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 -- Monad-laws (Category) monad-law-1-5 : {l : Level} {A : Set l} -> (m : Nat) (n : Nat) -> (ds : Delta (Delta A)) -> n-tail n (delta-bind ds (n-tail m)) ≡ delta-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) (delta-bind (mono ds) (n-tail O)) ≡⟨ refl ⟩ n-tail (S n) ds ≡⟨ refl ⟩ delta-bind (mono ds) (n-tail (S n)) ≡⟨ cong (\de -> delta-bind de (n-tail (S n))) (sym (tail-delta-to-mono (S n) ds)) ⟩ delta-bind (n-tail (S n) (mono ds)) (n-tail (S n)) ≡⟨ refl ⟩ delta-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) (delta-bind (delta d ds) (n-tail O)) ≡⟨ refl ⟩ n-tail (S n) (delta (headDelta d) (delta-bind ds tailDelta )) ≡⟨ cong (\t -> t (delta (headDelta d) (delta-bind ds tailDelta ))) (sym (n-tail-plus n)) ⟩ ((n-tail n) ∙ tailDelta) (delta (headDelta d) (delta-bind ds tailDelta )) ≡⟨ refl ⟩ (n-tail n) (delta-bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩ delta-bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩ delta-bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S n)) ≡⟨ cong (\t -> delta-bind (t (delta d ds)) (n-tail (S n))) (n-tail-plus n) ⟩ delta-bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩ delta-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 (delta-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 ⟩ delta-bind (mono (mono x)) (n-tail (S m + n)) ≡⟨ cong (\de -> delta-bind de (n-tail (S m + n))) (sym (tail-delta-to-mono n (mono x))) ⟩ delta-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 (delta-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 ⟩ delta-bind (mono (delta x ds)) (n-tail (S m + n)) ≡⟨ cong (\de -> delta-bind de (n-tail (S m + n))) (sym (tail-delta-to-mono n (delta x ds))) ⟩ delta-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 (delta-bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ (delta-bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ delta (headDelta ((n-tail (S m)) d)) (delta-bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩ delta-bind (delta d ds) (n-tail (S m)) ≡⟨ refl ⟩ delta-bind (n-tail O (delta d ds)) (n-tail (S m)) ≡⟨ cong (\n -> delta-bind (n-tail O (delta d ds)) (n-tail n)) (nat-add-right-zero (S m)) ⟩ delta-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) (delta-bind (delta d ds) (n-tail (S m))) ≡⟨ cong (\t -> t ((delta-bind (delta d ds) (n-tail (S m))))) (sym (n-tail-plus n)) ⟩ ((n-tail n) ∙ tailDelta) (delta-bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ ((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (delta-bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩ (n-tail n) (delta-bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩ (n-tail n) (delta-bind ds (n-tail (S (S m)))) ≡⟨ monad-law-1-5 (S (S m)) n ds ⟩ delta-bind ((n-tail n) ds) (n-tail (S (S (m + n)))) ≡⟨ cong (\nm -> delta-bind ((n-tail n) ds) (n-tail nm)) (sym (nat-right-increment (S m) n)) ⟩ delta-bind ((n-tail n) ds) (n-tail (S m + S n)) ≡⟨ refl ⟩ delta-bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S m + S n)) ≡⟨ cong (\t -> delta-bind (t (delta d ds)) (n-tail (S m + S n))) (n-tail-plus n) ⟩ delta-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) (delta-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) (delta-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) (delta-bind (delta d ds) (n-tail O))) ≡⟨ refl ⟩ headDelta (n-tail (S n) (delta-bind (delta d ds) id)) ≡⟨ refl ⟩ headDelta (n-tail (S n) (delta (headDelta d) (delta-bind ds tailDelta))) ≡⟨ cong (\t -> headDelta (t (delta (headDelta d) (delta-bind ds tailDelta)))) (sym (n-tail-plus n)) ⟩ headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta d) (delta-bind ds tailDelta))) ≡⟨ refl ⟩ headDelta (n-tail n (delta-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 (delta-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 (delta-bind (delta d ds) (n-tail (S m)))) ≡⟨ refl ⟩ headDelta (delta-bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ headDelta (delta (headDelta ((n-tail (S m) d))) (delta-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) (delta-bind (delta d ds) (n-tail (S m)))) ≡⟨ refl ⟩ headDelta (n-tail (S n) (delta (headDelta ((n-tail (S m)) d)) (delta-bind ds (tailDelta ∙ (n-tail (S m)))))) ≡⟨ cong (\t -> headDelta (t (delta (headDelta ((n-tail (S m)) d)) (delta-bind ds (tailDelta ∙ (n-tail (S m))))))) (sym (n-tail-plus n)) ⟩ headDelta ((((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (delta-bind ds (tailDelta ∙ (n-tail (S m))))))) ≡⟨ refl ⟩ headDelta (n-tail n (delta-bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩ headDelta (n-tail n (delta-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 (delta-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))) -> delta-bind (delta-fmap delta-mu d) (n-tail n) ≡ delta-bind (delta-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 delta-bind (delta-fmap delta-mu (delta d ds)) (n-tail O) ≡⟨ refl ⟩ delta-bind (delta (delta-mu d) (delta-fmap delta-mu ds)) (n-tail O) ≡⟨ refl ⟩ delta (headDelta (delta-mu d)) (delta-bind (delta-fmap delta-mu ds) tailDelta) ≡⟨ cong (\dx -> delta dx (delta-bind (delta-fmap delta-mu ds) tailDelta)) (monad-law-1-2 d) ⟩ delta (headDelta (headDelta d)) (delta-bind (delta-fmap delta-mu ds) tailDelta) ≡⟨ cong (\dx -> delta (headDelta (headDelta d)) dx) (monad-law-1-3 (S O) ds) ⟩ delta (headDelta (headDelta d)) (delta-bind (delta-bind ds tailDelta) tailDelta) ≡⟨ refl ⟩ delta-bind (delta (headDelta d) (delta-bind ds tailDelta)) (n-tail O) ≡⟨ refl ⟩ delta-bind (delta-bind (delta d ds) (n-tail O)) (n-tail O) ∎ monad-law-1-3 (S n) (mono (mono d)) = begin delta-bind (delta-fmap delta-mu (mono (mono d))) (n-tail (S n)) ≡⟨ refl ⟩ delta-bind (mono d) (n-tail (S n)) ≡⟨ refl ⟩ (n-tail (S n)) d ≡⟨ refl ⟩ delta-bind (mono d) (n-tail (S n)) ≡⟨ cong (\t -> delta-bind t (n-tail (S n))) (sym (tail-delta-to-mono (S n) d))⟩ delta-bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩ delta-bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩ delta-bind (delta-bind (mono (mono d)) (n-tail (S n))) (n-tail (S n)) ∎ monad-law-1-3 (S n) (mono (delta d ds)) = begin delta-bind (delta-fmap delta-mu (mono (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩ delta-bind (mono (delta-mu (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩ n-tail (S n) (delta-mu (delta d ds)) ≡⟨ refl ⟩ n-tail (S n) (delta (headDelta d) (delta-bind ds tailDelta)) ≡⟨ cong (\t -> t (delta (headDelta d) (delta-bind ds tailDelta))) (sym (n-tail-plus n)) ⟩ (n-tail n ∙ tailDelta) (delta (headDelta d) (delta-bind ds tailDelta)) ≡⟨ refl ⟩ n-tail n (delta-bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩ delta-bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩ delta-bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S n)) ≡⟨ cong (\t -> (delta-bind (t (delta d ds)) (n-tail (S n)))) (n-tail-plus n) ⟩ delta-bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩ delta-bind (delta-bind (mono (delta d ds)) (n-tail (S n))) (n-tail (S n)) ∎ monad-law-1-3 (S n) (delta (mono d) ds) = begin delta-bind (delta-fmap delta-mu (delta (mono d) ds)) (n-tail (S n)) ≡⟨ refl ⟩ delta-bind (delta (delta-mu (mono d)) (delta-fmap delta-mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ delta-bind (delta d (delta-fmap delta-mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ delta (headDelta ((n-tail (S n)) d)) (delta-bind (delta-fmap delta-mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ delta (headDelta ((n-tail (S n)) d)) (delta-bind (delta-fmap delta-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)) (delta-bind (delta-bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩ delta (headDelta ((n-tail (S n)) d)) (delta-bind (delta-bind ds (tailDelta ∙ (n-tail (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩ delta (headDelta ((n-tail (S n)) d)) (delta-bind (delta-bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ delta (headDelta ((n-tail (S n)) (headDelta (mono d)))) (delta-bind (delta-bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ cong (\de -> delta (headDelta ((n-tail (S n)) (headDelta de))) (delta-bind (delta-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))))) (delta-bind (delta-bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ delta-bind (delta (headDelta ((n-tail (S n)) (mono d))) (delta-bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩ delta-bind (delta-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 delta-bind (delta-fmap delta-mu (delta (delta d dd) ds)) (n-tail (S n)) ≡⟨ refl ⟩ delta-bind (delta (delta-mu (delta d dd)) (delta-fmap delta-mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ delta (headDelta ((n-tail (S n)) (delta-mu (delta d dd)))) (delta-bind (delta-fmap delta-mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ delta (headDelta ((n-tail (S n)) (delta (headDelta d) (delta-bind dd tailDelta)))) (delta-bind (delta-fmap delta-mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ cong (\t -> delta (headDelta (t (delta (headDelta d) (delta-bind dd tailDelta)))) (delta-bind (delta-fmap delta-mu ds) (tailDelta ∙ (n-tail (S n)))))(sym (n-tail-plus n)) ⟩ delta (headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta d) (delta-bind dd tailDelta)))) (delta-bind (delta-fmap delta-mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ delta (headDelta ((n-tail n) (delta-bind dd tailDelta))) (delta-bind (delta-fmap delta-mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ delta (headDelta ((n-tail n) (delta-bind dd tailDelta))) (delta-bind (delta-fmap delta-mu ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta ((n-tail n) (delta-bind dd tailDelta))) de) (monad-law-1-3 (S (S n)) ds) ⟩ delta (headDelta ((n-tail n) (delta-bind dd tailDelta))) (delta-bind (delta-bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta de ( (delta-bind (delta-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)))) (delta-bind (delta-bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩ delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (delta-bind (delta-bind ds (n-tail (S (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (delta-bind (delta-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))))) (delta-bind (delta-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))))) (delta-bind (delta-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))))) (delta-bind (delta-bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ delta-bind (delta (headDelta ((n-tail (S n)) (delta d dd))) (delta-bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩ delta-bind (delta-bind (delta (delta d dd) ds) (n-tail (S n))) (n-tail (S n)) ∎ -- monad-law-1 : join . delta-fmap join = join . join monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((delta-mu ∙ (delta-fmap delta-mu)) d) ≡ ((delta-mu ∙ delta-mu) d) monad-law-1 (mono d) = refl monad-law-1 (delta x d) = begin (delta-mu ∙ delta-fmap delta-mu) (delta x d) ≡⟨ refl ⟩ delta-mu (delta-fmap delta-mu (delta x d)) ≡⟨ refl ⟩ delta-mu (delta (delta-mu x) (delta-fmap delta-mu d)) ≡⟨ refl ⟩ delta (headDelta (delta-mu x)) (delta-bind (delta-fmap delta-mu d) tailDelta) ≡⟨ cong (\x -> delta x (delta-bind (delta-fmap delta-mu d) tailDelta)) (monad-law-1-2 x) ⟩ delta (headDelta (headDelta x)) (delta-bind (delta-fmap delta-mu d) tailDelta) ≡⟨ cong (\d -> delta (headDelta (headDelta x)) d) (monad-law-1-3 (S O) d) ⟩ delta (headDelta (headDelta x)) (delta-bind (delta-bind d tailDelta) tailDelta) ≡⟨ refl ⟩ delta-mu (delta (headDelta x) (delta-bind d tailDelta)) ≡⟨ refl ⟩ delta-mu (delta-mu (delta x d)) ≡⟨ refl ⟩ (delta-mu ∙ delta-mu) (delta x d) ∎ monad-law-2-1 : {l : Level} {A : Set l} -> (n : Nat) -> (d : Delta A) -> (delta-bind (delta-fmap delta-eta d) (n-tail n)) ≡ d monad-law-2-1 O (mono x) = refl monad-law-2-1 O (delta x d) = begin delta-bind (delta-fmap delta-eta (delta x d)) (n-tail O) ≡⟨ refl ⟩ delta-bind (delta (delta-eta x) (delta-fmap delta-eta d)) id ≡⟨ refl ⟩ delta (headDelta (delta-eta x)) (delta-bind (delta-fmap delta-eta d) tailDelta) ≡⟨ refl ⟩ delta x (delta-bind (delta-fmap delta-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 delta-bind (delta-fmap delta-eta (mono x)) (n-tail (S n)) ≡⟨ refl ⟩ delta-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 delta-bind (delta-fmap delta-eta (delta x d)) (n-tail (S n)) ≡⟨ refl ⟩ delta-bind (delta (delta-eta x) (delta-fmap delta-eta d)) (n-tail (S n)) ≡⟨ refl ⟩ delta (headDelta ((n-tail (S n) (delta-eta x)))) (delta-bind (delta-fmap delta-eta d) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ delta (headDelta ((n-tail (S n) (delta-eta x)))) (delta-bind (delta-fmap delta-eta d) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta (de)) (delta-bind (delta-fmap delta-eta d) (n-tail (S (S n))))) (tail-delta-to-mono (S n) x) ⟩ delta (headDelta (delta-eta x)) (delta-bind (delta-fmap delta-eta d) (n-tail (S (S n)))) ≡⟨ refl ⟩ delta x (delta-bind (delta-fmap delta-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 . delta-fmap return = join . return = id -- monad-law-2 join . delta-fmap return = join . return monad-law-2 : {l : Level} {A : Set l} -> (d : Delta A) -> (delta-mu ∙ delta-fmap delta-eta) d ≡ (delta-mu ∙ delta-eta) d monad-law-2 (mono x) = refl monad-law-2 (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 (mono x) (delta-fmap delta-eta d)) ≡⟨ refl ⟩ delta (headDelta (mono x)) (delta-bind (delta-fmap delta-eta d) tailDelta) ≡⟨ refl ⟩ delta x (delta-bind (delta-fmap delta-eta d) tailDelta) ≡⟨ cong (\d -> delta x d) (monad-law-2-1 (S O) d) ⟩ (delta x d) ≡⟨ refl ⟩ delta-mu (mono (delta x d)) ≡⟨ refl ⟩ delta-mu (delta-eta (delta x d)) ≡⟨ refl ⟩ (delta-mu ∙ delta-eta) (delta x d) ∎ -- monad-law-2' : join . return = id monad-law-2' : {l : Level} {A : Set l} -> (d : Delta A) -> (delta-mu ∙ delta-eta) d ≡ id d monad-law-2' d = refl -- monad-law-3 : return . f = delta-fmap f . return monad-law-3 : {l : Level} {A B : Set l} (f : A -> B) (x : A) -> (delta-eta ∙ f) x ≡ (delta-fmap f ∙ delta-eta) x monad-law-3 f x = refl monad-law-4-1 : {l : Level} {A B : Set l} -> (n : Nat) -> (f : A -> B) -> (ds : Delta (Delta A)) -> delta-bind (delta-fmap (delta-fmap f) ds) (n-tail n) ≡ delta-fmap f (delta-bind ds (n-tail n)) monad-law-4-1 O f (mono d) = refl monad-law-4-1 O f (delta d ds) = begin delta-bind (delta-fmap (delta-fmap f) (delta d ds)) (n-tail O) ≡⟨ refl ⟩ delta-bind (delta (delta-fmap f d) (delta-fmap (delta-fmap f) ds)) (n-tail O) ≡⟨ refl ⟩ delta (headDelta (delta-fmap f d)) (delta-bind (delta-fmap (delta-fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta de (delta-bind (delta-fmap (delta-fmap f) ds) tailDelta)) (head-delta-natural-transformation f d) ⟩ delta (f (headDelta d)) (delta-bind (delta-fmap (delta-fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f (headDelta d)) de) (monad-law-4-1 (S O) f ds) ⟩ delta (f (headDelta d)) (delta-fmap f (delta-bind ds tailDelta)) ≡⟨ refl ⟩ delta-fmap f (delta (headDelta d) (delta-bind ds tailDelta)) ≡⟨ refl ⟩ delta-fmap f (delta-bind (delta d ds) (n-tail O)) ∎ monad-law-4-1 (S n) f (mono d) = begin delta-bind (delta-fmap (delta-fmap f) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩ delta-bind (mono (delta-fmap f d)) (n-tail (S n)) ≡⟨ refl ⟩ n-tail (S n) (delta-fmap f d) ≡⟨ n-tail-natural-transformation (S n) f d ⟩ delta-fmap f (n-tail (S n) d) ≡⟨ refl ⟩ delta-fmap f (delta-bind (mono d) (n-tail (S n))) ∎ monad-law-4-1 (S n) f (delta d ds) = begin delta-bind (delta-fmap (delta-fmap f) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩ delta (headDelta (n-tail (S n) (delta-fmap f d))) (delta-bind (delta-fmap (delta-fmap f) ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ delta (headDelta (n-tail (S n) (delta-fmap f d))) (delta-bind (delta-fmap (delta-fmap f) ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta de) (delta-bind (delta-fmap (delta-fmap f) ds) (n-tail (S (S n))))) (n-tail-natural-transformation (S n) f d) ⟩ delta (headDelta (delta-fmap f ((n-tail (S n) d)))) (delta-bind (delta-fmap (delta-fmap f) ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta de (delta-bind (delta-fmap (delta-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))) (delta-bind (delta-fmap (delta-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))) (delta-fmap f (delta-bind ds (n-tail (S (S n))))) ≡⟨ refl ⟩ delta-fmap f (delta (headDelta (n-tail (S n) d)) (delta-bind ds (n-tail (S (S n))))) ≡⟨ refl ⟩ delta-fmap f (delta (headDelta (n-tail (S n) d)) (delta-bind ds (tailDelta ∙ (n-tail (S n))))) ≡⟨ refl ⟩ delta-fmap f (delta-bind (delta d ds) (n-tail (S n))) ∎ -- monad-law-4 : join . delta-fmap (delta-fmap f) = delta-fmap f . join monad-law-4 : {l : Level} {A B : Set l} (f : A -> B) (d : Delta (Delta A)) -> (delta-mu ∙ delta-fmap (delta-fmap f)) d ≡ (delta-fmap f ∙ delta-mu) d monad-law-4 f (mono d) = refl monad-law-4 f (delta (mono x) ds) = begin (delta-mu ∙ delta-fmap (delta-fmap f)) (delta (mono x) ds) ≡⟨ refl ⟩ delta-mu ( delta-fmap (delta-fmap f) (delta (mono x) ds)) ≡⟨ refl ⟩ delta-mu (delta (mono (f x)) (delta-fmap (delta-fmap f) ds)) ≡⟨ refl ⟩ delta (headDelta (mono (f x))) (delta-bind (delta-fmap (delta-fmap f) ds) tailDelta) ≡⟨ refl ⟩ delta (f x) (delta-bind (delta-fmap (delta-fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f x) de) (monad-law-4-1 (S O) f ds) ⟩ delta (f x) (delta-fmap f (delta-bind ds tailDelta)) ≡⟨ refl ⟩ delta-fmap f (delta x (delta-bind ds tailDelta)) ≡⟨ refl ⟩ delta-fmap f (delta (headDelta (mono x)) (delta-bind ds tailDelta)) ≡⟨ refl ⟩ delta-fmap f (delta-mu (delta (mono x) ds)) ≡⟨ refl ⟩ (delta-fmap f ∙ delta-mu) (delta (mono x) ds) ∎ monad-law-4 f (delta (delta x d) ds) = begin (delta-mu ∙ delta-fmap (delta-fmap f)) (delta (delta x d) ds) ≡⟨ refl ⟩ delta-mu (delta-fmap (delta-fmap f) (delta (delta x d) ds)) ≡⟨ refl ⟩ delta-mu (delta (delta (f x) (delta-fmap f d)) (delta-fmap (delta-fmap f) ds)) ≡⟨ refl ⟩ delta (headDelta (delta (f x) (delta-fmap f d))) (delta-bind (delta-fmap (delta-fmap f) ds) tailDelta) ≡⟨ refl ⟩ delta (f x) (delta-bind (delta-fmap (delta-fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f x) de) (monad-law-4-1 (S O) f ds) ⟩ delta (f x) (delta-fmap f (delta-bind ds tailDelta)) ≡⟨ refl ⟩ delta-fmap f (delta x (delta-bind ds tailDelta)) ≡⟨ refl ⟩ delta-fmap f (delta (headDelta (delta x d)) (delta-bind ds tailDelta)) ≡⟨ refl ⟩ delta-fmap f (delta-mu (delta (delta x d) ds)) ≡⟨ refl ⟩ (delta-fmap f ∙ delta-mu) (delta (delta x d) ds) ∎ delta-is-monad : {l : Level} -> Monad {l} Delta delta-is-functor delta-is-monad = record { eta = delta-eta; mu = delta-mu; return = delta-eta; bind = delta-bind; association-law = monad-law-1; left-unity-law = monad-law-2; right-unity-law = monad-law-2' } {- -- 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) ∎ -}