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) -- TODO: mu -- TODO: bind eta : {l : Level} {A : Set l} -> A -> Delta A eta x = mono x 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-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 = begin return a >>= k ≡⟨ refl ⟩ mu (fmap k (return a)) ≡⟨ refl ⟩ mu (return (k a)) ≡⟨ refl ⟩ (mu ∙ return) (k a) ≡⟨ refl ⟩ (mu ∙ eta) (k a) ≡⟨ (monad-law-2-2 (k a)) ⟩ id (k a) ≡⟨ refl ⟩ k a ∎ -- 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 (similar lx x ly y) = monad-law-2-1 (similar lx x ly y) -- 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 (similar lx x ly y) k h = begin ((similar lx x ly y) >>= (\x -> (k x) >>= h)) ≡⟨ refl ⟩ mu (fmap (\x -> k x >>= h) (similar lx x ly y)) ≡⟨ refl ⟩ (mu ∙ fmap (\x -> k x >>= h)) (similar lx x ly y) ≡⟨ refl ⟩ (mu ∙ fmap (\x -> mu (fmap h (k x)))) (similar lx x ly y) ≡⟨ refl ⟩ (mu ∙ fmap (mu ∙ (\x -> fmap h (k x)))) (similar lx x ly y) ≡⟨ refl ⟩ (mu ∙ (fmap mu ∙ (fmap (\x -> fmap h (k x))))) (similar lx x ly y) ≡⟨ refl ⟩ (mu ∙ (fmap mu)) ((fmap (\x -> fmap h (k x))) (similar lx x ly y)) ≡⟨ monad-law-1 (((fmap (\x -> fmap h (k x))) (similar lx x ly y))) ⟩ (mu ∙ mu) ((fmap (\x -> fmap h (k x))) (similar lx x ly y)) ≡⟨ refl ⟩ (mu ∙ (mu ∙ (fmap (\x -> fmap h (k x))))) (similar lx x ly y) ≡⟨ refl ⟩ (mu ∙ (mu ∙ (fmap ((fmap h) ∙ k)))) (similar lx x ly y) ≡⟨ refl ⟩ (mu ∙ (mu ∙ ((fmap (fmap h)) ∙ (fmap k)))) (similar lx x ly y) ≡⟨ refl ⟩ (mu ∙ (mu ∙ (fmap (fmap h)))) (fmap k (similar lx x ly y)) ≡⟨ refl ⟩ mu ((mu ∙ (fmap (fmap h))) (fmap k (similar lx x ly y))) ≡⟨ cong (\fx -> mu fx) (monad-law-4 h (fmap k (similar lx x ly y))) ⟩ mu (fmap h (mu (similar lx (k x) ly (k y)))) ≡⟨ refl ⟩ (mu ∙ fmap h) (mu (fmap k (similar lx x ly y))) ≡⟨ refl ⟩ mu (fmap h (mu (fmap k (similar lx x ly y)))) ≡⟨ refl ⟩ (mu (fmap k (similar lx x ly y))) >>= h ≡⟨ refl ⟩ ((similar lx x ly y) >>= k) >>= h ∎ -}