\subsection{Far-reaching dependencies do not impact
performance}\label{ssec:staticdeps:rob_proof}

\begin{definition}[Distance between instructions]
    Let $\left(I_p\right)_{0\leq p<n}$ be a trace of executed instructions.
    For $p<p'$, $\distance{I_p}{I_{p'}}$ is the overall number of decoded \uops{}
    for the subtrace $\left(I_r\right)_{p < r < p'}$.
\end{definition}

\begin{theorem}[Long distance dependencies]\label{thm.longdist}
    There exists $R \in \nat$, only dependent of microarchitectural parameters,
    such that the presence or absence of a dependency between two instructions
    that are separated by at least $R-1$ other \uops{} has no impact on the
    performance of this kernel.

    More formally, let $\kerK$ be a kernel of $n$ instructions. Let $(I_p)_{p
    \in \nat}$ be the trace of $\kerK$'s instructions executed in a loop. For
    any $p, q \in \nat$ such that $\distance{I_p}{I_q} \geq R-1$, $\cyc{\kerK}$ is
    invariant in the presence or absence of a dependency between the pairs of
    instructions $\left(I_{p+kn}, I_{q+kn}\right)_{k\in\nat}$.
\end{theorem}

To prove this assertion, we require a few postulates describing the functioning
of a CPU and, in particular, how \uops{} transit in (decoded) and out (retired)
the reorder buffer.

\begin{postulate}[Reorder buffer as a circular buffer]
    The reorder buffer is a circular buffer of size $R \in \nat^+$.
    It contains only decoded \uops{}.
    Let us denote $i_d$ the \uop{} at position $d$ in the reorder buffer.
    Assume $i_d$ just got decoded.
    We have that for every $q$ and $q'$ in $[0,R)$:
    \[
        (q-d-1) \% R<(q'-d-1) \% R\
            \iff \ i_q \text{ was decoded before } i_{q'}
    \]
\end{postulate}


If a \uop{} has not been retired yet (issued and executed), it cannot be
replaced in the ROB by any freshly decoded instruction. In other words, every
non-retired decoded \uop{} --~also called \emph{in-flight}~-- remains in the
reorder buffer. This is possible thanks to the notion of \emph{full reorder
buffer}:

\begin{postulate}[Full reorder buffer]
    Let us denote by $i_d$ the \uop{} that just got decoded.
    The reorder buffer is said to be full if for $q=(d+1) \% R$, \uop{} $i_q$ is not retired yet.

    If the reorder buffer is full, then instruction decoding is stalled.
\end{postulate}

Let $(I_p)_{0\le{} p<n}$ be a trace of executed instructions.
Each of these instructions are iteratively decoded, issued, and retired.
We will also denote by $(i_q)_{0\le q<m}$ the trace of decoded \uops{}.
To prove the theorem above, we need to state that any two in-flight \uops{} are distant of at most $R$ \uops{}.

For any instruction $I_p$, we denote as $Q_p$ the range of indices such that
$(i_q)_{q\in Q_p}$ are the \uops{} obtained from the decoding of $I_p$.

Note that in practice, we may not have $\bigcup{}_p Q_p = [0, n)$, as \eg{}
branch mispredictions may introduce unwanted \uops{} in the pipeline. However,
as the worst case for the lemma below occurs when no such ``spurious'' \uops{}
are present, we may safely ignore such occurrences.

\begin{lemma}[Distance of in-flight \uops{}]
    For any pair of instructions $(I_p,I_{p'})$, and two corresponding \uops{},
    $(i_q,i_{q'})$ such that $q \in Q_p, q' \in Q_{p'}$,
    \[
        \operatorname{inflight}(i_q) \wedge \operatorname{inflight}(i_{q'})
        \implies \distance{I_p}{I_{p'}} < R - 1
    \]
\end{lemma}

\begin{proof}
    The sets $(Q_p)$ are disjoint pairwise, and for any pair of instructions
    $(I_p,I_{p'})$, and any two corresponding \uops{},
    $(i_q,i_{q'})$ such that $q \in Q_p$, $q' \in Q_{p'}$,
    $p < p' \implies q < q'$.

    Thus, $\distance{I_p}{I_{p'}} < |q'-q|$.

Observe that at any time, the content of the ROB can be seen as a window of
length at most $R$ over $(i_q)_{0\le q<m}$.
Consequently, if both $i_q$  and $i_{q'}$ are in-flight then
$|q'-q|<R$, and thus $\distance{I_p}{I_{p'}} < R - 1$.
\end{proof}

\begin{postulate}[Issue delay]
    Reasons why the issue of a \uop{} $i$ is delayed can be:
    \begin{enumerate}
        \item $i$ is not yet in the reorder buffer
        \item $i$ depends on \uop{} $i'$ which is not retired yet
        \item ports on which $i$ can be mapped are all occupied
    \end{enumerate}
\end{postulate}

\begin{proof}[Proof of Long distance dependencies theorem]
The theorem above is now a direct consequence
of the previous observations. Let us consider two \uops{}, $i$ and $i'$,
respectively introduced by instructions $I_p$ and $I_q$. Assume a delayed issue
for \uop{} $i$ where the unique cause is a dependence from \uop{} $i'$, that
is:
\begin{enumerate}
    \item $i$ is already in the reorder buffer
    \item $i$ depends on \uop{} $i'$ which is not retired yet
    \item at least one port on which $i$ can be mapped is available
\end{enumerate}
Since $i'$ is not retired yet and $i'$ is ``before'' $i$, $i'$ is still in the
reorder buffer, \ie{} both $i$ and $i'$ are in the reorder buffer.

By the previous lemma, we have $\distance{I_p}{I_q} < R - 1$.

By contrapositive, for any two instructions $I_a, I_b$ such that
$\distance{I_a}{I_b} \geq R-1$, no \uop{} of $I_b$ may have its execution
delayed by a dependency between $I_a$ and $I_b$.
\end{proof}

\begin{remark}
    What we stated earlier is a direct consequence of this theorem: to detect
    meaningful dependencies over a kernel $\kerK$, it suffices to analyze the
    kernel unrolled enough times to obtain a sequence of $R + \card{\kerK}$
    instructions, as this yields a sequence where every instruction from the
    last occurrence of $\kerK$ is preceded by at least $R - 1$ instructions.
\end{remark}