open import list open import basic open import Level open import Relation.Binary.PropositionalEquality open ≡-Reasoning module similar where data Similar {l : Level} (A : Set l) : (Set (suc l)) where similar : List String -> A -> List String -> A -> Similar A fmap : {l ll : Level} {A : Set l} {B : Set ll} -> (A -> B) -> (Similar A) -> (Similar B) fmap f (similar xs x ys y) = similar xs (f x) ys (f y) mu : {l : Level} {A : Set l} -> Similar (Similar A) -> Similar A mu (similar lx (similar llx x _ _) ly (similar _ _ lly y)) = similar (lx ++ llx) x (ly ++ lly) y return : {l : Level} {A : Set l} -> A -> Similar A return x = similar [] x [] x returnS : {A : Set} -> A -> Similar A returnS x = similar [[ (show x) ]] x [[ (show x) ]] x returnSS : {A : Set} -> A -> A -> Similar A returnSS x y = similar [[ (show x) ]] x [[ (show y) ]] y --monad-law-1 : mu ∙ (fmap mu) ≡ mu ∙ mu monad-law-1 : {l : Level} {A : Set l} -> (s : Similar (Similar (Similar A))) -> ((mu ∙ (fmap mu)) s) ≡ ((mu ∙ mu) s) monad-law-1 (similar lx (similar llx (similar lllx x _ _) _ (similar _ _ _ _)) ly (similar _ (similar _ _ _ _) lly (similar _ _ llly y))) = begin similar (lx ++ (llx ++ lllx)) x (ly ++ (lly ++ llly)) y ≡⟨ cong (\left-list -> similar left-list x (ly ++ (lly ++ llly)) y) (list-associative lx llx lllx) ⟩ similar (lx ++ llx ++ lllx) x (ly ++ (lly ++ llly)) y ≡⟨ cong (\right-list -> similar (lx ++ llx ++ lllx) x right-list y ) (list-associative ly lly llly) ⟩ similar (lx ++ llx ++ lllx) x (ly ++ lly ++ llly) y ∎ --monad-law-2 : mu ∙ fmap return ≡ mu ∙ return ≡ id monad-law-2-1 : {l : Level} {A : Set l} -> (s : Similar A) -> (mu ∙ fmap return) s ≡ (mu ∙ return) s monad-law-2-1 (similar lx x ly y) = begin similar (lx ++ []) x (ly ++ []) y ≡⟨ cong (\left-list -> similar left-list x (ly ++ []) y) (empty-append lx)⟩ similar lx x (ly ++ []) y ≡⟨ cong (\right-list -> similar lx x right-list y) (empty-append ly) ⟩ similar lx x ly y ∎ {- monad-law-2-2 : mu ∙ return ≡ id monad-law-2-2 = {!!} monad-law-3 : ∀{f} -> return ∙ f ≡ fmap f ∙ return monad-law-3 = {!!} monad-law-4 : ∀{f} -> mu ∙ fmap (fmap f) ≡ fmap f ∙ mu monad-law-4 = {!!} -}