- Functions that perform IO operations have in their return type:
**IO**`putStr`**::****String****->****IO**()`getLine`**::****IO****String**`writeFile`**::****FilePath****->****String****->****IO**()`readFile`**::****FilePath****->****IO****String**

- We can build larger IO functions using the -notation:
**do**`copyFile`**::****FilePath****->****FilePath****->****IO**()`copyFile``from``to`**=****do**`text`**<-**`readFile``from``writeFile``to``text`

- We can not hide a call to an IO function inside a pure function.
- Pure functions have no side effects. Same argument ==> same result.

`prop_example`**::****Integer****->****Bool**`prop_example``n`**=**(`n``+`3)`^`2`==``n``^`2`+`6`*``n``+`9`quickCheck prop_example`

+++ OK, passed 100 tests.- QuickCheck generates 100 random integers. How is this done?

- QuickCheck uses a type for random test data generators and provides an overloaded function with instances available for most predefined types:
**Gen**`a` `arbitrary`**::****Arbitrary**`a`**=>****Gen**`a`- There are also functions that let you have a look at the random values generators produce:
`sample`**::****Show**`a`**=>****Gen**`a`**->****IO**()`sample'`**::****Gen**`a`**->****IO**[`a`]`generate`**::****Gen**`a`**->****IO**`a`

`sample' (arbitrary::Gen Integer)`

[0,2,-1,-6,2,4,4,-11,14,9,3]`sample' (arbitrary::Gen Integer)`

[0,2,-2,-3,5,4,-9,-5,16,-13,-6]- Same argument, but different results.

`sample' (arbitrary::Gen Bool)`

[True,False,True,False,False,True,True,False,False,True,True]- Two tests would be enough...

`sample (arbitrary::Gen [Integer])`

[] [2] [-3] [4] [7,-2,-3,-8,5,8,-5] [3,-4] [-10,0,7,1,2,5,-12,-3,8,-7,10,11] [-2,3,-5,-8,-3,-2,14,6,1,6,7,-2,-3,-12] [15,-13,15,-11,-11,-6,13,16,7,-13,-8,-2,-13,13,3,-13] [5,-15,5,13,12,1,-3,4,-13,-15,-7,18,-9,-18] [-8,-20,-10,-4,6,5,-15]- Short lists first, then longer lists.

- From the QuickCheck library:
`elements`**::**[`a`]**->****Gen**`a``oneof`**::**[**Gen**`a`]**->****Gen**`a``frequency`**::**[(**Int**,**Gen**`a`)]**->****Gen**`a``listOf`**::****Gen**`a`**->****Gen**[`a`]`vectorOf`**::****Int****->****Gen**`a`**->****Gen**[`a`]`choose`**::****Random**`a`**=>**(`a`,`a`)**->****Gen**`a`- uses functions from the module
`choose`

.**System.Random**

- Testing and
`choose``listOf` `sample' (choose ('a','z'))`

"ozlikagbygw"`sample' (listOf (choose ('a','z')))`

["","t","xviy","","hfpkft","","iiisswjvrkg","suoz","slfosefhofpgla","","l"]

- Remember the type

?**Suit****data****Suit****=****Spades****|****Hearts****|****Diamonds****|****Clubs****deriving**(**Eq**,**Show**)

- The function is perfect for generating random suits.
`elements``rSuit`**::****Gen****Suit**`rSuit`**=**`elements`[**Spades**,**Hearts**,**Diamonds**,**Clubs**]

**Rank**

?
**data****Rank****=****Numeric****Integer****|****Jack****|****Queen****|****King****|****Ace****deriving**(**Eq**,**Show**)- We could create a generator with and a list of all ranks. Here another way that gives us more flexibility:
`elements` `rNumeric`**::****Gen****Rank**`rNumeric`**=**`elements`[**Numeric**`n`**|**`n`**<-**[2**..**10]]`rFaceCard`**::****Gen****Rank**`rFaceCard`**=**`elements`[**Jack**,**Queen**,**King**,**Ace**]`rRank_v1`**::****Gen****Rank**`rRank_v1`**=**`oneof`[`rNumeric`,`rFaceCard`]`rRank`**::****Gen****Rank**`rRank`**=**`frequency`[(9,`rNumeric`),(4,`rFaceCard`)]- and
`rRank_v1`give different distributions...`rRank`

Card

**data****Card****=****Card****Rank****Suit****deriving**(**Eq**,**Show**)`rCard`**::****Gen****Card**`rCard`**=**(`??`)- Here we want to reuse and
`rSuit`. How can we do what?`rRank` - Are there some functions to combine generators in the QuickCheck library?

- The type is for functions that generate random values of type
**Gen**`a`.`a` - The type is for functions that performs IO operations and return values of type
**IO**`a`.`a` - But both are
*monads*and the-notation can be used with any monad!**do**- We can build larger generator functions from smaller ones in the same way as we build larger IO functions from smaller ones.

rCard::GenCardrCard=dor<-rRanks<-rSuitreturn(Cardrs)

- Other examples illustrating that and
**do**can be used with type`return`, not just with type**Gen**`a`.**IO**`a` `sample' (return 5::Gen Integer)`

[5,5,5,5,5,5,5,5,5,5,5]- Remember ?
`doTwice` `sample' (doTwice (arbitrary::Gen Integer))`

[(0,0),(2,-1),(1,1),(6,1),(-4,-1),(4,0), (-1,7),(4,11),(10,8),(18,-5),(11,14)]

`evenInteger`**::****Gen****Integer**`evenInteger`**=****do**`n`**<-**`arbitrary``return`(2`*``n`)`nonNegative`**::****Gen****Integer**`nonNegative`**=****do**`n`**<-**`arbitrary``return`(`abs``n`)`sample' evenInteger`

[0,2,-4,0,-2,6,-10,-22,22,-2,18]`sample' nonNegative`

[0,1,0,3,2,0,1,9,8,2,20]

**Hand**

?
**data****Hand****=****Empty****|****Add****Card****Hand****deriving**(**Eq**,**Show**)`rHand`**::****Gen****Hand**`rHand`**=**`oneof`[`return`**Empty**,**do**`c`**<-**`rCard``h`**<-**`rHand``return`(**Add**`c``h`)]- Note: is recursive!
`rHand` - Note: does it generate sensible hands?

- The type contains values that are not valid ranks.
**Rank**`validRank`**::****Rank****->****Bool**`validRank`(**Numeric**`n`)**=**2`<=``n``&&``n``<=`10`validRank`**_****=****True**

- We can use to test that
`validRank`is correct!`rRank`

- Using the function :
`forAll` `prop_all_validRank_1`**=**`forAll``rRank``validRank`- Using :
`arbitrary` **instance****Arbitrary****Rank****where**`arbitrary`**=**`rRank``prop_all_validRank_2``r`**=**`validRank``r`

**class****Arbitrary****where**`arbitrary`**::****Gen**`a``shrink`**::**`a`**->**[`a`]**instance****Arbitrary****Bool****instance****Arbitrary****Int****instance****Arbitrary****Integer**-- many more instances...- It specificies a
*default*test data generator for each type. - It lets us test functions without specifying generators explicitly.
- The function lets us use other generators.
`forAll`

- How do you know if a test has tested all the important cases?
`prop_all_validRank_3``r`**=**`collect``r`(`validRank``r`)`quickCheck prop_all_validRank_3`

+++ OK, passed 100 tests: 12% Numeric 10 11% Numeric 7 10% Ace 9% Numeric 6 9% Numeric 4 9% King ... 1% Jack- Using doesn't change the test, but collects the values of
`collect``r`that were tested.`r`

instanceArbitraryHandwherearbitrary=rHandsizeEmpty=0size(Addch)=1+sizehprop_Handh=collect(sizeh)True

`quickCheck prop_Hand`

+++ OK, passed 100 tests: 47% 0 20% 1 13% 2 11% 3 4% 4 2% 5 1% 7 1% 6 1% 12- In general 50% empty hands and only 25% hands with more than one card.

rHand::GenHandrHand=frequency[(1,returnEmpty), (4,doc<-rCardh<-rHandreturn(Addch))]

`quickCheck prop_Hand`

+++ OK, passed 100 tests: 19% 1 19% 0 13% 2 10% 4 8% 6 6% 3 5% 9 5% 5 3% 7 ...- In general 20% empty hands, 16% with one card, ...

- The function inserts a value at the right place in an ordered list
`insert` `insert 'c' "abdef"`

"abcdef"- The output list is ordered if the input list is ordered.
- How do we test it?

`prop_insert_1`**::****Integer****->**[**Integer**]**->****Bool**`prop_insert_1``x``xs`**=**`isOrdered`(`insert``x``xs`)`quickCheck prop_insert_1`

*** Failed! Falsifiable (after 3 tests and 4 shrinks): 0 [0,-1]- If the input list isn't ordered, the output list won't be ordered. Bad test.

`prop_insert_2`**::****Integer****->**[**Integer**]**->****Property**`prop_insert_2``x``xs`**=**`isOrdered``xs``==>``isOrdered`(`insert``x``xs`)`quickCheck prop_insert_2`

*** Gave up! Passed only 68 tests.- The probability that a random list is ordered is very low. (Can we test this?)
- We need a test generator for ordered lists!

- Fortunately, QuickCheck provides
`orderedList`**::**(**Arbitrary**`a`,**Ord**`a`)**=>****Gen**[`a`]- Two ways to use it
`prop_insert_3``x`**=**`forAll``orderedList`(**\**`xs`**->**`isOrdered`(`insert``x``xs`))`prop_insert_4``x`(**Ordered**`xs`)**=**`isOrdered`(`insert``x``xs`)

`prop_insert_4``x`(**Ordered**`xs`)**=**`isOrdered`(`insert``x``xs`)- This works, because QuickCheck also provides
**data****OrderedList**`a`**=****Ordered**[`a`]**instance**(**Ord**`a`,**Arbitrary**`a`)**=>****Arbitrary**(**OrderedList**`a`)**where**`arbitrary`**=****Ordered**`<$>``orderedList`

- Exercise: how would you define ?
`orderedList`- See self-study exercises, Week 4, assignment 6.

- We have seen how generate test data for QuickCheck.
- For our own data types (, etc)
**Card** - How to deal with invariants (ordered lists)
- More info: QuickCheck documentation

- For our own data types (
- We have also seen that
- and
**IO**are two examples of**Gen***monads*. - The -notation can be used for both
**do**and**IO**(and for other monads).**Gen** - Later we will see other examples of monads and how to define your own monads.