module Quox.Syntax.Subst

import public Quox.Syntax.Shift
import Quox.Syntax.Var
import Quox.Name

import Data.Nat
import Data.List
import Data.SnocVect
import Derive.Prelude

%default total
%language ElabReflection


public export
data Subst : (Nat -> Type) -> Nat -> Nat -> Type where
  Shift : Shift from to -> Subst env from to
  (:::) : (t : Lazy (env to)) -> Subst env from to -> Subst env (S from) to
%name Subst th, ph, ps

infixr 7 !:::
||| in case the automatic laziness insertion gets confused
public export
(!:::) : env to -> Subst env from to -> Subst env (S from) to
t !::: ts = t ::: ts


private
Repr : (Nat -> Type) -> Nat -> Type
Repr f to = (List (f to), Nat)

private
repr : Subst f from to -> Repr f to
repr (Shift by) = ([], by.nat)
repr (t ::: th) = let (ts, i) = repr th in (t::ts, i)


export Eq   (f to) => Eq   (Subst f from to) where (==) = (==) `on` repr
export Ord  (f to) => Ord  (Subst f from to) where compare = compare `on` repr
export Show (f to) => Show (Subst f from to) where show = show . repr


infixl 8 //
public export
interface FromVar term => CanSubstSelf term where
  (//) : term from -> Lazy (Subst term from to) -> term to


public export
getLoc : FromVar term => Subst term from to -> Var from -> Loc -> term to
getLoc (Shift by) i      loc = fromVarLoc (shift by i) loc
getLoc (t ::: th) VZ     _   = t
getLoc (t ::: th) (VS i) loc = getLoc th i loc


public export
CanSubstSelf Var where
  i    // Shift by   = shift by i
  VZ   // (t ::: th) = t
  VS i // (t ::: th) = i // th


public export %inline
shift : (by : Nat) -> Subst env from (by + from)
shift by = Shift $ fromNat by

public export %inline
shift0 : (by : Nat) -> Subst env 0 by
shift0 by = rewrite sym $ plusZeroRightNeutral by in Shift $ fromNat by


public export
(.) : CanSubstSelf f => Subst f from mid -> Subst f mid to -> Subst f from to
Shift by      . Shift bz   = Shift $ by . bz
Shift SZ      . ph         = ph
Shift (SS by) . (t ::: th) = Shift by . th
(t ::: th)    . ph         = (t // ph) ::: (th . ph)

public export %inline
id : Subst f n n
id = shift 0

public export
traverse : Applicative m =>
           (f to -> m (g to)) -> Subst f from to -> m (Subst g from to)
traverse f (Shift by) = pure $ Shift by
traverse f (t ::: th) = [|f t !::: traverse f th|]

-- not in terms of traverse because this map can maintain laziness better
public export
map : (f to -> g to) -> Subst f from to -> Subst g from to
map f (Shift by) = Shift by
map f (t ::: th) = f t ::: map f th


public export %inline
push : CanSubstSelf f => Subst f from to -> Subst f (S from) (S to)
push th = fromVar VZ ::: (th . shift 1)

-- [fixme] a better way to do this?
public export
pushN : CanSubstSelf f => (s : Nat) ->
        Subst f from to -> Subst f (s + from) (s + to)
pushN 0     th = th
pushN (S s) th =
  rewrite plusSuccRightSucc s from in
  rewrite plusSuccRightSucc s to in
  pushN s $ fromVar VZ ::: (th . shift 1)

public export
drop1 : Subst f (S from) to -> Subst f from to
drop1 (Shift by) = Shift $ ssDown by
drop1 (t ::: th) = th


public export
fromSnocVect : SnocVect s (f n) -> Subst f (s + n) n
fromSnocVect [<]       = id
fromSnocVect (xs :< x) = x ::: fromSnocVect xs

public export %inline
one : f n -> Subst f (S n) n
one x = fromSnocVect [< x]


||| whether two substitutions with the same codomain have the same shape
||| (the same number of terms and the same shift at the end). if so, they
||| also have the same domain
export
cmpShape : Subst env from1 to -> Subst env from2 to ->
           Either Ordering (from1 = from2)
cmpShape (Shift by) (Shift bz) = cmpLen by bz
cmpShape (Shift _)  (_ ::: _)  = Left LT
cmpShape (_ ::: _)  (Shift _)  = Left GT
cmpShape (_ ::: th) (_ ::: ph) = cong S <$> cmpShape th ph


public export
record WithSubst tm env n where
  constructor Sub
  term  : tm from
  subst : Lazy (Subst env from n)

export
(Eq (env n), forall n. Eq (tm n)) => Eq (WithSubst tm env n) where
  Sub t1 s1 == Sub t2 s2 =
    case cmpShape s1 s2 of
      Left  _    => False
      Right Refl => t1 == t2 && s1 == s2

export
(Ord (env n), forall n. Ord (tm n)) => Ord (WithSubst tm env n) where
  Sub t1 s1 `compare` Sub t2 s2 =
    case cmpShape s1 s2 of
      Left  o    => o
      Right Refl => compare (t1, s1) (t2, s2)

export %hint
ShowWithSubst : (Show (env n), forall n. Show (tm n)) =>
                Show (WithSubst tm env n)
ShowWithSubst = deriveShow