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{-# LANGUAGE DataKinds #-}
{-# LANGUAGE GADTs #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE TypeApplications #-}
{-# LANGUAGE TypeOperators #-}
{-# OPTIONS_GHC -fplugin GHC.TypeLits.Normalise #-}
{-# OPTIONS_GHC -fplugin GHC.TypeLits.KnownNat.Solver #-}
module Data.Array.Mixed.Lemmas where
import Data.Proxy
import Data.Type.Equality
import GHC.TypeLits
import Data.Array.Mixed.Shape
import Data.Array.Mixed.Types
lemRankApp :: forall sh1 sh2.
StaticShX sh1 -> StaticShX sh2
-> Rank (sh1 ++ sh2) :~: Rank sh1 + Rank sh2
lemRankApp ZKX _ = Refl
lemRankApp (_ :!% (ssh1 :: StaticShX sh1T)) ssh2
= lem2 (Proxy @(Rank sh1T)) Proxy Proxy $
lem (Proxy @(Rank sh2)) (Proxy @(Rank sh1T)) (Proxy @(Rank (sh1T ++ sh2))) $
lemRankApp ssh1 ssh2
where
lem :: proxy a -> proxy b -> proxy c
-> c :~: b + a
-> b + a :~: c
lem _ _ _ Refl = Refl
lem2 :: proxy a -> proxy b -> proxy c
-> (a + b :~: c)
-> c + 1 :~: (a + 1 + b)
lem2 _ _ _ Refl = Refl
lemRankAppComm :: StaticShX sh1 -> StaticShX sh2
-> Rank (sh1 ++ sh2) :~: Rank (sh2 ++ sh1)
lemRankAppComm _ _ = unsafeCoerceRefl -- TODO improve this
lemKnownNatRank :: IShX sh -> Dict KnownNat (Rank sh)
lemKnownNatRank ZSX = Dict
lemKnownNatRank (_ :$% sh) | Dict <- lemKnownNatRank sh = Dict
lemKnownNatRankSSX :: StaticShX sh -> Dict KnownNat (Rank sh)
lemKnownNatRankSSX ZKX = Dict
lemKnownNatRankSSX (_ :!% ssh) | Dict <- lemKnownNatRankSSX ssh = Dict
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