# HG changeset patch
# User Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
# Date 1429496002 -32400
# Node ID 7b22ac1b6a2cd7d5cf9a6f831ae00e716da96b2a
# Parent  75e49a25e9616e00d2c5d8e688a95c85fae347d8
Add description to example

diff -r 75e49a25e961 -r 7b22ac1b6a2c cfopm.pdf
Binary file cfopm.pdf has changed
diff -r 75e49a25e961 -r 7b22ac1b6a2c cfopm.tex
--- a/cfopm.tex	Mon Apr 20 10:31:10 2015 +0900
+++ b/cfopm.tex	Mon Apr 20 11:13:22 2015 +0900
@@ -161,21 +161,21 @@
 \begin{align*}
     f :: A \rightarrow B \\
     g :: B \rightarrow C \\
-    g \circ f :: A \rightarrow C \\
+    g . f :: A \rightarrow C \\
     \\
     h :: C \rightarrow D \\
     (h . g) . f = h . (g . f) :: A \rightarrow D
 \end{align*}
 
-Sum type is introduced using Haskel syntax(Table\ref{src:delta_data_definition}).
+Sum type is introduced using Haskel syntax(Table \ref{src:delta_data_definition}).
 
 \begin{table}[html]
 \begin{lstlisting}
 data Delta a = Mono a
              | Delta a (Delta a)
 \end{lstlisting}
+\caption{Definition of data type ``Delta`` in Haskell}
 \label{src:delta_data_definition}
-\caption{Definition of data type ``Delta`` in Haskell}
 \end{table}
 
 This is data type definition of sum type delta which has type variable {\tt a}.
@@ -188,12 +188,12 @@
 deltaFmap f (Mono x)    = Mono (f x)
 deltaFmap f (Delta x d) = Delta (f x) (deltaFmap f d)
 \end{lstlisting}
+\caption{Arrow mapping for data type ``Delta``}
 \label{src:delta_fmap}
-\caption{Arrow mapping for data type ``Delta``}
 \end{table}
 
 Arrow mapping in a functor satisfies identity law and distribution law.
-Data type can be accessed by pattern matching(Table\ref{src:head_delta}).
+Data type can be accessed by pattern matching(Table \ref{src:head_delta}).
 
 \begin{table}[html]
 \begin{lstlisting}
@@ -201,8 +201,8 @@
 headDelta (Mono  x)   = x
 headDelta (Delta x d) = x
 \end{lstlisting}
+\caption{Define function to Delta using pattern matching}
 \label{src:head_delta}
-\caption{Define function to Delta using pattern matching}
 \end{table}
 
 This is a natural transformation from functor Delta to identity functor.
@@ -211,8 +211,8 @@
 
 Monad in category A is $ triple (T, \eta, \mu) $.
 T is a functor from A to A.
-eta is a natural transformation from identity functor to T.
-mu is a natural transformation from TT to T.
+$ \eta $ is a natural transformation from identity functor to T.
+$ \mu $ is a natural transformation from TT to T.
 TT is a nested data structure of T.
 
 Monad also satisfies to laws below:
@@ -224,14 +224,14 @@
 
 Various meta computations represents by definition of triple.
 For each function $ f :: A \rightarrow B $ , there is a meta computation $ f^* :: A \rightarrow T B $.
-Combination of $ f^* $ $ g^*$ is defined as follows:
+Combination of $ f^* $ $ g^* $ $ h^* $ is defined as follows:
 
 \begin{align*}
-    f^* . g^* = \mu (deltaFmap f^* (\mu (deltaFmap g^*)))
+    (h^* . g^*) . f^* = h^* . (g^* . f^*)
 \end{align*}
 
 Association law of $ f^* $ is derived from Monad laws.
-In this way, for each Monad there is a new category of $ f^* $which is a well known Kleisli Category.
+In this way, for each Monad there is a new category of $ f^* $ which is a well known Kleisli Category.
 
 Normal function f has a meta function $ f^* $ which returns Monad T.
 
@@ -312,8 +312,8 @@
   (Delta x d) >>= f  = Delta (headDelta (f x))
                              (d >>= (tailDelta . f))
 \end{lstlisting}
+    \caption{Definition of Delta Monad in Haskell}
     \label{src:delta_monad}
-    \caption{Definition of Delta Monad in Haskell}
 \end{table}
 
 Modifications of values are stored as a list like structure.
@@ -325,6 +325,12 @@
 
 % FIXME: please check added an example
 \section{Example}
+We show an example using Delta Monad(Table \ref{src:example_of_delta}).
+A program contains only a function which calculate integer.
+In first version, function f is adding 2 to argument.
+In second version, modified function f multiplying 3 to argument.
+We can define {\tt f'} as a meta function contains both version.
+Meta function {\tt f'} outputs two results for value 100.
 
 \begin{table}[html]
 \begin{lstlisting}
@@ -344,8 +350,8 @@
 -- *Main> Mono 100 >>= f'
 -- Delta 102 (Mono 300)
 \end{lstlisting}
-\label{src:example_delta}
 \caption{An Example using Delta}
+\label{src:example_of_delta}
 \end{table}
 
 \section{Conclusion and Future Works}