\section{The Theory {\tt pre}}

The theory \ml{pre}\index{pre, theory@{\ptt pre}, theory} contains an
\adhoc\ collection of 
theorems about \ml{PRE}. It contains no new definitions.
The theorems that have been pre-proved in this theory include but are not
restricted to the following:
\begin{hol}
\index{SUC\_PRE@{\ptt SUC\_PRE}}
\index{PRE\_MONO@{\ptt PRE\_MONO}}
\index{PRE\_LESS\_EQ@{\ptt PRE\_LESS\_EQ}}
\index{PRE\_ADD@{\ptt PRE\_ADD}}
\index{PRE\_K\_K@{\ptt PRE\_K\_K}}
\begin{verbatim}
  SUC_PRE      |- !n. 0 < n ==> (SUC(PRE n) = n)

  PRE_MONO     |- !m n. (PRE m) < (PRE n) ==> m < n

  PRE_LESS_EQ  |- !n. (PRE n) <= n

  PRE_ADD      |- !n m. 0 < n ==> (PRE(n + m) = (PRE n) + m)

  PRE_K_K      |- !k. 0 < k ==> (PRE k) < k
\end{verbatim}\end{hol}
