mirror of
https://github.com/idris-lang/Idris2.git
synced 2024-12-24 20:23:11 +03:00
318 lines
9.3 KiB
Idris
318 lines
9.3 KiB
Idris
module Data.Colist1
|
|
|
|
import Data.Colist
|
|
import Data.List
|
|
import Data.List1
|
|
import Data.Nat
|
|
import Data.Stream
|
|
import public Data.Zippable
|
|
|
|
%default total
|
|
|
|
||| A possibly finite, non-empty Stream.
|
|
public export
|
|
data Colist1 : (a : Type) -> Type where
|
|
(:::) : a -> Colist a -> Colist1 a
|
|
|
|
--------------------------------------------------------------------------------
|
|
-- Creating Colist1
|
|
--------------------------------------------------------------------------------
|
|
|
|
||| Convert a `List1` to a `Colist1`.
|
|
public export
|
|
fromList1 : List1 a -> Colist1 a
|
|
fromList1 (h ::: t) = h ::: fromList t
|
|
|
|
||| Convert a stream to a `Colist1`.
|
|
public export
|
|
fromStream : Stream a -> Colist1 a
|
|
fromStream (x :: xs) = x ::: fromStream xs
|
|
|
|
||| Try to convert a `Colist` to a `Colist1`. Returns `Nothing` if
|
|
||| the given `Colist` is empty.
|
|
public export
|
|
fromColist : Colist a -> Maybe (Colist1 a)
|
|
fromColist Nil = Nothing
|
|
fromColist (x :: xs) = Just (x ::: xs)
|
|
|
|
||| Try to convert a list to a `Colist1`. Returns `Nothing` if
|
|
||| the given list is empty.
|
|
public export
|
|
fromList : List a -> Maybe (Colist1 a)
|
|
fromList = fromColist . fromList
|
|
|
|
||| Create a `Colist1` of only a single element.
|
|
public export
|
|
singleton : a -> Colist1 a
|
|
singleton a = a ::: Nil
|
|
|
|
||| An infinite `Colist1` of repetitions of the same element.
|
|
public export
|
|
repeat : a -> Colist1 a
|
|
repeat v = v ::: repeat v
|
|
|
|
||| Create a `Colist1` of `n` replications of the given element.
|
|
public export
|
|
replicate : (n : Nat) -> {auto 0 prf : IsSucc n} -> a -> Colist1 a
|
|
replicate 0 _ impossible
|
|
replicate (S k) x = x ::: replicate k x
|
|
|
|
||| Produce a `Colist1` by repeating a sequence
|
|
public export
|
|
cycle : List1 a -> Colist1 a
|
|
cycle (x ::: xs) = x ::: cycle (xs ++ [x])
|
|
|
|
||| Generate an infinite `Colist1` by repeatedly applying a function.
|
|
public export
|
|
iterate : (f : a -> a) -> a -> Colist1 a
|
|
iterate f a = a ::: iterate f (f a)
|
|
|
|
||| Generate a `Colist1` by repeatedly applying a function.
|
|
||| This stops once the function returns `Nothing`.
|
|
public export
|
|
iterateMaybe : (f : a -> Maybe a) -> a -> Colist1 a
|
|
iterateMaybe f a = a ::: iterateMaybe f (f a)
|
|
|
|
||| Generate a `Colist1` by repeatedly applying a function
|
|
||| to a seed value.
|
|
||| This stops once the function returns `Nothing`.
|
|
public export
|
|
unfold : (f : s -> Maybe (s,a)) -> s -> a -> Colist1 a
|
|
unfold f s a = a ::: unfold f s
|
|
|
|
--------------------------------------------------------------------------------
|
|
-- Basic Functions
|
|
--------------------------------------------------------------------------------
|
|
|
|
||| Convert a `Colist1` to a `Colist`
|
|
public export
|
|
forget : Colist1 a -> Colist a
|
|
forget (h ::: t) = h :: t
|
|
|
|
||| Convert an `Inf (Colist1 a)` to an `Inf (Colist a)`
|
|
public export
|
|
forgetInf : Inf (Colist1 a) -> Inf (Colist a)
|
|
forgetInf (h ::: t) = h :: t
|
|
|
|
||| Prepends an element to a `Colist1`.
|
|
public export
|
|
cons : (x : a) -> (xs : Colist1 a) -> Colist1 a
|
|
cons x xs = x ::: forget xs
|
|
|
|
||| Concatenate two `Colist1`s
|
|
public export
|
|
append : Colist1 a -> Colist1 a -> Colist1 a
|
|
append (h ::: t) ys = h ::: append t (forget ys)
|
|
|
|
||| Append a `Colist1` to a `List`.
|
|
public export
|
|
lappend : List a -> Colist1 a -> Colist1 a
|
|
lappend Nil ys = ys
|
|
lappend (x :: xs) ys = x ::: lappend xs (forget ys)
|
|
|
|
||| Append a `List` to a `Colist1`.
|
|
public export
|
|
appendl : Colist1 a -> List a -> Colist1 a
|
|
appendl (x ::: xs) ys = x ::: appendl xs ys
|
|
|
|
||| Take a `Colist1` apart
|
|
public export
|
|
uncons : Colist1 a -> (a, Colist a)
|
|
uncons (h ::: tl) = (h, tl)
|
|
|
|
||| Extract the first element from a `Colist1`
|
|
public export
|
|
head : Colist1 a -> a
|
|
head (h ::: _) = h
|
|
|
|
||| Drop the first element from a `Colist1`
|
|
public export
|
|
tail : Colist1 a -> Colist a
|
|
tail (_ ::: t) = t
|
|
|
|
||| Take up to `n` elements from a `Colist1`
|
|
public export
|
|
take : (n : Nat) -> {auto 0 prf : IsSucc n} -> Colist1 a -> List1 a
|
|
take 0 _ impossible
|
|
take (S k) (x ::: xs) = x ::: take k xs
|
|
|
|
||| Take elements from a `Colist1` up to and including the
|
|
||| first element, for which `p` returns `True`.
|
|
public export
|
|
takeUntil : (p : a -> Bool) -> Colist1 a -> Colist1 a
|
|
takeUntil p (x ::: xs) = if p x then singleton x else x ::: takeUntil p xs
|
|
|
|
||| Take elements from a `Colist1` up to (but not including) the
|
|
||| first element, for which `p` returns `True`.
|
|
public export
|
|
takeBefore : (p : a -> Bool) -> Colist1 a -> Colist a
|
|
takeBefore p = takeBefore p . forget
|
|
|
|
||| Take elements from a `Colist1` while the given predicate `p`
|
|
||| returns `True`.
|
|
public export
|
|
takeWhile : (p : a -> Bool) -> Colist1 a -> Colist a
|
|
takeWhile p = takeWhile p . forget
|
|
|
|
||| Extract all values wrapped in `Just` from the beginning
|
|
||| of a `Colist1`. This stops, once the first `Nothing` is encountered.
|
|
public export
|
|
takeWhileJust : Colist1 (Maybe a) -> Colist a
|
|
takeWhileJust = takeWhileJust . forget
|
|
|
|
||| Drop up to `n` elements from the beginning of the `Colist1`.
|
|
public export
|
|
drop : (n : Nat) -> Colist1 a -> Colist a
|
|
drop n = drop n . forget
|
|
|
|
||| Try to extract the `n`-th element from a `Colist1`.
|
|
public export
|
|
index : (n : Nat) -> Colist1 a -> Maybe a
|
|
index n = index n . forget
|
|
|
|
||| Produce a `Colist1` of left folds of prefixes of the given `Colist1`.
|
|
||| @ f the combining function
|
|
||| @ acc the initial value
|
|
||| @ xs the `Colist1` to process
|
|
export
|
|
scanl : (f : a -> b -> a) -> (acc : a) -> (xs : Colist1 b) -> Colist1 a
|
|
scanl f acc (x ::: xs) = acc ::: scanl f (f acc x) xs
|
|
|
|
--------------------------------------------------------------------------------
|
|
-- Interfaces
|
|
--------------------------------------------------------------------------------
|
|
|
|
public export
|
|
Semigroup (Colist1 a) where
|
|
(<+>) = append
|
|
|
|
public export
|
|
Functor Colist1 where
|
|
map f (x ::: xs) = f x ::: map f xs
|
|
|
|
public export
|
|
Applicative Colist1 where
|
|
pure = repeat
|
|
|
|
(f ::: fs) <*> (a ::: as) = f a ::: (fs <*> as)
|
|
|
|
public export
|
|
Zippable Colist1 where
|
|
zipWith f (x ::: xs) (y ::: ys) = f x y ::: zipWith f xs ys
|
|
|
|
zipWith3 f (x ::: xs) (y ::: ys) (z ::: zs) =
|
|
f x y z ::: zipWith3 f xs ys zs
|
|
|
|
unzip xs = (map fst xs, map snd xs)
|
|
|
|
unzip3 xs = ( map (\(a,_,_) => a) xs
|
|
, map (\(_,b,_) => b) xs
|
|
, map (\(_,_,c) => c) xs
|
|
)
|
|
|
|
unzipWith f = unzip . map f
|
|
|
|
unzipWith3 f = unzip3 . map f
|
|
|
|
--------------------------------------------------------------------------------
|
|
-- Interleavings
|
|
--------------------------------------------------------------------------------
|
|
|
|
-- zig, zag, and cantor are taken from the paper
|
|
-- Applications of Applicative Proof Search
|
|
-- by Liam O'Connor
|
|
|
|
public export
|
|
zig : List1 (Colist1 a) -> Colist (Colist1 a) -> Colist a
|
|
|
|
public export
|
|
zag : List1 a -> List (Colist1 a) -> Colist (Colist1 a) -> Colist a
|
|
|
|
zig xs = zag (head <$> xs) (mapMaybe (fromColist . tail) $ forget xs)
|
|
|
|
zag (x ::: []) [] [] = x :: []
|
|
zag (x ::: []) (z :: zs) [] = x :: zig (z ::: zs) []
|
|
zag (x ::: []) zs (l :: ls) = x :: zig (l ::: zs) ls
|
|
zag (x ::: (y :: xs)) zs ls = x :: zag (y ::: xs) zs ls
|
|
|
|
public export
|
|
cantor : Colist1 (Colist1 a) -> Colist1 a
|
|
cantor (xxs ::: []) = xxs
|
|
cantor ((x ::: xs) ::: (yys :: zzss))
|
|
= x ::: zig (yys ::: mapMaybe fromColist [xs]) zzss
|
|
|
|
namespace Colist
|
|
|
|
public export
|
|
cantor : List (Colist a) -> Colist a
|
|
cantor xs =
|
|
let Just (l ::: ls) = List.toList1' $ mapMaybe fromColist xs
|
|
| Nothing => []
|
|
in zig (l ::: []) (fromList ls)
|
|
|
|
-- Exploring the (truncated) Nat*Nat top right quadrant of the plane
|
|
-- using Cantor's zig-zag traversal:
|
|
example :
|
|
let nats : Colist Nat; nats = fromStream Stream.nats in
|
|
take 10 (Colist.cantor [ map (0,) nats
|
|
, map (1,) nats
|
|
, map (2,) nats
|
|
, map (3,) nats])
|
|
=== [ (0, 0)
|
|
, (1, 0), (0, 1)
|
|
, (2, 0), (1, 1), (0, 2)
|
|
, (3, 0), (2, 1), (1, 2), (0, 3)
|
|
]
|
|
example = Refl
|
|
|
|
namespace DPair
|
|
|
|
||| Explore the plane corresponding to all possible pairings
|
|
||| using Cantor's zig zag traversal
|
|
public export
|
|
planeWith : {0 p : a -> Type} ->
|
|
((x : a) -> p x -> c) ->
|
|
Colist1 a -> ((x : a) -> Colist1 (p x)) ->
|
|
Colist1 c
|
|
planeWith k as f = cantor (map (\ x => map (k x) (f x)) as)
|
|
|
|
||| Explore the plane corresponding to all possible pairings
|
|
||| using Cantor's zig zag traversal
|
|
public export
|
|
plane : {0 p : a -> Type} ->
|
|
Colist1 a -> ((x : a) -> Colist1 (p x)) ->
|
|
Colist1 (x : a ** p x)
|
|
plane = planeWith (\ x, prf => (x ** prf))
|
|
|
|
namespace Pair
|
|
|
|
||| Explore the plane corresponding to all possible pairings
|
|
||| using Cantor's zig zag traversal
|
|
public export
|
|
planeWith : (a -> b -> c) ->
|
|
Colist1 a -> (a -> Colist1 b) ->
|
|
Colist1 c
|
|
planeWith k as f = cantor (map (\ x => map (k x) (f x)) as)
|
|
|
|
||| Explore the plane corresponding to all possible pairings
|
|
||| using Cantor's zig zag traversal
|
|
public export
|
|
plane : Colist1 a -> (a -> Colist1 b) -> Colist1 (a, b)
|
|
plane = Pair.planeWith (,)
|
|
|
|
--------------------------------------------------------------------------------
|
|
-- Example
|
|
--------------------------------------------------------------------------------
|
|
|
|
-- Exploring the Nat*Nat top right quadrant of the plane
|
|
-- using Cantor's zig-zag traversal:
|
|
example :
|
|
let nats1 = fromStream Stream.nats in
|
|
Colist1.take 10 (Pair.plane nats1 (const nats1))
|
|
=== (0, 0) :::
|
|
[ (1, 0), (0, 1)
|
|
, (2, 0), (1, 1), (0, 2)
|
|
, (3, 0), (2, 1), (1, 2), (0, 3)
|
|
]
|
|
example = Refl
|