Mercurial > hg > Members > atton > delta_monad
changeset 94:bcd4fe52a504
Rewrite monad definitions for delta/deltaM
| author | Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 19 Jan 2015 17:10:29 +0900 |
| parents | 8d92ed54a94f |
| children | cf372fbcebd8 |
| files | agda/delta.agda agda/delta/monad.agda agda/deltaM.agda agda/laws.agda delta.hs |
| diffstat | 5 files changed, 232 insertions(+), 207 deletions(-) [+] |
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--- a/agda/delta.agda Mon Jan 19 15:21:29 2015 +0900 +++ b/agda/delta.agda Mon Jan 19 17:10:29 2015 +0900 @@ -39,30 +39,25 @@ -- Monad (Category) -eta : {l : Level} {A : Set l} -> A -> Delta A -eta x = mono x - -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)) +delta-eta : {l : Level} {A : Set l} -> A -> Delta A +delta-eta x = mono x -mu : {l : Level} {A : Set l} -> Delta (Delta A) -> Delta A -mu d = bind d id +delta-bind : {l : Level} {A B : Set l} -> (Delta A) -> (A -> Delta B) -> Delta B +delta-bind (mono x) f = f x +delta-bind (delta x d) f = delta (headDelta (f x)) (delta-bind d (tailDelta ∙ f)) -returnS : {l : Level} {A : Set l} -> A -> Delta A -returnS x = mono x +delta-mu : {l : Level} {A : Set l} -> Delta (Delta A) -> Delta A +delta-mu d = delta-bind d id -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 +delta-return : {l : Level} {A : Set l} -> A -> Delta A +delta-return = delta-eta _>>=_ : {l : Level} {A B : Set l} -> (x : Delta A) -> (f : A -> (Delta B)) -> (Delta B) -(mono x) >>= f = f x +(mono x) >>= f = f x (delta x d) >>= f = delta (headDelta (f x)) (d >>= (tailDelta ∙ f))
--- a/agda/delta/monad.agda Mon Jan 19 15:21:29 2015 +0900 +++ b/agda/delta/monad.agda Mon Jan 19 17:10:29 2015 +0900 @@ -15,36 +15,36 @@ -- 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)) + 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) (bind (mono ds) (n-tail O)) ≡⟨ refl ⟩ + n-tail (S n) (delta-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)) + 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) (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)) + 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 (bind (mono (mono x)) (n-tail (S m))) ≡⟨ refl ⟩ + 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 ⟩ - 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)) + 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 (bind (mono (delta x ds)) (n-tail (S m))) ≡⟨ refl ⟩ + 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) ⟩ @@ -53,186 +53,186 @@ ((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)) + 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 (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)) + 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) (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)) + 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) (bind dd (n-tail m))) ≡ headDelta ((n-tail (m + n)) (headDelta (n-tail n dd))) + 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) (bind (mono dd) (n-tail O))) ≡⟨ refl ⟩ + 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) (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 (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 (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 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 (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 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) (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 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 (mu d) ≡ (headDelta (headDelta d)) +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))) -> - bind (delta-fmap mu d) (n-tail n) ≡ bind (bind d (n-tail n)) (n-tail n) + 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 - 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) + 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 - bind (delta-fmap mu (mono (mono d))) (n-tail (S n)) ≡⟨ refl ⟩ - bind (mono d) (n-tail (S n)) ≡⟨ refl ⟩ + 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 ⟩ - 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)) + 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 - 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)) ⟩ - (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)) + 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 - 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 ⟩ - 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)) + 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 - 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 ⟩ - 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)) + 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))) -> ((mu ∙ (delta-fmap mu)) d) ≡ ((mu ∙ mu) d) +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 - (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 ⟩ - (mu ∙ mu) (delta x d) + (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) -> (bind (delta-fmap eta d) (n-tail n)) ≡ 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 - 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-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 - bind (delta-fmap eta (mono x)) (n-tail (S n)) ≡⟨ refl ⟩ - bind (mono (mono x)) (n-tail (S n)) ≡⟨ refl ⟩ + 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 - 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-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 ∎ @@ -240,91 +240,93 @@ -- 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 ∙ delta-fmap eta) d ≡ (mu ∙ eta) d + (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 - (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-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 ⟩ - mu (mono (delta x d)) ≡⟨ refl ⟩ - mu (eta (delta x d)) ≡⟨ refl ⟩ - (mu ∙ eta) (delta x d) + 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) -> (mu ∙ eta) d ≡ id d +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) -> (eta ∙ f) x ≡ (delta-fmap f ∙ eta) x +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 ll : Level} {A : Set l} {B : Set ll} -> (n : Nat) -> (f : A -> B) -> (ds : Delta (Delta A)) -> - bind (delta-fmap (delta-fmap f) ds) (n-tail n) ≡ delta-fmap f (bind ds (n-tail n)) + 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 - 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)) ∎ + 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 - bind (delta-fmap (delta-fmap f) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩ - bind (mono (delta-fmap f d)) (n-tail (S n)) ≡⟨ refl ⟩ + 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 (bind (mono d) (n-tail (S n))) + delta-fmap f (delta-bind (mono d) (n-tail (S n))) ∎ monad-law-4-1 (S n) f (delta d ds) = begin - 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))) ∎ + 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)) -> - (mu ∙ delta-fmap (delta-fmap f)) d ≡ (delta-fmap f ∙ mu) d + (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 - (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) ∎ + (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 - (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-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} {A : Set l} -> Monad {l} {A} Delta delta-is-functor -delta-is-monad = record { mu = mu; - eta = eta; +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' }
--- a/agda/deltaM.agda Mon Jan 19 15:21:29 2015 +0900 +++ b/agda/deltaM.agda Mon Jan 19 17:10:29 2015 +0900 @@ -1,5 +1,6 @@ open import Level +open import basic open import delta open import delta.functor open import nat @@ -55,6 +56,8 @@ checkOut {l} {A} {M} {functorM} {monadM} (S n) (deltaM (delta _ d)) = checkOut {l} {A} {M} {functorM} {monadM} n (deltaM d) + +-- functor definitions open Functor deltaM-fmap : {l : Level} {A B : Set l} {M : {l' : Level} -> Set l' -> Set l'} @@ -62,3 +65,25 @@ {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) + +-- monad definitions +open Monad +deltaM-eta : {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 -> (DeltaM M {functorM} {monadM} A) +deltaM-eta {_} {A} {_} {_} {_} {monadM} x = deltaM (mono (eta {_} {A} monadM x)) + +deltaM-bind : {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} + -> (DeltaM M {functorM} {monadM} A) -> (A -> DeltaM M {functorM} {monadM} B) -> DeltaM M {functorM} {monadM} B +deltaM-bind {l} {A} {B} {M} {functorM} {monadM} (deltaM (mono x)) f = deltaM (mono (bind {l} {A} monadM x (headDeltaM ∙ f))) +deltaM-bind {l} {A} {B} {M} {functorM} {monadM} (deltaM (delta x d)) f = appendDeltaM (deltaM (mono (bind {l} {A} monadM x (headDeltaM ∙ f)))) + (deltaM-bind (deltaM d) (tailDeltaM ∙ f)) + +deltaM-mu : {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} + -> (DeltaM M {functorM} {monadM} (DeltaM M {functorM} {monadM} A)) -> DeltaM M {functorM} {monadM} A +deltaM-mu d = deltaM-bind d id \ No newline at end of file
--- a/agda/laws.agda Mon Jan 19 15:21:29 2015 +0900 +++ b/agda/laws.agda Mon Jan 19 17:10:29 2015 +0900 @@ -32,12 +32,15 @@ (M : {ll : Level} -> Set ll -> Set ll) (functorM : Functor M) : Set (suc l) where - field - mu : {A : Set l} -> M (M A) -> M A - eta : {A : Set l} -> A -> M A - field + field -- category + mu : {A : Set l} -> M (M A) -> M A + eta : {A : Set l} -> A -> M A + field -- haskell + return : {A : Set l} -> A -> M A + bind : {A B : Set l} -> M A -> (A -> (M B)) -> M B + field -- category laws association-law : (x : (M (M (M A)))) -> (mu ∙ (fmap functorM mu)) x ≡ (mu ∙ mu) x left-unity-law : (x : M A) -> (mu ∙ (fmap functorM eta)) x ≡ id x right-unity-law : (x : M A) -> id x ≡ (mu ∙ eta) x -open Monad \ No newline at end of file +open Monad
--- a/delta.hs Mon Jan 19 15:21:29 2015 +0900 +++ b/delta.hs Mon Jan 19 17:10:29 2015 +0900 @@ -128,7 +128,7 @@ instance (Monad m) => Monad (DeltaM m) where return x = DeltaM $ Mono $ return x (DeltaM (Mono x)) >>= f = DeltaM $ Mono $ (x >>= headDeltaM . f) - (DeltaM (Delta x d)) >>= f = appendDeltaM ((DeltaM $ Mono x) >>= f) + (DeltaM (Delta x d)) >>= f = appendDeltaM (DeltaM $ Mono $ (x >>= (headDeltaM . f))) ((DeltaM d) >>= tailDeltaM . f)