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{-# language KindSignatures #-}
{-# language PolyKinds #-}
{-# language DataKinds #-}
{-# language TypeFamilies #-}
{-# language RankNTypes #-}
{-# language NoImplicitPrelude #-}
{-# language FlexibleContexts #-}
{-# language MultiParamTypeClasses #-}
{-# language GADTs #-}
{-# language ConstraintKinds #-}
{-# language FlexibleInstances #-}
{-# language TypeOperators #-}
{-# language ScopedTypeVariables #-}
{-# language DefaultSignatures #-}
{-# language FunctionalDependencies #-}
{-# language UndecidableSuperClasses #-}
{-# language UndecidableInstances #-}
{-# language TypeInType #-}
module T11523 where
import GHC.Types (Constraint, Type)
import qualified Prelude
type Cat i = i -> i -> Type
newtype Y (p :: i -> j -> Type) (a :: j) (b :: i) = Y { getY :: p b a }
class Vacuous (a :: i)
instance Vacuous a
class (Functor p, Dom p ~ Op p, Cod p ~ Nat p (->)) => Category (p :: Cat i) where
type Op p :: Cat i
type Op p = Y p
type Ob p :: i -> Constraint
type Ob p = Vacuous
class (Category (Dom f), Category (Cod f)) => Functor (f :: i -> j) where
type Dom f :: Cat i
type Cod f :: Cat j
class (Functor f, Dom f ~ p, Cod f ~ q) => Fun (p :: Cat i) (q :: Cat j) (f :: i -> j) | f -> p q
instance (Functor f, Dom f ~ p, Cod f ~ q) => Fun (p :: Cat i) (q :: Cat j) (f :: i -> j)
data Nat (p :: Cat i) (q :: Cat j) (f :: i -> j) (g :: i -> j)
instance (Category p, Category q) => Category (Nat p q) where
type Ob (Nat p q) = Fun p q
instance (Category p, Category q) => Functor (Nat p q) where
type Dom (Nat p q) = Y (Nat p q)
type Cod (Nat p q) = Nat (Nat p q) (->)
instance (Category p, Category q) => Functor (Nat p q f) where
type Dom (Nat p q f) = Nat p q
type Cod (Nat p q f) = (->)
instance Category (->)
instance Functor ((->) e) where
type Dom ((->) e) = (->)
type Cod ((->) e) = (->)
instance Functor (->) where
type Dom (->) = Y (->)
type Cod (->) = Nat (->) (->)
instance (Category p, Op p ~ Y p) => Category (Y p) where
type Op (Y p) = p
instance (Category p, Op p ~ Y p) => Functor (Y p a) where
type Dom (Y p a) = Y p
type Cod (Y p a) = (->)
instance (Category p, Op p ~ Y p) => Functor (Y p) where
type Dom (Y p) = p
type Cod (Y p) = Nat (Y p) (->)
{-
Given: Category p, Op p ~ Y p
--> Category p, Op p ~ Y p
Functor p, Dom p ~ Op p, Cod p ~ Nat p (->)
--> Category p, Op p ~ Y p
Functor p, Dom p ~ Op p, Cod p ~ Nat p (->)
Category (Dom p), Category (Cod p)
-}
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