open import Level open import Relation.Binary.PropositionalEquality open ≡-Reasoning open import basic open import delta open import delta.functor open import deltaM open import deltaM.functor open import laws module deltaM.monad where open Functor open NaturalTransformation open Monad postulate deltaM-mu-is-natural-transformation : {l : Level} {A : Set l} {M : {l' : Level} -> Set l' -> Set l'} -> {functorM : {l' : Level} -> Functor {l'} M} {monadM : {l' : Level} {A : Set l'} -> Monad {l'} {A} M (functorM ) } -> NaturalTransformation (\A -> DeltaM M (DeltaM M A)) (\A -> DeltaM M A) {deltaM-fmap ∙ deltaM-fmap} {deltaM-fmap {l}} (deltaM-mu {_} {_} {M} {functorM} {monadM}) headDeltaM-commute : {l : Level} {A B : Set l} {M : {l' : Level} -> Set l' -> Set l'} -> {functorM : {l' : Level} -> Functor {l'} M} -> {monadM : {l' : Level} {A : Set l'} -> Monad {l'} {A} M (functorM ) } -> (f : A -> B) -> (x : DeltaM M {functorM} {monadM} A) -> headDeltaM (deltaM-fmap f x) ≡ fmap functorM f (headDeltaM x) headDeltaM-commute f (deltaM (mono x)) = refl headDeltaM-commute f (deltaM (delta x d)) = refl headDeltaM-is-natural-transformation : {l : Level} {A : Set l} {M : {l' : Level} -> Set l' -> Set l'} -> {functorM : {l' : Level} -> Functor {l'} M} {monadM : {l' : Level} {A : Set l'} -> Monad {l'} {A} M functorM } -> NaturalTransformation {l} (\A -> DeltaM M {functorM} {monadM} A) M {\f d → deltaM (mono (headDeltaM (deltaM-fmap f d)))} {fmap functorM} headDeltaM -- {deltaM-fmap} {fmap (functorM {l} {A})} headDeltaM headDeltaM-is-natural-transformation = record { commute = headDeltaM-commute } deltaM-association-law : {l : Level} {A : Set l} {M : {l' : Level} -> Set l' -> Set l'} (functorM : {l' : Level} -> Functor {l'} M) (monadM : {l' : Level} {A : Set l'} -> Monad {l'} {A} M functorM) -> (d : DeltaM M (DeltaM M (DeltaM M {functorM} {monadM} A))) -> ((deltaM-mu ∙ (deltaM-fmap deltaM-mu)) d) ≡ ((deltaM-mu ∙ deltaM-mu) d) deltaM-association-law functorM monadM (deltaM x) = {!!} {- begin (deltaM-mu ∙ deltaM-fmap deltaM-mu) d ≡⟨ refl ⟩ deltaM-mu (deltaM-fmap deltaM-mu d) ≡⟨ {!!} ⟩ deltaM-mu (deltaM-mu d) ≡⟨ refl ⟩ (deltaM-mu ∙ deltaM-mu) d ∎ -} {- deltaM-association-law functorM monadM (deltaM (mono x)) = begin (deltaM-mu ∙ deltaM-fmap deltaM-mu) (deltaM (mono x)) ≡⟨ refl ⟩ deltaM-mu (deltaM-fmap deltaM-mu (deltaM (mono x))) ≡⟨ refl ⟩ deltaM-mu (deltaM (delta-fmap (fmap functorM deltaM-mu) (mono x))) ≡⟨ refl ⟩ deltaM-mu (deltaM (mono (fmap functorM deltaM-mu x))) ≡⟨ refl ⟩ deltaM-bind (deltaM (mono (fmap functorM deltaM-mu x))) id ≡⟨ refl ⟩ deltaM (mono (bind monadM (fmap functorM deltaM-mu x) (headDeltaM ∙ id))) ≡⟨ refl ⟩ deltaM (mono (bind monadM (fmap functorM deltaM-mu x) headDeltaM)) ≡⟨ {!!} ⟩ deltaM (mono (bind monadM (bind monadM x headDeltaM) headDeltaM)) ≡⟨ refl ⟩ deltaM (mono (bind monadM (bind monadM x (headDeltaM ∙ id)) (headDeltaM ∙ id))) ≡⟨ refl ⟩ deltaM-mu (deltaM (mono (bind monadM x (headDeltaM ∙ id)))) ≡⟨ refl ⟩ deltaM-mu (deltaM-mu (deltaM (mono x))) ≡⟨ refl ⟩ (deltaM-mu ∙ deltaM-mu) (deltaM (mono x)) ∎ deltaM-association-law functorM monadM (deltaM (delta x d)) = {!!} -} {- nya : {l : Level} {A B C : Set l} -> {M : {l' : Level} -> Set l' -> Set l'} {functorM : {l' : Level} -> Functor {l'} M } {monadM : {l' : Level} {A : Set l'} -> Monad {l'} {A} M functorM} (m : DeltaM M {functorM} {monadM} A) -> (f : A -> (DeltaM M {functorM} {monadM} B)) -> (g : B -> (DeltaM M C)) -> (x : M A) -> (deltaM (fmap delta-is-functor (\x -> bind {l} {A} monadM x (headDeltaM ∙ (\x -> deltaM-bind (f x) g))) (mono x))) ≡ (deltaM-bind (deltaM (fmap delta-is-functor (\x -> (bind {l} {A} monadM x (headDeltaM ∙ f))) (mono x))) g) nya = {!!} deltaM-monad-law-h-3 : {l : Level} {A B C : Set l} -> {M : {l' : Level} -> Set l' -> Set l'} {functorM : {l' : Level} -> Functor {l'} M } {monadM : {l' : Level} {A : Set l'} -> Monad {l'} {A} M functorM} (m : DeltaM M {functorM} {monadM} A) -> (f : A -> (DeltaM M B)) -> (g : B -> (DeltaM M C)) -> (deltaM-bind m (\x -> deltaM-bind (f x) g)) ≡ (deltaM-bind (deltaM-bind m f) g) {- deltaM-monad-law-h-3 {l} {A} {B} {C} {M} {functorM} {monadM} (deltaM (mono x)) f g = begin (deltaM-bind (deltaM (mono x)) (\x -> deltaM-bind (f x) g)) ≡⟨ refl ⟩ (deltaM (mono (bind {l} {A} monadM x (headDeltaM ∙ (\x -> deltaM-bind (f x) g))))) ≡⟨ {!!} ⟩ (deltaM-bind (deltaM (fmap delta-is-functor (\x -> (bind {l} {A} monadM x (headDeltaM ∙ f))) (mono x))) g) ≡⟨ refl ⟩ (deltaM-bind (deltaM (mono (bind {l} {A} monadM x (headDeltaM ∙ f)))) g) ≡⟨ refl ⟩ (deltaM-bind (deltaM-bind (deltaM (mono x)) f) g) ∎ -} deltaM-monad-law-h-3 {l} {A} {B} {C} {M} {functorM} {monadM} (deltaM (mono x)) f g = begin (deltaM-bind (deltaM (mono x)) (\x -> deltaM-bind (f x) g)) ≡⟨ refl ⟩ (deltaM (mono (bind {l} {A} monadM x (headDeltaM ∙ (\x -> deltaM-bind (f x) g))))) ≡⟨ {!!} ⟩ -- (deltaM (fmap delta-is-functor (\x -> bind {l} {A} monadM x (headDeltaM ∙ (\x -> deltaM-bind (f x) g))) (mono x))) ≡⟨ {!!} ⟩ deltaM (mono (bind {l} {B} monadM (bind {_} {A} monadM x (headDeltaM ∙ f)) (headDeltaM ∙ g))) ≡⟨ {!!} ⟩ deltaM (mono (bind {l} {B} monadM (bind {_} {A} monadM x (headDeltaM ∙ f)) (headDeltaM ∙ g))) ≡⟨ {!!} ⟩ (deltaM-bind (deltaM (mono (bind {l} {A} monadM x (headDeltaM ∙ f)))) g) ≡⟨ refl ⟩ (deltaM-bind (deltaM-bind (deltaM (mono x)) f) g) ∎ deltaM-monad-law-h-3 (deltaM (delta x d)) f g = {!!} {- begin (deltaM-bind m (\x -> deltaM-bind (f x) g)) ≡⟨ {!!} ⟩ (deltaM-bind (deltaM-bind m f) g) ∎ -} -}