1 files changed, 59 insertions, 0 deletions
diff --git a/src/Data/Array/Nested/Internal/Lemmas.hs b/src/Data/Array/Nested/Internal/Lemmas.hs
new file mode 100644
index 0000000..5ce5373
--- /dev/null
+++ b/src/Data/Array/Nested/Internal/Lemmas.hs
@@ -0,0 +1,59 @@
+{-# 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
+
+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