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LiquidHaskell Hackage Hackage-Deps Build Status


LiquidHaskell requires (in addition to the cabal dependencies)

  • SMTLIB2 compatible solver

How To Clone, Build and Install

See install instructions

How To Run

To verify a file called foo.hs at type

$ liquid foo.hs

How to Run inside GHCi

To run inside ghci e.g. when developing do:

$ stack ghci liquidhaskell
ghci> :m +Language.Haskell.Liquid.Liquid
ghci> liquid ["tests/pos/Abs.hs"]

How To Run Regression Tests

$ make test

To use threads to speed up the tests

$ make THREADS=30 test

Or your favorite number of threads, depending on cores etc.

You can directly extend and run the tests by modifying


To run the regression test and the benchmarks run

$ make all-test

How to Profile

  1. Build with profiling on

    $ make pdeps && make prof
  2. Run with profiling

    $ time liquid range.hs +RTS -hc -p
    $ time liquid range.hs +RTS -hy -p

    Followed by this which shows the stats file

    $ more

    or by this to see the graph

    $ hp2ps -e8in -c liquid.hp
    $ gv


How to Get Stack Traces On Exceptions

  1. Build with profiling on

    $ make pdeps && make prof
  2. Run with backtraces

    $ liquid +RTS -xc -RTS foo.hs

Working With Submodules

  • To update the liquid-fixpoint submodule, run

    cd ./liquid-fixpoint
    git fetch --all
    git checkout <remote>/<branch>
    cd ..

    This will update liquid-fixpoint to the latest version on <branch> (usually master) from <remote> (usually origin).

  • After updating liquid-fixpoint, make sure to include this change in a commit! Running

    git add ./liquid-fixpoint

    will save the current commit hash of liquid-fixpoint in your next commit to the liquidhaskell repository.

  • For the best experience, don't make changes directly to the ./liquid-fixpoint submodule, or else git may get confused. Do any liquid-fixpoint development inside a separate clone/copy elsewhere.

  • If something goes wrong, run

    rm -r ./liquid-fixpoint
    git submodule update --init

    to blow away your copy of the liquid-fixpoint submodule and revert to the last saved commit hash.

  • Want to work fully offline? git lets you add a local directory as a remote. Run

    cd ./liquid-fixpoint
    git remote add local /path/to/your/fixpoint/clone
    cd ..

    Then to update the submodule from your local clone, you can run

    cd ./liquid-fixpoint
    git fetch local
    git checkout local/<branch>
    cd ..
  • Updating prover submodule follows similarly

Command Line Options

LiquidHaskell supports several command line options, to configure the checking. Each option can be passed in at the command line, or directly added to the source file via:

{-@ LIQUID "option-within-quotes" @-}

for example, to disable termination checking (see below)

{-@ LIQUID "--notermination" @-}

You may also put command line options in the environment variable LIQUIDHASKELL_OPTS. For example, if you add the line:


to your .bashrc then, by default, all files will be incrementally checked unless you run with the overriding --full flag (see below).

Incremental Checking

LiquidHaskell supports incremental checking where each run only checks the part of the program that has been modified since the previous run.

$ liquid --diff foo.hs

Each run of liquid saves the file to a .bak file and the subsequent run + does a diff to see what has changed w.r.t. the .bak file + only generates constraints for the [CoreBind] corresponding to the changed top-level binders and their transitive dependencies.

The time savings are quite significant. For example:

$ time liquid --notermination -i . Data/ByteString.hs > log 2>&1

real    7m3.179s
user    4m18.628s
sys     0m21.549s

Now if you go and tweak the definition of spanEnd on line 1192 and re-run:

$ time liquid -d --notermination -i . Data/ByteString.hs > log 2>&1

real    0m11.584s
user    0m6.008s
sys     0m0.696s

The diff is only performed against code, i.e. if you only change specifications, qualifiers, measures, etc. liquid -d will not perform any checks. In this case, you may specify individual definitions to verify:

$ liquid -b bar -b baz foo.hs

This will verify bar and baz, as well as any functions they use.

If you always want to run a given file with diff-checking, add the pragma:

{-@ LIQUID "--diff" @-}

Full Checking (DEFAULT)

To force LiquidHaskell to check the whole file (DEFAULT), use:

$ liquid --full foo.hs

to the file. This will override any other --diff incantation elsewhere (e.g. inside the file.)

If you always want to run a given file with full-checking, add the pragma:

{-@ LIQUID "--full" @-}

Specifying Different SMT Solvers

By default, LiquidHaskell uses the SMTLIB2 interface for Z3.

To run a different solver (supporting SMTLIB2) do:

$ liquid --smtsolver=NAME foo.hs

Currently, LiquidHaskell supports

To use these solvers, you must install the corresponding binaries from the above web-pages into your PATH.

You can also build and link against the Z3 API (faster but requires more dependencies). If you do so, you can use that interface with:

$ liquid --smtsolver=z3mem foo.hs

Short Error Messages

By default, subtyping error messages will contain the inferred type, the expected type -- which is not a super-type, hence the error -- and a context containing relevant variables and their type to help you understand the error. If you don't want the above and instead, want only the source position of the error use:


Short (Unqualified) Module Names

By default, the inferred types will have fully qualified module names. To use unqualified names, much easier to read, use:


Totality Check

LiquidHaskell can prove the absence of pattern match failures. Use the totality flag to prove that all defined functions are total.

liquid --totality test.hs

For example, the definition

fromJust :: Maybe a -> a
fromJust (Just a) = a

is not total and it will create an error message. If we exclude Nothing from its domain, for example using the following specification

{-@ fromJust :: {v:Maybe a | (isJust v)} -> a @-}

fromJust will be safe.

Termination Check

By default a termination check is performed on all recursive functions.

Use the no-termination option to disable the check

liquid --no-termination test.hs

In recursive functions the first algebraic or integer argument should be decreasing.

The default decreasing measure for lists is length and Integers its value.

The user can specify the decreasing measure in data definitions:

{-@ data L [llen] a = Nil | Cons (x::a) (xs:: L a) @-}

Defines that llen is the decreasing measure (to be defined by the user).

For example, in the function foldl

foldl k acc N           = acc
foldl k acc (Cons x xs) = foldl k (x `k` acc) xs

by default the second argument (the first non-function argument) will be checked to be decreasing. However, the explicit hint

{-@ Decrease foo 3 @-}

tells LiquidHaskell to instead use the third argument.

Apart from specifying a specific decreasing measure for an Algebraic Data Type, the user can specify that the ADT follows the expected decreasing measure by

{-@ autosize L @-}

Then, LiquidHaskell will define an instance of the function autosize for L that decreases by 1 at each recursive call and use autosize at functions that recurse on L.

For example, autosize L will refine the data constroctors of L a with the autosize :: a -> Int information, such that

Nil  :: {v:L a | autosize v = 0}
Cons :: x:a -> xs:L a -> {v:L a | autosize v = 1 + autosize xs}

Also, an invariant that autosize is non negative will be generated

invariant  {v:L a| autosize v >= 0 }

This information is all LiquidHaskell needs to prove termination on functions that recurse on L a (on ADTs in general.)

To disable termination checking for foo that is, to assume that it is terminating (possibly for some complicated reason currently beyond the scope of LiquidHaskell) you can write

{-@ Lazy foo @-}

Some functions do not decrease on a single argument, but rather a combination of arguments, e.g. the Ackermann function.

ack m n
  | m == 0          = n + 1
  | m > 0 && n == 0 = ack (m-1) 1
  | m > 0 && n >  0 = ack (m-1) (ack m (n-1))

In all but one recursive call m decreases, in the final call m does not decrease but n does. We can capture this notion of "x normally decreases, but if it does not, y will" with an extended annotation

{-@ Decrease ack 1 2 @-}

An alternative way to express this specification is by annotating the function's type with the appropriate numeric decreasing expressions. As an example, you can give ack a type

{-@ ack :: m:Nat -> n:Nat -> Nat / [m,n] @-}

stating that the numeric expressions [m, n] are lexicographically decreasing.

Decreasing expressions can be arbitrary refinement expressions, e.g.,

{-@ merge :: Ord a => xs:[a] -> ys:[a] -> [a] / [(len xs) + (len ys)] @-}

states that at each recursive call of merge the sum of the lengths of its arguments will be decreased.

When dealing with mutually recursive functions you may run into a situation where the decreasing parameter must be measured across a series of invocations, e.g.

even 0 = True
even n = odd (n-1)

odd  n = not $ even n

In this case, you can introduce a ghost parameter that orders the functions

even 0 _ = True
even n _ = odd (n-1) 1

odd  n _ = not $ even n 0

thus recovering a decreasing measure for the pair of functions, the pair of arguments. This can be encoded with the lexicographic termination annotation {-@ Decrease even 1 2 @-} (see tests/pos/mutrec.hs for the full example).

Lazy Variables

A variable can be specified as LAZYVAR

{-@ LAZYVAR z @-}

With this annotation the definition of z will be checked at the points where it is used. For example, with the above annotation the following code is SAFE:

foo   = if x > 0 then z else x
    z = 42 `safeDiv` x
    x = choose 0

By default, all the variables starting with fail are marked as LAZY, to defer failing checks at the point where these variables are used.

Prune Unsorted Predicates

By default unsorted predicates are pruned away (yielding true for the corresponding refinement.) To disable this behaviour use the no-prune-unsorted flag.

liquid --no-prune-unsorted test.hs

Case Expansion

By default LiquidHaskell expands all data constructors to the case statements. For example, if F = A1 | A2 | .. | A10, then liquidHAskell will expand the code case f of {A1 -> True; _ -> False} to case f of {A1 -> True; A2 -> False; ...; A10 -> False}. This expansion can lead to more precise code analysis but it can get really expensive due to code explosion. The no-case-expand flag prevents this expansion and keeps the user provided cases for the case expression.

liquid --no-case-expand test.hs

Higher order logic

The flag --higherorder allows reasoning about higher order functions.

Restriction to Linear Arithmetic

When using z3 as the solver, LiquidHaskell allows for non-linear arithmetic: division and multiplication on integers are interpreted by z3. To treat division and multiplication as unintepreted functions use the linear flag

liquid --linear test.hs

Counter examples (Experimental!)

When given the --counter-examples flag, LiquidHaskell will attempt to produce counter-examples for the type errors it discovers. For example, see tests/neg/ListElem.hs

% liquid --counter-examples tests/neg/ListElem.hs


tests/neg/ListElem.hs:12:1-8: Error: Liquid Type Mismatch

 12 | listElem _ []      = False

   Inferred type
     VV : {VV : Bool | VV == True}
     VV = True

   not a subtype of Required type
     VV : {VV : Bool | Prop VV <=> Set_mem ?b (listElts ?a)}

   In Context
     ?a : {?a : [a] | len ?a >= 0}
     ?a = [1]

     ?b : a
     ?b = 0

The --counter-examples flag requires that each type in the context be an instance of GHC.Generics.Generic or Test.Targetable.Targetable (provided as part of LiquidHaskell). LiquidHaskell cannot generate counter-examples for polymorphic types, but will try (naively) to instantiate type variables with Int (as seen in the example above).

Writing Specifications

Modules WITHOUT code

When checking a file target.hs, you can specify an include directory by

liquid -i /path/to/include/  target.hs

Now, to write specifications for some module Foo.Bar.Baz for which you do not have the code, you can create a .spec file at:


See, for example, the contents of

(Note: The above directories are part of the LH prelude, and included by default when running liquid.)

Modules WITH code: Data

Write the specification directly into the .hs or .lhs file, above the data definition. See, for example, tests/pos/Map.hs

data Map k a <l :: k -> k -> Bool, r :: k -> k -> Bool>
  = Tip
  | Bin (sz    :: Size)
        (key   :: k)
        (value :: a)
        (left  :: Map <l, r> (k <l key>) a)
        (right :: Map <l, r> (k <r key>) a)
data Map k a = Tip
             | Bin Size k a (Map k a) (Map k a)

You can also write invariants for data type definitions together with the types. For example, see tests/pos/record0.hs

{-@ data LL a = BXYZ { size  :: {v: Int | v > 0 }
                     , elems :: {v: [a] | (len v) = size }

Finally you can specify the variance of type variables for data types. For example, see tests/pos/Variance.hs, where data type Foo has four type variables a, b, c, d, specified as invariant, bivariant, covariant and contravariant, respectively.

data Foo a b c d
{-@ data variance Foo invariant bivariant covariant contravariant @-}

Modules WITH code: Functions

Write the specification directly into the .hs or .lhs file, above the function definition. For example

{-@ incr :: x:{v: Int | v > 0} -> {v: Int | v > x} @-}
incr   :: Int -> Int
incr x = x + 1

Modules WITH code: Type Classes

Write the specification directly into the .hs or .lhs file, above the type class definition. For example

{-@ class Sized s where
      size :: forall a. x:s a -> {v:Int | v = (size x)}
class Sized s where
  size :: s a -> Int

Any measures used in the refined class definition will need to be generic (see Specifying Measures).

As an alternative, you can refine class instances. For example

instance Compare Int where

{-@ instance Compare Int where
    cmax :: Odd -> Odd -> Odd

cmax y x = if x >= y then x else y

When cmax method is used on Int, liquidHaskell will give it the refined type Odd -> Odd -> Odd.

Note that currently liquidHaskell does not allow refining instances of refined classes.

Refinement Type Aliases

Predicate Aliases

Often, the propositions in the refinements can get rather long and verbose. You can write predicate aliases like so:

{-@ predicate Lt X Y = X < Y        @-}
{-@ predicate Ge X Y = not (Lt X Y) @-}

and then use the aliases inside refinements, for example

{-@ incr :: x:{v:Int | (Pos v)} -> { v:Int | ((Pos v) && (Ge v x))} @-}
incr :: Int -> Int
incr x = x + 1

See Data.Map for a more substantial and compelling example.

Syntax: The key requirements for type aliases are:

  • Value parameters are specified in uppercase: X, Y, Z etc.

Type Aliases

Similarly, it is often quite tedious to keep writing

{v: Int | v > 0}

Thus, LiquidHaskell supports refinement-type aliases of the form:

{-@ type Gt      N = {v: Int | N <  v} @-}
{-@ type GeNum a N = {v: a   | N <= v} @-}


{-@ type SortedList a = [a]<{\fld v -> (v >= fld)}> @-}


{-@ type OMap k a = Map <{\root v -> v < root}, {\root v -> v > root}> k a @-}


{-@ type MinSPair a = (a, OSplay a) <\fld -> {v : Splay {v:a|v>fld} | 0=0}> @-}

and then use the above in signatures like:

{-@ incr: x: Int -> GeNum Int x @-}


{-@ incr: x: Int -> Gt x @-}


{-@ assert insert :: (Ord a) => a -> SortedList a -> SortedList a @-}

see tests/pos/ListSort.hs


{-@ assert insert :: (Ord k) => k -> a -> OMap k a -> OMap k a @-}

see tests/pos/Map.hs

Syntax: The key requirements for type aliases are:

  1. Type parameters are specified in lowercase: a, b, c etc.
  2. Value parameters are specified in uppercase: X, Y, Z etc.

Specifying Measures

Can be placed in .spec file or in .hs/.lhs file wrapped around {-@ @-}

Value measures: include/GHC/Base.spec

measure len :: forall a. [a] -> GHC.Types.Int
len ([])     = 0
len (y:ys)   = 1 + len(ys)

Propositional measures: tests/pos/LambdaEval.hs

measure isValue      :: Expr -> Bool
isValue (Const i)    = true
isValue (Lam x e)    = true
isValue (Var x)      = false
isValue (App e1 e2)  = false
isValue (Plus e1 e2) = false
isValue (Fst e)      = false
isValue (Snd e)      = false
isValue (Pair e1 e2) = ((? (isValue(e1))) && (? (isValue(e2))))

Raw measures: tests/pos/meas8.hs

{-@ measure rlen :: [a] -> Int
rlen ([])   = {v | v = 0}
rlen (y:ys) = {v | v = (1 + rlen(ys))}

Generic measures: tests/pos/Class.hs

{-@ class measure size :: a -> Int @-}
{-@ instance measure size :: [a] -> Int
    size ([])   = 0
    size (x:xs) = 1 + (size xs)
{-@ instance measure size :: Tree a -> Int
    size (Leaf)       = 0
    size (Node x l r) = 1 + (size l) + (size r)

Haskell Functions as Measures (beta): tests/pos/HaskellMeasure.hs

Inductive Haskell Functions from Data Types to some type can be lifted to logic

{-@ measure llen @-}
llen        :: [a] -> Int
llen []     = 0
llen (x:xs) = 1 + llen xs

The above definition

  • refines list's data constructors types with the llen information, and
  • specifies a singleton type for the haskell function llen :: xs:[a] -> {v:Int | v == llen xs} If the user specifies another type for llen, say llen :: xs:[a] -> {v:Int | llen xs >= 0} then the auto generated singleton type is overwriten.


Sometimes, we require specifications that allow inner components of a type to refer to the outer components, typically, to measure-based properties of outer components. For example, the following invariant about Maybe values

{-@ type IMaybe a = {v0 : Maybe {v : a | ((isJust v0) && v = (fromJust v0))} | 0 = 0 } @-}

states that the inner a enjoys the property that the outer container is definitely a Just and furthermore, the inner value is exactly the same as the fromJust property of the outer container.

As another example, suppose we have a measure:

measure listElts :: [a] -> (Set a)
listElts([])   = {v | (? Set_emp(v))}
listElts(x:xs) = {v | v = Set_cup(Set_sng(x), listElts(xs)) }

Now, all lists enjoy the property

{-@ type IList a = {v0 : List  {v : a | (Set_mem v (listElts v0)) } | true } @-}

which simply states that each inner element is indeed, a member of the set of the elements belonging to the entire list.

One often needs these circular or self invariants to connect different levels (or rather, to reify the connections between the two levels.) See this test for a simple example and hedgeUnion and Data.Map.Base for a complex one.


The bounds correspond to Horn implications between abstract refinements, which, as in the classical setting, correspond to subtyping constraints that must be satisfied by the concrete refinements used at any call-site.

See benchmarks/icfp15/pos/Overview.lhs for exaples on how to use bounds.


WARNING: Do not use this mechanism -- it is unsound and about to be replaced with something that is actually sound

There are two ways of specifying invariants in LiquidHaskell. First, there are global invariants that always hold for a data-type. For example, the length of a list cannot be negative

{-@ invariant {v:[a] | (len v >= 0)} @-}

LiquidHaskell can prove that this invariant holds, by proving that all List's constractos (ie., : and []) satisfy it.(TODO!) Then, LiquidHaskell assumes that each list element that is created satisfies this invariant.

Second, there are local invariants that one may use. For example, in tests/pos/StreamInvariants.hs every list is treated as a Stream. To establish this local invariant one can use the using declaration

{-@ using ([a]) as  {v:[a] | (len v > 0)} @-}

denoting that each list is not empty. Then, LiquidHaskell will prove that this invariant holds, by proving that all calls to List's constractos (ie., : and []) satisfy it, and will assume that each list element that is created satisfies this invariant.

With this, at the above test LiquidHaskell proves that taking the head of a list is safe. But, at tests/neg/StreamInvariants.hs the usage of [] falsifies this local invariant resulting in an "Invariant Check" error.

Formal Grammar of Refinement Predicates


c := 0, 1, 2, ...


v := x, y, z, ...


e := v                      -- variable
   | c                      -- constant
   | (e + e)                -- addition
   | (e - e)                -- subtraction
   | (c * e)                -- cmultiplication by constant
   | (v e1 e2 ... en)       -- uninterpreted function application
   | (if p then e else e)   -- if-then-else


r := ==               -- equality
   | /=               -- disequality
   | >=               -- greater than or equal
   | <=               -- less than or equal
   | >                -- greater than
   | <                -- less than


p := (e r e)          -- binary relation
   | (v e1 e2 ... en) -- predicate (or alias) application
   | (p && p)         -- and
   | (p || p)         -- or
   | (p => p)         -- implies
   | (not p)          -- negation
   | true
   | false

Specifying Qualifiers

There are several ways to specify qualifiers.

By Separate .hquals Files

You can write qualifier files e.g. Prelude.hquals

If a module is called or imports


Then the system automatically searches for


By Including .hquals Files

Additional qualifiers may be used by adding lines of the form:

{-@ include <path/to/file.hquals> @-}

to the Haskell source. See, this for example.

In Haskell Source or Spec Files

Finally, you can specifiers directly inside source (.hs or .lhs) or spec (.spec) files by writing as shown here

{-@ qualif Foo(v:Int, a: Int) : (v = a + 100)   @-}

Note In addition to these, LiquidHaskell scrapes qualifiers from all the specifications you write i.e.

  1. all imported type signatures,
  2. measure bodies and,
  3. data constructor definitions.

Generating HTML Output

The system produces HTML files with colorized source, and mouseover inferred type annotations, which are quite handy for debugging failed verification attempts.

  • Regular Haskell When you run: liquid foo.hs you get a file foo.hs.html with the annotations. The coloring is done using hscolour.

  • Markdown + Literate Haskell You can also feed in literate haskell files where the comments are in Pandoc markdown. In this case, the tool will run pandoc to generate the HTML from the comments. Of course, this requires that you have pandoc installed as a binary on your system. If not, hscolour is used to render the HTML.

    It is also possible to generate slide shows from the above. See the tutorial directory for an example.

Editor Integration

Command Line Options

To see all options, run liquid --help. Here are some common options:

  • --cabaldir will automatically find a .cabal file in the ancestor path from which the target file belongs, and then add the relevant source and dependencies to the paths searched for by LiquidHaskell.

    This means we don't have to manually do -i src etc. when checking large projects, which can be tedious e.g. within emacs.

  • --diff performs differential checking, i.e. only checks those binders that have transitively affected by edits since the previous check. Can speed things up greatly during editing.

  • --short-names prints out non-qualified names i.e. Int instead of GHC.Types.Int for inferred type annotations and error messages.

Pragmas are useful for embedding options directly within the source file, that is, somewhere in the file (perhaps at the top) put in:

{-@ LIQUID "--diff"        @-}
{-@ LIQUID "--short-names" @-}
{-@ LIQUID "--cabaldir"    @-}

to have the relevant option be used for that file.

Generating Performance Reports

We have set up infrastructure to generate performance reports using Gipeda.

Gipeda will generate a static webpage that tracks the peformance improvements and regressions between commits. To generate the site, first ensure you have the following dependencies available:

  • Git
  • Cabal >= 1.18
  • GHC
  • Make
  • Bash (installed at /bin/bash)

After ensuring all dependencies are available, from the Liquid Haskell directory, execute:

cd scripts/performance

This will download and install all the relevant repositories and files. Next, to generate the performance report, use the generate-site.bash script. This script has a few options:

  • -s [hash]: Do not attempt to generate performance reports for any commit older than the commit specified by the entered git hash
  • -e [hash]: Do not attempt to generate performance reports for any commit newer than the commit specified by the entered git hash
  • -f: The default behavior of generate-site.bash is to first check if logs have been created for a given hash. If logs already exist, generate-site.bash will not recreate them. Specify this option to skip this check and regenerate all logs.

You should expect this process to take a very long time. generate-site.bash will compile each commit, then run the entire test suite and benchmark suite for each commit. It is suggested to provide a managable range to generate-site.bash:

./generate-site.bash -s [starting hash] -e [ending hash]

...will generate reports for all commits between (inclusive) [starting hash] and [ending hash].

./generate-site.bash -s [starting hash]

... will generate reports for all commits newer than [starting hash]. This command can be the basis for some automated report generation process (i.e. a cron job).

Finally, to remove the Gipeda infrastructure from your computer, you may execute:


...which will remove any files created by deploy-gipeda.bash and generate-site.bash from your computer.

Configuration Management

It is very important that the version of Liquid Haskell be maintained properly.

Suppose that the current version of Liquid Haskell is A.B.C.D:

  • After a release to hackage is made, if any of the components B, C, or D are missing, they shall be added and set to 0. Then the D component of Liquid Haskell shall be incremented by 1. The version of Liquid Haskell is now A.B.C.(D + 1)

  • The first time a new function or type is exported from Liquid Haskell, if any of the components B, or C are missing, they shall be added and set to 0. Then the C component shall be incremented by 1, and the D component shall stripped. The version of Liquid Haskell is now A.B.(C + 1)

  • The first time the signature of an exported function or type is changed, or an exported function or type is removed (this includes functions or types that Liquid Haskell re-exports from its own dependencies), if the B component is missing, it shall be added and set to 0. Then the B component shall be incremented by 1, and the C and D components shall be stripped. The version of Liquid Haskell is now A.(B + 1)

  • The A component shall be updated at the sole discretion of the project owners.

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