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[ papers ] Type-based Divide & Conquer (#2860)
This commit is contained in:
G. Allais
GitHub
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@ -98,13 +98,19 @@
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* Changes `getNProcessors` to return the number of online processors rather than
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the number of configured processors.
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#### Contrib
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* Remove Data.List.HasLength from contrib library but add it to the base library
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with the type signature from the compiler codebase and some of the naming
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from the contrib library. The type ended up being `HasLength n xs` rather than
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`HasLength xs n`.
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#### Papers
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* In `Control.DivideAndConquer`: a port of the paper
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`A Type-Based Approach to Divide-And-Conquer Recursion in Coq'
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by Pedro Abreu, Benjamin Delaware, Alex Hubers, Christa Jenkins,
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J. Garret Morris, and Aaron Stump
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### Other Changes
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* The `data` subfolder of an installed or local dependency package is now automatically
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recognized as a "data" directory by Idris 2. See the
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408
libs/papers/Control/DivideAndConquer.idr
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408
libs/papers/Control/DivideAndConquer.idr
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@ -0,0 +1,408 @@
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||| The content of this module is based on the paper
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||| A Type-Based Approach to Divide-And-Conquer Recursion in Coq
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||| by Pedro Abreu, Benjamin Delaware, Alex Hubers, Christa Jenkins,
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||| J. Garret Morris, and Aaron Stump
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|||
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||| The original paper relies on Coq's impredicative Set axiom,
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||| something we don't have access to in Idris 2. We can however
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||| reproduce the results by ignoring the type levels
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module Control.DivideAndConquer
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%default total
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namespace Section4Sub1
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public export
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data ListF : (a, x : Type) -> Type where
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Nil : ListF a x
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(::) : a -> x -> ListF a x
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lengthAlg : ListF a Nat -> Nat
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lengthAlg [] = 0
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lengthAlg (_ :: n) = S n
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public export
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Functor (ListF a) where
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map f [] = []
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map f (x :: xs) = x :: f xs
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namespace Section6Sub1
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public export
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-- Only accepted because there is currently no universe check
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data Mu : (Type -> Type) -> Type where
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MkMu : forall r. (r -> Mu f) -> f r -> Mu f
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public export
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inMu : f (Mu f) -> Mu f
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inMu = MkMu id
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public export
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outMu : Functor f => Mu f -> f (Mu f)
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outMu (MkMu f d) = f <$> d
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public export
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fold : Functor f => (f a -> a) -> Mu f -> a
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fold alg (MkMu f r) = alg (assert_total (fold alg . f <$> r))
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namespace Section4Sub1
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list : Type -> Type
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list = Mu . ListF
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namespace Smart
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Nil : list a
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Nil = inMu []
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(::) : a -> list a -> list a
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x :: xs = inMu (x :: xs)
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fromList : List a -> list a
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fromList = foldr (::) []
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toList : list a -> List a
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toList = fold $ \case
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[] => []
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(x :: xs) => x :: xs
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namespace Section4Sub2
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public export
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KAlg : Type
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KAlg = (Type -> Type) -> Type
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public export
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0 Mono : (KAlg -> KAlg) -> Type
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Mono f
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= forall a, b.
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(forall x. a x -> b x) ->
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(forall x. f a x -> f b x)
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public export
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data Mu : (KAlg -> KAlg) -> KAlg where
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MkMu : (forall x. a x -> Mu f x) ->
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(forall x. f a x -> Mu f x)
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public export
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inMu : f (Mu f) x -> Mu f x
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inMu = MkMu id
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public export
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outMu : Mono f -> Mu f x -> f (Mu f) x
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outMu m (MkMu f d) = m f d
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parameters (0 f : Type -> Type)
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public export
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0 FoldT : KAlg -> Type -> Type
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FoldT a r
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= forall x.
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Functor x =>
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a x -> r -> x r
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public export
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0 SAlgF : KAlg -> (Type -> Type) -> Type
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SAlgF a x
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= forall p, r.
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(r -> p) ->
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FoldT a r ->
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(f r -> p) ->
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(r -> x r) ->
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f r -> x p
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public export
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0 SAlg : (Type -> Type) -> Type
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SAlg = Mu SAlgF
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public export
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0 AlgF : KAlg -> (Type -> Type) -> Type
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AlgF a x
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= forall r.
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FoldT a r ->
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FoldT SAlg r ->
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(r -> x r) ->
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f r -> x r
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public export
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0 Alg : (Type -> Type) -> Type
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Alg = Mu AlgF
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public export
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inSAlg : SAlgF SAlg x -> SAlg x
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inSAlg = inMu
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public export
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monoSAlgF : Mono SAlgF
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monoSAlgF f salg up sfo = salg up (sfo . f)
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public export
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outSAlg : SAlg x -> SAlgF SAlg x
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outSAlg = outMu monoSAlgF
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public export
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inAlg : AlgF Alg x -> Alg x
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inAlg = inMu
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public export
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monoAlgF : Mono AlgF
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monoAlgF f alg fo = alg (fo . f)
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public export
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outAlg : Alg x -> AlgF Alg x
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outAlg = outMu monoAlgF
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namespace Section6Sub2
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parameters (0 f : Type -> Type)
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public export
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0 DcF : Type -> Type
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DcF a = forall x. Functor x => Alg f x -> x a
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public export
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functorDcF : Functor DcF
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functorDcF = MkFunctor $ \ f, dc, alg => map f (dc alg)
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public export
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0 Dc : Type
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Dc = Mu DcF
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public export
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fold : FoldT f (Alg f) Dc
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fold alg dc = outMu @{functorDcF} dc alg
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public export
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record RevealT (x : Type -> Type) (r : Type) where
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constructor MkRevealT
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runRevealT : (r -> Dc) -> x Dc
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public export %hint
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functorRevealT : Functor (RevealT x)
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functorRevealT = MkFunctor $ \ f, t =>
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MkRevealT (\ g => runRevealT t (g . f))
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public export
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promote : Functor x => SAlg f x -> Alg f (RevealT x)
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promote salg
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= inAlg f $ \ fo, sfo, rec, fr =>
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MkRevealT $ \ reveal =>
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let abstIn := \ fr => inMu (\ alg => reveal <$> outAlg f alg fo sfo (fo alg) fr) in
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outSAlg f salg reveal sfo abstIn (sfo salg) fr
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public export
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sfold : FoldT f (SAlg f) Dc
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sfold salg dc = runRevealT (fold (promote salg) dc) id
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public export
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inDc : f Dc -> Dc
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inDc d = inMu (\ alg => outAlg f alg fold sfold (fold alg) d)
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out : Functor f => FoldT f (SAlg f) r -> r -> f r
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out sfo = sfo (inSAlg f (\ up, _, _, _ => map up))
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namespace Section5Sub1
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public export
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0 list : Type -> Type
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list a = Dc (ListF a)
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namespace Smart
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public export
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Nil : list a
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Nil = inDc (ListF a) []
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public export
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(::) : a -> list a -> list a
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x :: xs = inDc (ListF a) (x :: xs)
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public export
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fromList : List a -> list a
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fromList = foldr (::) []
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public export
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0 SpanF : Type -> Type -> Type
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SpanF a x = (List a, x)
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SpanSAlg : (a -> Bool) -> SAlg (ListF a) (SpanF a)
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SpanSAlg p = inSAlg (ListF a) $ \up, sfo, abstIn, span, xs =>
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case xs of
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[] => ([], abstIn xs)
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(x :: xs') =>
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if p x
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then let (r, s) = span xs' in (x :: r, up s)
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else ([], abstIn xs)
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export
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spanr : FoldT (ListF a) (SAlg (ListF a)) r ->
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(a -> Bool) -> (xs : r) -> SpanF a r
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spanr sfo p xs = sfo (SpanSAlg p) @{MkFunctor mapSnd} xs
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breakr : FoldT (ListF a) (SAlg (ListF a)) r ->
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(a -> Bool) -> (xs : r) -> SpanF a r
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breakr sfo p = spanr sfo (not . p)
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WordsByAlg : (a -> Bool) -> Alg (ListF a) (const (List (List a)))
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WordsByAlg p = inAlg (ListF a) $ \ fo, sfo, wordsBy, xs =>
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case xs of
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[] => []
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(hd :: tl) =>
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if p hd
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then wordsBy tl
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else
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let (w, rest) = breakr sfo p tl in
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(hd :: w) :: wordsBy rest
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wordsBy : (a -> Bool) -> (xs : List a) -> List (List a)
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wordsBy p = fold (ListF a) @{MkFunctor (const id)} (WordsByAlg p) . fromList
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namespace Section5Sub3
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data NatF x = Z | S x
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0 nat : Type
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nat = Dc NatF
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fromNat : Nat -> Section5Sub3.nat
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fromNat Z = inDc NatF Z
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fromNat (S n) = inDc NatF (S (fromNat n))
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toNat : Section5Sub3.nat -> Nat
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toNat = fold NatF @{MkFunctor (const id)} idAlg where
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idAlg : Alg NatF (const Nat)
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idAlg = inAlg NatF $ \ fo, sfo, toNat, n =>
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case n of
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Z => Z
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S n' => S (toNat n')
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zeroSAlg : SAlg NatF Prelude.id
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zeroSAlg = inSAlg NatF $ \ up, sfo, abstIn, zero, n =>
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case n of
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Z => abstIn n
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S p => up (zero (zero p))
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export
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zero : Nat -> Nat
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zero = toNat . sfold NatF @{MkFunctor id} zeroSAlg . fromNat
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namespace Section5Sub3
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data TreeF a x = Node a (List x)
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0 tree : Type -> Type
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tree a = Dc (TreeF a)
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node : a -> List (tree a) -> tree a
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node n ts = inDc (TreeF a) (Node n ts)
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mirrorAlg : SAlg (TreeF a) Prelude.id
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mirrorAlg = inSAlg (TreeF a) $ \ up, sfo, abstIn, mirror, t =>
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case t of Node a ts => abstIn (Node a $ map mirror (reverse ts))
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mirror : tree a -> tree a
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mirror = sfold (TreeF a) @{MkFunctor id} mirrorAlg
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namespace Section5Sub4
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0 MappedT : (a, b : Type) -> Type
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MappedT a b = forall r. FoldT (ListF a) (SAlg (ListF a)) r -> a -> r -> (b, r)
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MapThroughAlg : MappedT a b -> Alg (ListF a) (const (List b))
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MapThroughAlg f = inAlg (ListF a) $ \fo, sfo, mapThrough, xs =>
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case xs of
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[] => []
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hd :: tl =>
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let (b, rest) = f sfo hd tl in
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b :: mapThrough rest
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mapThrough : MappedT a b -> list a -> List b
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mapThrough f = fold (ListF a) (MapThroughAlg f) @{MkFunctor (const id)}
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compressSpan : Eq a => MappedT a (Nat, a)
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compressSpan sfo hd tl
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= let (pref, rest) = spanr sfo (hd ==) tl in
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((S (length pref), hd), rest)
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runLengthEncoding : Eq a => List a -> List (Nat, a)
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runLengthEncoding = mapThrough compressSpan . fromList
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namespace Section5Sub5
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K : Type -> Type
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K t = t -> Bool
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MatchT : Type -> Type
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MatchT t = K t -> Bool
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data Regex = Zero | Exact Char | Sum Regex Regex | Cat Regex Regex | Plus Regex
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matchi : (t -> Regex -> MatchT t) -> Regex -> Char -> t -> MatchT t
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matchi matcher Zero c cs k = False
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matchi matcher (Exact c') c cs k = (c == c') && k cs
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matchi matcher (Sum r1 r2) c cs k = matchi matcher r1 c cs k || matchi matcher r2 c cs k
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matchi matcher (Cat r1 r2) c cs k = matchi matcher r1 c cs (\ cs => matcher cs r2 k)
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matchi matcher (Plus r) c cs k = matchi matcher r c cs (\ cs => k cs || matcher cs (Plus r) k)
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MatcherF : Type -> Type
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MatcherF t = Regex -> MatchT t
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functorMatcherF : Functor MatcherF
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functorMatcherF = MkFunctor (\ f, t, r, p => t r (p . f))
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MatcherAlg : Alg (ListF Char) MatcherF
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MatcherAlg = inAlg (ListF Char) $ \ fo, sfo, matcher, s =>
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case s of
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[] => \ r, k => False
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(c :: cs) => \ r => matchi matcher r c cs
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match : Regex -> String -> Bool
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match r str = fold (ListF Char) MatcherAlg @{functorMatcherF} chars r isNil
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where
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isNil : Mu (DcF (ListF Char)) -> Bool
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isNil = fold (ListF Char) {x = const Bool} @{MkFunctor (const id)}
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$ inAlg (ListF Char)
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$ \fo, sfo, rec, xs => case xs of
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Nil => True
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(_ :: _) => False
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chars : Mu (DcF (ListF Char))
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chars = fromList (unpack str)
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export
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matchExample : Bool
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matchExample = match (Plus $ Cat (Sum (Exact 'a') (Exact 'b')) (Exact 'a')) "aabaaaba"
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namespace Section5Sub6
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parameters {0 a : Type} (ltA : a -> a -> Bool)
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0 PartF : Type -> Type
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PartF x = a -> (x, x)
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PartSAlg : SAlg (ListF a) PartF
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PartSAlg = inSAlg (ListF a) $ \up, sfo, abstIn, partition, d, pivot => case d of
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[] => let xs = abstIn d in (xs, xs)
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x :: xs => let (l, r) = partition xs pivot in
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if ltA x pivot then (abstIn (x :: l), up r)
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else (up l, abstIn (x :: r))
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partr : (sfo : FoldT (ListF a) (SAlg (ListF a)) r) -> r -> a -> (r, r)
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partr sfo = sfo @{MkFunctor $ \ f, p, x => bimap f f (p x)} PartSAlg
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QuickSortAlg : Alg (ListF a) (const (List a))
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QuickSortAlg = inAlg (ListF a) $ \ fo, sfo, qsort, xs => case xs of
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[] => []
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p :: xs => let (l, r) = partr sfo xs p in
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qsort l ++ p :: qsort r
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quicksort : List a -> List a
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quicksort = fold (ListF a) QuickSortAlg @{MkFunctor (const id)} . fromList
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export
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sortExample : String -> String
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sortExample = pack . quicksort (<=) . unpack
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@ -5,7 +5,9 @@ depends = contrib, linear
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options = "--ignore-missing-ipkg"
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modules = Data.Container,
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modules = Control.DivideAndConquer,
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Data.Container,
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Data.Description.Regular,
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Data.Description.Indexed,
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