\section{The Theory {\tt sub}}

The theory \ml{sub}\index{sub, theory@{\ptt sub}, theory} contains an
\adhoc\ collection of 
theorems about subtraction. 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{SUB\_LE\_ADD@{\ptt SUB\_LE\_ADD}}
\index{NOT\_0\_SUB@{\ptt NOT\_0\_SUB}}
\index{ADD\_SUB\_SYM@{\ptt ADD\_SUB\_SYM}}
\index{SUB\_ADD\_SELF@{\ptt SUB\_ADD\_SELF}}
\index{SUB\_BOTH\_SIDES@{\ptt SUB\_BOTH\_SIDES}}
\begin{verbatim}
  SUB_LE_ADD      |- !n m p. n <= p ==> (m <= (p - n) = (n + m) <= p)

  NOT_0_SUB       |- !m n. ~(m - n = 0) ==> ~(m = 0)

  ADD_SUB_SYM     |- !a c. (c + a) - c = a

  SUB_ADD_SELF    |- !m n. ~m < n ==> ((m - n) + n = m)

  SUB_BOTH_SIDES  |- !m n p. (m = n) ==> (m - p = n - p)
\end{verbatim}\end{hol}
