open import list open import basic open import nat 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 -> A headDelta (mono x) = x headDelta (delta x _) = x tailDelta : {l : Level} {A : Set l} -> Delta A -> Delta A tailDelta (mono x) = mono x tailDelta (delta _ d) = d n-tail : {l : Level} {A : Set l} -> Nat -> ((Delta A) -> (Delta A)) n-tail O = id n-tail (S n) = tailDelta ∙ (n-tail n) -- 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 = delta (headDelta (f x)) (bind d (tailDelta ∙ f)) 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 = delta (headDelta (f x)) (d >>= (tailDelta ∙ f)) -- proofs -- sub-proofs n-tail-plus : {l : Level} {A : Set l} -> (n : Nat) -> ((n-tail {l} {A} n) ∙ tailDelta) ≡ n-tail (S n) n-tail-plus O = refl n-tail-plus (S n) = begin n-tail (S n) ∙ tailDelta ≡⟨ refl ⟩ (tailDelta ∙ (n-tail n)) ∙ tailDelta ≡⟨ refl ⟩ tailDelta ∙ ((n-tail n) ∙ tailDelta) ≡⟨ cong (\t -> tailDelta ∙ t) (n-tail-plus n) ⟩ n-tail (S (S n)) ∎ n-tail-add : {l : Level} {A : Set l} {d : Delta A} -> (n m : Nat) -> (n-tail {l} {A} n) ∙ (n-tail m) ≡ n-tail (n + m) n-tail-add O m = refl n-tail-add (S n) O = begin n-tail (S n) ∙ n-tail O ≡⟨ refl ⟩ n-tail (S n) ≡⟨ cong (\n -> n-tail n) (nat-add-right-zero (S n))⟩ n-tail (S n + O) ∎ n-tail-add {l} {A} {d} (S n) (S m) = begin n-tail (S n) ∙ n-tail (S m) ≡⟨ refl ⟩ (tailDelta ∙ (n-tail n)) ∙ n-tail (S m) ≡⟨ refl ⟩ tailDelta ∙ ((n-tail n) ∙ n-tail (S m)) ≡⟨ cong (\t -> tailDelta ∙ t) (n-tail-add {l} {A} {d} n (S m)) ⟩ tailDelta ∙ (n-tail (n + (S m))) ≡⟨ refl ⟩ n-tail (S (n + S m)) ≡⟨ refl ⟩ n-tail (S n + S m) ∎ tail-delta-to-mono : {l : Level} {A : Set l} -> (n : Nat) -> (x : A) -> (n-tail n) (mono x) ≡ (mono x) tail-delta-to-mono O x = refl tail-delta-to-mono (S n) x = begin n-tail (S n) (mono x) ≡⟨ refl ⟩ tailDelta (n-tail n (mono x)) ≡⟨ refl ⟩ tailDelta (n-tail n (mono x)) ≡⟨ cong (\t -> tailDelta t) (tail-delta-to-mono n x) ⟩ tailDelta (mono x) ≡⟨ refl ⟩ mono x ∎ head-delta-natural-transformation : {l ll : Level} {A : Set l} {B : Set ll} -> (f : A -> B) -> (d : Delta A) -> headDelta (fmap f d) ≡ f (headDelta d) head-delta-natural-transformation f (mono x) = refl head-delta-natural-transformation f (delta x d) = refl n-tail-natural-transformation : {l ll : Level} {A : Set l} {B : Set ll} -> (n : Nat) -> (f : A -> B) -> (d : Delta A) -> n-tail n (fmap f d) ≡ fmap f (n-tail n d) n-tail-natural-transformation O f d = refl n-tail-natural-transformation (S n) f (mono x) = begin n-tail (S n) (fmap f (mono x)) ≡⟨ refl ⟩ n-tail (S n) (mono (f x)) ≡⟨ tail-delta-to-mono (S n) (f x) ⟩ (mono (f x)) ≡⟨ refl ⟩ fmap f (mono x) ≡⟨ cong (\d -> fmap f d) (sym (tail-delta-to-mono (S n) x)) ⟩ fmap f (n-tail (S n) (mono x)) ∎ n-tail-natural-transformation (S n) f (delta x d) = begin n-tail (S n) (fmap f (delta x d)) ≡⟨ refl ⟩ n-tail (S n) (delta (f x) (fmap f d)) ≡⟨ cong (\t -> t (delta (f x) (fmap f d))) (sym (n-tail-plus n)) ⟩ ((n-tail n) ∙ tailDelta) (delta (f x) (fmap f d)) ≡⟨ refl ⟩ n-tail n (fmap f d) ≡⟨ n-tail-natural-transformation n f d ⟩ fmap f (n-tail n d) ≡⟨ refl ⟩ fmap f (((n-tail n) ∙ tailDelta) (delta x d)) ≡⟨ cong (\t -> fmap f (t (delta x d))) (n-tail-plus n) ⟩ fmap f (n-tail (S n) (delta x d)) ∎ -- 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-1-5 : {l : Level} {A : Set l} -> (m : Nat) (n : Nat) -> (ds : Delta (Delta A)) -> n-tail n (bind ds (n-tail m)) ≡ bind (n-tail n ds) (n-tail (m + n)) monad-law-1-5 O O ds = refl monad-law-1-5 O (S n) (mono ds) = begin n-tail (S n) (bind (mono ds) (n-tail O)) ≡⟨ refl ⟩ n-tail (S n) ds ≡⟨ refl ⟩ bind (mono ds) (n-tail (S n)) ≡⟨ cong (\de -> bind de (n-tail (S n))) (sym (tail-delta-to-mono (S n) ds)) ⟩ bind (n-tail (S n) (mono ds)) (n-tail (S n)) ≡⟨ refl ⟩ bind (n-tail (S n) (mono ds)) (n-tail (O + S n)) ∎ monad-law-1-5 O (S n) (delta d ds) = begin n-tail (S n) (bind (delta d ds) (n-tail O)) ≡⟨ refl ⟩ n-tail (S n) (delta (headDelta d) (bind ds tailDelta )) ≡⟨ cong (\t -> t (delta (headDelta d) (bind ds tailDelta ))) (sym (n-tail-plus n)) ⟩ ((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta )) ≡⟨ refl ⟩ (n-tail n) (bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩ bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩ bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S n)) ≡⟨ cong (\t -> bind (t (delta d ds)) (n-tail (S n))) (n-tail-plus n) ⟩ bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩ bind (n-tail (S n) (delta d ds)) (n-tail (O + S n)) ∎ monad-law-1-5 (S m) n (mono (mono x)) = begin n-tail n (bind (mono (mono x)) (n-tail (S m))) ≡⟨ refl ⟩ n-tail n (n-tail (S m) (mono x)) ≡⟨ cong (\de -> n-tail n de) (tail-delta-to-mono (S m) x)⟩ n-tail n (mono x) ≡⟨ tail-delta-to-mono n x ⟩ mono x ≡⟨ sym (tail-delta-to-mono (S m + n) x) ⟩ (n-tail (S m + n)) (mono x) ≡⟨ refl ⟩ bind (mono (mono x)) (n-tail (S m + n)) ≡⟨ cong (\de -> bind de (n-tail (S m + n))) (sym (tail-delta-to-mono n (mono x))) ⟩ bind (n-tail n (mono (mono x))) (n-tail (S m + n)) ∎ monad-law-1-5 (S m) n (mono (delta x ds)) = begin n-tail n (bind (mono (delta x ds)) (n-tail (S m))) ≡⟨ refl ⟩ n-tail n (n-tail (S m) (delta x ds)) ≡⟨ cong (\t -> n-tail n (t (delta x ds))) (sym (n-tail-plus m)) ⟩ n-tail n (((n-tail m) ∙ tailDelta) (delta x ds)) ≡⟨ refl ⟩ n-tail n ((n-tail m) ds) ≡⟨ cong (\t -> t ds) (n-tail-add {d = ds} n m) ⟩ n-tail (n + m) ds ≡⟨ cong (\n -> n-tail n ds) (nat-add-sym n m) ⟩ n-tail (m + n) ds ≡⟨ refl ⟩ ((n-tail (m + n)) ∙ tailDelta) (delta x ds) ≡⟨ cong (\t -> t (delta x ds)) (n-tail-plus (m + n))⟩ n-tail (S (m + n)) (delta x ds) ≡⟨ refl ⟩ n-tail (S m + n) (delta x ds) ≡⟨ refl ⟩ bind (mono (delta x ds)) (n-tail (S m + n)) ≡⟨ cong (\de -> bind de (n-tail (S m + n))) (sym (tail-delta-to-mono n (delta x ds))) ⟩ bind (n-tail n (mono (delta x ds))) (n-tail (S m + n)) ∎ monad-law-1-5 (S m) O (delta d ds) = begin n-tail O (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩ bind (delta d ds) (n-tail (S m)) ≡⟨ refl ⟩ bind (n-tail O (delta d ds)) (n-tail (S m)) ≡⟨ cong (\n -> bind (n-tail O (delta d ds)) (n-tail n)) (nat-add-right-zero (S m)) ⟩ bind (n-tail O (delta d ds)) (n-tail (S m + O)) ∎ monad-law-1-5 (S m) (S n) (delta d ds) = begin n-tail (S n) (bind (delta d ds) (n-tail (S m))) ≡⟨ cong (\t -> t ((bind (delta d ds) (n-tail (S m))))) (sym (n-tail-plus n)) ⟩ ((n-tail n) ∙ tailDelta) (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ ((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩ (n-tail n) (bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩ (n-tail n) (bind ds (n-tail (S (S m)))) ≡⟨ monad-law-1-5 (S (S m)) n ds ⟩ bind ((n-tail n) ds) (n-tail (S (S (m + n)))) ≡⟨ cong (\nm -> bind ((n-tail n) ds) (n-tail nm)) (sym (nat-right-increment (S m) n)) ⟩ bind ((n-tail n) ds) (n-tail (S m + S n)) ≡⟨ refl ⟩ bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S m + S n)) ≡⟨ cong (\t -> bind (t (delta d ds)) (n-tail (S m + S n))) (n-tail-plus n) ⟩ bind (n-tail (S n) (delta d ds)) (n-tail (S m + S n)) ∎ monad-law-1-4 : {l : Level} {A : Set l} -> (m n : Nat) -> (dd : Delta (Delta A)) -> headDelta ((n-tail n) (bind dd (n-tail m))) ≡ headDelta ((n-tail (m + n)) (headDelta (n-tail n dd))) monad-law-1-4 O O (mono dd) = refl monad-law-1-4 O O (delta dd dd₁) = refl monad-law-1-4 O (S n) (mono dd) = begin headDelta (n-tail (S n) (bind (mono dd) (n-tail O))) ≡⟨ refl ⟩ headDelta (n-tail (S n) dd) ≡⟨ refl ⟩ headDelta (n-tail (S n) (headDelta (mono dd))) ≡⟨ cong (\de -> headDelta (n-tail (S n) (headDelta de))) (sym (tail-delta-to-mono (S n) dd)) ⟩ headDelta (n-tail (S n) (headDelta (n-tail (S n) (mono dd)))) ≡⟨ refl ⟩ headDelta (n-tail (O + S n) (headDelta (n-tail (S n) (mono dd)))) ∎ monad-law-1-4 O (S n) (delta d ds) = begin headDelta (n-tail (S n) (bind (delta d ds) (n-tail O))) ≡⟨ refl ⟩ headDelta (n-tail (S n) (bind (delta d ds) id)) ≡⟨ refl ⟩ headDelta (n-tail (S n) (delta (headDelta d) (bind ds tailDelta))) ≡⟨ cong (\t -> headDelta (t (delta (headDelta d) (bind ds tailDelta)))) (sym (n-tail-plus n)) ⟩ headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta))) ≡⟨ refl ⟩ headDelta (n-tail n (bind ds tailDelta)) ≡⟨ monad-law-1-4 (S O) n ds ⟩ headDelta (n-tail (S n) (headDelta (n-tail n ds))) ≡⟨ refl ⟩ headDelta (n-tail (S n) (headDelta (((n-tail n) ∙ tailDelta) (delta d ds)))) ≡⟨ cong (\t -> headDelta (n-tail (S n) (headDelta (t (delta d ds))))) (n-tail-plus n) ⟩ headDelta (n-tail (S n) (headDelta (n-tail (S n) (delta d ds)))) ≡⟨ refl ⟩ headDelta (n-tail (O + S n) (headDelta (n-tail (S n) (delta d ds)))) ∎ monad-law-1-4 (S m) n (mono dd) = begin headDelta (n-tail n (bind (mono dd) (n-tail (S m)))) ≡⟨ refl ⟩ headDelta (n-tail n ((n-tail (S m)) dd)) ≡⟨ cong (\t -> headDelta (t dd)) (n-tail-add {d = dd} n (S m)) ⟩ headDelta (n-tail (n + S m) dd) ≡⟨ cong (\n -> headDelta ((n-tail n) dd)) (nat-add-sym n (S m)) ⟩ headDelta (n-tail (S m + n) dd) ≡⟨ refl ⟩ headDelta (n-tail (S m + n) (headDelta (mono dd))) ≡⟨ cong (\de -> headDelta (n-tail (S m + n) (headDelta de))) (sym (tail-delta-to-mono n dd)) ⟩ headDelta (n-tail (S m + n) (headDelta (n-tail n (mono dd)))) ∎ monad-law-1-4 (S m) O (delta d ds) = begin headDelta (n-tail O (bind (delta d ds) (n-tail (S m)))) ≡⟨ refl ⟩ headDelta (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ headDelta (delta (headDelta ((n-tail (S m) d))) (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩ headDelta (n-tail (S m) d) ≡⟨ cong (\n -> headDelta ((n-tail n) d)) (nat-add-right-zero (S m)) ⟩ headDelta (n-tail (S m + O) d) ≡⟨ refl ⟩ headDelta (n-tail (S m + O) (headDelta (delta d ds))) ≡⟨ refl ⟩ headDelta (n-tail (S m + O) (headDelta (n-tail O (delta d ds)))) ∎ monad-law-1-4 (S m) (S n) (delta d ds) = begin headDelta (n-tail (S n) (bind (delta d ds) (n-tail (S m)))) ≡⟨ refl ⟩ headDelta (n-tail (S n) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m)))))) ≡⟨ cong (\t -> headDelta (t (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))))) (sym (n-tail-plus n)) ⟩ headDelta ((((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))))) ≡⟨ refl ⟩ headDelta (n-tail n (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩ headDelta (n-tail n (bind ds (n-tail (S (S m))))) ≡⟨ monad-law-1-4 (S (S m)) n ds ⟩ headDelta (n-tail ((S (S m) + n)) (headDelta (n-tail n ds))) ≡⟨ cong (\nm -> headDelta ((n-tail nm) (headDelta (n-tail n ds)))) (sym (nat-right-increment (S m) n)) ⟩ headDelta (n-tail (S m + S n) (headDelta (n-tail n ds))) ≡⟨ refl ⟩ headDelta (n-tail (S m + S n) (headDelta (((n-tail n) ∙ tailDelta) (delta d ds)))) ≡⟨ cong (\t -> headDelta (n-tail (S m + S n) (headDelta (t (delta d ds))))) (n-tail-plus n) ⟩ headDelta (n-tail (S m + S n) (headDelta (n-tail (S n) (delta d ds)))) ∎ monad-law-1-2 : {l : Level} {A : Set l} -> (d : Delta (Delta A)) -> headDelta (mu d) ≡ (headDelta (headDelta d)) monad-law-1-2 (mono _) = refl monad-law-1-2 (delta _ _) = refl monad-law-1-3 : {l : Level} {A : Set l} -> (n : Nat) -> (d : Delta (Delta (Delta A))) -> bind (fmap mu d) (n-tail n) ≡ bind (bind d (n-tail n)) (n-tail n) monad-law-1-3 O (mono d) = refl monad-law-1-3 O (delta d ds) = begin bind (fmap mu (delta d ds)) (n-tail O) ≡⟨ refl ⟩ bind (delta (mu d) (fmap mu ds)) (n-tail O) ≡⟨ refl ⟩ delta (headDelta (mu d)) (bind (fmap mu ds) tailDelta) ≡⟨ cong (\dx -> delta dx (bind (fmap mu ds) tailDelta)) (monad-law-1-2 d) ⟩ delta (headDelta (headDelta d)) (bind (fmap mu ds) tailDelta) ≡⟨ cong (\dx -> delta (headDelta (headDelta d)) dx) (monad-law-1-3 (S O) ds) ⟩ delta (headDelta (headDelta d)) (bind (bind ds tailDelta) tailDelta) ≡⟨ refl ⟩ bind (delta (headDelta d) (bind ds tailDelta)) (n-tail O) ≡⟨ refl ⟩ bind (bind (delta d ds) (n-tail O)) (n-tail O) ∎ monad-law-1-3 (S n) (mono (mono d)) = begin bind (fmap mu (mono (mono d))) (n-tail (S n)) ≡⟨ refl ⟩ bind (mono d) (n-tail (S n)) ≡⟨ refl ⟩ (n-tail (S n)) d ≡⟨ refl ⟩ bind (mono d) (n-tail (S n)) ≡⟨ cong (\t -> bind t (n-tail (S n))) (sym (tail-delta-to-mono (S n) d))⟩ bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩ bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩ bind (bind (mono (mono d)) (n-tail (S n))) (n-tail (S n)) ∎ monad-law-1-3 (S n) (mono (delta d ds)) = begin bind (fmap mu (mono (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩ bind (mono (mu (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩ n-tail (S n) (mu (delta d ds)) ≡⟨ refl ⟩ n-tail (S n) (delta (headDelta d) (bind ds tailDelta)) ≡⟨ cong (\t -> t (delta (headDelta d) (bind ds tailDelta))) (sym (n-tail-plus n)) ⟩ (n-tail n ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta)) ≡⟨ refl ⟩ n-tail n (bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩ bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩ bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S n)) ≡⟨ cong (\t -> (bind (t (delta d ds)) (n-tail (S n)))) (n-tail-plus n) ⟩ bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩ bind (bind (mono (delta d ds)) (n-tail (S n))) (n-tail (S n)) ∎ monad-law-1-3 (S n) (delta (mono d) ds) = begin bind (fmap mu (delta (mono d) ds)) (n-tail (S n)) ≡⟨ refl ⟩ bind (delta (mu (mono d)) (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ bind (delta d (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ delta (headDelta ((n-tail (S n)) d)) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ delta (headDelta ((n-tail (S n)) d)) (bind (fmap mu ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta ((n-tail (S n)) d)) de) (monad-law-1-3 (S (S n)) ds) ⟩ delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩ delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩ delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ delta (headDelta ((n-tail (S n)) (headDelta (mono d)))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ cong (\de -> delta (headDelta ((n-tail (S n)) (headDelta de))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n))))) (sym (tail-delta-to-mono (S n) d)) ⟩ delta (headDelta ((n-tail (S n)) (headDelta ((n-tail (S n)) (mono d))))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ bind (delta (headDelta ((n-tail (S n)) (mono d))) (bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩ bind (bind (delta (mono d) ds) (n-tail (S n))) (n-tail (S n)) ∎ monad-law-1-3 (S n) (delta (delta d dd) ds) = begin bind (fmap mu (delta (delta d dd) ds)) (n-tail (S n)) ≡⟨ refl ⟩ bind (delta (mu (delta d dd)) (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ delta (headDelta ((n-tail (S n)) (mu (delta d dd)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ delta (headDelta ((n-tail (S n)) (delta (headDelta d) (bind dd tailDelta)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ cong (\t -> delta (headDelta (t (delta (headDelta d) (bind dd tailDelta)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))))(sym (n-tail-plus n)) ⟩ delta (headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind dd tailDelta)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (fmap mu ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta ((n-tail n) (bind dd tailDelta))) de) (monad-law-1-3 (S (S n)) ds) ⟩ delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta de ( (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))))) (monad-law-1-4 (S O) n dd) ⟩ delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩ delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (bind (bind ds (n-tail (S (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ delta (headDelta ((n-tail (S n)) (headDelta (((n-tail n) ∙ tailDelta) (delta d dd))))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ cong (\t -> delta (headDelta ((n-tail (S n)) (headDelta (t (delta d dd))))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n))))) (n-tail-plus n) ⟩ delta (headDelta ((n-tail (S n)) (headDelta ((n-tail (S n)) (delta d dd))))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ bind (delta (headDelta ((n-tail (S n)) (delta d dd))) (bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩ bind (bind (delta (delta d dd) ds) (n-tail (S n))) (n-tail (S n)) ∎ -- 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 x d) = begin (mu ∙ fmap mu) (delta x d) ≡⟨ refl ⟩ mu (fmap mu (delta x d)) ≡⟨ refl ⟩ mu (delta (mu x) (fmap mu d)) ≡⟨ refl ⟩ delta (headDelta (mu x)) (bind (fmap mu d) tailDelta) ≡⟨ cong (\x -> delta x (bind (fmap mu d) tailDelta)) (monad-law-1-2 x) ⟩ delta (headDelta (headDelta x)) (bind (fmap mu d) tailDelta) ≡⟨ cong (\d -> delta (headDelta (headDelta x)) d) (monad-law-1-3 (S O) d) ⟩ delta (headDelta (headDelta x)) (bind (bind d tailDelta) tailDelta) ≡⟨ refl ⟩ mu (delta (headDelta x) (bind d tailDelta)) ≡⟨ refl ⟩ mu (mu (delta x d)) ≡⟨ refl ⟩ (mu ∙ mu) (delta x d) ∎ monad-law-2-1 : {l : Level} {A : Set l} -> (n : Nat) -> (d : Delta A) -> (bind (fmap eta d) (n-tail n)) ≡ d monad-law-2-1 O (mono x) = refl monad-law-2-1 O (delta x d) = begin bind (fmap eta (delta x d)) (n-tail O) ≡⟨ refl ⟩ bind (delta (eta x) (fmap eta d)) id ≡⟨ refl ⟩ delta (headDelta (eta x)) (bind (fmap eta d) tailDelta) ≡⟨ refl ⟩ delta x (bind (fmap eta d) tailDelta) ≡⟨ cong (\de -> delta x de) (monad-law-2-1 (S O) d) ⟩ delta x d ∎ monad-law-2-1 (S n) (mono x) = begin bind (fmap eta (mono x)) (n-tail (S n)) ≡⟨ refl ⟩ bind (mono (mono x)) (n-tail (S n)) ≡⟨ refl ⟩ n-tail (S n) (mono x) ≡⟨ tail-delta-to-mono (S n) x ⟩ mono x ∎ monad-law-2-1 (S n) (delta x d) = begin bind (fmap eta (delta x d)) (n-tail (S n)) ≡⟨ refl ⟩ bind (delta (eta x) (fmap eta d)) (n-tail (S n)) ≡⟨ refl ⟩ delta (headDelta ((n-tail (S n) (eta x)))) (bind (fmap eta d) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ delta (headDelta ((n-tail (S n) (eta x)))) (bind (fmap eta d) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta (de)) (bind (fmap eta d) (n-tail (S (S n))))) (tail-delta-to-mono (S n) x) ⟩ delta (headDelta (eta x)) (bind (fmap eta d) (n-tail (S (S n)))) ≡⟨ refl ⟩ delta x (bind (fmap eta d) (n-tail (S (S n)))) ≡⟨ cong (\d -> delta x d) (monad-law-2-1 (S (S n)) d) ⟩ delta x d ∎ -- monad-law-2 : join . fmap return = join . return = id -- monad-law-2 join . fmap return = join . return monad-law-2 : {l : Level} {A : Set l} -> (d : Delta A) -> (mu ∙ fmap eta) d ≡ (mu ∙ eta) d monad-law-2 (mono x) = refl monad-law-2 (delta x d) = begin (mu ∙ fmap eta) (delta x d) ≡⟨ refl ⟩ mu (fmap eta (delta x d)) ≡⟨ refl ⟩ mu (delta (mono x) (fmap eta d)) ≡⟨ refl ⟩ delta (headDelta (mono x)) (bind (fmap eta d) tailDelta) ≡⟨ refl ⟩ delta x (bind (fmap eta d) tailDelta) ≡⟨ cong (\d -> delta x d) (monad-law-2-1 (S O) d) ⟩ (delta x d) ≡⟨ refl ⟩ mu (mono (delta x d)) ≡⟨ refl ⟩ mu (eta (delta x d)) ≡⟨ refl ⟩ (mu ∙ eta) (delta x d) ∎ -- monad-law-2' : join . return = id monad-law-2' : {l : Level} {A : Set l} -> (d : Delta A) -> (mu ∙ eta) d ≡ id d monad-law-2' d = 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-4-1 : {l ll : Level} {A : Set l} {B : Set ll} -> (n : Nat) -> (f : A -> B) -> (ds : Delta (Delta A)) -> bind (fmap (fmap f) ds) (n-tail n) ≡ fmap f (bind ds (n-tail n)) monad-law-4-1 O f (mono d) = refl monad-law-4-1 O f (delta d ds) = begin bind (fmap (fmap f) (delta d ds)) (n-tail O) ≡⟨ refl ⟩ bind (delta (fmap f d) (fmap (fmap f) ds)) (n-tail O) ≡⟨ refl ⟩ delta (headDelta (fmap f d)) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta de (bind (fmap (fmap f) ds) tailDelta)) (head-delta-natural-transformation f d) ⟩ delta (f (headDelta d)) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f (headDelta d)) de) (monad-law-4-1 (S O) f ds) ⟩ delta (f (headDelta d)) (fmap f (bind ds tailDelta)) ≡⟨ refl ⟩ fmap f (delta (headDelta d) (bind ds tailDelta)) ≡⟨ refl ⟩ fmap f (bind (delta d ds) (n-tail O)) ∎ monad-law-4-1 (S n) f (mono d) = begin bind (fmap (fmap f) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩ bind (mono (fmap f d)) (n-tail (S n)) ≡⟨ refl ⟩ n-tail (S n) (fmap f d) ≡⟨ n-tail-natural-transformation (S n) f d ⟩ fmap f (n-tail (S n) d) ≡⟨ refl ⟩ fmap f (bind (mono d) (n-tail (S n))) ∎ monad-law-4-1 (S n) f (delta d ds) = begin bind (fmap (fmap f) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩ delta (headDelta (n-tail (S n) (fmap f d))) (bind (fmap (fmap f) ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ delta (headDelta (n-tail (S n) (fmap f d))) (bind (fmap (fmap f) ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta de) (bind (fmap (fmap f) ds) (n-tail (S (S n))))) (n-tail-natural-transformation (S n) f d) ⟩ delta (headDelta (fmap f ((n-tail (S n) d)))) (bind (fmap (fmap f) ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta de (bind (fmap (fmap f) ds) (n-tail (S (S n))))) (head-delta-natural-transformation f (n-tail (S n) d)) ⟩ delta (f (headDelta (n-tail (S n) d))) (bind (fmap (fmap f) ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (f (headDelta (n-tail (S n) d))) de) (monad-law-4-1 (S (S n)) f ds) ⟩ delta (f (headDelta (n-tail (S n) d))) (fmap f (bind ds (n-tail (S (S n))))) ≡⟨ refl ⟩ fmap f (delta (headDelta (n-tail (S n) d)) (bind ds (n-tail (S (S n))))) ≡⟨ refl ⟩ fmap f (delta (headDelta (n-tail (S n) d)) (bind ds (tailDelta ∙ (n-tail (S n))))) ≡⟨ refl ⟩ fmap f (bind (delta d ds) (n-tail (S n))) ∎ -- 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) (d : Delta (Delta A)) -> (mu ∙ fmap (fmap f)) d ≡ (fmap f ∙ mu) d monad-law-4 f (mono d) = refl monad-law-4 f (delta (mono x) ds) = begin (mu ∙ fmap (fmap f)) (delta (mono x) ds) ≡⟨ refl ⟩ mu ( fmap (fmap f) (delta (mono x) ds)) ≡⟨ refl ⟩ mu (delta (mono (f x)) (fmap (fmap f) ds)) ≡⟨ refl ⟩ delta (headDelta (mono (f x))) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ refl ⟩ delta (f x) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f x) de) (monad-law-4-1 (S O) f ds) ⟩ delta (f x) (fmap f (bind ds tailDelta)) ≡⟨ refl ⟩ fmap f (delta x (bind ds tailDelta)) ≡⟨ refl ⟩ fmap f (delta (headDelta (mono x)) (bind ds tailDelta)) ≡⟨ refl ⟩ fmap f (mu (delta (mono x) ds)) ≡⟨ refl ⟩ (fmap f ∙ mu) (delta (mono x) ds) ∎ monad-law-4 f (delta (delta x d) ds) = begin (mu ∙ fmap (fmap f)) (delta (delta x d) ds) ≡⟨ refl ⟩ mu (fmap (fmap f) (delta (delta x d) ds)) ≡⟨ refl ⟩ mu (delta (delta (f x) (fmap f d)) (fmap (fmap f) ds)) ≡⟨ refl ⟩ delta (headDelta (delta (f x) (fmap f d))) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ refl ⟩ delta (f x) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f x) de) (monad-law-4-1 (S O) f ds) ⟩ delta (f x) (fmap f (bind ds tailDelta)) ≡⟨ refl ⟩ fmap f (delta x (bind ds tailDelta)) ≡⟨ refl ⟩ fmap f (delta (headDelta (delta x d)) (bind ds tailDelta)) ≡⟨ refl ⟩ fmap f (mu (delta (delta x d) ds)) ≡⟨ refl ⟩ (fmap f ∙ mu) (delta (delta x d) ds) ∎ {- -- 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 = {!!} -}