# Difference between revisions of "User:PaoloMartini"

## Points-free contest

### Manual reductions

```\a b c -> a + (b*c)
\a b c -> (+) a ((*) b c)
\a b -> ((+) a) . ((*) b)
\a -> (((+) a) .) . (*)
\a -> (.) ((+ a) .) (*)
\a -> (flip (.)) (*) ((+ a) .)
\a -> (flip (.)) (*) (((.) (+ a))
(flip (.)) (*) . ((.) . (+))

<xerox> I think I deserve an award for that reduction.
<dons> xerox reaches PointFree Hacker, Level 7.
```
```<xerox> f . g . h = (\x -> f . x . h) g = (\x -> f . (x . h)) g = (\x -> (f .) ((.) x h)) g = ((f .) . (. h)) g
```
```pascal = iterate (ap (zipWith (+) . (++ [1])) ([1] ++)) [1]
-- > take 3 pascal
-- [[1],[2,2],[3,4,3]]
```

### First revision (W (a -> a))

```{-# OPTIONS_GHC -fglasgow-exts -fallow-undecidable-instances #-}
module VarArg
where

data W a = W { unW :: a } deriving Show

-- `c' is:
--
-- c f x = (f x)
-- c f x y = c (f x) y
--

class C a r | r -> a where
c :: (a -> a -> a) -> a -> r

instance C a (W (a -> a)) where
c f x = W (\y -> f x y)

r :: Int -> W (Int -> Int) -> Int
r x = (\$ x) . unW

instance C a r => C a (a -> r) where
c f x y = c f (f x y)

test1 = let t1 = c (+) 1
t2 = c (+) 1 2
t3 = c (+) 1 2 3
t4 = c (+) 1 2 3 4
in map (r 0) [t1,t2,t3,t4]

test2 = zipWith (==) [1, 1+2, sum [1,2,3], foldr (+) 0 [1,2,3,4]] test1

-- `d' is:
--
-- d f [    ] = f
-- d f (x:xs) = d (f x) xs
--
-- ..for which `c' is the only valid `f'.
--

class D a r | r -> a where
d :: (forall r. (C a r) => a -> r) -> [a] -> r

instance C a (W (a -> a)) => D a (W (a -> a)) where
d f (x:[]) = f x
d f (x:xs) = d (f x) xs

test3 = let t1 = d (c (^)) [2..3]
t2 = d (c (*)) [2..10]
t3 = d (c (+)) [2..100]
in map (r 1) [t1,t2,t3]

test4 = zipWith (==) [foldl1 (^) [2..3], foldl1 (*) [1..10], foldl1 (+) [1..100]] test3
```

### No more incoherent instances ((W a) and QuickCheck)

```{-# OPTIONS_GHC -fglasgow-exts #-}
module Apply
where

import Test.QuickCheck

data W a = W { reify :: a } deriving Show -- wrapper

class Apply a r | r -> a where
apply :: (a -> a -> a) -> a -> a -> r

instance Apply a (W a) where
apply f x y = W (f x y)

instance Apply a r => Apply a (a -> r) where
apply f x y z = apply f (f x y) z

-- test

plus_prop = quickCheck p2 >> quickCheck p3 >> quickCheck p4

where p2 :: Int -> Int -> Bool
p3 :: Int -> Int -> Int -> Bool
p4 :: Int -> Int -> Int -> Int -> Bool

p2 x y     = (reify \$ apply (+) x y)     == x + y
p3 x y z   = (reify \$ apply (+) x y z)   == x + y + z
p4 x y z w = (reify \$ apply (+) x y z w) == x + y + z + w
```

## Catamorfism on a binary tree

Replacing systematically the data constructors with an evaluation function.

```data Tree a = Leaf a
| Branch (Tree a) (Tree a)

cata :: (a -> r, r -> r -> r) -> Tree a -> r
cata (f1,f2) (Leaf x) = f1 x
cata (f1,f2) (Branch b1 b2) = f2 (cata (f1,f2) b1) (cata (f1,f2) b2)
```

## Fibonacci (type-)numbers

```{-# OPTIONS_GHC -fglasgow-exts -fallow-undecidable-instances #-}
module Fibonacci where

data Zero
data Succ n
type One = Succ Zero

class Add a b c | a b -> c where
add :: a -> b -> c

instance Add a b c => Add (Succ a) b (Succ c)

class Fib n m | n -> m where
fib :: n -> m

instance Fib Zero Zero
instance Fib One One
instance (Fib n a, Fib (Succ n) b, Add a b c) => Fib (Succ (Succ n)) c
```

## Points-free hylomorphism

```module Refold where

fold f n [    ] = n
fold f n (x:xs) = f x (fold f n xs)

unfold p f g x = if p x then [] else f x : unfold p f g (g x)

{-
refold c n p f g = fold c n . unfold p f g
refold c n p f = (fold c n .) . unfold p f
refold c n p = ((fold c n .) .) . unfold p
refold c n = (((fold c n .) .) .) . unfold
refold c n = (\x -> (((x .) .) .) . unfold) fold c n
refold c = (\x -> (((x .) .) .) . unfold) . fold c
refold = ((\x -> (((x .) .) .) . unfold) .) . fold
refold = ((\x -> (.) (((x .) .) .) unfold) .) . fold
refold = ((\x -> flip (.) unfold (((x .) .) .)) .) . fold
refold = ((\x -> flip (.) unfold . (\x -> (.) ((.) ((.) x))) \$ x) .) . fold
refold = ((flip (.) unfold . ((.) . (.) . (.))) .) . fold
-}

refold = ((flip (.) unfold . (.) . (.) . (.)) .) . fold
```

## Roman (type-)numerals

```{-# OPTIONS_GHC -fglasgow-exts #-}
module Romans where

class Roman t where
roman :: t -> Int

data O   -- 0
data I a -- 1
data V a -- 5
data X a -- 10
data L a -- 50
data C a -- 100
data D a -- 500
data M a -- 1000

instance                Roman O         where roman _ = 0
instance                Roman (I O)     where roman _ = 1
instance                Roman (V O)     where roman _ = 5
instance                Roman (X O)     where roman _ = 10

instance Roman (I a) => Roman (I (I a)) where roman _ = roman (undefined :: (I a)) + 1
instance Roman a     => Roman (I (V a)) where roman _ = roman (undefined :: a)     + 4
instance Roman a     => Roman (I (X a)) where roman _ = roman (undefined :: a)     + 9

instance Roman (I a) => Roman (V (I a)) where roman _ = roman (undefined :: (I a)) + 5
instance Roman (V a) => Roman (V (V a)) where roman _ = roman (undefined :: (V a)) + 5

instance Roman (I a) => Roman (X (I a)) where roman _ = roman (undefined :: (I a)) + 10
instance Roman (V a) => Roman (X (V a)) where roman _ = roman (undefined :: (V a)) + 10
instance Roman (X a) => Roman (X (X a)) where roman _ = roman (undefined :: (X a)) + 10
instance Roman a     => Roman (X (L a)) where roman _ = roman (undefined :: a)     + 40
instance Roman a     => Roman (X (C a)) where roman _ = roman (undefined :: a)     + 90
instance Roman a     => Roman (X (D a)) where roman _ = roman (undefined :: a)     + 490

instance Roman a     => Roman (L a)     where roman _ = roman (undefined :: a)     + 50

instance Roman (I a) => Roman (C (I a)) where roman _ = roman (undefined :: (I a)) + 100
instance Roman (V a) => Roman (C (V a)) where roman _ = roman (undefined :: (V a)) + 100
instance Roman (X a) => Roman (C (X a)) where roman _ = roman (undefined :: (X a)) + 100
instance Roman (L a) => Roman (C (L a)) where roman _ = roman (undefined :: (L a)) + 100
instance Roman (C a) => Roman (C (C a)) where roman _ = roman (undefined :: (C a)) + 100
instance Roman a     => Roman (C (D a)) where roman _ = roman (undefined :: a)     + 400
instance Roman a     => Roman (C (M a)) where roman _ = roman (undefined :: a)     + 900

instance Roman a     => Roman (D a)     where roman _ = roman (undefined :: a)     + 500

instance Roman a     => Roman (M a)     where roman _ = roman (undefined :: a)     + 1000

powersoftwo = [roman (undefined :: (I (I O))),
roman (undefined :: (I (V O))),
roman (undefined :: (V (I (I (I O))))),
roman (undefined :: (X (V (I O)))),
roman (undefined :: (X (X (X (I (I O)))))),
roman (undefined :: (L (X (I (V O))))),
roman (undefined :: (C (X (X (V (I (I (I O)))))))),
roman (undefined :: (C (C (L (V (I O)))))),
roman (undefined :: (D (X (I (I O))))),
roman (undefined :: (M (X (X (I (V O)))))),
roman (undefined :: (M (M (X (L (V (I (I (I O)))))))))]
```

## randoms

```randoms :: (RandomGen g) => g -> [Int]
randoms = fix ((. next) . uncurry . (. (:)) . flip (.))
```