### changeset 10:7c7659d4521d

Improve NaturalTransformation definition
author Yasutaka Higa Tue, 20 Jan 2015 10:51:33 +0900 4a0841123810 26e64661b969 sandbox/FunctorExample.agda 1 files changed, 63 insertions(+), 25 deletions(-) [+]
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```--- a/sandbox/FunctorExample.agda	Sun Jan 18 20:44:49 2015 +0900
+++ b/sandbox/FunctorExample.agda	Tue Jan 20 10:51:33 2015 +0900
@@ -8,18 +8,19 @@
id : {l : Level} {A : Set l} -> A -> A
id x = x

-_∙_ : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} -> (B -> C) -> (A -> B) -> (A -> C)
+_∙_ : {l : Level} {A B C : Set l} -> (B -> C) -> (A -> B) -> (A -> C)
f ∙ g = \x -> f (g x)

+record Functor {l : Level} {A : Set l} (F : {l' : Level} -> Set l' -> Set l') : Set (suc l) where
+  field
+    fmap : {A B : Set l} -> (A -> B) -> (F A) -> (F B)
+  field
+    preserve-id : {A : Set l} (x : F A) → fmap id x ≡ id x
+    covariant   : {A B C : Set l} (f : A -> B) -> (g : B -> C) -> (x : F A)
+                    -> fmap (g ∙ f) x ≡ ((fmap g) ∙ (fmap f)) x
+open Functor

-record Functor {l : Level} (F : Set l -> Set l) : (Set (suc l)) where
-  field
-    fmap : ∀{A B} -> (A -> B) -> (F A) -> (F B)
-  field
-    preserve-id : ∀{A} (Fa : F A) → fmap id Fa ≡ id Fa
-    covariant : ∀{A B C} (f : A → B) → (g : B → C) → (x : F A)
-      → fmap (g ∙ f) x ≡ fmap g (fmap f x)

data List {l : Level} (A : Set l) : (Set l) where
nil  : List A
@@ -33,20 +34,20 @@
list-preserve-id nil = refl
list-preserve-id (cons x xs) = cong (\li -> cons x li) (list-preserve-id xs)

-list-covariant : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} ->
+list-covariant : {l : Level} {A B C : Set l} ->
(f : A -> B) → (g : B -> C) → (x : List A) → list-fmap (g ∙ f) x ≡ list-fmap g (list-fmap f x)
list-covariant f g nil         = refl
list-covariant f g (cons x xs) = cong (\li -> cons (g (f x)) li) (list-covariant f g xs)

-list-is-functor : {l : Level} -> Functor (List {l})
+list-is-functor : {l : Level} {A : Set l}-> Functor {l} {A} List
list-is-functor = record { fmap        = list-fmap ;
preserve-id = list-preserve-id ;
covariant   = list-covariant}

-fmap-to-nest-list : {l ll : Level} {A : Set l} {B : Set l} {fl : Functor List}
+fmap-to-nest-list : {l : Level} {A B : Set l}
-> (A -> B) -> (List (List A)) -> (List (List B))
-fmap-to-nest-list {_} {_} {_} {_} {fl} f xs = Functor.fmap fl (Functor.fmap fl f)  xs
+fmap-to-nest-list {l} {A} {B} f xs = Functor.fmap (list-is-functor {l} {List A}) (Functor.fmap {l} {A} list-is-functor f)  xs

data Identity {l : Level} (A : Set l) : Set l where
identity : A -> Identity A
@@ -57,11 +58,11 @@
identity-preserve-id : {l : Level} {A : Set l} -> (x : Identity A) -> identity-fmap id x ≡ id x
identity-preserve-id (identity x) = refl

-identity-covariant : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} ->
+identity-covariant : {l : Level} {A B C : Set l}
(f : A -> B) → (g : B -> C) → (x : Identity A) → identity-fmap (g ∙ f) x ≡ identity-fmap g (identity-fmap f x)
identity-covariant f g (identity x) = refl

-identity-is-functor : {l : Level} -> Functor (Identity {l})
+identity-is-functor : {l : Level} {A : Set l} -> Functor {l} {A} Identity
identity-is-functor {l} = record { fmap        = identity-fmap {l};
preserve-id = identity-preserve-id ;
covariant   = identity-covariant }
@@ -69,24 +70,61 @@

-record NaturalTransformation {l ll : Level} (F G : Set l -> Set l)
-                                            (functorF : Functor F)
-                                            (functorG : Functor G) : Set (suc (l ⊔ ll)) where
+record NaturalTransformation {l : Level} (F G : {l' : Level} -> Set l' -> Set l')
+                                         {fmapF : {A B : Set l} -> (A -> B) -> (F A) -> (F B)}
+                                         {fmapG : {A B : Set l} -> (A -> B) -> (G A) -> (G B)}
+                                         (natural-transformation : {A : Set l}  -> F A -> G A)
+                             : Set (suc l) where
field
-    natural-transformation : {A : Set l}  -> F A -> G A
-  field
-    commute : ∀ {A B} -> (function : A -> B) -> (x : F A) ->
-              natural-transformation (Functor.fmap functorF function x) ≡  Functor.fmap functorG function (natural-transformation x)
+    commute : {A B : Set l} -> (f : A -> B) -> (x : F A) ->
+              natural-transformation (fmapF f x) ≡  fmapG f (natural-transformation x)
+open NaturalTransformation

tail : {l : Level} {A : Set l} -> List A -> List A
tail nil         = nil
tail (cons _ xs) = xs

-tail-commute : {l ll : Level} {A : Set l} {B : Set ll} -> (f : A -> B) -> (xs : List A) ->
+tail-commute : {l : Level} {A B : Set l} -> (f : A -> B) -> (xs : List A) ->
tail (list-fmap f xs) ≡ list-fmap f (tail xs)
tail-commute f nil = refl
tail-commute f (cons x xs) = refl

-tail-is-natural-transformation : {l ll : Level} -> NaturalTransformation  {l} {ll} List List list-is-functor list-is-functor
-tail-is-natural-transformation = record { natural-transformation = tail;
-                                          commute                = tail-commute}
+
+tail-is-natural-transformation : {l : Level} -> NaturalTransformation List List {list-fmap} {list-fmap {l}} tail
+tail-is-natural-transformation = record { commute                = tail-commute}
+
+
+append : {l : Level} {A : Set l} -> List A -> List A -> List A
+append nil ys = ys
+append (cons x xs) ys = cons x (append xs ys)
+
+append-with-fmap : {l : Level} {A B : Set l} -> (f : A -> B) -> (xs : List A) -> (ys : List A) ->
+  append (list-fmap f xs) (list-fmap f ys) ≡  list-fmap f (append xs ys)
+append-with-fmap f nil ys         = refl
+append-with-fmap f (cons x xs) ys = begin
+  append (list-fmap f (cons x xs)) (list-fmap f ys)     ≡⟨ refl ⟩
+  append (cons (f x) (list-fmap f xs)) (list-fmap f ys) ≡⟨ refl ⟩
+  cons (f x) (append (list-fmap f xs) (list-fmap f ys)) ≡⟨ cong (\li -> cons (f x) li) (append-with-fmap f xs ys) ⟩
+  list-fmap f (append (cons x xs) ys)                   ∎
+
+
+
+concat : {l : Level} {A : Set l} -> List (List A) -> List A
+concat nil         = nil
+concat (cons x xs) = append x (concat xs)
+
+concat-commute :  {l : Level} {A B : Set l} -> (f : A -> B) -> (xs : List (List A)) ->
+               concat ((list-fmap ∙ list-fmap) f xs) ≡ list-fmap  f (concat xs)
+concat-commute f nil         = refl
+concat-commute f (cons x xs) = begin
+  concat ((list-fmap ∙ list-fmap) f (cons x xs))                 ≡⟨ refl ⟩
+  concat (cons (list-fmap f x) ((list-fmap ∙ list-fmap) f xs))   ≡⟨ refl ⟩
+  append (list-fmap f x) (concat ((list-fmap ∙ list-fmap) f xs)) ≡⟨ cong (\li -> append (list-fmap f x) li) (concat-commute f xs) ⟩
+  append (list-fmap f x) (list-fmap f (concat xs))               ≡⟨ append-with-fmap f x (concat xs) ⟩
+  list-fmap f (append x (concat xs)) ≡⟨ refl ⟩
+  list-fmap f (concat (cons x xs))
+  ∎
+
+concat-is-natural-transformation : {l : Level} -> NaturalTransformation (\A -> List (List A)) List
+                                   {list-fmap ∙ list-fmap} {list-fmap {l}} concat
+concat-is-natural-transformation = record {commute = concat-commute}
\ No newline at end of file```