------------------------------------------------------------------------
-- Indexed monads
------------------------------------------------------------------------

-- Note that currently the monad laws are not included here.

module Category.Monad.Indexed where

open import Function
open import Category.Applicative.Indexed

record RawIMonad {I : Set} (M : IFun I) : Set₁ where
  infixl 1 _>>=_ _>>_
  infixr 1 _=<<_

  field
    return :  {i A}  A  M i i A
    _>>=_  :  {i j k A B}  M i j A  (A  M j k B)  M i k B

  _>>_ :  {i j k A B}  M i j A  M j k B  M i k B
  m₁ >> m₂ = m₁ >>= λ _  m₂

  _=<<_ :  {i j k A B}  (A  M j k B)  M i j A  M i k B
  f =<< c = c >>= f

  join :  {i j k A}  M i j (M j k A)  M i k A
  join m = m >>= id

  rawIApplicative : RawIApplicative M
  rawIApplicative = record
    { pure = return
    ; _⊛_  = λ f x  f >>= λ f'  x >>= λ x'  return (f' x')
    }

  open RawIApplicative rawIApplicative public

record RawIMonadZero {I : Set} (M : IFun I) : Set₁ where
  field
    monad : RawIMonad M
         :  {i j A}  M i j A

  open RawIMonad monad public

record RawIMonadPlus {I : Set} (M : IFun I) : Set₁ where
  infixr 3 _∣_
  field
    monadZero : RawIMonadZero M
    _∣_       :  {i j A}  M i j A  M i j A  M i j A

  open RawIMonadZero monadZero public