\documentclass{article}
\input{../../../../../lib.sty}

\title{\texttt{fn floor\_div}}
\author{Michael Shoemate}
\date{}

\begin{document}

\maketitle

\contrib
Proves soundness of the implementation of \rustdoc{measurements/noise/distribution/float/utilities/fn}{floor\_div} in \asOfCommit{mod.rs}{f5bb719}.

\section{Hoare Triple}
\subsection*{Precondition}
\subsubsection*{Compiler-Verified}
None

\subsubsection*{User-Verified}
None

\subsection*{Pseudocode}
In the following pseudocode, define $/$ as zero integer division, as used by the Rust Dashu dependency's IBig.
Zero integer division is real division that is then rounded towards zero.
This differs from Python's division, which converts to float,
and Python's integer division, which is the desired floor integer division.

\lstinputlisting[language=Python,firstline=2,escapechar=`]{./pseudocode/floor\_div.py}

\subsection*{Postcondition}
\begin{theorem}
    Return $floor(a / b)$, where $/$ denotes real division.
\end{theorem}

\begin{proof}
    Consider the following cases:
    \begin{itemize}
        \item If $a \geq 0$ then zero integer division and floor integer division are equivalent.
            Therefore the result is $floor(a / b)$.
        \item If $a < 0$ then $b - 1$ is subtracted before zero integer division.
            This shifts the problematic upper elements that would round up into the range that rounds down,
            while the lower elements that would round down now round up, resulting in the correct floor division.
            Therefore the result is $floor(a / b)$.
    \end{itemize}
    In all cases, the result is $floor(a / b)$.
\end{proof}
\end{document}
