Mercurial > hg > Members > atton > delta_monad
view agda/delta.agda @ 59:46b15f368905
Define bind and mu for Infinite Delta
| author | Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sat, 22 Nov 2014 16:03:40 +0900 |
| parents | dfcd72dc697e |
| children | 73bb981cb1c6 |
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open import list open import basic open import Level open import Relation.Binary.PropositionalEquality open ≡-Reasoning module delta where data Delta {l : Level} (A : Set l) : (Set (suc l)) where mono : A -> Delta A delta : A -> Delta A -> Delta A deltaAppend : {l : Level} {A : Set l} -> Delta A -> Delta A -> Delta A deltaAppend (mono x) d = delta x d deltaAppend (delta x d) ds = delta x (deltaAppend d ds) headDelta : {l : Level} {A : Set l} -> Delta A -> Delta A headDelta (mono x) = mono x headDelta (delta x _) = mono x tailDelta : {l : Level} {A : Set l} -> Delta A -> Delta A tailDelta (mono x) = mono x tailDelta (delta _ d) = d -- Functor fmap : {l ll : Level} {A : Set l} {B : Set ll} -> (A -> B) -> (Delta A) -> (Delta B) fmap f (mono x) = mono (f x) fmap f (delta x d) = delta (f x) (fmap f d) -- Monad (Category) eta : {l : Level} {A : Set l} -> A -> Delta A eta x = mono x bind : {l ll : Level} {A : Set l} {B : Set ll} -> (Delta A) -> (A -> Delta B) -> Delta B bind (mono x) f = f x bind (delta x d) f = deltaAppend (headDelta (f x)) (bind d (tailDelta ∙ f)) -- can not apply id. because different Level mu : {l : Level} {A : Set l} -> Delta (Delta A) -> Delta A mu d = bind d id returnS : {l : Level} {A : Set l} -> A -> Delta A returnS x = mono x returnSS : {l : Level} {A : Set l} -> A -> A -> Delta A returnSS x y = deltaAppend (returnS x) (returnS y) -- Monad (Haskell) return : {l : Level} {A : Set l} -> A -> Delta A return = eta _>>=_ : {l ll : Level} {A : Set l} {B : Set ll} -> (x : Delta A) -> (f : A -> (Delta B)) -> (Delta B) (mono x) >>= f = f x (delta x d) >>= f = deltaAppend (headDelta (f x)) (d >>= (tailDelta ∙ f)) -- proofs -- Functor-laws -- Functor-law-1 : T(id) = id' functor-law-1 : {l : Level} {A : Set l} -> (d : Delta A) -> (fmap id) d ≡ id d functor-law-1 (mono x) = refl functor-law-1 (delta x d) = cong (delta x) (functor-law-1 d) -- Functor-law-2 : T(f . g) = T(f) . T(g) functor-law-2 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} -> (f : B -> C) -> (g : A -> B) -> (d : Delta A) -> (fmap (f ∙ g)) d ≡ ((fmap f) ∙ (fmap g)) d functor-law-2 f g (mono x) = refl functor-law-2 f g (delta x d) = cong (delta (f (g x))) (functor-law-2 f g d) -- Monad-laws (Category) -- monad-law-1 : join . fmap join = join . join monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((mu ∙ (fmap mu)) d) ≡ ((mu ∙ mu) d) monad-law-1 (mono d) = refl monad-law-1 (delta (mono (mono x)) (mono (mono (mono xx)))) = refl monad-law-1 (delta (mono (mono x)) (mono (mono (delta xx d)))) = refl monad-law-1 (delta (mono (mono x)) (mono (delta (mono xx) (mono (mono xxx))))) = refl monad-law-1 (delta (mono (mono x)) (mono (delta (mono xx) (mono (delta xxx d))))) = refl monad-law-1 (delta (mono (mono x)) (mono (delta (mono xx) (delta (mono x₁) (mono (mono x₂)))))) = refl monad-law-1 (delta (mono (mono x)) (mono (delta (mono xx) (delta (mono x₁) (mono (delta x₂ (mono x₃))))))) = refl monad-law-1 (delta (mono (mono x)) (mono (delta (mono xx) (delta (mono x₁) (mono (delta x₂ (delta x₃ d₂))))))) = refl monad-law-1 (delta (mono (mono x)) (mono (delta (mono xx) (delta (mono x₁) (delta (mono x₂) (mono (mono x₃))))))) = refl monad-law-1 (delta (mono (mono x)) (mono (delta (mono xx) (delta (mono x₁) (delta (delta x₂ (mono x₃)) (mono (mono x₄))))))) = refl monad-law-1 (delta (mono (mono x)) (mono (delta (mono xx) (delta (mono x₁) (delta (delta x₂ (delta x₃ d₂)) (mono (mono x₄))))))) = refl monad-law-1 (delta (mono (mono x)) (mono (delta (mono xx) (delta (mono x₁) (delta d₂ (mono (delta x₂ d₃))))))) = refl monad-law-1 (delta (mono (mono x)) (mono (delta (mono xx) (delta (mono x₁) (delta d₂ (delta d₃ d₄)))))) = refl monad-law-1 (delta (mono (mono x)) (mono (delta (mono xx) (delta (delta x₁ d₁) d₂)))) = refl monad-law-1 (delta (mono (mono x)) (mono (delta (delta x₁ d) d₁))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (mono d₁)))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (mono x₂) (mono d₂))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (mono x₂) (delta (mono x₃) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (mono x₂) (delta (delta x₃ (mono x₄)) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (mono x₂) (delta (delta x₃ (delta x₄ d₂)) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (mono x₃)) (mono d₂))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (mono x₃)) (delta (mono x₄) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (mono x₃)) (delta (delta x₄ (mono x₅)) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (mono x₃)) (delta (delta x₄ (delta x₅ d₂)) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (mono x₄))) (mono d₂))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (mono x₄))) (delta (mono x₅) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (mono x₄))) (delta (delta x₅ (mono x₆)) (mono d₃)))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (mono x₄))) (delta (delta x₅ (delta x₆ d₂)) (mono d₃)))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (mono x₄))) (delta (delta x₅ (mono x₆)) (delta d₃ d₄)))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (mono x₄))) (delta (delta x₅ (delta x₆ d₂)) (delta d₃ d₄)))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ d₁))) (mono d₂))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (mono x₅)))) (delta (mono x₆) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (mono x₅)))) (delta (delta x₆ (mono x₇)) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (mono x₅)))) (delta (delta x₆ (delta x₇ d₂)) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (mono x₆))))) (delta (mono x₇) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (mono x₆))))) (delta (delta x₇ (mono x₈)) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (mono x₆))))) (delta (delta x₇ (delta x₈ d₂)) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (mono x₇)))))) (delta (mono x₈) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (mono x₇)))))) (delta (delta x₈ (mono x₉)) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (mono x₇)))))) (delta (delta x₈ (delta x₉ d₂)) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (mono x₈))))))) (delta (mono x₉) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (mono x₈))))))) (delta (delta x₉ (mono x₁₀)) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (mono x₈))))))) (delta (delta x₉ (delta x₁₀ d₂)) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (mono x₉)))))))) (delta (mono x₁₀) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (mono x₉)))))))) (delta (delta x₁₀ (mono x₁₁)) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (mono x₉)))))))) (delta (delta x₁₀ (delta x₁₁ d₂)) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (mono x₁₀))))))))) (delta (mono x₁₁) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (mono x₁₀))))))))) (delta (delta x₁₁ (mono x₁₂)) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (mono x₁₀))))))))) (delta (delta x₁₁ (delta x₁₂ d₂)) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (delta x₁₀ (mono x₁₁)))))))))) (delta (mono x₁₂) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (delta x₁₀ (mono x₁₁)))))))))) (delta (delta x₁₂ (mono x₁₃)) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (delta x₁₀ (mono x₁₁)))))))))) (delta (delta x₁₂ (delta x₁₃ d₂)) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (delta x₁₀ (delta x₁₁ (mono x₁₂))))))))))) (delta (mono x₁₃) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (delta x₁₀ (delta x₁₁ (mono x₁₂))))))))))) (delta (delta x₁₃ (mono x₁₄)) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (delta x₁₀ (delta x₁₁ (mono x₁₂))))))))))) (delta (delta x₁₃ (delta x₁₄ d₂)) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (delta x₁₀ (delta x₁₁ (delta x₁₂ (mono x₁₃)))))))))))) (delta (mono x₁₄) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (delta x₁₀ (delta x₁₁ (delta x₁₂ (mono x₁₃)))))))))))) (delta (delta x₁₄ (mono x₁₅)) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (delta x₁₀ (delta x₁₁ (delta x₁₂ (mono x₁₃)))))))))))) (delta (delta x₁₄ (delta x₁₅ d₂)) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (delta x₁₀ (delta x₁₁ (delta x₁₂ (delta x₁₃ (mono x₁₄))))))))))))) (delta (mono x₁₅) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (delta x₁₀ (delta x₁₁ (delta x₁₂ (delta x₁₃ (mono x₁₄))))))))))))) (delta (delta x₁₅ (mono x₁₆)) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (delta x₁₀ (delta x₁₁ (delta x₁₂ (delta x₁₃ (mono x₁₄))))))))))))) (delta (delta x₁₅ (delta x₁₆ d₂)) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (delta x₁₀ (delta x₁₁ (delta x₁₂ (delta x₁₃ (delta x₁₄ d₁))))))))))))) (delta (mono x₁₅) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (delta x₁₀ (delta x₁₁ (delta x₁₂ (delta x₁₃ (delta x₁₄ d₁))))))))))))) (delta (delta x₁₅ (mono x₁₆)) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (delta x₁₀ (delta x₁₁ (delta x₁₂ (delta x₁₃ (delta x₁₄ d₁))))))))))))) (delta (delta x₁₅ (delta x₁₆ d₂)) (mono d₃)))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (delta x₁₀ (delta x₁₁ (delta x₁₂ (delta x₁₃ (delta x₁₄ d₁))))))))))))) (delta (delta x₁₅ (delta x₁₆ d₂)) (delta d₃ d₄)))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ d)) (mono (mono (mono x₂))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ d)) (mono (mono (delta x₂ d₁))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ (mono x₂))) (mono (delta (mono x₃) (mono d₂))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ (mono x₂))) (mono (delta (mono x₃) (delta (mono x₄) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ (mono x₂))) (mono (delta (mono x₃) (delta (delta x₄ (mono x₅)) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ (mono x₂))) (mono (delta (mono x₃) (delta (delta x₄ (delta x₅ d₂)) d₃))))) = refl -- 6 goals monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ (mono x₂))) (mono (delta (delta x₃ (mono x₄)) (mono (mono x₅)))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ (mono x₂))) (mono (delta (delta x₃ (mono x₄)) (mono (delta x₅ d₂)))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ (mono x₂))) (mono (delta (delta x₃ (mono x₄)) (delta (mono x₅) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ (mono x₂))) (mono (delta (delta x₃ (mono x₄)) (delta (delta x₅ (mono x₆)) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ (mono x₂))) (mono (delta (delta x₃ (mono x₄)) (delta (delta x₅ (delta x₆ d₂)) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ (mono x₂))) (mono (delta (delta x₃ (delta x₄ (mono x₅))) (mono d₂))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ (mono x₂))) (mono (delta (delta x₃ (delta x₄ (mono x₅))) (delta (mono x₆) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ (mono x₂))) (mono (delta (delta x₃ (delta x₄ (mono x₅))) (delta (delta x₆ (mono x₇)) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ (mono x₂))) (mono (delta (delta x₃ (delta x₄ (mono x₅))) (delta (delta x₆ (delta x₇ d₂)) d₃))))) = refl monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ (mono x₂))) (mono (delta (delta x₃ (delta x₄ (delta x₅ d₁))) (mono d₂))))) = {!!} monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ (mono x₂))) (mono (delta (delta x₃ (delta x₄ (delta x₅ d₁))) (delta d₂ d₃))))) = {!!} monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ (delta x₂ d))) (mono (delta d₁ d₂)))) = {!!} monad-law-1 (delta (mono (mono x)) (delta (mono d) (delta d₁ d₂))) = {!!} monad-law-1 (delta (mono (mono x)) (delta (delta d d₁) d₂)) = {!!} -- monad-law-1 (delta (mono (delta x x₁)) d) = {!!} monad-law-1 (delta (delta x x₁) d) = {!!} {- -- monad-law-2-2 : join . return = id monad-law-2-2 : {l : Level} {A : Set l } -> (s : Delta A) -> (mu ∙ eta) s ≡ id s monad-law-2-2 (similar lx x ly y) = refl -- monad-law-3 : return . f = fmap f . return monad-law-3 : {l : Level} {A B : Set l} (f : A -> B) (x : A) -> (eta ∙ f) x ≡ (fmap f ∙ eta) x monad-law-3 f x = refl -- monad-law-4 : join . fmap (fmap f) = fmap f . join monad-law-4 : {l ll : Level} {A : Set l} {B : Set ll} (f : A -> B) (s : Delta (Delta A)) -> (mu ∙ fmap (fmap f)) s ≡ (fmap f ∙ mu) s monad-law-4 f (similar lx (similar llx x _ _) ly (similar _ _ lly y)) = refl -- Monad-laws (Haskell) -- monad-law-h-1 : return a >>= k = k a monad-law-h-1 : {l ll : Level} {A : Set l} {B : Set ll} -> (a : A) -> (k : A -> (Delta B)) -> (return a >>= k) ≡ (k a) monad-law-h-1 a k = refl -- monad-law-h-2 : m >>= return = m monad-law-h-2 : {l : Level}{A : Set l} -> (m : Delta A) -> (m >>= return) ≡ m monad-law-h-2 (mono x) = refl monad-law-h-2 (delta x d) = cong (delta x) (monad-law-h-2 d) -- monad-law-h-3 : m >>= (\x -> k x >>= h) = (m >>= k) >>= h monad-law-h-3 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} -> (m : Delta A) -> (k : A -> (Delta B)) -> (h : B -> (Delta C)) -> (m >>= (\x -> k x >>= h)) ≡ ((m >>= k) >>= h) monad-law-h-3 (mono x) k h = refl monad-law-h-3 (delta x d) k h = begin (delta x d) >>= (\x -> k x >>= h) ≡⟨ refl ⟩ -- (delta x d) >>= f = deltaAppend (headDelta (f x)) (d >>= (tailDelta ∙ f)) deltaAppend (headDelta ((\x -> k x >>= h) x)) (d >>= (tailDelta ∙ (\x -> k x >>= h))) ≡⟨ refl ⟩ deltaAppend (headDelta (k x >>= h)) (d >>= (tailDelta ∙ (\x -> k x >>= h))) ≡⟨ {!!} ⟩ ((delta x d) >>= k) >>= h ∎ -}