Mercurial > hg > Members > atton > delta_monad
changeset 90:55d11ce7e223
Unify levels on data type. only use suc to proofs
| author | Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 19 Jan 2015 12:11:38 +0900 |
| parents | 5411ce26d525 |
| children | f41682b53992 |
| files | agda/basic.agda agda/delta.agda agda/delta/functor.agda agda/delta/monad.agda agda/deltaM.agda agda/laws.agda |
| diffstat | 6 files changed, 111 insertions(+), 111 deletions(-) [+] |
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--- a/agda/basic.agda Mon Jan 19 11:48:41 2015 +0900 +++ b/agda/basic.agda Mon Jan 19 12:11:38 2015 +0900 @@ -5,7 +5,7 @@ id : {l : Level} {A : Set l} -> A -> A id x = x -_∙_ : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} -> (B -> C) -> (A -> B) -> (A -> C) +_∙_ : {l : Level} {A B C : Set l} -> (B -> C) -> (A -> B) -> (A -> C) f ∙ g = \x -> f (g x) postulate String : Set
--- a/agda/delta.agda Mon Jan 19 11:48:41 2015 +0900 +++ b/agda/delta.agda Mon Jan 19 12:11:38 2015 +0900 @@ -42,7 +42,7 @@ 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 : {l : Level} {A B : Set l} -> (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)) @@ -60,7 +60,7 @@ return : {l : Level} {A : Set l} -> A -> Delta A return = eta -_>>=_ : {l ll : Level} {A : Set l} {B : Set ll} -> +_>>=_ : {l : Level} {A B : Set l} -> (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))
--- a/agda/delta/functor.agda Mon Jan 19 11:48:41 2015 +0900 +++ b/agda/delta/functor.agda Mon Jan 19 12:11:38 2015 +0900 @@ -16,7 +16,7 @@ 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} -> +functor-law-2 : {l : Level} {A B C : Set l} -> (f : B -> C) -> (g : A -> B) -> (d : Delta A) -> (delta-fmap (f ∙ g)) d ≡ (delta-fmap f) (delta-fmap g d) functor-law-2 f g (mono x) = refl
--- a/agda/delta/monad.agda Mon Jan 19 11:48:41 2015 +0900 +++ b/agda/delta/monad.agda Mon Jan 19 12:11:38 2015 +0900 @@ -132,19 +132,19 @@ 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) + bind (delta-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) ⟩ + bind (delta-fmap mu (delta d ds)) (n-tail O) ≡⟨ refl ⟩ + bind (delta (mu d) (delta-fmap mu ds)) (n-tail O) ≡⟨ refl ⟩ + delta (headDelta (mu d)) (bind (delta-fmap mu ds) tailDelta) ≡⟨ cong (\dx -> delta dx (bind (delta-fmap mu ds) tailDelta)) (monad-law-1-2 d) ⟩ + delta (headDelta (headDelta d)) (bind (delta-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 (delta-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))⟩ @@ -153,7 +153,7 @@ 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 (delta-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)) ⟩ @@ -165,11 +165,11 @@ 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) ⟩ + bind (delta-fmap mu (delta (mono d) ds)) (n-tail (S n)) ≡⟨ refl ⟩ + bind (delta (mu (mono d)) (delta-fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ + bind (delta d (delta-fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ + delta (headDelta ((n-tail (S n)) d)) (bind (delta-fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ + delta (headDelta ((n-tail (S n)) d)) (bind (delta-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 ⟩ @@ -179,13 +179,13 @@ 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) ⟩ + bind (delta-fmap mu (delta (delta d dd) ds)) (n-tail (S n)) ≡⟨ refl ⟩ + bind (delta (mu (delta d dd)) (delta-fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ + delta (headDelta ((n-tail (S n)) (mu (delta d dd)))) (bind (delta-fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ + delta (headDelta ((n-tail (S n)) (delta (headDelta d) (bind dd tailDelta)))) (bind (delta-fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ cong (\t -> delta (headDelta (t (delta (headDelta d) (bind dd tailDelta)))) (bind (delta-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 (delta-fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ + delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (delta-fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ + delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (delta-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 ⟩ @@ -197,15 +197,15 @@ ∎ --- 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 : join . delta-fmap join = join . join +monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((mu ∙ (delta-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) ⟩ + (mu ∙ delta-fmap mu) (delta x d) ≡⟨ refl ⟩ + mu (delta-fmap mu (delta x d)) ≡⟨ refl ⟩ + mu (delta (mu x) (delta-fmap mu d)) ≡⟨ refl ⟩ + delta (headDelta (mu x)) (bind (delta-fmap mu d) tailDelta) ≡⟨ cong (\x -> delta x (bind (delta-fmap mu d) tailDelta)) (monad-law-1-2 x) ⟩ + delta (headDelta (headDelta x)) (bind (delta-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 ⟩ @@ -213,41 +213,41 @@ ∎ -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 : {l : Level} {A : Set l} -> (n : Nat) -> (d : Delta A) -> (bind (delta-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) ⟩ + bind (delta-fmap eta (delta x d)) (n-tail O) ≡⟨ refl ⟩ + bind (delta (eta x) (delta-fmap eta d)) id ≡⟨ refl ⟩ + delta (headDelta (eta x)) (bind (delta-fmap eta d) tailDelta) ≡⟨ refl ⟩ + delta x (bind (delta-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 (delta-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) ⟩ + bind (delta-fmap eta (delta x d)) (n-tail (S n)) ≡⟨ refl ⟩ + bind (delta (eta x) (delta-fmap eta d)) (n-tail (S n)) ≡⟨ refl ⟩ + delta (headDelta ((n-tail (S n) (eta x)))) (bind (delta-fmap eta d) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ + delta (headDelta ((n-tail (S n) (eta x)))) (bind (delta-fmap eta d) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta (de)) (bind (delta-fmap eta d) (n-tail (S (S n))))) (tail-delta-to-mono (S n) x) ⟩ + delta (headDelta (eta x)) (bind (delta-fmap eta d) (n-tail (S (S n)))) ≡⟨ refl ⟩ + delta x (bind (delta-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 : 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) -> - (mu ∙ fmap eta) d ≡ (mu ∙ eta) d + (mu ∙ delta-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) ⟩ + (mu ∙ delta-fmap eta) (delta x d) ≡⟨ refl ⟩ + mu (delta-fmap eta (delta x d)) ≡⟨ refl ⟩ + mu (delta (mono x) (delta-fmap eta d)) ≡⟨ refl ⟩ + delta (headDelta (mono x)) (bind (delta-fmap eta d) tailDelta) ≡⟨ refl ⟩ + delta x (bind (delta-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 ⟩ @@ -260,67 +260,67 @@ 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 : return . f = delta-fmap f . return +monad-law-3 : {l : Level} {A B : Set l} (f : A -> B) (x : A) -> (eta ∙ f) x ≡ (delta-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)) + bind (delta-fmap (delta-fmap f) ds) (n-tail n) ≡ delta-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)) ∎ + bind (delta-fmap (delta-fmap f) (delta d ds)) (n-tail O) ≡⟨ refl ⟩ + bind (delta (delta-fmap f d) (delta-fmap (delta-fmap f) ds)) (n-tail O) ≡⟨ refl ⟩ + delta (headDelta (delta-fmap f d)) (bind (delta-fmap (delta-fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta de (bind (delta-fmap (delta-fmap f) ds) tailDelta)) (head-delta-natural-transformation f d) ⟩ + delta (f (headDelta d)) (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 (bind ds tailDelta)) ≡⟨ refl ⟩ + delta-fmap f (delta (headDelta d) (bind ds tailDelta)) ≡⟨ refl ⟩ + delta-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))) + bind (delta-fmap (delta-fmap f) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩ + 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 (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))) ∎ + bind (delta-fmap (delta-fmap f) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩ + delta (headDelta (n-tail (S n) (delta-fmap f d))) (bind (delta-fmap (delta-fmap f) ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ + delta (headDelta (n-tail (S n) (delta-fmap f d))) (bind (delta-fmap (delta-fmap f) ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta de) (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)))) (bind (delta-fmap (delta-fmap f) ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta de (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))) (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 (bind ds (n-tail (S (S n))))) ≡⟨ refl ⟩ + delta-fmap f (delta (headDelta (n-tail (S n) d)) (bind ds (n-tail (S (S n))))) ≡⟨ refl ⟩ + delta-fmap f (delta (headDelta (n-tail (S n) d)) (bind ds (tailDelta ∙ (n-tail (S n))))) ≡⟨ refl ⟩ + delta-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 : 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)) -> + (mu ∙ delta-fmap (delta-fmap f)) d ≡ (delta-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) ∎ + (mu ∙ delta-fmap (delta-fmap f)) (delta (mono x) ds) ≡⟨ refl ⟩ + mu ( delta-fmap (delta-fmap f) (delta (mono x) ds)) ≡⟨ refl ⟩ + mu (delta (mono (f x)) (delta-fmap (delta-fmap f) ds)) ≡⟨ refl ⟩ + delta (headDelta (mono (f x))) (bind (delta-fmap (delta-fmap f) ds) tailDelta) ≡⟨ refl ⟩ + delta (f x) (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 (bind ds tailDelta)) ≡⟨ refl ⟩ + delta-fmap f (delta x (bind ds tailDelta)) ≡⟨ refl ⟩ + delta-fmap f (delta (headDelta (mono x)) (bind ds tailDelta)) ≡⟨ refl ⟩ + delta-fmap f (mu (delta (mono x) ds)) ≡⟨ refl ⟩ + (delta-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) ∎ + (mu ∙ delta-fmap (delta-fmap f)) (delta (delta x d) ds) ≡⟨ refl ⟩ + mu (delta-fmap (delta-fmap f) (delta (delta x d) ds)) ≡⟨ refl ⟩ + 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))) (bind (delta-fmap (delta-fmap f) ds) tailDelta) ≡⟨ refl ⟩ + delta (f x) (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 (bind ds tailDelta)) ≡⟨ refl ⟩ + delta-fmap f (delta x (bind ds tailDelta)) ≡⟨ refl ⟩ + delta-fmap f (delta (headDelta (delta x d)) (bind ds tailDelta)) ≡⟨ refl ⟩ + delta-fmap f (mu (delta (delta x d) ds)) ≡⟨ refl ⟩ + (delta-fmap f ∙ mu) (delta (delta x d) ds) ∎ delta-is-monad : {l : Level} {A : Set l} -> Monad {l} {A} Delta delta-is-functor delta-is-monad = record { mu = mu;
--- a/agda/deltaM.agda Mon Jan 19 11:48:41 2015 +0900 +++ b/agda/deltaM.agda Mon Jan 19 12:11:38 2015 +0900 @@ -54,11 +54,11 @@ checkOut (S n) (deltaM (mono x)) = x checkOut {l} {A} {M} {functorM} {monadM} (S n) (deltaM (delta _ d)) = checkOut {l} {A} {M} {functorM} {monadM} n (deltaM d) -{- -deltaM-fmap : {l ll : Level} {A : Set l} {B : Set ll} - {M : {l' : Level} -> Set l' -> Set l'} - {functorM : {l' : Level} -> Functor {l'} M} - {monadM : {l' : Level} {A : Set l'} -> Monad {l'} {A} M functorM} - -> (A -> B) -> DeltaM M {functorM} {monadM} A -> DeltaM M {functorM} {monadM} B -deltaM-fmap {l} {ll} {A} {B} {M} {functorM} f (deltaM d) = deltaM (Functor.fmap delta-is-functor (Functor.fmap functorM f) d) --} \ No newline at end of file + +open Functor +deltaM-fmap : {l : Level} {A B : Set l} + {M : {l' : Level} -> Set l' -> Set l'} + {functorM : {l' : Level} -> Functor {l'} M} + {monadM : {l' : Level} {A : Set l'} -> Monad {l'} {A} M functorM} + -> (A -> B) -> DeltaM M {functorM} {monadM} A -> DeltaM M {functorM} {monadM} B +deltaM-fmap {l} {A} {B} {M} {functorM} f (deltaM d) = deltaM (fmap delta-is-functor (fmap functorM f) d)
--- a/agda/laws.agda Mon Jan 19 11:48:41 2015 +0900 +++ b/agda/laws.agda Mon Jan 19 12:11:38 2015 +0900 @@ -6,18 +6,18 @@ record Functor {l : Level} (F : Set l -> Set l) : (Set (suc l)) where field - fmap : ∀{A B} -> (A -> B) -> (F A) -> (F B) + fmap : {A B : Set l} -> (A -> B) -> (F A) -> (F B) field - preserve-id : ∀{A} (x : F A) → fmap id x ≡ id x - covariant : ∀{A B C} (f : A -> B) -> (g : B -> C) -> (x : F A) - -> fmap (g ∙ f) x ≡ ((fmap g) ∙ (fmap f)) x + preserve-id : {A : Set l} (x : F A) → fmap id x ≡ id x + covariant : {A B C : Set l} (f : A -> B) -> (g : B -> C) -> (x : F A) + -> fmap (g ∙ f) x ≡ ((fmap g) ∙ (fmap f)) x open Functor -record NaturalTransformation {l ll : Level} (F G : Set l -> Set l) +record NaturalTransformation {l : Level} (F G : Set l -> Set l) (functorF : Functor F) - (functorG : Functor G) : Set (suc (l ⊔ ll)) where + (functorG : Functor G) : Set (suc l) where field natural-transformation : {A : Set l} -> F A -> G A field