Reasoning about Programs

The goal of Liquid Haskell is to move reasoning about Haskell programs into the programs themselves and to automate this reasoning as much as possible. It accomplishes this goal by extending the Haskell language with refinement types, which are checked by an external SMT solver.

{-@ LIQUID "--reflection" @-} {-@ LIQUID "--ple" @-} module Lists where import Prelude hiding (reverse, (++), length) {-@ infix : @-} import Language.Haskell.Liquid.Equational

Lightweight Reasoning

The power of SMT solving allows Liquid Haskell to prove certain properties entirely automatically, with no user input; we call these lightweight program properties.

  1. Linear Arithmetic

Many properties involving arithmetic can be proved automatically in this manner. For example, given the standard length function on lists

length :: [a] -> Int length [] = 0 length (_:xs) = 1 + length xs

we might find it useful to specify that the length of a list is never negative. Liquid Haskell extends the syntax of Haskell by interpreting comments of the form {-@ ... @-} as declarations, using which we can express this property by:

{-@ length :: [a] -> Nat @-}

Liquid Haskell is able to verify this specification automatically due to the standard refinement typing checking automated by the SMT solver:

  • In the first equation in the definition for length, the value \(v\) is \(0\), so the SMT solver determines that \(0 \leq v\).

  • In the second equation, the value \(v\) is \(1 + v'\), where \(v'\) is the result of the recursive call to length xs. From the refinement type of length, Liquid Haskell knows \(0 \leq v'\), and the SMT solver can deduce that \(0 \leq v\).

Proving that the length of a list is nonnegative is thus fully automated by the SMT solver. This is because SMT solvers can efficiently decide linear arithmetic queries, so verifying this kind of property is tractable. Note that the refinement type does not actually mention the recursive function length.

Measures In order to allow Haskell functions to appear in refinement types, we need to lift them to the refinement type level. Liquid Haskell provides a simple mechanism for performing this lifting on a particular restricted set of functions, called measures. Measures are functions which: take one parameter, which must be an algebraic data type; are defined by a single equation for each constructor; and whose body calls only primitive (e.g., arithmetic) functions and measures. For this restricted class of functions, refinement types can still be checked fully automatically.

For instance, length is a measure: it has one argument, is defined by one equation for each constructor, and calls only itself and the arithmetic operator (+). To allow length to appear in refinements, we declare it to be a measure:

{-@ measure length @-}

For example, we can now state that the length of two lists appended together is the sum of their lengths:

{-@ infix ++ @-} {-@ reflect ++ @-} {-@ (++) :: xs:[a] -> ys:[a] -> {os:[a] | length os == length xs + length ys} @-} (++) :: [a] -> [a] -> [a] [] ++ ys = ys (x:xs) ++ ys = x:(xs ++ ys)

Liquid Haskell checks this refinement type in two steps:

  • In the first equation in the definition of (++), the list xs is empty, thus its length is 0, and the SMT solver can discharge this case via linear arithmetic.

  • In the second equation case, the input list is known to be x:xs, thus its length is 1 + length xs. The recursive call additionally indicates that length (xs ++ ys) = length xs + length ys and the SMT solver can also discharge this case using linear arithmetic.

  1. Deep Reasoning

We saw that because length is a measure, it can be lifted to the refinement type level while retaining completely automatic reasoning. We cannot expect that for recursive functions in general, as quantifier instantiation leads to unpredictable performance.

The append function, for example, takes two arguments, and therefore is not a measure. If we lift it to the refinement type level, the SMT solver will not be able to automatically check refinements involving it. Liquid Haskell still allows reasoning about such functions, but this limitation means the user may have to supply the proofs. We call properties that the SMT solver cannot solve entirely automatically deep program properties.

For example, consider the following definition for the reverse function on lists in terms of the append function:

{-@ reverse :: is:[a] -> {os:[a] | length is == length os} @-} reverse :: [a] -> [a] reverse [] = [] reverse (x:xs) = reverse xs ++ [x]

Because the definition uses append, which is not a measure, the reverse function itself is not a measure, so reasoning about it will not be fully automatic.

Instead, Liquid Haskell can lift arbitrary Haskell functions into the refinement type level via the notion of reflection. Rather than using the straightforward translation available for measures, which completely describes the function to the SMT solver, reflection gives the SMT solver only the value of the function for the arguments on which it is actually called. Restricting the information available to the SMT solver in this way ensures that checking refinement types remains decidable.

To see this in action, we prove that reversing a singleton list does not change it, i.e., reverse [x] == [x]. We first declare reverse and the append function as reflected:

{-@ infix ++ @-} {-@ reflect ++ @-} {-@ reflect reverse @-}

We then introduce the function singletonP, whose refinement type expresses the desired result, and whose body provides the proof in equational reasoning style:

singletonP :: a -> Proof {-@ singletonP :: x:a -> { reverse [x] == [x] } @-} singletonP x = reverse [x] ==. reverse [] ++ [x] ==. [] ++ [x] ==. [x] *** QED

We can understand this function as mapping a value x to a proof of reverse [x] = [x]. The type Proof is simply Haskell's unit type (), and {reverse [x] = x} is syntactic sugar for {v:() | reverse [x] = x}, a refinement of the unit type. This syntax hides the irrelevant value v :: ().

Note that the body of the singletonP function looks very much like a typical pen-and-paper proof, such as the one in Graham's book. The correspondence is so close that we claim proving a property in Liquid Haskell can be just as easy as proving it on paper by equational reasoning --- but the proof in Liquid Haskell is machine-checked!

As always, Liquid Haskell uses an SMT solver to check this proof. Because the body of singletonP syntactically contains the three terms reverse [x], reverse [] and [] ++ [x], Liquid Haskell passes the corresponding equations
  reverse [x] = reverse [] ++ [x]
  reverse []  = []
  [] ++ [x]   = [x]
to the SMT solver, which then easily derives the desired property reverse [x] = [x]. Note that the following definition would constitute an equally valid proof:
  singletonP x =
    const () (reverse [x], reverse [], [] ++ [x])

But such a compressed ``proof'' is neither easy to come up with directly, nor is it readable or very insightful. Therefore, we use proof combinators to write readable equational-style proofs, where each reasoning step is checked.

Proof Combinators As already noted in the previous section, we use Haskell's unit type to represent a proof:
  type Proof = ()

This unit type is sufficient because a theorem is expressed as a refinement on the arguments of a function. In other words, the ``value'' of a theorem has no meaning.

Proof combinators themselves are then simply Haskell functions, defined in the Equational module that comes with Liquid Haskell. The most basic example is the *** function that takes any expression and ignores its value, returning a proof:
  data QED = QED

  (***) :: a -> QED -> Proof
  _ *** QED = ()
  infixl 2 ***

The QED argument serves a purely aesthetic purpose, allowing us to conclude proofs with *** QED.

Equational Reasoning The key combinator for equational reasoning is the operator (==.). Its refinement type ensures its arguments are equal, and it returns its second argument, so that multiple uses of (==.) can be chained together:
  {-@ (==.) :: x:a -> y:{a | x == y} ->
        {o:a | o == y && o == x} @-}

  (==.) :: a -> a -> a
  _ ==. x = x
Explanations Sometimes we need to refer to other theorems in our proofs. Because theorems are just Haskell functions, all we need is an operator that accepts an argument of type Proof, which is defined as follows:
  (?) :: a -> Proof -> a
  x ? _ = x
For example, we can invoke the theorem singletonP for the value 1 simply by mentioning singletonP 1 in a proof:
singleton1P :: Proof {-@ singleton1P :: { reverse [1] == [1] } @-} singleton1P = reverse [1] ? singletonP 1 ==. [1] *** QED

Note that although the ? operator is suggestively placed next to the equation that we want to justify, its placement in the proof is actually immaterial --- the body of a function equation is checked all at once.

Induction on Lists

Structural induction is a fundamental technique for proving properties of functional programs. For the list type in Haskell, the principle of induction states that to prove that a property holds for all (finite) lists, it suffices to show that:

  • It holds for the empty list [] (the base case), and
  • It holds for any non-empty list x:xs assuming it holds for the tail xs of the list (the inductive case).

Induction does not require a new proof combinator. Instead, proofs by induction can be expressed as recursive functions in Liquid Haskell. For example, let us prove that reverse is its own inverse, i.e., reverse (reverse xs) == xs. We express this property as the type of a function involutionP, whose body constitutes the proof:

{-@ involutionP :: xs:[a] -> {reverse (reverse xs) == xs } @-} involutionP :: [a] -> Proof involutionP [] = reverse (reverse []) -- applying inner reverse ==. reverse [] -- applying reverse ==. [] *** QED involutionP (x:xs) = reverse (reverse (x:xs)) -- applying inner reverse ==. reverse (reverse xs ++ [x]) ? distributivityP (reverse xs) [x] ==. reverse [x] ++ reverse (reverse xs) ? involutionP xs ==. reverse [x] ++ xs ? singletonP x ==. [x] ++ xs -- applying append on [] ==. x:([]++ xs) -- applying ++ ==. (x:xs) *** QED

Because involutionP is a recursive function, this constitutes a proof by induction. The two equations for involutionP correspond to the two cases of the principle of induction:

  • In the base case, because the body of the function contains the terms reverse (reverse []) and reverse [], the corresponding equations are passed to the SMT solver, which then proves that reverse (reverse []) = [].

  • In the inductive case, we need to show that reverse (reverse (x:xs)) = (x:xs), which proceeds in several steps. The validity of each step is checked by Liquid Haskell when verifying that the refinement type of (==.) is satisfied. Some of the steps follow directly from definitions, and we just add a comment for clarity. Other steps require external lemmas or the inductive hypothesis, which we invoke via the explanation operator (?).

We use the lemma distributivityP, which states that list reversal distributes (contravariantly) over list append:

distributivityP :: [a] -> [a] -> Proof {-@ distributivityP :: xs:[a] -> ys:[a] -> {reverse (xs ++ ys) == reverse ys ++ reverse xs} @-}

Again, we define distributivityP as a recursive function, as the property can be proven by induction:

distributivityP [] ys = reverse ([] ++ ys) ==. reverse ys ? leftIdP (reverse ys) ==. reverse ys ++ [] ==. reverse ys ++ reverse [] *** QED distributivityP (x:xs) ys = reverse ((x:xs) ++ ys) ==. reverse (x:(xs ++ ys)) ==. reverse (xs ++ ys) ++ [x] ? distributivityP xs ys ==. (reverse ys ++ reverse xs) ++ [x] ? assocP (reverse ys) (reverse xs) [x] ==. reverse ys ++ (reverse xs ++ [x]) ==. reverse ys ++ reverse (x:xs) *** QED

This proof itself requires additional lemmas about append, namely right identity (rightIdP) and associativity (assocP), which we tackle with further automation below.

Proof Automation

In the proofs presented so far, we explicitly wrote every step of a function's evaluation. For example, in the base case of involutionP we twice applied the function reverse to the empty list. Writing proofs explicitly in this way is often helpful (for instance, it makes clear that to prove that reverse is an involution we need to prove that it distributes over append) but it can quickly become tedious.

To simplify proofs, Liquid Haskell employs the complete and terminating proof technique of Proof By (Logical) Evaluation (PLE). Conceptually, PLE executes functions for as many steps as needed and automatically provides all the resulting equations to the SMT solver.

Without using this technique, we could prove that the empty list is append's right identity as follows:

{-@ rightIdP :: xs:[a] -> { xs ++ [] == xs } @-} rightIdP :: [a] -> Proof rightIdP [] = [] ++ [] ==. [] *** QED rightIdP (x:xs) = (x:xs) ++ [] ==. x : (xs ++ []) ? rightIdP xs ==. x : xs *** QED

However, we can activate PLE in the definition of rightIdP using the ple rightIdP annotation. This automates all the rewriting steps, and the proof can be simplified to:

{-@ ple leftIdP @-} leftIdP :: [a] -> Proof {-@ leftIdP :: xs:[a] -> { xs ++ [] == xs } @-} leftIdP [] = () leftIdP (_:xs) = leftIdP xs

That is, the base case is fully automated by PLE, while in the inductive case we must make a recursive call to get the induction hypothesis, but the rest is taken care of by PLE.

Using this technique we can also prove the remaining lemma, namely the associativity of append:
{-@ ple assocP @-} {-@ assocP :: xs:[a] -> ys:[a] -> zs:[a] -> { xs ++ (ys ++ zs) == (xs ++ ys) ++ zs } @-} assocP :: [a] -> [a] -> [a] -> () assocP [] _ _ = () assocP (x:xs) ys zs = assocP xs ys zs

Again, we only have to give the structure of the induction and the arguments to the recursive call, and the PLE machinery adds all the necessary equations to complete the proof.

PLE is a powerful tool that makes proofs shorter and easier to write. However, proofs using this technique are usually more difficult to read, as they hide the details of function expansion. For this reason, while we could apply PLE to simplify many of the proofs in this paper, we prefer to spell out each step. Doing so keeps our proofs easier to understand and in close correspondence with the pen-and-paper proofs we reference in Graham's book.