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

\title{\texttt{fn match\_truncations}}
\author{Michael Shoemate}
\begin{document}
\maketitle  


\contrib

Proves soundness of \rustdoc{transformations/make_stable_lazyframe/truncate/matching/fn}{match\_truncations} 
in \asOfCommit{mod.rs}{f5bb719}.

\section{Hoare Triple}
\subsubsection*{Compiler Verified}
Types matching pseudocode.

\subsection*{Precondition}
None 

\subsection*{Function}
\label{sec:python-pseudocode}
\lstinputlisting[language=Python,firstline=2,escapechar=|]{./pseudocode/match_truncations.py}

\subsection*{Postcondition}
\begin{theorem}[Postcondition]
    For any choice of LazyFrame plan,
    returns the plan with the truncations removed,
    the truncations that were removed in the order they are applied,
    and per-id bounds on row and/or group contributions.
\end{theorem}

\begin{proof}

    The algorithm maintains three invariants:
    \begin{itemize}
        \item \texttt{input} is the LazyFrame plan with truncations removed
        \item \texttt{truncations} is a list of truncations in reverse-order
        \item \texttt{bounds} is a list of per-id bounds on row and/or group contributions
    \end{itemize}

    In order to ensure that all per-id bounds remain valid after successive truncations,
    the group-by truncation may only be applied last in the truncation pipeline,
    as the group-by truncation rewrites all columns in the data,
    and potentially overwrites user identifiers.

    Since parsing the query plan happens in reverse order,
    the algorithm starts by attempting to parse a group-by truncation on line \ref{line:match_group_by_truncation}.
    By the postcondition of \rustdoc{transformations/make_stable_lazyframe/truncate/matching/fn}{match\_group\_by\_truncation},
    if the group-by truncation is present, it will be returned as a tuple of
    \texttt{input}, (the execution plan with the group-by truncation removed),
    \texttt{truncation}, (the group-by truncation) and 
    \texttt{per\_id\_bounds} (the per-id bounds on row contributions).
    The state of the algorithm is then updated on lines \ref{line:match_group_by_truncation_start}-\ref{line:match_group_by_truncation_end},
    maintaining the invariants on \texttt{input}, \texttt{truncations} and \texttt{bounds}.

    Another limitation of the group-by truncation is that bounds on row contributions when grouped by a given set of columns
    are no longer valid if those columns are changed in the group-by truncation.
    Therefore, \texttt{allowed\_keys} on line \ref{line:allowed_keys} contains columns that are preserved through the group-by truncation,
    by virtue of being part of the grouping columns.
    This limitation does not hinder expected use-cases, but is necessary to ensure that per-id bounds on contributions remain valid after the group-by truncation.

    If a group-by truncation is not present, no update is made to the state.
    
    The algorithm then attempts to repeatedly parse filter truncations on line \ref{line:match_filter_truncation}.
    By the postcondition of \rustdoc{transformations/make_stable_lazyframe/truncate/matching/fn}{match\_truncation\_predicate} on line \ref{line:match_truncation_predicate},
    the return is a list of per-id contribution bounds if the predicate consists solely of truncations, 
    otherwise none.
    This ensures that the algorithm rejects predicates that contain conditions that are not truncations.

    Line \ref{line:allowed_keys_check} checks that the truncation predicate is valid,
    by ensuring that the truncation predicate is a subset of the allowed keys.
    The algorithm then updates the state on lines \ref{line:match_filter_truncation_start}-\ref{line:match_filter_truncation_end},
    maintaining the invariants on \texttt{input}, \texttt{truncations} and \texttt{bounds}.
    Finally, since the descent through the query plan is in reverse order,
    line \ref{line:truncations_reverse} ensures that the truncation order is correct.
    Neither the check on line \ref{line:match_group_by_truncation_check} 
    nor the reversal on line \ref{line:bounds_reverse} are necessary to match the postcondition,
    but both are included to improve usability.

    Since the invariants on \texttt{input}, \texttt{truncations} and \texttt{bounds} are maintained,
    and the algorithm only matches through truncations,
    the postcondition is satisfied.
\end{proof}

\end{document}
