{-# LANGUAGE FlexibleInstances #-}
 
{-# LANGUAGE GeneralizedNewtypeDeriving #-}
 
-- {-# LANGUAGE DerivingVia #-}
 
 
-- Finite Field 6
 
newtype ToFF a = ToFF a deriving ( Eq, Integral, Real , Enum ,Ord )
 
newtype ToWM a = ToWM a deriving ( Eq, Enum, Real, Ord, Integral, Num ) -- ( Eq, Integral, Real , Enum ,Ord )
 
instance  ( Show a ) => Show ( ToWM a ) where
   show ( ToWM x ) = dropWhile ( == ' ' ) $ dropWhile (/=' ' ) $ show x
    --show ( ToWM x ) = show x
 
class WithModulus a where
    modulus :: a -> Integer
 
instance ( WithModulus a, Num a, Integral a ) => Num ( ToFF a ) where
    ( ToFF x ) + ( ToFF y ) = ToFF $ mod ( x + y ) ( fromInteger $ modulus x )
    ( ToFF x ) - ( ToFF y ) = ToFF $ mod ( x - y ) ( fromInteger $ modulus x )
    ( ToFF x ) * ( ToFF y ) = ToFF $ mod ( x * y ) ( fromInteger $ modulus x )
    signum ( ToFF x ) = ToFF $ signum x
    abs ( ToFF x ) = ToFF $ abs x
    fromInteger = ToFF . fromInteger
 
instance  ( WithModulus a, Show a ) => Show ( ToFF a ) where
   --show ( ToFF x ) = dropWhile ( == ' ' ) $ dropWhile (/=' ' ) $ show x
   show ( ToFF x ) =  show x
 
 
 
{-
ひとつの標数につ1個の整数型の型を用意する
用意する型は
( Eq, Num, Integral, Real, Ord, Enum ,Show )
のインスタンスであるとする
以下それをBTP型とする
ToWM BTP型にWithModulusクラスを継承する
その際メンバ関数modurusは引数によらす標数を返す関数として定義しておく
ToFF ( ToWM BTP )型が有限素体の計算を行う型として使える
これには( Eq, Num, Integral, Real, Ord, Enum ,Show )のインスタンスが自動的に付与される
-}
------ 𝔽₅------
newtype BT65537 = BT65537 Integer deriving ( Eq, Num, Integral, Real, Ord, Enum ,Show )
 
instance WithModulus ( ToWM BT65537 ) where
    modulus _ = 65537
 
newtype F65537 = F65537 ( ToFF ( ToWM BT65537 ) ) deriving ( Eq, Num, Integral, Real, Ord, Enum ,Show )
 
------ 𝔽₂₅₇------
newtype BT257= BT257 Int deriving ( Eq, Num, Integral, Real, Ord, Enum ,Show )
 
instance WithModulus ( ToWM BT257 ) where
    modulus _ = 257
 
newtype F257 = F257 ( ToFF ( ToWM BT257) ) deriving ( Eq, Num, Integral, Real, Ord, Enum ,Show )
 
------ 𝔽₆₅₅₃₇------
newtype BT5= BT5 Integer deriving ( Eq, Num, Integral, Real, Ord, Enum ,Show )
 
instance WithModulus ( ToWM BT5 ) where
    modulus _ = 5
 
newtype F5 = F5( ToFF ( ToWM BT5) ) deriving ( Eq, Num, Integral, Real, Ord, Enum ,Show )
 
main = do
    print $ ( F5 3 )^4
    print $ ( F257 3 )^256
    print $ product [ F257 1..F257 256 ]
    print $ product [ F65537 1..F65537 65536 ]
 

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