Prove right-unity-law on DeltaM
author Yasutaka Higa Thu, 29 Jan 2015 11:42:22 +0900 ebd0d6e2772c 0a3b6cb91a05
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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 nat
open import laws

open Functor
open NaturalTransformation

deltaM-right-unity-law : {l : Level} {A : Set l}
{M : {l' : Level} -> Set l' -> Set l'}
{functorM : {l' : Level} -> Functor {l'} M}
{n : Nat}
(d : DeltaM M {functorM} {monadM} A (S n)) ->
(deltaM-mu ∙ deltaM-eta) d ≡ id d
deltaM-right-unity-law {l} {A} {M} {fm} {mm} {O} (deltaM (mono x)) = begin
deltaM-mu (deltaM-eta (deltaM (mono x)))             ≡⟨ refl ⟩
deltaM-mu (deltaM (mono (eta mm (deltaM (mono x))))) ≡⟨ refl ⟩
deltaM (mono (mu mm (fmap fm (headDeltaM {M = M})(eta mm (deltaM (mono x))))))
≡⟨ cong (\de -> deltaM (mono (mu mm de))) (sym (eta-is-nt mm headDeltaM (deltaM (mono x)) )) ⟩
deltaM (mono (mu mm (eta mm ((headDeltaM {l} {A} {O} {M} {fm} {mm}) (deltaM (mono x)))))) ≡⟨ refl ⟩
deltaM (mono (mu mm (eta mm x))) ≡⟨ cong (\de -> deltaM (mono de)) (sym (right-unity-law mm x)) ⟩
deltaM (mono x)
∎
deltaM-right-unity-law {l} {A} {M} {fm} {mm} {S n} (deltaM (delta x d)) = begin
deltaM-mu (deltaM-eta (deltaM (delta x d)))
≡⟨ refl ⟩
deltaM-mu (deltaM (delta (eta mm (deltaM (delta x d))) (delta-eta (eta mm (deltaM (delta x d))))))
≡⟨ refl ⟩
appendDeltaM (deltaM (mono (mu mm (fmap fm (headDeltaM {monadM = mm}) (eta mm (deltaM (delta x d)))))))
(deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-eta (eta mm (deltaM (delta x d)))))))
≡⟨ cong (\de -> appendDeltaM (deltaM (mono (mu mm de)))
(deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-eta (eta mm (deltaM (delta x d))))))))
(sym (eta-is-nt mm headDeltaM (deltaM (delta x d)))) ⟩
appendDeltaM (deltaM (mono (mu mm (eta mm ((headDeltaM {monadM = mm}) (deltaM (delta x d)))))))
(deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-eta (eta mm (deltaM (delta x d)))))))
≡⟨ refl ⟩
appendDeltaM (deltaM (mono (mu mm (eta mm x))))
(deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-eta (eta mm (deltaM (delta x d)))))))
≡⟨ cong (\de -> appendDeltaM (deltaM (mono de)) (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-eta (eta mm (deltaM (delta x d))))))))
(sym (right-unity-law mm x)) ⟩
appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-eta (eta mm (deltaM (delta x d)))))))
≡⟨ refl ⟩
appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) (delta-eta (eta mm (deltaM (delta x d)))))))
≡⟨ cong (\de -> appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM de))) (sym (eta-is-nt delta-is-monad (fmap fm tailDeltaM) (eta mm (deltaM (delta x d))))) ⟩
appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM (delta-eta (fmap fm tailDeltaM (eta mm (deltaM (delta x d)))))))
≡⟨ cong (\de -> appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM (delta-eta de)))) (sym (eta-is-nt mm tailDeltaM (deltaM (delta x d)))) ⟩
appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM (delta-eta (eta mm (tailDeltaM (deltaM (delta x d)))))))
≡⟨ refl ⟩
appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM (delta-eta (eta mm (deltaM d)))))
≡⟨ refl ⟩
appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM-eta (deltaM d)))
≡⟨ cong (\de -> appendDeltaM (deltaM (mono x)) de) (deltaM-right-unity-law (deltaM d)) ⟩
appendDeltaM (deltaM (mono x)) (deltaM d)
≡⟨ refl ⟩
deltaM (delta x d)
∎

{-
deltaM-left-unity-law : {l : Level} {A : Set l}
{M : {l' : Level} -> Set l' -> Set l'}
(functorM : {l' : Level} -> Functor {l'} M)
(d : DeltaM M {functorM} {monadM} A (S O)) ->
(deltaM-mu ∙ (deltaM-fmap deltaM-eta)) d ≡ id d
deltaM-left-unity-law functorM monadM (deltaM (mono x)) = begin
(deltaM-mu ∙ deltaM-fmap deltaM-eta) (deltaM (mono x)) ≡⟨ refl ⟩
deltaM-mu (deltaM-fmap deltaM-eta (deltaM (mono x)))   ≡⟨ refl ⟩
deltaM-mu (deltaM (mono (fmap functorM deltaM-eta x))) ≡⟨ refl ⟩
deltaM (mono (mu monadM (fmap functorM headDeltaM (fmap functorM deltaM-eta x)))) ≡⟨ {!!} ⟩
deltaM (mono (mu monadM (fmap functorM headDeltaM (fmap functorM deltaM-eta x)))) ≡⟨ {!!} ⟩

id (deltaM (mono x))
∎
deltaM-left-unity-law functorM monadM (deltaM (delta x ()))
-}

deltaM-is-monad : {l : Level} {A : Set l} {n : Nat}
{M : {l' : Level} -> Set l' -> Set l'}
(functorM : {l' : Level}  -> Functor {l'} M)
Monad {l} (\A -> DeltaM M {functorM} {monadM} A (S n)) deltaM-is-functor
{ mu     = deltaM-mu;
eta    = deltaM-eta;
return = deltaM-eta;
bind   = deltaM-bind;
association-law = {!!};
left-unity-law = {!!};
right-unity-law = (\x -> (sym (deltaM-right-unity-law x))) ;
eta-is-nt = {!!}
}

{-
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 (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)))              ≡⟨ {!!} ⟩
deltaM-mu (deltaM-mu (deltaM (mono x)))                                         ≡⟨ 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))) ≡⟨ {!!} ⟩