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\begin{document}


\title[] % (optional, use only with long paper titles)
{Structures algébriques / Classes en Haskell}

\subtitle{INFO1 - Semaine 48}

\author[] % (optional, use only with lots of authors)
{Guillaume CONNAN }
% - Give the names in the same order as the appear in the paper.
% - Use the inst{?} command only if the authors have different
%   affiliation.

\institute{\textsc{IUT} de Nantes - Dpt d'informatique }% (optional, but mostly needed)

\logo{\includegraphics[scale=0.15]{logo_iut}}

%\logo{\includegraphics[scale=0.15]{big_connan}}

\date[] % (optional, should be abbreviation of conference name)
{Dernière mise à jour: \today{} à \now}
% - Either use conference name or its abbreviation.
% - Not really informative to the audience, more for people (including
%   yourself) who are reading the slides online

\subject{ }


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 \frametitle{Sommaire}
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\begin{multicols}{2} 
 \tableofcontents
\end{multicols}
}

 
 \end{frame}


\haskell


\section{Calcul modulaire sans arithmétique...}



\begin{frame}[fragile]
  \begin{lstlisting}
Prelude> data Romain = O | I | II | III | IV | V deriving (Enum, Bounded, Eq, Show) 
  \end{lstlisting}

\pause

  \begin{lstlisting}
Prelude> minBound :: Romain
O
  \end{lstlisting}


\pause


\begin{lstlisting}
Prelude> maxBound :: Romain
V
\end{lstlisting}


\pause


\begin{lstlisting}
Prelude> minBound :: Int
-9223372036854775808
Prelude> maxBound :: Int
9223372036854775807
\end{lstlisting}

\end{frame}



\begin{frame}[fragile]
  \begin{lstlisting}
Prelude> succ II
III
  \end{lstlisting}

\pause

\begin{lstlisting}
Prelude> pred V
IV
\end{lstlisting}


\pause


\begin{lstlisting}
Prelude> succ V
*** Exception: succ{Romain}: tried to take `succ' of last tag in enumeration
\end{lstlisting}



\pause

\begin{lstlisting}
Prelude> [II .. V]
[II,III,IV,V]
\end{lstlisting}
\end{frame}



\begin{frame}[fragile]


  \begin{lstlisting}
Prelude> :t fromEnum
fromEnum :: Enum a => a -> Int
  \end{lstlisting}

\pause

\begin{lstlisting}
Prelude> fromEnum II
2
\end{lstlisting}


\pause

\begin{lstlisting}
Prelude> toEnum 5 :: Romain
V
\end{lstlisting}

\end{frame}


\begin{frame}[fragile]
  \begin{lstlisting}
data ClasseMod t = (Eq t, Show t, Enum t, Bounded t) => Zn t 
  \end{lstlisting}


\pause


\begin{lstlisting}
data LesZ5 = O5 | I5 | II5 | III5 | IV5 deriving (Eq,Show, Bounded, Enum)

type Z5 = ClasseMod LesZ5
\end{lstlisting}

\pause

\begin{lstlisting}
instance (Show t) => Show (ClasseMod t) where
  show (Zn x) = show x
\end{lstlisting}


\end{frame}





\begin{frame}[fragile]

  \begin{lstlisting}
*CalMod> Zn II5
II5
  \end{lstlisting}


\end{frame}



\begin{frame}[fragile]

  \begin{lstlisting}
succZn :: (ClasseMod t) -> (ClasseMod t) 
succZn (Zn x)        = if (x == maxBound) then (Zn minBound) else (Zn (succ x))
  \end{lstlisting}


\pause

\begin{lstlisting}
*CalMod> succZn (Zn III5)
IV5
\end{lstlisting}


\pause

\begin{lstlisting}
*CalMod> succZn (Zn IV5)
O5
\end{lstlisting}


\pause 


\verb+predZn+ ??

\end{frame}






\begin{frame}[fragile]

  \begin{lstlisting}
instance (Eq t, Show t, Enum t, Bounded t) => Enum (ClasseMod t) where
  succ = succZn
  \end{lstlisting}


\pause


\begin{lstlisting}
*CalMod> succ (Zn III5)
IV5
\end{lstlisting}

\end{frame}








\begin{frame}[fragile]

  \begin{lstlisting}
instance (Eq t, Show t, Enum t, Bounded t) => Enum (ClasseMod t) where
  succ = succZn
  pred = predZn
  \end{lstlisting}


\pause



\begin{lstlisting}
*CalMod> pred (Zn III5)
II5
\end{lstlisting}


\end{frame}







\begin{frame}[fragile]

  \begin{lstlisting}
instance (Eq t, Show t, Enum t, Bounded t) => Enum (ClasseMod t) where
  succ = succZn
  pred = predZn
  fromEnum (Zn x) = fromEnum x
  \end{lstlisting}




\pause


\begin{lstlisting}
*CalMod> fromEnum IV5
4
\end{lstlisting}


\end{frame}






\begin{frame}[fragile]

  \begin{lstlisting}
instance (Eq t, Show t, Enum t, Bounded t) => Enum (ClasseMod t) where
  succ = succZn
  pred = predZn
  fromEnum (Zn x) = fromEnum x
  toEnum n = Zn (toEnum n)
  \end{lstlisting}


\pause

\begin{lstlisting}
*CalMod> toEnum 4 :: Z5
IV5
\end{lstlisting}


\end{frame}













\begin{frame}[fragile]

  \begin{lstlisting}
instance (Eq t) => Eq (ClasseMod t) where
  (Zn x) == (Zn y) = x == y
  \end{lstlisting}

\pause


\begin{lstlisting}
instance (Eq t, Show t, Enum t, Bounded t) => Bounded (ClasseMod t) where
  maxBound = Zn (maxBound)
  minBound = Zn (minBound)
\end{lstlisting}

\end{frame}


\begin{frame}[fragile]

  \begin{lstlisting}
plusZn  x y = 
  if (fromEnum x == 0) then y
  else
    succ (plusZn (pred x) y)
  \end{lstlisting}


\pause


Multiplication ??

\end{frame}



\begin{frame}[fragile]

  \begin{lstlisting}
foisZn x y =
  if (fromEnum x == 0) then (toEnum 0)
  else
    if (fromEnum x == 1) then y
    else plusZn y (foisZn (pred x) y) 
  \end{lstlisting}
  
\end{frame}





\begin{frame}[fragile]

  \begin{lstlisting}
instance (Eq t, Show t, Enum t, Bounded t) => Num (ClasseMod t) where
  \end{lstlisting}

\end{frame}








\begin{frame}[fragile]

  \begin{lstlisting}
instance (Eq t, Show t, Enum t, Bounded t) => Num (ClasseMod t) where
  (+) = plusZn
  \end{lstlisting}

\end{frame}













\begin{frame}[fragile]

  \begin{lstlisting}
instance (Eq t, Show t, Enum t, Bounded t) => Num (ClasseMod t) where
  (+) = plusZn
  (*) = foisZn
  \end{lstlisting}

\end{frame}





















\begin{frame}[fragile]

  \begin{lstlisting}
instance (Eq t, Show t, Enum t, Bounded t) => Num (ClasseMod t) where
  (+) = plusZn
  (*) = foisZn
  fromInteger  = classeOfInt 
  \end{lstlisting}

\end{frame}












\begin{frame}[fragile]

  \begin{lstlisting}
instance (Eq t, Show t, Enum t, Bounded t) => Num (ClasseMod t) where
  (+) = plusZn
  (*) = foisZn
  fromInteger  = classeOfInt 
  negate x =  toEnum (- (fromEnum x) + (fromEnum (maxBound :: ClasseMod t)) + 1)
  \end{lstlisting}

\end{frame}




\begin{frame}[fragile]

  \begin{lstlisting}
*CalMod> 2 :: Z5
II5
  \end{lstlisting}

\end{frame}




\begin{frame}[fragile]

  \begin{lstlisting}
*CalMod> 2 :: Z5
II5
*CalMod> 17 :: Z5
II5
  \end{lstlisting}

\end{frame}





\begin{frame}[fragile]

  \begin{lstlisting}
*CalMod> 2 :: Z5
II5
*CalMod> 17 :: Z5
II5
*CalMod> (-3) :: Z5
II5
  \end{lstlisting}

\end{frame}






\begin{frame}[fragile]

  \begin{lstlisting}
*CalMod> 2 :: Z5
II5
*CalMod> 17 :: Z5
II5
*CalMod> (-3) :: Z5
II5
*CalMod> (-3) + 9 :: Z5
I5
  \end{lstlisting}

\end{frame}




























\begin{frame}[fragile]

  \begin{lstlisting}
*CalMod> 2 :: Z5
II5
*CalMod> 17 :: Z5
II5
*CalMod> (-3) :: Z5
II5
*CalMod> (-3) + 9 :: Z5
I5
*CalMod> 3 * 7 :: Z5
I5
  \end{lstlisting}

\end{frame}





\begin{frame}[fragile]

  \begin{lstlisting}
*CalMod> 2 :: Z5
II5
*CalMod> 17 :: Z5
II5
*CalMod> (-3) :: Z5
II5
*CalMod> (-3) + 9 :: Z5
I5
*CalMod> 3 * 7 :: Z5
I5
*CalMod> 301 * 7157 :: Z5
II5
  \end{lstlisting}

\end{frame}



\section{Rationnels}


\end{document}
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