Mercurial > hg > Members > atton > delta_monad
view agda/delta/monad.agda @ 105:e6499a50ccbd
Retrying prove monad-laws for delta
| author | Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Tue, 27 Jan 2015 17:49:25 +0900 |
| parents | ebd0d6e2772c |
| children | caaf364f45ac |
line wrap: on
line source
open import basic open import delta open import delta.functor open import nat open import laws open import Level open import Relation.Binary.PropositionalEquality open ≡-Reasoning module delta.monad where delta-eta-is-nt : {l : Level} {A B : Set l} -> {n : Nat} (f : A -> B) -> (x : A) -> (delta-eta {n = n} ∙ f) x ≡ delta-fmap f (delta-eta x) delta-eta-is-nt {n = O} f x = refl delta-eta-is-nt {n = S O} f x = refl delta-eta-is-nt {n = S n} f x = begin (delta-eta ∙ f) x ≡⟨ refl ⟩ delta-eta (f x) ≡⟨ refl ⟩ delta (f x) (delta-eta (f x)) ≡⟨ cong (\de -> delta (f x) de) (delta-eta-is-nt f x) ⟩ delta (f x) (delta-fmap f (delta-eta x)) ≡⟨ refl ⟩ delta-fmap f (delta x (delta-eta x)) ≡⟨ refl ⟩ delta-fmap f (delta-eta x) ∎ delta-mu-is-nt : {l : Level} {A B : Set l} {n : Nat} -> (f : A -> B) -> (d : Delta (Delta A (S n)) (S n)) -> delta-mu (delta-fmap (delta-fmap f) d) ≡ delta-fmap f (delta-mu d) delta-mu-is-nt f d = {!!} hoge : {l : Level} {A : Set l} {n : Nat} -> (ds : Delta (Delta A (S (S n))) (S (S n))) -> (tailDelta {n = n} ∙ delta-mu {n = (S n)}) ds ≡ (((delta-mu {n = n}) ∙ (delta-fmap tailDelta)) ∙ tailDelta) ds hoge (delta ds ds₁) = refl -- Monad-laws (Category) -- monad-law-1 : join . delta-fmap join = join . join monad-law-1 : {l : Level} {A : Set l} {n : Nat} (d : Delta (Delta (Delta A (S n)) (S n)) (S n)) -> ((delta-mu ∙ (delta-fmap delta-mu)) d) ≡ ((delta-mu ∙ delta-mu) d) monad-law-1 {n = O} (mono d) = refl monad-law-1 {n = S O} (delta (delta (delta _ _) _) (mono (delta (delta _ (mono _)) (mono (delta _ (mono _)))))) = refl monad-law-1 {n = S n} (delta (delta (delta x d) dd) ds) = begin (delta-mu ∙ delta-fmap delta-mu) (delta (delta (delta x d) dd) ds) ≡⟨ refl ⟩ delta-mu (delta-fmap delta-mu (delta (delta (delta x d) dd) ds)) ≡⟨ refl ⟩ delta-mu (delta (delta-mu (delta (delta x d) dd)) (delta-fmap delta-mu ds)) ≡⟨ refl ⟩ delta-mu (delta (delta (headDelta (delta x d)) (delta-mu (delta-fmap tailDelta dd))) (delta-fmap delta-mu ds)) ≡⟨ refl ⟩ delta-mu (delta (delta x (delta-mu (delta-fmap tailDelta dd))) (delta-fmap delta-mu ds)) ≡⟨ refl ⟩ delta (headDelta (delta x (delta-mu (delta-fmap tailDelta dd)))) (delta-mu (delta-fmap tailDelta (delta-fmap delta-mu ds))) ≡⟨ refl ⟩ delta x (delta-mu (delta-fmap tailDelta (delta-fmap delta-mu ds))) ≡⟨ cong (\de -> delta x (delta-mu de)) (sym (functor-law-2 tailDelta delta-mu ds)) ⟩ delta x (delta-mu (delta-fmap (tailDelta {n = n} ∙ delta-mu {n = (S n)}) ds)) -- ≡⟨ cong (\ff -> delta x (delta-mu (delta-fmap ff ds))) hoge ⟩ ≡⟨ {!!} ⟩ delta x (delta-mu (delta-fmap (((delta-mu {n = n}) ∙ (delta-fmap tailDelta)) ∙ tailDelta) ds)) ≡⟨ cong (\de -> delta x (delta-mu de)) (functor-law-2 (delta-mu ∙ (delta-fmap tailDelta)) tailDelta ds ) ⟩ delta x (delta-mu (delta-fmap ((delta-mu {n = n}) ∙ (delta-fmap tailDelta)) (delta-fmap tailDelta ds))) ≡⟨ cong (\de -> delta x (delta-mu de)) (functor-law-2 delta-mu (delta-fmap tailDelta) (delta-fmap tailDelta ds)) ⟩ delta x (delta-mu (delta-fmap (delta-mu {n = n}) (delta-fmap (delta-fmap tailDelta) (delta-fmap tailDelta ds)))) ≡⟨ cong (\de -> delta x de) (monad-law-1 (delta-fmap (delta-fmap tailDelta) (delta-fmap tailDelta ds))) ⟩ delta x (delta-mu (delta-mu (delta-fmap (delta-fmap tailDelta) (delta-fmap tailDelta ds)))) ≡⟨ cong (\de -> delta x (delta-mu de)) (delta-mu-is-nt tailDelta (delta-fmap tailDelta ds)) ⟩ delta x (delta-mu (delta-fmap tailDelta (delta-mu (delta-fmap tailDelta ds)))) ≡⟨ refl ⟩ delta (headDelta (delta x d)) (delta-mu (delta-fmap tailDelta (delta-mu (delta-fmap tailDelta ds)))) ≡⟨ refl ⟩ delta-mu (delta (delta x d) (delta-mu (delta-fmap tailDelta ds))) ≡⟨ refl ⟩ delta-mu (delta (headDelta (delta (delta x d) dd)) (delta-mu (delta-fmap tailDelta ds))) ≡⟨ refl ⟩ delta-mu (delta-mu (delta (delta (delta x d) dd) ds)) ≡⟨ refl ⟩ (delta-mu ∙ delta-mu) (delta (delta (delta x d) dd) ds) ∎ {- begin (delta-mu ∙ delta-fmap delta-mu) (delta d ds) ≡⟨ refl ⟩ delta-mu (delta-fmap delta-mu (delta d ds)) ≡⟨ refl ⟩ delta-mu (delta (delta-mu d) (delta-fmap delta-mu ds)) ≡⟨ refl ⟩ delta (headDelta (delta-mu d)) (delta-mu (delta-fmap tailDelta (delta-fmap delta-mu ds))) ≡⟨ {!!} ⟩ delta (headDelta (headDelta d)) (delta-mu (delta-fmap tailDelta (delta-mu (delta-fmap tailDelta ds)))) ≡⟨ refl ⟩ delta-mu (delta (headDelta d) (delta-mu (delta-fmap tailDelta ds))) ≡⟨ refl ⟩ delta-mu (delta-mu (delta d ds)) ≡⟨ refl ⟩ (delta-mu ∙ delta-mu) (delta d ds) ∎ -} delta-right-unity-law : {l : Level} {A : Set l} {n : Nat} (d : Delta A (S n)) -> (delta-mu ∙ delta-eta) d ≡ id d delta-right-unity-law (mono x) = refl delta-right-unity-law (delta x d) = begin (delta-mu ∙ delta-eta) (delta x d) ≡⟨ refl ⟩ delta-mu (delta-eta (delta x d)) ≡⟨ refl ⟩ delta-mu (delta (delta x d) (delta-eta (delta x d))) ≡⟨ refl ⟩ delta (headDelta (delta x d)) (delta-mu (delta-fmap tailDelta (delta-eta (delta x d)))) ≡⟨ refl ⟩ delta x (delta-mu (delta-fmap tailDelta (delta-eta (delta x d)))) ≡⟨ cong (\de -> delta x (delta-mu de)) (sym (delta-eta-is-nt tailDelta (delta x d))) ⟩ delta x (delta-mu (delta-eta (tailDelta (delta x d)))) ≡⟨ refl ⟩ delta x (delta-mu (delta-eta d)) ≡⟨ cong (\de -> delta x de) (delta-right-unity-law d) ⟩ delta x d ≡⟨ refl ⟩ id (delta x d) ∎ delta-left-unity-law : {l : Level} {A : Set l} {n : Nat} -> (d : Delta A (S n)) -> (delta-mu ∙ (delta-fmap delta-eta)) d ≡ id d delta-left-unity-law (mono x) = refl delta-left-unity-law {n = (S n)} (delta x d) = begin (delta-mu ∙ delta-fmap delta-eta) (delta x d) ≡⟨ refl ⟩ delta-mu ( delta-fmap delta-eta (delta x d)) ≡⟨ refl ⟩ delta-mu (delta (delta-eta x) (delta-fmap delta-eta d)) ≡⟨ refl ⟩ delta (headDelta {n = S n} (delta-eta x)) (delta-mu (delta-fmap tailDelta (delta-fmap delta-eta d))) ≡⟨ refl ⟩ delta x (delta-mu (delta-fmap tailDelta (delta-fmap delta-eta d))) ≡⟨ cong (\de -> delta x (delta-mu de)) (sym (functor-law-2 tailDelta delta-eta d)) ⟩ delta x (delta-mu (delta-fmap (tailDelta ∙ delta-eta {n = S n}) d)) ≡⟨ refl ⟩ delta x (delta-mu (delta-fmap (delta-eta {n = n}) d)) ≡⟨ cong (\de -> delta x de) (delta-left-unity-law d) ⟩ delta x d ≡⟨ refl ⟩ id (delta x d) ∎ delta-is-monad : {l : Level} {n : Nat} -> Monad {l} (\A -> Delta A (S n)) delta-is-functor delta-is-monad = record { eta = delta-eta; mu = delta-mu; return = delta-eta; bind = delta-bind; eta-is-nt = delta-eta-is-nt; association-law = monad-law-1; left-unity-law = delta-left-unity-law ; right-unity-law = \x -> (sym (delta-right-unity-law x)) } {- -- Monad-laws (Haskell) -- monad-law-h-1 : return a >>= k = k a monad-law-h-1 : {l : Level} {A B : Set l} -> (a : A) -> (k : A -> (Delta B)) -> (delta-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 >>= delta-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 -> f x >>= g) = (m >>= f) >>= g monad-law-h-3 : {l : Level} {A B C : Set l} -> (m : Delta A) -> (f : A -> (Delta B)) -> (g : B -> (Delta C)) -> (delta-bind m (\x -> delta-bind (f x) g)) ≡ (delta-bind (delta-bind m f) g) monad-law-h-3 (mono x) f g = refl monad-law-h-3 (delta x d) f g = begin (delta-bind (delta x d) (\x -> delta-bind (f x) g)) ≡⟨ {!!} ⟩ (delta-bind (delta-bind (delta x d) f) g) ∎ -}