{-# LANGUAGE AllowAmbiguousTypes #-} module Proarrow.Category.Instance.Simplex where import Data.Kind (Type) import Prelude (type (~)) import Proarrow.Category.Monoidal (Monoidal (..), MonoidalProfunctor (..)) import Proarrow.Core (CAT, CategoryOf (..), Profunctor (..), Promonad (..), dimapDefault, obj, src, type (+->)) import Proarrow.Monoid (Monoid (..)) import Proarrow.Object.Initial (HasInitialObject (..)) import Proarrow.Object.Terminal (HasTerminalObject (..)) import Proarrow.Profunctor.Representable (Representable (..), dimapRep) type data Nat = Z | S Nat data SNat :: Nat -> Type where SZ :: SNat Z SS :: (IsNat n) => SNat (S n) class ((a + b) + c ~ a + (b + c)) => Assoc a b c instance ((a + b) + c ~ a + (b + c)) => Assoc a b c class (a + Z ~ a, forall b c. Assoc a b c) => IsNat (a :: Nat) where singNat :: SNat a instance IsNat Z where singNat :: SNat Z singNat = SNat Z SZ instance (IsNat a) => IsNat (S a) where singNat :: SNat (S a) singNat = SNat (S a) forall (a :: Nat). IsNat a => SNat (S a) SS type Simplex :: CAT Nat data Simplex a b where ZZ :: Simplex Z Z Y :: Simplex x y -> Simplex x (S y) X :: Simplex x (S y) -> Simplex (S x) (S y) suc :: Simplex a b -> Simplex (S a) (S b) suc :: forall (a :: Nat) (b :: Nat). Simplex a b -> Simplex (S a) (S b) suc = Simplex a (S b) -> Simplex (S a) (S b) forall (n :: Nat) (y :: Nat). Simplex n (S y) -> Simplex (S n) (S y) X (Simplex a (S b) -> Simplex (S a) (S b)) -> (Simplex a b -> Simplex a (S b)) -> Simplex a b -> Simplex (S a) (S b) forall b c a. (b -> c) -> (a -> b) -> a -> c forall {k} (p :: PRO k k) (b :: k) (c :: k) (a :: k). Promonad p => p b c -> p a b -> p a c . Simplex a b -> Simplex a (S b) forall (x :: Nat) (n :: Nat). Simplex x n -> Simplex x (S n) Y instance CategoryOf Nat where type (~>) = Simplex type Ob a = IsNat a instance Promonad Simplex where id :: forall (a :: Nat). Ob a => Simplex a a id @a = case forall (a :: Nat). IsNat a => SNat a singNat @a of SNat a SZ -> Simplex a a Simplex Z Z ZZ SNat a SS -> Simplex n n -> Simplex (S n) (S n) forall (a :: Nat) (b :: Nat). Simplex a b -> Simplex (S a) (S b) suc Simplex n n forall {k} (p :: PRO k k) (a :: k). (Promonad p, Ob a) => p a a forall (a :: Nat). Ob a => Simplex a a id Simplex b c ZZ . :: forall (b :: Nat) (c :: Nat) (a :: Nat). Simplex b c -> Simplex a b -> Simplex a c . Simplex a b f = Simplex a b Simplex a c f Y Simplex b y f . Simplex a b g = Simplex a y -> Simplex a (S y) forall (x :: Nat) (n :: Nat). Simplex x n -> Simplex x (S n) Y (Simplex b y f Simplex b y -> Simplex a b -> Simplex a y forall {k} (p :: PRO k k) (b :: k) (c :: k) (a :: k). Promonad p => p b c -> p a b -> p a c forall (b :: Nat) (c :: Nat) (a :: Nat). Simplex b c -> Simplex a b -> Simplex a c . Simplex a b g) X Simplex x (S y) f . Y Simplex a y g = Simplex x c Simplex x (S y) f Simplex x c -> Simplex a x -> Simplex a c forall {k} (p :: PRO k k) (b :: k) (c :: k) (a :: k). Promonad p => p b c -> p a b -> p a c forall (b :: Nat) (c :: Nat) (a :: Nat). Simplex b c -> Simplex a b -> Simplex a c . Simplex a x Simplex a y g X Simplex x (S y) f . X Simplex x (S y) g = Simplex x (S y) -> Simplex (S x) (S y) forall (n :: Nat) (y :: Nat). Simplex n (S y) -> Simplex (S n) (S y) X (Simplex x (S y) -> Simplex (S x) (S y) forall (n :: Nat) (y :: Nat). Simplex n (S y) -> Simplex (S n) (S y) X Simplex x (S y) f Simplex (S x) (S y) -> Simplex x (S x) -> Simplex x (S y) forall {k} (p :: PRO k k) (b :: k) (c :: k) (a :: k). Promonad p => p b c -> p a b -> p a c forall (b :: Nat) (c :: Nat) (a :: Nat). Simplex b c -> Simplex a b -> Simplex a c . Simplex x (S x) Simplex x (S y) g) instance Profunctor Simplex where dimap :: forall (c :: Nat) (a :: Nat) (b :: Nat) (d :: Nat). (c ~> a) -> (b ~> d) -> Simplex a b -> Simplex c d dimap = (c ~> a) -> (b ~> d) -> Simplex a b -> Simplex c d Simplex c a -> Simplex b d -> Simplex a b -> Simplex c d forall {k} (p :: PRO k k) (c :: k) (a :: k) (b :: k) (d :: k). Promonad p => p c a -> p b d -> p a b -> p c d dimapDefault (Ob a, Ob b) => r r \\ :: forall (a :: Nat) (b :: Nat) r. ((Ob a, Ob b) => r) -> Simplex a b -> r \\ Simplex a b ZZ = r (Ob a, Ob b) => r r (Ob a, Ob b) => r r \\ Y Simplex a y f = r (Ob a, Ob b) => r (Ob a, Ob y) => r r ((Ob a, Ob y) => r) -> Simplex a y -> r forall {j} {k} (p :: PRO j k) (a :: j) (b :: k) r. Profunctor p => ((Ob a, Ob b) => r) -> p a b -> r forall (a :: Nat) (b :: Nat) r. ((Ob a, Ob b) => r) -> Simplex a b -> r \\ Simplex a y f (Ob a, Ob b) => r r \\ X Simplex x (S y) f = r (Ob a, Ob b) => r (Ob x, Ob (S y)) => r r ((Ob x, Ob (S y)) => r) -> Simplex x (S y) -> r forall {j} {k} (p :: PRO j k) (a :: j) (b :: k) r. Profunctor p => ((Ob a, Ob b) => r) -> p a b -> r forall (a :: Nat) (b :: Nat) r. ((Ob a, Ob b) => r) -> Simplex a b -> r \\ Simplex x (S y) f instance HasInitialObject Nat where type InitialObject = Z initiate :: forall (a :: Nat). Ob a => InitialObject ~> a initiate @a = case forall (a :: Nat). IsNat a => SNat a singNat @a of SNat a SZ -> InitialObject ~> a Simplex Z Z ZZ SS @a' -> Simplex Z n -> Simplex Z (S n) forall (x :: Nat) (n :: Nat). Simplex x n -> Simplex x (S n) Y (forall k (a :: k). (HasInitialObject k, Ob a) => InitialObject ~> a initiate @_ @a') instance HasTerminalObject Nat where type TerminalObject = S Z terminate :: forall (a :: Nat). Ob a => a ~> TerminalObject terminate @a = case forall (a :: Nat). IsNat a => SNat a singNat @a of SNat a SZ -> Simplex Z Z -> Simplex Z (S Z) forall (x :: Nat) (n :: Nat). Simplex x n -> Simplex x (S n) Y Simplex Z Z ZZ SS @n -> Simplex n (S Z) -> Simplex (S n) (S Z) forall (n :: Nat) (y :: Nat). Simplex n (S y) -> Simplex (S n) (S y) X (forall k (a :: k). (HasTerminalObject k, Ob a) => a ~> TerminalObject terminate @_ @n) data Fin :: Nat -> Type where Fz :: Fin (S n) Fs :: Fin n -> Fin (S n) type Forget :: Nat +-> Type data Forget a b where Forget :: (Ob b) => {forall (b :: Nat) a. Forget a b -> a -> Fin b unForget :: a -> Fin b} -> Forget a b instance Profunctor Forget where dimap :: forall c a (b :: Nat) (d :: Nat). (c ~> a) -> (b ~> d) -> Forget a b -> Forget c d dimap = (c ~> a) -> (b ~> d) -> Forget a b -> Forget c d forall {j} {k} (p :: j +-> k) (a :: k) (b :: j) (c :: k) (d :: j). Representable p => (c ~> a) -> (b ~> d) -> p a b -> p c d dimapRep (Ob a, Ob b) => r r \\ :: forall a (b :: Nat) r. ((Ob a, Ob b) => r) -> Forget a b -> r \\ Forget a -> Fin b f = r (Ob a, Ob (Fin b)) => r (Ob a, Ob b) => r r ((Ob a, Ob (Fin b)) => r) -> (a -> Fin b) -> r forall a b r. ((Ob a, Ob b) => r) -> (a -> b) -> r forall {j} {k} (p :: PRO j k) (a :: j) (b :: k) r. Profunctor p => ((Ob a, Ob b) => r) -> p a b -> r \\ a -> Fin b f instance Representable Forget where type Forget % n = Fin n index :: forall a (b :: Nat). Forget a b -> a ~> (Forget % b) index = Forget a b -> a ~> (Forget % b) Forget a b -> a -> Fin b forall (b :: Nat) a. Forget a b -> a -> Fin b unForget tabulate :: forall (b :: Nat) a. Ob b => (a ~> (Forget % b)) -> Forget a b tabulate = (a ~> (Forget % b)) -> Forget a b (a -> Fin b) -> Forget a b forall (b :: Nat) a. Ob b => (a -> Fin b) -> Forget a b Forget repMap :: forall (a :: Nat) (b :: Nat). (a ~> b) -> (Forget % a) ~> (Forget % b) repMap a ~> b Simplex a b ZZ = (Forget % a) ~> (Forget % b) Fin Z -> Fin Z forall a. Ob a => a -> a forall {k} (p :: PRO k k) (a :: k). (Promonad p, Ob a) => p a a id repMap (Y Simplex a y f) = Fin y -> Fin (S y) forall (n :: Nat). Fin n -> Fin (S n) Fs (Fin y -> Fin (S y)) -> (Fin a -> Fin y) -> Fin a -> Fin (S y) forall b c a. (b -> c) -> (a -> b) -> a -> c forall {k} (p :: PRO k k) (b :: k) (c :: k) (a :: k). Promonad p => p b c -> p a b -> p a c . forall {j} {k} (p :: j +-> k) (a :: j) (b :: j). Representable p => (a ~> b) -> (p % a) ~> (p % b) forall (p :: Nat +-> Type) (a :: Nat) (b :: Nat). Representable p => (a ~> b) -> (p % a) ~> (p % b) repMap @Forget a ~> y Simplex a y f repMap (X Simplex x (S y) f) = \case Fin (S x) Fz -> Fin (S y) forall (n :: Nat). Fin (S n) Fz Fs Fin n n -> forall {j} {k} (p :: j +-> k) (a :: j) (b :: j). Representable p => (a ~> b) -> (p % a) ~> (p % b) forall (p :: Nat +-> Type) (a :: Nat) (b :: Nat). Representable p => (a ~> b) -> (p % a) ~> (p % b) repMap @Forget x ~> S y Simplex x (S y) f Fin n n type family (a :: Nat) + (b :: Nat) :: Nat where Z + b = b S a + b = S (a + b) instance MonoidalProfunctor Simplex where par0 :: Simplex Unit Unit par0 = Simplex Unit Unit Simplex Z Z ZZ Simplex x1 x2 ZZ par :: forall (x1 :: Nat) (x2 :: Nat) (y1 :: Nat) (y2 :: Nat). Simplex x1 x2 -> Simplex y1 y2 -> Simplex (x1 ** y1) (x2 ** y2) `par` Simplex y1 y2 g = Simplex y1 y2 Simplex (x1 ** y1) (x2 ** y2) g Y Simplex x1 y f `par` Simplex y1 y2 g = Simplex (x1 + y1) (y + y2) -> Simplex (x1 + y1) (S (y + y2)) forall (x :: Nat) (n :: Nat). Simplex x n -> Simplex x (S n) Y (Simplex x1 y f Simplex x1 y -> Simplex y1 y2 -> Simplex (x1 ** y1) (y ** y2) forall {j} {k} (p :: j +-> k) (x1 :: k) (x2 :: j) (y1 :: k) (y2 :: j). MonoidalProfunctor p => p x1 x2 -> p y1 y2 -> p (x1 ** y1) (x2 ** y2) forall (x1 :: Nat) (x2 :: Nat) (y1 :: Nat) (y2 :: Nat). Simplex x1 x2 -> Simplex y1 y2 -> Simplex (x1 ** y1) (x2 ** y2) `par` Simplex y1 y2 g) X Simplex x (S y) f `par` Simplex y1 y2 g = Simplex (x + y1) (S (y + y2)) -> Simplex (S (x + y1)) (S (y + y2)) forall (n :: Nat) (y :: Nat). Simplex n (S y) -> Simplex (S n) (S y) X (Simplex x (S y) f Simplex x (S y) -> Simplex y1 y2 -> Simplex (x ** y1) (S y ** y2) forall {j} {k} (p :: j +-> k) (x1 :: k) (x2 :: j) (y1 :: k) (y2 :: j). MonoidalProfunctor p => p x1 x2 -> p y1 y2 -> p (x1 ** y1) (x2 ** y2) forall (x1 :: Nat) (x2 :: Nat) (y1 :: Nat) (y2 :: Nat). Simplex x1 x2 -> Simplex y1 y2 -> Simplex (x1 ** y1) (x2 ** y2) `par` Simplex y1 y2 g) instance Monoidal Nat where type Unit = Z type a ** b = a + b withOb2 :: forall (a :: Nat) (b :: Nat) r. (Ob a, Ob b) => (Ob (a ** b) => r) -> r withOb2 @a @b Ob (a ** b) => r r = case forall (a :: Nat). IsNat a => SNat a singNat @a of SNat a SZ -> r Ob (a ** b) => r r SS @a' -> forall k (a :: k) (b :: k) r. (Monoidal k, Ob a, Ob b) => (Ob (a ** b) => r) -> r withOb2 @_ @a' @b r Ob (a ** b) => r Ob (n ** b) => r r leftUnitor :: forall (a :: Nat). Ob a => (Unit ** a) ~> a leftUnitor = (Unit ** a) ~> a Simplex a a forall {k} (p :: PRO k k) (a :: k). (Promonad p, Ob a) => p a a forall (a :: Nat). Ob a => Simplex a a id leftUnitorInv :: forall (a :: Nat). Ob a => a ~> (Unit ** a) leftUnitorInv = a ~> (Unit ** a) Simplex a a forall {k} (p :: PRO k k) (a :: k). (Promonad p, Ob a) => p a a forall (a :: Nat). Ob a => Simplex a a id rightUnitor :: forall (a :: Nat). Ob a => (a ** Unit) ~> a rightUnitor = (a ** Unit) ~> a Simplex a a forall {k} (p :: PRO k k) (a :: k). (Promonad p, Ob a) => p a a forall (a :: Nat). Ob a => Simplex a a id rightUnitorInv :: forall (a :: Nat). Ob a => a ~> (a ** Unit) rightUnitorInv = a ~> (a ** Unit) Simplex a a forall {k} (p :: PRO k k) (a :: k). (Promonad p, Ob a) => p a a forall (a :: Nat). Ob a => Simplex a a id associator :: forall (a :: Nat) (b :: Nat) (c :: Nat). (Ob a, Ob b, Ob c) => ((a ** b) ** c) ~> (a ** (b ** c)) associator @a @b @c = forall k (a :: k) (b :: k) r. (Monoidal k, Ob a, Ob b) => (Ob (a ** b) => r) -> r withOb2 @_ @a @b (forall k (a :: k) (b :: k) r. (Monoidal k, Ob a, Ob b) => (Ob (a ** b) => r) -> r withOb2 @_ @(a ** b) @c (forall {k} (p :: PRO k k) (a :: k). (Promonad p, Ob a) => p a a forall (p :: PRO Nat Nat) (a :: Nat). (Promonad p, Ob a) => p a a id @Simplex)) associatorInv :: forall (a :: Nat) (b :: Nat) (c :: Nat). (Ob a, Ob b, Ob c) => (a ** (b ** c)) ~> ((a ** b) ** c) associatorInv @a @b @c = forall k (a :: k) (b :: k) r. (Monoidal k, Ob a, Ob b) => (Ob (a ** b) => r) -> r withOb2 @_ @b @c (forall k (a :: k) (b :: k) r. (Monoidal k, Ob a, Ob b) => (Ob (a ** b) => r) -> r withOb2 @_ @a @(b ** c) (forall {k} (p :: PRO k k) (a :: k). (Promonad p, Ob a) => p a a forall (p :: PRO Nat Nat) (a :: Nat). (Promonad p, Ob a) => p a a id @Simplex)) instance Monoid (S Z) where mempty :: Unit ~> S Z mempty = Simplex Z Z -> Simplex Z (S Z) forall (x :: Nat) (n :: Nat). Simplex x n -> Simplex x (S n) Y Simplex Z Z ZZ mappend :: (S Z ** S Z) ~> S Z mappend = Simplex (S Z) (S Z) -> Simplex (S (S Z)) (S Z) forall (n :: Nat) (y :: Nat). Simplex n (S y) -> Simplex (S n) (S y) X (Simplex Z (S Z) -> Simplex (S Z) (S Z) forall (n :: Nat) (y :: Nat). Simplex n (S y) -> Simplex (S n) (S y) X (Simplex Z Z -> Simplex Z (S Z) forall (x :: Nat) (n :: Nat). Simplex x n -> Simplex x (S n) Y Simplex Z Z ZZ)) type Replicate :: k -> Nat +-> k data Replicate m a b where Replicate :: (Ob b) => a ~> (Replicate m % b) -> Replicate m a b instance (Monoid m) => Profunctor (Replicate m) where dimap :: forall (c :: j) (a :: j) (b :: Nat) (d :: Nat). (c ~> a) -> (b ~> d) -> Replicate m a b -> Replicate m c d dimap = (c ~> a) -> (b ~> d) -> Replicate m a b -> Replicate m c d forall {j} {k} (p :: j +-> k) (a :: k) (b :: j) (c :: k) (d :: j). Representable p => (c ~> a) -> (b ~> d) -> p a b -> p c d dimapRep (Ob a, Ob b) => r r \\ :: forall (a :: j) (b :: Nat) r. ((Ob a, Ob b) => r) -> Replicate m a b -> r \\ Replicate a ~> (Replicate m % b) f = r (Ob a, Ob (Replicate m % b)) => r (Ob a, Ob b) => r r ((Ob a, Ob (Replicate m % b)) => r) -> (a ~> (Replicate m % b)) -> r forall (a :: j) (b :: j) r. ((Ob a, Ob b) => r) -> (a ~> b) -> r forall {j} {k} (p :: PRO j k) (a :: j) (b :: k) r. Profunctor p => ((Ob a, Ob b) => r) -> p a b -> r \\ a ~> (Replicate m % b) f instance (Monoid m) => Representable (Replicate m) where type Replicate m % Z = Unit type Replicate m % S b = m ** (Replicate m % b) index :: forall (a :: k) (b :: Nat). Replicate m a b -> a ~> (Replicate m % b) index (Replicate a ~> (Replicate m % b) f) = a ~> (Replicate m % b) f tabulate :: forall (b :: Nat) (a :: k). Ob b => (a ~> (Replicate m % b)) -> Replicate m a b tabulate = (a ~> (Replicate m % b)) -> Replicate m a b forall {k} (b :: Nat) (a :: k) (m :: k). Ob b => (a ~> (Replicate m % b)) -> Replicate m a b Replicate repMap :: forall (a :: Nat) (b :: Nat). (a ~> b) -> (Replicate m % a) ~> (Replicate m % b) repMap a ~> b Simplex a b ZZ = Unit ~> Unit (Replicate m % a) ~> (Replicate m % b) forall {j} {k} (p :: j +-> k). MonoidalProfunctor p => p Unit Unit par0 repMap (Y Simplex a y f) = let g :: (Replicate m % a) ~> (Replicate m % y) g = forall {j} {k} (p :: j +-> k) (a :: j) (b :: j). Representable p => (a ~> b) -> (p % a) ~> (p % b) forall (p :: Nat +-> k) (a :: Nat) (b :: Nat). Representable p => (a ~> b) -> (p % a) ~> (p % b) repMap @(Replicate m) a ~> y Simplex a y f in (forall (m :: k). Monoid m => Unit ~> m forall {k} (m :: k). Monoid m => Unit ~> m mempty @m (Unit ~> m) -> ((Replicate m % a) ~> (Replicate m % y)) -> (Unit ** (Replicate m % a)) ~> (m ** (Replicate m % y)) forall (x1 :: k) (x2 :: k) (y1 :: k) (y2 :: k). (x1 ~> x2) -> (y1 ~> y2) -> (x1 ** y1) ~> (x2 ** y2) forall {j} {k} (p :: j +-> k) (x1 :: k) (x2 :: j) (y1 :: k) (y2 :: j). MonoidalProfunctor p => p x1 x2 -> p y1 y2 -> p (x1 ** y1) (x2 ** y2) `par` (Replicate m % a) ~> (Replicate m % y) g) ((Unit ** (Replicate m % a)) ~> (m ** (Replicate m % y))) -> ((Replicate m % a) ~> (Unit ** (Replicate m % a))) -> (Replicate m % a) ~> (m ** (Replicate m % y)) forall (b :: k) (c :: k) (a :: k). (b ~> c) -> (a ~> b) -> a ~> c forall {k} (p :: PRO k k) (b :: k) (c :: k) (a :: k). Promonad p => p b c -> p a b -> p a c . (Replicate m % a) ~> (Unit ** (Replicate m % a)) forall (a :: k). Ob a => a ~> (Unit ** a) forall k (a :: k). (Monoidal k, Ob a) => a ~> (Unit ** a) leftUnitorInv ((Ob (Replicate m % a), Ob (Replicate m % y)) => (Replicate m % a) ~> (m ** (Replicate m % y))) -> ((Replicate m % a) ~> (Replicate m % y)) -> (Replicate m % a) ~> (m ** (Replicate m % y)) forall (a :: k) (b :: k) r. ((Ob a, Ob b) => r) -> (a ~> b) -> r forall {j} {k} (p :: PRO j k) (a :: j) (b :: k) r. Profunctor p => ((Ob a, Ob b) => r) -> p a b -> r \\ (Replicate m % a) ~> (Replicate m % y) g repMap (X (Y Simplex x y f)) = forall (a :: k). (CategoryOf k, Ob a) => Obj a forall {k} (a :: k). (CategoryOf k, Ob a) => Obj a obj @m Obj m -> ((Replicate m % x) ~> (Replicate m % y)) -> (m ** (Replicate m % x)) ~> (m ** (Replicate m % y)) forall (x1 :: k) (x2 :: k) (y1 :: k) (y2 :: k). (x1 ~> x2) -> (y1 ~> y2) -> (x1 ** y1) ~> (x2 ** y2) forall {j} {k} (p :: j +-> k) (x1 :: k) (x2 :: j) (y1 :: k) (y2 :: j). MonoidalProfunctor p => p x1 x2 -> p y1 y2 -> p (x1 ** y1) (x2 ** y2) `par` forall {j} {k} (p :: j +-> k) (a :: j) (b :: j). Representable p => (a ~> b) -> (p % a) ~> (p % b) forall (p :: Nat +-> k) (a :: Nat) (b :: Nat). Representable p => (a ~> b) -> (p % a) ~> (p % b) repMap @(Replicate m) x ~> y Simplex x y f repMap (X (X @x Simplex x (S y) f)) = let g :: (Replicate m % S x) ~> (Replicate m % S y) g = forall {j} {k} (p :: j +-> k) (a :: j) (b :: j). Representable p => (a ~> b) -> (p % a) ~> (p % b) forall (p :: Nat +-> k) (a :: Nat) (b :: Nat). Representable p => (a ~> b) -> (p % a) ~> (p % b) repMap @(Replicate m) (Simplex x (S y) -> Simplex (S x) (S y) forall (n :: Nat) (y :: Nat). Simplex n (S y) -> Simplex (S n) (S y) X Simplex x (S y) f) b :: (Replicate m % x) ~> (Replicate m % x) b = forall {j} {k} (p :: j +-> k) (a :: j) (b :: j). Representable p => (a ~> b) -> (p % a) ~> (p % b) forall (p :: Nat +-> k) (a :: Nat) (b :: Nat). Representable p => (a ~> b) -> (p % a) ~> (p % b) repMap @(Replicate m) (Simplex x (S y) -> x ~> x forall {k1} {k2} (a :: k2) (b :: k1) (p :: PRO k2 k1). Profunctor p => p a b -> Obj a src Simplex x (S y) f) in (m ** (Replicate m % x)) ~> (m ** (Replicate m % y)) (Replicate m % S x) ~> (Replicate m % S y) g ((m ** (Replicate m % x)) ~> (m ** (Replicate m % y))) -> ((m ** (m ** (Replicate m % x))) ~> (m ** (Replicate m % x))) -> (m ** (m ** (Replicate m % x))) ~> (m ** (Replicate m % y)) forall (b :: k) (c :: k) (a :: k). (b ~> c) -> (a ~> b) -> a ~> c forall {k} (p :: PRO k k) (b :: k) (c :: k) (a :: k). Promonad p => p b c -> p a b -> p a c . (forall (m :: k). Monoid m => (m ** m) ~> m forall {k} (m :: k). Monoid m => (m ** m) ~> m mappend @m ((m ** m) ~> m) -> ((Replicate m % x) ~> (Replicate m % x)) -> ((m ** m) ** (Replicate m % x)) ~> (m ** (Replicate m % x)) forall (x1 :: k) (x2 :: k) (y1 :: k) (y2 :: k). (x1 ~> x2) -> (y1 ~> y2) -> (x1 ** y1) ~> (x2 ** y2) forall {j} {k} (p :: j +-> k) (x1 :: k) (x2 :: j) (y1 :: k) (y2 :: j). MonoidalProfunctor p => p x1 x2 -> p y1 y2 -> p (x1 ** y1) (x2 ** y2) `par` (Replicate m % x) ~> (Replicate m % x) b) (((m ** m) ** (Replicate m % x)) ~> (m ** (Replicate m % x))) -> ((m ** (m ** (Replicate m % x))) ~> ((m ** m) ** (Replicate m % x))) -> (m ** (m ** (Replicate m % x))) ~> (m ** (Replicate m % x)) forall (b :: k) (c :: k) (a :: k). (b ~> c) -> (a ~> b) -> a ~> c forall {k} (p :: PRO k k) (b :: k) (c :: k) (a :: k). Promonad p => p b c -> p a b -> p a c . forall k (a :: k) (b :: k) (c :: k). (Monoidal k, Ob a, Ob b, Ob c) => (a ** (b ** c)) ~> ((a ** b) ** c) associatorInv @_ @m @m @(Replicate m % x) ((Ob (Replicate m % x), Ob (Replicate m % x)) => (m ** (m ** (Replicate m % x))) ~> (m ** (Replicate m % y))) -> ((Replicate m % x) ~> (Replicate m % x)) -> (m ** (m ** (Replicate m % x))) ~> (m ** (Replicate m % y)) forall (a :: k) (b :: k) r. ((Ob a, Ob b) => r) -> (a ~> b) -> r forall {j} {k} (p :: PRO j k) (a :: j) (b :: k) r. Profunctor p => ((Ob a, Ob b) => r) -> p a b -> r \\ (Replicate m % x) ~> (Replicate m % x) b