{-# OPTIONS_GHC -Wno-orphans #-}

-- |
-- Module      : Data.HBifunctor.Tensor
-- Copyright   : (c) Justin Le 2019
-- License     : BSD3
--
-- Maintainer  : justin@jle.im
-- Stability   : experimental
-- Portability : non-portable
--
-- This module provides tools for working with binary functor combinators.
--
-- "Data.Functor.HFunctor" deals with /single/ functor combinators
-- (transforming a single functor).  This module provides tools for working
-- with combinators that combine and mix two functors "together".
--
-- The binary analog of 'HFunctor' is 'HBifunctor': we can map
-- a structure-transforming function over both of the transformed functors.
--
-- 'Tensor' gives some extra properties of your binary functor combinator:
-- associativity and identity (see docs for 'Tensor' for more details).
--
-- The binary analog of 'Interpret' is 'MonoidIn'.  If your combinator @t@
-- and target functor @f@ is an instance of @'MonoidIn' t f@, it means you
-- can "interpret" out of your tensored values, and also "generate" values
-- of @f@.
--
-- @
-- 'biretract' :: (f ':+:' f) a -> f a
-- 'pureT'     :: 'V1' a -> f a
--
-- biretract :: 'Plus' f => (f ':*:' f) a -> f a
-- pureT     :: Plus f => 'Proxy' a -> f a
--
-- biretract :: 'Applicative' f => 'Day' f f a -> f a
-- pureT     :: Applicative f => 'Identity' a -> f a
--
-- biretract :: 'Monad' f => 'Comp' f f a -> f a
-- pureT     :: Monad f => 'Identity' a -> f a
-- @
--
module Data.HBifunctor.Tensor (
  -- * 'Tensor'
    Tensor(..)
  , rightIdentity
  , leftIdentity
  , sumLeftIdentity
  , sumRightIdentity
  , prodLeftIdentity
  , prodRightIdentity
  -- * 'MonoidIn'
  , MonoidIn(..)
  , nilLB
  , consLB
  , unconsLB
  , retractLB
  , interpretLB
  -- ** Utility
  , inL
  , inR
  , outL
  , outR
  , prodOutL
  , prodOutR
  , WrapF(..)
  , WrapLB(..)
  -- * 'Matchable'
  , Matchable(..)
  , splittingNE
  , matchingLB
  ) where

import           Control.Applicative.Free
import           Control.Applicative.ListF
import           Control.Applicative.Step
import           Control.Monad.Freer.Church
import           Control.Monad.Trans.Compose
import           Control.Natural
import           Control.Natural.IsoF
import           Data.Bifunctor
import           Data.Coerce
import           Data.Data
import           Data.Function
import           Data.Functor.Apply.Free
import           Data.Functor.Bind
import           Data.Functor.Classes
import           Data.Functor.Contravariant
import           Data.Functor.Contravariant.Conclude
import           Data.Functor.Contravariant.Decide
import           Data.Functor.Contravariant.Divise
import           Data.Functor.Contravariant.Divisible
import           Data.Functor.Contravariant.Divisible.Free
import           Data.Functor.Contravariant.Night          (Night(..), Not(..))
import           Data.Functor.Day                          (Day(..))
import           Data.Functor.Identity
import           Data.Functor.Invariant
import           Data.Functor.Invariant.Internative
import           Data.Functor.Invariant.Inplicative
import           Data.Functor.Plus
import           Data.Functor.Product
import           Data.Functor.Sum
import           Data.Functor.These
import           Data.HBifunctor
import           Data.HBifunctor.Associative
import           Data.HBifunctor.Tensor.Internal
import           Data.HFunctor
import           Data.HFunctor.Chain.Internal
import           Data.HFunctor.Internal
import           Data.HFunctor.Interpret
import           Data.List.NonEmpty                        (NonEmpty(..))
import           Data.Void
import           GHC.Generics
import qualified Data.Bifunctor.Assoc                      as B
import qualified Data.Functor.Contravariant.Coyoneda       as CCY
import qualified Data.Functor.Contravariant.Day            as CD
import qualified Data.Functor.Contravariant.Night          as N
import qualified Data.Functor.Day                          as D
import qualified Data.Functor.Invariant.Day                as ID
import qualified Data.Functor.Invariant.Night              as IN
import qualified Data.Map.NonEmpty                         as NEM

-- | @f@ is isomorphic to @t f i@: that is, @i@ is the identity of @t@, and
-- leaves @f@ unchanged.
rightIdentity :: (Tensor t i, FunctorBy t f) => f <~> t f i
rightIdentity :: forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
(Tensor t i, FunctorBy t f) =>
f <~> t f i
rightIdentity = forall {k} (f :: k -> *) (g :: k -> *).
(f ~> g) -> (g ~> f) -> f <~> g
isoF forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
Tensor t i =>
f ~> t f i
intro1 forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
(Tensor t i, FunctorBy t f) =>
t f i ~> f
elim1

-- | @g@ is isomorphic to @t i g@: that is, @i@ is the identity of @t@, and
-- leaves @g@ unchanged.
leftIdentity  :: (Tensor t i, FunctorBy t g) => g <~> t i g
leftIdentity :: forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (g :: * -> *).
(Tensor t i, FunctorBy t g) =>
g <~> t i g
leftIdentity = forall {k} (f :: k -> *) (g :: k -> *).
(f ~> g) -> (g ~> f) -> f <~> g
isoF forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (g :: * -> *).
Tensor t i =>
g ~> t i g
intro2 forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (g :: * -> *).
(Tensor t i, FunctorBy t g) =>
t i g ~> g
elim2

-- | 'leftIdentity' ('intro1' and 'elim1') for ':+:' actually does not
-- require 'Functor'.  This is the more general version.
sumLeftIdentity :: f <~> V1 :+: f
sumLeftIdentity :: forall {k} (f :: k -> *). f <~> (V1 :+: f)
sumLeftIdentity = forall {k} (f :: k -> *) (g :: k -> *).
(f ~> g) -> (g ~> f) -> f <~> g
isoF forall k (f :: k -> *) (g :: k -> *) (p :: k). g p -> (:+:) f g p
R1 (forall {k} (a :: k) (f :: k -> *). V1 a -> f a
absurd1 forall {k} (f :: k -> *) (h :: k -> *) (g :: k -> *).
(f ~> h) -> (g ~> h) -> (f :+: g) ~> h
!+! forall a. a -> a
id)

-- | 'rightIdentity' ('intro2' and 'elim2') for ':+:' actually does not
-- require 'Functor'.  This is the more general version.
sumRightIdentity :: f <~> f :+: V1
sumRightIdentity :: forall {k} (f :: k -> *). f <~> (f :+: V1)
sumRightIdentity = forall {k} (f :: k -> *) (g :: k -> *).
(f ~> g) -> (g ~> f) -> f <~> g
isoF forall k (f :: k -> *) (g :: k -> *) (p :: k). f p -> (:+:) f g p
L1 (forall a. a -> a
id forall {k} (f :: k -> *) (h :: k -> *) (g :: k -> *).
(f ~> h) -> (g ~> h) -> (f :+: g) ~> h
!+! forall {k} (a :: k) (f :: k -> *). V1 a -> f a
absurd1)

-- | 'leftIdentity' ('intro1' and 'elim1') for ':*:' actually does not
-- require 'Functor'.  This is the more general version.
prodLeftIdentity :: f <~> Proxy :*: f
prodLeftIdentity :: forall {k} (f :: k -> *). f <~> (Proxy :*: f)
prodLeftIdentity = forall {k} (f :: k -> *) (g :: k -> *).
(f ~> g) -> (g ~> f) -> f <~> g
isoF (forall {k} (t :: k). Proxy t
Proxy forall k (f :: k -> *) (g :: k -> *) (p :: k).
f p -> g p -> (:*:) f g p
:*:) (\case Proxy x
_ :*: f x
y -> f x
y)

-- | 'rightIdentity' ('intro2' and 'elim2') for ':*:' actually does not
-- require 'Functor'.  This is the more general version.
prodRightIdentity :: g <~> g :*: Proxy
prodRightIdentity :: forall {k} (g :: k -> *). g <~> (g :*: Proxy)
prodRightIdentity = forall {k} (f :: k -> *) (g :: k -> *).
(f ~> g) -> (g ~> f) -> f <~> g
isoF (forall k (f :: k -> *) (g :: k -> *) (p :: k).
f p -> g p -> (:*:) f g p
:*: forall {k} (t :: k). Proxy t
Proxy) (\case g x
x :*: Proxy x
_ -> g x
x)

-- | A poly-kinded version of 'outL' for ':*:'.
prodOutL :: f :*: g ~> f
prodOutL :: forall {k} (f :: k -> *) (g :: k -> *). (f :*: g) ~> f
prodOutL (f x
x :*: g x
_) = f x
x

-- | A poly-kinded version of 'outR' for ':*:'.
prodOutR :: f :*: g ~> g
prodOutR :: forall {k} (f :: k -> *) (g :: k -> *). (f :*: g) ~> g
prodOutR (f x
_ :*: g x
y) = g x
y

-- | This class effectively gives us a way to generate a value of @f a@
-- based on an @i a@, for @'Tensor' t i@.  Having this ability makes a lot
-- of interesting functions possible when used with 'biretract' from
-- 'SemigroupIn' that weren't possible without it: it gives us a "base
-- case" for recursion in a lot of cases.
--
-- Essentially, we get an @i ~> f@, 'pureT', where we can introduce an @f
-- a@ as long as we have an @i a@.
--
-- Formally, if we have @'Tensor' t i@, we are enriching the category of
-- endofunctors with monoid structure, turning it into a monoidal category.
-- Different choices of @t@ give different monoidal categories.
--
-- A functor @f@ is known as a "monoid in the (monoidal) category
-- of endofunctors on @t@" if we can 'biretract':
--
-- @
-- t f f ~> f
-- @
--
-- and also 'pureT':
--
-- @
-- i ~> f
-- @
--
-- This gives us a few interesting results in category theory, which you
-- can stil reading about if you don't care:
--
-- *  /All/ functors are monoids in the monoidal category
--    on ':+:'
-- *  The class of functors that are monoids in the monoidal
--    category on ':*:' is exactly the functors that are instances of
--    'Plus'.
-- *  The class of functors that are monoids in the monoidal
--    category on 'Day' is exactly the functors that are instances of
--    'Applicative'.
-- *  The class of functors that are monoids in the monoidal
--    category on 'Comp' is exactly the functors that are instances of
--    'Monad'.
--
--    This is the meaning behind the common adage, "monads are just monoids
--    in the category of endofunctors".  It means that if you enrich the
--    category of endofunctors to be monoidal with 'Comp', then the class
--    of functors that are monoids in that monoidal category are exactly
--    what monads are.  However, the adage is a little misleading: there
--    are many other ways to enrich the category of endofunctors to be
--    monoidal, and 'Comp' is just one of them.  Similarly, the class of
--    functors that are monoids in the category of endofunctors enriched by
--    'Day' are 'Applicative'.
--
-- Note that instances of this class are /intended/ to be written with @t@
-- and @i@ to be fixed type constructors, and @f@ to be allowed to vary
-- freely:
--
-- @
-- instance Monad f => MonoidIn Comp Identity f
-- @
--
-- Any other sort of instance and it's easy to run into problems with type
-- inference.  If you want to write an instance that's "polymorphic" on
-- tensor choice, use the 'WrapHBF' and 'WrapF' newtype wrappers over type
-- variables, where the third argument also uses a type constructor:
--
-- @
-- instance MonoidIn (WrapHBF t) (WrapF i) (MyFunctor t i)
-- @
--
-- This will prevent problems with overloaded instances.
class (Tensor t i, SemigroupIn t f) => MonoidIn t i f where

    -- | If we have an @i@, we can generate an @f@ based on how it
    -- interacts with @t@.
    --
    -- Specialized (and simplified), this type is:
    --
    -- @
    -- 'pureT' \@'Day'   :: 'Applicative' f => 'Identity' a -> f a  -- 'pure'
    -- pureT \@'Comp'  :: 'Monad' f => Identity a -> f a        -- 'return'
    -- pureT \@(':*:') :: 'Plus' f => 'Proxy' a -> f a            -- 'zero'
    -- @
    --
    -- Note that because @t@ appears nowhere in the input or output types,
    -- you must always use this with explicit type application syntax (like
    -- @pureT \@Day@)
    --
    -- Along with 'biretract', this function makes @f@ a monoid in the
    -- category of endofunctors with respect to tensor @t@.
    pureT  :: i ~> f

    default pureT :: Interpret (ListBy t) f => i ~> f
    pureT  = forall {k} (t :: (k -> *) -> k -> *) (f :: k -> *).
Interpret t f =>
t f ~> f
retract forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall {k} (f :: k -> *) (g :: k -> *). (f <~> g) -> g ~> f
reviewF (forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
Tensor t i =>
ListBy t f <~> (i :+: t f (ListBy t f))
splittingLB @t) forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall k (f :: k -> *) (g :: k -> *) (p :: k). f p -> (:+:) f g p
L1

-- | An implementation of 'retract' that works for any instance of
-- @'MonoidIn' t i@ for @'ListBy' t@.
--
-- Can be useful as a default implementation if you already have 'MonoidIn'
-- implemented.
retractLB :: forall t i f. MonoidIn t i f => ListBy t f ~> f
retractLB :: forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
MonoidIn t i f =>
ListBy t f ~> f
retractLB = (forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
MonoidIn t i f =>
i ~> f
pureT @t forall (t :: (* -> *) -> (* -> *) -> * -> *) (h :: * -> *)
       (f :: * -> *) (g :: * -> *).
SemigroupIn t h =>
(f ~> h) -> (g ~> h) -> t f g ~> h
!*! forall (t :: (* -> *) -> (* -> *) -> * -> *) (f :: * -> *).
SemigroupIn t f =>
t f f ~> f
biretract @t forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall k (t :: (k -> *) -> (k -> *) -> k -> *) (g :: k -> *)
       (l :: k -> *) (f :: k -> *).
HBifunctor t =>
(g ~> l) -> t f g ~> t f l
hright (forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
MonoidIn t i f =>
ListBy t f ~> f
retractLB @t))
              forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
Tensor t i =>
ListBy t f ~> (i :+: t f (ListBy t f))
unconsLB @t

-- | An implementation of 'interpret' that works for any instance of
-- @'MonoidIn' t i@ for @'ListBy' t@.
--
-- Can be useful as a default implementation if you already have 'MonoidIn'
-- implemented.
interpretLB :: forall t i g f. MonoidIn t i f => (g ~> f) -> ListBy t g ~> f
interpretLB :: forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (g :: * -> *) (f :: * -> *).
MonoidIn t i f =>
(g ~> f) -> ListBy t g ~> f
interpretLB g ~> f
f = forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
MonoidIn t i f =>
ListBy t f ~> f
retractLB @t forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall {k} {k1} (t :: (k -> *) -> k1 -> *) (f :: k -> *)
       (g :: k -> *).
HFunctor t =>
(f ~> g) -> t f ~> t g
hmap g ~> f
f

-- | Convenient wrapper over 'intro1' that lets us introduce an arbitrary
-- functor @g@ to the right of an @f@.
--
-- You can think of this as an 'HBifunctor' analogue of 'inject'.
inL
    :: forall t i f g. MonoidIn t i g
    => f ~> t f g
inL :: forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *) (g :: * -> *).
MonoidIn t i g =>
f ~> t f g
inL = forall k (t :: (k -> *) -> (k -> *) -> k -> *) (g :: k -> *)
       (l :: k -> *) (f :: k -> *).
HBifunctor t =>
(g ~> l) -> t f g ~> t f l
hright (forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
MonoidIn t i f =>
i ~> f
pureT @t) forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
Tensor t i =>
f ~> t f i
intro1

-- | Convenient wrapper over 'intro2' that lets us introduce an arbitrary
-- functor @f@ to the right of a @g@.
--
-- You can think of this as an 'HBifunctor' analogue of 'inject'.
inR
    :: forall t i f g. MonoidIn t i f
    => g ~> t f g
inR :: forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *) (g :: * -> *).
MonoidIn t i f =>
g ~> t f g
inR = forall k (t :: (k -> *) -> (k -> *) -> k -> *) (f :: k -> *)
       (j :: k -> *) (g :: k -> *).
HBifunctor t =>
(f ~> j) -> t f g ~> t j g
hleft (forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
MonoidIn t i f =>
i ~> f
pureT @t) forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (g :: * -> *).
Tensor t i =>
g ~> t i g
intro2

-- | Convenient wrapper over 'elim1' that lets us drop one of the arguments
-- of a 'Tensor' for free, without requiring any extra constraints (like
-- for 'binterpret').
--
-- See 'prodOutL' for a version that does not require @'Functor' f@,
-- specifically for ':*:'.
outL
    :: (Tensor t Proxy, FunctorBy t f)
    => t f g ~> f
outL :: forall (t :: (* -> *) -> (* -> *) -> * -> *) (f :: * -> *)
       (g :: * -> *).
(Tensor t Proxy, FunctorBy t f) =>
t f g ~> f
outL = forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
(Tensor t i, FunctorBy t f) =>
t f i ~> f
elim1 forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall k (t :: (k -> *) -> (k -> *) -> k -> *) (g :: k -> *)
       (l :: k -> *) (f :: k -> *).
HBifunctor t =>
(g ~> l) -> t f g ~> t f l
hright forall {k} (f :: k -> *). f ~> Proxy
absorb

-- | Convenient wrapper over 'elim2' that lets us drop one of the arguments
-- of a 'Tensor' for free, without requiring any constraints (like for
-- 'binterpret').
--
-- See 'prodOutR' for a version that does not require @'Functor' g@,
-- specifically for ':*:'.
outR
    :: (Tensor t Proxy, FunctorBy t g)
    => t f g ~> g
outR :: forall (t :: (* -> *) -> (* -> *) -> * -> *) (g :: * -> *)
       (f :: * -> *).
(Tensor t Proxy, FunctorBy t g) =>
t f g ~> g
outR = forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (g :: * -> *).
(Tensor t i, FunctorBy t g) =>
t i g ~> g
elim2 forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall k (t :: (k -> *) -> (k -> *) -> k -> *) (f :: k -> *)
       (j :: k -> *) (g :: k -> *).
HBifunctor t =>
(f ~> j) -> t f g ~> t j g
hleft forall {k} (f :: k -> *). f ~> Proxy
absorb

-- | For some @t@, we have the ability to "statically analyze" the @'ListBy' t@
-- and pattern match and manipulate the structure without ever
-- interpreting or retracting.  These are 'Matchable'.
class Tensor t i => Matchable t i where
    -- | The inverse of 'splitNE'.  A consing of @f@ to @'ListBy' t f@ is
    -- non-empty, so it can be represented as an @'NonEmptyBy' t f@.
    --
    -- This is analogous to a function @'uncurry' ('Data.List.NonEmpty.:|')
    -- :: (a, [a]) -> 'Data.List.NonEmpty.NonEmpty' a@.
    unsplitNE :: FunctorBy t f => t f (ListBy t f) ~> NonEmptyBy t f

    -- | "Pattern match" on an @'ListBy' t f@: it is either empty, or it is
    -- non-empty (and so can be an @'NonEmptyBy' t f@).
    --
    -- This is analgous to a function @'Data.List.NonEmpty.nonEmpty' :: [a]
    -- -> Maybe ('Data.List.NonEmpty.NonEmpty' a)@.
    --
    -- Note that because @t@ cannot be inferred from the input or output
    -- type, you should use this with /-XTypeApplications/:
    --
    -- @
    -- 'matchLB' \@'Day' :: 'Ap' f a -> ('Identity' :+: 'Ap1' f) a
    -- @
    --
    -- Note that you can recursively "unroll" a 'ListBy' completely into
    -- a 'Data.HFunctor.Chain.Chain' by using
    -- 'Data.HFunctor.Chain.unrollLB'.
    matchLB   :: FunctorBy t f => ListBy t f ~> i :+: NonEmptyBy t f

-- | An @'NonEmptyBy' t f@ is isomorphic to an @f@ consed with an @'ListBy' t f@, like
-- how a @'Data.List.NonEmpty.NonEmpty' a@ is isomorphic to @(a, [a])@.
splittingNE
    :: (Matchable t i, FunctorBy t f)
    => NonEmptyBy t f <~> t f (ListBy t f)
splittingNE :: forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
(Matchable t i, FunctorBy t f) =>
NonEmptyBy t f <~> t f (ListBy t f)
splittingNE = forall {k} (f :: k -> *) (g :: k -> *).
(f ~> g) -> (g ~> f) -> f <~> g
isoF forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
Tensor t i =>
NonEmptyBy t f ~> t f (ListBy t f)
splitNE forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
(Matchable t i, FunctorBy t f) =>
t f (ListBy t f) ~> NonEmptyBy t f
unsplitNE

-- | An @'ListBy' t f@ is isomorphic to either the empty case (@i@) or the
-- non-empty case (@'NonEmptyBy' t f@), like how @[a]@ is isomorphic to @'Maybe'
-- ('Data.List.NonEmpty.NonEmpty' a)@.
matchingLB
    :: forall t i f. (Matchable t i, FunctorBy t f)
    => ListBy t f <~> i :+: NonEmptyBy t f
matchingLB :: forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
(Matchable t i, FunctorBy t f) =>
ListBy t f <~> (i :+: NonEmptyBy t f)
matchingLB = forall {k} (f :: k -> *) (g :: k -> *).
(f ~> g) -> (g ~> f) -> f <~> g
isoF (forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
(Matchable t i, FunctorBy t f) =>
ListBy t f ~> (i :+: NonEmptyBy t f)
matchLB @t) (forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
Tensor t i =>
i ~> ListBy t f
nilLB @t forall (t :: (* -> *) -> (* -> *) -> * -> *) (h :: * -> *)
       (f :: * -> *) (g :: * -> *).
SemigroupIn t h =>
(f ~> h) -> (g ~> h) -> t f g ~> h
!*! forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
Tensor t i =>
NonEmptyBy t f ~> ListBy t f
fromNE @t)

instance Tensor (:*:) Proxy where
    type ListBy (:*:) = ListF
    intro1 :: forall (f :: * -> *). f ~> (f :*: Proxy)
intro1 = (forall k (f :: k -> *) (g :: k -> *) (p :: k).
f p -> g p -> (:*:) f g p
:*: forall {k} (t :: k). Proxy t
Proxy)
    intro2 :: forall (g :: * -> *). g ~> (Proxy :*: g)
intro2 = (forall {k} (t :: k). Proxy t
Proxy forall k (f :: k -> *) (g :: k -> *) (p :: k).
f p -> g p -> (:*:) f g p
:*:)
    elim1 :: forall (f :: * -> *). FunctorBy (:*:) f => (f :*: Proxy) ~> f
elim1 (f x
x      :*: ~Proxy x
Proxy) = f x
x
    elim2 :: forall (g :: * -> *). FunctorBy (:*:) g => (Proxy :*: g) ~> g
elim2 (~Proxy x
Proxy :*: g x
y     ) = g x
y

    appendLB :: forall (f :: * -> *).
(ListBy (:*:) f :*: ListBy (:*:) f) ~> ListBy (:*:) f
appendLB (ListF [f x]
xs :*: ListF [f x]
ys) = forall {k} (f :: k -> *) (a :: k). [f a] -> ListF f a
ListF ([f x]
xs forall a. [a] -> [a] -> [a]
++ [f x]
ys)
    splitNE :: forall (f :: * -> *). NonEmptyBy (:*:) f ~> (f :*: ListBy (:*:) f)
splitNE     = forall {k} (f :: k -> *) (a :: k).
NonEmptyF f a -> (:*:) f (ListF f) a
nonEmptyProd
    splittingLB :: forall (f :: * -> *).
ListBy (:*:) f <~> (Proxy :+: (f :*: ListBy (:*:) f))
splittingLB = forall {k} (f :: k -> *) (g :: k -> *).
(f ~> g) -> (g ~> f) -> f <~> g
isoF forall {k} {f :: k -> *} {p :: k}.
ListF f p -> (:+:) Proxy (f :*: ListF f) p
to_ forall {k} {f :: k -> *} {a :: k}.
(:+:) Proxy (f :*: ListF f) a -> ListF f a
from_
      where
        to_ :: ListF f p -> (:+:) Proxy (f :*: ListF f) p
to_ = \case
          ListF []     -> forall k (f :: k -> *) (g :: k -> *) (p :: k). f p -> (:+:) f g p
L1 forall {k} (t :: k). Proxy t
Proxy
          ListF (f p
x:[f p]
xs) -> forall k (f :: k -> *) (g :: k -> *) (p :: k). g p -> (:+:) f g p
R1 (f p
x forall k (f :: k -> *) (g :: k -> *) (p :: k).
f p -> g p -> (:*:) f g p
:*: forall {k} (f :: k -> *) (a :: k). [f a] -> ListF f a
ListF [f p]
xs)
        from_ :: (:+:) Proxy (f :*: ListF f) a -> ListF f a
from_ = \case
          L1 ~Proxy a
Proxy           -> forall {k} (f :: k -> *) (a :: k). [f a] -> ListF f a
ListF []
          R1 (f a
x :*: ListF [f a]
xs) -> forall {k} (f :: k -> *) (a :: k). [f a] -> ListF f a
ListF (f a
xforall a. a -> [a] -> [a]
:[f a]
xs)

    toListBy :: forall (f :: * -> *). (f :*: f) ~> ListBy (:*:) f
toListBy (f x
x :*: f x
y) = forall {k} (f :: k -> *) (a :: k). [f a] -> ListF f a
ListF [f x
x, f x
y]

-- | Instances of 'Plus' are monoids in the monoidal category on
-- ':*:'.
instance Plus f => MonoidIn (:*:) Proxy f where
    pureT :: Proxy ~> f
pureT Proxy x
_        = forall (f :: * -> *) a. Plus f => f a
zero

instance Tensor Product Proxy where
    type ListBy Product = ListF
    intro1 :: forall (f :: * -> *). f ~> Product f Proxy
intro1 = (forall {k} (f :: k -> *) (g :: k -> *) (a :: k).
f a -> g a -> Product f g a
`Pair` forall {k} (t :: k). Proxy t
Proxy)
    intro2 :: forall (g :: * -> *). g ~> Product Proxy g
intro2 = (forall {k} (t :: k). Proxy t
Proxy forall {k} (f :: k -> *) (g :: k -> *) (a :: k).
f a -> g a -> Product f g a
`Pair`)
    elim1 :: forall (f :: * -> *). FunctorBy Product f => Product f Proxy ~> f
elim1 (Pair f x
x ~Proxy x
Proxy) = f x
x
    elim2 :: forall (g :: * -> *). FunctorBy Product g => Product Proxy g ~> g
elim2 (Pair ~Proxy x
Proxy g x
y) = g x
y

    appendLB :: forall (f :: * -> *).
Product (ListBy Product f) (ListBy Product f) ~> ListBy Product f
appendLB (ListF [f x]
xs `Pair` ListF [f x]
ys) = forall {k} (f :: k -> *) (a :: k). [f a] -> ListF f a
ListF ([f x]
xs forall a. [a] -> [a] -> [a]
++ [f x]
ys)
    splitNE :: forall (f :: * -> *).
NonEmptyBy Product f ~> Product f (ListBy Product f)
splitNE     = forall {k} (f :: k -> *) (g :: k -> *). (f <~> g) -> f ~> g
viewF forall {k} (f :: k -> *) (g :: k -> *). (f :*: g) <~> Product f g
prodProd forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall {k} (f :: k -> *) (a :: k).
NonEmptyF f a -> (:*:) f (ListF f) a
nonEmptyProd
    splittingLB :: forall (f :: * -> *).
ListBy Product f <~> (Proxy :+: Product f (ListBy Product f))
splittingLB = forall {k} (f :: k -> *) (g :: k -> *).
(f ~> g) -> (g ~> f) -> f <~> g
isoF forall {k} {f :: k -> *} {p :: k}.
ListF f p -> (:+:) Proxy (Product f (ListF f)) p
to_ forall {k} {f :: k -> *} {a :: k}.
(:+:) Proxy (Product f (ListF f)) a -> ListF f a
from_
      where
        to_ :: ListF f p -> (:+:) Proxy (Product f (ListF f)) p
to_ = \case
          ListF []     -> forall k (f :: k -> *) (g :: k -> *) (p :: k). f p -> (:+:) f g p
L1 forall {k} (t :: k). Proxy t
Proxy
          ListF (f p
x:[f p]
xs) -> forall k (f :: k -> *) (g :: k -> *) (p :: k). g p -> (:+:) f g p
R1 (f p
x forall {k} (f :: k -> *) (g :: k -> *) (a :: k).
f a -> g a -> Product f g a
`Pair` forall {k} (f :: k -> *) (a :: k). [f a] -> ListF f a
ListF [f p]
xs)
        from_ :: (:+:) Proxy (Product f (ListF f)) a -> ListF f a
from_ = \case
          L1 ~Proxy a
Proxy              -> forall {k} (f :: k -> *) (a :: k). [f a] -> ListF f a
ListF []
          R1 (f a
x `Pair` ListF [f a]
xs) -> forall {k} (f :: k -> *) (a :: k). [f a] -> ListF f a
ListF (f a
xforall a. a -> [a] -> [a]
:[f a]
xs)

    toListBy :: forall (f :: * -> *). Product f f ~> ListBy Product f
toListBy (Pair f x
x f x
y) = forall {k} (f :: k -> *) (a :: k). [f a] -> ListF f a
ListF [f x
x, f x
y]

-- | Instances of 'Plus' are monoids in the monoidal category on
-- 'Product'.
instance Plus f => MonoidIn Product Proxy f where
    pureT :: Proxy ~> f
pureT Proxy x
_         = forall (f :: * -> *) a. Plus f => f a
zero

instance Tensor Day Identity where
    type ListBy Day = Ap
    intro1 :: forall (f :: * -> *). f ~> Day f Identity
intro1   = forall (f :: * -> *). f ~> Day f Identity
D.intro2
    intro2 :: forall (g :: * -> *). g ~> Day Identity g
intro2   = forall (g :: * -> *). g ~> Day Identity g
D.intro1
    elim1 :: forall (f :: * -> *). FunctorBy Day f => Day f Identity ~> f
elim1    = forall (f :: * -> *) a. Functor f => Day f Identity a -> f a
D.elim2
    elim2 :: forall (g :: * -> *). FunctorBy Day g => Day Identity g ~> g
elim2    = forall (f :: * -> *) a. Functor f => Day Identity f a -> f a
D.elim1

    appendLB :: forall (f :: * -> *).
Day (ListBy Day f) (ListBy Day f) ~> ListBy Day f
appendLB (Day ListBy Day f b
x ListBy Day f c
y b -> c -> x
z) = b -> c -> x
z forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> ListBy Day f b
x forall (f :: * -> *) a b. Applicative f => f (a -> b) -> f a -> f b
<*> ListBy Day f c
y
    splitNE :: forall (f :: * -> *). NonEmptyBy Day f ~> Day f (ListBy Day f)
splitNE     = forall (f :: * -> *) a. Ap1 f a -> Day f (Ap f) a
ap1Day
    splittingLB :: forall (f :: * -> *).
ListBy Day f <~> (Identity :+: Day f (ListBy Day f))
splittingLB = forall {k} (f :: k -> *) (g :: k -> *).
(f ~> g) -> (g ~> f) -> f <~> g
isoF forall {f :: * -> *} {p}. Ap f p -> (:+:) Identity (Day f (Ap f)) p
to_ forall {f :: * -> *} {a}. (:+:) Identity (Day f (Ap f)) a -> Ap f a
from_
      where
        to_ :: Ap f p -> (:+:) Identity (Day f (Ap f)) p
to_ = \case
          Pure p
x  -> forall k (f :: k -> *) (g :: k -> *) (p :: k). f p -> (:+:) f g p
L1 (forall a. a -> Identity a
Identity p
x)
          Ap f a1
x Ap f (a1 -> p)
xs -> forall k (f :: k -> *) (g :: k -> *) (p :: k). g p -> (:+:) f g p
R1 (forall (f :: * -> *) (g :: * -> *) a b c.
f b -> g c -> (b -> c -> a) -> Day f g a
Day f a1
x Ap f (a1 -> p)
xs forall a b. a -> (a -> b) -> b
(&))
        from_ :: (:+:) Identity (Day f (Ap f)) a -> Ap f a
from_ = \case
          L1 (Identity a
x) -> forall a (f :: * -> *). a -> Ap f a
Pure a
x
          R1 (Day f b
x Ap f c
xs b -> c -> a
f) -> forall (f :: * -> *) a1 a. f a1 -> Ap f (a1 -> a) -> Ap f a
Ap f b
x (forall a b c. (a -> b -> c) -> b -> a -> c
flip b -> c -> a
f forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> Ap f c
xs)

    toListBy :: forall (f :: * -> *). Day f f ~> ListBy Day f
toListBy (Day f b
x f c
y b -> c -> x
z) = forall (f :: * -> *) a1 a. f a1 -> Ap f (a1 -> a) -> Ap f a
Ap f b
x (forall (f :: * -> *) a1 a. f a1 -> Ap f (a1 -> a) -> Ap f a
Ap f c
y (forall a (f :: * -> *). a -> Ap f a
Pure (forall a b c. (a -> b -> c) -> b -> a -> c
flip b -> c -> x
z)))

-- | Instances of 'Applicative' are monoids in the monoidal category on
-- the covariant 'Day'.
--
-- Note that because of typeclass constraints, this requires 'Apply' as
-- well as 'Applicative'.  But, you can get a "local" instance of 'Apply'
-- for any 'Applicative' using
-- 'Data.Functor.Combinators.Unsafe.unsafeApply'.
instance (Apply f, Applicative f) => MonoidIn Day Identity f where
    pureT :: Identity ~> f
pureT            = forall (f :: * -> *). Applicative f => Identity ~> f
generalize

-- | @since 0.3.0.0
instance Tensor CD.Day Proxy where
    type ListBy CD.Day = Div
    intro1 :: forall (f :: * -> *). f ~> Day f Proxy
intro1 = forall (f :: * -> *). f ~> Day f Proxy
CD.intro2
    intro2 :: forall (g :: * -> *). g ~> Day Proxy g
intro2 = forall (g :: * -> *). g ~> Day Proxy g
CD.intro1
    elim1 :: forall (f :: * -> *). FunctorBy Day f => Day f Proxy ~> f
elim1 = forall (f :: * -> *) (g :: * -> *) a.
Contravariant f =>
Day f g a -> f a
CD.day1
    elim2 :: forall (g :: * -> *). FunctorBy Day g => Day Proxy g ~> g
elim2 = forall (g :: * -> *) (f :: * -> *) a.
Contravariant g =>
Day f g a -> g a
CD.day2

    appendLB :: forall (f :: * -> *).
Day (ListBy Day f) (ListBy Day f) ~> ListBy Day f
appendLB (CD.Day ListBy Day f b
x ListBy Day f c
y x -> (b, c)
z) = forall (f :: * -> *) a b c.
Divisible f =>
(a -> (b, c)) -> f b -> f c -> f a
divide x -> (b, c)
z ListBy Day f b
x ListBy Day f c
y
    splitNE :: forall (f :: * -> *). NonEmptyBy Day f ~> Day f (ListBy Day f)
splitNE = forall {f :: * -> *} {f :: * -> *} {p}.
(:*:) (Coyoneda f) (ListF (Coyoneda f)) p -> Day f (Div f) p
go forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
Tensor t i =>
NonEmptyBy t f ~> t f (ListBy t f)
splitNE @(:*:) forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall {k} (f :: k -> *) (a :: k). NonEmpty (f a) -> NonEmptyF f a
NonEmptyF forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (f :: * -> *) a. Div1 f a -> NonEmpty (Coyoneda f a)
unDiv1
      where
        go :: (:*:) (Coyoneda f) (ListF (Coyoneda f)) p -> Day f (Div f) p
go (CCY.Coyoneda p -> b
f f b
x :*: ListF [Coyoneda f p]
xs) = forall (f :: * -> *) (g :: * -> *) a b c.
f b -> g c -> (a -> (b, c)) -> Day f g a
CD.Day f b
x (forall (f :: * -> *) a. [Coyoneda f a] -> Div f a
Div [Coyoneda f p]
xs) (\p
y -> (p -> b
f p
y, p
y))
    splittingLB :: forall (f :: * -> *).
ListBy Day f <~> (Proxy :+: Day f (ListBy Day f))
splittingLB = forall {k} (f :: k -> *) (g :: k -> *).
(f ~> g) -> (g ~> f) -> f <~> g
isoF (forall {k} (f :: k -> *) (a :: k). [f a] -> ListF f a
ListF forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (f :: * -> *) a. Div f a -> [Coyoneda f a]
unDiv) (forall (f :: * -> *) a. [Coyoneda f a] -> Div f a
Div forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall {k} (f :: k -> *) (a :: k). ListF f a -> [f a]
runListF) forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
Tensor t i =>
ListBy t f <~> (i :+: t f (ListBy t f))
splittingLB @(:*:) forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall {k} (f :: k -> *) (g :: k -> *).
(f ~> g) -> (g ~> f) -> f <~> g
isoF (forall k (t :: (k -> *) -> (k -> *) -> k -> *) (g :: k -> *)
       (l :: k -> *) (f :: k -> *).
HBifunctor t =>
(g ~> l) -> t f g ~> t f l
hright forall {f :: * -> *} {f :: * -> *} {p}.
(:*:) (Coyoneda f) (ListF (Coyoneda f)) p -> Day f (Div f) p
to_) (forall k (t :: (k -> *) -> (k -> *) -> k -> *) (g :: k -> *)
       (l :: k -> *) (f :: k -> *).
HBifunctor t =>
(g ~> l) -> t f g ~> t f l
hright forall {f :: * -> *} {f :: * -> *} {a}.
Day f (Div f) a -> (:*:) (Coyoneda f) (ListF (Coyoneda f)) a
from_)
      where
        to_ :: (:*:) (Coyoneda f) (ListF (Coyoneda f)) p -> Day f (Div f) p
to_   (CCY.Coyoneda p -> b
f f b
x :*: ListF [Coyoneda f p]
xs) = forall (f :: * -> *) (g :: * -> *) a b c.
f b -> g c -> (a -> (b, c)) -> Day f g a
CD.Day f b
x (forall (f :: * -> *) a. [Coyoneda f a] -> Div f a
Div [Coyoneda f p]
xs) (\p
y -> (p -> b
f p
y, p
y))
        from_ :: Day f (Div f) a -> (:*:) (Coyoneda f) (ListF (Coyoneda f)) a
from_ (CD.Day f b
x (Div [Coyoneda f c]
xs) a -> (b, c)
f) = forall a b (f :: * -> *). (a -> b) -> f b -> Coyoneda f a
CCY.Coyoneda (forall a b. (a, b) -> a
fst forall b c a. (b -> c) -> (a -> b) -> a -> c
. a -> (b, c)
f) f b
x forall k (f :: k -> *) (g :: k -> *) (p :: k).
f p -> g p -> (:*:) f g p
:*: forall (f :: * -> *) a' a.
Contravariant f =>
(a' -> a) -> f a -> f a'
contramap (forall a b. (a, b) -> b
snd forall b c a. (b -> c) -> (a -> b) -> a -> c
. a -> (b, c)
f) (forall {k} (f :: k -> *) (a :: k). [f a] -> ListF f a
ListF [Coyoneda f c]
xs)

    toListBy :: forall (f :: * -> *). Day f f ~> ListBy Day f
toListBy (CD.Day f b
x f c
y x -> (b, c)
f) = forall (f :: * -> *) a. [Coyoneda f a] -> Div f a
Div forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall {k} (f :: k -> *) (a :: k). ListF f a -> [f a]
runListF forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
Tensor t i =>
t f f ~> ListBy t f
toListBy forall a b. (a -> b) -> a -> b
$
        forall a b (f :: * -> *). (a -> b) -> f b -> Coyoneda f a
CCY.Coyoneda (forall a b. (a, b) -> a
fst forall b c a. (b -> c) -> (a -> b) -> a -> c
. x -> (b, c)
f) f b
x forall k (f :: k -> *) (g :: k -> *) (p :: k).
f p -> g p -> (:*:) f g p
:*: forall a b (f :: * -> *). (a -> b) -> f b -> Coyoneda f a
CCY.Coyoneda (forall a b. (a, b) -> b
snd forall b c a. (b -> c) -> (a -> b) -> a -> c
. x -> (b, c)
f) f c
y

-- | Instances of 'Divisible' are monoids in the monoidal category on
-- contravariant 'CD.Day'.
--
-- Note that because of typeclass constraints, this requires 'Divise' as
-- well as 'Divisible'.  But, you can get a "local" instance of 'Divise'
-- for any 'Divisible' using
-- 'Data.Functor.Combinators.Unsafe.unsafeDivise'.
--
-- @since 0.3.0.0
instance (Divise f, Divisible f) => MonoidIn CD.Day Proxy f where
    pureT :: Proxy ~> f
pureT Proxy x
_ = forall (f :: * -> *) a. Divisible f => f a
conquer

instance Tensor ID.Day Identity where
    type ListBy ID.Day = DivAp

    intro1 :: forall (f :: * -> *). f ~> Day f Identity
intro1 = forall (f :: * -> *). f ~> Day f Identity
ID.intro2
    intro2 :: forall (g :: * -> *). g ~> Day Identity g
intro2 = forall (g :: * -> *). g ~> Day Identity g
ID.intro1
    elim1 :: forall (f :: * -> *). FunctorBy Day f => Day f Identity ~> f
elim1 = forall (f :: * -> *) a. Invariant f => Day f Identity a -> f a
ID.elim2
    elim2 :: forall (g :: * -> *). FunctorBy Day g => Day Identity g ~> g
elim2 = forall (f :: * -> *) a. Invariant f => Day Identity f a -> f a
ID.elim1

    appendLB :: forall (f :: * -> *).
Day (ListBy Day f) (ListBy Day f) ~> ListBy Day f
appendLB (ID.Day (DivAp Chain Day Identity f b
xs) (DivAp Chain Day Identity f c
ys) b -> c -> x
f x -> (b, c)
g) = forall (f :: * -> *) a. Chain Day Identity f a -> DivAp f a
DivAp forall a b. (a -> b) -> a -> b
$ case Chain Day Identity f b
xs of
      Done (Identity b
x)      -> forall (f :: * -> *) a b.
Invariant f =>
(a -> b) -> (b -> a) -> f a -> f b
invmap (b -> c -> x
f b
x) (forall a b. (a, b) -> b
snd forall b c a. (b -> c) -> (a -> b) -> a -> c
. x -> (b, c)
g) Chain Day Identity f c
ys
      More (ID.Day f b
z Chain Day Identity f c
zs b -> c -> b
h b -> (b, c)
j) -> forall {k} {k1} (t :: k -> (k1 -> *) -> k1 -> *) (i :: k1 -> *)
       (f :: k) (a :: k1).
t f (Chain t i f) a -> Chain t i f a
More forall a b. (a -> b) -> a -> b
$ forall (f :: * -> *) (g :: * -> *) a b c.
f b -> g c -> (b -> c -> a) -> (a -> (b, c)) -> Day f g a
ID.Day
        f b
z
        (forall (f :: * -> *) a. DivAp f a -> Chain Day Identity f a
unDivAp forall a b. (a -> b) -> a -> b
$ forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
Tensor t i =>
t (ListBy t f) (ListBy t f) ~> ListBy t f
appendLB (forall (f :: * -> *) (g :: * -> *) a b c.
f b -> g c -> (b -> c -> a) -> (a -> (b, c)) -> Day f g a
ID.Day (forall (f :: * -> *) a. Chain Day Identity f a -> DivAp f a
DivAp Chain Day Identity f c
zs) (forall (f :: * -> *) a. Chain Day Identity f a -> DivAp f a
DivAp Chain Day Identity f c
ys) (,) forall a. a -> a
id))
        (\b
q (c
r, c
s) -> b -> c -> x
f (b -> c -> b
h b
q c
r) c
s)
        (forall (p :: * -> * -> *) a b c.
Assoc p =>
p (p a b) c -> p a (p b c)
B.assoc forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (p :: * -> * -> *) a b c.
Bifunctor p =>
(a -> b) -> p a c -> p b c
first b -> (b, c)
j forall b c a. (b -> c) -> (a -> b) -> a -> c
. x -> (b, c)
g)

    splitNE :: forall (f :: * -> *). NonEmptyBy Day f ~> Day f (ListBy Day f)
splitNE = coerce :: forall a b. Coercible a b => a -> b
coerce forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
Tensor t i =>
Chain1 t f ~> t f (Chain t i f)
splitChain1
    splittingLB :: forall (f :: * -> *).
ListBy Day f <~> (Identity :+: Day f (ListBy Day f))
splittingLB = forall {k} (f :: k -> *) (g :: k -> *).
(forall (x :: k). Coercible (f x) (g x),
 forall (x :: k). Coercible (g x) (f x)) =>
f <~> g
coercedF forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall {k1} {k2} (t :: k1 -> (k2 -> *) -> k2 -> *) (i :: k2 -> *)
       (f :: k1).
Chain t i f <~> (i :+: t f (Chain t i f))
splittingChain forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall {k} (f :: k -> *) (g :: k -> *).
(forall (x :: k). Coercible (f x) (g x),
 forall (x :: k). Coercible (g x) (f x)) =>
f <~> g
coercedF

    toListBy :: forall (f :: * -> *). Day f f ~> ListBy Day f
toListBy = forall (f :: * -> *) a. Chain Day Identity f a -> DivAp f a
DivAp forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall {k} {k1} (t :: k -> (k1 -> *) -> k1 -> *) (i :: k1 -> *)
       (f :: k) (a :: k1).
t f (Chain t i f) a -> Chain t i f a
More forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall k (t :: (k -> *) -> (k -> *) -> k -> *) (g :: k -> *)
       (l :: k -> *) (f :: k -> *).
HBifunctor t =>
(g ~> l) -> t f g ~> t f l
hright (forall (f :: * -> *) a. DivAp f a -> Chain Day Identity f a
unDivAp forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall {k} (t :: (k -> *) -> k -> *) (f :: k -> *).
Inject t =>
f ~> t f
inject)

instance Matchable ID.Day Identity where
    unsplitNE :: forall (f :: * -> *).
FunctorBy Day f =>
Day f (ListBy Day f) ~> NonEmptyBy Day f
unsplitNE = coerce :: forall a b. Coercible a b => a -> b
coerce forall (f :: * -> *).
Invariant f =>
Day f (Chain Day Identity f) ~> Chain1 Day f
unsplitNEIDay_
    matchLB :: forall (f :: * -> *).
FunctorBy Day f =>
ListBy Day f ~> (Identity :+: NonEmptyBy Day f)
matchLB = coerce :: forall a b. Coercible a b => a -> b
coerce forall (f :: * -> *).
Invariant f =>
Chain Day Identity f ~> (Identity :+: Chain1 Day f)
matchLBIDay_

unsplitNEIDay_ :: Invariant f => ID.Day f (Chain ID.Day Identity f) ~> Chain1 ID.Day f
unsplitNEIDay_ :: forall (f :: * -> *).
Invariant f =>
Day f (Chain Day Identity f) ~> Chain1 Day f
unsplitNEIDay_ (ID.Day f b
x Chain Day Identity f c
xs b -> c -> x
g x -> (b, c)
f) = case Chain Day Identity f c
xs of
  Done (Identity c
r) -> forall {k} (t :: (k -> *) -> (k -> *) -> k -> *) (f :: k -> *)
       (a :: k).
f a -> Chain1 t f a
Done1 forall a b. (a -> b) -> a -> b
$ forall (f :: * -> *) a b.
Invariant f =>
(a -> b) -> (b -> a) -> f a -> f b
invmap (b -> c -> x
`g` c
r) (forall a b. (a, b) -> a
fst forall b c a. (b -> c) -> (a -> b) -> a -> c
. x -> (b, c)
f) f b
x
  More Day f (Chain Day Identity f) c
ys           -> forall {k} (t :: (k -> *) -> (k -> *) -> k -> *) (f :: k -> *)
       (a :: k).
t f (Chain1 t f) a -> Chain1 t f a
More1 forall a b. (a -> b) -> a -> b
$ forall (f :: * -> *) (g :: * -> *) a b c.
f b -> g c -> (b -> c -> a) -> (a -> (b, c)) -> Day f g a
ID.Day f b
x (forall (f :: * -> *).
Invariant f =>
Day f (Chain Day Identity f) ~> Chain1 Day f
unsplitNEIDay_ Day f (Chain Day Identity f) c
ys) b -> c -> x
g x -> (b, c)
f

matchLBIDay_ :: Invariant f => Chain ID.Day Identity f ~> (Identity :+: Chain1 ID.Day f)
matchLBIDay_ :: forall (f :: * -> *).
Invariant f =>
Chain Day Identity f ~> (Identity :+: Chain1 Day f)
matchLBIDay_ = \case
  Done Identity x
x  -> forall k (f :: k -> *) (g :: k -> *) (p :: k). f p -> (:+:) f g p
L1 Identity x
x
  More Day f (Chain Day Identity f) x
xs -> forall k (f :: k -> *) (g :: k -> *) (p :: k). g p -> (:+:) f g p
R1 forall a b. (a -> b) -> a -> b
$ forall (f :: * -> *).
Invariant f =>
Day f (Chain Day Identity f) ~> Chain1 Day f
unsplitNEIDay_ Day f (Chain Day Identity f) x
xs

instance Inplicative f => MonoidIn ID.Day Identity f where
    pureT :: Identity ~> f
pureT (Identity x
x) = forall (f :: * -> *) a. Inplicative f => a -> f a
knot x
x

instance Tensor IN.Night IN.Not where
    type ListBy IN.Night = DecAlt

    intro1 :: forall (f :: * -> *). f ~> Night f Not
intro1 = forall (f :: * -> *). f ~> Night f Not
IN.intro2
    intro2 :: forall (g :: * -> *). g ~> Night Not g
intro2 = forall (g :: * -> *). g ~> Night Not g
IN.intro1
    elim1 :: forall (f :: * -> *). FunctorBy Night f => Night f Not ~> f
elim1 = forall (f :: * -> *). Invariant f => Night f Not ~> f
IN.elim2
    elim2 :: forall (g :: * -> *). FunctorBy Night g => Night Not g ~> g
elim2 = forall (g :: * -> *). Invariant g => Night Not g ~> g
IN.elim1

    appendLB :: forall (f :: * -> *).
Night (ListBy Night f) (ListBy Night f) ~> ListBy Night f
appendLB (IN.Night (DecAlt Chain Night Not f b1
xs) (DecAlt Chain Night Not f c1
ys) b1 -> x
f c1 -> x
g x -> Either b1 c1
h) = forall (f :: * -> *) a. Chain Night Not f a -> DecAlt f a
DecAlt forall a b. (a -> b) -> a -> b
$ case Chain Night Not f b1
xs of
      Done Not b1
r      -> forall (f :: * -> *) a b.
Invariant f =>
(a -> b) -> (b -> a) -> f a -> f b
invmap c1 -> x
g (forall a c b. (a -> c) -> (b -> c) -> Either a b -> c
either (forall a. Void -> a
absurd forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. Not a -> a -> Void
refute Not b1
r) forall a. a -> a
id forall b c a. (b -> c) -> (a -> b) -> a -> c
. x -> Either b1 c1
h) Chain Night Not f c1
ys
      More (IN.Night f b1
z Chain Night Not f c1
zs b1 -> b1
j c1 -> b1
k b1 -> Either b1 c1
l) -> forall {k} {k1} (t :: k -> (k1 -> *) -> k1 -> *) (i :: k1 -> *)
       (f :: k) (a :: k1).
t f (Chain t i f) a -> Chain t i f a
More forall a b. (a -> b) -> a -> b
$ forall (a :: * -> *) b1 (b :: * -> *) c1 c.
a b1
-> b c1
-> (b1 -> c)
-> (c1 -> c)
-> (c -> Either b1 c1)
-> Night a b c
IN.Night
        f b1
z
        (forall (f :: * -> *) a. DecAlt f a -> Chain Night Not f a
unDecAlt forall a b. (a -> b) -> a -> b
$ forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
Tensor t i =>
t (ListBy t f) (ListBy t f) ~> ListBy t f
appendLB (forall (a :: * -> *) b1 (b :: * -> *) c1 c.
a b1
-> b c1
-> (b1 -> c)
-> (c1 -> c)
-> (c -> Either b1 c1)
-> Night a b c
IN.Night (forall (f :: * -> *) a. Chain Night Not f a -> DecAlt f a
DecAlt Chain Night Not f c1
zs) (forall (f :: * -> *) a. Chain Night Not f a -> DecAlt f a
DecAlt Chain Night Not f c1
ys) forall a b. a -> Either a b
Left forall a b. b -> Either a b
Right forall a. a -> a
id))
        (b1 -> x
f forall b c a. (b -> c) -> (a -> b) -> a -> c
. b1 -> b1
j)
        (forall a c b. (a -> c) -> (b -> c) -> Either a b -> c
either (b1 -> x
f forall b c a. (b -> c) -> (a -> b) -> a -> c
. c1 -> b1
k) c1 -> x
g)
        (forall (p :: * -> * -> *) a b c.
Assoc p =>
p (p a b) c -> p a (p b c)
B.assoc forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (p :: * -> * -> *) a b c.
Bifunctor p =>
(a -> b) -> p a c -> p b c
first b1 -> Either b1 c1
l forall b c a. (b -> c) -> (a -> b) -> a -> c
. x -> Either b1 c1
h)
    splitNE :: forall (f :: * -> *).
NonEmptyBy Night f ~> Night f (ListBy Night f)
splitNE = coerce :: forall a b. Coercible a b => a -> b
coerce forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
Tensor t i =>
Chain1 t f ~> t f (Chain t i f)
splitChain1
    splittingLB :: forall (f :: * -> *).
ListBy Night f <~> (Not :+: Night f (ListBy Night f))
splittingLB = forall {k} (f :: k -> *) (g :: k -> *).
(forall (x :: k). Coercible (f x) (g x),
 forall (x :: k). Coercible (g x) (f x)) =>
f <~> g
coercedF forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall {k1} {k2} (t :: k1 -> (k2 -> *) -> k2 -> *) (i :: k2 -> *)
       (f :: k1).
Chain t i f <~> (i :+: t f (Chain t i f))
splittingChain forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall {k} (f :: k -> *) (g :: k -> *).
(forall (x :: k). Coercible (f x) (g x),
 forall (x :: k). Coercible (g x) (f x)) =>
f <~> g
coercedF

    toListBy :: forall (f :: * -> *). Night f f ~> ListBy Night f
toListBy = forall (f :: * -> *) a. Chain Night Not f a -> DecAlt f a
DecAlt forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall {k} {k1} (t :: k -> (k1 -> *) -> k1 -> *) (i :: k1 -> *)
       (f :: k) (a :: k1).
t f (Chain t i f) a -> Chain t i f a
More forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall k (t :: (k -> *) -> (k -> *) -> k -> *) (g :: k -> *)
       (l :: k -> *) (f :: k -> *).
HBifunctor t =>
(g ~> l) -> t f g ~> t f l
hright (forall (f :: * -> *) a. DecAlt f a -> Chain Night Not f a
unDecAlt forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall {k} (t :: (k -> *) -> k -> *) (f :: k -> *).
Inject t =>
f ~> t f
inject)

instance Matchable IN.Night Not where
    unsplitNE :: forall (f :: * -> *).
FunctorBy Night f =>
Night f (ListBy Night f) ~> NonEmptyBy Night f
unsplitNE = coerce :: forall a b. Coercible a b => a -> b
coerce forall (f :: * -> *).
Invariant f =>
Night f (Chain Night Not f) ~> Chain1 Night f
unsplitNEINight_
    matchLB :: forall (f :: * -> *).
FunctorBy Night f =>
ListBy Night f ~> (Not :+: NonEmptyBy Night f)
matchLB = coerce :: forall a b. Coercible a b => a -> b
coerce forall (f :: * -> *).
Invariant f =>
Chain Night Not f ~> (Not :+: Chain1 Night f)
matchLBINight_

unsplitNEINight_ :: Invariant f => IN.Night f (Chain IN.Night Not f) ~> Chain1 IN.Night f
unsplitNEINight_ :: forall (f :: * -> *).
Invariant f =>
Night f (Chain Night Not f) ~> Chain1 Night f
unsplitNEINight_ (IN.Night f b1
x Chain Night Not f c1
xs b1 -> x
f c1 -> x
g x -> Either b1 c1
h) = case Chain Night Not f c1
xs of
  Done Not c1
r  -> forall {k} (t :: (k -> *) -> (k -> *) -> k -> *) (f :: k -> *)
       (a :: k).
f a -> Chain1 t f a
Done1 forall a b. (a -> b) -> a -> b
$ forall (f :: * -> *) a b.
Invariant f =>
(a -> b) -> (b -> a) -> f a -> f b
invmap b1 -> x
f (forall a c b. (a -> c) -> (b -> c) -> Either a b -> c
either forall a. a -> a
id (forall a. Void -> a
absurd forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. Not a -> a -> Void
refute Not c1
r) forall b c a. (b -> c) -> (a -> b) -> a -> c
. x -> Either b1 c1
h) f b1
x
  More Night f (Chain Night Not f) c1
ys -> forall {k} (t :: (k -> *) -> (k -> *) -> k -> *) (f :: k -> *)
       (a :: k).
t f (Chain1 t f) a -> Chain1 t f a
More1 forall a b. (a -> b) -> a -> b
$ forall (a :: * -> *) b1 (b :: * -> *) c1 c.
a b1
-> b c1
-> (b1 -> c)
-> (c1 -> c)
-> (c -> Either b1 c1)
-> Night a b c
IN.Night f b1
x (forall (f :: * -> *).
Invariant f =>
Night f (Chain Night Not f) ~> Chain1 Night f
unsplitNEINight_ Night f (Chain Night Not f) c1
ys) b1 -> x
f c1 -> x
g x -> Either b1 c1
h

matchLBINight_ :: Invariant f => Chain IN.Night Not f ~> (Not :+: Chain1 IN.Night f)
matchLBINight_ :: forall (f :: * -> *).
Invariant f =>
Chain Night Not f ~> (Not :+: Chain1 Night f)
matchLBINight_ = \case
  Done Not x
x  -> forall k (f :: k -> *) (g :: k -> *) (p :: k). f p -> (:+:) f g p
L1 Not x
x
  More Night f (Chain Night Not f) x
xs -> forall k (f :: k -> *) (g :: k -> *) (p :: k). g p -> (:+:) f g p
R1 forall a b. (a -> b) -> a -> b
$ forall (f :: * -> *).
Invariant f =>
Night f (Chain Night Not f) ~> Chain1 Night f
unsplitNEINight_ Night f (Chain Night Not f) x
xs

-- | @since 0.4.0.0
instance Inplus f => MonoidIn IN.Night IN.Not f where
    pureT :: Not ~> f
pureT (Not x -> Void
x) = forall (f :: * -> *) a. Inplus f => (a -> Void) -> f a
reject x -> Void
x

-- | @since 0.3.0.0
instance Tensor Night Not where
    type ListBy Night = Dec
    intro1 :: forall (f :: * -> *). f ~> Night f Not
intro1 = forall (f :: * -> *). f ~> Night f Not
N.intro2
    intro2 :: forall (g :: * -> *). g ~> Night Not g
intro2 = forall (g :: * -> *). g ~> Night Not g
N.intro1
    elim1 :: forall (f :: * -> *). FunctorBy Night f => Night f Not ~> f
elim1 = forall (f :: * -> *). Contravariant f => Night f Not ~> f
N.elim2
    elim2 :: forall (g :: * -> *). FunctorBy Night g => Night Not g ~> g
elim2 = forall (g :: * -> *). Contravariant g => Night Not g ~> g
N.elim1

    appendLB :: forall (f :: * -> *).
Night (ListBy Night f) (ListBy Night f) ~> ListBy Night f
appendLB (Night ListBy Night f b1
x ListBy Night f c1
y x -> Either b1 c1
z) = forall (f :: * -> *) a b c.
Decide f =>
(a -> Either b c) -> f b -> f c -> f a
decide x -> Either b1 c1
z ListBy Night f b1
x ListBy Night f c1
y
    splitNE :: forall (f :: * -> *).
NonEmptyBy Night f ~> Night f (ListBy Night f)
splitNE (Dec1 x -> Either b1 c
f f b1
x Dec f c
xs) = forall (a :: * -> *) b1 (b :: * -> *) c1 c.
a b1 -> b c1 -> (c -> Either b1 c1) -> Night a b c
Night f b1
x Dec f c
xs x -> Either b1 c
f
    splittingLB :: forall (f :: * -> *).
ListBy Night f <~> (Not :+: Night f (ListBy Night f))
splittingLB = forall {k} (f :: k -> *) (g :: k -> *).
(f ~> g) -> (g ~> f) -> f <~> g
isoF forall {a :: * -> *} {p}. Dec a p -> (:+:) Not (Night a (Dec a)) p
to_ forall {a :: * -> *} {b}. (:+:) Not (Night a (Dec a)) b -> Dec a b
from_
      where
        to_ :: Dec a p -> (:+:) Not (Night a (Dec a)) p
to_ = \case
          Lose   p -> Void
f      -> forall k (f :: k -> *) (g :: k -> *) (p :: k). f p -> (:+:) f g p
L1 (forall a. (a -> Void) -> Not a
Not p -> Void
f)
          Choose p -> Either b1 c
f a b1
x Dec a c
xs -> forall k (f :: k -> *) (g :: k -> *) (p :: k). g p -> (:+:) f g p
R1 (forall (a :: * -> *) b1 (b :: * -> *) c1 c.
a b1 -> b c1 -> (c -> Either b1 c1) -> Night a b c
Night a b1
x Dec a c
xs p -> Either b1 c
f)
        from_ :: (:+:) Not (Night a (Dec a)) b -> Dec a b
from_ = \case
          L1 (Not b -> Void
f)    -> forall b (a :: * -> *). (b -> Void) -> Dec a b
Lose b -> Void
f
          R1 (Night a b1
x Dec a c1
xs b -> Either b1 c1
f) -> forall b b1 c (a :: * -> *).
(b -> Either b1 c) -> a b1 -> Dec a c -> Dec a b
Choose b -> Either b1 c1
f a b1
x Dec a c1
xs

    toListBy :: forall (f :: * -> *). Night f f ~> ListBy Night f
toListBy (Night f b1
x f c1
y x -> Either b1 c1
z) = forall b b1 c (a :: * -> *).
(b -> Either b1 c) -> a b1 -> Dec a c -> Dec a b
Choose x -> Either b1 c1
z f b1
x (forall {k} (t :: (k -> *) -> k -> *) (f :: k -> *).
Inject t =>
f ~> t f
inject f c1
y)

-- | Instances of 'Conclude' are monoids in the monoidal category on 'Night'.
instance Conclude f => MonoidIn Night Not f where
    pureT :: Not ~> f
pureT (Not x -> Void
x) = forall (f :: * -> *) a. Conclude f => (a -> Void) -> f a
conclude x -> Void
x

instance Tensor (:+:) V1 where
    type ListBy (:+:) = Step
    intro1 :: forall (f :: * -> *). f ~> (f :+: V1)
intro1 = forall k (f :: k -> *) (g :: k -> *) (p :: k). f p -> (:+:) f g p
L1
    intro2 :: forall (g :: * -> *). g ~> (V1 :+: g)
intro2 = forall k (f :: k -> *) (g :: k -> *) (p :: k). g p -> (:+:) f g p
R1

    elim1 :: forall (f :: * -> *). FunctorBy (:+:) f => (f :+: V1) ~> f
elim1 = \case
      L1 f x
x -> f x
x
      R1 V1 x
y -> forall {k} (a :: k) (f :: k -> *). V1 a -> f a
absurd1 V1 x
y
    elim2 :: forall (g :: * -> *). FunctorBy (:+:) g => (V1 :+: g) ~> g
elim2 = \case
      L1 V1 x
x -> forall {k} (a :: k) (f :: k -> *). V1 a -> f a
absurd1 V1 x
x
      R1 g x
y -> g x
y

    appendLB :: forall (f :: * -> *).
(ListBy (:+:) f :+: ListBy (:+:) f) ~> ListBy (:+:) f
appendLB    = forall a. a -> a
id forall (t :: (* -> *) -> (* -> *) -> * -> *) (h :: * -> *)
       (f :: * -> *) (g :: * -> *).
SemigroupIn t h =>
(f ~> h) -> (g ~> h) -> t f g ~> h
!*! forall {k} (f :: k -> *). (f :+: Step f) ~> Step f
stepUp forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall k (f :: k -> *) (g :: k -> *) (p :: k). g p -> (:+:) f g p
R1
    splitNE :: forall (f :: * -> *). NonEmptyBy (:+:) f ~> (f :+: ListBy (:+:) f)
splitNE     = forall {k} (f :: k -> *). Step f ~> (f :+: Step f)
stepDown
    splittingLB :: forall (f :: * -> *).
ListBy (:+:) f <~> (V1 :+: (f :+: ListBy (:+:) f))
splittingLB = forall {k} (f :: k -> *). Step f <~> (f :+: Step f)
stepping forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall {k} (f :: k -> *). f <~> (V1 :+: f)
sumLeftIdentity

    toListBy :: forall (f :: * -> *). (f :+: f) ~> ListBy (:+:) f
toListBy  = \case
      L1 f x
x -> forall {k} (f :: k -> *) (a :: k). Natural -> f a -> Step f a
Step Natural
0 f x
x
      R1 f x
x -> forall {k} (f :: k -> *) (a :: k). Natural -> f a -> Step f a
Step Natural
1 f x
x

-- | All functors are monoids in the monoidal category on ':+:'.
instance MonoidIn (:+:) V1 f where
    pureT :: V1 ~> f
pureT = forall {k} (a :: k) (f :: k -> *). V1 a -> f a
absurd1

instance Tensor Sum V1 where
    type ListBy Sum = Step
    intro1 :: forall (f :: * -> *). f ~> Sum f V1
intro1 = forall {k} (f :: k -> *) (g :: k -> *) (a :: k). f a -> Sum f g a
InL
    intro2 :: forall (g :: * -> *). g ~> Sum V1 g
intro2 = forall {k} (f :: k -> *) (g :: k -> *) (a :: k). g a -> Sum f g a
InR

    elim1 :: forall (f :: * -> *). FunctorBy Sum f => Sum f V1 ~> f
elim1 = \case
      InL f x
x -> f x
x
      InR V1 x
y -> forall {k} (a :: k) (f :: k -> *). V1 a -> f a
absurd1 V1 x
y
    elim2 :: forall (g :: * -> *). FunctorBy Sum g => Sum V1 g ~> g
elim2 = \case
      InL V1 x
x -> forall {k} (a :: k) (f :: k -> *). V1 a -> f a
absurd1 V1 x
x
      InR g x
y -> g x
y

    appendLB :: forall (f :: * -> *).
Sum (ListBy Sum f) (ListBy Sum f) ~> ListBy Sum f
appendLB    = forall a. a -> a
id forall (t :: (* -> *) -> (* -> *) -> * -> *) (h :: * -> *)
       (f :: * -> *) (g :: * -> *).
SemigroupIn t h =>
(f ~> h) -> (g ~> h) -> t f g ~> h
!*! forall {k} (f :: k -> *). (f :+: Step f) ~> Step f
stepUp forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall k (f :: k -> *) (g :: k -> *) (p :: k). g p -> (:+:) f g p
R1
    splitNE :: forall (f :: * -> *). NonEmptyBy Sum f ~> Sum f (ListBy Sum f)
splitNE     = forall {k} (f :: k -> *) (g :: k -> *). (f <~> g) -> f ~> g
viewF forall {k} (f :: k -> *) (g :: k -> *). (f :+: g) <~> Sum f g
sumSum forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall {k} (f :: k -> *). Step f ~> (f :+: Step f)
stepDown
    splittingLB :: forall (f :: * -> *).
ListBy Sum f <~> (V1 :+: Sum f (ListBy Sum f))
splittingLB = forall {k} (f :: k -> *). Step f <~> (f :+: Step f)
stepping
                forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall {k} (f :: k -> *). f <~> (V1 :+: f)
sumLeftIdentity
                forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall {k} (t :: (k -> *) -> (k -> *) -> k -> *) (f :: k -> *)
       (f' :: k -> *) (g :: k -> *) (g' :: k -> *).
HBifunctor t =>
(f <~> f') -> (g <~> g') -> t f g <~> t f' g'
overHBifunctor forall a. a -> a
id forall {k} (f :: k -> *) (g :: k -> *). (f :+: g) <~> Sum f g
sumSum

    toListBy :: forall (f :: * -> *). Sum f f ~> ListBy Sum f
toListBy  = \case
      InL f x
x -> forall {k} (f :: k -> *) (a :: k). Natural -> f a -> Step f a
Step Natural
0 f x
x
      InR f x
x -> forall {k} (f :: k -> *) (a :: k). Natural -> f a -> Step f a
Step Natural
1 f x
x

-- | All functors are monoids in the monoidal category on 'Sum'.
instance MonoidIn Sum V1 f where
    pureT :: V1 ~> f
pureT = forall {k} (a :: k) (f :: k -> *). V1 a -> f a
absurd1


instance Tensor These1 V1 where
    type ListBy These1 = Steps
    intro1 :: forall (f :: * -> *). f ~> These1 f V1
intro1 = forall (f :: * -> *) (g :: * -> *) a. f a -> These1 f g a
This1
    intro2 :: forall (g :: * -> *). g ~> These1 V1 g
intro2 = forall (f :: * -> *) (g :: * -> *) a. g a -> These1 f g a
That1
    elim1 :: forall (f :: * -> *). FunctorBy These1 f => These1 f V1 ~> f
elim1  = \case
      This1  f x
x   -> f x
x
      That1    V1 x
y -> forall {k} (a :: k) (f :: k -> *). V1 a -> f a
absurd1 V1 x
y
      These1 f x
_ V1 x
y -> forall {k} (a :: k) (f :: k -> *). V1 a -> f a
absurd1 V1 x
y
    elim2 :: forall (g :: * -> *). FunctorBy These1 g => These1 V1 g ~> g
elim2 = \case
      This1  V1 x
x   -> forall {k} (a :: k) (f :: k -> *). V1 a -> f a
absurd1 V1 x
x
      That1    g x
y -> g x
y
      These1 V1 x
x g x
_ -> forall {k} (a :: k) (f :: k -> *). V1 a -> f a
absurd1 V1 x
x

    appendLB :: forall (f :: * -> *).
These1 (ListBy These1 f) (ListBy These1 f) ~> ListBy These1 f
appendLB    = \case
      This1  ListBy These1 f x
x   -> ListBy These1 f x
x
      That1    ListBy These1 f x
y -> forall (f :: * -> *). These1 f (Steps f) ~> Steps f
stepsUp forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (f :: * -> *) (g :: * -> *) a. g a -> These1 f g a
That1 forall a b. (a -> b) -> a -> b
$ ListBy These1 f x
y
      These1 ListBy These1 f x
x ListBy These1 f x
y -> ListBy These1 f x
x forall a. Semigroup a => a -> a -> a
<> ListBy These1 f x
y
    splitNE :: forall (f :: * -> *).
NonEmptyBy These1 f ~> These1 f (ListBy These1 f)
splitNE     = forall (f :: * -> *). Steps f ~> These1 f (Steps f)
stepsDown forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall {k} (f :: k -> *) (a :: k). Flagged f a -> f a
flaggedVal forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (f :: (* -> *) -> * -> *) (g :: (* -> *) -> * -> *)
       (m :: * -> *) a.
ComposeT f g m a -> f (g m) a
getComposeT
    splittingLB :: forall (f :: * -> *).
ListBy These1 f <~> (V1 :+: These1 f (ListBy These1 f))
splittingLB = forall (f :: * -> *). Steps f <~> These1 f (Steps f)
steppings forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall {k} (f :: k -> *). f <~> (V1 :+: f)
sumLeftIdentity

    toListBy :: forall (f :: * -> *). These1 f f ~> ListBy These1 f
toListBy  = \case
      This1  f x
x   -> forall {k} (f :: k -> *) (a :: k). NEMap Natural (f a) -> Steps f a
Steps forall a b. (a -> b) -> a -> b
$ forall k a. k -> a -> NEMap k a
NEM.singleton Natural
0 f x
x
      That1    f x
y -> forall {k} (f :: k -> *) (a :: k). NEMap Natural (f a) -> Steps f a
Steps forall a b. (a -> b) -> a -> b
$ forall k a. k -> a -> NEMap k a
NEM.singleton Natural
1 f x
y
      These1 f x
x f x
y -> forall {k} (f :: k -> *) (a :: k). NEMap Natural (f a) -> Steps f a
Steps forall a b. (a -> b) -> a -> b
$ forall k a. NonEmpty (k, a) -> NEMap k a
NEM.fromDistinctAscList ((Natural
0, f x
x) forall a. a -> [a] -> NonEmpty a
:| [(Natural
1, f x
y)])

instance Alt f => MonoidIn These1 V1 f where
    pureT :: V1 ~> f
pureT = forall {k} (a :: k) (f :: k -> *). V1 a -> f a
absurd1

instance Tensor Comp Identity where
    type ListBy Comp = Free

    intro1 :: forall (f :: * -> *). f ~> Comp f Identity
intro1 = (forall {k} (f :: * -> *) (g :: k -> *) (a :: k) x.
f x -> (x -> g a) -> Comp f g a
:>>= forall a. a -> Identity a
Identity)
    intro2 :: forall (g :: * -> *). g ~> Comp Identity g
intro2 = (forall a. a -> Identity a
Identity () forall {k} (f :: * -> *) (g :: k -> *) (a :: k) x.
f x -> (x -> g a) -> Comp f g a
:>>=) forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a b. a -> b -> a
const

    elim1 :: forall (f :: * -> *). FunctorBy Comp f => Comp f Identity ~> f
elim1 (f x
x :>>= x -> Identity x
y) = forall a. Identity a -> a
runIdentity forall b c a. (b -> c) -> (a -> b) -> a -> c
. x -> Identity x
y forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> f x
x
    elim2 :: forall (g :: * -> *). FunctorBy Comp g => Comp Identity g ~> g
elim2 (Identity x
x :>>= x -> g x
y) = x -> g x
y (forall a. Identity a -> a
runIdentity Identity x
x)

    appendLB :: forall (f :: * -> *).
Comp (ListBy Comp f) (ListBy Comp f) ~> ListBy Comp f
appendLB (ListBy Comp f x
x :>>= x -> ListBy Comp f x
y) = ListBy Comp f x
x forall (m :: * -> *) a b. Monad m => m a -> (a -> m b) -> m b
>>= x -> ListBy Comp f x
y
    splitNE :: forall (f :: * -> *). NonEmptyBy Comp f ~> Comp f (ListBy Comp f)
splitNE             = forall (f :: * -> *). Free1 f ~> Comp f (Free f)
free1Comp
    splittingLB :: forall (f :: * -> *).
ListBy Comp f <~> (Identity :+: Comp f (ListBy Comp f))
splittingLB = forall {k} (f :: k -> *) (g :: k -> *).
(f ~> g) -> (g ~> f) -> f <~> g
isoF forall (f :: * -> *). Free f ~> (Identity :+: Comp f (Free f))
to_ forall (f :: * -> *). (Identity :+: Comp f (Free f)) ~> Free f
from_
      where
        to_ :: Free f ~> Identity :+: Comp f (Free f)
        to_ :: forall (f :: * -> *). Free f ~> (Identity :+: Comp f (Free f))
to_ = forall a r (f :: * -> *).
(a -> r) -> (forall s. f s -> (s -> r) -> r) -> Free f a -> r
foldFree' (forall k (f :: k -> *) (g :: k -> *) (p :: k). f p -> (:+:) f g p
L1 forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. a -> Identity a
Identity) forall a b. (a -> b) -> a -> b
$ \f s
y s -> (:+:) Identity (Comp f (Free f)) x
n -> forall k (f :: k -> *) (g :: k -> *) (p :: k). g p -> (:+:) f g p
R1 forall a b. (a -> b) -> a -> b
$
            f s
y forall {k} (f :: * -> *) (g :: k -> *) (a :: k) x.
f x -> (x -> g a) -> Comp f g a
:>>= (forall (f :: * -> *). (Identity :+: Comp f (Free f)) ~> Free f
from_ forall b c a. (b -> c) -> (a -> b) -> a -> c
. s -> (:+:) Identity (Comp f (Free f)) x
n)
        from_ :: Identity :+: Comp f (Free f) ~> Free f
        from_ :: forall (f :: * -> *). (Identity :+: Comp f (Free f)) ~> Free f
from_ = forall (f :: * -> *). Applicative f => Identity ~> f
generalize
            forall (t :: (* -> *) -> (* -> *) -> * -> *) (h :: * -> *)
       (f :: * -> *) (g :: * -> *).
SemigroupIn t h =>
(f ~> h) -> (g ~> h) -> t f g ~> h
!*! (\case f x
x :>>= x -> Free f x
f -> forall (f :: * -> *). f ~> Free f
liftFree f x
x forall (m :: * -> *) a b. Monad m => m a -> (a -> m b) -> m b
>>= x -> Free f x
f)

    toListBy :: forall (f :: * -> *). Comp f f ~> ListBy Comp f
toListBy (f x
x :>>= x -> f x
y) = forall (f :: * -> *). f ~> Free f
liftFree f x
x forall (m :: * -> *) a b. Monad m => m a -> (a -> m b) -> m b
>>= (forall {k} (t :: (k -> *) -> k -> *) (f :: k -> *).
Inject t =>
f ~> t f
inject forall b c a. (b -> c) -> (a -> b) -> a -> c
. x -> f x
y)

-- | Instances of 'Monad' are monoids in the monoidal category on
-- 'Comp'.
--
-- This instance is the "proof" that "monads are the monoids in the
-- category of endofunctors (enriched with 'Comp')"
--
-- Note that because of typeclass constraints, this requires 'Bind' as
-- well as 'Monad'.  But, you can get a "local" instance of 'Apply'
-- for any 'Monad' using
-- 'Data.Functor.Combinators.Unsafe.unsafeBind'.
instance (Bind f, Monad f) => MonoidIn Comp Identity f where
    pureT :: Identity ~> f
pureT           = forall (f :: * -> *). Applicative f => Identity ~> f
generalize

instance Matchable (:*:) Proxy where
    unsplitNE :: forall (f :: * -> *).
FunctorBy (:*:) f =>
(f :*: ListBy (:*:) f) ~> NonEmptyBy (:*:) f
unsplitNE = forall {k} (f :: k -> *) (a :: k).
(:*:) f (ListF f) a -> NonEmptyF f a
ProdNonEmpty
    matchLB :: forall (f :: * -> *).
FunctorBy (:*:) f =>
ListBy (:*:) f ~> (Proxy :+: NonEmptyBy (:*:) f)
matchLB   = forall {k} (f :: k -> *). ListF f ~> (Proxy :+: NonEmptyF f)
fromListF

instance Matchable Product Proxy where
    unsplitNE :: forall (f :: * -> *).
FunctorBy Product f =>
Product f (ListBy Product f) ~> NonEmptyBy Product f
unsplitNE = forall {k} (f :: k -> *) (a :: k).
(:*:) f (ListF f) a -> NonEmptyF f a
ProdNonEmpty forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall {k} (f :: k -> *) (g :: k -> *). (f <~> g) -> g ~> f
reviewF forall {k} (f :: k -> *) (g :: k -> *). (f :*: g) <~> Product f g
prodProd
    matchLB :: forall (f :: * -> *).
FunctorBy Product f =>
ListBy Product f ~> (Proxy :+: NonEmptyBy Product f)
matchLB   = forall {k} (f :: k -> *). ListF f ~> (Proxy :+: NonEmptyF f)
fromListF

instance Matchable Day Identity where
    unsplitNE :: forall (f :: * -> *).
FunctorBy Day f =>
Day f (ListBy Day f) ~> NonEmptyBy Day f
unsplitNE = forall (f :: * -> *) a. Day f (Ap f) a -> Ap1 f a
DayAp1
    matchLB :: forall (f :: * -> *).
FunctorBy Day f =>
ListBy Day f ~> (Identity :+: NonEmptyBy Day f)
matchLB   = forall (f :: * -> *). Ap f ~> (Identity :+: Ap1 f)
fromAp

-- | Instances of 'Conclude' are monoids in the monoidal category on 'Night'.
--
-- @since 0.3.0.0
instance Matchable CD.Day Proxy where
    unsplitNE :: forall (f :: * -> *).
FunctorBy Day f =>
Day f (ListBy Day f) ~> NonEmptyBy Day f
unsplitNE (CD.Day f b
x (Div [Coyoneda f c]
xs) x -> (b, c)
f) = forall (f :: * -> *) a. NonEmpty (Coyoneda f a) -> Div1 f a
Div1 forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall {k} (f :: k -> *) (a :: k). NonEmptyF f a -> NonEmpty (f a)
runNonEmptyF forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
(Matchable t i, FunctorBy t f) =>
t f (ListBy t f) ~> NonEmptyBy t f
unsplitNE forall a b. (a -> b) -> a -> b
$
      forall a b (f :: * -> *). (a -> b) -> f b -> Coyoneda f a
CCY.Coyoneda (forall a b. (a, b) -> a
fst forall b c a. (b -> c) -> (a -> b) -> a -> c
. x -> (b, c)
f) f b
x forall k (f :: k -> *) (g :: k -> *) (p :: k).
f p -> g p -> (:*:) f g p
:*: forall (f :: * -> *) a' a.
Contravariant f =>
(a' -> a) -> f a -> f a'
contramap (forall a b. (a, b) -> b
snd forall b c a. (b -> c) -> (a -> b) -> a -> c
. x -> (b, c)
f) (forall {k} (f :: k -> *) (a :: k). [f a] -> ListF f a
ListF [Coyoneda f c]
xs)
    matchLB :: forall (f :: * -> *).
FunctorBy Day f =>
ListBy Day f ~> (Proxy :+: NonEmptyBy Day f)
matchLB = forall k (t :: (k -> *) -> (k -> *) -> k -> *) (g :: k -> *)
       (l :: k -> *) (f :: k -> *).
HBifunctor t =>
(g ~> l) -> t f g ~> t f l
hright (forall (f :: * -> *) a. NonEmpty (Coyoneda f a) -> Div1 f a
Div1 forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall {k} (f :: k -> *) (a :: k). NonEmptyF f a -> NonEmpty (f a)
runNonEmptyF) forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
(Matchable t i, FunctorBy t f) =>
ListBy t f ~> (i :+: NonEmptyBy t f)
matchLB @(:*:) forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall {k} (f :: k -> *) (a :: k). [f a] -> ListF f a
ListF forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (f :: * -> *) a. Div f a -> [Coyoneda f a]
unDiv

-- | @since 0.3.0.0
instance Matchable Night Not where
    unsplitNE :: forall (f :: * -> *).
FunctorBy Night f =>
Night f (ListBy Night f) ~> NonEmptyBy Night f
unsplitNE (Night f b1
x ListBy Night f c1
xs x -> Either b1 c1
f) = forall b b1 c (a :: * -> *).
(b -> Either b1 c) -> a b1 -> Dec a c -> Dec1 a b
Dec1 x -> Either b1 c1
f f b1
x ListBy Night f c1
xs
    matchLB :: forall (f :: * -> *).
FunctorBy Night f =>
ListBy Night f ~> (Not :+: NonEmptyBy Night f)
matchLB = \case
      Lose   x -> Void
f      -> forall k (f :: k -> *) (g :: k -> *) (p :: k). f p -> (:+:) f g p
L1 (forall a. (a -> Void) -> Not a
Not x -> Void
f)
      Choose x -> Either b1 c
f f b1
x Dec f c
xs -> forall k (f :: k -> *) (g :: k -> *) (p :: k). g p -> (:+:) f g p
R1 (forall b b1 c (a :: * -> *).
(b -> Either b1 c) -> a b1 -> Dec a c -> Dec1 a b
Dec1 x -> Either b1 c
f f b1
x Dec f c
xs)

instance Matchable (:+:) V1 where
    unsplitNE :: forall (f :: * -> *).
FunctorBy (:+:) f =>
(f :+: ListBy (:+:) f) ~> NonEmptyBy (:+:) f
unsplitNE   = forall {k} (f :: k -> *). (f :+: Step f) ~> Step f
stepUp
    matchLB :: forall (f :: * -> *).
FunctorBy (:+:) f =>
ListBy (:+:) f ~> (V1 :+: NonEmptyBy (:+:) f)
matchLB     = forall k (f :: k -> *) (g :: k -> *) (p :: k). g p -> (:+:) f g p
R1

instance Matchable Sum V1 where
    unsplitNE :: forall (f :: * -> *).
FunctorBy Sum f =>
Sum f (ListBy Sum f) ~> NonEmptyBy Sum f
unsplitNE   = forall {k} (f :: k -> *). (f :+: Step f) ~> Step f
stepUp forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall {k} (f :: k -> *) (g :: k -> *). (f <~> g) -> g ~> f
reviewF forall {k} (f :: k -> *) (g :: k -> *). (f :+: g) <~> Sum f g
sumSum
    matchLB :: forall (f :: * -> *).
FunctorBy Sum f =>
ListBy Sum f ~> (V1 :+: NonEmptyBy Sum f)
matchLB     = forall k (f :: k -> *) (g :: k -> *) (p :: k). g p -> (:+:) f g p
R1

-- We can't write this until we get an isomorphism between MF These1 and SF These1
-- instance Matchable These1 where
--     unsplitNE = stepsUp
--     matchLB   = R1

-- | A newtype wrapper meant to be used to define polymorphic 'MonoidIn'
-- instances.  See documentation for 'MonoidIn' for more information.
--
-- Please do not ever define an instance of 'MonoidIn' "naked" on the
-- third parameter:
--
-- @
-- instance MonidIn (WrapHBF t) (WrapF i) f
-- @
--
-- As that would globally ruin everything using 'WrapHBF'.
newtype WrapF f a = WrapF { forall {k} (f :: k -> *) (a :: k). WrapF f a -> f a
unwrapF :: f a }
  deriving (Int -> WrapF f a -> ShowS
forall a.
(Int -> a -> ShowS) -> (a -> String) -> ([a] -> ShowS) -> Show a
forall k (f :: k -> *) (a :: k).
Show (f a) =>
Int -> WrapF f a -> ShowS
forall k (f :: k -> *) (a :: k). Show (f a) => [WrapF f a] -> ShowS
forall k (f :: k -> *) (a :: k). Show (f a) => WrapF f a -> String
showList :: [WrapF f a] -> ShowS
$cshowList :: forall k (f :: k -> *) (a :: k). Show (f a) => [WrapF f a] -> ShowS
show :: WrapF f a -> String
$cshow :: forall k (f :: k -> *) (a :: k). Show (f a) => WrapF f a -> String
showsPrec :: Int -> WrapF f a -> ShowS
$cshowsPrec :: forall k (f :: k -> *) (a :: k).
Show (f a) =>
Int -> WrapF f a -> ShowS
Show, ReadPrec [WrapF f a]
ReadPrec (WrapF f a)
ReadS [WrapF f a]
forall a.
(Int -> ReadS a)
-> ReadS [a] -> ReadPrec a -> ReadPrec [a] -> Read a
forall k (f :: k -> *) (a :: k). Read (f a) => ReadPrec [WrapF f a]
forall k (f :: k -> *) (a :: k). Read (f a) => ReadPrec (WrapF f a)
forall k (f :: k -> *) (a :: k).
Read (f a) =>
Int -> ReadS (WrapF f a)
forall k (f :: k -> *) (a :: k). Read (f a) => ReadS [WrapF f a]
readListPrec :: ReadPrec [WrapF f a]
$creadListPrec :: forall k (f :: k -> *) (a :: k). Read (f a) => ReadPrec [WrapF f a]
readPrec :: ReadPrec (WrapF f a)
$creadPrec :: forall k (f :: k -> *) (a :: k). Read (f a) => ReadPrec (WrapF f a)
readList :: ReadS [WrapF f a]
$creadList :: forall k (f :: k -> *) (a :: k). Read (f a) => ReadS [WrapF f a]
readsPrec :: Int -> ReadS (WrapF f a)
$creadsPrec :: forall k (f :: k -> *) (a :: k).
Read (f a) =>
Int -> ReadS (WrapF f a)
Read, WrapF f a -> WrapF f a -> Bool
forall a. (a -> a -> Bool) -> (a -> a -> Bool) -> Eq a
forall k (f :: k -> *) (a :: k).
Eq (f a) =>
WrapF f a -> WrapF f a -> Bool
/= :: WrapF f a -> WrapF f a -> Bool
$c/= :: forall k (f :: k -> *) (a :: k).
Eq (f a) =>
WrapF f a -> WrapF f a -> Bool
== :: WrapF f a -> WrapF f a -> Bool
$c== :: forall k (f :: k -> *) (a :: k).
Eq (f a) =>
WrapF f a -> WrapF f a -> Bool
Eq, WrapF f a -> WrapF f a -> Bool
WrapF f a -> WrapF f a -> Ordering
WrapF f a -> WrapF f a -> WrapF f a
forall a.
Eq a
-> (a -> a -> Ordering)
-> (a -> a -> Bool)
-> (a -> a -> Bool)
-> (a -> a -> Bool)
-> (a -> a -> Bool)
-> (a -> a -> a)
-> (a -> a -> a)
-> Ord a
forall {k} {f :: k -> *} {a :: k}. Ord (f a) => Eq (WrapF f a)
forall k (f :: k -> *) (a :: k).
Ord (f a) =>
WrapF f a -> WrapF f a -> Bool
forall k (f :: k -> *) (a :: k).
Ord (f a) =>
WrapF f a -> WrapF f a -> Ordering
forall k (f :: k -> *) (a :: k).
Ord (f a) =>
WrapF f a -> WrapF f a -> WrapF f a
min :: WrapF f a -> WrapF f a -> WrapF f a
$cmin :: forall k (f :: k -> *) (a :: k).
Ord (f a) =>
WrapF f a -> WrapF f a -> WrapF f a
max :: WrapF f a -> WrapF f a -> WrapF f a
$cmax :: forall k (f :: k -> *) (a :: k).
Ord (f a) =>
WrapF f a -> WrapF f a -> WrapF f a
>= :: WrapF f a -> WrapF f a -> Bool
$c>= :: forall k (f :: k -> *) (a :: k).
Ord (f a) =>
WrapF f a -> WrapF f a -> Bool
> :: WrapF f a -> WrapF f a -> Bool
$c> :: forall k (f :: k -> *) (a :: k).
Ord (f a) =>
WrapF f a -> WrapF f a -> Bool
<= :: WrapF f a -> WrapF f a -> Bool
$c<= :: forall k (f :: k -> *) (a :: k).
Ord (f a) =>
WrapF f a -> WrapF f a -> Bool
< :: WrapF f a -> WrapF f a -> Bool
$c< :: forall k (f :: k -> *) (a :: k).
Ord (f a) =>
WrapF f a -> WrapF f a -> Bool
compare :: WrapF f a -> WrapF f a -> Ordering
$ccompare :: forall k (f :: k -> *) (a :: k).
Ord (f a) =>
WrapF f a -> WrapF f a -> Ordering
Ord, forall a b. a -> WrapF f b -> WrapF f a
forall a b. (a -> b) -> WrapF f a -> WrapF f b
forall (f :: * -> *) a b. Functor f => a -> WrapF f b -> WrapF f a
forall (f :: * -> *) a b.
Functor f =>
(a -> b) -> WrapF f a -> WrapF f b
forall (f :: * -> *).
(forall a b. (a -> b) -> f a -> f b)
-> (forall a b. a -> f b -> f a) -> Functor f
<$ :: forall a b. a -> WrapF f b -> WrapF f a
$c<$ :: forall (f :: * -> *) a b. Functor f => a -> WrapF f b -> WrapF f a
fmap :: forall a b. (a -> b) -> WrapF f a -> WrapF f b
$cfmap :: forall (f :: * -> *) a b.
Functor f =>
(a -> b) -> WrapF f a -> WrapF f b
Functor, forall a. Eq a => a -> WrapF f a -> Bool
forall a. Num a => WrapF f a -> a
forall a. Ord a => WrapF f a -> a
forall m. Monoid m => WrapF f m -> m
forall a. WrapF f a -> Bool
forall a. WrapF f a -> Int
forall a. WrapF f a -> [a]
forall a. (a -> a -> a) -> WrapF f a -> a
forall m a. Monoid m => (a -> m) -> WrapF f a -> m
forall b a. (b -> a -> b) -> b -> WrapF f a -> b
forall a b. (a -> b -> b) -> b -> WrapF f a -> b
forall (f :: * -> *) a.
(Foldable f, Eq a) =>
a -> WrapF f a -> Bool
forall (f :: * -> *) a. (Foldable f, Num a) => WrapF f a -> a
forall (f :: * -> *) a. (Foldable f, Ord a) => WrapF f a -> a
forall (f :: * -> *) m. (Foldable f, Monoid m) => WrapF f m -> m
forall (f :: * -> *) a. Foldable f => WrapF f a -> Bool
forall (f :: * -> *) a. Foldable f => WrapF f a -> Int
forall (f :: * -> *) a. Foldable f => WrapF f a -> [a]
forall (f :: * -> *) a.
Foldable f =>
(a -> a -> a) -> WrapF f a -> a
forall (f :: * -> *) m a.
(Foldable f, Monoid m) =>
(a -> m) -> WrapF f a -> m
forall (f :: * -> *) b a.
Foldable f =>
(b -> a -> b) -> b -> WrapF f a -> b
forall (f :: * -> *) a b.
Foldable f =>
(a -> b -> b) -> b -> WrapF f a -> b
forall (t :: * -> *).
(forall m. Monoid m => t m -> m)
-> (forall m a. Monoid m => (a -> m) -> t a -> m)
-> (forall m a. Monoid m => (a -> m) -> t a -> m)
-> (forall a b. (a -> b -> b) -> b -> t a -> b)
-> (forall a b. (a -> b -> b) -> b -> t a -> b)
-> (forall b a. (b -> a -> b) -> b -> t a -> b)
-> (forall b a. (b -> a -> b) -> b -> t a -> b)
-> (forall a. (a -> a -> a) -> t a -> a)
-> (forall a. (a -> a -> a) -> t a -> a)
-> (forall a. t a -> [a])
-> (forall a. t a -> Bool)
-> (forall a. t a -> Int)
-> (forall a. Eq a => a -> t a -> Bool)
-> (forall a. Ord a => t a -> a)
-> (forall a. Ord a => t a -> a)
-> (forall a. Num a => t a -> a)
-> (forall a. Num a => t a -> a)
-> Foldable t
product :: forall a. Num a => WrapF f a -> a
$cproduct :: forall (f :: * -> *) a. (Foldable f, Num a) => WrapF f a -> a
sum :: forall a. Num a => WrapF f a -> a
$csum :: forall (f :: * -> *) a. (Foldable f, Num a) => WrapF f a -> a
minimum :: forall a. Ord a => WrapF f a -> a
$cminimum :: forall (f :: * -> *) a. (Foldable f, Ord a) => WrapF f a -> a
maximum :: forall a. Ord a => WrapF f a -> a
$cmaximum :: forall (f :: * -> *) a. (Foldable f, Ord a) => WrapF f a -> a
elem :: forall a. Eq a => a -> WrapF f a -> Bool
$celem :: forall (f :: * -> *) a.
(Foldable f, Eq a) =>
a -> WrapF f a -> Bool
length :: forall a. WrapF f a -> Int
$clength :: forall (f :: * -> *) a. Foldable f => WrapF f a -> Int
null :: forall a. WrapF f a -> Bool
$cnull :: forall (f :: * -> *) a. Foldable f => WrapF f a -> Bool
toList :: forall a. WrapF f a -> [a]
$ctoList :: forall (f :: * -> *) a. Foldable f => WrapF f a -> [a]
foldl1 :: forall a. (a -> a -> a) -> WrapF f a -> a
$cfoldl1 :: forall (f :: * -> *) a.
Foldable f =>
(a -> a -> a) -> WrapF f a -> a
foldr1 :: forall a. (a -> a -> a) -> WrapF f a -> a
$cfoldr1 :: forall (f :: * -> *) a.
Foldable f =>
(a -> a -> a) -> WrapF f a -> a
foldl' :: forall b a. (b -> a -> b) -> b -> WrapF f a -> b
$cfoldl' :: forall (f :: * -> *) b a.
Foldable f =>
(b -> a -> b) -> b -> WrapF f a -> b
foldl :: forall b a. (b -> a -> b) -> b -> WrapF f a -> b
$cfoldl :: forall (f :: * -> *) b a.
Foldable f =>
(b -> a -> b) -> b -> WrapF f a -> b
foldr' :: forall a b. (a -> b -> b) -> b -> WrapF f a -> b
$cfoldr' :: forall (f :: * -> *) a b.
Foldable f =>
(a -> b -> b) -> b -> WrapF f a -> b
foldr :: forall a b. (a -> b -> b) -> b -> WrapF f a -> b
$cfoldr :: forall (f :: * -> *) a b.
Foldable f =>
(a -> b -> b) -> b -> WrapF f a -> b
foldMap' :: forall m a. Monoid m => (a -> m) -> WrapF f a -> m
$cfoldMap' :: forall (f :: * -> *) m a.
(Foldable f, Monoid m) =>
(a -> m) -> WrapF f a -> m
foldMap :: forall m a. Monoid m => (a -> m) -> WrapF f a -> m
$cfoldMap :: forall (f :: * -> *) m a.
(Foldable f, Monoid m) =>
(a -> m) -> WrapF f a -> m
fold :: forall m. Monoid m => WrapF f m -> m
$cfold :: forall (f :: * -> *) m. (Foldable f, Monoid m) => WrapF f m -> m
Foldable, forall (t :: * -> *).
Functor t
-> Foldable t
-> (forall (f :: * -> *) a b.
    Applicative f =>
    (a -> f b) -> t a -> f (t b))
-> (forall (f :: * -> *) a. Applicative f => t (f a) -> f (t a))
-> (forall (m :: * -> *) a b.
    Monad m =>
    (a -> m b) -> t a -> m (t b))
-> (forall (m :: * -> *) a. Monad m => t (m a) -> m (t a))
-> Traversable t
forall {f :: * -> *}. Traversable f => Functor (WrapF f)
forall {f :: * -> *}. Traversable f => Foldable (WrapF f)
forall (f :: * -> *) (m :: * -> *) a.
(Traversable f, Monad m) =>
WrapF f (m a) -> m (WrapF f a)
forall (f :: * -> *) (f :: * -> *) a.
(Traversable f, Applicative f) =>
WrapF f (f a) -> f (WrapF f a)
forall (f :: * -> *) (m :: * -> *) a b.
(Traversable f, Monad m) =>
(a -> m b) -> WrapF f a -> m (WrapF f b)
forall (f :: * -> *) (f :: * -> *) a b.
(Traversable f, Applicative f) =>
(a -> f b) -> WrapF f a -> f (WrapF f b)
forall (f :: * -> *) a b.
Applicative f =>
(a -> f b) -> WrapF f a -> f (WrapF f b)
sequence :: forall (m :: * -> *) a. Monad m => WrapF f (m a) -> m (WrapF f a)
$csequence :: forall (f :: * -> *) (m :: * -> *) a.
(Traversable f, Monad m) =>
WrapF f (m a) -> m (WrapF f a)
mapM :: forall (m :: * -> *) a b.
Monad m =>
(a -> m b) -> WrapF f a -> m (WrapF f b)
$cmapM :: forall (f :: * -> *) (m :: * -> *) a b.
(Traversable f, Monad m) =>
(a -> m b) -> WrapF f a -> m (WrapF f b)
sequenceA :: forall (f :: * -> *) a.
Applicative f =>
WrapF f (f a) -> f (WrapF f a)
$csequenceA :: forall (f :: * -> *) (f :: * -> *) a.
(Traversable f, Applicative f) =>
WrapF f (f a) -> f (WrapF f a)
traverse :: forall (f :: * -> *) a b.
Applicative f =>
(a -> f b) -> WrapF f a -> f (WrapF f b)
$ctraverse :: forall (f :: * -> *) (f :: * -> *) a b.
(Traversable f, Applicative f) =>
(a -> f b) -> WrapF f a -> f (WrapF f b)
Traversable, Typeable, forall a.
(forall x. a -> Rep a x) -> (forall x. Rep a x -> a) -> Generic a
forall k (f :: k -> *) (a :: k) x. Rep (WrapF f a) x -> WrapF f a
forall k (f :: k -> *) (a :: k) x. WrapF f a -> Rep (WrapF f a) x
$cto :: forall k (f :: k -> *) (a :: k) x. Rep (WrapF f a) x -> WrapF f a
$cfrom :: forall k (f :: k -> *) (a :: k) x. WrapF f a -> Rep (WrapF f a) x
Generic, WrapF f a -> DataType
WrapF f a -> Constr
forall a.
Typeable a
-> (forall (c :: * -> *).
    (forall d b. Data d => c (d -> b) -> d -> c b)
    -> (forall g. g -> c g) -> a -> c a)
-> (forall (c :: * -> *).
    (forall b r. Data b => c (b -> r) -> c r)
    -> (forall r. r -> c r) -> Constr -> c a)
-> (a -> Constr)
-> (a -> DataType)
-> (forall (t :: * -> *) (c :: * -> *).
    Typeable t =>
    (forall d. Data d => c (t d)) -> Maybe (c a))
-> (forall (t :: * -> * -> *) (c :: * -> *).
    Typeable t =>
    (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c a))
-> ((forall b. Data b => b -> b) -> a -> a)
-> (forall r r'.
    (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> a -> r)
-> (forall r r'.
    (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> a -> r)
-> (forall u. (forall d. Data d => d -> u) -> a -> [u])
-> (forall u. Int -> (forall d. Data d => d -> u) -> a -> u)
-> (forall (m :: * -> *).
    Monad m =>
    (forall d. Data d => d -> m d) -> a -> m a)
-> (forall (m :: * -> *).
    MonadPlus m =>
    (forall d. Data d => d -> m d) -> a -> m a)
-> (forall (m :: * -> *).
    MonadPlus m =>
    (forall d. Data d => d -> m d) -> a -> m a)
-> Data a
forall {k} {f :: k -> *} {a :: k}.
(Typeable a, Typeable f, Typeable k, Data (f a)) =>
Typeable (WrapF f a)
forall k (f :: k -> *) (a :: k).
(Typeable a, Typeable f, Typeable k, Data (f a)) =>
WrapF f a -> DataType
forall k (f :: k -> *) (a :: k).
(Typeable a, Typeable f, Typeable k, Data (f a)) =>
WrapF f a -> Constr
forall k (f :: k -> *) (a :: k).
(Typeable a, Typeable f, Typeable k, Data (f a)) =>
(forall b. Data b => b -> b) -> WrapF f a -> WrapF f a
forall k (f :: k -> *) (a :: k) u.
(Typeable a, Typeable f, Typeable k, Data (f a)) =>
Int -> (forall d. Data d => d -> u) -> WrapF f a -> u
forall k (f :: k -> *) (a :: k) u.
(Typeable a, Typeable f, Typeable k, Data (f a)) =>
(forall d. Data d => d -> u) -> WrapF f a -> [u]
forall k (f :: k -> *) (a :: k) r r'.
(Typeable a, Typeable f, Typeable k, Data (f a)) =>
(r -> r' -> r)
-> r -> (forall d. Data d => d -> r') -> WrapF f a -> r
forall k (f :: k -> *) (a :: k) r r'.
(Typeable a, Typeable f, Typeable k, Data (f a)) =>
(r' -> r -> r)
-> r -> (forall d. Data d => d -> r') -> WrapF f a -> r
forall k (f :: k -> *) (a :: k) (m :: * -> *).
(Typeable a, Typeable f, Typeable k, Data (f a), Monad m) =>
(forall d. Data d => d -> m d) -> WrapF f a -> m (WrapF f a)
forall k (f :: k -> *) (a :: k) (m :: * -> *).
(Typeable a, Typeable f, Typeable k, Data (f a), MonadPlus m) =>
(forall d. Data d => d -> m d) -> WrapF f a -> m (WrapF f a)
forall k (f :: k -> *) (a :: k) (c :: * -> *).
(Typeable a, Typeable f, Typeable k, Data (f a)) =>
(forall b r. Data b => c (b -> r) -> c r)
-> (forall r. r -> c r) -> Constr -> c (WrapF f a)
forall k (f :: k -> *) (a :: k) (c :: * -> *).
(Typeable a, Typeable f, Typeable k, Data (f a)) =>
(forall d b. Data d => c (d -> b) -> d -> c b)
-> (forall g. g -> c g) -> WrapF f a -> c (WrapF f a)
forall k (f :: k -> *) (a :: k) (t :: * -> *) (c :: * -> *).
(Typeable a, Typeable f, Typeable k, Data (f a), Typeable t) =>
(forall d. Data d => c (t d)) -> Maybe (c (WrapF f a))
forall k (f :: k -> *) (a :: k) (t :: * -> * -> *) (c :: * -> *).
(Typeable a, Typeable f, Typeable k, Data (f a), Typeable t) =>
(forall d e. (Data d, Data e) => c (t d e))
-> Maybe (c (WrapF f a))
forall (c :: * -> *).
(forall b r. Data b => c (b -> r) -> c r)
-> (forall r. r -> c r) -> Constr -> c (WrapF f a)
forall (c :: * -> *).
(forall d b. Data d => c (d -> b) -> d -> c b)
-> (forall g. g -> c g) -> WrapF f a -> c (WrapF f a)
gmapMo :: forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> WrapF f a -> m (WrapF f a)
$cgmapMo :: forall k (f :: k -> *) (a :: k) (m :: * -> *).
(Typeable a, Typeable f, Typeable k, Data (f a), MonadPlus m) =>
(forall d. Data d => d -> m d) -> WrapF f a -> m (WrapF f a)
gmapMp :: forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> WrapF f a -> m (WrapF f a)
$cgmapMp :: forall k (f :: k -> *) (a :: k) (m :: * -> *).
(Typeable a, Typeable f, Typeable k, Data (f a), MonadPlus m) =>
(forall d. Data d => d -> m d) -> WrapF f a -> m (WrapF f a)
gmapM :: forall (m :: * -> *).
Monad m =>
(forall d. Data d => d -> m d) -> WrapF f a -> m (WrapF f a)
$cgmapM :: forall k (f :: k -> *) (a :: k) (m :: * -> *).
(Typeable a, Typeable f, Typeable k, Data (f a), Monad m) =>
(forall d. Data d => d -> m d) -> WrapF f a -> m (WrapF f a)
gmapQi :: forall u. Int -> (forall d. Data d => d -> u) -> WrapF f a -> u
$cgmapQi :: forall k (f :: k -> *) (a :: k) u.
(Typeable a, Typeable f, Typeable k, Data (f a)) =>
Int -> (forall d. Data d => d -> u) -> WrapF f a -> u
gmapQ :: forall u. (forall d. Data d => d -> u) -> WrapF f a -> [u]
$cgmapQ :: forall k (f :: k -> *) (a :: k) u.
(Typeable a, Typeable f, Typeable k, Data (f a)) =>
(forall d. Data d => d -> u) -> WrapF f a -> [u]
gmapQr :: forall r r'.
(r' -> r -> r)
-> r -> (forall d. Data d => d -> r') -> WrapF f a -> r
$cgmapQr :: forall k (f :: k -> *) (a :: k) r r'.
(Typeable a, Typeable f, Typeable k, Data (f a)) =>
(r' -> r -> r)
-> r -> (forall d. Data d => d -> r') -> WrapF f a -> r
gmapQl :: forall r r'.
(r -> r' -> r)
-> r -> (forall d. Data d => d -> r') -> WrapF f a -> r
$cgmapQl :: forall k (f :: k -> *) (a :: k) r r'.
(Typeable a, Typeable f, Typeable k, Data (f a)) =>
(r -> r' -> r)
-> r -> (forall d. Data d => d -> r') -> WrapF f a -> r
gmapT :: (forall b. Data b => b -> b) -> WrapF f a -> WrapF f a
$cgmapT :: forall k (f :: k -> *) (a :: k).
(Typeable a, Typeable f, Typeable k, Data (f a)) =>
(forall b. Data b => b -> b) -> WrapF f a -> WrapF f a
dataCast2 :: forall (t :: * -> * -> *) (c :: * -> *).
Typeable t =>
(forall d e. (Data d, Data e) => c (t d e))
-> Maybe (c (WrapF f a))
$cdataCast2 :: forall k (f :: k -> *) (a :: k) (t :: * -> * -> *) (c :: * -> *).
(Typeable a, Typeable f, Typeable k, Data (f a), Typeable t) =>
(forall d e. (Data d, Data e) => c (t d e))
-> Maybe (c (WrapF f a))
dataCast1 :: forall (t :: * -> *) (c :: * -> *).
Typeable t =>
(forall d. Data d => c (t d)) -> Maybe (c (WrapF f a))
$cdataCast1 :: forall k (f :: k -> *) (a :: k) (t :: * -> *) (c :: * -> *).
(Typeable a, Typeable f, Typeable k, Data (f a), Typeable t) =>
(forall d. Data d => c (t d)) -> Maybe (c (WrapF f a))
dataTypeOf :: WrapF f a -> DataType
$cdataTypeOf :: forall k (f :: k -> *) (a :: k).
(Typeable a, Typeable f, Typeable k, Data (f a)) =>
WrapF f a -> DataType
toConstr :: WrapF f a -> Constr
$ctoConstr :: forall k (f :: k -> *) (a :: k).
(Typeable a, Typeable f, Typeable k, Data (f a)) =>
WrapF f a -> Constr
gunfold :: forall (c :: * -> *).
(forall b r. Data b => c (b -> r) -> c r)
-> (forall r. r -> c r) -> Constr -> c (WrapF f a)
$cgunfold :: forall k (f :: k -> *) (a :: k) (c :: * -> *).
(Typeable a, Typeable f, Typeable k, Data (f a)) =>
(forall b r. Data b => c (b -> r) -> c r)
-> (forall r. r -> c r) -> Constr -> c (WrapF f a)
gfoldl :: forall (c :: * -> *).
(forall d b. Data d => c (d -> b) -> d -> c b)
-> (forall g. g -> c g) -> WrapF f a -> c (WrapF f a)
$cgfoldl :: forall k (f :: k -> *) (a :: k) (c :: * -> *).
(Typeable a, Typeable f, Typeable k, Data (f a)) =>
(forall d b. Data d => c (d -> b) -> d -> c b)
-> (forall g. g -> c g) -> WrapF f a -> c (WrapF f a)
Data)

instance Show1 f => Show1 (WrapF f) where
    liftShowsPrec :: forall a.
(Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> WrapF f a -> ShowS
liftShowsPrec Int -> a -> ShowS
sp [a] -> ShowS
sl Int
d (WrapF f a
x) = forall a. (Int -> a -> ShowS) -> String -> Int -> a -> ShowS
showsUnaryWith (forall (f :: * -> *) a.
Show1 f =>
(Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> f a -> ShowS
liftShowsPrec Int -> a -> ShowS
sp [a] -> ShowS
sl) String
"WrapF" Int
d f a
x

instance Eq1 f => Eq1 (WrapF f) where
    liftEq :: forall a b. (a -> b -> Bool) -> WrapF f a -> WrapF f b -> Bool
liftEq a -> b -> Bool
eq (WrapF f a
x) (WrapF f b
y) = forall (f :: * -> *) a b.
Eq1 f =>
(a -> b -> Bool) -> f a -> f b -> Bool
liftEq a -> b -> Bool
eq f a
x f b
y

instance Ord1 f => Ord1 (WrapF f) where
    liftCompare :: forall a b.
(a -> b -> Ordering) -> WrapF f a -> WrapF f b -> Ordering
liftCompare a -> b -> Ordering
c (WrapF f a
x) (WrapF f b
y) = forall (f :: * -> *) a b.
Ord1 f =>
(a -> b -> Ordering) -> f a -> f b -> Ordering
liftCompare a -> b -> Ordering
c f a
x f b
y

instance Tensor t i => Tensor (WrapHBF t) (WrapF i) where
    type ListBy (WrapHBF t) = ListBy t

    intro1 :: forall (f :: * -> *). f ~> WrapHBF t f (WrapF i)
intro1 = forall {k} {k1} {k2} (t :: k -> k1 -> k2 -> *) (f :: k) (g :: k1)
       (a :: k2).
t f g a -> WrapHBF t f g a
WrapHBF forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall k (t :: (k -> *) -> (k -> *) -> k -> *) (g :: k -> *)
       (l :: k -> *) (f :: k -> *).
HBifunctor t =>
(g ~> l) -> t f g ~> t f l
hright forall {k} (f :: k -> *) (a :: k). f a -> WrapF f a
WrapF forall b c a. (b -> c) -> (a -> b) -> a -> c
.  forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
Tensor t i =>
f ~> t f i
intro1
    intro2 :: forall (g :: * -> *). g ~> WrapHBF t (WrapF i) g
intro2 = forall {k} {k1} {k2} (t :: k -> k1 -> k2 -> *) (f :: k) (g :: k1)
       (a :: k2).
t f g a -> WrapHBF t f g a
WrapHBF forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall k (t :: (k -> *) -> (k -> *) -> k -> *) (f :: k -> *)
       (j :: k -> *) (g :: k -> *).
HBifunctor t =>
(f ~> j) -> t f g ~> t j g
hleft forall {k} (f :: k -> *) (a :: k). f a -> WrapF f a
WrapF forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (g :: * -> *).
Tensor t i =>
g ~> t i g
intro2
    elim1 :: forall (f :: * -> *).
FunctorBy (WrapHBF t) f =>
WrapHBF t f (WrapF i) ~> f
elim1 = forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
(Tensor t i, FunctorBy t f) =>
t f i ~> f
elim1 forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall k (t :: (k -> *) -> (k -> *) -> k -> *) (g :: k -> *)
       (l :: k -> *) (f :: k -> *).
HBifunctor t =>
(g ~> l) -> t f g ~> t f l
hright forall {k} (f :: k -> *) (a :: k). WrapF f a -> f a
unwrapF forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall {k1} {k2} {k3} (t :: k1 -> k2 -> k3 -> *) (f :: k1)
       (g :: k2) (a :: k3).
WrapHBF t f g a -> t f g a
unwrapHBF
    elim2 :: forall (g :: * -> *).
FunctorBy (WrapHBF t) g =>
WrapHBF t (WrapF i) g ~> g
elim2 = forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (g :: * -> *).
(Tensor t i, FunctorBy t g) =>
t i g ~> g
elim2 forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall k (t :: (k -> *) -> (k -> *) -> k -> *) (f :: k -> *)
       (j :: k -> *) (g :: k -> *).
HBifunctor t =>
(f ~> j) -> t f g ~> t j g
hleft forall {k} (f :: k -> *) (a :: k). WrapF f a -> f a
unwrapF forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall {k1} {k2} {k3} (t :: k1 -> k2 -> k3 -> *) (f :: k1)
       (g :: k2) (a :: k3).
WrapHBF t f g a -> t f g a
unwrapHBF
    appendLB :: forall (f :: * -> *).
WrapHBF t (ListBy (WrapHBF t) f) (ListBy (WrapHBF t) f)
~> ListBy (WrapHBF t) f
appendLB = forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
Tensor t i =>
t (ListBy t f) (ListBy t f) ~> ListBy t f
appendLB forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall {k1} {k2} {k3} (t :: k1 -> k2 -> k3 -> *) (f :: k1)
       (g :: k2) (a :: k3).
WrapHBF t f g a -> t f g a
unwrapHBF
    splitNE :: forall (f :: * -> *).
NonEmptyBy (WrapHBF t) f ~> WrapHBF t f (ListBy (WrapHBF t) f)
splitNE = forall {k} {k1} {k2} (t :: k -> k1 -> k2 -> *) (f :: k) (g :: k1)
       (a :: k2).
t f g a -> WrapHBF t f g a
WrapHBF forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
Tensor t i =>
NonEmptyBy t f ~> t f (ListBy t f)
splitNE
    splittingLB :: forall (f :: * -> *).
ListBy (WrapHBF t) f
<~> (WrapF i :+: WrapHBF t f (ListBy (WrapHBF t) f))
splittingLB = forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
Tensor t i =>
ListBy t f <~> (i :+: t f (ListBy t f))
splittingLB @t
                forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall {k} (t :: (k -> *) -> (k -> *) -> k -> *) (f :: k -> *)
       (f' :: k -> *) (g :: k -> *) (g' :: k -> *).
HBifunctor t =>
(f <~> f') -> (g <~> g') -> t f g <~> t f' g'
overHBifunctor (forall {k} (f :: k -> *) (g :: k -> *).
(f ~> g) -> (g ~> f) -> f <~> g
isoF forall {k} (f :: k -> *) (a :: k). f a -> WrapF f a
WrapF forall {k} (f :: k -> *) (a :: k). WrapF f a -> f a
unwrapF) (forall {k} (f :: k -> *) (g :: k -> *).
(f ~> g) -> (g ~> f) -> f <~> g
isoF forall {k} {k1} {k2} (t :: k -> k1 -> k2 -> *) (f :: k) (g :: k1)
       (a :: k2).
t f g a -> WrapHBF t f g a
WrapHBF forall {k1} {k2} {k3} (t :: k1 -> k2 -> k3 -> *) (f :: k1)
       (g :: k2) (a :: k3).
WrapHBF t f g a -> t f g a
unwrapHBF)
    toListBy :: forall (f :: * -> *). WrapHBF t f f ~> ListBy (WrapHBF t) f
toListBy = forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
Tensor t i =>
t f f ~> ListBy t f
toListBy forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall {k1} {k2} {k3} (t :: k1 -> k2 -> k3 -> *) (f :: k1)
       (g :: k2) (a :: k3).
WrapHBF t f g a -> t f g a
unwrapHBF
    fromNE :: forall (f :: * -> *).
NonEmptyBy (WrapHBF t) f ~> ListBy (WrapHBF t) f
fromNE = forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
Tensor t i =>
NonEmptyBy t f ~> ListBy t f
fromNE @t

-- | Any @'ListBy' t f@ is a @'SemigroupIn' t@ and a @'MonoidIn' t i@, if we
-- have @'Tensor' t i@. This newtype wrapper witnesses that fact.  We
-- require a newtype wrapper to avoid overlapping instances.
newtype WrapLB t f a = WrapLB { forall (t :: (* -> *) -> (* -> *) -> * -> *) (f :: * -> *) a.
WrapLB t f a -> ListBy t f a
unwrapLB :: ListBy t f a }

instance Functor (ListBy t f) => Functor (WrapLB t f) where
    fmap :: forall a b. (a -> b) -> WrapLB t f a -> WrapLB t f b
fmap a -> b
f (WrapLB ListBy t f a
x) = forall (t :: (* -> *) -> (* -> *) -> * -> *) (f :: * -> *) a.
ListBy t f a -> WrapLB t f a
WrapLB (forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap a -> b
f ListBy t f a
x)

-- | @since 0.3.0.0
instance Contravariant (ListBy t f) => Contravariant (WrapLB t f) where
    contramap :: forall a' a. (a' -> a) -> WrapLB t f a -> WrapLB t f a'
contramap a' -> a
f (WrapLB ListBy t f a
x) = forall (t :: (* -> *) -> (* -> *) -> * -> *) (f :: * -> *) a.
ListBy t f a -> WrapLB t f a
WrapLB (forall (f :: * -> *) a' a.
Contravariant f =>
(a' -> a) -> f a -> f a'
contramap a' -> a
f ListBy t f a
x)

-- | @since 0.3.0.0
instance Invariant (ListBy t f) => Invariant (WrapLB t f) where
    invmap :: forall a b. (a -> b) -> (b -> a) -> WrapLB t f a -> WrapLB t f b
invmap a -> b
f b -> a
g (WrapLB ListBy t f a
x) = forall (t :: (* -> *) -> (* -> *) -> * -> *) (f :: * -> *) a.
ListBy t f a -> WrapLB t f a
WrapLB (forall (f :: * -> *) a b.
Invariant f =>
(a -> b) -> (b -> a) -> f a -> f b
invmap a -> b
f b -> a
g ListBy t f a
x)

instance (Tensor t i, FunctorBy t f, FunctorBy t (WrapLB t f)) => SemigroupIn (WrapHBF t) (WrapLB t f) where
    biretract :: WrapHBF t (WrapLB t f) (WrapLB t f) ~> WrapLB t f
biretract = forall (t :: (* -> *) -> (* -> *) -> * -> *) (f :: * -> *) a.
ListBy t f a -> WrapLB t f a
WrapLB forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
Tensor t i =>
t (ListBy t f) (ListBy t f) ~> ListBy t f
appendLB forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall k (t :: (k -> *) -> (k -> *) -> k -> *) (f :: k -> *)
       (j :: k -> *) (g :: k -> *) (l :: k -> *).
HBifunctor t =>
(f ~> j) -> (g ~> l) -> t f g ~> t j l
hbimap forall (t :: (* -> *) -> (* -> *) -> * -> *) (f :: * -> *) a.
WrapLB t f a -> ListBy t f a
unwrapLB forall (t :: (* -> *) -> (* -> *) -> * -> *) (f :: * -> *) a.
WrapLB t f a -> ListBy t f a
unwrapLB forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall {k1} {k2} {k3} (t :: k1 -> k2 -> k3 -> *) (f :: k1)
       (g :: k2) (a :: k3).
WrapHBF t f g a -> t f g a
unwrapHBF
    binterpret :: forall (g :: * -> *) (h :: * -> *).
(g ~> WrapLB t f)
-> (h ~> WrapLB t f) -> WrapHBF t g h ~> WrapLB t f
binterpret g ~> WrapLB t f
f h ~> WrapLB t f
g = forall (t :: (* -> *) -> (* -> *) -> * -> *) (f :: * -> *).
SemigroupIn t f =>
t f f ~> f
biretract forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall k (t :: (k -> *) -> (k -> *) -> k -> *) (f :: k -> *)
       (j :: k -> *) (g :: k -> *) (l :: k -> *).
HBifunctor t =>
(f ~> j) -> (g ~> l) -> t f g ~> t j l
hbimap g ~> WrapLB t f
f h ~> WrapLB t f
g

instance (Tensor t i, FunctorBy t f, FunctorBy t (WrapLB t f)) => MonoidIn (WrapHBF t) (WrapF i) (WrapLB t f) where
    pureT :: WrapF i ~> WrapLB t f
pureT = forall (t :: (* -> *) -> (* -> *) -> * -> *) (f :: * -> *) a.
ListBy t f a -> WrapLB t f a
WrapLB forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (t :: (* -> *) -> (* -> *) -> * -> *) (i :: * -> *)
       (f :: * -> *).
Tensor t i =>
i ~> ListBy t f
nilLB @t forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall {k} (f :: k -> *) (a :: k). WrapF f a -> f a
unwrapF