Fcf.Alg.Tree

Fcf.Alg.Tree

Type-level TreeF and BTreeF to be used with Cata, Ana and Hylo. This also provides some algorithms: general purpose sorting with Qsort, Size of an Tree, Fibonaccis.

Documentation

data TreeF a b

TreeF is functor for Trees. TreeF has Map-instance (on structure).

Constructors

NodeF a [b]

Instances

type Eval (FSum (NodeF a2 (b ': bs)) :: Nat -> Type)

Defined in Fcf.Alg.Tree

type Eval (FSum (NodeF a2 (b ': bs)) :: Nat -> Type) = Eval (Sum (b ': bs))

type Eval (FSum (NodeF a2 ([] :: [Nat])) :: Nat -> Type)

Instances to make TreeF to be a foldable sum. After this one, we can write the Sizes example.

Defined in Fcf.Alg.Tree

type Eval (FSum (NodeF a2 ([] :: [Nat])) :: Nat -> Type) = 0

type Eval (TreeToFix (Node a2 (b ': bs)) :: Fix (TreeF a1) -> Type)

Defined in Fcf.Alg.Tree

type Eval (TreeToFix (Node a2 (b ': bs)) :: Fix (TreeF a1) -> Type) = Fix (NodeF a2 (Eval (Map (TreeToFix :: Tree a1 -> Fix (TreeF a1) -> Type) (b ': bs))))

type Eval (TreeToFix (Node a2 ([] :: [Tree a1])) :: Fix (TreeF a1) -> Type)

Defined in Fcf.Alg.Tree

type Eval (TreeToFix (Node a2 ([] :: [Tree a1])) :: Fix (TreeF a1) -> Type) = Fix (NodeF a2 ([] :: [Fix (TreeF a1)]))

type Eval (BuildFibTreeCoA n :: TreeF Nat Nat -> Type)

Defined in Fcf.Alg.Tree

type Eval (BuildFibTreeCoA n :: TreeF Nat Nat -> Type) = If (Eval (n >= 2)) (NodeF 0 ((n - 1) ': ((n - 2) ': ([] :: [Nat])))) (NodeF n ([] :: [Nat]))

type Eval (BuildNodeCoA n :: TreeF Nat Nat -> Type)

Defined in Fcf.Alg.Tree

type Eval (BuildNodeCoA n :: TreeF Nat Nat -> Type) = If (Eval (((2 * n) + 1) >= 8)) (NodeF n ([] :: [Nat])) (NodeF n ((2 * n) ': (((2 * n) + 1) ': ([] :: [Nat]))))

type Eval (Map f (NodeF a3 (b2 ': bs)) :: TreeF a2 b1 -> Type)

Defined in Fcf.Alg.Tree

type Eval (Map f (NodeF a3 (b2 ': bs)) :: TreeF a2 b1 -> Type) = NodeF a3 (Eval (Map f (b2 ': bs)))

type Eval (Map f (NodeF a3 ([] :: [a1])) :: TreeF a2 b -> Type)

Defined in Fcf.Alg.Tree

type Eval (Map f (NodeF a3 ([] :: [a1])) :: TreeF a2 b -> Type) = NodeF a3 ([] :: [b])

data TreeToFix :: Tree a -> Exp (Fix (TreeF a))

A function to transform a Tree into fixed structure that can be used by Cata and Ana.

See the implementation of Size for an example.

Instances

type Eval (TreeToFix (Node a2 (b ': bs)) :: Fix (TreeF a1) -> Type)

Defined in Fcf.Alg.Tree

type Eval (TreeToFix (Node a2 (b ': bs)) :: Fix (TreeF a1) -> Type) = Fix (NodeF a2 (Eval (Map (TreeToFix :: Tree a1 -> Fix (TreeF a1) -> Type) (b ': bs))))

type Eval (TreeToFix (Node a2 ([] :: [Tree a1])) :: Fix (TreeF a1) -> Type)

Defined in Fcf.Alg.Tree

type Eval (TreeToFix (Node a2 ([] :: [Tree a1])) :: Fix (TreeF a1) -> Type) = Fix (NodeF a2 ([] :: [Fix (TreeF a1)]))

data SumNodesAlg :: Algebra (TreeF Nat) Nat

Sum the nodes of TreeF containing Nats.

See the implementation of Fib for an example.

Instances

type Eval (SumNodesAlg (NodeF x (b ': bs)) :: Nat -> Type)

Defined in Fcf.Alg.Tree

type Eval (SumNodesAlg (NodeF x (b ': bs)) :: Nat -> Type) = x + Eval (Sum (b ': bs))

type Eval (SumNodesAlg (NodeF x ([] :: [Nat])) :: Nat -> Type)

Defined in Fcf.Alg.Tree

type Eval (SumNodesAlg (NodeF x ([] :: [Nat])) :: Nat -> Type) = x

data CountNodesAlg :: Algebra (TreeF a) Nat

Count the nodes of TreeF.

See the Size for an example.

Instances

type Eval (CountNodesAlg (NodeF x (b ': bs)) :: Nat -> Type)

Defined in Fcf.Alg.Tree

type Eval (CountNodesAlg (NodeF x (b ': bs)) :: Nat -> Type) = 1 + Eval (Sum (b ': bs))

type Eval (CountNodesAlg (NodeF x ([] :: [Nat])) :: Nat -> Type)

Defined in Fcf.Alg.Tree

type Eval (CountNodesAlg (NodeF x ([] :: [Nat])) :: Nat -> Type) = 1

data Size :: Tree a -> Exp Nat

Size of the Tree is the number of nodes in it.

Example

Size is defined as Cata CountNodesAlg =<< TreeToFix tr and can be used with the following.

>>> data BuildNode :: Nat -> Exp (Nat,[Nat])
>>> :{
  type instance Eval (BuildNode x) =
      If (Eval ((2 TL.* x TL.+ 1) >= 8))
          '(x, '[])
          '(x, '[ 2 TL.* x, (2 TL.* x) TL.+ 1 ])
:}

>>> :kind! Eval (Size =<< UnfoldTree BuildNode 1)
Eval (Size =<< UnfoldTree BuildNode 1) :: Nat
= 7

Instances

type Eval (Size tr :: Nat -> Type)

Defined in Fcf.Alg.Tree

type Eval (Size tr :: Nat -> Type) = Eval (Cata (CountNodesAlg :: TreeF a Nat -> Nat -> Type) =<< TreeToFix tr)

data BuildNodeCoA :: CoAlgebra (TreeF Nat) Nat

CoAlgebra to build TreeF's. This is an example from containers-package. See Size and example in there.

:kind! Eval (Ana BuildNodeCoA 1) :kind! Eval (Hylo CountNodesAlg BuildNodeCoA 1)

Instances

type Eval (BuildNodeCoA n :: TreeF Nat Nat -> Type)

Defined in Fcf.Alg.Tree

type Eval (BuildNodeCoA n :: TreeF Nat Nat -> Type) = If (Eval (((2 * n) + 1) >= 8)) (NodeF n ([] :: [Nat])) (NodeF n ((2 * n) ': (((2 * n) + 1) ': ([] :: [Nat]))))

data BuildFibTreeCoA :: CoAlgebra (TreeF Nat) Nat

CoAlgebra for the Fib-function.

Instances

type Eval (BuildFibTreeCoA n :: TreeF Nat Nat -> Type)

Defined in Fcf.Alg.Tree

type Eval (BuildFibTreeCoA n :: TreeF Nat Nat -> Type) = If (Eval (n >= 2)) (NodeF 0 ((n - 1) ': ((n - 2) ': ([] :: [Nat])))) (NodeF n ([] :: [Nat]))

data FibHylo :: Nat -> Exp Nat

Fibonaccis with Hylo, not efficient

Example

>>> :kind! Eval (FibHylo 10)
Eval (FibHylo 10) :: Nat
= 55

Instances

type Eval (FibHylo n :: Nat -> Type)

Defined in Fcf.Alg.Tree

type Eval (FibHylo n :: Nat -> Type) = Eval (Hylo SumNodesAlg BuildFibTreeCoA n)

data BTreeF a b

BTreeF is a btree functor. At the moment, it is used to build sorting algorithms.

Constructors

BEmptyF

BNodeF a b b

Instances

type Eval (PartHlp smaller (h ': t) :: BTreeF a [a] -> Type)

Defined in Fcf.Alg.Sort

type Eval (PartHlp smaller (h ': t) :: BTreeF a [a] -> Type) = BNodeF h (Eval (Filter (smaller h) t)) (Eval (Filter (Not <=< smaller h) t))

type Eval (PartHlp _ ([] :: [a]) :: BTreeF a [a] -> Type)

Defined in Fcf.Alg.Sort

type Eval (PartHlp _ ([] :: [a]) :: BTreeF a [a] -> Type) = (BEmptyF :: BTreeF a [a])

type Eval (PartCmp cmp coalg :: BTreeF a [a] -> Type)

Defined in Fcf.Alg.Sort

type Eval (PartCmp cmp coalg :: BTreeF a [a] -> Type) = Eval (PartHlp (Flip cmp) coalg)

type Eval (Map f (BNodeF a4 b1 b2) :: BTreeF a3 a2 -> Type)

Defined in Fcf.Alg.Tree

type Eval (Map f (BNodeF a4 b1 b2) :: BTreeF a3 a2 -> Type) = BNodeF a4 (Eval (f b1)) (Eval (f b2))

type Eval (Map f (BEmptyF :: BTreeF a2 a1) :: BTreeF a2 b -> Type)

BTreeF is a functor

Defined in Fcf.Alg.Tree

type Eval (Map f (BEmptyF :: BTreeF a2 a1) :: BTreeF a2 b -> Type) = (BEmptyF :: BTreeF a2 b)

data FSum :: f a -> Exp a

A kind of foldable sum class. Pun may or may not be intended.

Instances

type Eval (FSum (NodeF a2 (b ': bs)) :: Nat -> Type)

Defined in Fcf.Alg.Tree

type Eval (FSum (NodeF a2 (b ': bs)) :: Nat -> Type) = Eval (Sum (b ': bs))

type Eval (FSum (NodeF a2 ([] :: [Nat])) :: Nat -> Type)

Instances to make TreeF to be a foldable sum. After this one, we can write the Sizes example.

Defined in Fcf.Alg.Tree

type Eval (FSum (NodeF a2 ([] :: [Nat])) :: Nat -> Type) = 0

data Sizes :: Fix f -> Exp (Ann f Nat)

Instances

type Eval (Sizes fx :: Ann f Nat -> Type)

Sizes example from Recursion Schemes by example, Tim Williams. This annotes each node with the size of its subtree.

Example

>>> :kind! Eval (Sizes =<< Ana BuildNodeCoA 1)
Eval (Sizes =<< Ana BuildNodeCoA 1) :: Fix (AnnF (TreeF Nat) Nat)
= 'Fix
    ('AnnF
       '( 'NodeF
            1
            '[ 'Fix
                 ('AnnF
                    '( 'NodeF
                         2
                         '[ 'Fix ('AnnF '( 'NodeF 4 '[], 1)),
                            'Fix ('AnnF '( 'NodeF 5 '[], 1))],
                       3)),
               'Fix
                 ('AnnF
                    '( 'NodeF
                         3
                         '[ 'Fix ('AnnF '( 'NodeF 6 '[], 1)),
                            'Fix ('AnnF '( 'NodeF 7 '[], 1))],
                       3))],
          7))

Defined in Fcf.Alg.Tree

type Eval (Sizes fx :: Ann f Nat -> Type) = Eval (Synthesize ((+) 1 <=< (FSum :: f Nat -> Nat -> Type)) fx)

data NatF r

A NatF functor that can be used with different morphisms. This tree-module is probably a wrong place to this one. Now it is here for the Fibonacci example.

Constructors

Succ r

Zero

Instances

type Eval (FibAlgebra (Succ (Fix (AnnF ((,) (Succ (Fix (AnnF ((,) _ n)))) m)))) :: Nat -> Type)

Defined in Fcf.Alg.Tree

type Eval (FibAlgebra (Succ (Fix (AnnF ((,) (Succ (Fix (AnnF ((,) _ n)))) m)))) :: Nat -> Type) = Eval (n + m)

type Eval (FibAlgebra (Succ (Fix (AnnF ((,) (Zero :: NatF (Fix (AnnF NatF Nat))) _)))) :: Nat -> Type)

Defined in Fcf.Alg.Tree

type Eval (FibAlgebra (Succ (Fix (AnnF ((,) (Zero :: NatF (Fix (AnnF NatF Nat))) _)))) :: Nat -> Type) = 1

type Eval (FibAlgebra (Zero :: NatF (Ann NatF Nat)))

Defined in Fcf.Alg.Tree

type Eval (FibAlgebra (Zero :: NatF (Ann NatF Nat))) = 0

type Eval (RecNTF n :: Fix NatF -> Type)

Defined in Fcf.Alg.Tree

type Eval (RecNTF n :: Fix NatF -> Type) = Fix (Succ (Eval (NatToFix n)))

type Eval (NatToFix n :: Fix NatF -> Type)

Defined in Fcf.Alg.Tree

type Eval (NatToFix n :: Fix NatF -> Type) = Eval (If (Eval (n < 1)) (Pure (Fix (Zero :: NatF (Fix NatF)))) (RecNTF =<< (n - 1)))

type Eval (Map f (Succ r2) :: NatF r1 -> Type)

Defined in Fcf.Alg.Tree

type Eval (Map f (Succ r2) :: NatF r1 -> Type) = Succ (Eval (f r2))

type Eval (Map f (Zero :: NatF a) :: NatF b -> Type)

NatF has to have functor-instances so that morphisms will work.

Defined in Fcf.Alg.Tree

type Eval (Map f (Zero :: NatF a) :: NatF b -> Type) = (Zero :: NatF b)

data NatToFix :: Nat -> Exp (Fix NatF)

We want to be able to build NatF Fix-structures out of Nat's.

Instances

type Eval (NatToFix n :: Fix NatF -> Type)

Defined in Fcf.Alg.Tree

type Eval (NatToFix n :: Fix NatF -> Type) = Eval (If (Eval (n < 1)) (Pure (Fix (Zero :: NatF (Fix NatF)))) (RecNTF =<< (n - 1)))

data RecNTF :: Nat -> Exp (Fix NatF)

Instances

type Eval (RecNTF n :: Fix NatF -> Type)

Defined in Fcf.Alg.Tree

type Eval (RecNTF n :: Fix NatF -> Type) = Fix (Succ (Eval (NatToFix n)))

data FibAlgebra :: NatF (Ann NatF Nat) -> Exp Nat

Efficient Fibonacci algebra from Recursion Schemes by example, Tim Williams.

Instances

type Eval (FibAlgebra (Succ (Fix (AnnF ((,) (Succ (Fix (AnnF ((,) _ n)))) m)))) :: Nat -> Type)

Defined in Fcf.Alg.Tree

type Eval (FibAlgebra (Succ (Fix (AnnF ((,) (Succ (Fix (AnnF ((,) _ n)))) m)))) :: Nat -> Type) = Eval (n + m)

type Eval (FibAlgebra (Succ (Fix (AnnF ((,) (Zero :: NatF (Fix (AnnF NatF Nat))) _)))) :: Nat -> Type)

Defined in Fcf.Alg.Tree

type Eval (FibAlgebra (Succ (Fix (AnnF ((,) (Zero :: NatF (Fix (AnnF NatF Nat))) _)))) :: Nat -> Type) = 1

type Eval (FibAlgebra (Zero :: NatF (Ann NatF Nat)))

Defined in Fcf.Alg.Tree

type Eval (FibAlgebra (Zero :: NatF (Ann NatF Nat))) = 0

data FibHisto :: Nat -> Exp Nat

Efficient Fibonacci type-level function (from Recursion Schemes by example, Tim Williams). Compare this to FibHylo.

Example

>>> :kind! Eval (FibHisto 100)
Eval (FibHisto 100) :: Nat
= 354224848179261915075

Instances

type Eval (FibHisto n :: Nat -> Type)

Defined in Fcf.Alg.Tree

type Eval (FibHisto n :: Nat -> Type) = Eval (Histo FibAlgebra =<< NatToFix n)