## QuickCheck: Type-directed Property Testing

`module QuickCheck where`

In this lecture, we will look at QuickCheck, a technique that cleverly exploits typeclasses and monads to deliver a powerful automatic testing methodology.

Quickcheck was developed by Koen Claessen and John Hughes more than ten years ago, and has since been ported to other languages and is currently used, among other things to find subtle concurrency bugs in telecommunications code. In 2010, it received the most influential paper award for the ICFP 2000 conference.

The key idea on which QuickCheck is founded is *property-based testing*. That is, instead of writing individual test cases (eg unit tests corresponding to input-output pairs for particular functions) one should write *properties* that are desired of the functions, and then *automatically* generate *random* tests which can be run to verify (or rather, falsify) the property.

By emphasizing the importance of specifications, QuickCheck yields several benefits:

The developer is forced to think about what the code

*should do*,The tool finds corner-cases where the specification is violated, which leads to either the code or the specification getting fixed,

The specifications live on as rich, machine-checkable documentation about how the code should behave.

To use the QuickCheck library, you need to first install it with cabal.

`cabal install quickcheck`

```
import Test.QuickCheck
import Control.Monad(liftM,liftM2,liftM3)
import Data.List(sort,insert)
import Data.Maybe (fromMaybe)
```

```
import Data.Map(Map)
import qualified Data.Map as Map
```

# Properties

A QuickCheck property is essentially a function whose output is a boolean. A standard "hello-world" QC property might be something about common functions on lists.

```
prop_revapp :: [Int] -> [Int] -> Bool
prop_revapp xs ys = reverse (xs ++ ys) == reverse xs ++ reverse ys
```

That is, a property looks a bit like a mathematical theorem that the programmer believes is true. A QC convention is to use the prefix `"prop_"`

for QC properties. Note that the type signature for the property is not the usual polymorphic signature; we have given the concrete type `Int`

for the elements of the list. This is because QC uses the types to generate random inputs, and hence is restricted to monomorphic properties (those that don't contain type variables.)

To *check* a property, we simply invoke the quickCheck action with the property. Note that only certain types of properties can be tested, these properties are all in the 'Testable' type class.

```
quickCheck :: (Testable prop) => prop -> IO ()
-- Defined in Test.QuickCheck.Test
```

`[Int] -> [Int] -> Bool`

is a Testable property, so let's try quickCheck on our example property above

`*Main> quickCheck prop_revapp `

What's that ?! Well, let's run the *property* function on the two inputs

`*Main> prop_revapp [0] [1] `

QC has found a sample input for which the property function *fails* ie, returns `False`

. Of course, those of you who are paying attention will realize there was a bug in our property, namely it should be

```
prop_revapp_ok :: [Int] -> [Int] -> Bool
prop_revapp_ok xs ys = reverse (xs ++ ys) == reverse ys ++ reverse xs
```

because `reverse`

will flip the order of the two parts `xs`

and `ys`

of `xs ++ ys`

. Now, when we run

`*Main> quickCheck prop_revapp_ok`

That is, Haskell generated 100 test inputs and for all of those, the property held. You can up the stakes a bit by changing the number of tests you want to run

`quickCheckN n = quickCheckWith $ stdArgs { maxSuccess = n }`

and then do

`*Main> quickCheckN 1000 prop_revapp_ok`

## QuickCheck QuickSort

Let's look at a slightly more interesting example. Here is the canonical implementation of *quicksort* in Haskell.

```
qsort [] = []
qsort (x:xs) = qsort lhs ++ [x] ++ qsort rhs
where lhs = [y | y <- xs, y < x] -- this is a "list comprehension"
rhs = [z | z <- xs, z > x]
```

Really doesn't need much explanation! Let's run it "by hand" on a few inputs

```
*Main> [10,9..1]
*Main> qsort [10,9..1]
*Main> [2,4..20] ++ [1,3..11]
*Main> qsort $ [2,4..20] ++ [1,3..11]
```

Looks good -- let's try to test that the output is in fact sorted. We need a function that checks that a list is ordered

```
isOrdered :: Ord a => [a] -> Bool
isOrdered [] = True
isOrdered [x] = True
isOrdered (x:y:xs) = x <= y && isOrdered (y:xs)
```

and then we can use the above to write a property saying that the result of qsort is an ordered list.

```
prop_qsort_isOrdered :: [Int] -> Bool
prop_qsort_isOrdered = isOrdered . qsort
```

Let's test it!

`*Main> quickCheckN 1000 prop_qsort_isOrdered `

## Conditional Properties

Here are several other properties that we might want. First, repeated `qsorting`

should not change the list. That is,

```
prop_qsort_idemp :: [Int] -> Bool
prop_qsort_idemp xs = qsort (qsort xs) == qsort xs
```

Second, the head of the result is the minimum element of the input

```
prop_qsort_min :: [Int] -> Bool
prop_qsort_min xs = head (qsort xs) == minimum xs
```

`*Main> quickCheck prop_qsort_min `

However, when we run this, we run into a glitch.

But of course! The earlier properties held *for all inputs* while this property makes no sense if the input list is empty! This is why thinking about specifications and properties has the benefit of clarifying the *preconditions* under which a given piece of code is supposed to work.

In this case we want a *conditional properties* where we only want the output to satisfy to satisfy the spec *if* the input meets the precondition that it is non-empty.

```
prop_qsort_nn_min :: [Int] -> Property
prop_qsort_nn_min xs =
not (null xs) ==> head (qsort xs) == minimum xs
prop_qsort_nn_max :: [Int] -> Property
prop_qsort_nn_max xs =
not (null xs) ==> head (reverse (qsort xs)) == maximum xs
```

We can write a similar property for the maximum element too.

```
*Main> quickCheckN 100 prop_qsort_nn_min
*Main> quickCheckN 100 prop_qsort_nn_max
```

This time around, both the properties hold.

Note that now, instead of just being a `Bool`

the output of the function is a `Property`

a special type built into the QC library. Similarly the *implies* combinator `==>`

is one of many QC combinators that allow the construction of rich properties.

## Testing Against a Model Implementation

We could keep writing different properties that capture various aspects of the desired functionality of `qsort`

. Another approach for validation is to test that our `qsort`

is *behaviorally* identical to a trusted *reference implementation* which itself may be too inefficient or otherwise unsuitable for deployment. In this case, let's use the standard library's `sort`

function

```
prop_qsort_sort :: [Int] -> Bool
prop_qsort_sort xs = qsort xs == sort xs
```

which we can put to the test

`*Main> quickCheckN 1000 prop_qsort_sort`

Say, what?!

`*Main> qsort [-1,-1]`

Ugh! So close, and yet ... Can you spot the bug in our code?

```
qsort [] = []
qsort (x:xs) = qsort lhs ++ [x] ++ qsort rhs
where lhs = [y | y <- xs, y <= x]
rhs = [z | z <- xs, z > x]
```

We're assuming that the *only* occurrence of (the value) `x`

is itself! That is, if there are any *copies* of `x`

in the tail, they will not appear in either `lhs`

or `rhs`

and hence they get thrown out of the output.

Is this a bug in the code? What *is* a bug anyway? Perhaps the fact that all duplicates are eliminated is a *feature*! At any rate there is an inconsistency between our mental model of how the code *should* behave as articulated in `prop_qsort_sort`

and the actual behavior of the code itself.

We can rectify matters by stipulating that the `qsort`

produces lists of distinct elements

```
isDistinct :: Eq a => [a] -> Bool
isDistinct [] = True
isDistinct (x:xs) = not (x `elem` xs) && isDistinct xs
prop_qsort_distinct :: [Int] -> Bool
prop_qsort_distinct = isDistinct . qsort
```

and then, weakening the equivalence to only hold on inputs that are duplicate-free

```
prop_qsort_distinct_sort :: [Int] -> Property
prop_qsort_distinct_sort xs =
(isDistinct xs) ==> qsort xs == sort xs
```

QuickCheck happily checks the modified properties

```
*Main> quickCheck prop_qsort_distinct
*Main> quickCheck prop_qsort_distinct_sort
```

## The Perils of Conditional Testing

Well, we managed to *fix* the `qsort`

property, but beware! Adding preconditions leads one down a slippery slope. In fact, if we paid closer attention to the above runs, we would notice something

```
*Main> quickCheckN 10000 prop_qsort_distinct_sort
...
(5012 tests; 248 discarded)
...
+++ OK, passed 10000 tests.
```

The bit about some tests being *discarded* is ominous. In effect, when the property is constructed with the `==>`

combinator, QC discards the randomly generated tests on which the precondition is false. In the above case QC grinds away on the remainder until it can meet its target of `10000`

valid tests. This is because the probability of a randomly generated list meeting the precondition (having distinct elements) is high enough. This may not always be the case.

The following code is (a simplified version of) the `insert`

function from the standard library

```
insert x [] = [x]
insert x (y:ys) | x > y = x : y : ys
| otherwise = y : insert x ys
```

Given an element `x`

and a list `xs`

, the function walks along `xs`

till it finds the first element greater than `x`

and it places `x`

to the left of that element. Thus

`*Main> insert 8 ([1..3] ++ [10..13])`

Indeed, the following is the well known insertion-sort algorithm

```
isort :: Ord a => [a] -> [a]
isort = foldr insert []
```

We could write our own tests, but why do something a machine can do better?!

```
prop_isort_sort :: [Int] -> Bool
prop_isort_sort xs = isort xs == sort xs
```

`*Main> quickCheckN 1000 prop_isort_sort `

Now, the reason that the above works is that the `insert`

routine *preserves* sorted-ness. That is while of course the property

```
prop_insert_ordered' :: Int -> [Int] -> Bool
prop_insert_ordered' x xs = isOrdered (insert x xs)
```

is bogus

```
*Main> quickCheckN 1000 prop_insert_ordered'
*Main> insert 0 [0, -1]
```

the output *is* ordered if the input was ordered to begin with

```
prop_insert_ordered :: Int -> [Int] -> Property
prop_insert_ordered x xs =
isOrdered xs ==> isOrdered (insert x xs)
```

Notice that now, the precondition is more *complex* -- the property requires that the input list be ordered. If we QC the property

`*Main> quickCheckN 1000 prop_insert_ordered`

Ugh! The ordered lists are so *sparsely* distributed among random lists, that QC timed out well before it found 1000 valid inputs!

*Aside* the above example also illustrates the benefit of writing the property as `p ==> q`

instead of using the boolean operator `||`

to write `not p || q`

. In the latter case, there is a flat predicate, and QC doesn't know what the precondition is, so a property may hold *vacuously*. For example consider the variant

```
prop_insert_ordered_vacuous :: Int -> [Int] -> Bool
prop_insert_ordered_vacuous x xs =
not (isOrdered xs) || isOrdered (insert x xs)
```

QC will happily check it for us

`*Main> quickCheckN 1000 prop_insert_ordered_vacuous`

Unfortunately, in the above, the tests passed *vacuously* only because their inputs were *not* ordered, and one should use `==>`

to avoid the false sense of security delivered by vacuity.

QC provides us with some combinators for guarding against vacuity by allowing us to investigate the *distribution* of test cases

```
collect :: Show a => a -> Property -> Property
classify :: Bool -> String -> Property -> Property
```

We may use these to write a property that looks like

```
prop_insert_ordered_vacuous' :: Int -> [Int] -> Property
prop_insert_ordered_vacuous' x xs =
collect (length xs) $
classify (isOrdered xs) "ord" $
classify (not (isOrdered xs)) "not-ord" $
not (isOrdered xs) || isOrdered (insert x xs)
```

When we run this, as before we get a detailed breakdown of the 100 passing tests

`*Main> quickCheck prop_insert_ordered_vacuous'`

where a line `P% N, COND`

means that `p`

percent of the inputs had length `N`

and satisfied the predicate denoted by the string `COND`

. Thus, as we see from the above, a paltry 13% of the tests were ordered and that was because they were either empty (`2% 0, ord`

) or had one (`9% 1, ord`

). or two elements (`2% 2, ord`

). The odds of randomly stumbling upon a beefy list that is ordered are rather small indeed!

# Generating Data

Before we start discussing how QC generates data (and how we can help it generate data meeting some pre-conditions), we must ask ourselves a basic question: how does QC behave *randomly* in the first place?!

```
*Main> quickCheck prop_insert_ordered'
*Main> quickCheck prop_insert_ordered'
```

Eh? This seems most *impure* -- same inputs yielding two totally different outputs! Well, this should give you a clue as to one of the key techniques underlying QC -- **monads!**

The QC library defines a type

Gen a

of "generators for values of type a". The parameterized type 'Gen' is an instance of the monad type class. What this means (for today) is that there are a number of monadic operations available for it.

```
-- from the class Monad
--
return :: a -> Gen a
(>>=) :: Gen a -> (a -> Gen b) -> Gen b
-- from the library Control.Monad
--
liftM :: (a -> b) -> Gen a -> Gen b
liftM2 :: (a -> b -> c) -> Gen a -> Gen b -> Gen c
liftM3 :: (a -> b -> c -> d) -> Gen a -> Gen b -> Gen c -> Gen d
```

Note, `liftM`

above has another name---`fmap`

. That's right, every monad is also a functor.

We will cover what it exactly means for Gen to be a monad in a future lecture.

## The Arbitrary Typeclass

QC uses the above to define a typeclass for types for which random values can be generated!

```
class Arbitrary a where
arbitrary :: Gen a
```

Thus, to have QC work with (ie generate random tests for) values of type `a`

we need only make `a`

an instance of `Arbitrary`

by defining an appropriate `arbitrary`

function for it. QC defines instances for base types like `Int`

, `Float`

, lists etc and lifts them to compound types.

```
instance (Arbitrary a, Arbitrary b, Arbitrary c) => Arbitrary (a,b,c) where
arbitrary = liftM3 (,,) (arbitrary :: Gen a) (arbitrary :: Gen b) (arbitrary :: Gen c)
```

## Generator Combinators

QC comes loaded with a set of combinators that allow us to create custom instances for our own types.

The first of these combinators is `choose`

`choose :: (System.Random.Random a) => (a, a) -> Gen a`

which takes an *interval* and returns an random element from that interval. (The typeclass `System.Random.Random`

describes types which can be *sampled*. For example, the following is a randomly chosen set of numbers between `0`

and `3`

.

`*Main> sample $ choose (0, 3)`

A second useful combinator is `elements`

`data Day = Monday | Tuesday | Wednesday deriving (Show)`

`elements :: [a] -> Gen a`

which returns a generator that produces values drawn from the input list

`*Main> sample $ elements [10, 20..100]`

A third combinator is `oneof`

`oneof :: [Gen a] -> Gen a`

which allows us to randomly choose between multiple generators

`*Main> sample $ oneof [elements [10,20,30], choose (0,3)]`

and finally, the above is generalized into the `frequency`

combinator

`frequency :: [(Int, Gen a)] -> Gen a`

which allows us to build weighted combinations of individual generators.

## Generating Ordered Lists

We can use the above combinators to write generators for lists

```
genList1 :: (Arbitrary a) => Gen [a]
genList1 = liftM2 (:) arbitrary genList1
```

`*Main> sample (genList1 :: Gen [Int])`

Can you spot a problem in the above? It only generates infinite lists! Hmm. Let's try again,

```
genList2 :: (Arbitrary a) => Gen [a]
genList2 = oneof [ return []
, liftM2 (:) arbitrary genList2]
```

`*Main> sample (genList2 :: Gen [Int])`

This is not bad, but we may want to give the generator a higher chance of not finishing off with the empty list, so let's use

```
genList3 :: (Arbitrary a) => Gen [a]
genList3 = frequency [ (1, return [])
, (7, liftM2 (:) arbitrary genList3) ]
```

`*Main> sample (genList3 :: Gen [Int])`

We can use the above to build a custom generator that always returns *ordered lists* by piping the generated list into the `sort`

function

```
genOrdList :: (Arbitrary a, Ord a) => Gen [a]
genOrdList = genList3 >>= return . sort
```

`*Main> sample (genOrdList :: Gen [Int])`

To *check* the output of a custom generator we can use the `forAll`

combinator

`forAll :: (Show a, Testable prop) => Gen a -> (a -> prop) -> Property`

For example, we can check that in fact, the combinator only produces ordered lists

`*Main> quickCheck $ forAll genOrdList isOrdered `

and now, we can properly test the `insert`

property

```
prop_insert :: Int -> Property
prop_insert x = forAll genOrdList $ \xs -> isOrdered xs && isOrdered (insert x xs)
```

`*Main> quickCheck prop_insert `

# Case study: Checking Compiler Optimizations

Next, let's look at how QC can be used to generate structured data, by doing a small case study on program expressions.

The language we will use has arithmetic expressions only. Variables and values are called "atomic expressions", whereas the last case lets us build larger bits of arithmetic.

```
data Expression =
Var Variable
| Val Value
| Op Bop Expression Expression
deriving (Eq, Ord)
```

Variables are just strings, but instead of using strings directly, we create a newtype.

`newtype Variable = V String deriving (Eq, Ord)`

Values include literal int or boolean values.

```
data Value =
IntVal Int
| BoolVal Bool
deriving (Eq, Ord)
```

The language also includes several built-in boolean operators.

```
data Bop =
Plus -- + :: Int -> Int -> Int
| Minus -- - :: Int -> Int -> Int
| Times -- * :: Int -> Int -> Int
| Gt -- > :: Int -> Int -> Bool
| Ge -- >= :: Int -> Int -> Bool
| Lt -- < :: Int -> Int -> Bool
| Le -- <= :: Int -> Int -> Bool
deriving (Eq, Ord)
```

## Showing expressions

Before we go any further with expressions, we'd like a way to look at them. We could just add "deriving (Show)" to the datatypes above, but that will lead to fairly large examples. Instead, let's show them as traditional mathematical expressions.

```
instance Show Variable where
show (V x) = x
```

```
instance Show Value where
show (IntVal x) = show x
show (BoolVal x) = show x
```

```
instance Show Bop where
show b = case b of
Plus -> "+"
Minus -> "-"
Times -> "*"
Gt -> ">"
Ge -> ">="
Lt -> "<"
Le -> "<="
```

For nonatomic expressions, we'll use a little operator precedence to conditionally add parens where necessary. So instead of defining 'show', we will define 'showsPrec' in the Show instance for Expressions.

```
instance Show Expression where
showsPrec d (Var x) = showsPrec d x
showsPrec d (Val x) = showsPrec d x
showsPrec d (Op bop e1 e2) = showParen (d > op_prec) $
showsPrec (op_prec + 1) e1 .
showsPrec d bop .
showsPrec (op_prec + 1) e2
where op_prec = precedence bop
```

```
precedence :: Bop -> Int
precedence Times = 8
precedence Plus = 7
precedence Minus = 7
precedence Gt = 6
precedence Ge = 6
precedence Lt = 6
precedence Le = 6
```

*Main> show (Op Plus (Val (IntVal 1)) (Op Times (Val (IntVal 2)) (Val (IntVal 3))))

*Main> show (Op Times (Val (IntVal 1)) (Op Plus (Val (IntVal 2)) (Val (IntVal 3))))

## Semantics

To compute the value of an expression, we need a Store---a map from a map from variables to values. Finite Maps (AKA dictionaries) are part of the standard library Data.Map.

`type Store = Map Variable Value`

With this store, we can determine the value of any expression. If there is some sort of error in the evaluation (i.e. the variable is not in the map, or the wrong type of values are given to an operator) we'll just return null, er, IntVal 0.

```
evalBop :: Bop -> Value -> Value -> Value
evalBop Plus (IntVal v1) (IntVal v2) = IntVal (v1 + v2)
evalBop Times (IntVal v1) (IntVal v2) = IntVal (v1 * v2)
evalBop Minus (IntVal v1) (IntVal v2) = IntVal (v1 - v2)
evalBop Gt (IntVal v1) (IntVal v2) = BoolVal (v1 > v2)
evalBop Ge (IntVal v1) (IntVal v2) = BoolVal (v1 >= v2)
evalBop Lt (IntVal v1) (IntVal v2) = BoolVal (v1 < v2)
evalBop Le (IntVal v1) (IntVal v2) = BoolVal (v1 <= v2)
evalBop _ _ _ = IntVal 0
```

```
eval :: Expression -> Store -> Value
eval (Var x) m = Map.findWithDefault (IntVal 0) x m
eval (Val v) m = v
eval (Op bop e1 e2) m = evalBop bop (eval e1 m) (eval e2 m)
```

## Generating Expressions

We could painstakingly write manual test cases, but instead let's write some simple generators for expressions, so that we can then check interesting properties of the programs and the evaluator.

First, let's write a generator for variables.

```
instance Arbitrary Variable where
-- arbitrary :: Gen Variable
arbitrary = elements (map (\ c -> V [c]) [ 'A' .. 'Z' ])
```

thus, we assume that the programs are over variables drawn from the uppercase alphabet characters. That is, our test programs range over 26 variables (you can change the above if you like.)

Second, we can write a generator for constant values (that can appear in expressions). Our generator simply chooses between randomly generated `Bool`

and `Int`

values.

```
instance Arbitrary Value where
arbitrary = oneof [ liftM IntVal arbitrary, liftM BoolVal arbitrary ]
```

Third, we define a generator for `Expression`

and `Bop`

which selects from the different cases.

```
instance Arbitrary Bop where
arbitrary = elements [Plus, Times, Minus, Gt, Ge, Lt, Le]
```

For expressions, we'll use frequency, choosing binary operators five times more often than the atomic expressions.

```
{-
instance Arbitrary Expression where
arbitrary = frequency [ (1, liftM Val arbitrary),
(1, liftM Var arbitrary),
(5, liftM3 Op arbitrary arbitrary arbitrary) ]
-}
```

Finally, we need to write a generator for `Store`

so that we can evaluate the expression with some arbitrary store.

```
instance (Ord a, Arbitrary a, Arbitrary b) => Arbitrary (Map a b) where
arbitrary = liftM Map.fromList arbitrary
```

In the above, we randomly generate a list of key-value tuples, which is turned into a `Map`

by the `fromList`

function.

## Expression Equivalence

Let `e1`

and `e2`

be two expressions. We say that `e1`

is *equivalent to* `e2`

if for all stores `st`

the value resulting from `e1`

with `st`

is the same as that obtained by evaluating `e2`

from `st`

. Formally,

```
(===) :: Expression -> Expression -> Property
e1 === e2 = forAll arbitrary $ \st -> eval e1 st == eval e2 st
```

## Checking An Optimization: Zero-Add-Elimination

Excellent! Let's take our generators our for a spin, by checking some *compiler optimizations*. Intuitively, a compiler optimization (or transformation) can be viewed as a *pair* of programs -- the input program `p_in`

and a transformed program `p_out`

. A transformation `(p_in, p_out)`

is *correct* iff `p_in`

is equivalent to `p_out`

.

Here's are some simple *sanity* check properties that correspond to optimizations.

```
prop_add_zero_elim :: Variable -> Expression -> Property
prop_add_zero_elim x e =
(Op Plus e $ Val (IntVal 0)) === e
prop_sub_zero_elim :: Variable -> Expression -> Property
prop_sub_zero_elim x e =
(Op Minus e $ Val (IntVal 0)) === e
```

Let's check the properties!

`*Main> quickCheck prop_add_zero_elim `

Uh? whats going on? Well, let's look at the generator for expressions. It generates some pretty big ones. Can we tone this down a bit without getting too many atomic expressions?

```
arbnE :: Int -> Gen Expression
arbnE n = frequency [ (1, liftM Var arbitrary),
(1, liftM Val arbitrary),
(n, liftM3 Op arbitrary (arbnE (n `div` 2)) (arbnE (n `div` 2))) ]
```

In the above, we keep *halving* the number of allowed nodes, and when that number goes to `0`

we just return an atomic expression (either a variable or a constant.) We can now update the generator for expressions to

```
{-
instance Arbitrary Expression where
arbitrary = intE
-}
```

And now, let's check the property again

`*Main> quickCheck prop_add_zero_elim `

whoops! Forgot about those pesky boolean expressions! If you think about it,

`True + 0`

evaluates to 0, whereas

`True `

Is just True.

Ok, let's limit ourselves to *Integer* expressions (using a slightly different way of controlling the size):

```
intE :: Gen Expression
intE = sized arbnEI
where arbnEI 0 = oneof [ liftM Var arbitrary
, liftM (Val . IntVal) arbitrary ]
arbnEI n = oneof [ liftM Var arbitrary
, liftM (Val . IntVal) arbitrary
, liftM2 (Op Plus) (arbnEI n_by_2) (arbnEI n_by_2)
, liftM2 (Op Times) (arbnEI n_by_2) (arbnEI n_by_2)
, liftM2 (Op Minus) (arbnEI n_by_2) (arbnEI n_by_2)
]
where n_by_2 = n `div` 2
```

using which, we can tweak the property to limit ourselves to integer expressions

```
prop_add_zero_elim' :: Property
prop_add_zero_elim' =
forAll intE $ \e -> (Op Plus e $ Val (IntVal 0)) === e
```

O, Quickcheck, what say you now?

`*Main> quickCheck prop_add_zero_elim'`

Of course! in the input state where `N`

has the value `True`

, the result of evaluating `N`

is quite different from executing `N + 0`

. Oh well, so much for that optimization, I guess we need some type information before we can eliminate the additions-to-zero!

## Checking An Optimization: Constant Folding (sort of)

Well, that first one ran aground because expressions are untyped (tsk tsk.) and so adding a zero can cause problems if the expression is a boolean. Let's look at another optimization that is not plagued by the int-v-bool conflict.

Suppose we evaluate e to a value v, and then name that value x in the store. Then evaluating x should be the same as evaluating e, right? This behavior is sort of like constant propagation---replacing this code:

```
X := e
Y := e
```

with this this one:

```
X := e
Y := X
```

Let's see how we might express the correctness of this transformation as a QC property

```
prop_const_prop :: Variable -> Expression -> Store -> Bool
prop_const_prop x e s = eval (Var x) s' == eval e s' where
s' = Map.insert x (eval e s) s
```

Mighty QC, do you agree ?

`*Main> quickCheck prop_const_prop `

## Shrinking

Holy transfer function!! It fails?!! And what is that bizarre test? It seems rather difficult to follow. Turns out, QC comes with a *test shrinking* mechanism; all we need do is add to the `Arbitrary`

instance a function of type

`shrink :: a -> [a]`

which will take a candidate and generate a list of *smaller* candidates that QC will systematically crunch through till it finds a minimally failing test!

```
instance Arbitrary Expression where
arbitrary = sized arbnE
shrink (Var _) = []
shrink (Val _) = []
shrink (Op bop e1 e2) = [e1, e2] ++ (map (\ x -> Op bop x e2) (shrink e1))
++ (map (Op bop e1) (shrink e2))
```

Let's try it again to see if we can figure it out!

```
*Main> quickCheckN 1000 prop_const_prop
*** Failed! Falsifiable (after 26 tests and 4 shrinks):
D
U
A + D
fromList [(D,-638),(G,256),(H,False),(K,False),(O,True),(R,True),(S,-81),(T,926)]
```

Aha! Consider the two programs

```
D := A + D;
U := A + D
```

and

```
D := A + D;
U := D
```

are they equivalent? Pretty subtle, eh.

Well, I hope I've convinced you that QuickCheck is pretty awesome. The astonishing thing about it is its sheer simplicity -- a few fistfuls of typeclasses and a tiny pinch of monads and lo! a shocking useful testing technique that can find a bunch of subtle bugs or inconsistencies in your code.

Moral of the story -- types can go a long way towards making your code *obviously correct*, but not the whole distance. Make up the difference by writing properties, and have the machine crank out tests for you!

There is a lot of literature on QuickCheck on the web. It is used for a variety of commercial applications, both in Haskell and in pretty much every modern language, including Perl. Even if you don't implement a system in Haskell, you can use QuickCheck to test it, by just using the nifty data generation facilities.

Credit: lecture is adapted from UCSD 12.

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