module Category.Monad.Indexed where
open import Category.Applicative.Indexed
open import Function
open import Level
record RawIMonad {i f} {I : Set i} (M : IFun I f) :
Set (i ⊔ suc f) 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 f} {I : Set i} (M : IFun I f) :
Set (i ⊔ suc f) where
field
monad : RawIMonad M
∅ : ∀ {i j A} → M i j A
open RawIMonad monad public
record RawIMonadPlus {i f} {I : Set i} (M : IFun I f) :
Set (i ⊔ suc f) 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