--- a/agda/delta/functor.agda	Mon Jan 26 23:00:05 2015 +0900
+++ b/agda/delta/functor.agda	Tue Jan 27 17:49:25 2015 +0900
@@ -1,32 +1,43 @@
 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} {n : Rev} ->  (d : Delta A n) -> (delta-fmap id) d ≡ id d
+functor-law-1 :  {l : Level} {A : Set l} {n : Nat} ->  (d : Delta A (S 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} {n : Rev} {A B C : Set l} -> 
-                (f : B -> C) -> (g : A -> B) -> (d : Delta A n) ->
+functor-law-2 : {l : Level} {n : Nat} {A B C : Set l} ->
+                (f : B -> C) -> (g : A -> B) -> (d : Delta A (S 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} {n : Rev} -> Functor {l} (\A -> Delta A n)
+
+
+delta-is-functor : {l : Level} {n : Nat} -> Functor {l} (\A -> Delta A (S n))
 delta-is-functor = record {  fmap = delta-fmap ;
                              preserve-id = functor-law-1;
                              covariant  = \f g -> functor-law-2 g f}
+
+
+open ≡-Reasoning
+delta-fmap-equiv : {l : Level} {A B : Set l} {n : Nat} 
+                   (f g : A -> B) (eq : f ≡ g) (d : Delta A (S n)) ->
+                 delta-fmap f d ≡ delta-fmap g d
+delta-fmap-equiv f g eq (mono x) = begin
+  mono (f x) ≡⟨ cong (\h -> (mono (h x))) eq ⟩
+  mono (g x) ∎
+delta-fmap-equiv f g eq (delta x d) = begin
+  delta (f x) (delta-fmap f d) ≡⟨ cong (\h -> (delta (h x) (delta-fmap f d))) eq ⟩
+  delta (g x) (delta-fmap f d) ≡⟨ cong (\fx -> (delta (g x) fx)) (delta-fmap-equiv f g eq d) ⟩
+  delta (g x) (delta-fmap g d)   ∎