Copyright	(c) gspia 2020-
License	BSD
Maintainer	gspia
Safe Haskell	Safe
Language	Haskell2010

Fcf.Alg.Tree

Description

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.

Synopsis

data TreeF a b = NodeF a [b]
data TreeToFix :: Tree a -> Exp (Fix (TreeF a))
data SumNodesAlg :: Algebra (TreeF Nat) Nat
data CountNodesAlg :: Algebra (TreeF a) Nat
data Size :: Tree a -> Exp Nat
data BuildNodeCoA :: CoAlgebra (TreeF Nat) Nat
data BuildFibTreeCoA :: CoAlgebra (TreeF Nat) Nat
data Fib :: Nat -> Exp Nat
data BTreeF a b
- = BEmptyF
- | BNodeF a b b

Documentation

data TreeF a b Source #

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

Constructors

NodeF a [b]

Instances

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

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) Source #
Instance details 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) Source #
Instance details 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 Source #

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) Source #
Instance details 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) Source #
Instance details Defined in Fcf.Alg.Tree type Eval (SumNodesAlg (NodeF x ([] :: [Nat])) :: Nat -> Type) = x

data CountNodesAlg :: Algebra (TreeF a) Nat Source #

Count the nodes of TreeF.

See the Size for an example.

Instances

type Eval (CountNodesAlg (NodeF x (b ': bs)) :: Nat -> Type) Source #
Instance details 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) Source #
Instance details Defined in Fcf.Alg.Tree type Eval (CountNodesAlg (NodeF x ([] :: [Nat])) :: Nat -> Type) = 1

data Size :: Tree a -> Exp Nat Source #

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) Source #
Instance details 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 Source #

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) Source #
Instance details 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 Source #

CoAlgebra for the Fib-function.

Instances

type Eval (BuildFibTreeCoA n :: TreeF Nat Nat -> Type) Source #
Instance details 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 Fib :: Nat -> Exp Nat Source #

Fibonaccis with Hylo, not efficient

Example

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

Instances

type Eval (Fib n :: Nat -> Type) Source #
Instance details Defined in Fcf.Alg.Tree type Eval (Fib n :: Nat -> Type) = Eval (Hylo SumNodesAlg BuildFibTreeCoA n)

data BTreeF a b Source #

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) Source #
Instance details 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) Source #
Instance details 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) Source #
Instance details 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) Source #
Instance details 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) Source #
Instance details Defined in Fcf.Alg.Tree type Eval (Map f (BEmptyF :: BTreeF a2 a1) :: BTreeF a2 b -> Type) = (BEmptyF :: BTreeF a2 b)