Main Web site

• Splash page with examples and link to blog
• 120 minute workshop with more examples
• Long ish Tutorial

Questions

If you have any questions

• Join the Liquid Haskell slack channel
• Mail the users mailing list
• Create a github issue

Contributing Guide

Please see the contributing guide

Requirements

LiquidHaskell requires (in addition to the cabal dependencies)

• SMTLIB2 compatible solver installed and its executable found on PATH

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"]

See this file for instructions on running inside a custom nix-shell.

How To Run Regression Tests

$ stack 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

tests/test.hs

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 liquid.prof
   

   or by this to see the graph

   $ hp2ps -e8in -c liquid.hp
   $ gv liquid.ps
   

   etc.

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 ..
  

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 "--no-termination" @-}

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

    LIQUIDHASKELL_OPTS="--diff"

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

Theorem Proving

To enable theorem proving, e.g. as described here use the option

    {-@ LIQUID "--reflection" @-}

To additionally turn on proof by logical evaluation use the option

    {-@ LIQUID "--reflection" @-}

You can see many examples of proofs by logical evaluation in benchmarks/popl18/ple/pos

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

• CVC4
• MathSat

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-errors

Short (Unqualified) Module Names

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

--short-names

Totality Check

LiquidHaskell proves the absence of pattern match failures.

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.

Use the no-totality flag to disable totality checking.

liquid --no-totality test.hs

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.

Default Termination Metrics

The user can specify the size of a data-type in the data definition

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

In the above, the measure llen, which needs to be defined by the user (see below), is defined as the default metric for the type L a. LH will use this default metric to automatically prove that the following terminates:

    append :: L a -> L a -> L a  
    append N           ys = ys
    append (Cons x xs) ys = Cons x (append xs ys)

as, by default the first (non-function) argument with an associated size metric is checked to be strictly terminating and non-negative at each recursive call.

A default termination metric is a Haskell function that is proved terminating using structural induction. To deactivate structional induction check on the termination metric, use the --trust-sizes flag.

Explicit Termination Metrics

However, consider the function reverse

    reverseAcc :: L a -> L a -> L a  
    reverseAcc acc N           = acc
    reverseAcc acc (Cons x xs) = reverseAcc (Cons x acc) xs

Here, the first argument does not decrease, instead the second does. We can tell LH to use the second argument using the explicit termination metric

    reverseAcc :: L a -> xs:L a -> L a / [llen xs]  

which tells LH that the llen of the second argument xs is what decreases at each recursive call.

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

    {-@ merge :: Ord a => xs:L a -> ys:L a -> L a / [llen xs + llen ys] @-}

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

Lexicographic Termination Metrics

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

    {-@ ack :: m:Int -> n:Int -> Nat / [m, n] @-}
    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 m normally decreases, but if it does not, n will decrease with a lexicographic termination metric [m, n].

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.

Mutually Recursive Functions

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 :: Int -> Bool
    even 0 = True
    even n = odd (n-1)

    odd :: Int -> Bool
    odd  n = not (even n)

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

    {-@ isEven :: n:Nat -> z:{v:Int | v = 0} -> Bool / [n, z] @-}
    isEven :: Int -> Int -> Bool
    isEven 0 _ = True
    isEven n _ = isOdd (n-1) 1

    {-@ isOdd :: n:Nat -> z:{v:Int | v = 1} -> Bool / [n, z] @-}
    isOdd :: Int -> Int -> Bool
    isOdd  n _ = not (isEven 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 as shown above. See tests/pos/mutrec.hs for the full example.

Automatic Termination Metrics

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, LH 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 constructors 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.)

Disabling Termination Checking

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

    {-@ lazy foo @-}

Total Haskell

LiquidHaskell provides a total Haskell flag that checks both totallity and termination of the program, overriding a potential no-termination flag.

liquid --total-Haskell test.hs

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
  where
    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.

No measure fields

When a data type is refined, Liquid Haskell automatically turns the data constructor fields into measures. For example,

{-@ data L a = N | C {hd :: a, tl :: L a} @-}

will automatically create two measures hd and td. To deactivate this automatic measure definition, and speed up verification, you can use the no-measure-fields flag.

liquid --no-measure-fields test.hs

Prune Unsorted Predicates

The --prune-unsorted flag is needed when using measures over specialized instances of ADTs.

For example, consider a measure over lists of integers

    sum :: [Int] -> Int
    sum [] = 0
    sum (x:xs) = 1 + sum xs

This measure will translate into strengthening the types of list constructors

    [] :: {v:[Int] | sum v = 0 }
    (:) :: x:Int -> xs:[Int] -> {v:[Int] | sum v = x + sum xs}

But what if our list is polymorphic [a] and later instantiate to list of ints? The workaround we have right now is to strengthen the polymorphic list with the sum information

    [] :: {v:[a] | sum v = 0 }
    (:) :: x:a -> xs:[a] -> {v:[a] | sum v = x + sum xs}

But for non numeric as, refinements like x + sum xs are ill-sorted!

We use the flag --prune-unsorted to prune away unsorted expressions (like x + sum xs) inside refinements.

    liquid --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 uninterpreted 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 external module Foo.Bar.Baz for which you do not have the code, you can create a .spec file at:

/path/to/include/Foo/Bar/Baz.spec

See, for example, the contents of

• include/Prelude.spec
• include/Data/List.spec
• include/Data/Vector.spec

Note:

• The above directories are part of the LH prelude, and included by default when running liquid.
• The .spec mechanism is only for external modules* without code, see below for standalone specifications for internal or home modules.

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 -> Prop, r :: k -> k -> Prop>
  = 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.

Modules WITH code: QuasiQuotation

Instead of writing both a Haskell type signature and a LiquidHaskell specification for a function, the lq quasiquoter in the LiquidHaskell module can be used to generate both from just the LiquidHaskell specification.

module Nats (nats) where

{-@ nats :: [{v:Int | 0 <= v}] @-}
nats :: [Int]
nats = [1,2,3]

can be written as

{-# LANGUAGE QuasiQuotes #-}
module Nats (nats) where

import LiquidHaskell

[lq| nats :: [{v:Int | 0 <= v}] |]
nats = [1,2,3]

and the lq quasiquoter will generate the plain nats :: [Int] when GHC compiles the module.

Refined type aliases (see the next section) can also be written inside lq; for example:

{-# LANGUAGE QuasiQuoters #-}
module Nats (Nat, nats) where

[lq| type Nat = {v:Int | 0 <= v} |]

[lq| nats :: [Nat] |]
nats = [1,2,3]

Here, the lq quasiquoter will generate a plain Haskell type synonym for Nat as well as the refined one.

Note that this is still an experimental feature, and currently requires that one depend on LiquidHaskell as a build dependency for your project; the quasiquoter will be split out eventually into an independent, dependency-light package. Also, at this time, writing a type inside lq which refers to a refined type alias for which there is not a plain Haskell type synonym of the same name will result in a "not in scope" error from GHC.

Standalone Specifications for Internal Modules

Recall that the .spec mechanism is only for modules whose code is absent; if code is present then there can be multiple, possibly conflicting specifications. Nevertheless, you may want, for one reason or another, to write (assumed) specifications outside the file implementing the module.

You can do this as follows.

Lib.hs

module Lib (foo) where

foo a = a

now, instead of a .spec file, just use a haskell module, e.g. LibSpec.hs

module LibSpec ( module Lib ) where

import Lib

-- Don't forget to qualify the name!

{-@ Lib.foo :: {v:a | false} -> a @-}

and then here's Client.hs

module Client where

import Lib      -- use this if you DON'T want the spec
import LibSpec  -- use this if you DO want the spec, in addition to OR instead of the previous import.

bar = foo 1     -- if you `import LibSpec` then this call is rejected by LH

Inductive Predicates

Very Experimental

LH recently added support for Inductive Predicates in the style of Isabelle, Coq etc. These are encoded simply as plain Haskell GADTs but suitably refined.

Apologies for the minimal documentation; see the following examples for details:

• Even and Odd
• Permutations
• Transitive Closure

Implicit Arguments

Experimental

There is experimental support for implicit arguments, solved for with congruence closure. For example, consider Implicit1.hs:

{-@ type IntN N = {v:Int | v = N} @-}

{-@ foo :: n:Int ~> (() -> IntN n) -> IntN {n+1} @-}
foo f = 1 + f ()

{-@ test1 :: IntN 11 @-}
test1 = foo (\_ -> 10)

Here, the refinement on (\_ -> 10) :: Int -> { v:Int | v = 10 } allows us to solve for n = 10, the implicit argument to foo.

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} @-}

or

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

or

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

or

{-@ 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 @-}

or

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

and:

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

see tests/pos/ListSort.hs

and:

{-@ 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.

Infix Logic Operators

You can define infix operators in logic, following Haskell's infix notation. For example, if (+++) is defined as a measure or reflected function, you can use it infix by declaring

{-@ infixl 9 +++ @-}

Note: infix operators cannot contain the dot character ..

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)
@-}

Note: Measure names do not have to be the same as field name, e.g. we could call the measure sz in the above as shown in tests/pos/Class2.hs.

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 overwritten.

Inlines

The inline lets you use a Haskell function in a type specification.

{-@ inline max @-}
{-@ max :: Int -> Int -> Int @-}
max :: Int -> Int -> Int
max x y = if x > y then x else y

For example, if you write the above you can then write a function:

{-@ floor :: x:Int -> {v:Int | max 0 x} @-}
floor :: Int -> Int
floor x 
  | x <= 0    = 0
  | otherwise = x

That is, you can use the haskell max in the refinement type and it will automatically get “expanded” out to the full definition. This makes it useful e.g. to reuse plain Haskell code to compose specifications, and to share definitions common to refinements and code.

However, as they are expanded at compile time, inline functions cannot be recursive. The can call other (non-recursive) inline functions.

If you want to talk about arbitrary (recursive) functions inside your types, then you need to use reflect described [in the blog] (https://ucsd-progsys.github.io/liquidhaskell-blog/tags/reflection.html)

Self-Invariants

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.

Abstract and Bounded Refinements

This is probably the best example of the abstract refinement syntax:

• Abstract Refinements
• Bounded Refinements

Unfortunately, the best documentation for these two advanced features is the relevant papers at

• ESOP 2013
• ICFP 2015

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.

Dependent Pairs

Dependent Pairs are expressed by binding the initial tuples of the pair. For example incrPair defines an increasing pair.

{-@ incrPair :: Int -> (x::Int, {v:Int | x <= v}) @-}
incrPair i = (i, i+1)

Internally dependent pairs are implemented using abstract refinement types. That is (x::a, {v:b | p x}) desugars to (a,b)<\x -> {v:b | p x}>.

Invariants

LH lets you locally associate invariants with specific data types.

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 constructors (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.

WARNING: There is an older global invariant mechanism that attaches a refinement to a datatype globally. Do not use this mechanism -- it is unsound and about to deprecated in favor of something that is actually sound

Forexample, 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 constructors (ie., : and []) satisfy it.(TODO!) Then, LiquidHaskell assumes that each list element that is created satisfies this invariant.

Formal Grammar of Refinement Predicates

(C)onstants

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

(V)ariables

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

(E)xpressions

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

(R)elations

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

(P)redicates

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

Foo.Bar.Baz

Then the system automatically searches for

include/Foo/Bar/Baz.hquals

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 slides directory for an example.

Editor Integration

• Emacs/Flycheck
• Vim/Syntastic

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 performance 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
./deploy-gipeda.bash

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 manageable 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:

./cleanup-gipeda.bash

...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.

Proof Automation

The liquidinstances automatically generates proof terms using symbolic evaluation. See.

{-@ LIQUID "--automatic-instances=liquidinstances" @-}

This flag is global and will symbolically evaluation all the terms that appear in the specifications.

As an alternative, the liquidinstanceslocal flag has local behavior. See

{-@ LIQUID "--automatic-instances=liquidinstanceslocal" @-}

will only evaluate terms appearing in the specifications of the function theorem, in the function theorem is annotated for automatic instantiation using the following liquid annotation

{-@ automatic-instances theorem @-}