view agda/deltaM/monad.agda @ 118:53cb21845dea

Prove association-law for DeltaM
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Mon, 02 Feb 2015 11:54:23 +0900
parents 6f86b55bf8b4
children 48b44bd85056
line wrap: on
line source

open import Level
open import Relation.Binary.PropositionalEquality
open ≡-Reasoning

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

module deltaM.monad where
open Functor
open NaturalTransformation
open Monad


-- sub proofs 

deconstruct-id : {l : Level} {A : Set l} {n : Nat}
                          {M : Set l -> Set l} {fm : Functor M} {mm : Monad M fm}
                          (d : DeltaM M {fm} {mm} A (S n)) -> deltaM (unDeltaM d) ≡ d
deconstruct-id {n = O} (deltaM x)   = refl
deconstruct-id {n = S n} (deltaM x) = refl


headDeltaM-with-appendDeltaM : {l : Level} {A : Set l} {n m : Nat} 
                                    {M : Set l -> Set l} {fm : Functor M} {mm : Monad M fm}
                                  (d : DeltaM M {fm} {mm} A (S n)) -> (ds : DeltaM M {fm} {mm} A (S m)) -> 
                                  headDeltaM (appendDeltaM d ds) ≡ headDeltaM d
headDeltaM-with-appendDeltaM {l} {A} {n = O}     {O} (deltaM (mono _))    (deltaM _) = refl
headDeltaM-with-appendDeltaM {l} {A} {n = O}   {S m} (deltaM (mono _))    (deltaM _) = refl
headDeltaM-with-appendDeltaM {l} {A} {n = S n}   {O} (deltaM (delta _ _)) (deltaM _) = refl
headDeltaM-with-appendDeltaM {l} {A} {n = S n} {S m} (deltaM (delta _ _)) (deltaM _) = refl

fmap-headDeltaM-with-deltaM-eta : {l : Level} {A : Set l} {n : Nat}
                                  {M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM}
  (x : M A) ->  (fmap functorM ((headDeltaM {l} {A} {n} {M} {functorM} {monadM}) ∙ deltaM-eta) x) ≡ fmap functorM (eta monadM) x
fmap-headDeltaM-with-deltaM-eta {l} {A} {O} {M} {fm} {mm}    x = refl
fmap-headDeltaM-with-deltaM-eta {l} {A} {S n} {M} {fm} {mm} x  = refl


fmap-tailDeltaM-with-deltaM-eta : {l : Level} {A : Set l} {n : Nat}
                   {M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM}
                   (d : DeltaM M {functorM} {monadM} A (S n)) ->
       deltaM-fmap ((tailDeltaM {n = n} {monadM = monadM} )  ∙ deltaM-eta) d ≡ deltaM-fmap (deltaM-eta) d
fmap-tailDeltaM-with-deltaM-eta {n = O} d = refl
fmap-tailDeltaM-with-deltaM-eta {n = S n} d = refl

fmap-headDeltaM-with-deltaM-mu : {l : Level} {A : Set l} {n : Nat}
                   {M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM}
                   (x : M (DeltaM M (DeltaM M {functorM} {monadM} A (S n)) (S n))) ->
                   fmap functorM (headDeltaM ∙ deltaM-mu) x ≡ fmap functorM (((mu monadM) ∙ (fmap functorM headDeltaM)) ∙ headDeltaM) x
fmap-headDeltaM-with-deltaM-mu {n = O}   x = refl
fmap-headDeltaM-with-deltaM-mu {n = S n} x = refl


fmap-tailDeltaM-with-deltaM-mu : {l : Level} {A : Set l} {n : Nat}
                   {M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM}
                   (d : DeltaM M {functorM} {monadM} (DeltaM M {functorM} {monadM} (DeltaM M A (S (S n))) (S (S n))) (S n)) ->
                   deltaM-fmap (tailDeltaM ∙ deltaM-mu) d ≡ deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) d
fmap-tailDeltaM-with-deltaM-mu {n = O} (deltaM (mono x)) = refl
fmap-tailDeltaM-with-deltaM-mu {n = S n} (deltaM d)      = refl





-- main proofs

postulate deltaM-eta-is-nt : {l : Level} {A B : Set l} {n : Nat}
                   {M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM} 
                   (f : A -> B) -> (x : A) ->
                   ((deltaM-eta {l} {B} {n} {M} {functorM} {monadM} )∙ f) x ≡ deltaM-fmap f (deltaM-eta x)
{-
deltaM-eta-is-nt {l} {A} {B} {O} {M} {fm} {mm} f x   = begin
  deltaM-eta {n = O} (f x)              ≡⟨ refl ⟩
  deltaM (mono (eta mm (f x)))          ≡⟨ cong (\de -> deltaM (mono de)) (eta-is-nt mm f x) ⟩
  deltaM (mono (fmap fm f (eta mm x)))  ≡⟨ refl ⟩
  deltaM-fmap f (deltaM-eta {n = O} x)  ∎
deltaM-eta-is-nt {l} {A} {B} {S n} {M} {fm} {mm} f x = begin
  deltaM-eta {n = S n} (f x) ≡⟨ refl ⟩
  deltaM (delta-eta {n = S n} (eta mm (f x))) ≡⟨ refl ⟩
  deltaM (delta (eta mm (f x)) (delta-eta (eta mm (f x))))
  ≡⟨ cong (\de -> deltaM (delta de (delta-eta de))) (eta-is-nt mm f x) ⟩
  deltaM (delta (fmap fm f (eta mm x)) (delta-eta (fmap fm f (eta mm x))))
  ≡⟨ cong (\de ->  deltaM (delta (fmap fm f (eta mm x)) de)) (eta-is-nt delta-is-monad (fmap fm f) (eta mm x)) ⟩
  deltaM (delta (fmap fm f (eta mm x)) (delta-fmap (fmap fm f) (delta-eta (eta mm x))))
  ≡⟨ refl ⟩
  deltaM-fmap f (deltaM-eta {n = S n} x)

-}

postulate deltaM-mu-is-nt : {l : Level} {A B : Set l} {n : Nat}
                  {T : Set l -> Set l} {F : Functor T} {M : Monad T F}
                  (f : A -> B) ->
                  (d : DeltaM T {F} {M} (DeltaM T A (S n)) (S n)) ->
                  deltaM-fmap f (deltaM-mu d) ≡ deltaM-mu (deltaM-fmap (deltaM-fmap f) d)

postulate  deltaM-right-unity-law : {l : Level} {A : Set l}
                         {M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM} {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)

-}





postulate deltaM-left-unity-law : {l : Level} {A : Set l}
                        {M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM}
                        {n : Nat}
                        (d : DeltaM M {functorM} {monadM} A (S n)) ->
                              (deltaM-mu ∙ (deltaM-fmap deltaM-eta)) d ≡ id d
{-
deltaM-left-unity-law {l} {A} {M} {fm} {mm} {O} (deltaM (mono x))      = begin
  deltaM-mu (deltaM-fmap deltaM-eta (deltaM (mono x)))
  ≡⟨ refl ⟩
  deltaM-mu (deltaM (delta-fmap (fmap fm deltaM-eta) (mono x)))
  ≡⟨ refl ⟩
  deltaM-mu (deltaM (mono (fmap fm deltaM-eta x)))
  ≡⟨ refl ⟩
  deltaM (mono (mu mm (fmap fm (headDeltaM {l} {A} {O} {M}) (fmap fm deltaM-eta x))))
  ≡⟨ cong (\de -> deltaM (mono (mu mm de))) (sym (covariant fm deltaM-eta headDeltaM x)) ⟩
  deltaM (mono (mu mm (fmap fm ((headDeltaM {l} {A} {O} {M} {fm} {mm}) ∙ deltaM-eta) x)))
  ≡⟨ cong (\de -> deltaM (mono (mu mm de))) (fmap-headDeltaM-with-deltaM-eta {l} {A} {O} {M} {fm} {mm} x) ⟩
  deltaM (mono (mu mm (fmap fm (eta mm) x)))
  ≡⟨ cong (\de -> deltaM (mono de)) (left-unity-law mm x) ⟩
  deltaM (mono x)

deltaM-left-unity-law {l} {A} {M} {fm} {mm} {S n} (deltaM (delta x d)) = begin
  deltaM-mu (deltaM-fmap deltaM-eta (deltaM (delta x d)))
  ≡⟨ refl ⟩
  deltaM-mu (deltaM (delta-fmap (fmap fm deltaM-eta) (delta x d)))
  ≡⟨ refl ⟩
  deltaM-mu (deltaM (delta (fmap fm deltaM-eta x) (delta-fmap (fmap fm deltaM-eta) d)))
  ≡⟨ refl  ⟩
  appendDeltaM (deltaM (mono (mu mm (fmap fm (headDeltaM {l} {A} {S n} {M} {fm} {mm}) (fmap fm deltaM-eta x)))))
               (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-eta) d))))
  ≡⟨ cong (\de -> appendDeltaM (deltaM (mono (mu mm de)))
                                (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-eta) d)))))
           (sym (covariant fm deltaM-eta headDeltaM x)) ⟩
  appendDeltaM (deltaM (mono (mu mm (fmap fm ((headDeltaM {l} {A} {S n} {M} {fm} {mm}) ∙ deltaM-eta) x))))
               (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-eta) d))))
  ≡⟨ cong (\de -> appendDeltaM (deltaM (mono (mu mm de)))
                                (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-eta) d)))))
           (fmap-headDeltaM-with-deltaM-eta {l} {A} {S n} {M} {fm} {mm} x) ⟩
  appendDeltaM (deltaM (mono (mu mm (fmap fm (eta mm) x))))
               (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-eta) d))))

  ≡⟨ cong (\de -> (appendDeltaM (deltaM (mono de)) (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-eta) d))))))
           (left-unity-law mm x) ⟩
  appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-eta) d))))
  ≡⟨ refl ⟩
  appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM-fmap (tailDeltaM {n = n})(deltaM-fmap deltaM-eta (deltaM d))))
  ≡⟨ cong (\de -> appendDeltaM (deltaM (mono x)) (deltaM-mu de)) (sym (covariant deltaM-is-functor deltaM-eta tailDeltaM (deltaM d))) ⟩
  appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM-fmap ((tailDeltaM {n = n}) ∙ deltaM-eta) (deltaM d)))
  ≡⟨ cong (\de -> appendDeltaM (deltaM (mono x)) (deltaM-mu de)) (fmap-tailDeltaM-with-deltaM-eta (deltaM d)) ⟩
  appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM-fmap deltaM-eta (deltaM d)))
  ≡⟨ cong (\de -> appendDeltaM (deltaM (mono x)) de) (deltaM-left-unity-law (deltaM d)) ⟩
  appendDeltaM (deltaM (mono x)) (deltaM d)
  ≡⟨ refl ⟩
  deltaM (delta x d)


-}
postulate nya : {l : Level} {A : Set l}
                         (M : Set l -> Set l) (fm : Functor M) (mm : Monad M fm)
                         (d : DeltaM M {fm} {mm} (DeltaM M {fm} {mm} (DeltaM M {fm} {mm} A (S O)) (S O))  (S O)) ->
                         deltaM-mu (deltaM-fmap deltaM-mu d) ≡ deltaM-mu (deltaM-mu d)


  

                


deltaM-association-law : {l : Level} {A : Set l} {n : Nat}
                         (M : Set l -> Set l) (fm : Functor M) (mm : Monad M fm)
                         (d : DeltaM M {fm} {mm} (DeltaM M {fm} {mm} (DeltaM M {fm} {mm} A (S n)) (S n))  (S n)) ->
                         deltaM-mu (deltaM-fmap deltaM-mu d) ≡ deltaM-mu (deltaM-mu d)
deltaM-association-law {l} {A} {O} M fm mm (deltaM (mono x))    = nya {l} {A} M fm mm (deltaM (mono x))
{-
begin
  deltaM-mu (deltaM-fmap deltaM-mu (deltaM (mono x))) ≡⟨ refl ⟩
  deltaM-mu (deltaM (mono (fmap fm deltaM-mu x))) ≡⟨ refl ⟩
  deltaM (mono (mu mm (fmap fm headDeltaM (headDeltaM {A = DeltaM M A (S O)} {monadM = mm} (deltaM (mono (fmap fm deltaM-mu x))))))) ≡⟨ refl ⟩
  deltaM (mono (mu mm (fmap fm headDeltaM (fmap fm deltaM-mu x)))) ≡⟨ refl ⟩
  deltaM (mono (mu mm (fmap fm (headDeltaM {A = A} {monadM = mm}) (fmap fm 
    (\d -> (deltaM (mono (mu mm (fmap fm headDeltaM ((headDeltaM {l} {DeltaM M A (S O)} {monadM = mm}) d)))))) x)))) 
  ≡⟨ cong (\de -> deltaM (mono (mu mm de))) 
           (sym (covariant fm (\d -> (deltaM (mono (mu mm (fmap fm headDeltaM ((headDeltaM {l} {DeltaM M A (S O)} {monadM = mm}) d))))))  headDeltaM x)) ⟩
  deltaM (mono (mu mm (fmap fm ((headDeltaM {A = A} {monadM = mm}) ∙ 
    (\d -> (deltaM (mono (mu mm (fmap fm headDeltaM ((headDeltaM {l} {DeltaM M A (S O)} {monadM = mm}) d))))))) x))) 
  ≡⟨ refl ⟩
  deltaM (mono (mu mm (fmap fm (\d -> (headDeltaM {A = A} {monadM = mm} (deltaM (mono (mu mm (fmap fm headDeltaM ((headDeltaM {l} {DeltaM M A (S O)} {monadM = mm}) d))))))) x)))
  ≡⟨ refl ⟩
  deltaM (mono (mu mm (fmap fm (\d -> (mu mm (fmap fm headDeltaM ((headDeltaM {l} {DeltaM M A (S O)} {monadM = mm}) d)))) x)))
  ≡⟨ refl ⟩
  deltaM (mono (mu mm (fmap fm ((mu mm) ∙  (((fmap fm headDeltaM)) ∙  ((headDeltaM {l} {DeltaM M A (S O)} {monadM = mm})))) x)))
  ≡⟨ cong (\de -> deltaM (mono (mu mm de))) (covariant fm ((fmap fm headDeltaM) ∙ (headDeltaM)) (mu mm) x )⟩ 
  deltaM (mono (mu mm (((fmap fm (mu mm)) ∙ (fmap fm ((fmap fm headDeltaM) ∙  (headDeltaM {l} {DeltaM M A (S O)} {monadM = mm})))) x)))
  ≡⟨ refl ⟩
  deltaM (mono (mu mm (fmap fm (mu mm) ((fmap fm ((fmap fm headDeltaM) ∙ (headDeltaM {l} {DeltaM M A (S O)} {monadM = mm}))) x))))
  ≡⟨ cong (\de -> deltaM (mono (mu mm (fmap fm (mu mm) de)))) (covariant fm headDeltaM (fmap fm headDeltaM) x) ⟩
  deltaM (mono (mu mm (fmap fm (mu mm) (((fmap fm (fmap fm headDeltaM)) ∙ (fmap fm (headDeltaM {l} {DeltaM M A (S O)} {monadM = mm}))) x))))
  ≡⟨ refl ⟩ 
  deltaM (mono (mu mm (fmap fm (mu mm) (fmap fm (fmap fm headDeltaM) (fmap fm headDeltaM x)))))
  ≡⟨ cong (\de ->   deltaM (mono de)) (association-law mm (fmap fm (fmap fm headDeltaM) (fmap fm headDeltaM x))) ⟩
  deltaM (mono (mu mm (mu mm (fmap fm (fmap fm headDeltaM) (fmap fm headDeltaM x)))))  
  ≡⟨ cong (\de -> deltaM (mono (mu mm de))) (mu-is-nt mm headDeltaM (fmap fm headDeltaM x)) ⟩
  deltaM (mono (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x)))))  ≡⟨ refl ⟩
  deltaM (mono (mu mm (fmap fm headDeltaM (headDeltaM {A = DeltaM M A (S O)} {monadM = mm} (deltaM (mono (mu mm (fmap fm headDeltaM x))))))))  ≡⟨ refl ⟩
  deltaM-mu (deltaM (mono (mu mm (fmap fm headDeltaM x))))  ≡⟨ refl ⟩
  deltaM-mu (deltaM (mono (mu mm (fmap fm headDeltaM (headDeltaM  {A = DeltaM M (DeltaM M A (S O)) (S O)} {monadM = mm} (deltaM (mono x)))))))  ≡⟨ refl ⟩
  deltaM-mu (deltaM-mu (deltaM (mono x)))  ∎
-}
deltaM-association-law {l} {A} {S n} M fm mm (deltaM (delta x d)) = begin
  deltaM-mu (deltaM-fmap deltaM-mu (deltaM (delta x d)))
  ≡⟨ refl ⟩
  deltaM-mu (deltaM (delta (fmap fm deltaM-mu x) (delta-fmap (fmap fm deltaM-mu) d)))
  ≡⟨ refl ⟩
  deltaM (delta (mu mm (fmap fm (headDeltaM {A = A} {monadM = mm}) (headDeltaM {A = DeltaM M A (S (S n))} {monadM = mm} (deltaM (delta (fmap fm deltaM-mu x) (delta-fmap (fmap fm deltaM-mu) d))))))
                (unDeltaM {A = A} {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM (tailDeltaM (deltaM (delta (fmap fm deltaM-mu x) (delta-fmap (fmap fm deltaM-mu) d))))))))

  ≡⟨ refl ⟩
  deltaM (delta (mu mm (fmap fm (headDeltaM {A = A} {monadM = mm}) (fmap fm deltaM-mu x)))
                (unDeltaM {A = A} {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-mu) d))))))
  ≡⟨ cong (\de -> deltaM (delta (mu mm de) (unDeltaM {A = A} {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-mu) d)))))))
           (sym (covariant fm deltaM-mu headDeltaM x)) ⟩
  deltaM (delta (mu mm (fmap fm ((headDeltaM {A = A} {monadM = mm}) ∙  deltaM-mu) x))
                (unDeltaM {A = A} {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-mu) d))))))
  ≡⟨ cong (\de -> deltaM (delta (mu mm de)
                          (unDeltaM {A = A} {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-mu) d)))))))
           (fmap-headDeltaM-with-deltaM-mu {A = A} {monadM = mm} x) ⟩
  deltaM (delta (mu mm (fmap fm (((mu mm) ∙ (fmap fm headDeltaM)) ∙ headDeltaM) x))
                (unDeltaM {A = A} {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-mu) d))))))
  ≡⟨ refl ⟩
  deltaM (delta (mu mm (fmap fm (((mu mm) ∙ (fmap fm headDeltaM)) ∙ headDeltaM) x))
                (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM-fmap deltaM-mu (deltaM d))))))
  ≡⟨ cong (\de -> deltaM (delta (mu mm (fmap fm (((mu mm) ∙ (fmap fm headDeltaM)) ∙ headDeltaM) x))
                                 (unDeltaM {monadM = mm} (deltaM-mu de))))
           (sym (deltaM-covariant fm tailDeltaM deltaM-mu (deltaM d))) ⟩
  deltaM (delta (mu mm (fmap fm (((mu mm) ∙ (fmap fm headDeltaM)) ∙ headDeltaM) x))
                (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap (tailDeltaM ∙ deltaM-mu) (deltaM d)))))
  ≡⟨ cong (\de -> deltaM (delta (mu mm (fmap fm (((mu mm) ∙ (fmap fm headDeltaM)) ∙ headDeltaM) x))
                                 (unDeltaM {monadM = mm} (deltaM-mu de))))
           (fmap-tailDeltaM-with-deltaM-mu (deltaM d))  ⟩
  deltaM (delta (mu mm (fmap fm (((mu mm) ∙ (fmap fm headDeltaM)) ∙ headDeltaM) x))
                (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) (deltaM d)))))
  ≡⟨ cong (\de -> deltaM (delta (mu mm de)
                          (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) (deltaM d))))))
          (covariant fm headDeltaM ((mu mm) ∙ (fmap fm headDeltaM)) x) ⟩
  deltaM (delta (mu mm (((fmap fm ((mu mm) ∙ (fmap fm headDeltaM))) ∙ (fmap fm headDeltaM)) x))
                (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) (deltaM d)))))
  ≡⟨ refl ⟩
  deltaM (delta (mu mm (((fmap fm ((mu mm) ∙ (fmap fm headDeltaM))) (fmap fm headDeltaM x))))
                (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) (deltaM d)))))
  ≡⟨ cong (\de -> deltaM (delta (mu mm de) 
                (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) (deltaM d))))))
           (covariant fm (fmap fm headDeltaM)  (mu mm) (fmap fm headDeltaM x)) ⟩

  deltaM (delta (mu mm ((((fmap fm (mu mm)) ∙ (fmap fm (fmap fm headDeltaM))) (fmap fm headDeltaM x))))
                (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) (deltaM d)))))
  ≡⟨ refl ⟩
  deltaM (delta (mu mm (fmap fm (mu mm) (fmap fm (fmap fm headDeltaM) (fmap fm headDeltaM x))))
                (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) (deltaM d)))))
  ≡⟨ cong (\de -> deltaM (delta de (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) (deltaM d))))))
           (association-law mm (fmap fm (fmap fm headDeltaM) (fmap fm headDeltaM x))) ⟩
  deltaM (delta (mu mm (mu mm (fmap fm (fmap fm headDeltaM) (fmap fm headDeltaM x))))
                (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) (deltaM d)))))
  ≡⟨ cong (\de -> deltaM (delta (mu mm de) (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) (deltaM d))))))
           (mu-is-nt mm headDeltaM (fmap fm headDeltaM x)) ⟩
  deltaM (delta (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x))))
                (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) (deltaM d)))))
  ≡⟨ cong (\de ->   deltaM (delta (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x)))) (unDeltaM {monadM = mm} (deltaM-mu de))))
           (deltaM-covariant fm (deltaM-mu ∙ (deltaM-fmap tailDeltaM)) tailDeltaM (deltaM d)) ⟩
  deltaM (delta (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x))))
                (unDeltaM {monadM = mm} (deltaM-mu (((deltaM-fmap (deltaM-mu ∙ (deltaM-fmap tailDeltaM))  ∙ (deltaM-fmap tailDeltaM)) (deltaM d))))))
  ≡⟨ refl ⟩
  deltaM (delta (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x))))
                (unDeltaM {monadM = mm} (deltaM-mu (((deltaM-fmap (deltaM-mu ∙ (deltaM-fmap tailDeltaM)) (deltaM-fmap tailDeltaM (deltaM d))))))))
  ≡⟨ cong (\de -> deltaM (delta (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x)))) (unDeltaM {monadM = mm} (deltaM-mu de))))
           (deltaM-covariant fm deltaM-mu (deltaM-fmap tailDeltaM) (deltaM-fmap tailDeltaM (deltaM d)))  ⟩
  deltaM (delta (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x))))
                (unDeltaM {monadM = mm} (deltaM-mu (((deltaM-fmap deltaM-mu) ∙ (deltaM-fmap (deltaM-fmap tailDeltaM))) (deltaM-fmap tailDeltaM (deltaM d))))))
  ≡⟨ refl ⟩
  deltaM (delta (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x))))
                (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap deltaM-mu (deltaM-fmap (deltaM-fmap tailDeltaM) (deltaM-fmap tailDeltaM (deltaM d)))))))
  ≡⟨ cong (\de -> deltaM (delta (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x)))) (unDeltaM {monadM = mm} de)))
           (deltaM-association-law M fm mm (deltaM-fmap (deltaM-fmap tailDeltaM) (deltaM-fmap tailDeltaM (deltaM d)))) ⟩
  deltaM (delta (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x))))
                (unDeltaM {monadM = mm} (deltaM-mu (deltaM-mu (deltaM-fmap (deltaM-fmap tailDeltaM) (deltaM-fmap tailDeltaM (deltaM d)))))))

  ≡⟨ cong (\de -> deltaM (delta (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x))))
                                 (unDeltaM {monadM = mm} (deltaM-mu de))))
           (sym (deltaM-mu-is-nt tailDeltaM (deltaM-fmap tailDeltaM (deltaM d)))) ⟩
  deltaM (delta (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x))))
                (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d)))))))
  ≡⟨ cong (\de -> deltaM (delta (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x))))
                                 (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM de)))))
           (sym (deconstruct-id (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d))))) ⟩
  deltaM (delta (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x))))
                (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM 
                  (deltaM (unDeltaM {A = DeltaM M A (S (S n))} {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d)))))))))


  ≡⟨ refl ⟩
  deltaM (delta (mu mm (fmap fm headDeltaM (headDeltaM {monadM = mm} ((deltaM (delta (mu mm (fmap fm headDeltaM x))
                           (unDeltaM {A = DeltaM M A (S (S n))} {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d))))))))))
                (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM (tailDeltaM ((deltaM (delta (mu mm (fmap fm headDeltaM x))
                           (unDeltaM {A = DeltaM M A (S (S n))} {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d))))))))))))


  ≡⟨ refl ⟩
  deltaM-mu (deltaM (delta (mu mm (fmap fm headDeltaM x))
                           (unDeltaM {A = DeltaM M A (S (S n))} {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d))))))
  ≡⟨ refl ⟩
  deltaM-mu (deltaM (delta (mu mm (fmap fm headDeltaM (headDeltaM {monadM = mm} (deltaM (delta x d)))))
                           (unDeltaM {A = DeltaM M A (S (S n))} {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM (tailDeltaM (deltaM (delta x d))))))))
  ≡⟨ refl ⟩
  deltaM-mu (deltaM-mu (deltaM (delta x d)))

{-
deltaM-association-law {l} {A} {S n} M fm mm (deltaM (delta x d)) = begin
  deltaM-mu (deltaM-fmap deltaM-mu (deltaM (delta x d))) ≡⟨ refl ⟩
  deltaM-mu (deltaM (delta-fmap (fmap fm deltaM-mu) (delta x d))) ≡⟨ refl ⟩
  deltaM-mu (deltaM (delta (fmap fm deltaM-mu x) (delta-fmap (fmap fm deltaM-mu) d))) ≡⟨ refl ⟩
  appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM (fmap fm deltaM-mu x)))))
               (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-mu) d)))) ≡⟨ refl ⟩
  appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM (fmap fm deltaM-mu x)))))
               (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) (delta-fmap (fmap fm deltaM-mu) d))))
  ≡⟨ cong (\de -> appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM (fmap fm deltaM-mu x))))) de) 
           (sym (deconstruct-id (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) (delta-fmap (fmap fm deltaM-mu) d)))))) ⟩

  appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM (fmap fm deltaM-mu x)))))
               (deltaM (deconstruct {A = A} {mm = mm} (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) (delta-fmap (fmap fm deltaM-mu) d))))))
  ≡⟨ refl ⟩
  deltaM (deltaAppend (mono (mu mm (fmap fm headDeltaM (fmap fm deltaM-mu x))))
                      (deconstruct {A = A} {mm = mm} (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) (delta-fmap (fmap fm deltaM-mu) d))))))
  ≡⟨ refl ⟩
  deltaM (delta (mu mm (fmap fm headDeltaM (fmap fm deltaM-mu x)))
                (deconstruct {A = A} {mm = mm} (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) (delta-fmap (fmap fm deltaM-mu) d))))))
  ≡⟨ {!!} ⟩
  appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x))))))
               (deltaM-mu (deltaM-fmap tailDeltaM (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) d)))))
  ≡⟨ cong (\de -> appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x))))))
               (deltaM-mu (deltaM-fmap tailDeltaM de))) 
           (sym (deconstruct-id (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) d))))) ⟩
  appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x))))))
               (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (deconstruct {A = DeltaM M A (S (S n))} {mm = mm} (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) d)))))))

  ≡⟨ refl ⟩
  appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x))))))
               (deltaM-mu (deltaM-fmap tailDeltaM (tailDeltaM ( (deltaM (delta (mu mm (fmap fm headDeltaM x))
                          (deconstruct {A = DeltaM M A (S (S n))} {mm = mm} (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) d))))))))))
  ≡⟨ refl ⟩
  appendDeltaM (deltaM (mono (mu mm (fmap fm (headDeltaM {monadM = mm}) (headDeltaM {monadM = mm} ((deltaM (delta (mu mm (fmap fm headDeltaM x))
                           (deconstruct {A = DeltaM M A (S (S n))} {mm = mm} (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) d))))))))))))
               (deltaM-mu (deltaM-fmap tailDeltaM (tailDeltaM ( (deltaM (delta (mu mm (fmap fm headDeltaM x))
                           (deconstruct {A = DeltaM M A (S (S n))} {mm = mm} (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) d))))))))))
  ≡⟨ refl ⟩
  deltaM-mu (deltaM (delta (mu mm (fmap fm headDeltaM x))
                           (deconstruct {A = DeltaM M A (S (S n))} {mm = mm} (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) d))))))
  ≡⟨ refl ⟩
  deltaM-mu (deltaM (deltaAppend (mono (mu mm (fmap fm headDeltaM x)))
                                 (deconstruct {A = DeltaM M A (S (S n))} {mm = mm} (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) d))))))
  ≡⟨ refl ⟩
  deltaM-mu (appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM x)))) 
                          (deltaM (deconstruct {A = DeltaM M A (S (S n))} {mm = mm} (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) d))))))
  ≡⟨ cong (\de -> deltaM-mu (appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM x)))) de))
           (deconstruct-id (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) d)))) ⟩
  deltaM-mu (appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM x)))) 
                          (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) d)))) 
  ≡⟨ refl ⟩
  deltaM-mu (appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM x)))) 
                          (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d))))≡⟨ refl ⟩
  deltaM-mu (deltaM-mu (deltaM (delta x d)))

-}



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