0,2],[2,0],[3]]
  , ...
  ]

Normalizes tiers by removing up to 12 empty tiers from the end of a list of tiers.

normalizeT [xs0,xs1,...,xsN,[]]     =  [xs0,xs1,...,xsN]
normalizeT [xs0,xs1,...,xsN,[],[]]  =  [xs0,xs1,...,xsN]

The arbitrary limit of 12 tiers is necessary as this function would loop if there is an infinite trail of empty tiers.

Delete the first occurence of an element in a tier.

For normalized lists-of-tiers without repetitions, the following holds:

deleteT x  =  normalizeT . (`suchThat` (/= x))

Takes the product of N lists of tiers, producing lists of length N.

Alternatively, takes as argument a list of lists of tiers of elements; returns lists combining elements of each list of tiers.

products [xss]  =  mapT (:[]) xss
products [xss,yss]  =  mapT (\(x,y) -> [x,y]) (xss >< yss)
products [xss,yss,zss]  =  product3With (\x y z -> [x,y,z]) xss yss zss

Takes as argument tiers of element values; returns tiers of lists of elements.

listsOf [[]]  =  [[[]]]

listsOf [[x]]  =  [ [[]]
                  , [[x]]
                  , [[x,x]]
                  , [[x,x,x]]
                  , ...
                  ]

listsOf [[x],[y]]  =  [ [[]]
                      , [[x]]
                      , [[x,x],[y]]
                      , [[x,x,x],[x,y],[y,x]]
                      , ...
                      ]

Take the product of lists of tiers by a function returning a Maybe value discarding Nothing values.

Like productWith, but over 3 lists of tiers.

Given a constructor that takes a list with no duplicate elements, return tiers of applications of this constructor.

Given a constructor that takes a map of elements (encoded as a list), lists tiers of applications of this constructor

So long as the underlying Listable enumerations have no repetitions, this will generate no repetitions.

This allows defining an efficient implementation of tiers that does not repeat maps given by:

  tiers  =  mapCons fromList

Given a constructor that takes a set of elements (as a list), lists tiers of applications of this constructor.

A naive Listable instance for the Set (of Data.Set) would read:

instance Listable a => Listable (Set a) where
  tiers  =  cons0 empty \/ cons2 insert

The above instance has a problem: it generates repeated sets. A more efficient implementation that does not repeat sets is given by:

  tiers  =  setCons fromList

Alternatively, you can use setsOf direclty.

Given a constructor that takes a bag of elements (as a list), lists tiers of applications of this constructor.

For example, a Bag represented as a list.

bagCons Bag

Derives a Listable instance for a given type Name cascading derivation of type arguments as well.

Consider the following series of datatypes:

data Position  =  CEO | Manager | Programmer

data Person  =  Person
             {  name :: String
             ,  age :: Int
             ,  position :: Position
             }

data Company  =  Company
              {  name :: String
              ,  employees :: [Person]
              }

Writing

deriveListableCascading ''Company

will automatically derive the following three Listable instances:

instance Listable Position where
  tiers  =  cons0 CEO \/ cons0 Manager \/ cons0 Programmer

instance Listable Person where
  tiers  =  cons3 Person

instance Listable Company where
  tiers  =  cons2 Company

Derives a Listable instance for a given type Name.

Consider the following Stack datatype:

data Stack a  =  Stack a (Stack a) | Empty

Writing

deriveListable ''Stack

will automatically derive the following Listable instance:

instance Listable a => Listable (Stack a) where
  tiers  =  cons2 Stack \/ cons0 Empty

Warning: if the values in your type need to follow a data invariant, the derived instance won't respect it. Use this only on "free" datatypes.

Needs the TemplateHaskell extension.

Adds to the weight of a constructor or tiers.

instance Listable <Type> where
  tiers  =  ...
         \/ cons<N> <Cons>  `addWeight`  <W>
         \/ ...

Typically used as an infix operator when defining Listable instances:

> [ xs, ys, zs, ... ] `addWeight` 1
[ [], xs, ys, zs, ... ]

> [ xs, ys, zs, ... ] `addWeight` 2
[ [], [], xs, ys, zs, ... ]

> [ [], xs, ys, zs, ... ] `addWeight` 3
[ [], [], [], [], xs, ys, zs, ... ]

`addWeight` n is equivalent to n applications of delay.

Resets the weight of a constructor or tiers.

> [ [], [], ..., xs, ys, zs, ... ] `ofWeight` 1
[ [], xs, ys, zs, ... ]

> [ xs, ys, zs, ... ] `ofWeight` 2
[ [], [], xs, ys, zs, ... ]

> [ [], xs, ys, zs, ... ] `ofWeight` 3
[ [], [], [], xs, ys, zs, ... ]

Typically used as an infix operator when defining Listable instances:

instance Listable <Type> where
  tiers  =  ...
         \/ cons<N> <Cons>  `ofWeight`  <W>
         \/ ...

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

`ofWeight` n is equivalent to reset followed by n applications of delay.

Returns tiers of applications of a 12-argument constructor.

Returns tiers of applications of a 11-argument constructor.

Returns tiers of applications of a 10-argument constructor.

Returns tiers of applications of a 9-argument constructor.

Returns tiers of applications of a 8-argument constructor.

Returns tiers of applications of a 7-argument constructor.

Returns tiers of applications of a 6-argument constructor.

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.

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

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

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.

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)

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"]]

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)

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

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

Append tiers --- sum of two tiers enumerations.

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

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 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.

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).

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>)
         \/ ...

Returns tiers of applications of a 5-argument constructor.

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

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 3-argument constructor.

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

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

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 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>
         \/ ...

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)

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)

filter tiers

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

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

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]

Tiers of Floating values. This can be used as the implementation of tiers for Floating types.

This function is equivalent to tiersFractional with positive and negative infinities included: 10 and -10.

tiersFloating :: [[Float]]  =
  [ [0.0]
  , [1.0]
  , [-1.0, Infinity]
  , [ 0.5,  2.0, -Infinity]
  , [-0.5, -2.0]
  , [ 0.33333334,  3.0]
  , [-0.33333334, -3.0]
  , [ 0.25,  0.6666667,  1.5,  4.0]
  , [-0.25, -0.6666667, -1.5, -4.0]
  , [ 0.2,  5.0]
  , [-0.2, -5.0]
  , [ 0.16666667,  0.4,  0.75,  1.3333334,  2.5,  6.0]
  , [-0.16666667, -0.4, -0.75, -1.3333334, -2.5, -6.0]
  , ...
  ]

NaN and -0 are excluded from this enumeration.

Tiers of Fractional values. This can be used as the implementation of tiers for Fractional types.

tiersFractional :: [[Rational]]  =
  [ [  0  % 1]
  , [  1  % 1]
  , [(-1) % 1]
  , [  1  % 2,   2  % 1]
  , [(-1) % 2, (-2) % 1]
  , [  1  % 3,   3  % 1]
  , [(-1) % 3, (-3) % 1]
  , [  1  % 4,   2  % 3,   3  % 2,   4  % 1]
  , [(-1) % 4, (-2) % 3, (-3) % 2, (-4) % 1]
  , [  1  % 5,   5  % 1]
  , [(-1) % 5, (-5) % 1]
  , [  1  % 6,   2 % 5,    3  % 4,   4  % 3,   5  % 2,   6  % 1]
  , [(-1) % 6, (-2) % 5, (-3) % 4, (-4) % 3, (-5) % 2, (-6) % 1]
  , ...
  ]

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.

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.

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.