# HG changeset patch # User Yasutaka Higa # Date 1422348565 -32400 # Node ID e6499a50ccbdd29a819d4f69b381a8683d616980 # Parent ebd0d6e2772c00faa73ae7b839757cd474d3c44d Retrying prove monad-laws for delta diff -r ebd0d6e2772c -r e6499a50ccbd agda/delta.agda --- a/agda/delta.agda Mon Jan 26 23:00:05 2015 +0900 +++ b/agda/delta.agda Tue Jan 27 17:49:25 2015 +0900 @@ -1,7 +1,6 @@ open import list open import basic open import nat -open import revision open import laws open import Level @@ -10,42 +9,47 @@ module delta where -data Delta {l : Level} (A : Set l) : (Rev -> (Set l)) where - mono : A -> Delta A init - delta : {v : Rev} -> A -> Delta A v -> Delta A (commit v) +data Delta {l : Level} (A : Set l) : (Nat -> (Set l)) where + mono : A -> Delta A (S O) + delta : {n : Nat} -> A -> Delta A (S n) -> Delta A (S (S n)) -deltaAppend : {l : Level} {A : Set l} {n m : Rev} -> Delta A n -> Delta A m -> Delta A (merge n m) +deltaAppend : {l : Level} {A : Set l} {n m : Nat} -> Delta A (S n) -> Delta A (S m) -> Delta A ((S n) + (S m)) deltaAppend (mono x) d = delta x d deltaAppend (delta x d) ds = delta x (deltaAppend d ds) -headDelta : {l : Level} {A : Set l} {n : Rev} -> Delta A n -> A +headDelta : {l : Level} {A : Set l} {n : Nat} -> Delta A (S n) -> A headDelta (mono x) = x headDelta (delta x _) = x -tailDelta : {l : Level} {A : Set l} {n : Rev} -> Delta A (commit n) -> Delta A n +tailDelta : {l : Level} {A : Set l} {n : Nat} -> Delta A (S (S n)) -> Delta A (S n) tailDelta (delta _ d) = d -- Functor -delta-fmap : {l : Level} {A B : Set l} {n : Rev} -> (A -> B) -> (Delta A n) -> (Delta B n) +delta-fmap : {l : Level} {A B : Set l} {n : Nat} -> (A -> B) -> (Delta A (S n)) -> (Delta B (S n)) delta-fmap f (mono x) = mono (f x) delta-fmap f (delta x d) = delta (f x) (delta-fmap f d) -- Monad (Category) -delta-eta : {l : Level} {A : Set l} {v : Rev} -> A -> Delta A v -delta-eta {v = init} x = mono x -delta-eta {v = commit v} x = delta x (delta-eta {v = v} x) +delta-eta : {l : Level} {A : Set l} {n : Nat} -> A -> Delta A (S n) +delta-eta {n = O} x = mono x +delta-eta {n = (S n)} x = delta x (delta-eta {n = n} x) + + + -delta-bind : {l : Level} {A B : Set l} {n : Rev} -> (Delta A n) -> (A -> Delta B n) -> Delta B n -delta-bind (mono x) f = f x -delta-bind (delta x d) f = delta (headDelta (f x)) (tailDelta (f x)) +delta-mu : {l : Level} {A : Set l} {n : Nat} -> (Delta (Delta A (S n)) (S n)) -> Delta A (S n) +delta-mu (mono x) = x +delta-mu (delta x d) = delta (headDelta x) (delta-mu (delta-fmap tailDelta d)) -delta-mu : {l : Level} {A : Set l} {n : Rev} -> (Delta (Delta A n) n) -> Delta A n -delta-mu d = delta-bind d id +delta-bind : {l : Level} {A B : Set l} {n : Nat} -> (Delta A (S n)) -> (A -> Delta B (S n)) -> Delta B (S n) +delta-bind d f = delta-mu (delta-fmap f d) +--delta-bind (mono x) f = f x +--delta-bind (delta x d) f = delta (headDelta (f x)) (tailDelta (f x)) {- diff -r ebd0d6e2772c -r e6499a50ccbd agda/delta/functor.agda --- a/agda/delta/functor.agda Mon Jan 26 23:00:05 2015 +0900 +++ b/agda/delta/functor.agda Tue Jan 27 17:49:25 2015 +0900 @@ -1,32 +1,43 @@ open import Level open import Relation.Binary.PropositionalEquality - open import basic open import delta open import laws open import nat -open import revision - - module delta.functor where -- Functor-laws -- Functor-law-1 : T(id) = id' -functor-law-1 : {l : Level} {A : Set l} {n : Rev} -> (d : Delta A n) -> (delta-fmap id) d ≡ id d +functor-law-1 : {l : Level} {A : Set l} {n : Nat} -> (d : Delta A (S n)) -> (delta-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 : Level} {n : Rev} {A B C : Set l} -> - (f : B -> C) -> (g : A -> B) -> (d : Delta A n) -> +functor-law-2 : {l : Level} {n : Nat} {A B C : Set l} -> + (f : B -> C) -> (g : A -> B) -> (d : Delta A (S n)) -> (delta-fmap (f ∙ g)) d ≡ ((delta-fmap f) ∙ (delta-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) -delta-is-functor : {l : Level} {n : Rev} -> Functor {l} (\A -> Delta A n) + + +delta-is-functor : {l : Level} {n : Nat} -> Functor {l} (\A -> Delta A (S n)) delta-is-functor = record { fmap = delta-fmap ; preserve-id = functor-law-1; covariant = \f g -> functor-law-2 g f} + + +open ≡-Reasoning +delta-fmap-equiv : {l : Level} {A B : Set l} {n : Nat} + (f g : A -> B) (eq : f ≡ g) (d : Delta A (S n)) -> + delta-fmap f d ≡ delta-fmap g d +delta-fmap-equiv f g eq (mono x) = begin + mono (f x) ≡⟨ cong (\h -> (mono (h x))) eq ⟩ + mono (g x) ∎ +delta-fmap-equiv f g eq (delta x d) = begin + delta (f x) (delta-fmap f d) ≡⟨ cong (\h -> (delta (h x) (delta-fmap f d))) eq ⟩ + delta (g x) (delta-fmap f d) ≡⟨ cong (\fx -> (delta (g x) fx)) (delta-fmap-equiv f g eq d) ⟩ + delta (g x) (delta-fmap g d) ∎ diff -r ebd0d6e2772c -r e6499a50ccbd agda/delta/monad.agda --- a/agda/delta/monad.agda Mon Jan 26 23:00:05 2015 +0900 +++ b/agda/delta/monad.agda Tue Jan 27 17:49:25 2015 +0900 @@ -3,7 +3,6 @@ open import delta.functor open import nat open import laws -open import revision open import Level @@ -12,335 +11,130 @@ 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-1-5 : {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 : 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) ∎ -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) +{- +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) ∎ -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))) ∎ +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) ∎ --- 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-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) ∎ --} --- monad-law-1 : join . delta-fmap join = join . join -monad-law-1 : {l : Level} {A : Set l} {a : Rev} -> (d : Delta (Delta (Delta A a) a) a) -> - ((delta-mu ∙ (delta-fmap delta-mu)) d) ≡ ((delta-mu ∙ delta-mu) d) -monad-law-1 (mono d) = refl -monad-law-1 (delta d ds) = {!!} + -delta-is-monad : {l : Level} {v : Rev} -> Monad {l} (\A -> Delta A v) delta-is-functor +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; - association-law = monad-law-1 } --- left-unity-law = monad-law-2; --- right-unity-law = monad-law-2' } + 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)) } @@ -350,7 +144,7 @@ -- Monad-laws (Haskell) -- monad-law-h-1 : return a >>= k = k a -monad-law-h-1 : {l : Level} {A B : Set l} -> +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 @@ -365,7 +159,7 @@ -- monad-law-h-3 : m >>= (\x -> f x >>= g) = (m >>= f) >>= g -monad-law-h-3 : {l : Level} {A B C : Set l} -> +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 @@ -376,4 +170,4 @@ --} \ No newline at end of file +-}