updated cata

categories/30_How/300_Catamorphisms/030.md

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+κατα-morphism: fold generalization
+----------------------------------
+
+Typically, you need a recursive type (list, trees, ...)
+
+Str = Cons Char Str
+
+1st: replace the recursive type by another type.
+StrChar a = Cons Char a
+Str' a = Mu StrChar
+
+Str' = InF { outF :: StrChar (Mu ListElement) }
+Str' = InF { outF :: StrChar (Str') }
+
+Clearly isomorph to List.
+
+~~~
+type Algebra f a = f a -> a
+data Mu f = InF { outF :: f (Mu f) }
+
+cata :: Functor f => Algebra f a -> Mu f -> a
+cata f = f . fmap (cata f) . outF
+
+-- Example with natural fold on String
+data StrF x = Cons Char x | Nil
+type Str = Mu StrF
+instance Functor StrF where
+  fmap f (Cons a as) = Cons a (f as)
+  fmap _ Nil = Nil
+
+length' :: Str -> Int
+length' = cata phi where
+  phi (Cons a b) = 1 + b
+  phi Nil = 0
+
+toMu :: [Char] -> Str
+toMu (x:xs) = InF { outF = Cons x (toMu xs) }
+toMu [] = InF { outF = Nil }
+~~~

categories/30_How/300_Catamorphisms/040.md

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+κατα-morphism: example with strings
+-----------------------------------
+
+~~~
+type Algebra f a = f a -> a
+data Mu f = InF { outF :: f (Mu f) }
+
+cata :: Functor f => Algebra f a -> Mu f -> a
+cata f = f . fmap (cata f) . outF
+
+-- Example with natural fold on String
+data StrF x = Cons Char x | Nil
+type Str = Mu StrF
+instance Functor StrF where
+  fmap f (Cons a as) = Cons a (f as)
+  fmap _ Nil = Nil
+
+length' :: Str -> Int
+length' = cata phi where
+  phi (Cons a b) = 1 + b
+  phi Nil = 0
+
+toMu :: [Char] -> Str
+toMu (x:xs) = InF { outF = Cons x (toMu xs) }
+toMu [] = InF { outF = Nil }
+~~~

categories/30_How/300_Catamorphisms/050.md

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+κατα-morphism: example with strings
+-----------------------------------
+
+~~~
+type Algebra f a = f a -> a
+data Mu f = InF { outF :: f (Mu f) }
+
+cata :: Functor f => Algebra f a -> Mu f -> a
+cata f = f . fmap (cata f) . outF
+
+type Tree = Mu TreeF
+data TreeF x = Node Int [x]
+
+instance Functor TreeF where
+  fmap f (Node e xs) = Node e (fmap f xs)
+
+depth = cata phi where
+  phi (Node x sons) = 1 + foldr max 0 sons
+~~~