------------------------------------------------------------------------
-- The Agda standard library
--
-- The reader monad
------------------------------------------------------------------------

{-# OPTIONS --without-K --safe #-}

open import Level

module Category.Monad.Reader {r} (R : Set r) (a : Level) where

open import Function
open import Function.Identity.Categorical as Id using (Identity)
open import Category.Applicative.Indexed
open import Category.Monad.Indexed
open import Category.Monad
open import Data.Unit

private
  variable
     : Level
    A B I : Set 

------------------------------------------------------------------------
-- Indexed reader

IReaderT : IFun I (r  a)  IFun I (r  a)
IReaderT M i j A = R  M i j A

module _ {M : IFun I (r  a)} where

  ------------------------------------------------------------------------
  -- Indexed reader applicative

  ReaderTIApplicative : RawIApplicative M  RawIApplicative (IReaderT M)
  ReaderTIApplicative App = record
    { pure = λ x r  pure x
    ; _⊛_ = λ m n r  m r  n r
    } where open RawIApplicative App

  ReaderTIApplicativeZero : RawIApplicativeZero M 
                            RawIApplicativeZero (IReaderT M)
  ReaderTIApplicativeZero App = record
    { applicative = ReaderTIApplicative applicative
    ;  = const 
    } where open RawIApplicativeZero App

  ReaderTIAlternative : RawIAlternative M  RawIAlternative (IReaderT M)
  ReaderTIAlternative Alt = record
    { applicativeZero = ReaderTIApplicativeZero applicativeZero
    ; _∣_ = λ m n r  m r  n r
    } where open RawIAlternative Alt

  ------------------------------------------------------------------------
  -- Indexed reader monad

  ReaderTIMonad : RawIMonad M  RawIMonad (IReaderT M)
  ReaderTIMonad Mon = record
    { return = λ x r  return x
    ; _>>=_ = λ m f r  m r >>= flip f r
    } where open RawIMonad Mon

  ReaderTIMonadZero : RawIMonadZero M  RawIMonadZero (IReaderT M)
  ReaderTIMonadZero Mon = record
    { monad = ReaderTIMonad monad
    ; applicativeZero = ReaderTIApplicativeZero applicativeZero
    } where open RawIMonadZero Mon

  ReaderTIMonadPlus : RawIMonadPlus M  RawIMonadPlus (IReaderT M)
  ReaderTIMonadPlus Mon = record
    { monad = ReaderTIMonad monad
    ; alternative = ReaderTIAlternative alternative
    } where open RawIMonadPlus Mon

------------------------------------------------------------------------
-- Reader monad operations

record RawIMonadReader {I : Set } (M : IFun I (r  a))
                       : Set (  suc (r  a)) where
  field
    monad  : RawIMonad M
    reader :  {i}  (R  A)  M i i A
    local  :  {i j}  (R  R)  M i j A  M i j A

  open RawIMonad monad public

  ask :  {i}  M i i (Lift (r  a) R)
  ask = reader lift

  asks :  {i}  (R  A)  M i i A
  asks = reader

ReaderTIMonadReader : {I : Set } {M : IFun I (r  a)} 
                      RawIMonad M  RawIMonadReader (IReaderT M)
ReaderTIMonadReader Mon = record
  { monad = ReaderTIMonad Mon
  ; reader = λ f r  return (f r)
  ; local = λ f m  m  f
  } where open RawIMonad Mon

------------------------------------------------------------------------
-- Ordinary reader monads

RawMonadReader : (M : Set (r  a)  Set (r  a))  Set (suc (r  a))
RawMonadReader M = RawIMonadReader {I = }  _ _  M)

module RawMonadReader {M} (Mon : RawMonadReader M) where
  open RawIMonadReader Mon public

ReaderT : (M : Set (r  a)  Set (r  a))  Set _  Set _
ReaderT M = IReaderT {I = }  _ _  M) _ _

ReaderTMonad :  {M}  RawMonad M  RawMonad (ReaderT M)
ReaderTMonad = ReaderTIMonad

ReaderTMonadReader :  {M}  RawMonad M  RawMonadReader (ReaderT M)
ReaderTMonadReader = ReaderTIMonadReader

ReaderTMonadZero :  {M}  RawMonadZero M  RawMonadZero (ReaderT M)
ReaderTMonadZero = ReaderTIMonadZero

ReaderTMonadPlus :  {M}  RawMonadPlus M  RawMonadPlus (ReaderT M)
ReaderTMonadPlus = ReaderTIMonadPlus

Reader : Set (r  a)  Set (r  a)
Reader = ReaderT Identity

ReaderMonad : RawMonad Reader
ReaderMonad = ReaderTIMonad Id.monad

ReaderMonadReader : RawMonadReader Reader
ReaderMonadReader = ReaderTIMonadReader Id.monad