\section{Arith}\label{Arith}

The {\tt Arith} library deals with various arithmetical notions and 
their properties.

\subsection*{Standard {\tt Arith} library}

The following files are automatically loaded by {\tt Require Arith}.

\begin{itemize}

\item {\tt Le.v} states and proves properties of the large order {\tt le}.

\item {\tt Lt.v} states and proves properties of the strict order {\tt 
lt} (especially, the relationship with {\tt le}).

\item {\tt Plus.v} states and proves properties on the addition.

\item {\tt Gt.v} states and proves properties on the strict order {\tt gt}.

\item {\tt Minus.v} defines the difference on 
{\tt nat} and proves properties of it. On {\tt nat}, {\tt (minus n p)} is 
{\tt O} if {\tt n} $<$ {\tt p}.

\item {\tt Mult.v} states and proves properties on the multiplication.

\item {\tt Between.v} defines modalities on {\tt nat} and proves properties 
of them.

\end{itemize}

\subsection*{Additional {\tt Arith} library}

\begin{itemize}

\item {\tt Compare.v}, {\tt Compare\_dec.v} and {\tt Peano\_dec.v} state
and prove various decidability results on {\tt nat}.

\item {\tt Wf\_nat.v} states and proves various induction and recursion 
principles on {\tt nat}. Especially, recursion for objects measurable by 
a natural number and recursion on {\tt nat * nat} are provided.

\item {\tt Min.v} defines the minimum of two natural numbers and proves 
properties of it. 

\item {\tt Eqnat.v} defines a specific equality on {\tt nat} and shows 
the equivalence with Leibniz' equality.

\item {\tt Euclid.v} proves that the euclidean
division specification is realisable. Conversely, {\tt Div.v} exhibits
two different algorithms and semi-automatically reconstruct the proof of
their correctness. These files emphasize the extraction of program vs
reconstruction of proofs paradigm.

\end{itemize}