`> {-# LANGUAGE NoMonomorphismRestriction, FlexibleInstances, TypeSynonymInstances #-}`

```
> import Test.QuickCheck
> import Control.Monad
> import Data.List
> import qualified Data.Map as M
> import Control.Monad.State hiding (when)
> import Control.Applicative ((<$>), (<*>))
```

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.

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.

A QuickCheck property is essentially a function whose output is a boolean. The standard “hello-world” QC property is

```
> 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 (that don’t contain type variables.)

To *check* a property, we simply invoke the function

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

lets try it on our example property above

```
ghci> quickCheck prop_revapp
*** Failed! Falsifiable (after 2 tests and 1 shrink):
[0]
[1]
```

Whats that ?! Well, lets run the *property* function on the two inputs

```
ghci> prop_revapp [0] [1]
False
```

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
+++ OK, passed 100 tests.
```

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 10000 prop_revapp_ok
+++ OK, passed 10000 tests.
```

Lets 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]
> rhs = [z | z <- xs, z > x]
```

Really doesn’t need much explanation! Lets run it “by hand” on a few inputs

```
ghci> [10,9..1]
[10,9,8,7,6,5,4,3,2,1]
ghci> qsort [10,9..1]
[1,2,3,4,5,6,7,8,9,10]
ghci> [2,4..20] ++ [1,3..11]
[2,4,6,8,10,12,14,16,18,20,1,3,5,7,9,11]
ghci> qsort $ [2,4..20] ++ [1,3..11]
[1,2,3,4,5,6,7,8,9,10,11,12,14,16,18,20]
```

Looks good – lets try to test that the output is in fact sorted. We need a function that checks that a list is ordered

```
> isOrdered (x1:x2:xs) = x1 <= x2 && isOrdered (x2:xs)
> isOrdered _ = True
```

and then we can use the above to write a property

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

Lets test it!

```
ghci> quickCheckN 10000 prop_qsort_isOrdered
+++ OK, passed 10000 tests.
```

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
```

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

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

```
ghci> quickCheck prop_qsort_min
*** Failed! Exception: 'Prelude.head: empty list' (after 1 test):
[]
```

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. This time around, both the properties hold

```
ghci> quickCheckN 1000 prop_qsort_nn_min
+++ OK, passed 1000 tests.
ghci> quickCheckN 1000 prop_qsort_nn_max
+++ OK, passed 1000 tests.
```

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 on of many QC combinators that allow the construction of rich properties.

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 *behaviourally* identical to a trusted *reference implementation* which itself may be too inefficient or otherwise unsuitable for deployment. In this case, lets 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

```
ghci> quickCheckN 1000 prop_qsort_sort
*** Failed! Falsifiable (after 4 tests and 1 shrink):
[-1,-1]
```

Say, what?!

```
ghci> qsort [-1,-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 (x:xs) = not (x `elem` xs) && isDistinct xs
> isDistinct _ = True
>
> 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] -> Gen Prop
> prop_qsort_distinct_sort xs =
> (isDistinct xs) ==> (qsort xs == sort xs)
```

QuickCheck happily checks the modified properties

```
ghci> quickCheck prop_qsort_distinct
+++ OK, passed 100 tests.
ghci> quickCheck prop_qsort_distinct_sort
+++ OK, passed 100 tests.
```

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

```
ghci> 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

```
ghci> insert 8 ([1..3] ++ [10..13])
[1,2,3,8,10,11,12,13]
```

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

`> 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
```

```
ghci> quickCheckN 10000 prop_isort_sort
+++ OK, passed 10000 tests.
```

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

```
ghci> quickCheckN 10000 prop_insert_ordered'
*** Failed! Falsifiable (after 4 tests and 1 shrink):
0
[0,-1]
ghci> insert 0 [0, -1]
[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

```
ghci> quickCheckN 10000 prop_insert_ordered
*** Gave up! Passed only 35 tests.
```

Ugh! The ordered lists are so *sparsely* distributed among random lists, that QC timed out well before it found 10000 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

```
ghci> quickCheckN 1000 prop_insert_ordered_vacuous
+++ OK, passed 10000 tests.
```

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

```
ghci> quickCheck prop_insert_ordered_vacuous'
+++ OK, passed 100 tests:
9% 1, ord
2% 0, ord
2% 2, ord
5% 8, not-ord
4% 7, not-ord
4% 5, not-ord
...
```

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!

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?!

```
ghci> quickCheck prop_insert_ordered'
*** Failed! Falsifiable (after 4 tests and 2 shrinks):
0
[0,-1]
ghci> quickCheck prop_insert_ordered'
*** Failed! Falsifiable (after 5 tests and 5 shrinks):
0
[1,0]
```

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!**

A Haskell term that generates a (random value) of type `a`

has the type `Gen a`

which is defined as

`newtype Gen a = MkGen { unGen :: StdGen -> Int -> a }`

In effect, the term is a function that takes as input a random number generator `StdGen`

and a seed `Int`

and returns an `a`

value. One can easily (and we shall see, profitably!) turn `Gen`

into a `Monad`

by

```
instance Monad Gen where
return x =
MkGen (\_ _ -> x)
MkGen m >>= k =
MkGen (\r n ->
let (r1, r2) = split r
MkGen m' = k (m r1 n)
in m' r2 n
)
```

The function `split`

simply *forks* the random number generator into two parts; which are used by the left and right parameters of the bind operator `>>=`

. (*Aside* you should be able to readily spot the similarity between random number generators and the `ST`

monad – in both cases the basic action is to grab some value and transition the *state* to the next-value. For more details see Chapter 14, RWH)

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

```
class Show a where
show :: a -> String
class Arbitrary a where
arbitrary :: Gen a
```

`> gimmeInts = sample' arbitrary`

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 much like we did for `JSON`

a few lectures back

```
instance (Arbitrary a, Arbitrary b) => Arbitrary (a,b) where
arbitrary = do x <- arbitrary
y <- arbitrary
return (x,y)
```

or more simply

```
instance (Arbitrary a, Arbitrary b) => Arbitrary (a,b) where
arbitrary = liftM2 (,) arbitrary arbitrary
do x <- mx
y <- my
return $ f x y
```

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`

.

```
ghci> sample $ choose (0, 3)
2
1
0
2
1
0
2
3
0
0
```

What is a plausible type for `sample`

?

`Gen a -> [a]`

`Gen a -> Gen [a]`

`Gen a -> IO [a]`

`Gen a -> IO a`

`a -> Gen [a]`

A second useful combinator is `elements`

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

fmap :: (a -> b) -> (m a) -> (m b) fmap f m = do x <- m return (f x)

elements xs = do i <- choose (0, (length xs) - 1) return (xs !! i)

elements xs = (xs !!) <$> choose (0, length xs - 1)

oneOf :: [Gen a] -> Gen a oneOf gs = do g <- elements gs x <- g return x

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

```
ghci> sample $ elements [10, 20..100]
60
70
30
50
30
20
20
10
100
80
10
```

A third combinator is `oneof`

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

which allows us to randomly choose between multiple generators

```
ghci> sample $ oneof [elements [10,20,30], choose (0,3)]
10
0
10
1
30
1
20
2
20
3
30
```

Lets try to figure out the **implementation** of `oneOf`

```
oneOf :: [Gen a] -> Gen a
oneOf = error "LETS DO THIS IN CLASS"
```

Finally, `oneOf`

is generalized into the `frequency`

combinator

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

which allows us to build weighted combinations of individual generators.

We can use the above combinators to write generators for lists

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

Can you spot a problem in the above?

```
-- btw, don't freak out, remember that
liftM2 f mx my = do x <- m1
y <- m2
return $ f x y
-- So the above is the same as
genList1 = do x <- arbitrary
xs <- gentList1
return $ x : xs
```

**Problem**: `genList1`

only generates infinite lists! Hmm. Lets try again,

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

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

```
> genList3 :: (Arbitrary a) => Gen [a]
> genList3 = frequency [ (1, return [])
> , (7, liftM2 (:) arbitrary genList2) ]
```

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

function

`> genOrdList = sort <$> genList3 `

```
-- again, remember that, <$> is just `fmap` where:
fmap f m = do {x <- m; return (f x)}
-- so really the above is the same as:
genOrdList = do { x <- genList3 ; return (sort x) }
```

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

```
ghci> quickCheckN 1000 $ forAll genOrdList isOrdered
+++ OK, passed 1000 tests.
```

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)
```

```
ghci> quickCheckN 1000 prop_insert
+++ OK, passed 1000 tests.
```

Next, lets look at how QC can be used to generate structured data, by doing a small case-study on checking a compiler optimization.

Recall the small *While* language that you wrote an evaluator for in HW2.

The languages had arithmetic expressions

```
> data Expression
> = Var Variable
> | Val Value
> | Plus Expression Expression
> | Minus Expression Expression
> deriving (Eq, Ord)
```

where the atomic expressions were either variables or values

```
> data Variable = V String deriving (Eq, Ord)
>
> data Value
> = IntVal Int
> | BoolVal Bool
> deriving (Eq, Ord)
```

We used the expressions to define imperative *statements* which are either assignments, if-then-else, a sequence of two statements, or a while-loop.

```
> data Statement
> = Assign Variable Expression
> | If Expression Statement Statement
> | While Expression Statement
> | Sequence Statement Statement
> | Skip
> deriving (Eq, Ord)
```

The behavior of *While* programs was given using a *state* which is simply a map from variables to values. Intuitively, a statement will *update* the state by modifying the values of the variables appropriately.

`> type WState = M.Map Variable Value`

Your assignment was to (use the `State`

monad to) write an evaluator (aka interpreter) for the language that took as input a program and a starting state and returned the final state.

```
> execute :: WState -> Statement -> WState
> execute env = flip execState env . evalS
```

Since you wrote the code for the HW (you **DID** didn’t you?) we won’t go into the details now – scroll down to the bottom to see how `evalS`

is implemented.

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

First, lets write a generator for variables.

```
> instance Arbitrary Variable where
> arbitrary = do x <- elements ['A'..'Z']
> return $ V [x]
```

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 [ IntVal <$> arbitrary
> , BoolVal <$> arbitrary ]
```

Third, we define a generator for `Expression`

and `Statement`

which selects from the different cases.

```
> -- instance Arbitrary Expression where
> -- arbitrary = sized arbnE
>
> -- arbE = frequency [ (1, Var <$> arbitrary)
> -- , (1, Val <$> arbitrary)
> -- , (5, Plus <$> arbitrary <*> arbitrary)
> -- , (5, Minus <$> arbitrary <*> arbitrary) ]
```

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

so that we can run the *While* program from some arbitrary input configuration.

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

In the above, `xvs`

is a randomly generated list of key-value tuples, which is turned into a `Map`

by the `fromList`

function.

Let `p1`

and `p2`

be two *While* programs. We say that `p1`

is *equivalent to* `p2`

if for all input configurations `st`

the configuration resulting from executing `p1`

from `st`

is the same as that obtained by executing `p2`

from `st`

. Formally,

```
> (===) :: Statement -> Statement -> Property
> p1 === p2 = forAll arbitrary $ \st -> execute st p1 == execute st p2
```

Excellent! Lets 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 x e =
> (x `Assign` (e `Plus` Val (IntVal 0))) === (x `Assign` e)
>
> prop_sub_zero_elim x e =
> (x `Assign` (e `Minus` Val (IntVal 0))) === (x `Assign` e)
```

Lets check the properties!

```
ghci> quickCheck prop_add_zero_elim
(0 tests)
...
```

Uh? whats going on? Well, lets look at the generator for expressions.

```
arbE = frequency [ (1, liftM Var arbitrary)
, (1, liftM Val arbitrary)
, (5, liftM2 Plus arbitrary arbitrary)
, (5, liftM2 Minus arbitrary arbitrary) ]
```

in effect, its will generate infinite expressions with high probability! (do the math!) So we need some way to control the size, either by biasing the `Var`

and `Val`

constructors (which terminate the generation) or by looking at the *size* of the structure during generation. We can do this with the combinator

`sized :: (Int -> Gen a) -> Gen a`

which lets us write functions that parameterize the generator with an integer (and then turn that into a flat generator.)

```
> arbnE 0 = oneof [ liftM Var arbitrary
> , liftM Val arbitrary ]
> arbnE n = frequency [ (1, liftM Var arbitrary)
> , (1, liftM Val arbitrary)
> , (5, liftM2 Plus (arbnE n_by_2) (arbnE n_by_2))
> , (5, liftM2 Minus (arbnE n_by_2) (arbnE n_by_2)) ]
> where n_by_2 = 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 = sized arbnE
```

And now, lets check the property again

```
ghci> quickCheck prop_add_zero_elim
*** Failed! Falsifiable (after 10 tests):
W
True
fromList [(R,False)]
```

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

`X := True + 0`

will assign `0`

to the variable while

`X := True `

will assign `True`

to the variable! Urgh. Ok, lets limit ourselves to *Integer* expressions

```
> 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 Plus (arbnEI n_by_2) (arbnEI n_by_2)
> , liftM2 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' x =
> forAll intE $ \e -> (x `Assign` (e `Plus` Val (IntVal 0))) === (x `Assign` e)
```

O, Quickcheck, what say you now?

```
ghci> quickCheck prop_add_zero_elim'
*** Failed! Falsifiable (after 16 tests):
V
N
fromList [(A,-36),(B,True),(D,False),(E,44),(L,-32),(M,22),(N,True),(O,False),(Q,True),(S,True),(W,50)]
```

Of course! in the input state where `N`

has the value `True`

, the result of executing `V := N`

is quite different from executing `V := N + 0`

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

Well, that first one ran aground because *While* was untyped (tsk tsk.) and so adding a zero can cause problems if the expression is a boolean. Lets look at another optimization that is not plagued by the int-v-bool conflict. Suppose you have two back-to-back assignments

```
X := E
Y := E
```

It is silly to recompute `E`

twice, since the result is already stored in `X`

. So, we should be able to optimize the above code to

```
X := E
Y := X
```

Lets see how we might express the correctness of this transformation as a QC property

```
> prop_const_prop x y e =
> ((x `Assign` e) `Sequence` (y `Assign` e))
> ===
> ((x `Assign` e) `Sequence` (y `Assign` Var x))
```

Mighty QC, do you agree ?

```
*Main> quickCheck prop_const_prop
*** Failed! Falsifiable (after 35 tests):
Z
P
O + A + J + G + C + False + O + K + 6965 + True + True + W + T + K + 5266 + J + Z + R + -1588 + P + R + 3667 + B + Q + T + Y + Z + X + M + False + -1191 + 6124 + H + B + 1351 + S + T + E + R + 6969
fromList [(B,True),(D,3714),(E,True),(P,2455),(Q,True),(Y,-7341)]
```

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 (Plus e1 e2) = [e1, e2]
> shrink (Minus e1 e2) = [e1, e2]
> shrink _ = []
```

Lets try it again to see if we can figure it out!

```
ghci> quickCheck 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 shockingly 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 thousands of 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.

We don’t have exceptions, so if a variable is not found, return value 0

```
> evalE :: Expression -> State WState Value
> evalE (Var x) = get >>= return . M.findWithDefault (IntVal 0) x
> evalE (Val v) = return v
> evalE (Plus e1 e2) = return (intOp (+) 0 IntVal) `ap` evalE e1 `ap` evalE e2
> evalE (Minus e1 e2) = return (intOp (-) 0 IntVal) `ap` evalE e1 `ap` evalE e2
>
> evalS :: Statement -> State WState ()
> evalS w@(While e s) = evalS (If e (Sequence s w) Skip)
> evalS Skip = return ()
> evalS (Sequence s1 s2) = evalS s1 >> evalS s2
> evalS (Assign x e ) = do v <- evalE e
> m <- get
> put $ M.insert x v m
> return ()
> evalS (If e s1 s2) = do v <- evalE e
> case v of
> BoolVal True -> evalS s1
> BoolVal False -> evalS s2
> _ -> return ()
```

Return `0`

for arithmetic operations over a `Bool`

value.

```
> intOp :: (Int -> Int -> a) -> a -> (a -> Value) -> Value -> Value -> Value
> intOp op _ c (IntVal x) (IntVal y) = c $ x `op` y
> intOp _ d c _ _ = c d
```

```
> blank n = replicate n ' '
>
> instance Show Variable where
> show (V x) = x
>
> instance Show Value where
> show (IntVal i) = show i
> show (BoolVal b) = show b
>
> instance Show Expression where
> show (Var v) = show v
> show (Val v) = show v
> show (Plus e1 e2) = show e1 ++ " + " ++ show e2
> show (Minus e1 e2) = show e1 ++ " + " ++ show e2
>
> instance Show Statement where
> show = showi 0
>
> showi n (Skip) = blank n ++ "skip"
> showi n (Assign x e) = blank n ++ show x ++ " := " ++ show e
> showi n (If e s1 s2) = blank n ++ "if " ++ show e ++ " then\n" ++
> showi (n+2) s1 ++
> blank n ++ "else\n" ++
> showi (n+2) s2 ++
> blank n ++ "endif"
>
> showi n (While e s) = blank n ++ "while " ++ show e ++ " do\n" ++
> showi (n+2) s
> showi n (Sequence s1 s2) = showi n s1 ++ "\n" ++ showi n s2
```

```
> instance Arbitrary Statement where
> arbitrary = oneof [ liftM2 Assign arbitrary arbitrary
> , liftM3 If arbitrary arbitrary arbitrary
> , liftM2 While arbitrary arbitrary
> , liftM2 Sequence arbitrary arbitrary
> , return Skip ]
```