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

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

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

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

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