### changeset 104:ebd0d6e2772c

Trying redenition Delta with length constraints
author Yasutaka Higa Mon, 26 Jan 2015 23:00:05 +0900 a271f3ff1922 e6499a50ccbd agda/delta.agda agda/delta/functor.agda agda/delta/monad.agda agda/deltaM.agda agda/deltaM/functor.agda agda/deltaM/monad.agda agda/revision.agda 7 files changed, 197 insertions(+), 134 deletions(-) [+]
line wrap: on
line diff
```--- a/agda/delta.agda	Mon Jan 26 14:08:46 2015 +0900
+++ b/agda/delta.agda	Mon Jan 26 23:00:05 2015 +0900
@@ -1,6 +1,7 @@
open import list
open import basic
open import nat
+open import revision
open import laws

open import Level
@@ -9,59 +10,56 @@

module delta where

+data Delta {l : Level} (A : Set l) : (Rev -> (Set l)) where
+  mono    : A -> Delta A init
+  delta   : {v : Rev} -> A -> Delta A v -> Delta A (commit v)

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

-tailDelta : {l : Level} {A : Set l} -> Delta A -> Delta A
-tailDelta (mono x)    = mono x
+tailDelta : {l : Level} {A : Set l} {n : Rev} -> Delta A (commit n) -> Delta A n
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
-delta-fmap : {l : Level} {A B : Set l} -> (A -> B) -> (Delta A) -> (Delta B)
+delta-fmap : {l : Level} {A B : Set l} {n : Rev} -> (A -> B) -> (Delta A n) -> (Delta B n)
delta-fmap f (mono x)    = mono  (f x)
delta-fmap f (delta x d) = delta (f x) (delta-fmap f d)

-delta-eta : {l : Level} {A : Set l} -> A -> Delta A
-delta-eta x = mono x
+delta-eta : {l : Level} {A : Set l} {v : Rev} -> A -> Delta A v
+delta-eta {v = init} x     = mono x
+delta-eta {v = commit v} x = delta x (delta-eta {v = v} x)

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

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

+{-
-delta-return : {l : Level} {A : Set l} -> A -> Delta A
+delta-return : {l : Level} {A : Set l} -> A -> Delta A (S O)
delta-return = delta-eta

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

+-}

-
+{-
-- proofs

-- sub-proofs
@@ -122,3 +120,4 @@
delta-fmap f (n-tail n d)                               ≡⟨ refl ⟩
delta-fmap f (((n-tail n) ∙ tailDelta) (delta x d))     ≡⟨ cong (\t -> delta-fmap f (t (delta x d))) (n-tail-plus n) ⟩
delta-fmap f (n-tail (S n) (delta x d))                 ∎
+-}```
```--- a/agda/delta/functor.agda	Mon Jan 26 14:08:46 2015 +0900
+++ b/agda/delta/functor.agda	Mon Jan 26 23:00:05 2015 +0900
@@ -1,28 +1,32 @@
-open import delta
-open import basic
-open import laws
-
open import Level
open import Relation.Binary.PropositionalEquality

+open import basic
+open import delta
+open import laws
+open import nat
+open import revision
+
+
+
module delta.functor where

-- Functor-laws

-- Functor-law-1 : T(id) = id'
-functor-law-1 :  {l : Level} {A : Set l} ->  (d : Delta A) -> (delta-fmap id) d ≡ id d
+functor-law-1 :  {l : Level} {A : Set l} {n : Rev} ->  (d : Delta A n) -> (delta-fmap id) d ≡ id d
functor-law-1 (mono x)    = refl
functor-law-1 (delta x d) = cong (delta x) (functor-law-1 d)

-- Functor-law-2 : T(f . g) = T(f) . T(g)
-functor-law-2 : {l : Level} {A B C : Set l} ->
-                (f : B -> C) -> (g : A -> B) -> (d : Delta A) ->
+functor-law-2 : {l : Level} {n : Rev} {A B C : Set l} ->
+                (f : B -> C) -> (g : A -> B) -> (d : Delta A n) ->
(delta-fmap (f ∙ g)) d ≡ ((delta-fmap f) ∙ (delta-fmap g)) d
functor-law-2 f g (mono x)    = refl
functor-law-2 f g (delta x d) = cong (delta (f (g x))) (functor-law-2 f g d)

-delta-is-functor : {l : Level} -> Functor {l} Delta
+delta-is-functor : {l : Level} {n : Rev} -> Functor {l} (\A -> Delta A n)
delta-is-functor = record {  fmap = delta-fmap ;
preserve-id = functor-law-1;
covariant  = \f g -> functor-law-2 g f}```
```--- a/agda/delta/monad.agda	Mon Jan 26 14:08:46 2015 +0900
+++ b/agda/delta/monad.agda	Mon Jan 26 23:00:05 2015 +0900
@@ -3,6 +3,7 @@
open import delta.functor
open import nat
open import laws
+open import revision

open import Level
@@ -13,6 +14,7 @@

+{-

monad-law-1-5 : {l : Level} {A : Set l} -> (m : Nat) (n : Nat) -> (ds : Delta (Delta A)) ->
n-tail n (delta-bind ds (n-tail m))  ≡ delta-bind (n-tail n ds) (n-tail (m + n))
@@ -322,14 +324,25 @@
delta-fmap f (delta-mu (delta (delta x d) ds))                                              ≡⟨ refl ⟩
(delta-fmap f ∙ delta-mu) (delta (delta x d) ds) ∎

+-}
+-- monad-law-1 : join . delta-fmap join = join . join
+monad-law-1 : {l : Level} {A : Set l} {a : Rev} -> (d : Delta (Delta (Delta A a) a) a) ->
+            ((delta-mu ∙ (delta-fmap delta-mu)) d) ≡ ((delta-mu ∙ delta-mu) d)
+monad-law-1 (delta d ds) = {!!}
+
+delta-is-monad : {l : Level} {v : Rev} -> Monad {l} (\A -> Delta A v)  delta-is-functor
+
+
delta-is-monad = record { eta    = delta-eta;
mu     = delta-mu;
return = delta-eta;
bind   = delta-bind;
+
+

```
```--- a/agda/deltaM.agda	Mon Jan 26 14:08:46 2015 +0900
+++ b/agda/deltaM.agda	Mon Jan 26 23:00:05 2015 +0900
@@ -4,6 +4,7 @@
open import delta
open import delta.functor
open import nat
+open import revision
open import laws

module deltaM where
@@ -15,83 +16,76 @@
{functorM : {l' : Level} -> Functor {l'} M}
{monadM : {l' : Level} {A : Set l'} -> Monad {l'} M functorM}
(A : Set l)
-            : Set l where
-   deltaM : Delta (M A) -> DeltaM M {functorM} {monadM} A
+            : (Rev -> Set l) where
+   deltaM : {v : Rev} -> Delta (M A) v -> DeltaM M {functorM} {monadM} A v

-- DeltaM utils

-headDeltaM : {l : Level} {A : Set l}
+headDeltaM : {l : Level} {A : Set l} {v : Rev}
{M : {l' : Level} -> Set l' -> Set l'}
{functorM : {l' : Level} -> Functor {l'} M}
-             -> DeltaM M {functorM} {monadM} A -> M A
+             -> DeltaM M {functorM} {monadM} A v -> M A

-tailDeltaM :  {l : Level} {A : Set l}
-             {M : {l' : Level} -> Set l' -> Set l'}
-             {functorM : {l' : Level} -> Functor {l'} M}
-             -> DeltaM M {functorM} {monadM} A -> DeltaM M {functorM} {monadM} A
-tailDeltaM (deltaM d)    = deltaM (tailDelta d)
-
-
-appendDeltaM : {l : Level} {A : Set l}
+tailDeltaM :  {l : Level} {A : Set l} {v : Rev}
{M : {l' : Level} -> Set l' -> Set l'}
{functorM : {l' : Level} -> Functor {l'} M}
-             -> DeltaM M {functorM} {monadM} A -> DeltaM M {functorM} {monadM} A -> DeltaM M {functorM} {monadM} A
-appendDeltaM (deltaM d) (deltaM dd) = deltaM (deltaAppend d dd)
+             -> DeltaM {l} M {functorM} {monadM} A (commit v) -> DeltaM M {functorM} {monadM} A v
+tailDeltaM {_} {n} (deltaM d) = deltaM (tailDelta d)

-checkOut : {l : Level} {A : Set l}
-           {M : {l' : Level} -> Set l' -> Set l'}
-           {functorM : {l' : Level} -> Functor {l'} M}
-         -> Nat -> DeltaM M {functorM} {monadM} A -> M A
-checkOut O     (deltaM (mono x))    = x
-checkOut O     (deltaM (delta x _)) = x
-checkOut (S n) (deltaM (mono x))    = x
-checkOut {l} {A} {M} {functorM} {monadM} (S n) (deltaM (delta _ d)) = checkOut {l} {A} {M} {functorM} {monadM} n (deltaM d)
+appendDeltaM : {l : Level} {A : Set l} {n m : Rev}
+             {M : {l' : Level} -> Set l' -> Set l'}
+             {functorM : {l' : Level} -> Functor {l'} M}
+             -> DeltaM M {functorM} {monadM} A  n -> DeltaM M {functorM} {monadM} A m -> DeltaM M {functorM} {monadM} A (merge n m)
+appendDeltaM (deltaM d) (deltaM dd) = deltaM (deltaAppend d dd)
+

-- functor definitions
open Functor
-deltaM-fmap : {l : Level} {A B : Set l}
+deltaM-fmap : {l : Level} {A B : Set l} {n : Rev}
{M : {l' : Level} -> Set l' -> Set l'}
{functorM : {l' : Level} -> Functor {l'} M}
-              -> (A -> B) -> DeltaM M {functorM} {monadM} A -> DeltaM M {functorM} {monadM} B
-deltaM-fmap {l} {A} {B} {M} {functorM} f (deltaM d) = deltaM (fmap delta-is-functor (fmap functorM f) d)
+              -> (A -> B) -> DeltaM M {functorM} {monadM} A n -> DeltaM M {functorM} {monadM} B n
+deltaM-fmap {l} {A} {B} {n} {M} {functorM} f (deltaM d) = deltaM (fmap delta-is-functor (fmap functorM f) d)
+
+
+

-deltaM-eta : {l : Level} {A : Set l} {M : {l' : Level} -> Set l' -> Set l'}
-                                        {functorM : {l' : Level} -> Functor {l'} M}
-            -> A -> (DeltaM M {functorM} {monadM} A)
-deltaM-eta {_} {A} {_} {_} {monadM} x = deltaM (mono (eta monadM x))

-deltaM-mu : {l : Level} {A : Set l} {M : {l' : Level} -> Set l' -> Set l'}
-                                        {functorM : {l' : Level} -> Functor {l'} M}
-            -> (DeltaM M {functorM} {monadM} (DeltaM M {functorM} {monadM} A)) -> DeltaM M {functorM} {monadM} A
-deltaM-mu {l} {A} {M} {functorM} {monadM} (deltaM (mono x))               = deltaM (mono (mu monadM (fmap functorM headDeltaM x)))
-deltaM-mu {l} {A} {M} {functorM} {monadM} (deltaM (delta x (mono xx)))    = appendDeltaM (deltaM (mono (bind  monadM x headDeltaM)))
-                                                                                         (deltaM-mu (deltaM (mono xx)))
-deltaM-mu {l} {A} {M} {functorM} {monadM} (deltaM (delta x (delta xx d))) = appendDeltaM (deltaM (mono (bind {l}  monadM x headDeltaM)))
-                                                                                         (deltaM-mu (deltaM  d))
--- original deltaM-mu definitions. but it's cannot termination checking.
--- manually expand nested delta for delete tailDelta in argument to recursive deltaM-mu.
-{-
-deltaM-mu {l} {A} {M} {functorM} {monadM} (deltaM (delta x d)) =  appendDeltaM (deltaM (mono (bind monadM x headDeltaM)))
-                                                                               (deltaM-mu (deltaM (tailDelta d)))
--}
+deltaM-eta : {l : Level} {A : Set l} {v : Rev}
+                         {M : {l' : Level} -> Set l' -> Set l'}
+                         {functorM : {l' : Level} -> Functor {l'} M}
+            -> A -> (DeltaM M {functorM} {monadM} A v)
+deltaM-eta {v = init} {monadM = mm} x = deltaM (mono (eta mm x))
+deltaM-eta {v = (commit v)} {monadM = mm} x = appendDeltaM (deltaM (mono (eta mm x)))
+                                                           (deltaM-eta {v = v} x)

-deltaM-bind : {l : Level} {A B : Set l} {M : {l' : Level} -> Set l' -> Set l'}
+
+deltaM-bind : {l : Level} {A B : Set l} {v : Rev}
+                                        {M : {l' : Level} -> Set l' -> Set l'}
{functorM : {l' : Level} -> Functor {l'} M}
-            -> (DeltaM M {functorM} {monadM} A) -> (A -> DeltaM M {functorM} {monadM} B) -> DeltaM M {functorM} {monadM} B
-deltaM-bind {l} {A} {B} {M} {functorM} {monadM} d    f = deltaM-mu (deltaM-fmap f d)
+            -> (DeltaM M {functorM} {monadM} A v) -> (A -> DeltaM M {functorM} {monadM} B v) -> DeltaM M {functorM} {monadM} B v
+deltaM-bind {v = init}     {monadM = mm} (deltaM (mono x))    f = deltaM (mono (bind mm x (headDeltaM ∙ f)))
+deltaM-bind {v = commit v} {monadM = mm} (deltaM (delta x d)) f = appendDeltaM (deltaM (mono (bind mm x (headDeltaM ∙ f))))
+                                                                               (deltaM-bind (deltaM d) (tailDeltaM ∙ f))
+
+
+deltaM-mu : {l : Level} {A : Set l} {v : Rev}
+                        {M : {l' : Level} -> Set l' -> Set l'}
+                        {functorM : {l' : Level} -> Functor {l'} M}
+            -> (DeltaM M {functorM} {monadM} (DeltaM M {functorM} {monadM} A v) v) -> DeltaM M {functorM} {monadM} A v
+deltaM-mu d = deltaM-bind d id```
```--- a/agda/deltaM/functor.agda	Mon Jan 26 14:08:46 2015 +0900
+++ b/agda/deltaM/functor.agda	Mon Jan 26 23:00:05 2015 +0900
@@ -6,17 +6,19 @@
open import delta
open import delta.functor
open import deltaM
+open import nat
+open import revision
open import laws
open Functor

module deltaM.functor where

-deltaM-preserve-id :  {l : Level} {A : Set l}
+deltaM-preserve-id :  {l : Level} {A : Set l} {n : Rev}
{M : {l' : Level} -> Set l' -> Set l'}
(functorM : {l' : Level} -> Functor {l'} M)
-                      -> (d : DeltaM M {functorM} {monadM} A) -> deltaM-fmap id d ≡ id d
+                      -> (d : DeltaM M {functorM} {monadM} A n) -> deltaM-fmap id d ≡ id d
deltaM-preserve-id functorM (deltaM (mono x))  = begin
deltaM-fmap id (deltaM (mono x))                           ≡⟨ refl ⟩
deltaM (fmap delta-is-functor (fmap functorM id) (mono x)) ≡⟨ refl ⟩
@@ -42,11 +44,11 @@
∎

-deltaM-covariant : {l : Level} {A B C : Set l} ->
+deltaM-covariant : {l : Level} {A B C : Set l} {n : Rev} ->
{M : {l' : Level} -> Set l' -> Set l'}
(functorM : {l' : Level} -> Functor {l'} M)
-                   (f : B -> C) -> (g : A -> B) -> (d : DeltaM M {functorM} {monadM} A) ->
+                   (f : B -> C) -> (g : A -> B) -> (d : DeltaM M {functorM} {monadM} A n) ->
(deltaM-fmap (f ∙ g)) d ≡ ((deltaM-fmap f) ∙ (deltaM-fmap g)) d
deltaM-covariant functorM f g (deltaM (mono x))    = begin
deltaM-fmap (f ∙ g) (deltaM (mono x))                     ≡⟨ refl ⟩
@@ -79,10 +81,11 @@
∎

-deltaM-is-functor : {l : Level} {M : {l' : Level} -> Set l' -> Set l'}
+deltaM-is-functor : {l : Level} {n : Rev}
+                                {M : {l' : Level} -> Set l' -> Set l'}
{functorM : {l' : Level} -> Functor {l'} M }
-                    -> Functor {l} (DeltaM M {functorM} {monadM})
-deltaM-is-functor {_} {_} {functorM} = record { fmap        = deltaM-fmap ;
-                             preserve-id  = deltaM-preserve-id functorM ;
-                             covariant    = (\f g -> deltaM-covariant functorM g f)}
\ No newline at end of file
+                    -> Functor {l} (\A -> DeltaM M {functorM} {monadM} A n)
+deltaM-is-functor {_} {_} {_} {functorM} = record { fmap        = deltaM-fmap ;
+                                                    preserve-id  = deltaM-preserve-id functorM ;
+                                                    covariant    = (\f g -> deltaM-covariant functorM g f)}```
```--- a/agda/deltaM/monad.agda	Mon Jan 26 14:08:46 2015 +0900
+++ b/agda/deltaM/monad.agda	Mon Jan 26 23:00:05 2015 +0900
@@ -7,6 +7,7 @@
open import delta.functor
open import deltaM
open import deltaM.functor
+open import nat
open import laws

@@ -15,67 +16,91 @@

-postulate deltaM-mu-is-natural-transformation : {l : Level} {A : Set l}
-                                                  {M : {l' : Level} -> Set l' -> Set l'} ->
-                                                  {functorM :  {l' : Level} -> Functor {l'}  M}
-                                                  {monadM   : {l' : Level} -> Monad {l'} M (functorM ) } ->
-                                                  NaturalTransformation (\A -> DeltaM M (DeltaM M A)) (\A -> DeltaM M A)
-                                                                        {deltaM-fmap ∙ deltaM-fmap} {deltaM-fmap {l}}
-                                                  (deltaM-mu {_} {_} {M} {functorM} {monadM})
-
-headDeltaM-commute : {l : Level} {A B : Set l}
+headDeltaM-commute : {l : Level} {A B : Set l} {n : Nat}
{M : {l' : Level} -> Set l' -> Set l'} ->
{functorM :  {l' : Level}  -> Functor {l'}  M} ->
{monadM   : {l' : Level} -> Monad {l'} M (functorM ) } ->
-                                 (f : A -> B) -> (x : DeltaM M {functorM} {monadM} A) ->
+                                 (f : A -> B) -> (x : DeltaM M {functorM} {monadM} A n) ->
headDeltaM-commute f (deltaM (mono x))    = refl
headDeltaM-commute f (deltaM (delta x d)) = refl

-headDeltaM-is-natural-transformation : {l : Level} {A : Set l}
-                                                  {M : {l' : Level} -> Set l' -> Set l'} ->
-                                                  {functorM :  {l' : Level} -> Functor {l'} M}
-                                                  {monadM   : {l' : Level} -> Monad {l'} M functorM } ->
-                                                  NaturalTransformation {l} (\A -> DeltaM M {functorM} {monadM} A) M
-                                                                            {\f d → deltaM (mono (headDeltaM (deltaM-fmap f d)))} {fmap functorM} headDeltaM
---                                                                      {deltaM-fmap} {fmap (functorM {l} {A})} headDeltaM
+{-
+headDeltaM-is-natural-transformation : {l : Level} {A : Set l} {n : Nat}
+                                       {M : {l' : Level} -> Set l' -> Set l'} ->
+                                       {functorM :  {l' : Level} -> Functor {l'} M}
+                                       {monadM   : {l' : Level} -> Monad {l'} M functorM } ->
+                                       NaturalTransformation {l} (\A -> DeltaM M {functorM} {monadM} A n) M
+                                                                 {\f d → deltaM (mono (headDeltaM (deltaM-fmap f d)))}
+
+-}

-deltaM-right-unity-law : {l : Level} {A : Set l}
+deltaM-right-unity-law : {l : Level} {A : Set l}
{M : {l' : Level} -> Set l' -> Set l'}
(functorM : {l' : Level} -> Functor {l'} M)
-                         (d : DeltaM M {functorM} {monadM} A) ->
+                         (d : DeltaM M {functorM} {monadM} A (S O)) ->
(deltaM-mu ∙ deltaM-eta) d ≡ id d
-deltaM-right-unity-law {l} {A} {M} functorM monadM (deltaM (mono x)) = begin
+deltaM-right-unity-law {l} {A} {M} functorM monadM (deltaM (mono x))    = begin
(deltaM-mu ∙ deltaM-eta) (deltaM (mono x))                ≡⟨ refl ⟩
deltaM-mu (deltaM-eta (deltaM (mono x)))                  ≡⟨ refl ⟩
deltaM-mu (deltaM (mono (eta monadM (deltaM (mono x)))))  ≡⟨ refl ⟩
-  deltaM (mono (mu monadM (fmap functorM (headDeltaM {l} {A} {M}) (eta monadM (deltaM (mono x))))))  ≡⟨ refl ⟩
-  ≡⟨ cong (\de -> deltaM (mono (mu monadM de))) (sym (eta-is-nt monadM headDeltaM (deltaM (mono x)))) ⟩
-  ≡⟨ refl ⟩
-  ≡⟨ cong (\x -> deltaM (mono x)) (sym (right-unity-law monadM x)) ⟩
+  deltaM (mono (mu monadM (fmap functorM (headDeltaM {l} {A} {S O} {M}) (eta monadM (deltaM (mono x))))))
+  ≡⟨ cong (\de -> deltaM (mono (mu monadM (de)))) (sym (eta-is-nt monadM headDeltaM (deltaM (mono x)))) ⟩
+  ≡⟨ refl ⟩
+  ≡⟨ cong (\de -> deltaM (mono de)) (sym (right-unity-law monadM x ) )⟩
deltaM (mono x)
≡⟨ refl ⟩
id (deltaM (mono x))
∎
-deltaM-right-unity-law {l} {A} {M} functorM monadM (deltaM (delta x d)) = begin
-  (deltaM-mu ∙ deltaM-eta) (deltaM (delta x d))  ≡⟨ refl ⟩
-  deltaM-mu (deltaM-eta (deltaM (delta x d)))    ≡⟨ refl ⟩
-  deltaM-mu (deltaM (mono (eta monadM (deltaM (delta x d)))))    ≡⟨ refl ⟩
-  ≡⟨ cong (\de -> deltaM (mono (mu monadM de))) (sym (eta-is-nt monadM headDeltaM (deltaM (delta x d)))) ⟩
-  ≡⟨ refl ⟩
-  ≡⟨ {!!} ⟩
-  id (deltaM (delta x d))
-  ∎
+deltaM-right-unity-law functorM monadM (deltaM (delta x ()))
+-- cannot apply (mu ∙ eta) for over 2 version delta.
+
+
+{-
+deltaM-left-unity-law : {l : Level} {A : Set l}
+                        {M : {l' : Level} -> Set l' -> Set l'}
+                        (functorM : {l' : Level} -> Functor {l'} M)
+                        (d : DeltaM M {functorM} {monadM} A (S O)) ->
+                              (deltaM-mu ∙ (deltaM-fmap deltaM-eta)) d ≡ id d
+deltaM-left-unity-law functorM monadM (deltaM (mono x)) = begin
+   (deltaM-mu ∙ deltaM-fmap deltaM-eta) (deltaM (mono x)) ≡⟨ refl ⟩
+   deltaM-mu (deltaM-fmap deltaM-eta (deltaM (mono x)))   ≡⟨ refl ⟩
+   deltaM-mu (deltaM (mono (fmap functorM deltaM-eta x))) ≡⟨ refl ⟩
+   deltaM (mono (mu monadM (fmap functorM headDeltaM (fmap functorM deltaM-eta x)))) ≡⟨ {!!} ⟩
+   deltaM (mono (mu monadM (fmap functorM headDeltaM (fmap functorM deltaM-eta x)))) ≡⟨ {!!} ⟩
+
+   id (deltaM (mono x))
+   ∎
+deltaM-left-unity-law functorM monadM (deltaM (delta x ()))
+-}
+
+deltaM-is-monad : {l : Level} {A : Set l} {n : Nat}
+                              {M : {l' : Level} -> Set l' -> Set l'}
+                              (functorM : {l' : Level}  -> Functor {l'} M)
+               Monad {l} (\A -> DeltaM M {functorM} {monadM} A n) deltaM-is-functor
+                                    { mu = deltaM-mu;
+                                      eta = deltaM-eta;
+                                    return = {!!};
+                                    bind = {!!};
+                                    association-law = {!!};
+                                    left-unity-law = {!!};
+                                    right-unity-law = {!!};
+                                    eta-is-nt = {!!}
+                                    }
+
+
+
+

{-```
```--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/agda/revision.agda	Mon Jan 26 23:00:05 2015 +0900
@@ -0,0 +1,25 @@
+open import Relation.Binary.PropositionalEquality
+module revision where
+
+data Rev : Set where
+  init   : Rev
+  commit : Rev -> Rev
+
+merge : Rev -> Rev -> Rev
+merge init b       = commit b
+merge (commit a) b = commit (merge a b)
+
+tip : Rev -> Rev -> Rev
+tip init init             = init
+tip init (commit b)       = commit b
+tip (commit a) init       = commit a
+tip (commit a) (commit b) = commit (tip a b)
+
+open ≡-Reasoning
+
+same-tip : (a : Rev) -> tip a a ≡ a
+same-tip init       = refl
+same-tip (commit v) = begin
+     tip (commit v) (commit v) ≡⟨ refl ⟩
+     commit (tip v v)          ≡⟨ cong commit (same-tip v) ⟩
+     commit v                  ∎```