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(-) [+]
line wrap: on
line diff
--- 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 = {!!}
+
+-}
+