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\begin{background}[lattice]
    A \textbf{lattice} describes a real coordinate space $\mathbb{R}^n$ that satisfies: 
    \begin{itemize}
        \item {
            Addition / subtraction between two points always produce another point in lattice -- i.e., closed under addition / subtraction.
        }
        \item {
            Lattice points are separated by bounded distances in some range $(0, \mathrm{max}]$.
        }
    \end{itemize}
\end{background}

Define a lattice over which \textit{semantic equivalence relations} for spec. and impl. verification is defined.

\begin{definition}[discrimination measure]
    One equivalence relation $\equiv$ is \textbf{finer} / \textbf{more discriminating} than another $\sim$ if each $\equiv$-eq. class is a subset of a $\sim$-eq. class. In other words,
    \begin{align*}
            \ &p \equiv q \implies p \sim q \\
        \iff\ &\equiv\ \mathrm{finer\ than}\ \sim
    \end{align*}
\end{definition}

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