[英]Church encoding for dependent types: from Coq to Haskell
在Coq中,我可以為長度為n的列表定義一個Church編碼:
Definition listn (A : Type) : nat -> Type :=
fun m => forall (X : nat -> Type), X 0 -> (forall m, A -> X m -> X (S m)) -> X m.
Definition niln (A : Type) : listn A 0 :=
fun X n c => n.
Definition consn (A : Type) (m : nat) (a : A) (l : listn A m) : listn A (S m) :=
fun X n c => c m a (l X n c).
Haskell的類型系統(包括其擴展)是否足以容納這些定義? 如果有,怎么樣?
當然是啦:
{-# LANGUAGE RankNTypes #-}
{-# LANGUAGE KindSignatures #-}
{-# LANGUAGE DataKinds #-}
{-# LANGUAGE GADTs #-}
import Data.Kind -- Needed for `Type`
data Nat = Z | S Nat -- Roll your own...
type List (a :: Type) (n :: Nat) =
forall (x :: Nat -> Type). x Z -> (forall (m :: Nat). a -> x m -> x (S m)) -> x n
niln :: List a Z
niln = \z _ -> z
consn :: a -> List a n -> List a (S n)
consn a l = \n c -> c a (l n c)
使用通常的GADT公式進一步證明(對懷疑論者)同構性:
data List' (a :: Type) (n :: Nat) where
Nil :: List' a Z
Cons :: a -> List' a m -> List' a (S m)
to :: List' a n -> List a n
to Nil = niln
to (Cons a l) = consn a (to l)
from :: List a n -> List' a n
from l = l Nil Cons
聲明:本站的技術帖子網頁,遵循CC BY-SA 4.0協議,如果您需要轉載,請注明本站網址或者原文地址。任何問題請咨詢:yoyou2525@163.com.