open import Level

open import basic
open import delta
open import delta.functor
open import nat
open import laws

module deltaM where

-- DeltaM definitions

data DeltaM {l : Level}
            (M : Set l -> Set l)
            {functorM : Functor M}
            {monadM : Monad M functorM}
            (A : Set l)
            : (Nat -> Set l) where
   deltaM : {n : Nat} -> Delta (M A) (S n) -> DeltaM M {functorM} {monadM} A (S n)


-- DeltaM utils

headDeltaM : {l : Level} {A : Set l} {n : Nat}
             {M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM}
             -> DeltaM M {functorM} {monadM} A (S n) -> M A
headDeltaM (deltaM d) = headDelta d


tailDeltaM :  {l : Level} {A : Set l} {n : Nat}
              {M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM}
             -> DeltaM {l} M {functorM} {monadM} A (S (S n)) -> DeltaM M {functorM} {monadM} A (S n)
tailDeltaM {_} {n} (deltaM d) = deltaM (tailDelta d)


appendDeltaM : {l : Level} {A : Set l} {n m : Nat}
             {M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM} ->
             DeltaM M {functorM} {monadM} A (S n) -> 
             DeltaM M {functorM} {monadM} A (S m) -> 
             DeltaM M {functorM} {monadM} A ((S n) + (S m))
appendDeltaM (deltaM d) (deltaM dd) = deltaM (deltaAppend d dd)




-- functor definitions
open Functor
deltaM-fmap : {l : Level} {A B : Set l} {n : Nat}
              {M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM}
              -> (A -> B) -> DeltaM M {functorM} {monadM} A (S n) -> DeltaM M {functorM} {monadM} B (S n)
deltaM-fmap {l} {A} {B} {n} {M} {functorM} f (deltaM d) = deltaM (fmap delta-is-functor (fmap functorM f) d)




-- monad definitions
open Monad

deltaM-eta : {l : Level} {A : Set l} {n : Nat}
                         {M : Set l -> Set l} {functorM : Functor M} {monadM   : Monad M functorM} ->
            A -> (DeltaM M {functorM} {monadM} A (S n))
deltaM-eta {n = n} {monadM = mm} x = deltaM (delta-eta {n = n} (eta mm x))

deltaM-mu : {l : Level} {A : Set l} {n : Nat} 
                        {M : Set l -> Set l} {functorM : Functor  M} {monadM   : Monad  M functorM} ->
                        DeltaM M {functorM} {monadM} (DeltaM M {functorM} {monadM} A (S n)) (S n)  ->
                        DeltaM M {functorM} {monadM} A (S n)
deltaM-mu {n = O}   {functorM = fm} {monadM = mm} (deltaM (mono x))    = deltaM (mono (mu mm (fmap fm headDeltaM x)))
deltaM-mu {n = S n} {functorM = fm} {monadM = mm} (deltaM (delta x d)) = appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM x))))
                                                                                      (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d)))
                                                  

deltaM-bind : {l : Level} {A B : Set l} 
                          {n : Nat} 
                          {M : Set l -> Set l}
                          {functorM : Functor M}
                          {monadM   : Monad M functorM} ->
            (DeltaM M {functorM} {monadM} A (S n)) -> 
            (A -> DeltaM M {functorM} {monadM} B (S n)) 
            -> DeltaM M {functorM} {monadM} B (S n)
deltaM-bind {n = O}   {monadM = mm} (deltaM (mono x))    f = deltaM (mono (bind mm x (headDeltaM ∙ f)))
deltaM-bind {n = S n} {monadM = mm} (deltaM (delta x d)) f = appendDeltaM (deltaM (mono (bind mm x (headDeltaM ∙ f))))
                                                                               (deltaM-bind (deltaM d) (tailDeltaM ∙ f))