{-# LANGUAGE DataKinds #-}
{-# LANGUAGE GADTs #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE TypeApplications #-}
{-# LANGUAGE TypeOperators #-}
module Data.Array.Nested.Internal.Lemmas where

import Data.Proxy
import Data.Type.Equality
import GHC.TypeLits

import Data.Array.Mixed.Lemmas
import Data.Array.Mixed.Permutation
import Data.Array.Mixed.Shape
import Data.Array.Mixed.Types
import Data.Array.Nested.Internal.Shape


lemRankMapJust :: ShS sh -> Rank (MapJust sh) :~: Rank sh
lemRankMapJust ZSS = Refl
lemRankMapJust (_ :$$ sh') | Refl <- lemRankMapJust sh' = Refl

lemMapJustApp :: ShS sh1 -> Proxy sh2
              -> MapJust (sh1 ++ sh2) :~: MapJust sh1 ++ MapJust sh2
lemMapJustApp ZSS _ = Refl
lemMapJustApp (_ :$$ sh) p | Refl <- lemMapJustApp sh p = Refl

lemTakeLenMapJust :: Perm is -> ShS sh -> TakeLen is (MapJust sh) :~: MapJust (TakeLen is sh)
lemTakeLenMapJust PNil _ = Refl
lemTakeLenMapJust (_ `PCons` is) (_ :$$ sh) | Refl <- lemTakeLenMapJust is sh = Refl
lemTakeLenMapJust (_ `PCons` _) ZSS = error "TakeLen of empty"

lemDropLenMapJust :: Perm is -> ShS sh -> DropLen is (MapJust sh) :~: MapJust (DropLen is sh)
lemDropLenMapJust PNil _ = Refl
lemDropLenMapJust (_ `PCons` is) (_ :$$ sh) | Refl <- lemDropLenMapJust is sh = Refl
lemDropLenMapJust (_ `PCons` _) ZSS = error "DropLen of empty"

lemIndexMapJust :: SNat i -> ShS sh -> Index i (MapJust sh) :~: Just (Index i sh)
lemIndexMapJust SZ (_ :$$ _) = Refl
lemIndexMapJust (SS (i :: SNat i')) ((_ :: SNat n) :$$ (sh :: ShS sh'))
  | Refl <- lemIndexMapJust i sh
  , Refl <- lemIndexSucc (Proxy @i') (Proxy @(Just n)) (Proxy @(MapJust sh'))
  , Refl <- lemIndexSucc (Proxy @i') (Proxy @n) (Proxy @sh')
  = Refl
lemIndexMapJust _ ZSS = error "Index of empty"

lemPermuteMapJust :: Perm is -> ShS sh -> Permute is (MapJust sh) :~: MapJust (Permute is sh)
lemPermuteMapJust PNil _ = Refl
lemPermuteMapJust (i `PCons` is) sh
  | Refl <- lemPermuteMapJust is sh
  , Refl <- lemIndexMapJust i sh
  = Refl

lemKnownMapJust :: forall sh. KnownShS sh => Proxy sh -> Dict KnownShX (MapJust sh)
lemKnownMapJust _ = lemKnownShX (go (knownShS @sh))
  where
    go :: ShS sh' -> StaticShX (MapJust sh')
    go ZSS = ZKX
    go (n :$$ sh) = SKnown n :!% go sh