module Data.Universe.Helpers (
  -- | This module is for functions that are useful for writing instances,
  -- but not necessarily for using them (and hence are not exported by the
  -- main module to avoid cluttering up the namespace).

  -- * Building lists
  universeDef,
  interleave,
  diagonal,
  diagonals,
  (+++),
  cartesianProduct,
  (+*+),
  (<+*+>),
  choices,

  -- * Building cardinalities
  -- | These functions are handy for inheriting the definition of
  -- 'Data.Universe.Class.cardinality' in a newtype instance. For example,
  -- one might write
  --
  -- > newtype Foo = Foo Bar
  -- > instance Finite Foo where cardinality = retagWith Foo cardinality
  retagWith,
  retag,
  Tagged (..),
  Natural,

  -- * Debugging
  -- | These functions exist primarily as a specification to test against.
  unfairCartesianProduct,
  unfairChoices
  ) where

import Data.List
import Data.Tagged (Tagged (..), retag)
import Numeric.Natural (Natural)

-- | For many types, the 'universe' should be @[minBound .. maxBound]@;
-- 'universeDef' makes it easy to make such types an instance of 'Universe' via
-- the snippet
--
-- > instance Universe Foo where universe = universeDef
universeDef :: (Bounded a, Enum a) => [a]
universeDef = [minBound .. maxBound]

-- | Fair n-way interleaving: given a finite number of (possibly infinite)
-- lists, produce a single list such that whenever @v@ has finite index in one
-- of the input lists, @v@ also has finite index in the output list. No list's
-- elements occur more frequently (on average) than another's.
interleave :: [[a]] -> [a]
interleave = concat . transpose

-- | Unfair n-way interleaving: given a possibly infinite number of (possibly
-- infinite) lists, produce a single list such that whenever @v@ has finite
-- index in an input list at finite index, @v@ also has finite index in the
-- output list. Elements from lists at lower index occur more frequently, but
-- not exponentially so.
diagonal :: [[a]] -> [a]
diagonal = concat . diagonals

-- | Like 'diagonal', but expose a tiny bit more (non-semantic) information:
-- if you lay out the input list in two dimensions, each list in the result
-- will be one of the diagonals of the input. In particular, each element of
-- the output will be a list whose elements are each from a distinct input
-- list.
diagonals :: [[a]] -> [[a]]
diagonals = tail . go [] where
  -- it is critical for some applications that we start producing answers
  -- before inspecting es_
  go b es_ = [h | h:_ <- b] : case es_ of
    []   -> transpose ts
    e:es -> go (e:ts) es
    where ts = [t | _:t <- b]

-- | Fair 2-way interleaving.
(+++) :: [a] -> [a] -> [a]
xs +++ ys = interleave [xs,ys]

-- | Slightly unfair 2-way Cartesian product: given two (possibly infinite)
-- lists, produce a single list such that whenever @v@ and @w@ have finite
-- indices in the input lists, @(v,w)@ has finite index in the output list.
-- Lower indices occur as the @fst@ part of the tuple more frequently, but not
-- exponentially so.
cartesianProduct :: (a -> b -> c) -> [a] -> [b] -> [c]
-- special case: don't want to construct an infinite list of empty lists to pass to diagonal
cartesianProduct _ []   _  = []
cartesianProduct f xs  ys  = diagonal [[f x y | x <- xs] | y <- ys]

-- | @'cartesianProduct' (,)@
(+*+) :: [a] -> [b] -> [(a,b)]
(+*+) = cartesianProduct (,)

-- | A '+*+' with application.
--
-- @'cartesianProduct' ($)@
(<+*+>) :: [a -> b] -> [a] -> [b]
(<+*+>) = cartesianProduct ($)

-- | Slightly unfair n-way Cartesian product: given a finite number of
-- (possibly infinite) lists, produce a single list such that whenever @vi@ has
-- finite index in list i for each i, @[v1, ..., vn]@ has finite index in the
-- output list.
choices :: [[a]] -> [[a]]
choices = foldr (cartesianProduct (:)) [[]]

retagWith :: (a -> b) -> Tagged a x -> Tagged b x
retagWith _ (Tagged n) = Tagged n

-- | Very unfair 2-way Cartesian product: same guarantee as the slightly unfair
-- one, except that lower indices may occur as the @fst@ part of the tuple
-- exponentially more frequently.
unfairCartesianProduct :: (a -> b -> c) -> [a] -> [b] -> [c]
unfairCartesianProduct _ _  [] = [] -- special case: don't want to walk down xs forever hoping one of them will produce a nonempty thing
unfairCartesianProduct f xs ys = go xs ys where
  go (x:xs) ys = map (f x) ys +++ go xs ys
  go []     ys = []

-- | Very unfair n-way Cartesian product: same guarantee as the slightly unfair
-- one, but not as good in the same sense that the very unfair 2-way product is
-- worse than the slightly unfair 2-way product.
unfairChoices :: [[a]] -> [[a]]
unfairChoices = foldr ((map (uncurry (:)) .) . unfairCartesianProduct (,)) [[]]