{-# LANGUAGE DataKinds #-}
{-# LANGUAGE DeriveTraversable #-}
{-# LANGUAGE GADTs #-}
{-# LANGUAGE PolyKinds #-}
{-# LANGUAGE QuantifiedConstraints #-}
{-# LANGUAGE RankNTypes #-}
{-# LANGUAGE StandaloneDeriving #-}
{-# LANGUAGE TypeOperators #-}
module Data where

import Lemmas (Append)


data SList f l where
  SNil :: SList f '[]
  SCons :: f a -> SList f l -> SList f (a : l)
deriving instance (forall a. Show (f a)) => Show (SList f l)
infixr `SCons`

slistMap :: (forall t. f t -> g t) -> SList f list -> SList g list
slistMap _ SNil = SNil
slistMap f (SCons x list) = SCons (f x) (slistMap f list)

sappend :: SList f l1 -> SList f l2 -> SList f (Append l1 l2)
sappend SNil l = l
sappend (SCons x xs) l = SCons x (sappend xs l)

data Nat = Z | S Nat
  deriving (Show, Eq, Ord)

data SNat n where
  SZ :: SNat Z
  SS :: SNat n -> SNat (S n)
deriving instance Show (SNat n)

fromSNat :: SNat n -> Int
fromSNat SZ = 0
fromSNat (SS n) = succ (fromSNat n)

class KnownNat n where knownNat :: SNat n
instance KnownNat Z where knownNat = SZ
instance KnownNat n => KnownNat (S n) where knownNat = SS knownNat

data Vec n t where
  VNil :: Vec Z t
  (:<) :: t -> Vec n t -> Vec (S n) t
deriving instance Show t => Show (Vec n t)
deriving instance Functor (Vec n)
deriving instance Foldable (Vec n)
deriving instance Traversable (Vec n)

vecLength :: Vec n t -> SNat n
vecLength VNil = SZ
vecLength (_ :< v) = SS (vecLength v)