wording, proof, algo

1 files changed, 5 insertions, 5 deletions

diff --git a/appendix.tex b/appendix.tex
index 682fce2..bf163b4 100644
--- a/appendix.tex
+++ b/appendix.tex
@@ -476,12 +476,12 @@ Using Lemma~\ref{lemma:barrier} concludes the proof of the running time.\qed
 
 \section{Proof of Theorem~\ref{thm:main}}\label{sec:proofofmainthm}
 
-We now present the proof of Theorem~\ref{thm:main}. $\delta$-truthfulness and
-individual rationality follow from $\delta$-monotonicity and threshold
-payments. $\delta$-monotonicity and budget feasibility follow the same steps as the
-analysis of \citeN{chen}; for the sake of completeness, we restate their proof
-here.
+We now present the proof of Theorem~\ref{thm:main}. We use the notation $OPT_{-i^*}$ to denote the optimal value of \EDP{} when the maximum value element $i^*$ is  excluded. We also use $OPT'_{-i^*}$ the approximation computed by the $\delta$-decreasing, $\epsilon$-accurate approximation of $L^*_{c_{-i^*}}$, as defined in Algorithm~\ref{mechanism}.
 
+The properties of $\delta$-truthfulness and
+individual rationality follow from $\delta$-monotonicity and threshold
+payments. $\delta$-monotonicity and budget feasibility follow  similar steps as the
+analysis of \citeN{chen}, adapted to account for $\delta$-monotonicity:
 \begin{lemma}\label{lemma:monotone}
 Our mechanism for \EDP{} is $\delta$-monotone and budget feasible.
 \end{lemma}