\section{The Theory {\tt ineq}}

The theory \ml{ineq}\index{ineq, theory@{\ptt ineq}, theory} contains an \adhoc\ collection of
theorems about the inequality operators. It contains no new definitions.
Below is a selection of the theorems that have been pre-proved in this theory.
\begin{hol}
\index{EQ\_LESS\_EQ@{\ptt EQ\_LESS\_EQ}}
\index{NOT\_EQ\_LESS\_EQ@{\ptt NOT\_EQ\_LESS\_EQ}}
\index{LESS\_EQ\_LESS\_EQ\_EQ@{\ptt LESS\_EQ\_LESS\_EQ\_EQ}}
\begin{verbatim}
  EQ_LESS_EQ          |- !a b. (a = b) = a <= b /\ b <= a

  NOT_EQ_LESS_EQ      |- !a b. ~(a = b) = a < b \/ b < a

  LESS_EQ_LESS_EQ_EQ  |- !m n. m <= n /\ n <= m = (m = n)
\end{verbatim}\end{hol}
