问如何创建带约束的类型

EN

module Test(MyType,x,y,z,createMyType) where

data MyType = MT { x :: Int, y :: Int, z :: Int }

createMyType :: Int -> Int -> MyType
createMyType myX myY = MT { x = myX, y = myY, z = 5 - myX - myY }

{-# LANGUAGE GADTs, TypeFamilies, RankNTypes, StandaloneDeriving, UndecidableInstances, TypeOperators #-}

data Z = Z
data S n = S n

data Nat n where
  Zero :: Nat Z
  Suc  :: Nat n -> Nat (S n)

deriving instance Show (Nat n)

type family (:+) a b :: *
type instance (:+) Z b = b
type instance (:+) (S a) b = S (a :+ b)

plus :: Nat x -> Nat y -> Nat (x :+ y)
plus Zero y = y
plus (Suc x) y = Suc (x `plus` y)

type family (:*) a b :: *
type instance (:*) Z b = Z
type instance (:*) (S a) b = b :+ (a :* b)

times :: Nat x -> Nat y -> Nat (x :* y)
times Zero y = Zero
times (Suc x) y = y `plus` (x `times` y)

data (:==) a b where
  Refl :: a :== a

deriving instance Show (a :== b)

cong :: a :== b -> f a :== f b
cong Refl = Refl

data Triple where
  Triple :: Nat x -> Nat y -> Nat z -> (z :== (x :+ y)) -> Triple

deriving instance Show Triple

-- Half a decision procedure
equal :: Nat x -> Nat y -> Maybe (x :== y)
equal Zero Zero = Just Refl
equal (Suc x) Zero = Nothing
equal Zero (Suc y) = Nothing
equal (Suc x) (Suc y) = cong `fmap` equal x y

triple' :: Nat x -> Nat y -> Nat z -> Maybe Triple
triple' x y z = fmap (Triple x y z) $ equal z (x `plus` y)

toNat :: (forall n. Nat n -> r) -> Integer -> r
toNat f n | n < 0 = error "why can't we have a natural type?"
toNat f 0 = f Zero
toNat f n = toNat (f . Suc) (n - 1)

triple :: Integer -> Integer -> Integer -> Maybe Triple
triple x y z = toNat (\x' -> toNat (\y' -> toNat (\z' -> triple' x' y' z') z) y) x

data Yatima where
  Yatima :: Nat x -> Nat y -> Nat z -> ((x :* x) :+ (y :* y) :+ (z :* z) :== S (S (S (S (S Z))))) -> Yatima

deriving instance Show Yatima

yatima' :: Nat x -> Nat y -> Nat z -> Maybe Yatima
yatima' x y z = 
  fmap (Yatima x y z) $ equal ((x `times` x) `plus` (y `times` y) `plus` (z `times` z)) (Suc (Suc (Suc (Suc (Suc Zero)))))

yatima :: Integer -> Integer -> Integer -> Maybe Yatima
yatima x y z = toNat (\x' -> toNat (\y' -> toNat (\z' -> yatima' x' y' z') z) y) x


{-
λ> triple 3 4 5
Nothing
λ> triple 3 4 7
Just (Triple (Suc (Suc (Suc Zero))) (Suc (Suc (Suc (Suc Zero)))) Refl (Suc (Suc (Suc (Suc (Suc (Suc (Suc Zero))))))))

λ> yatima 0 1 2 
Just (Yatima Zero (Suc Zero) (Suc (Suc Zero)) Refl)
λ> yatima 1 1 2 
Nothing
-}

open import Relation.Binary.PropositionalEquality (_≡_)
open import Data.Nat (ℕ; _+_)
open import Data.Product (Σ; ×; _,_)

FiveTriple : Set
FiveTriple = Σ (ℕ × ℕ × ℕ) (λ{ (x , y , z) → x + y + z ≡ 5 })

someFiveTriple : FiveTriple
someFiveTriple = (0 , 2 , 5) , refl