Mercurial > hg > Members > atton > delta_monad
changeset 88:526186c4f298
Split monad-proofs into delta.monad
| author | Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 19 Jan 2015 11:10:58 +0900 |
| parents | 6789c65a75bc |
| children | 5411ce26d525 |
| files | agda/delta.agda agda/delta/monad.agda |
| diffstat | 2 files changed, 357 insertions(+), 338 deletions(-) [+] |
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--- a/agda/delta.agda Mon Jan 19 11:00:34 2015 +0900 +++ b/agda/delta.agda Mon Jan 19 11:10:58 2015 +0900 @@ -132,341 +132,3 @@ -{- --- Monad-laws (Category) - -monad-law-1-5 : {l : Level} {A : Set l} -> (m : Nat) (n : Nat) -> (ds : Delta (Delta A)) -> - n-tail n (bind ds (n-tail m)) ≡ bind (n-tail n ds) (n-tail (m + n)) -monad-law-1-5 O O ds = refl -monad-law-1-5 O (S n) (mono ds) = begin - n-tail (S n) (bind (mono ds) (n-tail O)) ≡⟨ refl ⟩ - n-tail (S n) ds ≡⟨ refl ⟩ - bind (mono ds) (n-tail (S n)) ≡⟨ cong (\de -> bind de (n-tail (S n))) (sym (tail-delta-to-mono (S n) ds)) ⟩ - bind (n-tail (S n) (mono ds)) (n-tail (S n)) ≡⟨ refl ⟩ - bind (n-tail (S n) (mono ds)) (n-tail (O + S n)) - ∎ -monad-law-1-5 O (S n) (delta d ds) = begin - n-tail (S n) (bind (delta d ds) (n-tail O)) ≡⟨ refl ⟩ - n-tail (S n) (delta (headDelta d) (bind ds tailDelta )) ≡⟨ cong (\t -> t (delta (headDelta d) (bind ds tailDelta ))) (sym (n-tail-plus n)) ⟩ - ((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta )) ≡⟨ refl ⟩ - (n-tail n) (bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩ - bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩ - bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S n)) ≡⟨ cong (\t -> bind (t (delta d ds)) (n-tail (S n))) (n-tail-plus n) ⟩ - bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩ - bind (n-tail (S n) (delta d ds)) (n-tail (O + S n)) - ∎ -monad-law-1-5 (S m) n (mono (mono x)) = begin - n-tail n (bind (mono (mono x)) (n-tail (S m))) ≡⟨ refl ⟩ - n-tail n (n-tail (S m) (mono x)) ≡⟨ cong (\de -> n-tail n de) (tail-delta-to-mono (S m) x)⟩ - n-tail n (mono x) ≡⟨ tail-delta-to-mono n x ⟩ - mono x ≡⟨ sym (tail-delta-to-mono (S m + n) x) ⟩ - (n-tail (S m + n)) (mono x) ≡⟨ refl ⟩ - bind (mono (mono x)) (n-tail (S m + n)) ≡⟨ cong (\de -> bind de (n-tail (S m + n))) (sym (tail-delta-to-mono n (mono x))) ⟩ - bind (n-tail n (mono (mono x))) (n-tail (S m + n)) - ∎ -monad-law-1-5 (S m) n (mono (delta x ds)) = begin - n-tail n (bind (mono (delta x ds)) (n-tail (S m))) ≡⟨ refl ⟩ - n-tail n (n-tail (S m) (delta x ds)) ≡⟨ cong (\t -> n-tail n (t (delta x ds))) (sym (n-tail-plus m)) ⟩ - n-tail n (((n-tail m) ∙ tailDelta) (delta x ds)) ≡⟨ refl ⟩ - n-tail n ((n-tail m) ds) ≡⟨ cong (\t -> t ds) (n-tail-add {d = ds} n m) ⟩ - n-tail (n + m) ds ≡⟨ cong (\n -> n-tail n ds) (nat-add-sym n m) ⟩ - n-tail (m + n) ds ≡⟨ refl ⟩ - ((n-tail (m + n)) ∙ tailDelta) (delta x ds) ≡⟨ cong (\t -> t (delta x ds)) (n-tail-plus (m + n))⟩ - n-tail (S (m + n)) (delta x ds) ≡⟨ refl ⟩ - n-tail (S m + n) (delta x ds) ≡⟨ refl ⟩ - bind (mono (delta x ds)) (n-tail (S m + n)) ≡⟨ cong (\de -> bind de (n-tail (S m + n))) (sym (tail-delta-to-mono n (delta x ds))) ⟩ - bind (n-tail n (mono (delta x ds))) (n-tail (S m + n)) - ∎ -monad-law-1-5 (S m) O (delta d ds) = begin - n-tail O (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ - (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ - delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩ - bind (delta d ds) (n-tail (S m)) ≡⟨ refl ⟩ - bind (n-tail O (delta d ds)) (n-tail (S m)) ≡⟨ cong (\n -> bind (n-tail O (delta d ds)) (n-tail n)) (nat-add-right-zero (S m)) ⟩ - bind (n-tail O (delta d ds)) (n-tail (S m + O)) - ∎ -monad-law-1-5 (S m) (S n) (delta d ds) = begin - n-tail (S n) (bind (delta d ds) (n-tail (S m))) ≡⟨ cong (\t -> t ((bind (delta d ds) (n-tail (S m))))) (sym (n-tail-plus n)) ⟩ - ((n-tail n) ∙ tailDelta) (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ - ((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩ - (n-tail n) (bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩ - (n-tail n) (bind ds (n-tail (S (S m)))) ≡⟨ monad-law-1-5 (S (S m)) n ds ⟩ - bind ((n-tail n) ds) (n-tail (S (S (m + n)))) ≡⟨ cong (\nm -> bind ((n-tail n) ds) (n-tail nm)) (sym (nat-right-increment (S m) n)) ⟩ - bind ((n-tail n) ds) (n-tail (S m + S n)) ≡⟨ refl ⟩ - bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S m + S n)) ≡⟨ cong (\t -> bind (t (delta d ds)) (n-tail (S m + S n))) (n-tail-plus n) ⟩ - bind (n-tail (S n) (delta d ds)) (n-tail (S m + S n)) - ∎ - -monad-law-1-4 : {l : Level} {A : Set l} -> (m n : Nat) -> (dd : Delta (Delta A)) -> - headDelta ((n-tail n) (bind dd (n-tail m))) ≡ headDelta ((n-tail (m + n)) (headDelta (n-tail n dd))) -monad-law-1-4 O O (mono dd) = refl -monad-law-1-4 O O (delta dd dd₁) = refl -monad-law-1-4 O (S n) (mono dd) = begin - headDelta (n-tail (S n) (bind (mono dd) (n-tail O))) ≡⟨ refl ⟩ - headDelta (n-tail (S n) dd) ≡⟨ refl ⟩ - headDelta (n-tail (S n) (headDelta (mono dd))) ≡⟨ cong (\de -> headDelta (n-tail (S n) (headDelta de))) (sym (tail-delta-to-mono (S n) dd)) ⟩ - headDelta (n-tail (S n) (headDelta (n-tail (S n) (mono dd)))) ≡⟨ refl ⟩ - headDelta (n-tail (O + S n) (headDelta (n-tail (S n) (mono dd)))) - ∎ -monad-law-1-4 O (S n) (delta d ds) = begin - headDelta (n-tail (S n) (bind (delta d ds) (n-tail O))) ≡⟨ refl ⟩ - headDelta (n-tail (S n) (bind (delta d ds) id)) ≡⟨ refl ⟩ - headDelta (n-tail (S n) (delta (headDelta d) (bind ds tailDelta))) ≡⟨ cong (\t -> headDelta (t (delta (headDelta d) (bind ds tailDelta)))) (sym (n-tail-plus n)) ⟩ - headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta))) ≡⟨ refl ⟩ - headDelta (n-tail n (bind ds tailDelta)) ≡⟨ monad-law-1-4 (S O) n ds ⟩ - headDelta (n-tail (S n) (headDelta (n-tail n ds))) ≡⟨ refl ⟩ - headDelta (n-tail (S n) (headDelta (((n-tail n) ∙ tailDelta) (delta d ds)))) ≡⟨ cong (\t -> headDelta (n-tail (S n) (headDelta (t (delta d ds))))) (n-tail-plus n) ⟩ - headDelta (n-tail (S n) (headDelta (n-tail (S n) (delta d ds)))) ≡⟨ refl ⟩ - headDelta (n-tail (O + S n) (headDelta (n-tail (S n) (delta d ds)))) - ∎ -monad-law-1-4 (S m) n (mono dd) = begin - headDelta (n-tail n (bind (mono dd) (n-tail (S m)))) ≡⟨ refl ⟩ - headDelta (n-tail n ((n-tail (S m)) dd)) ≡⟨ cong (\t -> headDelta (t dd)) (n-tail-add {d = dd} n (S m)) ⟩ - headDelta (n-tail (n + S m) dd) ≡⟨ cong (\n -> headDelta ((n-tail n) dd)) (nat-add-sym n (S m)) ⟩ - headDelta (n-tail (S m + n) dd) ≡⟨ refl ⟩ - headDelta (n-tail (S m + n) (headDelta (mono dd))) ≡⟨ cong (\de -> headDelta (n-tail (S m + n) (headDelta de))) (sym (tail-delta-to-mono n dd)) ⟩ - headDelta (n-tail (S m + n) (headDelta (n-tail n (mono dd)))) - ∎ -monad-law-1-4 (S m) O (delta d ds) = begin - headDelta (n-tail O (bind (delta d ds) (n-tail (S m)))) ≡⟨ refl ⟩ - headDelta (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ - headDelta (delta (headDelta ((n-tail (S m) d))) (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩ - headDelta (n-tail (S m) d) ≡⟨ cong (\n -> headDelta ((n-tail n) d)) (nat-add-right-zero (S m)) ⟩ - headDelta (n-tail (S m + O) d) ≡⟨ refl ⟩ - headDelta (n-tail (S m + O) (headDelta (delta d ds))) ≡⟨ refl ⟩ - headDelta (n-tail (S m + O) (headDelta (n-tail O (delta d ds)))) - ∎ -monad-law-1-4 (S m) (S n) (delta d ds) = begin - headDelta (n-tail (S n) (bind (delta d ds) (n-tail (S m)))) ≡⟨ refl ⟩ - headDelta (n-tail (S n) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m)))))) ≡⟨ cong (\t -> headDelta (t (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))))) (sym (n-tail-plus n)) ⟩ - headDelta ((((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))))) ≡⟨ refl ⟩ - headDelta (n-tail n (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩ - headDelta (n-tail n (bind ds (n-tail (S (S m))))) ≡⟨ monad-law-1-4 (S (S m)) n ds ⟩ - headDelta (n-tail ((S (S m) + n)) (headDelta (n-tail n ds))) ≡⟨ cong (\nm -> headDelta ((n-tail nm) (headDelta (n-tail n ds)))) (sym (nat-right-increment (S m) n)) ⟩ - headDelta (n-tail (S m + S n) (headDelta (n-tail n ds))) ≡⟨ refl ⟩ - headDelta (n-tail (S m + S n) (headDelta (((n-tail n) ∙ tailDelta) (delta d ds)))) ≡⟨ cong (\t -> headDelta (n-tail (S m + S n) (headDelta (t (delta d ds))))) (n-tail-plus n) ⟩ - headDelta (n-tail (S m + S n) (headDelta (n-tail (S n) (delta d ds)))) - ∎ - -monad-law-1-2 : {l : Level} {A : Set l} -> (d : Delta (Delta A)) -> headDelta (mu d) ≡ (headDelta (headDelta d)) -monad-law-1-2 (mono _) = refl -monad-law-1-2 (delta _ _) = refl - -monad-law-1-3 : {l : Level} {A : Set l} -> (n : Nat) -> (d : Delta (Delta (Delta A))) -> - bind (fmap mu d) (n-tail n) ≡ bind (bind d (n-tail n)) (n-tail n) -monad-law-1-3 O (mono d) = refl -monad-law-1-3 O (delta d ds) = begin - bind (fmap mu (delta d ds)) (n-tail O) ≡⟨ refl ⟩ - bind (delta (mu d) (fmap mu ds)) (n-tail O) ≡⟨ refl ⟩ - delta (headDelta (mu d)) (bind (fmap mu ds) tailDelta) ≡⟨ cong (\dx -> delta dx (bind (fmap mu ds) tailDelta)) (monad-law-1-2 d) ⟩ - delta (headDelta (headDelta d)) (bind (fmap mu ds) tailDelta) ≡⟨ cong (\dx -> delta (headDelta (headDelta d)) dx) (monad-law-1-3 (S O) ds) ⟩ - delta (headDelta (headDelta d)) (bind (bind ds tailDelta) tailDelta) ≡⟨ refl ⟩ - bind (delta (headDelta d) (bind ds tailDelta)) (n-tail O) ≡⟨ refl ⟩ - bind (bind (delta d ds) (n-tail O)) (n-tail O) - ∎ -monad-law-1-3 (S n) (mono (mono d)) = begin - bind (fmap mu (mono (mono d))) (n-tail (S n)) ≡⟨ refl ⟩ - bind (mono d) (n-tail (S n)) ≡⟨ refl ⟩ - (n-tail (S n)) d ≡⟨ refl ⟩ - bind (mono d) (n-tail (S n)) ≡⟨ cong (\t -> bind t (n-tail (S n))) (sym (tail-delta-to-mono (S n) d))⟩ - bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩ - bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩ - bind (bind (mono (mono d)) (n-tail (S n))) (n-tail (S n)) - ∎ -monad-law-1-3 (S n) (mono (delta d ds)) = begin - bind (fmap mu (mono (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩ - bind (mono (mu (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩ - n-tail (S n) (mu (delta d ds)) ≡⟨ refl ⟩ - n-tail (S n) (delta (headDelta d) (bind ds tailDelta)) ≡⟨ cong (\t -> t (delta (headDelta d) (bind ds tailDelta))) (sym (n-tail-plus n)) ⟩ - (n-tail n ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta)) ≡⟨ refl ⟩ - n-tail n (bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩ - bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩ - bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S n)) ≡⟨ cong (\t -> (bind (t (delta d ds)) (n-tail (S n)))) (n-tail-plus n) ⟩ - bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩ - bind (bind (mono (delta d ds)) (n-tail (S n))) (n-tail (S n)) - ∎ -monad-law-1-3 (S n) (delta (mono d) ds) = begin - bind (fmap mu (delta (mono d) ds)) (n-tail (S n)) ≡⟨ refl ⟩ - bind (delta (mu (mono d)) (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ - bind (delta d (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ - delta (headDelta ((n-tail (S n)) d)) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ - delta (headDelta ((n-tail (S n)) d)) (bind (fmap mu ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta ((n-tail (S n)) d)) de) (monad-law-1-3 (S (S n)) ds) ⟩ - delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩ - delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩ - delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ - delta (headDelta ((n-tail (S n)) (headDelta (mono d)))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ cong (\de -> delta (headDelta ((n-tail (S n)) (headDelta de))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n))))) (sym (tail-delta-to-mono (S n) d)) ⟩ - delta (headDelta ((n-tail (S n)) (headDelta ((n-tail (S n)) (mono d))))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ - bind (delta (headDelta ((n-tail (S n)) (mono d))) (bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩ - bind (bind (delta (mono d) ds) (n-tail (S n))) (n-tail (S n)) - ∎ -monad-law-1-3 (S n) (delta (delta d dd) ds) = begin - bind (fmap mu (delta (delta d dd) ds)) (n-tail (S n)) ≡⟨ refl ⟩ - bind (delta (mu (delta d dd)) (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ - delta (headDelta ((n-tail (S n)) (mu (delta d dd)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ - delta (headDelta ((n-tail (S n)) (delta (headDelta d) (bind dd tailDelta)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ cong (\t -> delta (headDelta (t (delta (headDelta d) (bind dd tailDelta)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))))(sym (n-tail-plus n)) ⟩ - delta (headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind dd tailDelta)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ - delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ - delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (fmap mu ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta ((n-tail n) (bind dd tailDelta))) de) (monad-law-1-3 (S (S n)) ds) ⟩ - delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta de ( (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))))) (monad-law-1-4 (S O) n dd) ⟩ - delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩ - delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (bind (bind ds (n-tail (S (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ - delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ - delta (headDelta ((n-tail (S n)) (headDelta (((n-tail n) ∙ tailDelta) (delta d dd))))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ cong (\t -> delta (headDelta ((n-tail (S n)) (headDelta (t (delta d dd))))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n))))) (n-tail-plus n) ⟩ - delta (headDelta ((n-tail (S n)) (headDelta ((n-tail (S n)) (delta d dd))))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ - bind (delta (headDelta ((n-tail (S n)) (delta d dd))) (bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩ - bind (bind (delta (delta d dd) ds) (n-tail (S n))) (n-tail (S n)) - ∎ - - --- monad-law-1 : join . fmap join = join . join -monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((mu ∙ (fmap mu)) d) ≡ ((mu ∙ mu) d) -monad-law-1 (mono d) = refl -monad-law-1 (delta x d) = begin - (mu ∙ fmap mu) (delta x d) ≡⟨ refl ⟩ - mu (fmap mu (delta x d)) ≡⟨ refl ⟩ - mu (delta (mu x) (fmap mu d)) ≡⟨ refl ⟩ - delta (headDelta (mu x)) (bind (fmap mu d) tailDelta) ≡⟨ cong (\x -> delta x (bind (fmap mu d) tailDelta)) (monad-law-1-2 x) ⟩ - delta (headDelta (headDelta x)) (bind (fmap mu d) tailDelta) ≡⟨ cong (\d -> delta (headDelta (headDelta x)) d) (monad-law-1-3 (S O) d) ⟩ - delta (headDelta (headDelta x)) (bind (bind d tailDelta) tailDelta) ≡⟨ refl ⟩ - mu (delta (headDelta x) (bind d tailDelta)) ≡⟨ refl ⟩ - mu (mu (delta x d)) ≡⟨ refl ⟩ - (mu ∙ mu) (delta x d) - ∎ - - -monad-law-2-1 : {l : Level} {A : Set l} -> (n : Nat) -> (d : Delta A) -> (bind (fmap eta d) (n-tail n)) ≡ d -monad-law-2-1 O (mono x) = refl -monad-law-2-1 O (delta x d) = begin - bind (fmap eta (delta x d)) (n-tail O) ≡⟨ refl ⟩ - bind (delta (eta x) (fmap eta d)) id ≡⟨ refl ⟩ - delta (headDelta (eta x)) (bind (fmap eta d) tailDelta) ≡⟨ refl ⟩ - delta x (bind (fmap eta d) tailDelta) ≡⟨ cong (\de -> delta x de) (monad-law-2-1 (S O) d) ⟩ - delta x d ∎ -monad-law-2-1 (S n) (mono x) = begin - bind (fmap eta (mono x)) (n-tail (S n)) ≡⟨ refl ⟩ - bind (mono (mono x)) (n-tail (S n)) ≡⟨ refl ⟩ - n-tail (S n) (mono x) ≡⟨ tail-delta-to-mono (S n) x ⟩ - mono x ∎ -monad-law-2-1 (S n) (delta x d) = begin - bind (fmap eta (delta x d)) (n-tail (S n)) ≡⟨ refl ⟩ - bind (delta (eta x) (fmap eta d)) (n-tail (S n)) ≡⟨ refl ⟩ - delta (headDelta ((n-tail (S n) (eta x)))) (bind (fmap eta d) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ - delta (headDelta ((n-tail (S n) (eta x)))) (bind (fmap eta d) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta (de)) (bind (fmap eta d) (n-tail (S (S n))))) (tail-delta-to-mono (S n) x) ⟩ - delta (headDelta (eta x)) (bind (fmap eta d) (n-tail (S (S n)))) ≡⟨ refl ⟩ - delta x (bind (fmap eta d) (n-tail (S (S n)))) ≡⟨ cong (\d -> delta x d) (monad-law-2-1 (S (S n)) d) ⟩ - delta x d - ∎ - - --- monad-law-2 : join . fmap return = join . return = id --- monad-law-2 join . fmap return = join . return -monad-law-2 : {l : Level} {A : Set l} -> (d : Delta A) -> - (mu ∙ fmap eta) d ≡ (mu ∙ eta) d -monad-law-2 (mono x) = refl -monad-law-2 (delta x d) = begin - (mu ∙ fmap eta) (delta x d) ≡⟨ refl ⟩ - mu (fmap eta (delta x d)) ≡⟨ refl ⟩ - mu (delta (mono x) (fmap eta d)) ≡⟨ refl ⟩ - delta (headDelta (mono x)) (bind (fmap eta d) tailDelta) ≡⟨ refl ⟩ - delta x (bind (fmap eta d) tailDelta) ≡⟨ cong (\d -> delta x d) (monad-law-2-1 (S O) d) ⟩ - (delta x d) ≡⟨ refl ⟩ - mu (mono (delta x d)) ≡⟨ refl ⟩ - mu (eta (delta x d)) ≡⟨ refl ⟩ - (mu ∙ eta) (delta x d) - ∎ - - --- monad-law-2' : join . return = id -monad-law-2' : {l : Level} {A : Set l} -> (d : Delta A) -> (mu ∙ eta) d ≡ id d -monad-law-2' d = refl - - --- monad-law-3 : return . f = fmap f . return -monad-law-3 : {l : Level} {A B : Set l} (f : A -> B) (x : A) -> (eta ∙ f) x ≡ (fmap f ∙ eta) x -monad-law-3 f x = refl - - -monad-law-4-1 : {l ll : Level} {A : Set l} {B : Set ll} -> (n : Nat) -> (f : A -> B) -> (ds : Delta (Delta A)) -> - bind (fmap (fmap f) ds) (n-tail n) ≡ fmap f (bind ds (n-tail n)) -monad-law-4-1 O f (mono d) = refl -monad-law-4-1 O f (delta d ds) = begin - bind (fmap (fmap f) (delta d ds)) (n-tail O) ≡⟨ refl ⟩ - bind (delta (fmap f d) (fmap (fmap f) ds)) (n-tail O) ≡⟨ refl ⟩ - delta (headDelta (fmap f d)) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta de (bind (fmap (fmap f) ds) tailDelta)) (head-delta-natural-transformation f d) ⟩ - delta (f (headDelta d)) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f (headDelta d)) de) (monad-law-4-1 (S O) f ds) ⟩ - delta (f (headDelta d)) (fmap f (bind ds tailDelta)) ≡⟨ refl ⟩ - fmap f (delta (headDelta d) (bind ds tailDelta)) ≡⟨ refl ⟩ - fmap f (bind (delta d ds) (n-tail O)) ∎ -monad-law-4-1 (S n) f (mono d) = begin - bind (fmap (fmap f) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩ - bind (mono (fmap f d)) (n-tail (S n)) ≡⟨ refl ⟩ - n-tail (S n) (fmap f d) ≡⟨ n-tail-natural-transformation (S n) f d ⟩ - fmap f (n-tail (S n) d) ≡⟨ refl ⟩ - fmap f (bind (mono d) (n-tail (S n))) - ∎ -monad-law-4-1 (S n) f (delta d ds) = begin - bind (fmap (fmap f) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩ - delta (headDelta (n-tail (S n) (fmap f d))) (bind (fmap (fmap f) ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ - delta (headDelta (n-tail (S n) (fmap f d))) (bind (fmap (fmap f) ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta de) (bind (fmap (fmap f) ds) (n-tail (S (S n))))) (n-tail-natural-transformation (S n) f d) ⟩ - delta (headDelta (fmap f ((n-tail (S n) d)))) (bind (fmap (fmap f) ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta de (bind (fmap (fmap f) ds) (n-tail (S (S n))))) (head-delta-natural-transformation f (n-tail (S n) d)) ⟩ - delta (f (headDelta (n-tail (S n) d))) (bind (fmap (fmap f) ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (f (headDelta (n-tail (S n) d))) de) (monad-law-4-1 (S (S n)) f ds) ⟩ - delta (f (headDelta (n-tail (S n) d))) (fmap f (bind ds (n-tail (S (S n))))) ≡⟨ refl ⟩ - fmap f (delta (headDelta (n-tail (S n) d)) (bind ds (n-tail (S (S n))))) ≡⟨ refl ⟩ - fmap f (delta (headDelta (n-tail (S n) d)) (bind ds (tailDelta ∙ (n-tail (S n))))) ≡⟨ refl ⟩ - fmap f (bind (delta d ds) (n-tail (S n))) ∎ - - --- monad-law-4 : join . fmap (fmap f) = fmap f . join -monad-law-4 : {l ll : Level} {A : Set l} {B : Set ll} (f : A -> B) (d : Delta (Delta A)) -> - (mu ∙ fmap (fmap f)) d ≡ (fmap f ∙ mu) d -monad-law-4 f (mono d) = refl -monad-law-4 f (delta (mono x) ds) = begin - (mu ∙ fmap (fmap f)) (delta (mono x) ds) ≡⟨ refl ⟩ - mu ( fmap (fmap f) (delta (mono x) ds)) ≡⟨ refl ⟩ - mu (delta (mono (f x)) (fmap (fmap f) ds)) ≡⟨ refl ⟩ - delta (headDelta (mono (f x))) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ refl ⟩ - delta (f x) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f x) de) (monad-law-4-1 (S O) f ds) ⟩ - delta (f x) (fmap f (bind ds tailDelta)) ≡⟨ refl ⟩ - fmap f (delta x (bind ds tailDelta)) ≡⟨ refl ⟩ - fmap f (delta (headDelta (mono x)) (bind ds tailDelta)) ≡⟨ refl ⟩ - fmap f (mu (delta (mono x) ds)) ≡⟨ refl ⟩ - (fmap f ∙ mu) (delta (mono x) ds) ∎ -monad-law-4 f (delta (delta x d) ds) = begin - (mu ∙ fmap (fmap f)) (delta (delta x d) ds) ≡⟨ refl ⟩ - mu (fmap (fmap f) (delta (delta x d) ds)) ≡⟨ refl ⟩ - mu (delta (delta (f x) (fmap f d)) (fmap (fmap f) ds)) ≡⟨ refl ⟩ - delta (headDelta (delta (f x) (fmap f d))) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ refl ⟩ - delta (f x) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f x) de) (monad-law-4-1 (S O) f ds) ⟩ - delta (f x) (fmap f (bind ds tailDelta)) ≡⟨ refl ⟩ - fmap f (delta x (bind ds tailDelta)) ≡⟨ refl ⟩ - fmap f (delta (headDelta (delta x d)) (bind ds tailDelta)) ≡⟨ refl ⟩ - fmap f (mu (delta (delta x d) ds)) ≡⟨ refl ⟩ - (fmap f ∙ mu) (delta (delta x d) ds) ∎ - - - -{- --- Monad-laws (Haskell) --- monad-law-h-1 : return a >>= k = k a -monad-law-h-1 : {l ll : Level} {A : Set l} {B : Set ll} -> - (a : A) -> (k : A -> (Delta B)) -> - (return a >>= k) ≡ (k a) -monad-law-h-1 a k = refl - - - --- monad-law-h-2 : m >>= return = m -monad-law-h-2 : {l : Level}{A : Set l} -> (m : Delta A) -> (m >>= return) ≡ m -monad-law-h-2 (mono x) = refl -monad-law-h-2 (delta x d) = cong (delta x) (monad-law-h-2 d) - - --- monad-law-h-3 : m >>= (\x -> k x >>= h) = (m >>= k) >>= h -monad-law-h-3 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} -> - (m : Delta A) -> (k : A -> (Delta B)) -> (h : B -> (Delta C)) -> - (m >>= (\x -> k x >>= h)) ≡ ((m >>= k) >>= h) -monad-law-h-3 (mono x) k h = refl -monad-law-h-3 (delta x d) k h = {!!} - --} --}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/agda/delta/monad.agda Mon Jan 19 11:10:58 2015 +0900 @@ -0,0 +1,357 @@ +open import basic +open import delta +open import delta.functor +open import nat +open import laws + + +open import Level +open import Relation.Binary.PropositionalEquality +open ≡-Reasoning + +module delta.monad where + + +-- Monad-laws (Category) + +monad-law-1-5 : {l : Level} {A : Set l} -> (m : Nat) (n : Nat) -> (ds : Delta (Delta A)) -> + n-tail n (bind ds (n-tail m)) ≡ bind (n-tail n ds) (n-tail (m + n)) +monad-law-1-5 O O ds = refl +monad-law-1-5 O (S n) (mono ds) = begin + n-tail (S n) (bind (mono ds) (n-tail O)) ≡⟨ refl ⟩ + n-tail (S n) ds ≡⟨ refl ⟩ + bind (mono ds) (n-tail (S n)) ≡⟨ cong (\de -> bind de (n-tail (S n))) (sym (tail-delta-to-mono (S n) ds)) ⟩ + bind (n-tail (S n) (mono ds)) (n-tail (S n)) ≡⟨ refl ⟩ + bind (n-tail (S n) (mono ds)) (n-tail (O + S n)) + ∎ +monad-law-1-5 O (S n) (delta d ds) = begin + n-tail (S n) (bind (delta d ds) (n-tail O)) ≡⟨ refl ⟩ + n-tail (S n) (delta (headDelta d) (bind ds tailDelta )) ≡⟨ cong (\t -> t (delta (headDelta d) (bind ds tailDelta ))) (sym (n-tail-plus n)) ⟩ + ((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta )) ≡⟨ refl ⟩ + (n-tail n) (bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩ + bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩ + bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S n)) ≡⟨ cong (\t -> bind (t (delta d ds)) (n-tail (S n))) (n-tail-plus n) ⟩ + bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩ + bind (n-tail (S n) (delta d ds)) (n-tail (O + S n)) + ∎ +monad-law-1-5 (S m) n (mono (mono x)) = begin + n-tail n (bind (mono (mono x)) (n-tail (S m))) ≡⟨ refl ⟩ + n-tail n (n-tail (S m) (mono x)) ≡⟨ cong (\de -> n-tail n de) (tail-delta-to-mono (S m) x)⟩ + n-tail n (mono x) ≡⟨ tail-delta-to-mono n x ⟩ + mono x ≡⟨ sym (tail-delta-to-mono (S m + n) x) ⟩ + (n-tail (S m + n)) (mono x) ≡⟨ refl ⟩ + bind (mono (mono x)) (n-tail (S m + n)) ≡⟨ cong (\de -> bind de (n-tail (S m + n))) (sym (tail-delta-to-mono n (mono x))) ⟩ + bind (n-tail n (mono (mono x))) (n-tail (S m + n)) + ∎ +monad-law-1-5 (S m) n (mono (delta x ds)) = begin + n-tail n (bind (mono (delta x ds)) (n-tail (S m))) ≡⟨ refl ⟩ + n-tail n (n-tail (S m) (delta x ds)) ≡⟨ cong (\t -> n-tail n (t (delta x ds))) (sym (n-tail-plus m)) ⟩ + n-tail n (((n-tail m) ∙ tailDelta) (delta x ds)) ≡⟨ refl ⟩ + n-tail n ((n-tail m) ds) ≡⟨ cong (\t -> t ds) (n-tail-add {d = ds} n m) ⟩ + n-tail (n + m) ds ≡⟨ cong (\n -> n-tail n ds) (nat-add-sym n m) ⟩ + n-tail (m + n) ds ≡⟨ refl ⟩ + ((n-tail (m + n)) ∙ tailDelta) (delta x ds) ≡⟨ cong (\t -> t (delta x ds)) (n-tail-plus (m + n))⟩ + n-tail (S (m + n)) (delta x ds) ≡⟨ refl ⟩ + n-tail (S m + n) (delta x ds) ≡⟨ refl ⟩ + bind (mono (delta x ds)) (n-tail (S m + n)) ≡⟨ cong (\de -> bind de (n-tail (S m + n))) (sym (tail-delta-to-mono n (delta x ds))) ⟩ + bind (n-tail n (mono (delta x ds))) (n-tail (S m + n)) + ∎ +monad-law-1-5 (S m) O (delta d ds) = begin + n-tail O (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ + (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ + delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩ + bind (delta d ds) (n-tail (S m)) ≡⟨ refl ⟩ + bind (n-tail O (delta d ds)) (n-tail (S m)) ≡⟨ cong (\n -> bind (n-tail O (delta d ds)) (n-tail n)) (nat-add-right-zero (S m)) ⟩ + bind (n-tail O (delta d ds)) (n-tail (S m + O)) + ∎ +monad-law-1-5 (S m) (S n) (delta d ds) = begin + n-tail (S n) (bind (delta d ds) (n-tail (S m))) ≡⟨ cong (\t -> t ((bind (delta d ds) (n-tail (S m))))) (sym (n-tail-plus n)) ⟩ + ((n-tail n) ∙ tailDelta) (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ + ((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩ + (n-tail n) (bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩ + (n-tail n) (bind ds (n-tail (S (S m)))) ≡⟨ monad-law-1-5 (S (S m)) n ds ⟩ + bind ((n-tail n) ds) (n-tail (S (S (m + n)))) ≡⟨ cong (\nm -> bind ((n-tail n) ds) (n-tail nm)) (sym (nat-right-increment (S m) n)) ⟩ + bind ((n-tail n) ds) (n-tail (S m + S n)) ≡⟨ refl ⟩ + bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S m + S n)) ≡⟨ cong (\t -> bind (t (delta d ds)) (n-tail (S m + S n))) (n-tail-plus n) ⟩ + bind (n-tail (S n) (delta d ds)) (n-tail (S m + S n)) + ∎ + +monad-law-1-4 : {l : Level} {A : Set l} -> (m n : Nat) -> (dd : Delta (Delta A)) -> + headDelta ((n-tail n) (bind dd (n-tail m))) ≡ headDelta ((n-tail (m + n)) (headDelta (n-tail n dd))) +monad-law-1-4 O O (mono dd) = refl +monad-law-1-4 O O (delta dd dd₁) = refl +monad-law-1-4 O (S n) (mono dd) = begin + headDelta (n-tail (S n) (bind (mono dd) (n-tail O))) ≡⟨ refl ⟩ + headDelta (n-tail (S n) dd) ≡⟨ refl ⟩ + headDelta (n-tail (S n) (headDelta (mono dd))) ≡⟨ cong (\de -> headDelta (n-tail (S n) (headDelta de))) (sym (tail-delta-to-mono (S n) dd)) ⟩ + headDelta (n-tail (S n) (headDelta (n-tail (S n) (mono dd)))) ≡⟨ refl ⟩ + headDelta (n-tail (O + S n) (headDelta (n-tail (S n) (mono dd)))) + ∎ +monad-law-1-4 O (S n) (delta d ds) = begin + headDelta (n-tail (S n) (bind (delta d ds) (n-tail O))) ≡⟨ refl ⟩ + headDelta (n-tail (S n) (bind (delta d ds) id)) ≡⟨ refl ⟩ + headDelta (n-tail (S n) (delta (headDelta d) (bind ds tailDelta))) ≡⟨ cong (\t -> headDelta (t (delta (headDelta d) (bind ds tailDelta)))) (sym (n-tail-plus n)) ⟩ + headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta))) ≡⟨ refl ⟩ + headDelta (n-tail n (bind ds tailDelta)) ≡⟨ monad-law-1-4 (S O) n ds ⟩ + headDelta (n-tail (S n) (headDelta (n-tail n ds))) ≡⟨ refl ⟩ + headDelta (n-tail (S n) (headDelta (((n-tail n) ∙ tailDelta) (delta d ds)))) ≡⟨ cong (\t -> headDelta (n-tail (S n) (headDelta (t (delta d ds))))) (n-tail-plus n) ⟩ + headDelta (n-tail (S n) (headDelta (n-tail (S n) (delta d ds)))) ≡⟨ refl ⟩ + headDelta (n-tail (O + S n) (headDelta (n-tail (S n) (delta d ds)))) + ∎ +monad-law-1-4 (S m) n (mono dd) = begin + headDelta (n-tail n (bind (mono dd) (n-tail (S m)))) ≡⟨ refl ⟩ + headDelta (n-tail n ((n-tail (S m)) dd)) ≡⟨ cong (\t -> headDelta (t dd)) (n-tail-add {d = dd} n (S m)) ⟩ + headDelta (n-tail (n + S m) dd) ≡⟨ cong (\n -> headDelta ((n-tail n) dd)) (nat-add-sym n (S m)) ⟩ + headDelta (n-tail (S m + n) dd) ≡⟨ refl ⟩ + headDelta (n-tail (S m + n) (headDelta (mono dd))) ≡⟨ cong (\de -> headDelta (n-tail (S m + n) (headDelta de))) (sym (tail-delta-to-mono n dd)) ⟩ + headDelta (n-tail (S m + n) (headDelta (n-tail n (mono dd)))) + ∎ +monad-law-1-4 (S m) O (delta d ds) = begin + headDelta (n-tail O (bind (delta d ds) (n-tail (S m)))) ≡⟨ refl ⟩ + headDelta (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ + headDelta (delta (headDelta ((n-tail (S m) d))) (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩ + headDelta (n-tail (S m) d) ≡⟨ cong (\n -> headDelta ((n-tail n) d)) (nat-add-right-zero (S m)) ⟩ + headDelta (n-tail (S m + O) d) ≡⟨ refl ⟩ + headDelta (n-tail (S m + O) (headDelta (delta d ds))) ≡⟨ refl ⟩ + headDelta (n-tail (S m + O) (headDelta (n-tail O (delta d ds)))) + ∎ +monad-law-1-4 (S m) (S n) (delta d ds) = begin + headDelta (n-tail (S n) (bind (delta d ds) (n-tail (S m)))) ≡⟨ refl ⟩ + headDelta (n-tail (S n) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m)))))) ≡⟨ cong (\t -> headDelta (t (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))))) (sym (n-tail-plus n)) ⟩ + headDelta ((((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))))) ≡⟨ refl ⟩ + headDelta (n-tail n (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩ + headDelta (n-tail n (bind ds (n-tail (S (S m))))) ≡⟨ monad-law-1-4 (S (S m)) n ds ⟩ + headDelta (n-tail ((S (S m) + n)) (headDelta (n-tail n ds))) ≡⟨ cong (\nm -> headDelta ((n-tail nm) (headDelta (n-tail n ds)))) (sym (nat-right-increment (S m) n)) ⟩ + headDelta (n-tail (S m + S n) (headDelta (n-tail n ds))) ≡⟨ refl ⟩ + headDelta (n-tail (S m + S n) (headDelta (((n-tail n) ∙ tailDelta) (delta d ds)))) ≡⟨ cong (\t -> headDelta (n-tail (S m + S n) (headDelta (t (delta d ds))))) (n-tail-plus n) ⟩ + headDelta (n-tail (S m + S n) (headDelta (n-tail (S n) (delta d ds)))) + ∎ + +monad-law-1-2 : {l : Level} {A : Set l} -> (d : Delta (Delta A)) -> headDelta (mu d) ≡ (headDelta (headDelta d)) +monad-law-1-2 (mono _) = refl +monad-law-1-2 (delta _ _) = refl + +monad-law-1-3 : {l : Level} {A : Set l} -> (n : Nat) -> (d : Delta (Delta (Delta A))) -> + bind (fmap mu d) (n-tail n) ≡ bind (bind d (n-tail n)) (n-tail n) +monad-law-1-3 O (mono d) = refl +monad-law-1-3 O (delta d ds) = begin + bind (fmap mu (delta d ds)) (n-tail O) ≡⟨ refl ⟩ + bind (delta (mu d) (fmap mu ds)) (n-tail O) ≡⟨ refl ⟩ + delta (headDelta (mu d)) (bind (fmap mu ds) tailDelta) ≡⟨ cong (\dx -> delta dx (bind (fmap mu ds) tailDelta)) (monad-law-1-2 d) ⟩ + delta (headDelta (headDelta d)) (bind (fmap mu ds) tailDelta) ≡⟨ cong (\dx -> delta (headDelta (headDelta d)) dx) (monad-law-1-3 (S O) ds) ⟩ + delta (headDelta (headDelta d)) (bind (bind ds tailDelta) tailDelta) ≡⟨ refl ⟩ + bind (delta (headDelta d) (bind ds tailDelta)) (n-tail O) ≡⟨ refl ⟩ + bind (bind (delta d ds) (n-tail O)) (n-tail O) + ∎ +monad-law-1-3 (S n) (mono (mono d)) = begin + bind (fmap mu (mono (mono d))) (n-tail (S n)) ≡⟨ refl ⟩ + bind (mono d) (n-tail (S n)) ≡⟨ refl ⟩ + (n-tail (S n)) d ≡⟨ refl ⟩ + bind (mono d) (n-tail (S n)) ≡⟨ cong (\t -> bind t (n-tail (S n))) (sym (tail-delta-to-mono (S n) d))⟩ + bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩ + bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩ + bind (bind (mono (mono d)) (n-tail (S n))) (n-tail (S n)) + ∎ +monad-law-1-3 (S n) (mono (delta d ds)) = begin + bind (fmap mu (mono (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩ + bind (mono (mu (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩ + n-tail (S n) (mu (delta d ds)) ≡⟨ refl ⟩ + n-tail (S n) (delta (headDelta d) (bind ds tailDelta)) ≡⟨ cong (\t -> t (delta (headDelta d) (bind ds tailDelta))) (sym (n-tail-plus n)) ⟩ + (n-tail n ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta)) ≡⟨ refl ⟩ + n-tail n (bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩ + bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩ + bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S n)) ≡⟨ cong (\t -> (bind (t (delta d ds)) (n-tail (S n)))) (n-tail-plus n) ⟩ + bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩ + bind (bind (mono (delta d ds)) (n-tail (S n))) (n-tail (S n)) + ∎ +monad-law-1-3 (S n) (delta (mono d) ds) = begin + bind (fmap mu (delta (mono d) ds)) (n-tail (S n)) ≡⟨ refl ⟩ + bind (delta (mu (mono d)) (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ + bind (delta d (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ + delta (headDelta ((n-tail (S n)) d)) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ + delta (headDelta ((n-tail (S n)) d)) (bind (fmap mu ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta ((n-tail (S n)) d)) de) (monad-law-1-3 (S (S n)) ds) ⟩ + delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩ + delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩ + delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ + delta (headDelta ((n-tail (S n)) (headDelta (mono d)))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ cong (\de -> delta (headDelta ((n-tail (S n)) (headDelta de))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n))))) (sym (tail-delta-to-mono (S n) d)) ⟩ + delta (headDelta ((n-tail (S n)) (headDelta ((n-tail (S n)) (mono d))))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ + bind (delta (headDelta ((n-tail (S n)) (mono d))) (bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩ + bind (bind (delta (mono d) ds) (n-tail (S n))) (n-tail (S n)) + ∎ +monad-law-1-3 (S n) (delta (delta d dd) ds) = begin + bind (fmap mu (delta (delta d dd) ds)) (n-tail (S n)) ≡⟨ refl ⟩ + bind (delta (mu (delta d dd)) (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ + delta (headDelta ((n-tail (S n)) (mu (delta d dd)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ + delta (headDelta ((n-tail (S n)) (delta (headDelta d) (bind dd tailDelta)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ cong (\t -> delta (headDelta (t (delta (headDelta d) (bind dd tailDelta)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))))(sym (n-tail-plus n)) ⟩ + delta (headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind dd tailDelta)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ + delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ + delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (fmap mu ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta ((n-tail n) (bind dd tailDelta))) de) (monad-law-1-3 (S (S n)) ds) ⟩ + delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta de ( (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))))) (monad-law-1-4 (S O) n dd) ⟩ + delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩ + delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (bind (bind ds (n-tail (S (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ + delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ + delta (headDelta ((n-tail (S n)) (headDelta (((n-tail n) ∙ tailDelta) (delta d dd))))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ cong (\t -> delta (headDelta ((n-tail (S n)) (headDelta (t (delta d dd))))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n))))) (n-tail-plus n) ⟩ + delta (headDelta ((n-tail (S n)) (headDelta ((n-tail (S n)) (delta d dd))))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ + bind (delta (headDelta ((n-tail (S n)) (delta d dd))) (bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩ + bind (bind (delta (delta d dd) ds) (n-tail (S n))) (n-tail (S n)) + ∎ + + +-- monad-law-1 : join . fmap join = join . join +monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((mu ∙ (fmap mu)) d) ≡ ((mu ∙ mu) d) +monad-law-1 (mono d) = refl +monad-law-1 (delta x d) = begin + (mu ∙ fmap mu) (delta x d) ≡⟨ refl ⟩ + mu (fmap mu (delta x d)) ≡⟨ refl ⟩ + mu (delta (mu x) (fmap mu d)) ≡⟨ refl ⟩ + delta (headDelta (mu x)) (bind (fmap mu d) tailDelta) ≡⟨ cong (\x -> delta x (bind (fmap mu d) tailDelta)) (monad-law-1-2 x) ⟩ + delta (headDelta (headDelta x)) (bind (fmap mu d) tailDelta) ≡⟨ cong (\d -> delta (headDelta (headDelta x)) d) (monad-law-1-3 (S O) d) ⟩ + delta (headDelta (headDelta x)) (bind (bind d tailDelta) tailDelta) ≡⟨ refl ⟩ + mu (delta (headDelta x) (bind d tailDelta)) ≡⟨ refl ⟩ + mu (mu (delta x d)) ≡⟨ refl ⟩ + (mu ∙ mu) (delta x d) + ∎ + + +monad-law-2-1 : {l : Level} {A : Set l} -> (n : Nat) -> (d : Delta A) -> (bind (fmap eta d) (n-tail n)) ≡ d +monad-law-2-1 O (mono x) = refl +monad-law-2-1 O (delta x d) = begin + bind (fmap eta (delta x d)) (n-tail O) ≡⟨ refl ⟩ + bind (delta (eta x) (fmap eta d)) id ≡⟨ refl ⟩ + delta (headDelta (eta x)) (bind (fmap eta d) tailDelta) ≡⟨ refl ⟩ + delta x (bind (fmap eta d) tailDelta) ≡⟨ cong (\de -> delta x de) (monad-law-2-1 (S O) d) ⟩ + delta x d ∎ +monad-law-2-1 (S n) (mono x) = begin + bind (fmap eta (mono x)) (n-tail (S n)) ≡⟨ refl ⟩ + bind (mono (mono x)) (n-tail (S n)) ≡⟨ refl ⟩ + n-tail (S n) (mono x) ≡⟨ tail-delta-to-mono (S n) x ⟩ + mono x ∎ +monad-law-2-1 (S n) (delta x d) = begin + bind (fmap eta (delta x d)) (n-tail (S n)) ≡⟨ refl ⟩ + bind (delta (eta x) (fmap eta d)) (n-tail (S n)) ≡⟨ refl ⟩ + delta (headDelta ((n-tail (S n) (eta x)))) (bind (fmap eta d) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ + delta (headDelta ((n-tail (S n) (eta x)))) (bind (fmap eta d) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta (de)) (bind (fmap eta d) (n-tail (S (S n))))) (tail-delta-to-mono (S n) x) ⟩ + delta (headDelta (eta x)) (bind (fmap eta d) (n-tail (S (S n)))) ≡⟨ refl ⟩ + delta x (bind (fmap eta d) (n-tail (S (S n)))) ≡⟨ cong (\d -> delta x d) (monad-law-2-1 (S (S n)) d) ⟩ + delta x d + ∎ + + +-- monad-law-2 : join . fmap return = join . return = id +-- monad-law-2 join . fmap return = join . return +monad-law-2 : {l : Level} {A : Set l} -> (d : Delta A) -> + (mu ∙ fmap eta) d ≡ (mu ∙ eta) d +monad-law-2 (mono x) = refl +monad-law-2 (delta x d) = begin + (mu ∙ fmap eta) (delta x d) ≡⟨ refl ⟩ + mu (fmap eta (delta x d)) ≡⟨ refl ⟩ + mu (delta (mono x) (fmap eta d)) ≡⟨ refl ⟩ + delta (headDelta (mono x)) (bind (fmap eta d) tailDelta) ≡⟨ refl ⟩ + delta x (bind (fmap eta d) tailDelta) ≡⟨ cong (\d -> delta x d) (monad-law-2-1 (S O) d) ⟩ + (delta x d) ≡⟨ refl ⟩ + mu (mono (delta x d)) ≡⟨ refl ⟩ + mu (eta (delta x d)) ≡⟨ refl ⟩ + (mu ∙ eta) (delta x d) + ∎ + + +-- monad-law-2' : join . return = id +monad-law-2' : {l : Level} {A : Set l} -> (d : Delta A) -> (mu ∙ eta) d ≡ id d +monad-law-2' d = refl + + +-- monad-law-3 : return . f = fmap f . return +monad-law-3 : {l : Level} {A B : Set l} (f : A -> B) (x : A) -> (eta ∙ f) x ≡ (fmap f ∙ eta) x +monad-law-3 f x = refl + + +monad-law-4-1 : {l ll : Level} {A : Set l} {B : Set ll} -> (n : Nat) -> (f : A -> B) -> (ds : Delta (Delta A)) -> + bind (fmap (fmap f) ds) (n-tail n) ≡ fmap f (bind ds (n-tail n)) +monad-law-4-1 O f (mono d) = refl +monad-law-4-1 O f (delta d ds) = begin + bind (fmap (fmap f) (delta d ds)) (n-tail O) ≡⟨ refl ⟩ + bind (delta (fmap f d) (fmap (fmap f) ds)) (n-tail O) ≡⟨ refl ⟩ + delta (headDelta (fmap f d)) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta de (bind (fmap (fmap f) ds) tailDelta)) (head-delta-natural-transformation f d) ⟩ + delta (f (headDelta d)) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f (headDelta d)) de) (monad-law-4-1 (S O) f ds) ⟩ + delta (f (headDelta d)) (fmap f (bind ds tailDelta)) ≡⟨ refl ⟩ + fmap f (delta (headDelta d) (bind ds tailDelta)) ≡⟨ refl ⟩ + fmap f (bind (delta d ds) (n-tail O)) ∎ +monad-law-4-1 (S n) f (mono d) = begin + bind (fmap (fmap f) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩ + bind (mono (fmap f d)) (n-tail (S n)) ≡⟨ refl ⟩ + n-tail (S n) (fmap f d) ≡⟨ n-tail-natural-transformation (S n) f d ⟩ + fmap f (n-tail (S n) d) ≡⟨ refl ⟩ + fmap f (bind (mono d) (n-tail (S n))) + ∎ +monad-law-4-1 (S n) f (delta d ds) = begin + bind (fmap (fmap f) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩ + delta (headDelta (n-tail (S n) (fmap f d))) (bind (fmap (fmap f) ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ + delta (headDelta (n-tail (S n) (fmap f d))) (bind (fmap (fmap f) ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta de) (bind (fmap (fmap f) ds) (n-tail (S (S n))))) (n-tail-natural-transformation (S n) f d) ⟩ + delta (headDelta (fmap f ((n-tail (S n) d)))) (bind (fmap (fmap f) ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta de (bind (fmap (fmap f) ds) (n-tail (S (S n))))) (head-delta-natural-transformation f (n-tail (S n) d)) ⟩ + delta (f (headDelta (n-tail (S n) d))) (bind (fmap (fmap f) ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (f (headDelta (n-tail (S n) d))) de) (monad-law-4-1 (S (S n)) f ds) ⟩ + delta (f (headDelta (n-tail (S n) d))) (fmap f (bind ds (n-tail (S (S n))))) ≡⟨ refl ⟩ + fmap f (delta (headDelta (n-tail (S n) d)) (bind ds (n-tail (S (S n))))) ≡⟨ refl ⟩ + fmap f (delta (headDelta (n-tail (S n) d)) (bind ds (tailDelta ∙ (n-tail (S n))))) ≡⟨ refl ⟩ + fmap f (bind (delta d ds) (n-tail (S n))) ∎ + + +-- monad-law-4 : join . fmap (fmap f) = fmap f . join +monad-law-4 : {l ll : Level} {A : Set l} {B : Set ll} (f : A -> B) (d : Delta (Delta A)) -> + (mu ∙ fmap (fmap f)) d ≡ (fmap f ∙ mu) d +monad-law-4 f (mono d) = refl +monad-law-4 f (delta (mono x) ds) = begin + (mu ∙ fmap (fmap f)) (delta (mono x) ds) ≡⟨ refl ⟩ + mu ( fmap (fmap f) (delta (mono x) ds)) ≡⟨ refl ⟩ + mu (delta (mono (f x)) (fmap (fmap f) ds)) ≡⟨ refl ⟩ + delta (headDelta (mono (f x))) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ refl ⟩ + delta (f x) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f x) de) (monad-law-4-1 (S O) f ds) ⟩ + delta (f x) (fmap f (bind ds tailDelta)) ≡⟨ refl ⟩ + fmap f (delta x (bind ds tailDelta)) ≡⟨ refl ⟩ + fmap f (delta (headDelta (mono x)) (bind ds tailDelta)) ≡⟨ refl ⟩ + fmap f (mu (delta (mono x) ds)) ≡⟨ refl ⟩ + (fmap f ∙ mu) (delta (mono x) ds) ∎ +monad-law-4 f (delta (delta x d) ds) = begin + (mu ∙ fmap (fmap f)) (delta (delta x d) ds) ≡⟨ refl ⟩ + mu (fmap (fmap f) (delta (delta x d) ds)) ≡⟨ refl ⟩ + mu (delta (delta (f x) (fmap f d)) (fmap (fmap f) ds)) ≡⟨ refl ⟩ + delta (headDelta (delta (f x) (fmap f d))) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ refl ⟩ + delta (f x) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f x) de) (monad-law-4-1 (S O) f ds) ⟩ + delta (f x) (fmap f (bind ds tailDelta)) ≡⟨ refl ⟩ + fmap f (delta x (bind ds tailDelta)) ≡⟨ refl ⟩ + fmap f (delta (headDelta (delta x d)) (bind ds tailDelta)) ≡⟨ refl ⟩ + fmap f (mu (delta (delta x d) ds)) ≡⟨ refl ⟩ + (fmap f ∙ mu) (delta (delta x d) ds) ∎ + +delta-is-monad : {l : Level} {A : Set l} -> Monad {l} {A} Delta delta-is-functor +delta-is-monad = record { mu = mu; + eta = eta; + association-law = monad-law-1; + left-unity-law = monad-law-2; + right-unity-law = monad-law-2' } + + +{- +-- Monad-laws (Haskell) +-- monad-law-h-1 : return a >>= k = k a +monad-law-h-1 : {l ll : Level} {A : Set l} {B : Set ll} -> + (a : A) -> (k : A -> (Delta B)) -> + (return a >>= k) ≡ (k a) +monad-law-h-1 a k = refl + + + +-- monad-law-h-2 : m >>= return = m +monad-law-h-2 : {l : Level}{A : Set l} -> (m : Delta A) -> (m >>= return) ≡ m +monad-law-h-2 (mono x) = refl +monad-law-h-2 (delta x d) = cong (delta x) (monad-law-h-2 d) + + +-- monad-law-h-3 : m >>= (\x -> k x >>= h) = (m >>= k) >>= h +monad-law-h-3 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} -> + (m : Delta A) -> (k : A -> (Delta B)) -> (h : B -> (Delta C)) -> + (m >>= (\x -> k x >>= h)) ≡ ((m >>= k) >>= h) +monad-law-h-3 (mono x) k h = refl +monad-law-h-3 (delta x d) k h = {!!} + +-} +