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Speculate automatically discovers laws about Haskell functions. Give Speculate a bunch of Haskell functions and it will discover laws like:

  • equations, such as id x == x;
  • relations of order, such as 0 <= x * x;
  • conditional equations, such as x <= 0 ==> x + abs x == 0.

Speculate is similar to, and inspired by, QuickSpec.

Installing Speculate

To install the latest Speculate version from Hackage, just:

$ cabal update
$ cabal install speculate

Pre-requisites are cmdargs, express and leancheck. They should be automatically resolved and installed by Cabal.

Using Speculate

Speculate is used as a library: import it, then call the function speculate with relevant arguments. The following program Speculates about the functions (+) and abs:

import Test.Speculate

main :: IO ()
main = speculate args
  { constants =
      [ showConstant (0::Int)
      , showConstant (1::Int)
      , constant "+"   ((+)  :: Int -> Int -> Int)
      , constant "abs" (abs  :: Int -> Int)

when run, it prints the following:

_ :: Int  (holes: Int)
0 :: Int
1 :: Int
(+) :: Int -> Int -> Int
abs :: Int -> Int

    abs (abs x) == abs x
          x + 0 == x
          x + y == y + x
    (x + y) + z == x + (y + z)
abs (x + abs x) == x + abs x
  abs x + abs x == abs (x + x)
abs (1 + abs x) == 1 + abs x

x <= abs x
0 <= abs x
x <= x + 1

Now, if we add <= and < as background constants on args

  , constants =
      [ showConstant (0::Int)
      , showConstant (1::Int)
      , constant "+"   ((+)  :: Int -> Int -> Int)
      , constant "abs" (abs  :: Int -> Int)
      , background
      , constant "<="  ((<=) :: Int -> Int -> Bool)
      , constant "<"   ((<)  :: Int -> Int -> Bool)

then run again, we get the following as well:

    y <= x ==> abs (x + abs y) == x + abs y
    x <= 0 ==>       x + abs x == 0
abs x <= y ==>     abs (x + y) == x + y
abs y <= x ==>     abs (x + y) == x + y

For more examples, see the eg folder.

(One can use the TypeApplications to simplify the above examples: ((+) @ Int) instead of ((+) :: Int -> Int -> Int)). I have chosen to keep the example Haskell 98 compliant.)

Supported types

Speculate works for virtually any type. However, if you would like to produce equations, comparisons and variables of any given type this type must be respectively an instance of the Eq, Ord, Listable and Name typeclasses.

By default, Speculate will produce equations, comparison and variables to a few types in the Haskell 2010 Language Report. If you would like expand that to more types, you need to pass reified instances to Speculate explicitly by using reifyInstances on instances = of speculate’s args like so:

main = speculate args
  { instances = [ reifyInstances (undefined :: <Type1>)
                , reifyInstances (undefined :: <Type2>)
                , reifyInstances (undefined :: <Type3>)
                , ...
  , constants = ...
  , ...

To use reifyInstances, your type must be an instance of Eq, Ord, Listable and Name.

  • Eq is needed for equations between values of the type;

  • Ord is needed for comparisons between values of the type;

  • Listable is needed for involving variables of the type. This is needed in order for Speculate to be able to generate values of your type to replace any variables. LeanCheck comes with Listable instances for virtually all types in the Haskell 2010 Language Report.

  • Name is needed for cosmetic puposes: if there are any variables of your type, Name allows you to tell Speculate how to call your variables. For example, if you have an User type, you can define your name instance as:

      instance Name (User) where
      	name u  =  "usr"

    This way, variables of your User type will be called: usr, usr1, usr2, usr3, etc.

It is also fine to have only one, two or three of the above instances. In that case, instead of reifyInstances you can use reifyEq, reifyOrd, reifyListable and reifyName accordingly. If you do not provide a Name implementation, your variables will default to being x, y and z. This may cause confusion as you involve more and more types, compare the following two identical equations:

[x,y] `areOwnedBy` z  ==  z `owns` x && z `owns` y
[tckt,tckt1] `areOwnedBy` user  ==  usr `owns` tckt && user `owns tckt1`

The second is clearer. So, I recomment you add a Name instance. It is simple enough.

You also have to do this for any user defined types you are using or even for newtypes.

Speculate comes with a few examples illustrating the use of reifyInstances: on the eg folder: eg/algebraic-graphs.hs, eg/binarytree0.hs, eg/binarytree.hs, eg/colour.hs, eg/digraphs.hs, eg/fun.hs, eg/monad.hs, eg/pretty-compact.hs, eg/pretty.hs, eg/regexes.hs, eg/sets.hs, eg/speculate-reason.hs, eg/string.hs, eg/tauts.hs, eg/tuples.hs, eg/zip.hs.

Not having the reified instances for a given type will cause the following warnings to be printed:

Warning: no Listable instance for <YourTypeHere>, variables of this type will not be considered
Warning: no Listable instance for <YourTypeHere>, variables of this type will not be considered
Warning: no Eq instance for <YourTypeHere>, equations of this type will not be considered
Warning: no Eq instance for <YourTypeHere>, equations of this type will not be considered
Warning: no Ord instance for <YourTypeHere>, inequations of this type will not be considered
Warning: no Ord instance for <YourTypeHere>, inequations of this type will not be considered

You can silence the above warnings by following the instructions above. However, it may be the case that you don’t want variables, equations or comparisons for a given type. If that is so, you can ignore these warnings.

Similarities and Differences to QuickSpec

Speculate is inspired by QuickSpec. Like QuickSpec, Speculate uses testing to speculate equational laws about given Haskell functions. There are some differences:

  • Speculate tests enumeratively using LeanCheck, QuickSpec tests randomly using QuickCheck;
  • Speculate is able to report comparisons directly;
  • QuickSpec allows polymorphism, Speculate does not;
  • For most examples, Speculate runs slower than QuickSpec 2 but faster than QuickSpec 1.

More documentation

For more examples, see the eg and bench folders.

Speculate has been subject to a paper, see the Speculate Paper on Haskell Symposium 2017. Speculate is also subject to a chapter in a PhD Thesis (2017).


Changelog for Speculate


  • Test.Speculate.Reason: add doubleCheck;
  • add lowtests benchmark;
  • bump express requirement to v1.0.0;
  • fix parallel compilation when using the Makefile.


  • bump express requirement to v0.2.0
  • add this changelog


  • no changes in the actual Speculate library
  • cleanup build files
  • remove uneeded typeable derivations on examples and tests


  • no changes in the actual Speculate library
  • refactor build scripts
  • use GitHub Workflows as the CI
  • fix compilation of some examples under the new LeanCheck


  • Test.Speculate: export reifyName;
  • “internal” modules:
    • Test.Speculate.Args: remove compareExpr;
    • Test.Speculate.Engine: add three new wrappers for “theory and representatives”;
    • Test.Speculate.Expr.Core: rename functions to compareLexicographicallyBy and compareComplexityThenIndex;
    • Test.Speculate.Function: add Function.A10 and A100 and A1000;
    • Test.Speculate.Reason: export isRootNormal and isRootNormalE;
  • add trilean benchmark;
  • improve order tests.

Earlier versions

Please refer to the git commit history.