quox/lib/Quox/Context.idr

module Quox.Context

import Quox.Syntax.Shift
import Quox.Pretty

import Data.DPair
import Data.Nat
import Data.SnocList
import Data.Vect
import Control.Monad.Identity

%default total


||| a sequence of bindings under an existing context. each successive element
||| has one more bound variable, which correspond to all previous elements
||| as well as the surrounding context.
public export
data Telescope : (tm : Nat -> Type) -> (from, to : Nat) -> Type where
  Lin  : Telescope tm from from
  (:<) : Telescope tm from to -> tm to -> Telescope tm from (S to)
%name Telescope tel

public export
Telescope' : (a : Type) -> (from, to : Nat) -> Type
Telescope' a = Telescope (\_ => a)

||| a top level context is actually just a telescope over no existing bindings
public export
Context : (tm : Nat -> Type) -> (len : Nat) -> Type
Context tm len = Telescope tm 0 len

public export
Context' : (a : Type) -> (len : Nat) -> Type
Context' a = Context (\_ => a)


public export
tail : Context tm (S n) -> Context tm n
tail (tel :< _) = tel

export
toSnocList : Telescope tm _ _ -> SnocList (Exists tm)
toSnocList [<]        = [<]
toSnocList (tel :< t) = toSnocList tel :< Evidence _ t

export %inline
toList : Telescope tm _ _ -> List (Exists tm)
toList tel = toListAcc tel [] where
  toListAcc : Telescope tm _ _ -> List (Exists tm) -> List (Exists tm)
  toListAcc [<]        acc = acc
  toListAcc (tel :< t) acc = toListAcc tel (Evidence _ t :: acc)

export %inline
toSnocList' : Telescope' a _ _ -> SnocList a
toSnocList' = map snd . toSnocList

export %inline
toList' : Telescope' a _ _ -> List a
toList' = map snd . toList


public export
(.) : Telescope tm from mid -> Telescope tm mid to -> Telescope tm from to
tel1 . [<]         = tel1
tel1 . (tel2 :< s) = (tel1 . tel2) :< s

export
(<><) : Telescope' a from to -> Vect n a -> Telescope' a from (n + to)
(<><) tel [] = tel
(<><) tel (x :: xs) {n = S n} =
  rewrite plusSuccRightSucc n to in
  (tel :< x) <>< xs


public export
getShiftWith : (forall from, to. tm from -> Shift from to -> tm to) ->
               Shift len out -> Context tm len -> Var len -> tm out
getShiftWith shft by (ctx :< t) VZ     = t `shft` ssDown by
getShiftWith shft by (ctx :< t) (VS i) = getShiftWith shft (ssDown by) ctx i

public export %inline
getShift : CanShift tm => Shift len out -> Context tm len -> Var len -> tm out
getShift = getShiftWith (//)

public export %inline
getWith : (forall from, to. tm from -> Shift from to -> tm to) ->
          Context tm len -> Var len -> tm len
getWith shft = getShiftWith shft SZ

infixl 8 !!
public export %inline
(!!) : CanShift tm => Context tm len -> Var len -> tm len
(!!) = getWith (//)

infixl 8 !!!
public export %inline
(!!!) : Context' tm len -> Var len -> tm
(!!!) = getWith const

public export
find : Alternative f =>
       (forall n. tm n -> Bool) -> Context tm len -> f (Var len)
find p [<]        = empty
find p (ctx :< x) = (guard (p x) $> VZ) <|> (VS <$> find p ctx)


export
0 telescopeLTE : Telescope _ from to -> from `LTE` to
telescopeLTE [<]        = reflexive
telescopeLTE (tel :< _) = lteSuccRight $ telescopeLTE tel

export
(gt : from `GT` to) => Uninhabited (Telescope _ from to) where
  uninhabited tel = void $ LTEImpliesNotGT (telescopeLTE tel) gt

export %hint
0 succGT : S n `GT` n
succGT = LTESucc reflexive


parameters {auto _ : Applicative f}
  export
  traverse : (forall n. tm1 n -> f (tm2 n)) ->
             Telescope tm1 from to -> f (Telescope tm2 from to)
  traverse f [<]        = pure [<]
  traverse f (tel :< x) = [|traverse f tel :< f x|]

  infixl 3 `app`
  ||| like `(<*>)` but with effects
  export
  app : Telescope (\n => tm1 n -> f (tm2 n)) from to ->
        Telescope tm1 from to -> f (Telescope tm2 from to)
  app [<]         [<]         = pure [<]
  app (ftel :< f) (xtel :< x) = [|app ftel xtel :< f x|]
  app [<]         (xtel :< _) = void $ uninhabited xtel
  app (ftel :< _) [<]         = void $ uninhabited ftel

  export %inline
  sequence : Telescope (f . tm) from to -> f (Telescope tm from to)
  sequence = traverse id

parameters {0 tm1, tm2 : Nat -> Type}
           (f : forall n. tm1 n -> tm2 n)
  export %inline
  map : Telescope tm1 from to -> Telescope tm2 from to
  map = runIdentity . traverse (pure . f)

  export %inline
  (<$>) : Telescope tm1 from to -> Telescope tm2 from to
  (<$>) = map

export %inline
(<*>) : Telescope (\n => tm1 n -> tm2 n) from to ->
        Telescope tm1 from to -> Telescope tm2 from to
ftel <*> xtel = runIdentity $ (pure .) <$> ftel `app` xtel
-- ...but can't write pure without `from,to` being ω, so no idiom brackets ☹

export %inline
(<$) : (forall n. tm1 n) -> Telescope tm2 from to -> Telescope tm1 from to
x <$ tel = const x <$> tel


export %inline
zipWith : (forall n. tm1 n -> tm2 n -> tm3 n) ->
          Telescope tm1 from to -> Telescope tm2 from to ->
          Telescope tm3 from to
zipWith f tel1 tel2 = f <$> tel1 <*> tel2

export %inline
zipWithLazy : forall tm1, tm2, tm3.
              (forall n. tm1 n -> tm2 n -> tm3 n) ->
              Telescope tm1 from to -> Telescope tm2 from to ->
              Telescope (\n => Lazy (tm3 n)) from to
zipWithLazy f = zipWith $ delay .: f

export
unzip : Telescope (\n => (tm1 n, tm2 n)) from to ->
        (Telescope tm1 from to, Telescope tm2 from to)
unzip [<]             = ([<], [<])
unzip (tel :< (x, y)) = let (xs, ys) = unzip tel in (xs :< x, ys :< y)


export
lengthPrf : Telescope _ from to -> Subset Nat (\len => len + from = to)
lengthPrf [<]        = Element 0 Refl
lengthPrf (tel :< _) =
  let len = lengthPrf tel in Element (S len.fst) (cong S len.snd)

export
lengthPrf0 : Context _ to -> Subset Nat (\len => len = to)
lengthPrf0 ctx =
  let len = lengthPrf ctx in
  Element len.fst (rewrite sym $ plusZeroRightNeutral len.fst in len.snd)

public export %inline
length : Telescope {} -> Nat
length = fst . lengthPrf


export
foldl : {0 acc : Nat -> Type} ->
        (f : forall n. acc n -> tm (n + from) -> acc (S n)) ->
        (z : acc 0) -> (tel : Telescope tm from to) -> acc (length tel)
foldl f z [<]        = z
foldl f z (tel :< t) = f (foldl f z tel) (rewrite (lengthPrf tel).snd in t)

export %inline
foldMap : Monoid a => (forall n. tm n -> a) -> Telescope tm from to -> a
foldMap f = foldl (\acc, tm => acc <+> f tm) neutral

export %inline
fold : Monoid a => Telescope' a from to -> a
fold = foldMap id

||| like `fold` but calculate the elements only when actually appending
export %inline
foldLazy : Monoid a => Telescope' (Lazy a) from to -> a
foldLazy = foldMap force


export %inline
(forall n. Eq (tm n)) => Eq (Telescope tm from to) where
  (==) = foldLazy @{All} .: zipWithLazy (==)

export %inline
(forall n. Ord (tm n)) => Ord (Telescope tm from to) where
  compare = foldLazy .: zipWithLazy compare

export %inline
(forall n. Show (tm n)) => Show (Telescope tm from to) where
  showPrec d = showPrec d . toSnocList
    where Show (Exists tm) where showPrec d t = showPrec d t.snd

export %inline
(forall n. PrettyHL (tm n)) => PrettyHL (Telescope tm from to) where
  prettyM tel = separate (hl Delim ";") <$> traverse prettyM (toList tel)