### view agda/delta.agda @ 87:6789c65a75bc

Split functor-proofs into delta.functor
author Yasutaka Higa Mon, 19 Jan 2015 11:00:34 +0900 fc5cd8c50312 526186c4f298
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line source
```
open import list
open import basic
open import nat
open import laws

open import Level
open import Relation.Binary.PropositionalEquality
open ≡-Reasoning

module delta where

data Delta {l : Level} (A : Set l) : (Set l) where
mono    : A -> Delta A
delta   : A -> Delta A -> Delta A

deltaAppend : {l : Level} {A : Set l} -> Delta A -> Delta A -> Delta A
deltaAppend (mono x) d     = delta x d
deltaAppend (delta x d) ds = delta x (deltaAppend d ds)

headDelta : {l : Level} {A : Set l} -> Delta A -> A
headDelta (delta x _) = x

tailDelta : {l : Level} {A : Set l} -> Delta A -> Delta A
tailDelta (mono x)    = mono x
tailDelta (delta _ d) = d

n-tail : {l : Level} {A : Set l} -> Nat -> ((Delta A) -> (Delta A))
n-tail O = id
n-tail (S n) =  tailDelta ∙ (n-tail n)

-- Functor
fmap : {l ll : Level} {A : Set l} {B : Set ll} -> (A -> B) -> (Delta A) -> (Delta B)
fmap f (mono x)    = mono  (f x)
fmap f (delta x d) = delta (f x) (fmap f d)

eta : {l : Level} {A : Set l} -> A -> Delta A
eta x = mono x

bind : {l ll : Level} {A : Set l} {B : Set ll} -> (Delta A) -> (A -> Delta B) -> Delta B
bind (mono x)    f = f x
bind (delta x d) f = delta (headDelta (f x)) (bind d (tailDelta ∙ f))

mu : {l : Level} {A : Set l} -> Delta (Delta A) -> Delta A
mu d = bind d id

returnS : {l : Level} {A : Set l} -> A -> Delta A
returnS x = mono x

returnSS : {l : Level} {A : Set l} -> A -> A -> Delta A
returnSS x y = deltaAppend (returnS x) (returnS y)

return : {l : Level} {A : Set l} -> A -> Delta A
return = eta

_>>=_ : {l ll : Level} {A : Set l} {B : Set ll} ->
(x : Delta A) -> (f : A -> (Delta B)) -> (Delta B)
(mono x) >>= f    = f x
(delta x d) >>= f = delta (headDelta (f x)) (d >>= (tailDelta ∙ f))

-- proofs

-- sub-proofs

n-tail-plus : {l : Level} {A : Set l} -> (n : Nat) -> ((n-tail {l} {A} n) ∙ tailDelta) ≡ n-tail (S n)
n-tail-plus O     = refl
n-tail-plus (S n) = begin
n-tail (S n) ∙ tailDelta             ≡⟨ refl ⟩
(tailDelta ∙ (n-tail n)) ∙ tailDelta ≡⟨ refl ⟩
tailDelta ∙ ((n-tail n) ∙ tailDelta) ≡⟨ cong (\t -> tailDelta ∙ t) (n-tail-plus n) ⟩
n-tail (S (S n))
∎

n-tail-add : {l : Level} {A : Set l} {d : Delta A} -> (n m : Nat) -> (n-tail {l} {A} n) ∙ (n-tail m) ≡ n-tail (n + m)
n-tail-add (S n) O = begin
n-tail (S n) ∙ n-tail O  ≡⟨ refl ⟩
n-tail (S n)             ≡⟨ cong (\n -> n-tail n) (nat-add-right-zero (S n))⟩
n-tail (S n + O)
∎
n-tail-add {l} {A} {d} (S n) (S m) =      begin
n-tail (S n) ∙ n-tail (S m)             ≡⟨ refl ⟩
(tailDelta ∙ (n-tail n)) ∙ n-tail (S m) ≡⟨ refl ⟩
tailDelta ∙ ((n-tail n) ∙ n-tail (S m)) ≡⟨ cong (\t -> tailDelta ∙ t) (n-tail-add {l} {A} {d} n (S m)) ⟩
tailDelta ∙ (n-tail (n + (S m)))        ≡⟨ refl ⟩
n-tail (S (n + S m))                    ≡⟨ refl ⟩
n-tail (S n + S m)                      ∎

tail-delta-to-mono : {l : Level} {A : Set l} -> (n : Nat) -> (x : A) ->
(n-tail n) (mono x) ≡ (mono x)
tail-delta-to-mono O x     = refl
tail-delta-to-mono (S n) x =      begin
n-tail (S n) (mono x)           ≡⟨ refl ⟩
tailDelta (n-tail n (mono x))   ≡⟨ refl ⟩
tailDelta (n-tail n (mono x))   ≡⟨ cong (\t -> tailDelta t) (tail-delta-to-mono n x) ⟩
tailDelta (mono x)              ≡⟨ refl ⟩
mono x                          ∎

head-delta-natural-transformation : {l ll : Level} {A : Set l} {B : Set ll}
-> (f : A -> B) -> (d : Delta A) -> headDelta (fmap f d) ≡ f (headDelta d)
head-delta-natural-transformation f (mono x)    = refl
head-delta-natural-transformation f (delta x d) = refl

n-tail-natural-transformation  : {l ll : Level} {A : Set l} {B : Set ll}
-> (n : Nat) -> (f : A -> B) -> (d : Delta A) ->  n-tail n (fmap f d) ≡ fmap f (n-tail n d)
n-tail-natural-transformation O f d            = refl
n-tail-natural-transformation (S n) f (mono x) = begin
n-tail (S n) (fmap f (mono x))  ≡⟨ refl ⟩
n-tail (S n) (mono (f x))       ≡⟨ tail-delta-to-mono (S n) (f x) ⟩
(mono (f x))                    ≡⟨ refl ⟩
fmap f (mono x)                 ≡⟨ cong (\d -> fmap f d) (sym (tail-delta-to-mono (S n) x)) ⟩
fmap f (n-tail (S n) (mono x))  ∎
n-tail-natural-transformation (S n) f (delta x d) = begin
n-tail (S n) (fmap f (delta x d))                 ≡⟨ refl ⟩
n-tail (S n) (delta (f x) (fmap f d))             ≡⟨ cong (\t -> t (delta (f x) (fmap f d))) (sym (n-tail-plus n)) ⟩
((n-tail n) ∙ tailDelta) (delta (f x) (fmap f d)) ≡⟨ refl ⟩
n-tail n (fmap f d)                               ≡⟨ n-tail-natural-transformation n f d ⟩
fmap f (n-tail n d)                               ≡⟨ refl ⟩
fmap f (((n-tail n) ∙ tailDelta) (delta x d))     ≡⟨ cong (\t -> fmap f (t (delta x d))) (n-tail-plus n) ⟩
fmap f (n-tail (S n) (delta x d))                 ∎

{-

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) ∙ 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)) ⟩
∎
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 (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 (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 (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 (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 (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-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-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-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-2 : m >>= return  =  m
monad-law-h-2 : {l : Level}{A : Set l} -> (m : Delta A) -> (m >>= return)  ≡ m