如何使用范围内的约束系列来证明表达式主体内的实例？

Question

这是的跟进。我在那里收到了一些很好的答案，但是，由于我简化实际问题的方式，我认为我误导了回答者，我希望在这里纠正这个问题。

TL;DR 我有一个来自 constrained-categories 的类型类 Category 它限制了 domain/codomain 的 Category 使用名为 Object 的 TypeFamily（“约束系列”）。我想做一个免费的 Category，但我正在努力获得表达式满足 Object 约束的证据。

考虑我的 Free 数据类型及其 constrained-categories.Category 实例：

data Free p a b where
  Id :: Object p a => Free p a a
  Comp :: Object p b => Free p b c -> Free p a b -> Free p a c

instance Category (Free p) where
  type Object (Free p) a = Object p a -- passes through the constraints from instantiated `p`
  id = Id
  (.) = Comp

为了解释得更详细一些，让我们也考虑一个非免费的 Category 我想 eval 至：

newtype MyArr a b = MyArr (a -> b)

instance Category MyArr where
  type Object MyArr a = () -- trivial, no constraints needed
  id = MyArr P.id
  MyArr g . MyArr f = MyArr (g . f)

我想写一个这样的表达式（这显然比我将在实践中写的表达式简单得多）：

-- error: Could not deduce: Object p Int arising from a use of ‘Id’
e0 :: Free p Int Int
e0 = Id

我可以用显而易见的方式解决这个问题，但对于较大的表达式来说，这会变得冗长。想象一下组合元组的函数，并且需要为组合中的每个中间步骤显式提供一个 Object p _ 实例：

e1 :: Object p Int => Free p Int Int
e1 = Id

我可以选择不抽象 Category p，这很有效：

e2 :: Free MyArr Int Int
e2 = Id

但我想将其抽象化。我应该认为添加约束 Category p 应该有效，并将其约束 type Object (Free p) a = Object p a 纳入范围，并给我任何我需要的 Object p _ 实例，但是，它不起作用。

-- error: Could not deduce: Object p Int arising from a use of ‘Id’
e3 :: Category p => Free p Int Int
e3 = Id

在我看来 QuantifiedConstraints 可能会有所帮助，例如 forall a. Object (Free p) a => Object p a，但谓词的头部不能有 TypeFamily。

我也考虑过使用 type Object p :: * -> Constraint，就像 Conal Elliot 在那个库中的 concat library, but, that would require a different library, a different Category class, and, last time I used categories constrained that way, it seemed a bit more cumbersome and I'm not even sure if that would solve my boilerplate issue. A comment 表明 QuantifiedConstraints 的用处，但是，我现在很困惑不知往哪个方向艰难前行

可运行

{-# LANGUAGE GADTs #-}
{-# LANGUAGE ConstraintKinds #-}
{-# LANGUAGE TypeFamilies #-}

module ConstrainedCategoryFreeArrow3 where

import Prelude hiding (id, (.))
import qualified Prelude as P
import Control.Category.Constrained


-- * A Free 'Category'

data Free p a b where
  Id :: Object p a => Free p a a
  Comp :: Object p b => Free p b c -> Free p a b -> Free p a c

instance Category (Free p) where
  type Object (Free p) a = Object p a -- passes through the constraints from instantiated `p`
  id = Id
  (.) = Comp

eval :: (Category p, Object p a, Object p b) => Free p a b -> p a b
eval Id = id
eval (Comp g f) = eval g . eval f


-- * A specific (trivial) 'Category'

newtype MyArr a b = MyArr (a -> b)

instance Category MyArr where
  type Object MyArr a = () -- trivial, no constraints needed
  id = MyArr P.id
  MyArr g . MyArr f = MyArr (g . f)


-- * A generic expression

-- -- error: Could not deduce: Object p Int arising from a use of ‘Id’
-- e0 :: Free p Int Int
-- e0 = Id


-- works, but becomes verbose. Imagine for instance building an expression with
-- functions of tuples; you would need to provide an incredible amount of
-- boilerplate constraints for each permutation of types in tuples used anywhere
-- in the body. The problem is a little worse than this once you account for
-- `Cartesian` which has a Constraint Family called `PairObjects` where you need
-- to prove that each tuple is an allowed product in the current constrained
-- category.
e1 :: Object p Int => Free p Int Int
e1 = Id


-- works, but is no longer abstract in the category `p`
e2 :: Free MyArr Int Int
e2 = Id


-- -- ideal solution, but alas
-- -- error: Could not deduce: Object p Int arising from a use of ‘Id’
-- e3 :: Category p => Free p Int Int
-- e3 = Id

Answer 1

{-# LANGUAGE GADTs, RankNTypes, TypeApplications, AllowAmbiguousTypes #-}

恐怕让对象约束都与p类别“同步”，但同时在该类别中又是抽象的，难免会很麻烦。

我建议您使用自己的对象层次结构来避免这种情况，它比 p 层次结构更强大。然后，您需要一个 class 来将 translation/weakening 从您自己的层次结构实施到目标类别的层次结构中。

最简单的情况是 p 确实没有约束，即 ∀ a. Object (Free p) a 也可以表达什么。这将是 class

class UnconstrainedCat p where
  proveUnconstrained :: ∀ a r . (Object p a => r) -> r

...但这很不现实；如果是这种情况，那么您一开始就不需要使用 constrained-categories。

假设您在表达式中需要的对象是 Int、() 以及它们的嵌套元组。（可以很容易地扩展到其他类型，或原子类型列表。）然后我们可以在值级别跟踪我们实际处理的是什么类型：

data IntTupleFlavour t where
  UnitCase :: IntTupleFlavour ()
  IntCase :: IntTupleFlavour Int
  TupleCase :: IntTupleFlavour a -> IntTupleFlavour b -> IntTupleFlavour (a,b)

class IntTuple t where
  intTupleFlavour :: IntTupleFlavour t
instance IntTuple () where
  intTupleFlavour = UnitCase
instance IntTuple Int where
  intTupleFlavour = IntCase
instance (IntTuple a, IntTuple b) => IntTuple (a,b) where
  intTupleFlavour = TupleCase intTupleFlavour intTupleFlavour

data IntFree a b where
  IntId :: IntTupleFlavour a -> IntFree a a
  IntCompo :: IntTupleFlavour b -> IntFree b c -> IntFree a b -> IntFree a c

instance Category IntFree where
  type Object IntFree a = IntTuple a
  id = IntId intTupleFlavour
  (.) = IntCompo intTupleFlavour

现在很容易编写这个自由类别的表达式 – 编译器知道这里的对象是什么，p甚至还没有提到。

e4 :: IntFree Int Int
e4 = id

p 只有在您最终翻译到该类别时才会出现。

class IntMonoiCat p where
  proveIntTupleInstance :: IntTupleFlavour t -> (Object p t => r) -> r

instance IntMonoiCat MyArr where
  proveIntTupleInstance _ q = q
    -- trivial, because `MyArr` doesn't even have a constraint! In general,
    -- you would need to pattern-match on the `IntTupleFlavour` here.

然后

instance EnhancedCat MyArr IntFree where
  arr (IntId itf) = proveIntTupleInstance itf id
  arr (IntCompo f g) = ...
        -- here you still need to extract the `IntTupleFlavours`
        -- for the `a` and `c` types. That requires digging into
        -- the free-category structure with a helper function.

如何使用范围内的约束系列来证明表达式主体内的实例？

How can I use a Constraint Family that's in scope to prove instances within the body of an expression?

haskell

typeclass

type-families

gadt

quantified-constraints