diffSq :: Integer -> Integer -> Integer
diffSq x y = (x – y) * (x + y)
diffSq 2 1
X -> Y -> Z
diffSqV3a, diffSqV3b :: Integer -> Integer -> Integer
diffSqV3a x y =
let minus = x – y
plus = x + y
in minus * plus
diffSqV3b x y = minus * plus
where
minus = x – y
plus = x + y
let in x=5 in x+1)
2 * (let
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,g.e ,detcepxe si noisserpxe na reverehw desu eb nac ,noisserpxe na si ¡± – ¡° :ecnereffid ehT
.erutcel retal a ni
ekil sepyt no yroeht erom tib a si erehT :esu ot woH
:rewsna regetni na sevig dna sretemarap regetni owt sekat elpmaxe sihT
sretemarap elpitlum fo snoitcnuF
.elgna wofl-lortnoc eht naht srennigeb rof retteb si elgna arbegla eht tub ,em ekil kaeps ot evah t’nod uoy ,tcerroc era selgna htoB .¡±noitcnuf siht llac¡° ton )ni gniggulp( ¡±retemarap taht ot noitcnuf siht ylppa¡° ,¡±epyt nruter¡° ton¡±niamodoc¡°,¡±01snruter¡°ton¡±01sevig/slauqe)4(f¡°yasemraehlliwuoy,sihtfoesuaceB
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sh.scisaBlleksaH ni
:snoitinfied lacol fo syaw owT
stcurtsnoc noitinfied lacoL
sh.scisaBlleksaH ni
dnim fo emarF scisaB lleksaH
where
2 * (x+1 where x=5)
y = … where x=5
slowFib2
slowFactorial :: Integer -> Integer
slowFactorial 0 = 1
slowFactorial n = n * slowFactorial (n – 1)
— Slow Fibonacci (yawn) to show you can have more patterns.
— “This Fibonacci joke is as bad as the last two you heard combined.”
— https://twitter.com/sigfpe/status/776420034419658752
slowFib :: Integer -> Integer
slowFib 0 = 0
slowFib 1 = 1
slowFib n = slowFib (n-1) + slowFib (n-2)
— More fundamental form—using a “case-of” expression:
slowFib2 :: Integer -> Integer
slowFib2 x = case x of
0 -> 0
1 -> 1
n -> slowFib2 n1 + slowFib2 n2
where
n1 = n – 1
n2 = n – 2
— The other example of “where”. This one is part of “n -> …”.
where
f (x*2 + 1) :: Integer
f (x*2 + 1) :: Integer
^^^^^^^^^^^ ^^^^^^^
term type
h . g :: Char -> Bool
^^^^^ ^^^^^^^^^^^^
term type
f (x*2 + 1)
Integer
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,¡±
dna edoc lautca htob ni desu si tI .¡±
.esorp noitanalpxe si fo epyt eht¡° rof noitaton si sihT
:emit eht lla siht ekil gnihtemos ees lliw uoY
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:elpmaxe erom enO
:dellac era sgniht owt esohT
slowFactorial 0 = 1
5+4
n! = n*(n-1)!
= n * slowFactorial (n-1) by I.H.
slowFactorial n = n * slowFactorial (n-1)
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!n = n lairotcaFwols :PTW !)1-n( = )1-n( lairotcaFwols :sisehtopyh noitcudnI .nevig eb 1 ¡Ý n larutan teL
!snwonknu eht rof evlos ot woh tuo dnfi uoy foorp eht gnirud tub ha ,snwonknu sniatnoc llits taht gnihtemos evorp ot noitcudni esU = noitcudni dneterP :)edoc evisrucer etirw I woh( weiv sisehtnyS
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noitaulave sv/dna sisehtnyS
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ecitoN
sisehtnyS
slowFactorial 3
3 * slowFactorial (3 – 1)
3 * slowFactorial 2
3 * (2 * slowFactorial (2 – 1))
3 * (2 * slowFactorial 1)
3 * (2 * (1 * slowFactorial (1 – 1))) 3 * (2 * (1 * slowFactorial 0))
3 * (2 * (1 * 1))
3 * (2 * 1)
3* 2
6
[Integer] [Bool] [] Integer [] Bool []
[]
[4, 1, 6]
4 : (1 : (6 : []))
4 : 1 : 6 : []
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:fo eno si tsil a :)63BCSC ni sa noitinfied evisrucer( yllamroF .gnihctam nrettap ni desu eb nac smrof xatnys esehT
:siht ekil ,lanoitpo era sesehtnerap esohT
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noitaulavE
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insert
insert 4 [1,3,5,8,9,10] = [1,3,4,5,8,9,10]
insert :: Integer -> [Integer] -> [Integer]
— Structural induction on xs.
— Base case: xs is empty. Answer is [e] aka e:[]
insert e [] = [e] — e : []
— Induction step: Suppose xs has the form x:xt (and xt is shorter than xs).
— E.g., xs = [1,3,5,8], x = 1, xt = [3,5,8].
— Induction hypothesis: insert e xt = put e into the right place in xt.
insert e xs@(x:xt)
— xs@(x:xt) is an “as-pattern”, “xs as (x:xt)”,
— so xs refers to the whole list, and it also matches x:xt
—
— If e <= x, then e should be put before x, in fact all of xs, and be done.
-- E.g., insert 1 (10 : xs) = 1 : (10 : xs)
| e <= x = e : xs
-- Otherwise, the answer should go like:
-- x, followed by whatever is putting e into the right place in xt.
-- i.e.,
-- x, followed by insert e xt (because IH)
-- E.g., insert 25 (10 : xt) = 10 : (insert 25 xt)
| otherwise = x : insert e xt
insert 15 (10 : 20 : [])
insert
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.meht no noitcudni esu ot nrut ruoY :esicrexE .selyts owt gnitcefler ,snoisrev owt era ereH .mhtirogla gnitros eht etelpmoc nac uoy gnisU
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,.g.E .detros llits si elohw eht os sx ni ¡±ecalp thgir¡° eht otni e tuP .detros neeb evah ot demussa si sx .sx tsil dna e tnemele ekaT
: noitcnuf repleh a evaH :ygetartS :tros noitresnI
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:noisrev noisrucer tceriD
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)psiL morf( ¡±llec snoc¡° ,edon tsil >ereh tsil a< : >ereh meti na<
sh.scisaBlleksaH ni
insertionSort :: [Integer] -> [Integer]
insertionSort [] = []
insertionSort (x:xt) = insert x (insertionSort xt)
acc
insertionSortAcc :: [Integer] -> [Integer]
insertionSortAcc xs = go xs []
where
— Specification for go (when you prove go correct by induction):
— for all xs, for all acc:
— Assume acc is in sorted order (precondition).
— go xs acc = add elements of xs to acc but in sorted order
go (x:xt) acc = let acc2 = insert x acc
go
go [] acc = acc
in go xt acc2
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.elbairav retalumucca na sledom retemarap dnoces sti ;pool-elihw a sledom noitcnuf repleh ehT :elbairav rotalumucca na dna pool-elihw a sledom noisrev rehto sihT
sh.scisaBlleksaH ni
sh.scisaBlleksaH ni