Proof. Let \(s := \mathsf{slot}(t)\), \(s' := \mathsf{slot}(t')\), and \(b'\) any block such that \(b' \preceq b\). Then we can proceed as follows. \[\begin{align} \setkeys{indicatorfromblock}{from=b',to=s'-1,val=v,when=t',chkp={C '}} {{{\mathit{P}}}}_{\indicatorfromblock@From}^{\indicatorfromblock@To,\indicatorfromblock@Val,\indicatorfromblock@When,\indicatorfromblock@ValSetChkp} \setkeys{indicatorfromblock}{from=,to=,val=,when=,chkp=} &= \frac{ \setkeys{attsetweightfromblock}{from=b', to=s'-1,val=v',when=t', chkp={C '}} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b', to=s'-1, chkp={C '}} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} && \ifthenelse{\equal{7cm}{}}{ \text{--- By definition.} }{ \begin{minipage}[t]{7cm} \begin{itemize}[label=---,nosep,left=0pt] \item By definition. \end{itemize} \end{minipage} } \\ &= \frac{ \setkeys{attsetweightfromblock}{from=b', to=s'-1,val=v',when=t', chkp={C}} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b', to=s'-1, chkp={C}} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} && \ifthenelse{\equal{7cm}{}}{ \text{--- As, per Assumption~\ref{assum:no-change-to-the-validator-set}, the effective balance of honest validators never changes.} }{ \begin{minipage}[t]{7cm} \begin{itemize}[label=---,nosep,left=0pt] \item As, per Assumption~\ref{assum:no-change-to-the-validator-set}, the effective balance of honest validators never changes. \end{itemize} \end{minipage} } \\ &= \frac{ \setkeys{attsetweightfromblock}{from=b', to=s'-1, val=v, when=t', chkp={C}} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=} }{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b', to=s'-1, chkp={C}} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} } && \ifthenelse{\equal{7cm}{}}{ \text{--- Given that \mathsf{st}(s'-1)\geq\mathsf{st}(s-1)\geq\mathbb{GST} and t' \geq \mathsf{st}(s'), any honest attestation for slots up to s'-1 received by v' at time t', it is also received by v' by the same time t'.} }{ \begin{minipage}[t]{7cm} \begin{itemize}[label=---,nosep,left=0pt] \item Given that \mathsf{st}(s'-1)\geq\mathsf{st}(s-1)\geq\mathbb{GST} and t' \geq \mathsf{st}(s'), any honest attestation for slots up to s'-1 received by v' at time t', it is also received by v' by the same time t'. \end{itemize} \end{minipage} } \\ &= \frac{ \setkeys{attsetweightfromblock}{from=b', to=s-1, val=v, when=t, chkp={C}} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=} + \setkeys{weightofset}{chkp=C} \left|{ \setkeys{commfromslot}{from=,to=,val=,when=} \setkeys{commfromslot}{from=s, to=s'-1} \overline{\mathcal{J}}_{\commfromslot@From}^{\commfromslot@To \ifthenelse{\equal{\commfromslot@Val}{}}{}{,\commfromslot@Val} \ifthenelse{\equal{\commfromslot@When}{}}{}{,\commfromslot@When} } \setminus \setkeys{attsetfromblockunfiltered}{from=b', to=s-1, val=v, when=t} {\mathcal{H}}_{\attsetfromblockunfiltered@From}^{\attsetfromblockunfiltered@To,\attsetfromblockunfiltered@Val,\attsetfromblockunfiltered@When} \setkeys{attsetfromblockunfiltered}{from=,to=,val=,when=} }\right|^{\weightofset@Chkp} }{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b', to=s'-1, chkp={C}} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} } && \ifthenelse{\equal{7cm}{}}{ \text{---\setkeys{attsetfromblockunfiltered}{from=b', to=s'-1, val=v, when=t'} {\mathcal{H}}_{\attsetfromblockunfiltered@From}^{\attsetfromblockunfiltered@To,\attsetfromblockunfiltered@Val,\attsetfromblockunfiltered@When} \setkeys{attsetfromblockunfiltered}{from=,to=,val=,when=} corresponds to the union of the honest validators whose \textsf{GHOST} votes in support of b and for slots up to s-1 are in the view of validator v at time t with the honest validators in the committees between slot s ans slot s'-1, as we assume that any of these validators has \textsf{GHOST} voted in support of b and \mathsf{st}(\mathsf{slot}(s))\geq \mathbb{GST}.} }{ \begin{minipage}[t]{7cm} \begin{itemize}[label=---,nosep,left=0pt] \item\setkeys{attsetfromblockunfiltered}{from=b', to=s'-1, val=v, when=t'} {\mathcal{H}}_{\attsetfromblockunfiltered@From}^{\attsetfromblockunfiltered@To,\attsetfromblockunfiltered@Val,\attsetfromblockunfiltered@When} \setkeys{attsetfromblockunfiltered}{from=,to=,val=,when=} corresponds to the union of the honest validators whose \textsf{GHOST} votes in support of b and for slots up to s-1 are in the view of validator v at time t with the honest validators in the committees between slot s ans slot s'-1, as we assume that any of these validators has \textsf{GHOST} voted in support of b and \mathsf{st}(\mathsf{slot}(s))\geq \mathbb{GST}. \end{itemize} \end{minipage} } \\ &= \frac{ \setkeys{attsetweightfromblock}{from=b', to=s-1, val=v,when=t, chkp={C}} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=} + \setkeys{weightofset}{chkp=C} \left|{ \setkeys{commfromslot}{from=,to=,val=,when=} \setkeys{commfromslot}{from=s, to=s'-1} \overline{\mathcal{J}}_{\commfromslot@From}^{\commfromslot@To \ifthenelse{\equal{\commfromslot@Val}{}}{}{,\commfromslot@Val} \ifthenelse{\equal{\commfromslot@When}{}}{}{,\commfromslot@When} } \setminus \setkeys{attsetfromblockunfiltered}{from=b', to=s-1, val=v, when=t} {\mathcal{H}}_{\attsetfromblockunfiltered@From}^{\attsetfromblockunfiltered@To,\attsetfromblockunfiltered@Val,\attsetfromblockunfiltered@When} \setkeys{attsetfromblockunfiltered}{from=,to=,val=,when=} }\right|^{\weightofset@Chkp} }{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b', to=s-1, chkp={C}} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} + \setkeys{weightofset}{chkp=C} \left|{ \setkeys{commfromslot}{from=,to=,val=,when=} \setkeys{commfromslot}{from=s, to=s'-1} \overline{\mathcal{J}}_{\commfromslot@From}^{\commfromslot@To \ifthenelse{\equal{\commfromslot@Val}{}}{}{,\commfromslot@Val} \ifthenelse{\equal{\commfromslot@When}{}}{}{,\commfromslot@When} } \setminus \setkeys{commfromblock}{from=,to=,val=,when=} \setkeys{commfromblock}{from=b', to=s-1} {\mathcal{J}}_{\commfromblock@From}^{\commfromblock@To \ifthenelse{\equal{\commfromblock@Val}{}}{}{,\commfromblock@Val} \ifthenelse{\equal{\commfromblock@When}{}}{}{,\commfromblock@When} } }\right|^{\weightofset@Chkp} } && \ifthenelse{\equal{7cm}{}}{ \text{--- By definition} }{ \begin{minipage}[t]{7cm} \begin{itemize}[label=---,nosep,left=0pt] \item By definition \end{itemize} \end{minipage} } \\ &\geq \frac{ \setkeys{attsetweightfromblock}{from=b', to=s-1, val=v, when=t, chkp={C}} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=} }{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b', to=s-1, chkp={C}} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} } && \ifthenelse{\equal{7cm}{}}{ \text{--- From,\setkeys{attsetfromblockunfiltered}{from=b', to=s-1, val=v, when=t} {\mathcal{H}}_{\attsetfromblockunfiltered@From}^{\attsetfromblockunfiltered@To,\attsetfromblockunfiltered@Val,\attsetfromblockunfiltered@When} \setkeys{attsetfromblockunfiltered}{from=,to=,val=,when=} \subseteq \setkeys{commfromblock}{from=,to=,val=,when=} \setkeys{commfromblock}{from=b', to=s-1} {\mathcal{J}}_{\commfromblock@From}^{\commfromblock@To \ifthenelse{\equal{\commfromblock@Val}{}}{}{,\commfromblock@Val} \ifthenelse{\equal{\commfromblock@When}{}}{}{,\commfromblock@When} } and the fact that \frac{a+x}{b+y} \geq \frac{a}{b}, if a \leq b \land x \geq y. } }{ \begin{minipage}[t]{7cm} \begin{itemize}[label=---,nosep,left=0pt] \item From,\setkeys{attsetfromblockunfiltered}{from=b', to=s-1, val=v, when=t} {\mathcal{H}}_{\attsetfromblockunfiltered@From}^{\attsetfromblockunfiltered@To,\attsetfromblockunfiltered@Val,\attsetfromblockunfiltered@When} \setkeys{attsetfromblockunfiltered}{from=,to=,val=,when=} \subseteq \setkeys{commfromblock}{from=,to=,val=,when=} \setkeys{commfromblock}{from=b', to=s-1} {\mathcal{J}}_{\commfromblock@From}^{\commfromblock@To \ifthenelse{\equal{\commfromblock@Val}{}}{}{,\commfromblock@Val} \ifthenelse{\equal{\commfromblock@When}{}}{}{,\commfromblock@When} } and the fact that \frac{a+x}{b+y} \geq \frac{a}{b}, if a \leq b \land x \geq y. \end{itemize} \end{minipage} } \\ &= \setkeys{indicatorfromblock}{from=b',to=s-1,val=v,when=t,chkp={C}} {{{\mathit{P}}}}_{\indicatorfromblock@From}^{\indicatorfromblock@To,\indicatorfromblock@Val,\indicatorfromblock@When,\indicatorfromblock@ValSetChkp} \setkeys{indicatorfromblock}{from=,to=,val=,when=,chkp=} \end{align}\] ◻

Proof. We want to prove that \(b \preceq \setkeys{fcparam}{fc=\textsf{LMD-GHOST},balf=\mathsf{GJ},val=v} \ifthenelse{\equal{\fcparam@Val}{}}{{\fcparam@FC}_{\fcparam@BalF}}{{\fcparam@FC}_{\fcparam@BalF}^{\fcparam@Val}}(t)\).

Let \(b_i\) be the value of the variable \(b\) at the end of the \(i\)-th iteration of the while loop of Algorithm 1, with \(b_0\) corresponding to the value of the variable \(b\) at the beginning of the first iteration. We now prove by induction on \(i\) that either \(b_i \succeq b\) or \(b_i \preceq b\).

Base case: \(i=0\).

Trivial as the while loop in Algorithm 2 starts with variable \(b\) set to \(b_{\texttt{gen}}\preceq b\).

Inductive step.

By the inductive hypothesis, we assume that \(b_i \succeq b \lor b_i \preceq b\) and prove that \(b_{i+1} \succeq b \lor b_{i+1} \preceq b\). By line [ln:alg:GHOST:argmax] of Algorithm 1, \(b_{i+1}\) is the descendant of \(b_i\) with the heaviest total weight. Let us proceed by cases.

Case \(b_i \succeq b\).

This immediately implies that \(b_{i+1} \succeq b\).

Case \(b_i \prec b\).

Let \(b_c\) be the child of \(b_i\) in the chain of \(b\), i.e., \(b_c \preceq b \land \mathsf{parent}(b_c) = b_i\), and let \(b'\) be any child of \(b_i\). Let \(\textsf{FIL}_\textsf{LMD-GHOST}(\mathcal{V}, t) := \textsf{FIL}_{\mathsf{lmd}}(\textsf{FIL}_{\lnot \mathsf{valid}}(\textsf{FIL}_{\mathsf{cur}}(\textsf{FIL}_{\mathsf{eq}}(\mathcal{V}),t)))\) and note that, for \(\textsf{LMD-GHOST}^{\mathsf{GJ}}\), the argument of \(\mathop{\mathrm{arg\,max}}\) at line [ln:alg:GHOST:argmax] of Algorithm 1 corresponds to \(\setkeys{weightofset}{chkp={ \setkeys{chkpattime}{time=t,val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \left|{ \setkeys{ghostvoters}{block=b',view={\textsf{FIL}_\textsf{LMD-GHOST}(\mathcal{V}), t}} {GS}(\ghostvoters@Block,\ghostvoters@View) }\right|^{\weightofset@Chkp}\). Due to \(\textsf{FIL}_{\lnot \mathsf{valid}}\) and \(\textsf{FIL}_{\mathsf{cur}}\), the maximum value that such expression can evaluate to corresponds to the weight of the the committees between slot \(\mathsf{slot}(b_i)+1\) and slot \(\mathsf{slot}(t)-1\) plus, potentially, the proposer boost weight, i.e., \(\setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from={b},to={\mathsf{slot}(t)-1},chkp={ \setkeys{chkpattime}{time=t,val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} + \setkeys{boostweight}{chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} W_p^{\boostweight@Chkp}\). Given that honest validators never equivocate, we have that \[\setkeys{weightofset}{chkp={ \setkeys{chkpattime}{time=t,val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \left|{ \setkeys{ghostvoters}{block=b_c,view={\textsf{FIL}_\textsf{LMD-GHOST}(\mathcal{V}, t)}} {GS}(\ghostvoters@Block,\ghostvoters@View) }\right|^{\weightofset@Chkp} \geq \setkeys{attsetweightfromblock}{from=b_c,to=\mathsf{slot}(t)-1,val=v,when=t,chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=} > \frac{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b,to=\mathsf{slot}(t)-1,chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}+ \setkeys{boostweight}{chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} W_p^{\boostweight@Chkp} }{2}\] Take any \(b' \neq b_c\). This means that \(b'\) and \(b_c\) conflict which implies that

\[\begin{align} \setkeys{weightofset}{chkp={ \setkeys{chkpattime}{time=t,val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \left|{ \setkeys{ghostvoters}{block=b',view={\textsf{FIL}_\textsf{LMD-GHOST}(\mathcal{V}, t)}} {GS}(\ghostvoters@Block,\ghostvoters@View) }\right|^{\weightofset@Chkp} &< \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from={b},to={\mathsf{slot}(t)-1},chkp={ \setkeys{chkpattime}{time=t,val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} + \setkeys{boostweight}{chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} W_p^{\boostweight@Chkp} - \setkeys{weightofset}{chkp={ \setkeys{chkpattime}{time=t,val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \left|{ \setkeys{ghostvoters}{block=b_c,view={\textsf{FIL}_\textsf{LMD-GHOST}(\mathcal{V}, t)}} {GS}(\ghostvoters@Block,\ghostvoters@View) }\right|^{\weightofset@Chkp}\\ &\leq \frac{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b,to=\mathsf{slot}(t)-1,chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}+ \setkeys{boostweight}{chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} W_p^{\boostweight@Chkp} }{2} \end{align}\]

This furhter implies that \(b_c = b_{i+1}\) and hence \(b_{i+1} \preceq b\).

Note that any block in \(\mathsf{chain}(b)\) has at least one child, except potentially for \(b\). Note also that the while loop continues till it finds a block that either is for a slot higher than \(\mathsf{slot}(t)\) or that has no valid children. Given that we assume \(\mathsf{slot}(b) \leq \mathsf{slot}(t)\), honest validators never GHOST vote for an invalid block and that above we have established that \(\setkeys{fcparam}{fc=\textsf{LMD-GHOST},balf=\mathsf{GJ},val=v} \ifthenelse{\equal{\fcparam@Val}{}}{{\fcparam@FC}_{\fcparam@BalF}}{{\fcparam@FC}_{\fcparam@BalF}^{\fcparam@Val}}(t) \succeq b \lor \setkeys{fcparam}{fc=\textsf{LMD-GHOST},balf=\mathsf{GJ},val=v} \ifthenelse{\equal{\fcparam@Val}{}}{{\fcparam@FC}_{\fcparam@BalF}}{{\fcparam@FC}_{\fcparam@BalF}^{\fcparam@Val}}(t) \preceq b\), we can conclude the proof for this Lemma. ◻

Proof. Let \(b'\) by any block such that \(b' \preceq b\). Now we can proceed as follows. \[\begin{align} \setkeys{attsetweightfromblock}{from=b',to=s,val=v,when=t,chkp=C} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=} &= \setkeys{indicatorfromblock}{from=b',to=s,val=v,when=t,chkp=C} {{{\mathit{P}}}}_{\indicatorfromblock@From}^{\indicatorfromblock@To,\indicatorfromblock@Val,\indicatorfromblock@When,\indicatorfromblock@ValSetChkp} \setkeys{indicatorfromblock}{from=,to=,val=,when=,chkp=} \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} && \ifthenelse{\equal{}{}}{ \text{--- By definition} }{ \begin{minipage}[t]{} \begin{itemize}[label=---,nosep,left=0pt] \item By definition \end{itemize} \end{minipage} } \\ &> \frac{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}}{2(1-\beta)} \left( 1+ \frac{ \setkeys{boostweight}{chkp=C} W_p^{\boostweight@Chkp}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} \right) && \ifthenelse{\equal{}{}}{ \text{--- By expanding the condition on \setkeys{indicatorfromblock}{from=b',to=s,val=v,when=t,chkp=C} {{{\mathit{P}}}}_{\indicatorfromblock@From}^{\indicatorfromblock@To,\indicatorfromblock@Val,\indicatorfromblock@When,\indicatorfromblock@ValSetChkp} \setkeys{indicatorfromblock}{from=,to=,val=,when=,chkp=}} }{ \begin{minipage}[t]{} \begin{itemize}[label=---,nosep,left=0pt] \item By expanding the condition on \setkeys{indicatorfromblock}{from=b',to=s,val=v,when=t,chkp=C} {{{\mathit{P}}}}_{\indicatorfromblock@From}^{\indicatorfromblock@To,\indicatorfromblock@Val,\indicatorfromblock@When,\indicatorfromblock@ValSetChkp} \setkeys{indicatorfromblock}{from=,to=,val=,when=,chkp=} \end{itemize} \end{minipage} } \\ &\geq \frac{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}(1-\beta)}{2(1-\beta)} \left( 1+ \frac{ \setkeys{boostweight}{chkp=C} W_p^{\boostweight@Chkp}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} \right) && \ifthenelse{\equal{}{}}{ \text{--- As, due to Assumption~\ref{assum:beta},\setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} \geq \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}(1-\beta)} }{ \begin{minipage}[t]{} \begin{itemize}[label=---,nosep,left=0pt] \item As, due to Assumption~\ref{assum:beta},\setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} \geq \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}(1-\beta) \end{itemize} \end{minipage} } \\ &= \frac{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} + \setkeys{boostweight}{chkp=C} W_p^{\boostweight@Chkp}}{2} \end{align}\] ◻

Proof. We proceed as follows.
\[\begin{align} \setkeys{indicatorfromblock}{from=b',to=s,val=v,when=t,chkp=C} {{{\mathit{P}}}}_{\indicatorfromblock@From}^{\indicatorfromblock@To,\indicatorfromblock@Val,\indicatorfromblock@When,\indicatorfromblock@ValSetChkp} \setkeys{indicatorfromblock}{from=,to=,val=,when=,chkp=} &= \frac{ \setkeys{attsetweightfromblock}{from=b',to=s,val=v,when=t,chkp=C} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} && \ifthenelse{\equal{9cm}{}}{ \text{--- By definition.} }{ \begin{minipage}[t]{9cm} \begin{itemize}[label=---,nosep,left=0pt] \item By definition. \end{itemize} \end{minipage} } \\ &\geq \frac{ \setkeys{attsetweightfromblock}{from=b',to=s,val=v,when=t,chkp=C} {\addweight{\mathcal{S}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=}- \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{A}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} && \ifthenelse{\equal{9cm}{}}{ \text{--- By definition,\setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{A}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}= \setkeys{valsetfromblock}{from=b',to=s,chkp=C} {\mathcal{W}}_{\valsetfromblock@From}^{\valsetfromblock@To,\valsetfromblock@Chkp} \setminus \mathcal{J}.} }{ \begin{minipage}[t]{9cm} \begin{itemize}[label=---,nosep,left=0pt] \item By definition,\setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{A}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}= \setkeys{valsetfromblock}{from=b',to=s,chkp=C} {\mathcal{W}}_{\valsetfromblock@From}^{\valsetfromblock@To,\valsetfromblock@Chkp} \setminus \mathcal{J}. \end{itemize} \end{minipage} } \\ &= \frac{ \setkeys{attsetweightfromblock}{from=b',to=s,val=v,when=t,chkp=C} {\addweight{\mathcal{S}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=}- \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{A}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} - \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{A}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} && \ifthenelse{\equal{9cm}{}}{ \text{--- By definition,\setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} = \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} + \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{A}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}.} }{ \begin{minipage}[t]{9cm} \begin{itemize}[label=---,nosep,left=0pt] \item By definition,\setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} = \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} + \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{A}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}. \end{itemize} \end{minipage} } \\ &\geq \frac{ \setkeys{attsetweightfromblock}{from=b',to=s,val=v,when=t,chkp=C} {\addweight{\mathcal{S}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=}- \beta \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} - \beta \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} && \ifthenelse{\equal{9cm}{}}{ \text{--- By Assumption~\ref{assum:beta},\setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{A}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}\leq \beta \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}, and, given that\setkeys{attsetweightfromblock}{from=b',to=s,val=v,when=t,chkp=C} {\addweight{\mathcal{S}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=} \leq \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}, the function g(x)=\frac{ \setkeys{attsetweightfromblock}{from=b',to=s,val=v,when=t,chkp=C} {\addweight{\mathcal{S}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=}-x}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}-x} is monotone decreasing in [0, \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}] } }{ \begin{minipage}[t]{9cm} \begin{itemize}[label=---,nosep,left=0pt] \item By Assumption~\ref{assum:beta},\setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{A}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}\leq \beta \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}, and, given that\setkeys{attsetweightfromblock}{from=b',to=s,val=v,when=t,chkp=C} {\addweight{\mathcal{S}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=} \leq \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}, the function g(x)=\frac{ \setkeys{attsetweightfromblock}{from=b',to=s,val=v,when=t,chkp=C} {\addweight{\mathcal{S}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=}-x}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}-x} is monotone decreasing in [0, \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}] \end{itemize} \end{minipage} } \\ &= \frac{ \setkeys{attsetweightfromblock}{from=b',to=s,val=v,when=t,chkp=C} {\addweight{\mathcal{S}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=}- \beta \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} \left( \frac{1}{1-\beta} \right) && \ifthenelse{\equal{9cm}{}}{ \text{--- Simplification.} }{ \begin{minipage}[t]{9cm} \begin{itemize}[label=---,nosep,left=0pt] \item Simplification. \end{itemize} \end{minipage} } \\ &= \left( \setkeys{indicatorfromblock}{from=b',to=s,val=v,when=t,chkp=C} {{{\mathit{Q}}}}_{\indicatorfromblock@From}^{\indicatorfromblock@To,\indicatorfromblock@Val,\indicatorfromblock@When,\indicatorfromblock@ValSetChkp} \setkeys{indicatorfromblock}{from=,to=,val=,when=,chkp=} -\beta\right)\left(\frac{1}{1-\beta}\right) && \ifthenelse{\equal{9cm}{}}{ \text{--- Simplification.} }{ \begin{minipage}[t]{9cm} \begin{itemize}[label=---,nosep,left=0pt] \item Simplification. \end{itemize} \end{minipage} } \\ &> \frac{1}{2(1-\beta)} \left( 1+ \frac{ \setkeys{boostweight}{chkp=C} W_p^{\boostweight@Chkp}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} \right) && \ifthenelse{\equal{9cm}{}}{ \text{--- By applying the condition on\setkeys{indicatorfromblock}{from=b',to=s,val=v,when=t,chkp=C} {{{\mathit{Q}}}}_{\indicatorfromblock@From}^{\indicatorfromblock@To,\indicatorfromblock@Val,\indicatorfromblock@When,\indicatorfromblock@ValSetChkp} \setkeys{indicatorfromblock}{from=,to=,val=,when=,chkp=}.} }{ \begin{minipage}[t]{9cm} \begin{itemize}[label=---,nosep,left=0pt] \item By applying the condition on\setkeys{indicatorfromblock}{from=b',to=s,val=v,when=t,chkp=C} {{{\mathit{Q}}}}_{\indicatorfromblock@From}^{\indicatorfromblock@To,\indicatorfromblock@Val,\indicatorfromblock@When,\indicatorfromblock@ValSetChkp} \setkeys{indicatorfromblock}{from=,to=,val=,when=,chkp=}. \end{itemize} \end{minipage} } \end{align}\] ◻

Proof. We can apply Lemmas 4 and 3, in this order, to conclude that \(\setkeys{attsetweightfromblock}{from=b,to=\mathsf{slot}(t)-1,val=v,when=t,chkp={ \setkeys{chkpattime}{time=t,val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=} > \frac{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b,to=\mathsf{slot}(t)-1,chkp={ \setkeys{chkpattime}{time=t,val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} + \setkeys{boostweight}{chkp={ \setkeys{chkpattime}{time=t,val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} W_p^{\boostweight@Chkp}}{2}\). This implies that at least one honest validator \(v' \in \setkeys{commfromslot}{from=,to=,val=,when=} \setkeys{commfromslot}{from=n(b),to=\mathsf{slot}(t)-1} \overline{\mathcal{J}}_{\commfromslot@From}^{\commfromslot@To \ifthenelse{\equal{\commfromslot@Val}{}}{}{,\commfromslot@Val} \ifthenelse{\equal{\commfromslot@When}{}}{}{,\commfromslot@When} }\) has GHOST voted in support of \(b\). This implies that block \(b\) was in the view of validator \(v'\) by the time it voted in a slot \(s \leq \mathsf{slot}(t)-1\) as by definition of LMD-GHOST, honest validators only GHOST vote for blocks that are in their view. This further implies that \(v'\) broadcast block \(b\) no later than the time \(t\) it voted in slot \(s \leq \mathsf{slot}(t)-1\) as honest validators immediately broadcast any message that they receive. Then \(b\) is in the view of any honest validator by time \(\mathsf{st}(\mathsf{slot}(t))\).

Also, given that \(v'\) GHOST votes in support of \(b\), Algorithm 1 implies that \(\mathsf{slot}(b) \leq \mathsf{slot}(t)\). ◻

Proof. We proceed by induction on \(t'\).

Base case.

This is a strong induction quantified over \(t'\), so there is no need for a base case. Alternatively, we can take \(t' < \mathsf{st}(\mathsf{slot}(t))\) as base case for which the Lemma is vacuously true.

Inductive step: \(t' \geq \mathsf{st}(\mathsf{slot}(t))\).

Let \(s := \mathsf{slot}(t)\), \(s':= \mathsf{slot}(t')\), \(v'\) be any honest validator, \(\mathsf{GJ}:= \setkeys{chkpattime}{time=t,val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}\), \(\mathsf{GJ}' := \setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}\) and \(b'\) be any block such that \(b' \preceq b\). We assume that the Lemma holds for any time \(t''\) such that \(t'' < t'\) and we prove that it holds at time \(t'\) as well.

Given that, as described in Section 2.2.2, honest validators always GHOST vote for the block returned by the fork-choice function executed at the time of voting, any honest validator in the committees between slot \(s\) and slot \(s'-1\) has GHOST voted in support of \(b\) and, consequently, in support of \(b'\).

Also, note that due condition 4 of the Lemma’s statement we can conclude that \(\setkeys{boostweight}{chkp={\mathsf{GJ}'}} W_p^{\boostweight@Chkp} \leq \setkeys{boostweight}{chkp={\mathsf{GJ}}} W_p^{\boostweight@Chkp}\).

Then, we can apply Lemma 5 to conclude that \(b\) is in the view of \(v'\) at time \(t'\) and that \(\mathsf{slot}(b)\leq \mathsf{slot}(t)\).

With all of the above in mind, we can now proceed by cases.

Case \(\setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b', to=s'-1, chkp={\mathsf{GJ}'}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} \geq \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b', to=s-1, chkp={\mathsf{GJ}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}\).

\[\begin{align} \setkeys{indicatorfromblock}{from=b',to=s'-1,val=v',when=t,chkp={\mathsf{GJ}'}} {{{\mathit{P}}}}_{\indicatorfromblock@From}^{\indicatorfromblock@To,\indicatorfromblock@Val,\indicatorfromblock@When,\indicatorfromblock@ValSetChkp} \setkeys{indicatorfromblock}{from=,to=,val=,when=,chkp=} &\geq \setkeys{indicatorfromblock}{from=b',to=s-1,val=v,when=t,chkp={\mathsf{GJ}}} {{{\mathit{P}}}}_{\indicatorfromblock@From}^{\indicatorfromblock@To,\indicatorfromblock@Val,\indicatorfromblock@When,\indicatorfromblock@ValSetChkp} \setkeys{indicatorfromblock}{from=,to=,val=,when=,chkp=} && \ifthenelse{\equal{7cm}{}}{ \text{--- By Lemma~\ref{lem:lmd-p-monotonic}. } }{ \begin{minipage}[t]{7cm} \begin{itemize}[label=---,nosep,left=0pt] \item By Lemma~\ref{lem:lmd-p-monotonic}. \end{itemize} \end{minipage} } \\ &> \frac{1}{2} \left( 1+ \frac{ \setkeys{boostweight}{chkp={\mathsf{GJ}}} W_p^{\boostweight@Chkp}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s-1,chkp={\mathsf{GJ}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} \right) && \ifthenelse{\equal{7cm}{}}{ \text{--- By condition 2 of the Lemma's statement and Lemma~\ref{lem:lmd-cond-on-q-implies-cond-on-p}. } }{ \begin{minipage}[t]{7cm} \begin{itemize}[label=---,nosep,left=0pt] \item By condition 2 of the Lemma's statement and Lemma~\ref{lem:lmd-cond-on-q-implies-cond-on-p}. \end{itemize} \end{minipage} } \\ &\geq \frac{1}{2} \left( 1+ \frac{ \setkeys{boostweight}{chkp={\mathsf{GJ}'}} W_p^{\boostweight@Chkp}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s'-1,chkp={\mathsf{GJ}'}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} \right) && \ifthenelse{\equal{7cm}{}}{ \text{--- As we assume\setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b', to=s'-1, chkp={\mathsf{GJ}'}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} \geq \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b', to=s-1, chkp={\mathsf{GJ}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} and have established above that\setkeys{boostweight}{chkp={\mathsf{GJ}'}} W_p^{\boostweight@Chkp} \leq \setkeys{boostweight}{chkp={\mathsf{GJ}}} W_p^{\boostweight@Chkp} . } }{ \begin{minipage}[t]{7cm} \begin{itemize}[label=---,nosep,left=0pt] \item As we assume\setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b', to=s'-1, chkp={\mathsf{GJ}'}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} \geq \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b', to=s-1, chkp={\mathsf{GJ}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} and have established above that\setkeys{boostweight}{chkp={\mathsf{GJ}'}} W_p^{\boostweight@Chkp} \leq \setkeys{boostweight}{chkp={\mathsf{GJ}}} W_p^{\boostweight@Chkp} . \end{itemize} \end{minipage} } \end{align}\]

From here, we can apply Lemmas 2 and 3 to conclude the proof for this case.

Case \(\setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b', to=s'-1, chkp={\mathsf{GJ}'}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} < \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b', to=s-1, chkp={\mathsf{GJ}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}\).

\[\begin{align} \setkeys{attsetweightfromblock}{from=b', to=s'-1, val=v', when=t', chkp={\mathsf{GJ}'}} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=} &= \setkeys{weightofset}{chkp={\mathsf{GJ}'}} \left|{ \setkeys{attsetfromblockunfiltered}{from=b', to=s-1, val=v', when=t'} {\mathcal{H}}_{\attsetfromblockunfiltered@From}^{\attsetfromblockunfiltered@To,\attsetfromblockunfiltered@Val,\attsetfromblockunfiltered@When} \setkeys{attsetfromblockunfiltered}{from=,to=,val=,when=} \cup \setkeys{commfromslot}{from=,to=,val=,when=} \setkeys{commfromslot}{from=s, to=s'-1} \overline{\mathcal{J}}_{\commfromslot@From}^{\commfromslot@To \ifthenelse{\equal{\commfromslot@Val}{}}{}{,\commfromslot@Val} \ifthenelse{\equal{\commfromslot@When}{}}{}{,\commfromslot@When} } }\right|^{\weightofset@Chkp} && \ifthenelse{\equal{5cm}{}}{ \text{--- As, by the inductive hypothesis, all honest validators in the committees between slot s and slot s'-1 \textsf{GHOST} vote in support of b.} }{ \begin{minipage}[t]{5cm} \begin{itemize}[label=---,nosep,left=0pt] \item As, by the inductive hypothesis, all honest validators in the committees between slot s and slot s'-1 \textsf{GHOST} vote in support of b. \end{itemize} \end{minipage} } \\ \\ &\geq \setkeys{attsetweightfromblock}{from=b', to=s-1, val=v', when=t, chkp={\mathsf{GJ}'}} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=} \\ &= \setkeys{attsetweightfromblock}{from=b', to=s-1, val=v', when=t, chkp={\mathsf{GJ}}} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=} && \ifthenelse{\equal{5cm}{}}{ \text{--- Due to Assumption~\ref{assum:no-change-to-the-validator-set}.} }{ \begin{minipage}[t]{5cm} \begin{itemize}[label=---,nosep,left=0pt] \item Due to Assumption~\ref{assum:no-change-to-the-validator-set}. \end{itemize} \end{minipage} } \\ &\geq \setkeys{attsetweightfromblock}{from=b', to=s-1, when=t, chkp={\mathsf{GJ}}} {\addweight{\mathcal{S}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=} - \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b', to=s-1, chkp={\mathsf{GJ}}} \addweight{\mathcal{A}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} && \ifthenelse{\equal{}{}}{ \text{--- By definition.} }{ \begin{minipage}[t]{} \begin{itemize}[label=---,nosep,left=0pt] \item By definition. \end{itemize} \end{minipage} } \\ &= \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b', to=s-1, chkp={\mathsf{GJ}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} \setkeys{indicatorfromblock}{from=b',to=s-1,val=v,when=t,chkp={\mathsf{GJ}}} {{{\mathit{Q}}}}_{\indicatorfromblock@From}^{\indicatorfromblock@To,\indicatorfromblock@Val,\indicatorfromblock@When,\indicatorfromblock@ValSetChkp} \setkeys{indicatorfromblock}{from=,to=,val=,when=,chkp=} - \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b', to=s-1, chkp={\mathsf{GJ}}} \addweight{\mathcal{A}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} && \ifthenelse{\equal{}{}}{ \text{--- By definition of\setkeys{indicatorfromblock}{from=b',to=s-1,val=v,when=t,chkp={\mathsf{GJ}}} {{{\mathit{Q}}}}_{\indicatorfromblock@From}^{\indicatorfromblock@To,\indicatorfromblock@Val,\indicatorfromblock@When,\indicatorfromblock@ValSetChkp} \setkeys{indicatorfromblock}{from=,to=,val=,when=,chkp=}.} }{ \begin{minipage}[t]{} \begin{itemize}[label=---,nosep,left=0pt] \item By definition of\setkeys{indicatorfromblock}{from=b',to=s-1,val=v,when=t,chkp={\mathsf{GJ}}} {{{\mathit{Q}}}}_{\indicatorfromblock@From}^{\indicatorfromblock@To,\indicatorfromblock@Val,\indicatorfromblock@When,\indicatorfromblock@ValSetChkp} \setkeys{indicatorfromblock}{from=,to=,val=,when=,chkp=}. \end{itemize} \end{minipage} } \\ &> \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b', to=s-1, chkp={\mathsf{GJ}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} \left( \frac{1}{2} \left( 1+ \frac{ \setkeys{boostweight}{chkp={\mathsf{GJ}}} W_p^{\boostweight@Chkp}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s-1,chkp={\mathsf{GJ}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} \right) + \beta \right) - \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b', to=s-1, chkp={\mathsf{GJ}}} \addweight{\mathcal{A}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} && \ifthenelse{\equal{5cm}{}}{ \text{--- By applying condition 2 of the Lemma's statement.} }{ \begin{minipage}[t]{5cm} \begin{itemize}[label=---,nosep,left=0pt] \item By applying condition 2 of the Lemma's statement. \end{itemize} \end{minipage} } \\ &= \frac{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b', to=s-1, chkp={\mathsf{GJ}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} + \setkeys{boostweight}{chkp={\mathsf{GJ}}} W_p^{\boostweight@Chkp}}{2} + \beta \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b', to=s-1, chkp={\mathsf{GJ}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} - \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b', to=s-1, chkp={\mathsf{GJ}}} \addweight{\mathcal{A}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} && \ifthenelse{\equal{5cm}{}}{ \text{--- By simplifications.} }{ \begin{minipage}[t]{5cm} \begin{itemize}[label=---,nosep,left=0pt] \item By simplifications. \end{itemize} \end{minipage} } \\ &\geq \frac{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b', to=s-1, chkp={\mathsf{GJ}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} + \setkeys{boostweight}{chkp={\mathsf{GJ}}} W_p^{\boostweight@Chkp}}{2} && \ifthenelse{\equal{5cm}{}}{ \text{--- As, due to Assumption~\ref{assum:beta}, \beta \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b', to=s-1, chkp={\mathsf{GJ}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} \geq \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b', to=s-1, chkp={\mathsf{GJ}}} \addweight{\mathcal{A}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}. } }{ \begin{minipage}[t]{5cm} \begin{itemize}[label=---,nosep,left=0pt] \item As, due to Assumption~\ref{assum:beta}, \beta \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b', to=s-1, chkp={\mathsf{GJ}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} \geq \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b', to=s-1, chkp={\mathsf{GJ}}} \addweight{\mathcal{A}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}. \end{itemize} \end{minipage} } \\ &> \frac{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b', to=s'-1, chkp={\mathsf{GJ}'}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} + \setkeys{boostweight}{chkp={\mathsf{GJ}'}} W_p^{\boostweight@Chkp}}{2} && \ifthenelse{\equal{5cm}{}}{ \text{--- As we assume\setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b', to=s'-1, chkp={\mathsf{GJ}'}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} < \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b', to=s-1, chkp={\mathsf{GJ}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} and and have established above that\setkeys{boostweight}{chkp={\mathsf{GJ}'}} W_p^{\boostweight@Chkp} \leq \setkeys{boostweight}{chkp={\mathsf{GJ}}} W_p^{\boostweight@Chkp}.} }{ \begin{minipage}[t]{5cm} \begin{itemize}[label=---,nosep,left=0pt] \item As we assume\setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b', to=s'-1, chkp={\mathsf{GJ}'}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} < \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b', to=s-1, chkp={\mathsf{GJ}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} and and have established above that\setkeys{boostweight}{chkp={\mathsf{GJ}'}} W_p^{\boostweight@Chkp} \leq \setkeys{boostweight}{chkp={\mathsf{GJ}}} W_p^{\boostweight@Chkp}. \end{itemize} \end{minipage} } \\ \end{align}\] Now we can apply Lemma 2 to conclude the proof for this case.

 ◻

Proof. The condition \(\mathrm{isLMDConfirmed}_v(b,t)\) implies that there exists a slot \(s \in [\mathsf{first\_slot}(epoch(t)-1)+1,\mathsf{slot}(t)]\) such that \(\setkeys{varforvalattime}{val=v} \ifthenelse {\equal{\varforvalattime@Time} {}} {{\mathit{isLMDGHOSTSafe}}_{\varforvalattime@Val}} {{\mathit{isLMDGHOSTSafe}}_{\varforvalattime@Val}^{\varforvalattime@Time}} \setkeys{varforvalattime}{time=,val=}(b, \setkeys{chkpattime}{time=\mathsf{st}(s),val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}},\mathsf{st}(s))\). Given that \(s\leq\mathsf{slot}(t)\) and \(\mathsf{st}(s-1)\geq\mathsf{st}(\mathsf{epoch}(t)-1)\geq\mathbb{GST}\), due to 3, we can apply 6 to conclude that \(b\) is canonical in the view of any honest validator from time \(\mathsf{st}(\mathsf{slot}(t))\) and thereafter. ◻

Proof. Let \(t'\) be any time \(t' \geq \mathsf{st}(e+1)\). Given that, as described in Section 2.2.2, honest validators always GHOST vote for the block returned by the fork-choice function executed at the time of voting, then any honest validator in the committees of epoch \(e\) GHOST votes in support of \(b\). Note that as per 1, honest validators only GHOST vote in support of blocks that are from previous slots. Therefore, \(\mathsf{slot}(b) < \mathsf{st}(e)\leq \mathsf{epoch}(t')-1\). Hence, we can proceed as follows.

\[\begin{align} \setkeys{indicatorfromblock}{from=b',to=\mathsf{slot}(t')-1,val=v,when=t',chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {{{\mathit{Q}}}}_{\indicatorfromblock@From}^{\indicatorfromblock@To,\indicatorfromblock@Val,\indicatorfromblock@When,\indicatorfromblock@ValSetChkp} \setkeys{indicatorfromblock}{from=,to=,val=,when=,chkp=} &= \frac{ \setkeys{attsetweightfromblock}{from=b',to=\mathsf{slot}(t')-1,val=v,when=t',chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {\addweight{\mathcal{S}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=\mathsf{slot}(t')-1,chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} \\ &\geq \frac{ \setkeys{weightofset}{chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \left|{ \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=e} {\hat{\overline{\mathcal{J}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } }\right|^{\weightofset@Chkp}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=\mathsf{slot}(t')-1,chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} && \ifthenelse{\equal{6cm}{}}{ \text{--- As, all honest validators \textsf{GHOST} vote in support of b' during epoch e.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item As, all honest validators \textsf{GHOST} vote in support of b' during epoch e. \end{itemize} \end{minipage} } \\ &= \frac{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=\mathsf{slot}(t')-1,chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=\mathsf{slot}(t')-1,chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} && \ifthenelse{\equal{6cm}{}}{ \text{--- As, given \Cref{assum:no-change-to-the-validator-set} and that \mathsf{slot}(b)<\mathsf{epoch}(t')-1,\setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=e} {\hat{\overline{\mathcal{J}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} }includes all of the honest validators in any possible committee.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item As, given \Cref{assum:no-change-to-the-validator-set} and that \mathsf{slot}(b)<\mathsf{epoch}(t')-1,\setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=e} {\hat{\overline{\mathcal{J}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} }includes all of the honest validators in any possible committee. \end{itemize} \end{minipage} } \\ &\geq (1-\beta) && \ifthenelse{\equal{6cm}{}}{ \text{--- By applying Assumption~\ref{assum:beta}.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item By applying Assumption~\ref{assum:beta}. \end{itemize} \end{minipage} } \\ &= (1-2\beta+\beta) \\ &> \frac{1}{2} \left( 1+ \frac{p}{{E}} \right) + \beta && \ifthenelse{\equal{6cm}{}}{ \text{--- By applying the condition \beta < \frac{1}{4}\left(1-\frac{p}{{E}}\right), from Assumption~\ref{assum:beta-lmd-monotonicity}, to 2\beta.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item By applying the condition \beta < \frac{1}{4}\left(1-\frac{p}{{E}}\right), from Assumption~\ref{assum:beta-lmd-monotonicity}, to 2\beta. \end{itemize} \end{minipage} } \\ &= \frac{1}{2} \left(1+ \frac{ \setkeys{boostweight}{chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} W_p^{\boostweight@Chkp}}{ \setkeys{totvalsetweight}{chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}}\right)+\beta && \ifthenelse{\equal{6cm}{}}{ \text{--- As, by definition,\setkeys{boostweight}{chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} W_p^{\boostweight@Chkp} = \setkeys{totvalsetweight}{chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \frac{p}{{E}} .} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item As, by definition,\setkeys{boostweight}{chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} W_p^{\boostweight@Chkp} = \setkeys{totvalsetweight}{chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \frac{p}{{E}} . \end{itemize} \end{minipage} } \\ &\geq \frac{1}{2}\left(1+\frac{ \setkeys{boostweight}{chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} W_p^{\boostweight@Chkp}}{ \setkeys{weightofset}{chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \left|{ \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=e} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } }\right|^{\weightofset@Chkp}}\right)+\beta && \ifthenelse{\equal{6cm}{}}{ \text{--- As, by \Cref{assum:no-change-to-the-validator-set,assum:no-slashing},\setkeys{totvalsetweight}{chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} = \setkeys{totvalsetweight}{chkp={b_{\texttt{gen}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}= \setkeys{weightofset}{chkp={b_{\texttt{gen}}}} \left|{ \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=\mathsf{epoch}(b_{\texttt{gen}})} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } }\right|^{\weightofset@Chkp}= \setkeys{weightofset}{chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \left|{ \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=e} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } }\right|^{\weightofset@Chkp}.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item As, by \Cref{assum:no-change-to-the-validator-set,assum:no-slashing},\setkeys{totvalsetweight}{chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} = \setkeys{totvalsetweight}{chkp={b_{\texttt{gen}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}= \setkeys{weightofset}{chkp={b_{\texttt{gen}}}} \left|{ \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=\mathsf{epoch}(b_{\texttt{gen}})} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } }\right|^{\weightofset@Chkp}= \setkeys{weightofset}{chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \left|{ \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=e} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } }\right|^{\weightofset@Chkp}. \end{itemize} \end{minipage} } \\ &= \frac{1}{2}\left(1+\frac{ \setkeys{boostweight}{chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} W_p^{\boostweight@Chkp}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=\mathsf{slot}(t')-1,chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}}\right)+\beta && \ifthenelse{\equal{6cm}{}}{ \text{--- As \mathsf{slot}(b) < \mathsf{st}(e)\leq \mathsf{epoch}(t')-1 which implies that \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=\mathsf{slot}(t')-1,chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} contains validators in all committees of epoch e, \textit{i}.\textit{e}., \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=\mathsf{slot}(t')-1,chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}= \setkeys{weightofset}{chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \left|{ \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=e} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } }\right|^{\weightofset@Chkp}.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item As \mathsf{slot}(b) < \mathsf{st}(e)\leq \mathsf{epoch}(t')-1 which implies that \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=\mathsf{slot}(t')-1,chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} contains validators in all committees of epoch e, \textit{i}.\textit{e}., \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=\mathsf{slot}(t')-1,chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}= \setkeys{weightofset}{chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \left|{ \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=e} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } }\right|^{\weightofset@Chkp}. \end{itemize} \end{minipage} } \end{align}\] ◻

Proof. Condition \(\mathrm{isConfirmed}_v(b,t)\) implies that there exists a slot \(s \in [\mathsf{first\_slot}(epoch(t)-1)+1,\mathsf{slot}(t)]\) such that \(\setkeys{varforvalattime}{val=v} \ifthenelse {\equal{\varforvalattime@Time} {}} {{\mathit{isLMDGHOSTSafe}}_{\varforvalattime@Val}} {{\mathit{isLMDGHOSTSafe}}_{\varforvalattime@Val}^{\varforvalattime@Time}} \setkeys{varforvalattime}{time=,val=}(b, \setkeys{chkpattime}{time=t,val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}},\mathsf{st}(s))\). Given that \(\mathsf{st}(s-1)\geq\mathsf{st}(\mathsf{epoch}(t)-1)\geq\mathbb{GST}\), 7 implies that \(b\) is canonical in the view of any honest validator from time \(\mathsf{st}(s)\) and thereafter.

Now, let \(b' := \setkeys{varforvalattime}{val=v} \ifthenelse {\equal{\varforvalattime@Time} {}} {{\mathrm{highestConfirmedSinceEpoch}}_{\varforvalattime@Val}} {{\mathrm{highestConfirmedSinceEpoch}}_{\varforvalattime@Val}^{\varforvalattime@Time}} \setkeys{varforvalattime}{time=,val=}(\mathsf{epoch}(t')-1,t')\). Then, there exists a slot \(s' \in [\mathsf{first\_slot}(\mathsf{epoch}(t')-1)+1,\mathsf{slot}(t')]\) such that \(\setkeys{varforvalattime}{val=v} \ifthenelse {\equal{\varforvalattime@Time} {}} {{\mathit{isLMDGHOSTSafe}}_{\varforvalattime@Val}} {{\mathit{isLMDGHOSTSafe}}_{\varforvalattime@Val}^{\varforvalattime@Time}} \setkeys{varforvalattime}{time=,val=}(b', \setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}},\mathsf{st}(s'))\). Thanks to 7, this also implies that \(b'\) is canonical for any honest validator at time \(\mathsf{st}(\mathsf{slot}(t'))\).

To show \(\mathrm{isConfirmed}_v(b,t')\), by Algorithm 4, we need to prove \(b\preceq b'\). We can now proceed by cases.

Case 1: \(s \in [\mathsf{first\_slot}(\mathsf{epoch}(t')-1)+1,\mathsf{slot}(t')]\).

This implies that \(\mathsf{slot}(b')\geq\mathsf{slot}(b)\). Given that \(b'\) is also canonical at time \(t'\), we can conclude that \(b\preceq b'\).

Case 2: \(s \notin [\mathsf{first\_slot}(\mathsf{epoch}(t')-1)+1,\mathsf{slot}(t')]\)

This case implies that \(\mathsf{st}(s)\leq\mathsf{st}(\mathsf{epoch}(t')-1)\). Hence, given that \(b\) is canonical in the view of any honest validator from time \(\mathsf{st}(s)\) and thereafter, this further implies that \(b\) has been canonical in the view of any honest validator for the entire epoch \(\mathsf{epoch}(t')-1\). Then, 8 implies that \(\setkeys{varforvalattime}{val=v} \ifthenelse {\equal{\varforvalattime@Time} {}} {{\mathit{isLMDGHOSTSafe}}_{\varforvalattime@Val}} {{\mathit{isLMDGHOSTSafe}}_{\varforvalattime@Val}^{\varforvalattime@Time}} \setkeys{varforvalattime}{time=,val=}(b', \setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}},\mathsf{st}(\mathsf{slot}(t')))\). Given that \(\mathsf{slot}(t') \in [\mathsf{first\_slot}(\mathsf{epoch}(t')-1)+1,\mathsf{slot}(t')]\), this implies that \(\mathsf{slot}(b')\geq\mathsf{slot}(b)\). Becuase \(b'\) is also canonical at time \(t'\), we can conclude that \(b\preceq b'\).

 ◻

Proof. By applying [lem:lmd-algo-safety,lem:lmd-monotoncity]. ◻

Proof. Because of 1, condition 4.2 implies that, for any validator \(v'' \in \setkeys{commfromslot}{from=,to=,val=,when=} \setkeys{commfromslot}{from=\mathsf{slot}(t),to=\mathsf{slot}(t')} \overline{\mathcal{J}}_{\commfromslot@From}^{\commfromslot@To \ifthenelse{\equal{\commfromslot@Val}{}}{}{,\commfromslot@Val} \ifthenelse{\equal{\commfromslot@When}{}}{}{,\commfromslot@When} }\) and time \(t''\) such that \(t \leq t'' \leq t'\), \(\setkeys{totvalsetweight}{chkp={ \setkeys{chkpattime}{time=t'',val=v''} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \leq \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}\). Then, given that the only difference between LMD-GHOST and LMD-GHOST-HFC is the application of \(\textsf{FIL}_{\mathsf{hfc}}\) and condition 4.1 of the Lemma’s statement, the proof for this Lemma is identical to the proof of Lemma 6. ◻

Proof. Let \(v'\) be any validator and \(t'\) be any time such that \(\mathsf{epoch}(t') = \mathsf{epoch}(t)\). Let us now proceed by cases.

Case 1: \(epoch\left( \setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}\right) = \mathsf{epoch}(\mathsf{C}(b))\).

By the Lemma’s assumptions, we know that there exists a block \(b' \succeq b\) such that \(\mathsf{C}(b) \in \mathsf{AU}(b')\). Let \(b''\) be any block \(b'' \succeq b'\). Given that \(\mathsf{epoch}(b') < \mathsf{epoch}(t)\), \(\mathsf{epoch}(\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b'',time=t'} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} ))\geq\mathsf{epoch}(\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b',time=t'} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )) \geq \mathsf{C}(b)\). By Property [prop:gasper-basic:only-one-justified-per-epoch] and the definition of \(\setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}\) (3), we have that \(\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b'',time=t'} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} ) = \mathsf{C}(b) = \setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}\). By [prop:gasper-basic:just-succ-finalization], this also implies that \(b'' \succeq \setkeys{finattime}{time=t',val=v'} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}\). Given that clearly \(b' \succeq \mathsf{block}(\mathsf{C}(b))\) we have that \(b' \in \setkeys{filtered}{time=t',val=v'} \mathsf{filt}_{\mathsf{hfc}}^{\filtered@Time\ifthenelse{\equal{\filtered@Val}{}}{}{,\filtered@Val}}\), from which it follows that \(b \in \setkeys{filtered}{time=t',val=v'} \mathsf{filt}_{\mathsf{hfc}}^{\filtered@Time\ifthenelse{\equal{\filtered@Val}{}}{}{,\filtered@Val}}\).

Case 2: \(epoch\left( \setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}\right) > \mathsf{epoch}(\mathsf{C}(b))\).

By Property [prop:gasper-basic:highest-justified-not-from-current-epoch], we know that \(epoch\left( \setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}\right) \in [\mathsf{epoch}(b)+1,\mathsf{epoch}(t) -1]\). Hence, by the Lemma’s assumptions, \(\setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}\) does not conflict with \(b\) which, given that in this case we assume \(epoch\left( \setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}\right) > \mathsf{epoch}(\mathsf{C}(b)) = \mathsf{epoch}(b)\), implies that \(b \prec \setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}\), from which we can conclude that \(b \in \setkeys{filtered}{time=t',val=v'} \mathsf{filt}_{\mathsf{hfc}}^{\filtered@Time\ifthenelse{\equal{\filtered@Val}{}}{}{,\filtered@Val}}\).

Case 3: \(epoch\left( \setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}\right) < \mathsf{C}(b)\).

Given that there exists a block \(b' \succeq b\) such that \(\mathsf{C}(b) \in \mathsf{AU}(b')\), we have that \(\mathsf{epoch}(\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b',time=t'} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )) \geq \mathsf{epoch}(\mathsf{C}(b))\). Hence, the definition of \(\setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}\) (Definition 3) implies that \(epoch\left( \setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}\right) \geq \mathsf{epoch}(\mathsf{C}(b))\) meaning that this case is not possible.

 ◻

Proof. First, we proceed by induction on \(t' \geq \mathsf{st}(\mathsf{slot}(t))\) to show that all of the following inductive conditions hold

1. there exists no checkpoint \(C'\) with \(\mathsf{epoch}(C') \in [\mathsf{epoch}(b), \mathsf{epoch}(t')]\) which conflicts with \(b\).

2. for any honest validator \(v''\) and time \(t''\) such that \(\mathsf{st}(\mathsf{slot}(t)) \leq t'' < \mathsf{st}(\mathsf{epoch}(t')+1)\)

   1. \(b \in \setkeys{filtered}{val=v'',time=t''} \mathsf{filt}_{\mathsf{hfc}}^{\filtered@Time\ifthenelse{\equal{\filtered@Val}{}}{}{,\filtered@Val}}\)

   2. \(\setkeys{chkpattime}{time=t'',val=v''} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}} \succeq C\)

Base Case: \(\mathsf{epoch}(t') < \mathsf{epoch}(b)+2\).

All inductive hypothesis are trivially implied by [def:induction-conditions:all-validators] and [def:induction-conditions:no-conflicting].

Inductive Step: \(\mathsf{epoch}(t') \geq \mathsf{epoch}(b)+2\).

Assume that all the inductive hypotheses hold at any time \(t_i\) up to \(\mathsf{epoch}(t_i) \leq \mathsf{epoch}(t')-1\) and prove that they hold at time \(t'\) as well. Let \(v'\) be any honest validator.

Induction hypothesis [lem:ffg-safety-from-sir:safety-cond-1] and [def:induction-conditions:ub] allow us to apply Lemma 11 to conclude that \(b \in \setkeys{filtered}{time=t',val=v'} \mathsf{filt}_{\mathsf{hfc}}^{\filtered@Time\ifthenelse{\equal{\filtered@Val}{}}{}{,\filtered@Val}}\), i.e., \(b\) does not get filtered out by any honest validator in epoch \(\mathsf{epoch}(t')\). This proves induction hypothesis [lem:ffg-safety-from-sir:safety-cond-2461] holds at time \(t'\) as well.

Also, induction hypothesis [lem:ffg-safety-from-sir:safety-cond-1], [def:induction-conditions:is-lmd-confirmed], [def:induction-conditions:ub], Property [prop:gasper-basic:highest-justified-not-from-current-epoch] and the definition of \(\setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}\) (Definition 3) imply that \(\setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}} \succeq \mathsf{C}(b) \succeq C\) which proves inductive condition [lem:ffg-safety-from-sir:safety-cond-2462] for \(t'\).

Given that \(t'\geq\mathsf{st}(\mathsf{slot}(t))\) and that above we have proved that inductive condition [lem:ffg-safety-from-sir:safety-cond-2] is satisfied for time \(t'\), thanks to [def:induction-conditions:is-lmd-confirmed], we can apply Lemma 10 to conclude that \(b\) is always canonical in the view of all honest validators at any time during epoch \(\mathsf{epoch}(t')\).

By Properties [prop:gasper-basic:ldm-vote-for-b-is-ffg-vote-for-cb] and [prop:gasper-basic:no-conflicting-if-all-honest-votes-in-support-of-b], this immediately implies that no checkpoint conflicting with \(b\) can be justified in epoch \(\mathsf{epoch}(t')\), which concludes the proof for the inductive hypothesis [lem:ffg-safety-from-sir:safety-cond-1].

Given that we have just established that the inductive condition [lem:ffg-safety-from-sir:safety-cond-2] hold for any time \(t' \geq \mathsf{st}(\mathsf{slot}(t))\), thanks to [def:induction-conditions:is-lmd-confirmed], we can apply Lemma 10 to complete the proof. ◻

Proof. Let \(C_b := \mathsf{C}(b,e)\) and \(\mathit{VS}_b := \setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b,time=e} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )\), \(t'\) be any time such that \(t' \geq \mathsf{st}(\mathsf{epoch}(b)+1)\), and \(b'\) be any block such that \(b' \succeq \mathsf{C}(b,e)\).

We can now proceed as follows to prove the Lemma. \[\begin{align} \setkeys{weightofset}{chkp=b'} \left|{ \setkeys{ffgvalsetallsentraw}{source=,target=C_b,time=t'} \overset{T}{\mathcal{F}}{}_{\ffgvalsetallsentraw@Source \to \ffgvalsetallsentraw@Target}^{\ffgvalsetallsentraw@Time}\setminus \setkeys{slashedset}{chkp=b'} \mathcal{D}^{\slashedset@Chkp}}\right|^{\weightofset@Chkp} &\geq \setkeys{weightofset}{chkp=b'} \left|{ \left( \setkeys{ffgvalsettoslot}{to=s-1,source=\mathit{VS}_b,target=C_b,time=t,val=v} \mathcal{F}_{\ffgvalsettoslot@Source \to \ffgvalsettoslot@Target}^{\ffgvalsettoslot@To,\ffgvalsettoslot@Val,\ffgvalsettoslot@Time} \sqcup \setkeys{commfromslot}{from=,to=,val=,when=} \setkeys{commfromslot}{from=s,to=\mathsf{last\_slot}(e)} \overline{\mathcal{J}}_{\commfromslot@From}^{\commfromslot@To \ifthenelse{\equal{\commfromslot@Val}{}}{}{,\commfromslot@Val} \ifthenelse{\equal{\commfromslot@When}{}}{}{,\commfromslot@When} }\right) \setminus \setkeys{slashedset}{chkp=b'} \mathcal{D}^{\slashedset@Chkp} }\right|^{\weightofset@Chkp} && \ifthenelse{\equal{6cm}{}}{ \text{--- Given that t' \geq \mathsf{st}(e+1), by time t' every honest validator in slots [s,\mathsf{last\_slot}(e)] has \textsf{GHOST} voted for a block b'' \succeq b, which, by Property~\ref{prop:gasper-basic:ldm-vote-for-b-is-ffg-vote-for-cb} equates to an FFG vote for\setkeys{ffgvote}{from=\mathit{VS}_b, to=C_b} {\ffgvote@From}\to{\ffgvote@To} . To this, we add the validators whose \textsf{GHOST} votes have already been received at time t. } }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item Given that t' \geq \mathsf{st}(e+1), by time t' every honest validator in slots [s,\mathsf{last\_slot}(e)] has \textsf{GHOST} voted for a block b'' \succeq b, which, by Property~\ref{prop:gasper-basic:ldm-vote-for-b-is-ffg-vote-for-cb} equates to an FFG vote for\setkeys{ffgvote}{from=\mathit{VS}_b, to=C_b} {\ffgvote@From}\to{\ffgvote@To} . To this, we add the validators whose \textsf{GHOST} votes have already been received at time t. \end{itemize} \end{minipage} } \\ &= \setkeys{weightofset}{chkp=C_b} \left|{ \left( \setkeys{ffgvalsettoslot}{to=s-1,source=\mathit{VS}_b,target=C_b,time=t,val=v} \mathcal{F}_{\ffgvalsettoslot@Source \to \ffgvalsettoslot@Target}^{\ffgvalsettoslot@To,\ffgvalsettoslot@Val,\ffgvalsettoslot@Time} \sqcup \setkeys{commfromslot}{from=,to=,val=,when=} \setkeys{commfromslot}{from=s,to=\mathsf{last\_slot}(e)} \overline{\mathcal{J}}_{\commfromslot@From}^{\commfromslot@To \ifthenelse{\equal{\commfromslot@Val}{}}{}{,\commfromslot@Val} \ifthenelse{\equal{\commfromslot@When}{}}{}{,\commfromslot@When} } \right) \setminus \setkeys{slashedset}{chkp=b'} \mathcal{D}^{\slashedset@Chkp} }\right|^{\weightofset@Chkp} && \ifthenelse{\equal{6cm}{}}{ \text{--- The only difference in effective balances between b' and C_b is represented by those validators\setkeys{slashedset}{chkp=b'} \mathcal{D}^{\slashedset@Chkp} \setminus \setkeys{slashedset}{chkp=C_b} \mathcal{D}^{\slashedset@Chkp} that are slashed between C_b and b'.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item The only difference in effective balances between b' and C_b is represented by those validators\setkeys{slashedset}{chkp=b'} \mathcal{D}^{\slashedset@Chkp} \setminus \setkeys{slashedset}{chkp=C_b} \mathcal{D}^{\slashedset@Chkp} that are slashed between C_b and b'. \end{itemize} \end{minipage} } \\ &= \setkeys{weightofset}{chkp=C_b} \left|{ \left( \setkeys{ffgvalsettoslot}{to=s-1,source=\mathit{VS}_b,target=C_b,time=t,val=v} \mathcal{F}_{\ffgvalsettoslot@Source \to \ffgvalsettoslot@Target}^{\ffgvalsettoslot@To,\ffgvalsettoslot@Val,\ffgvalsettoslot@Time} \setminus \setkeys{slashedset}{chkp=b'} \mathcal{D}^{\slashedset@Chkp} \right) \sqcup \setkeys{commfromslot}{from=,to=,val=,when=} \setkeys{commfromslot}{from=s,to=\mathsf{last\_slot}(e)} \overline{\mathcal{J}}_{\commfromslot@From}^{\commfromslot@To \ifthenelse{\equal{\commfromslot@Val}{}}{}{,\commfromslot@Val} \ifthenelse{\equal{\commfromslot@When}{}}{}{,\commfromslot@When} } }\right|^{\weightofset@Chkp} && \ifthenelse{\equal{6cm}{}}{ \text{--- As honest validators never get slashed.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item As honest validators never get slashed. \end{itemize} \end{minipage} } \\ &= \setkeys{weightofset}{chkp=C_b} \left|{ \left( \setkeys{ffgvalsettoslot}{to=s-1,source=\mathit{VS}_b,target=C_b,time=t,val=v} \mathcal{F}_{\ffgvalsettoslot@Source \to \ffgvalsettoslot@Target}^{\ffgvalsettoslot@To,\ffgvalsettoslot@Val,\ffgvalsettoslot@Time} \setminus \left( \setkeys{ffgvalsettoslot}{to=s-1,source=\mathit{VS}_b,target=C_b,time=t,val=v} \mathcal{F}_{\ffgvalsettoslot@Source \to \ffgvalsettoslot@Target}^{\ffgvalsettoslot@To,\ffgvalsettoslot@Val,\ffgvalsettoslot@Time} \cap \setkeys{slashedset}{chkp=b'} \mathcal{D}^{\slashedset@Chkp} \right)\right) \sqcup \setkeys{commfromslot}{from=,to=,val=,when=} \setkeys{commfromslot}{from=s,to=\mathsf{last\_slot}(e)} \overline{\mathcal{J}}_{\commfromslot@From}^{\commfromslot@To \ifthenelse{\equal{\commfromslot@Val}{}}{}{,\commfromslot@Val} \ifthenelse{\equal{\commfromslot@When}{}}{}{,\commfromslot@When} } }\right|^{\weightofset@Chkp} \\ &= \setkeys{weightofset}{chkp=C_b} \left|{ \setkeys{ffgvalsettoslot}{to=s-1,source=\mathit{VS}_b,target=C_b,time=t,val=v} \mathcal{F}_{\ffgvalsettoslot@Source \to \ffgvalsettoslot@Target}^{\ffgvalsettoslot@To,\ffgvalsettoslot@Val,\ffgvalsettoslot@Time} }\right|^{\weightofset@Chkp} - \setkeys{weightofset}{chkp=C_b} \left|{ \setkeys{ffgvalsettoslot}{to=s-1,source=\mathit{VS}_b,target=C_b,time=t,val=v} \mathcal{F}_{\ffgvalsettoslot@Source \to \ffgvalsettoslot@Target}^{\ffgvalsettoslot@To,\ffgvalsettoslot@Val,\ffgvalsettoslot@Time} \cap \setkeys{slashedset}{chkp=b'} \mathcal{D}^{\slashedset@Chkp} }\right|^{\weightofset@Chkp} + \setkeys{commweightfromslot}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromslot}{from=s,to=\mathsf{last\_slot}(e),chkp=C_b} \overline{{\addweight{\mathcal{J}}}}_{\commweightfromslot@From}^{\commweightfromslot@To, \ifthenelse{\equal{\commweightfromslot@Val}{}}{}{\commweightfromslot@Val,} \ifthenelse{\equal{\commweightfromslot@When}{}}{}{\commweightfromslot@When,} \commweightfromslot@ValsetChkp} \\ &\geq \begin{aligned}[t] & \setkeys{weightofset}{chkp=C_b} \left|{ \setkeys{ffgvalsettoslot}{to=s-1,source=\mathit{VS}_b,target=C_b,time=t,val=v} \mathcal{F}_{\ffgvalsettoslot@Source \to \ffgvalsettoslot@Target}^{\ffgvalsettoslot@To,\ffgvalsettoslot@Val,\ffgvalsettoslot@Time} }\right|^{\weightofset@Chkp} - \setkeys{weightofset}{chkp=C_b} \left|{ \setkeys{ffgvalsettoslot}{to=s-1,source=\mathit{VS}_b,target=C_b,time=t,val=v} \mathcal{F}_{\ffgvalsettoslot@Source \to \ffgvalsettoslot@Target}^{\ffgvalsettoslot@To,\ffgvalsettoslot@Val,\ffgvalsettoslot@Time} \cap \setkeys{slashedset}{chkp=b'} \mathcal{D}^{\slashedset@Chkp} }\right|^{\weightofset@Chkp}\\ &+ (1-\beta) \setkeys{commweightfromslot}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromslot}{from=s,to=\mathsf{last\_slot}(e),chkp=C_b} \overline{{\addweight{\mathcal{W}}}}_{\commweightfromslot@From}^{\commweightfromslot@To, \ifthenelse{\equal{\commweightfromslot@Val}{}}{}{\commweightfromslot@Val,} \ifthenelse{\equal{\commweightfromslot@When}{}}{}{\commweightfromslot@When,} \commweightfromslot@ValsetChkp} \end{aligned}\\ &\geq \begin{align}[t] &\frac{2}{3} \setkeys{totvalsetweight}{chkp=C_b} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} + \min\left( W_e, \beta \setkeys{totvalsetweight}{chkp=C_b} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \right) - (1-\beta) \setkeys{valsetweightfromslot}{from=s,to=\mathsf{last\_slot}(e),chkp=C_b} \overline{{\addweight{\mathcal{W}}}}_{\valsetweightfromslot@From}^{\valsetweightfromslot@To,\valsetweightfromslot@ValsetChkp\ifthenelse{\equal{\valsetweightfromslot@WeightChkp}{}}{}{,\valsetweightfromslot@WeightChkp}}\\ &- \setkeys{weightofset}{chkp=C_b} \left|{ \setkeys{ffgvalsettoslot}{to=s-1,source=\mathit{VS}_b,target=C_b,time=t,val=v} \mathcal{F}_{\ffgvalsettoslot@Source \to \ffgvalsettoslot@Target}^{\ffgvalsettoslot@To,\ffgvalsettoslot@Val,\ffgvalsettoslot@Time} \cap \setkeys{slashedset}{chkp=b'} \mathcal{D}^{\slashedset@Chkp} }\right|^{\weightofset@Chkp} + (1-\beta) \setkeys{commweightfromslot}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromslot}{from=s,to=\mathsf{last\_slot}(e),chkp=C_b} \overline{{\addweight{\mathcal{W}}}}_{\commweightfromslot@From}^{\commweightfromslot@To, \ifthenelse{\equal{\commweightfromslot@Val}{}}{}{\commweightfromslot@Val,} \ifthenelse{\equal{\commweightfromslot@When}{}}{}{\commweightfromslot@When,} \commweightfromslot@ValsetChkp} \end{align} && \ifthenelse{\equal{6cm}{}}{ \text{--- By applying the condition on \setkeys{weightofset}{chkp=C_b} \left|{ \setkeys{ffgvalsettoslot}{to=s-1,source=\mathit{VS}_b,target=C_b,time=t,val=v} \mathcal{F}_{\ffgvalsettoslot@Source \to \ffgvalsettoslot@Target}^{\ffgvalsettoslot@To,\ffgvalsettoslot@Val,\ffgvalsettoslot@Time} }\right|^{\weightofset@Chkp} as per the Lemma's statement.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item By applying the condition on \setkeys{weightofset}{chkp=C_b} \left|{ \setkeys{ffgvalsettoslot}{to=s-1,source=\mathit{VS}_b,target=C_b,time=t,val=v} \mathcal{F}_{\ffgvalsettoslot@Source \to \ffgvalsettoslot@Target}^{\ffgvalsettoslot@To,\ffgvalsettoslot@Val,\ffgvalsettoslot@Time} }\right|^{\weightofset@Chkp} as per the Lemma's statement. \end{itemize} \end{minipage} } \\ &\geq \frac{2}{3} \setkeys{totvalsetweight}{chkp=C_b} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} && \ifthenelse{\equal{6cm}{}}{ \text{--- As, due to Assumption~\ref{assum:beta} and~\ref{assum:ffg-assumptions:max-slashing}, and the fact that honest validators never commit slashing offences,\min\left( W_e, \beta \setkeys{totvalsetweight}{chkp=C_b} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \right) \geq \setkeys{weightofset}{chkp=C_b} \left|{ \setkeys{ffgvalsettoslot}{to=s-1,source=\mathit{VS}_b,target=C_b,time=t,val=v} \mathcal{F}_{\ffgvalsettoslot@Source \to \ffgvalsettoslot@Target}^{\ffgvalsettoslot@To,\ffgvalsettoslot@Val,\ffgvalsettoslot@Time} \cap \setkeys{slashedset}{chkp=b'} \mathcal{D}^{\slashedset@Chkp} }\right|^{\weightofset@Chkp} } }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item As, due to Assumption~\ref{assum:beta} and~\ref{assum:ffg-assumptions:max-slashing}, and the fact that honest validators never commit slashing offences,\min\left( W_e, \beta \setkeys{totvalsetweight}{chkp=C_b} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \right) \geq \setkeys{weightofset}{chkp=C_b} \left|{ \setkeys{ffgvalsettoslot}{to=s-1,source=\mathit{VS}_b,target=C_b,time=t,val=v} \mathcal{F}_{\ffgvalsettoslot@Source \to \ffgvalsettoslot@Target}^{\ffgvalsettoslot@To,\ffgvalsettoslot@Val,\ffgvalsettoslot@Time} \cap \setkeys{slashedset}{chkp=b'} \mathcal{D}^{\slashedset@Chkp} }\right|^{\weightofset@Chkp} \end{itemize} \end{minipage} } \\ &\geq \frac{2}{3} \setkeys{totvalsetweight}{chkp=b'} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} && \ifthenelse{\equal{6cm}{}}{ \text{--- By \Cref{assum:no-change-to-the-validator-set}, given that b'\succeq C_b,\setkeys{totvalsetweight}{chkp=C_b} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}\geq \setkeys{totvalsetweight}{chkp=b'} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp},} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item By \Cref{assum:no-change-to-the-validator-set}, given that b'\succeq C_b,\setkeys{totvalsetweight}{chkp=C_b} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}\geq \setkeys{totvalsetweight}{chkp=b'} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}, \end{itemize} \end{minipage} } \end{align}\] Note that if \(e<\mathsf{epoch}(s)\), then, some of the conditions above are vacuously true (e.g., all honest validators in slots \([s,\mathsf{last\_slot}(e)] =\emptyset\) GHOST vote in support of \(b\)), but the reasoning above still works. This concludes the proof. ◻

Proof. Let be any block \(b'\) such that \(b'\succeq b \land \mathsf{epoch}(b)\leq\mathsf{epoch}(t)\). First, property [prop:gasper-basic:highest-justified-not-from-current-epoch] implies that \(\mathsf{epoch}(\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b',time=t} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )) = \mathsf{epoch}(t)-1 = \mathsf{epoch}( \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}})\), then Property [prop:gasper-basic:only-one-justified-per-epoch] implies that the highest justified checkpoint in the view of \(v\) at time \(t\) is \(\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b',time=t} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )\), i.e., \(\setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}=\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b',time=t} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )\). [prop:gasper-basic:just-succ-finalization] also implies that \(b' \succeq \setkeys{finattime}{time=t,val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}\). Given that clarly, \(b' \succeq \mathsf{block}( \setkeys{chkpattime}{time=t,val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}})\), it follows that \(b \in \setkeys{filtered}{time=t,val=v} \mathsf{filt}_{\mathsf{hfc}}^{\filtered@Time\ifthenelse{\equal{\filtered@Val}{}}{}{,\filtered@Val}}\). ◻

Proof. Let \(b'\) be any block such that \(b' \in \setkeys{viewattime}{time=t,val=v} \mathcal{V}^{\viewattime@Val,\viewattime@Time},\: b \preceq b' \land \mathsf{epoch}(b')\leq\mathsf{epoch}(t)\land\mathsf{epoch}(\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b',time=t} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )) \geq \mathsf{epoch}(t) - 2\) and let \(b''\) be any block such that \(b'' \succeq b' \land \mathsf{epoch}(b'')\leq\mathsf{epoch}(t)\). Property [prop:gasper-basic:highest-justified-not-from-current-epoch] implies that \(\mathsf{epoch}(t) - 2 \leq \mathsf{epoch}( \setkeys{chkpattime}{time=t,val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}) \leq \mathsf{epoch}(t) -1\). We now proceed by cases to show that \(b'' \in \setkeys{filtered}{time=t,val=v} \mathsf{filt}_{\mathsf{hfc}}^{\filtered@Time\ifthenelse{\equal{\filtered@Val}{}}{}{,\filtered@Val}}\) which implies \(b \in \setkeys{filtered}{time=t,val=v} \mathsf{filt}_{\mathsf{hfc}}^{\filtered@Time\ifthenelse{\equal{\filtered@Val}{}}{}{,\filtered@Val}}\).

Case 1: \(\mathsf{epoch}( \setkeys{chkpattime}{time=t,val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}) = \mathsf{epoch}(t) -2\).

Property [prop:gasper-basic:highest-justified-not-from-current-epoch] implies that \(\mathsf{epoch}(\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b'',time=t} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )) = \mathsf{epoch}( \setkeys{chkpattime}{time=t,val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}})\). Then, due to Property [prop:gasper-basic:only-one-justified-per-epoch], \(\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b'',time=t} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} ) = \setkeys{chkpattime}{time=t,val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}\). [prop:gasper-basic:just-succ-finalization] also implies that \(b'' \succeq \setkeys{finattime}{time=t,val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}\). Given that clarly, \(b'' \succeq \mathsf{block}( \setkeys{chkpattime}{time=t,val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}})\), it follows that \(b'' \in \setkeys{filtered}{time=t,val=v} \mathsf{filt}_{\mathsf{hfc}}^{\filtered@Time\ifthenelse{\equal{\filtered@Val}{}}{}{,\filtered@Val}}\).

Case 2: \(\mathsf{epoch}( \setkeys{chkpattime}{time=t,val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}) = \mathsf{epoch}(t) -1\).

Due to condition 3, \(\setkeys{chkpattime}{time=t,val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}} = \mathsf{C}(b)\). Hence, due to [prop:gasper-basic:just-succ-finalization], \(b''\succeq \mathsf{C}(b)= \setkeys{chkpattime}{time=t,val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}\succeq \setkeys{finattime}{time=t,val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}\). Also, \(\mathsf{epoch}(\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b'',time=t} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )) \geq\mathsf{epoch}(\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b',time=t} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )) \geq \mathsf{epoch}(t) -2\). Given that clearly \(b'' \succeq \mathsf{block}( \setkeys{chkpattime}{time=t,val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}})\), we have that \(b'' \in \setkeys{filtered}{time=t,val=v} \mathsf{filt}_{\mathsf{hfc}}^{\filtered@Time\ifthenelse{\equal{\filtered@Val}{}}{}{,\filtered@Val}}\).

 ◻

Proof. Condition [def:induction-conditions:is-lmd-confirmed] is trivially satisfied given the Lemma’s assumption.

We now proceed to prove the remaining conditions by bounded induction on \(\mathsf{epoch}(t')\). Let \(v'\) be any honest validator. Lemma 5 implies that at time \(t'\), block \(b\) is in the view of \(v'\).

Base Case: \(\mathsf{epoch}(t') = \mathsf{epoch}(t) \land t' \geq \mathsf{st}(\mathsf{slot}(t))\).

We can apply Lemma 14 to conclude condition [def:induction-conditions:all-validators:not-filtered-out] for this case, i.e., that \(b \in \setkeys{filtered}{time=t',val=v'} \mathsf{filt}_{\mathsf{hfc}}^{\filtered@Time\ifthenelse{\equal{\filtered@Val}{}}{}{,\filtered@Val}}\).

Then, from line [ln:ffg:gjblock-from-previous-epoch], Properties [prop:gasper-basic:only-one-justified-per-epoch] and [prop:gasper-basic:highest-justified-not-from-current-epoch], we have that \(\setkeys{chkpattime}{val=v',time=t'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}} = \mathsf{GJ}(b) = \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}\) which proves [def:induction-conditions:all-validators:gj-succ] for this case.

Hence, we can apply Lemma 10 to conclude that \(b\) is canonical for any validator at any time \(t''\) such that \(\mathsf{epoch}(t'') = \mathsf{epoch}(t') \land t'' \geq \mathsf{st}(\mathsf{slot}(t))\).

Then, thanks to [ln:ffg:call-will-chkp-be-justified-current-epoch], we can apply Lemma 13 and Property [prop:gasper-basic:sufficient-condition-for-no-conflicting-chkps] to conclude no conflicting checkpoint for epoch \(\mathsf{epoch}(b)\) can ever be justified which corresponds to proving condition [def:induction-conditions:no-conflicting] for this case.

[def:induction-conditions:ub] is vacuously satisfied in this case.

Inductive Case: \(\mathsf{epoch}(t') = \mathsf{epoch}(t) + 1\).

Given that no checkpoint for epoch \(\mathsf{epoch}(t')-1\) conflicting with \(b\) can ever be justified, line [ln:ffg:gjblock-from-previous-epoch], Properties [prop:gasper-basic:only-one-justified-per-epoch] and [prop:gasper-basic:highest-justified-not-from-current-epoch] imply that \(\setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}} \in \{\mathsf{GJ}(b), C(b)\}\). This implies that \(\setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}} \succeq \mathsf{GJ}(b)\) as, by definition, \(\mathsf{C}(b) \succeq \mathsf{GJ}(b)\). This proves [def:induction-conditions:all-validators:gj-succ]. From the above, we can also conclude that \(\mathsf{epoch}( \setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}) = \mathsf{epoch}(t) \implies \setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}} = C(b)\). Then, we can apply Lemma 15 to conclude that \(b\) does not get filtered out at any point in epoch \(\mathsf{epoch}(t)\) which concludes the proof for [def:induction-conditions:all-validators:not-filtered-out].

We now have all of the conditions required to apply Lemma 10 to conclude that \(b\) is canonical for any validator at any time during \(\mathsf{epoch}(t')\).

By Properties [prop:gasper-basic:ldm-vote-for-b-is-ffg-vote-for-cb] and [prop:gasper-basic:no-conflicting-if-all-honest-votes-in-support-of-b], this immediately implies that no checkpoint for epoch \(\mathsf{epoch}(t)+1\) conflicting with \(b\) can be ever be justified, which concludes the proof for condition [def:induction-conditions:no-conflicting] as well.

Then, given that \(\mathsf{st}(\mathsf{epoch}(\mathsf{C}(b))+1)=\mathsf{st}(\mathsf{epoch}(b)+1)=\mathsf{st}(\mathsf{epoch}(t)+1)\geq\mathsf{st}(\mathsf{slot}(t)-1)\geq\mathbb{GST}\), due to [ln:ffg:call-will-chkp-be-justified-current-epoch], we can apply Lemma 13 and [assum:ffg-assumptions:justified-checkpoint-next-epoch] to conclude condition [def:induction-conditions:ub] as well.

 ◻

Proof. Let \(v'\) be any honest validator and \(t'\) be any time such that \(\mathsf{epoch}(t') = \mathsf{epoch}(t)\). Lemma 5 implies that \(b\) is in the view of validator \(v'\) at time \(t'\). Due to line [ln:ffg:call-will-chkp-be-justified-prev-epoch], we can apply 13 and [prop:gasper-basic:sufficient-condition-for-no-conflicting-chkps] to conclude that no checkpoint for epoch \(\mathsf{epoch}(t)-1\) conflicting with \(\mathsf{C}(b)\) could ever be justified.

From , we know that there exists a block \(b' \in \setkeys{viewattime}{time=\mathsf{st}(\mathsf{slot}(t)-1),val=v} \mathcal{V}^{\viewattime@Val,\viewattime@Time}\) such that \(b\preceq b' \land \mathsf{epoch}(\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b',time=t} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )) \geq \mathsf{epoch}(t)-2\). Hence, we can apply Lemma 15 to conclude condition [def:induction-conditions:all-validators:not-filtered-out], i.e., \(b \in \setkeys{filtered}{time=t',val=v'} \mathsf{filt}_{\mathsf{hfc}}^{\filtered@Time\ifthenelse{\equal{\filtered@Val}{}}{}{,\filtered@Val}}\). [ln:ffg:check-first-slot] imply that \(t'\geq\mathsf{st}(\mathsf{epoch}(t))\geq\mathsf{st}(\mathsf{first\_slot}(\mathsf{epoch}(t)))\geq\mathsf{st}(\mathsf{slot}(t))\). Then, given that \(b' \in \setkeys{viewattime}{time=\mathsf{st}(\mathsf{slot}(t)-1),val=v} \mathcal{V}^{\viewattime@Val,\viewattime@Time}\) and \(\mathsf{st}(\mathsf{slot}(t)-1)\geq\mathbb{GST}\), we can conclude that \(b\) is in the view of validator \(v'\) at time \(t'\).

Then, because no checkpoint for epoch \(\mathsf{epoch}(t)-1\) conflicting with \(\mathsf{C}(b)\) could ever be justified, Property [prop:gasper-basic:highest-justified-not-from-current-epoch] and the definition of \(\setkeys{chkpattime}{time=t',val=v''} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}\) (Definition 3) imply that \(\setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}} \succeq \setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b',time=t} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )\) proving [def:induction-conditions:all-validators:gj-succ].

Hence, we can now apply Lemma 10 to conclude that \(b\) is canonical in the view of any honest validator at any time during epoch \(\mathsf{epoch}(t)\).

By Properties [prop:gasper-basic:ldm-vote-for-b-is-ffg-vote-for-cb] and [prop:gasper-basic:no-conflicting-if-all-honest-votes-in-support-of-b], the above immediately implies that no checkpoint for epoch \(\mathsf{epoch}(t)\) conflicting with \(b\) can be ever be justified, which concludes the proof for condition [def:induction-conditions:no-conflicting] as well. Finally, given that \(b\) is canonical in the view of any honest validator during the entire epoch \(\mathsf{epoch}(t)\) and that \(\mathsf{st}(\mathsf{epoch}(\mathsf{C}(b))+1)=\mathsf{st}(\mathsf{epoch}(b)+1)=\mathsf{st}(\mathsf{epoch}(t))\geq\mathbb{GST}\), [ln:ffg:call-will-chkp-be-justified-prev-epoch], 13 and [assum:ffg-assumptions:justified-checkpoint-next-epoch] prove condition [def:induction-conditions:ub]. Given that [def:induction-conditions:is-lmd-confirmed] is directly implied by the Lemma’s statement, the proof is concluded. ◻

Proof. We can apply either Lemma 16 or Lemma 17 to conclude that there exists a checkpoint \(C\) such that the Safety Induction Requirements (Definition 9) are satisfied for block \(b\), time \(t\) and a checkpoint \(C\). Then, from this, we apply Lemma 12 to conclude the proof. ◻

Proof. From \(\mathrm{isConfirmed}_v(b,t)\), we know that there exists a block \(b' \succeq b\) and a slot \(s' \in [\mathsf{first\_slot}(\mathsf{epoch}(t)-1)+1,\mathsf{slot}(t)]\) such that \(\setkeys{varforvalattime}{val=v} \ifthenelse {\equal{\varforvalattime@Time} {}} {{\mathrm{isConfirmedNoCaching}}_{\varforvalattime@Val}} {{\mathrm{isConfirmedNoCaching}}_{\varforvalattime@Val}^{\varforvalattime@Time}} \setkeys{varforvalattime}{time=,val=}(b',\mathsf{st}(s'))\).

Given that \(\mathsf{st}(s'-1)\geq\mathsf{st}(\mathsf{epoch}(t)-1)\geq\mathbb{GST}\), we can apply 18 to conclude that \(b'\) is canonical in the view of any honest validator at time \(\mathsf{st}(\mathsf{slot}(t))\geq\mathsf{st}(s)\) and thereafter, which, given that \(b \preceq b'\), implies that \(b\) is also canonical in the view of any honest validator at time \(\mathsf{st}(\mathsf{slot}(t))\) and thereafter. ◻

Proof. \[\begin{align} \frac{2}{3}+\beta\; &= \frac{1}{2}\left(1 + \frac{1}{3}\right) + \beta \\ &= \frac{1}{2}\left(1 + \frac{5}{18\frac{5}{6}}\right) + \beta \\ &> \frac{1}{2}\left(1 + \frac{p}{{E}(1-\beta)}\right) + \beta && \ifthenelse{\equal{5cm}{}}{ \text{--- By applying \Cref{prop:ffg-stronger-justification:p-e} and \Cref{assum:ffg-stronger-justification:beta}, we get \frac{5}{6}<(1-\beta).} }{ \begin{minipage}[t]{5cm} \begin{itemize}[label=---,nosep,left=0pt] \item By applying \Cref{prop:ffg-stronger-justification:p-e} and \Cref{assum:ffg-stronger-justification:beta}, we get \frac{5}{6}<(1-\beta). \end{itemize} \end{minipage} } \end{align}\]  ◻

Proof. The proof is by induction on \(t'\geq t\). We assume that the Lemma is satisfied for all times \(t_p<t'\), and we show that the the Lemma also holds at any time \(t'\) as well. We proceed directly with the inductive argument as this is a total induction and therefore it does not necessitate of an analysis of the base case.

Given that we assume \(\setkeys{varforvalattime}{val=v} \ifthenelse {\equal{\varforvalattime@Time} {}} {{\mathrm{isConfirmed}}_{\varforvalattime@Val}} {{\mathrm{isConfirmed}}_{\varforvalattime@Val}^{\varforvalattime@Time}} \setkeys{varforvalattime}{time=,val=}(b,t_p)\), we know that there exists a block \(b_{t_p}\succeq b\) and a slot \(s_{t_p} \in [\mathsf{first\_slot}(\mathsf{epoch}(t_p)-1)+1,\mathsf{slot}(t_p)]\) such that \(\setkeys{varforvalattime}{val=v} \ifthenelse {\equal{\varforvalattime@Time} {}} {{\mathrm{isConfirmedNoCaching}}_{\varforvalattime@Val}} {{\mathrm{isConfirmedNoCaching}}_{\varforvalattime@Val}^{\varforvalattime@Time}} \setkeys{varforvalattime}{time=,val=}(b_{t_p},\mathsf{st}(s_{t_p}))\).

Now, let \(b' := \setkeys{varforvalattime}{val=v} \ifthenelse {\equal{\varforvalattime@Time} {}} {{\mathrm{highestConfirmedSinceEpoch}}_{\varforvalattime@Val}} {{\mathrm{highestConfirmedSinceEpoch}}_{\varforvalattime@Val}^{\varforvalattime@Time}} \setkeys{varforvalattime}{time=,val=}(\mathsf{epoch}(t')-1,t')\). Then, there exists a slot \(s' \in [\mathsf{first\_slot}(\mathsf{epoch}(t')-1)+1,\mathsf{slot}(t')]\) such that \(\setkeys{varforvalattime}{val=v} \ifthenelse {\equal{\varforvalattime@Time} {}} {{\mathrm{isConfirmedNoCaching}}_{\varforvalattime@Val}} {{\mathrm{isConfirmedNoCaching}}_{\varforvalattime@Val}^{\varforvalattime@Time}} \setkeys{varforvalattime}{time=,val=}(b',\mathsf{st}(s'))\).

Given that \(t_p \geq t\) and \(\mathsf{st}(\mathsf{epoch}(t)-1) \geq \mathbb{GST}\), we can apply 18 to conclude that \(b_{t_p}\) is canonical in the view of any honest validator starting from \(\mathsf{st}(s_{t_p})\), and that both \(b_{t_p}\) and \(b'\) are canonical at time \(\mathsf{st}(\mathsf{slot}(t'))\).

Let us now proceed by cases keeping in mind that, by definition, \(\mathsf{epoch}(t') \in \{\mathsf{epoch}(t_p),\mathsf{epoch}(t_p)+1\}\).

Case 1: \(s_{t_p} \in [\mathsf{first\_slot}(\mathsf{epoch}(t')-1)+1,\mathsf{slot}(t')]\).

This implies that \(\mathsf{slot}(b')\geq\mathsf{slot}(b_{t_p})\). Given that both blocks are canonical for any honest validator at time \(\mathsf{st}(\mathsf{slot}(t'))\), we can conclude that \(b\preceq b_{t_p}\preceq b'\).

Case 2: \(s_{t_p} \notin [\mathsf{first\_slot}(\mathsf{epoch}(t')-1)+1,\mathsf{slot}(t')]\).

This case implies that \(\mathsf{st}(s_{t_p})\leq\mathsf{st}(\mathsf{epoch}(t')-1)\) which further implies that \(\mathsf{epoch}(t_p) = \mathsf{epoch}(t')-1\). Also, given that \(\mathsf{slot}(b_{t_p})<s_{t_p}\), we also have that \(\mathsf{epoch}(b_{t_p}) < \mathsf{epoch}(t')-1 = \mathsf{epoch}(t_p)\). Hence, since \(b_{t_p}\) is canonical in the view of any honest validator from time \(\mathsf{st}(s_{t_p})\) and thereafter, this implies that \(b_{t_p}\) has been canonical in the view of any honest validator for the entirity of epoch \(\mathsf{epoch}(t')-1\).

Given [ln:ffg:call-will-chkp-be-justified-prev-epoch], we can apply 13 to conclude that for any block \(b'' \succeq \mathsf{C}(b_{t_p},\mathsf{epoch}(t_p)-1)\) and time \(t'' \geq \mathsf{st}(\mathsf{epoch}(t_p))\), \(\setkeys{weightofset}{chkp=b''} \left|{ \setkeys{ffgvalsetallsentraw}{source=,target=\mathsf{C}(b_{t_p}),time=t''} \overset{T}{\mathcal{F}}{}_{\ffgvalsetallsentraw@Source \to \ffgvalsetallsentraw@Target}^{\ffgvalsetallsentraw@Time}\setminus \setkeys{slashedset}{chkp=b''} \mathcal{D}^{\slashedset@Chkp}}\right|^{\weightofset@Chkp} \geq \frac{2}{3} \setkeys{totvalsetweight}{chkp=b''} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}\). Hence, we can apply Assumption [assum:ffg-stronger-justification:next-chkp-justified] to conclude that there exists a checkpoint \(C\) such that

1. \(C \succeq b_{t_p}\)

2. \(\mathsf{epoch}(C) = \mathsf{epoch}(t_p) = \mathsf{epoch}(t')-1\)

3. by time \(\mathsf{st}(\mathsf{epoch}(t_p)+1) = \mathsf{st}(\mathsf{epoch}(t'))\), \(\displaystyle \frac{ \setkeys{weightofset}{chkp=C} \left|{ \setkeys{ffgvalsetraw}{source={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{chkp=C} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}, target=C,time=\mathsf{st}(\mathsf{epoch}(t')),val=v} \mathcal{F}_{\ffgvalsetraw@Source \to \ffgvalsetraw@Target}^{\ffgvalsetraw@Val,\ffgvalsetraw@Time}\cap\mathcal{J}}\right|^{\weightofset@Chkp}}{ \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{J}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}} > \mathsf{honFFGratio}(\beta)\),
   where \(\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{chkp=C} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )=\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=\mathsf{block}(C),time=\mathsf{epoch}(C)} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )\)

4. by time \(\mathsf{st}(\mathsf{last\_slot}(\mathsf{epoch}(t_p))) = \mathsf{st}(\mathsf{last\_slot}(\mathsf{epoch}(t')-1))\) the view of validator \(v\) includes a block \(b''\) such that \(b''\succeq C \land \mathsf{epoch}(b'') < \mathsf{epoch}(t') \land \mathsf{C}(b_{t_p},\mathsf{epoch}(t_p)-1) \in \mathsf{AU}(b'')\).

Now we want to show that are satisfied for \(\setkeys{varforvalattime}{val=v} \ifthenelse {\equal{\varforvalattime@Time} {}} {{\mathrm{isConfirmedNoChaching}}_{\varforvalattime@Val}} {{\mathrm{isConfirmedNoChaching}}_{\varforvalattime@Val}^{\varforvalattime@Time}} \setkeys{varforvalattime}{time=,val=}(\mathsf{block}(C),\mathsf{st}(\mathsf{slot}(t')))\).

[ln:ffg:check-first-slot].

The condition at this line follows from \(\mathsf{epoch}(t')=\mathsf{epoch}(t_p)+1\) and \(\mathsf{slot}(t') = \mathsf{slot}(t_p)+1\).

[ln:ffg:call-will-chkp-be-justified-prev-epoch].

Note that \(C = (\mathsf{block}(C),\mathsf{epoch}(t')-1)\). Then proceed as follows. \[\begin{align} \setkeys{weightofset}{chkp=C} \left|{ \setkeys{ffgvalsetraw}{source=, target=C,time=\mathsf{st}(\mathsf{epoch}(t')),val=v} \mathcal{F}_{\ffgvalsetraw@Source \to \ffgvalsetraw@Target}^{\ffgvalsetraw@Val,\ffgvalsetraw@Time}}\right|^{\weightofset@Chkp} &> \mathsf{honFFGratio}(\beta) \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{J}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} && \ifthenelse{\equal{7cm}{}}{ \text{--- By applying condition~\ref{itm:lem:ffg-motonocity:weight-ffg} above.} }{ \begin{minipage}[t]{7cm} \begin{itemize}[label=---,nosep,left=0pt] \item By applying condition~\ref{itm:lem:ffg-motonocity:weight-ffg} above. \end{itemize} \end{minipage} } \\ &\geq \mathsf{honFFGratio}(\beta)(1-\beta) \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} && \ifthenelse{\equal{7cm}{}}{ \text{--- As\setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{J}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}\geq(1-\beta) \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}.} }{ \begin{minipage}[t]{7cm} \begin{itemize}[label=---,nosep,left=0pt] \item As\setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{J}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}\geq(1-\beta) \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}. \end{itemize} \end{minipage} } \\ &\geq \frac{2}{3} \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} + \beta \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} && \ifthenelse{\equal{7cm}{}}{ \text{--- By expanding \mathsf{honFFGratio}(\beta).} }{ \begin{minipage}[t]{7cm} \begin{itemize}[label=---,nosep,left=0pt] \item By expanding \mathsf{honFFGratio}(\beta). \end{itemize} \end{minipage} } \\ &\geq \frac{2}{3} \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} + \min\left( W_e, \beta \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \right) \\ &= \begin{aligned}[t] &\frac{2}{3} \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} + \min\left( W_e, \beta \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \right)\\ &- (1-\beta) \setkeys{commweightfromslot}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromslot}{from=\mathsf{slot}(t'),to=\mathsf{last\_slot}(\mathsf{epoch}(C)),chkp=C} \overline{{\addweight{\mathcal{W}}}}_{\commweightfromslot@From}^{\commweightfromslot@To, \ifthenelse{\equal{\commweightfromslot@Val}{}}{}{\commweightfromslot@Val,} \ifthenelse{\equal{\commweightfromslot@When}{}}{}{\commweightfromslot@When,} \commweightfromslot@ValsetChkp} \end{aligned} && \ifthenelse{\equal{7cm}{}}{ \text{--- As, given that \mathsf{slot}(t')>\mathsf{last\_slot}(\mathsf{epoch}(b)),\setkeys{commweightfromslot}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromslot}{from=\mathsf{slot}(t'),to=\mathsf{last\_slot}(\mathsf{epoch}(C)),chkp=C} \overline{{\addweight{\mathcal{W}}}}_{\commweightfromslot@From}^{\commweightfromslot@To, \ifthenelse{\equal{\commweightfromslot@Val}{}}{}{\commweightfromslot@Val,} \ifthenelse{\equal{\commweightfromslot@When}{}}{}{\commweightfromslot@When,} \commweightfromslot@ValsetChkp} =0.} }{ \begin{minipage}[t]{7cm} \begin{itemize}[label=---,nosep,left=0pt] \item As, given that \mathsf{slot}(t')>\mathsf{last\_slot}(\mathsf{epoch}(b)),\setkeys{commweightfromslot}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromslot}{from=\mathsf{slot}(t'),to=\mathsf{last\_slot}(\mathsf{epoch}(C)),chkp=C} \overline{{\addweight{\mathcal{W}}}}_{\commweightfromslot@From}^{\commweightfromslot@To, \ifthenelse{\equal{\commweightfromslot@Val}{}}{}{\commweightfromslot@Val,} \ifthenelse{\equal{\commweightfromslot@When}{}}{}{\commweightfromslot@When,} \commweightfromslot@ValsetChkp} =0. \end{itemize} \end{minipage} } \end{align}\]

.

Condition [itm:lem:ffg-motonocity:justied-chkp] above implies that there exists a block \(b'' \in \setkeys{viewattime}{val=v,time=\mathsf{st}(\mathsf{epoch}(t'))} \mathcal{V}^{\viewattime@Val,\viewattime@Time}\) such that \(b''\succeq C \land \mathsf{epoch}(b'') < \mathsf{epoch}(t') \land \mathsf{C}(b_{t_p},\mathsf{epoch}(t_p)-1) \in \mathsf{AU}(b'')\). This also implies that \(\mathsf{epoch}(\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b'',time=t'} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )) \geq \mathsf{epoch}(t_p)-1 = \mathsf{epoch}(t')-2\). Note also that \(b'' \succeq C\) and \(\mathsf{block}(C) \succeq b_{t_p}\) imply \(b''\succeq b_{t_p}\). Hence, conditions at are satisfied for \(b''\).

[ln:ffg:is-lmd-confimed-no-caching-prev-epoch].

Let \(b'''\) be any block such that \(b''' \preceq \mathsf{block}(C) \land b''' \neq b_{\texttt{gen}}\). \[\begin{align} \setkeys{indicatorfromblock}{from=b''',to=\mathsf{slot}(t')-1,val=v,when=t',chkp={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b'',time=t'} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}} {{{\mathit{Q}}}}_{\indicatorfromblock@From}^{\indicatorfromblock@To,\indicatorfromblock@Val,\indicatorfromblock@When,\indicatorfromblock@ValSetChkp} \setkeys{indicatorfromblock}{from=,to=,val=,when=,chkp=} &= \frac{ \setkeys{attsetweightfromblock}{from=b''',to=\mathsf{slot}(t')-1,val=v,when=t',chkp={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b'',time=t'} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}} {\addweight{\mathcal{S}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b''',to=\mathsf{slot}(t')-1,chkp={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b'',time=t'} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} \\ &\geq \frac{ \setkeys{weightofset}{chkp={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b'',time=t'} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}} \left|{ \setkeys{ffgvalsetraw}{source={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{chkp=C} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}, target=C,time=\mathsf{st}(\mathsf{epoch}(t')),val=v} \mathcal{F}_{\ffgvalsetraw@Source \to \ffgvalsetraw@Target}^{\ffgvalsetraw@Val,\ffgvalsetraw@Time}\cap\mathcal{J}}\right|^{\weightofset@Chkp}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b''',to=\mathsf{slot}(t')-1,chkp={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b'',time=t'} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} && \ifthenelse{\equal{7cm}{}}{ \text{--- By \Cref{prop:gasper-basic:ffg-vote-for-c-ghost-vote-for-b-succ-c}.} }{ \begin{minipage}[t]{7cm} \begin{itemize}[label=---,nosep,left=0pt] \item By \Cref{prop:gasper-basic:ffg-vote-for-c-ghost-vote-for-b-succ-c}. \end{itemize} \end{minipage} } \\ &\geq \frac{ \setkeys{weightofset}{chkp={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b'',time=t'} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}} \left|{ \setkeys{ffgvalsetraw}{source={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{chkp=C} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}, target=C,time=\mathsf{st}(\mathsf{epoch}(t')),val=v} \mathcal{F}_{\ffgvalsetraw@Source \to \ffgvalsetraw@Target}^{\ffgvalsetraw@Val,\ffgvalsetraw@Time}\cap\mathcal{J}}\right|^{\weightofset@Chkp}(1-\beta)}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b''',to=\mathsf{slot}(t')-1,chkp={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b'',time=t'} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} && \ifthenelse{\equal{7cm}{}}{ \text{--- As, by Assumption~\ref{assum:beta},\setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b''',to=s',chkp={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b'',time=t'} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} \geq \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b''',to=s',chkp={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b'',time=t'} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}(1-\beta).} }{ \begin{minipage}[t]{7cm} \begin{itemize}[label=---,nosep,left=0pt] \item As, by Assumption~\ref{assum:beta},\setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b''',to=s',chkp={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b'',time=t'} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} \geq \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b''',to=s',chkp={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b'',time=t'} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}(1-\beta). \end{itemize} \end{minipage} } \\ &= \frac{ \setkeys{weightofset}{chkp=C} \left|{ \setkeys{ffgvalsetraw}{source={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{chkp=C} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}, target=C,time=\mathsf{st}(\mathsf{epoch}(t')),val=v} \mathcal{F}_{\ffgvalsetraw@Source \to \ffgvalsetraw@Target}^{\ffgvalsetraw@Val,\ffgvalsetraw@Time}\cap\mathcal{J}}\right|^{\weightofset@Chkp}(1-\beta)}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b''',to=\mathsf{slot}(t')-1,chkp=C} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} && \ifthenelse{\equal{7cm}{}}{ \text{--- As honest validators' effective balance never changes.} }{ \begin{minipage}[t]{7cm} \begin{itemize}[label=---,nosep,left=0pt] \item As honest validators' effective balance never changes. \end{itemize} \end{minipage} } \\ &= \frac{ \setkeys{weightofset}{chkp=C} \left|{ \setkeys{ffgvalsetraw}{source={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{chkp=C} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}, target=C,time=\mathsf{st}(\mathsf{epoch}(t')),val=v} \mathcal{F}_{\ffgvalsetraw@Source \to \ffgvalsetraw@Target}^{\ffgvalsetraw@Val,\ffgvalsetraw@Time}\cap\mathcal{J}}\right|^{\weightofset@Chkp}(1-\beta)}{ \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{J}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}} && \ifthenelse{\equal{7cm}{}}{ \text{--- Given that \mathsf{slot}(b''')\leq\mathsf{slot}(b_{t_p})<\mathsf{first\_slot}(\mathsf{epoch}(t_p))=\mathsf{first\_slot}(\mathsf{epoch}(t')-1)\leq\mathsf{last\_slot}(\mathsf{epoch}(t')-1)\leq\mathsf{slot}(t')-1,\setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=\mathsf{epoch}(t_p)} {\hat{\overline{\mathcal{J}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } \subseteq \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b''',to=s',chkp=C} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}. \Cref{assum:no-change-to-the-validator-set} then implies that\setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b''',to=s',chkp=C} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} = \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{J}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}.} }{ \begin{minipage}[t]{7cm} \begin{itemize}[label=---,nosep,left=0pt] \item Given that \mathsf{slot}(b''')\leq\mathsf{slot}(b_{t_p})<\mathsf{first\_slot}(\mathsf{epoch}(t_p))=\mathsf{first\_slot}(\mathsf{epoch}(t')-1)\leq\mathsf{last\_slot}(\mathsf{epoch}(t')-1)\leq\mathsf{slot}(t')-1,\setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=\mathsf{epoch}(t_p)} {\hat{\overline{\mathcal{J}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } \subseteq \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b''',to=s',chkp=C} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}. \Cref{assum:no-change-to-the-validator-set} then implies that\setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b''',to=s',chkp=C} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} = \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{J}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}. \end{itemize} \end{minipage} } \\ &> \mathsf{honFFGratio}(\beta)(1-\beta) && \ifthenelse{\equal{7cm}{}}{ \text{--- By appling condition~\ref{itm:lem:ffg-motonocity:weight-ffg} above.} }{ \begin{minipage}[t]{7cm} \begin{itemize}[label=---,nosep,left=0pt] \item By appling condition~\ref{itm:lem:ffg-motonocity:weight-ffg} above. \end{itemize} \end{minipage} } \\ &\geq \frac{1}{2}\left(1+\frac{p}{(1-\beta){E}}\right)+\beta && \ifthenelse{\equal{7cm}{}}{ \text{--- By appling Lemma~\ref{lem:ffg-hon-ffg-ratio-implies-q}.} }{ \begin{minipage}[t]{7cm} \begin{itemize}[label=---,nosep,left=0pt] \item By appling Lemma~\ref{lem:ffg-hon-ffg-ratio-implies-q}. \end{itemize} \end{minipage} } \\ &= \frac{1}{2}\left(1+\frac{ \setkeys{boostweight}{chkp={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b'',time=t'} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}} W_p^{\boostweight@Chkp}}{(1-\beta) \setkeys{totvalsetweight}{chkp={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b'',time=t'} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}}\right)+\beta && \ifthenelse{\equal{7cm}{}}{ \text{--- As, by definition,\setkeys{boostweight}{chkp={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b'',time=t'} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}} W_p^{\boostweight@Chkp} = \setkeys{totvalsetweight}{chkp={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b'',time=t'} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \frac{p}{{E}} .} }{ \begin{minipage}[t]{7cm} \begin{itemize}[label=---,nosep,left=0pt] \item As, by definition,\setkeys{boostweight}{chkp={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b'',time=t'} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}} W_p^{\boostweight@Chkp} = \setkeys{totvalsetweight}{chkp={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b'',time=t'} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \frac{p}{{E}} . \end{itemize} \end{minipage} } \\ &\geq \frac{1}{2}\left(1+\frac{ \setkeys{boostweight}{chkp={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b'',time=t'} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}} W_p^{\boostweight@Chkp}}{ \setkeys{weightofset}{chkp={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b'',time=t'} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}} \left|{ \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=\mathsf{epoch}(t_p)} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } }\right|^{\weightofset@Chkp}}\right)+\beta && \ifthenelse{\equal{7cm}{}}{ \text{--- As, due to \Cref{assum:no-change-to-the-validator-set},\setkeys{weightofset}{chkp={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b'',time=t'} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}} \left|{ \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=\mathsf{epoch}(t_p)} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } }\right|^{\weightofset@Chkp} \geq (1-\beta) \setkeys{weightofset}{chkp={b_{\texttt{gen}}}} \left|{ \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=b_{\texttt{gen}}} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } }\right|^{\weightofset@Chkp} \geq (1-\beta) \setkeys{totvalsetweight}{chkp={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b'',time=t'} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} } }{ \begin{minipage}[t]{7cm} \begin{itemize}[label=---,nosep,left=0pt] \item As, due to \Cref{assum:no-change-to-the-validator-set},\setkeys{weightofset}{chkp={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b'',time=t'} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}} \left|{ \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=\mathsf{epoch}(t_p)} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } }\right|^{\weightofset@Chkp} \geq (1-\beta) \setkeys{weightofset}{chkp={b_{\texttt{gen}}}} \left|{ \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=b_{\texttt{gen}}} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } }\right|^{\weightofset@Chkp} \geq (1-\beta) \setkeys{totvalsetweight}{chkp={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b'',time=t'} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}\end{itemize} \end{minipage} } \\ &= \frac{1}{2}\left(1+\frac{ \setkeys{boostweight}{chkp={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b'',time=t'} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}} W_p^{\boostweight@Chkp}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b''',to=\mathsf{slot}(t')-1,chkp={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b'',time=t'} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}}\right)+\beta && \ifthenelse{\equal{7cm}{}}{ \text{--- Given that \mathsf{slot}(b''')<\mathsf{first\_slot}(\mathsf{epoch}(t_p))\leq\mathsf{last\_slot}(\mathsf{epoch}(t_p))\leq\mathsf{slot}(t')-1,\setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=\mathsf{epoch}(t_p)} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } \subseteq \setkeys{commfromblock}{from=,to=,val=,when=} \setkeys{commfromblock}{from=b''',to=\mathsf{slot}(t')-1} {\mathcal{W}}_{\commfromblock@From}^{\commfromblock@To \ifthenelse{\equal{\commfromblock@Val}{}}{}{,\commfromblock@Val} \ifthenelse{\equal{\commfromblock@When}{}}{}{,\commfromblock@When} }.} }{ \begin{minipage}[t]{7cm} \begin{itemize}[label=---,nosep,left=0pt] \item Given that \mathsf{slot}(b''')<\mathsf{first\_slot}(\mathsf{epoch}(t_p))\leq\mathsf{last\_slot}(\mathsf{epoch}(t_p))\leq\mathsf{slot}(t')-1,\setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=\mathsf{epoch}(t_p)} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } \subseteq \setkeys{commfromblock}{from=,to=,val=,when=} \setkeys{commfromblock}{from=b''',to=\mathsf{slot}(t')-1} {\mathcal{W}}_{\commfromblock@From}^{\commfromblock@To \ifthenelse{\equal{\commfromblock@Val}{}}{}{,\commfromblock@Val} \ifthenelse{\equal{\commfromblock@When}{}}{}{,\commfromblock@When} }. \end{itemize} \end{minipage} } \end{align}\] Hence, \(\setkeys{varforvalattime}{val=v} \ifthenelse {\equal{\varforvalattime@Time} {}} {{\mathit{isLMDConfirmedNoCaching}}_{\varforvalattime@Val}} {{\mathit{isLMDConfirmedNoCaching}}_{\varforvalattime@Val}^{\varforvalattime@Time}} \setkeys{varforvalattime}{time=,val=}(\mathsf{block}(C),\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b'',time=t'} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} ),t')\) which satisfies line [ln:ffg:is-lmd-confimed-no-caching-prev-epoch].

Above we have show that \(\setkeys{varforvalattime}{val=v} \ifthenelse {\equal{\varforvalattime@Time} {}} {{\mathrm{isConfirmedNoChaching}}_{\varforvalattime@Val}} {{\mathrm{isConfirmedNoChaching}}_{\varforvalattime@Val}^{\varforvalattime@Time}} \setkeys{varforvalattime}{time=,val=}(\mathsf{block}(C),\mathsf{st}(\mathsf{slot}(t'))) =\mathrm{\small True}\) which, by 19, also implies that \(\mathsf{block}(C)\) is canonical for any honest valdiator at time \(\mathsf{st}(\mathsf{slot}(t'))\). Given that \(\setkeys{varforvalattime}{val=v} \ifthenelse {\equal{\varforvalattime@Time} {}} {{\mathrm{isConfirmedNoChaching}}_{\varforvalattime@Val}} {{\mathrm{isConfirmedNoChaching}}_{\varforvalattime@Val}^{\varforvalattime@Time}} \setkeys{varforvalattime}{time=,val=}(b',\mathsf{st}(\mathsf{slot}(t'))) =\mathrm{\small True}\), the above implies that \(\mathsf{slot}(b') \geq \mathsf{slot}(\mathsf{block}(C))\). Then, because \(b'\) is also canonical for any honest validator at time \(\mathsf{st}(\mathsf{slot}(t'))\), we have that \(b \preceq b_{t_p}\preceq \mathsf{block}(C)\preceq b'\).

 ◻

Proof. Note that 6 implies 5. Hence, we can apply [lem:ffg-motonocity,lem:ffg-overall-safety] to conclude the proof. ◻

Proof. Under these assumptions, Gasper ensures that (i) there are never forks, (ii) a block is proposed in each slot, and (iii) any block \(b'\) receives \(\textsf{GHOST}\) votes from the entire committee of \(\mathsf{slot}(b')\). [prop:gasper-basic:ldm-vote-for-b-is-ffg-vote-for-cb] implies then that any checkpoint \(\mathsf{C}(b', \mathsf{epoch}(b'))\) receives FFG votes from the entire validator set and 1 implies that no validator’s effective balance ever changes. Also, as per the Ethereum’s Gasper implementation [6] \(\left(1+\frac{ \setkeys{boostweight}{chkp=C} W_p^{\boostweight@Chkp}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b,to=s,chkp=C} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}}\right)\) is upper bounded by \(0.7\). From this, follows that \(\setkeys{varforvalattime}{val=v} \ifthenelse {\equal{\varforvalattime@Time} {}} {{\mathrm{isConfirmed(b,\mathsf{st}(\mathsf{slot}(b)+1))}}_{\varforvalattime@Val}} {{\mathrm{isConfirmed(b,\mathsf{st}(\mathsf{slot}(b)+1))}}_{\varforvalattime@Val}^{\varforvalattime@Time}} \setkeys{varforvalattime}{time=,val=}\). ◻

Proof. \[\begin{align} \setkeys{totvalsetweight}{chkp=C_{e+1}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} &= \setkeys{weightofset}{chkp=C_{e+1}} \left|{ \setkeys{totvalset}{chkp=C_{e+1}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp}}\right|^{\weightofset@Chkp}\\ &= \setkeys{weightofset}{chkp=C_{e+1}} \left|{ \left( \setkeys{totvalset}{chkp=C_{e+1}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \setminus \setkeys{totvalset}{chkp=C_{e}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \right) \sqcup \left( \setkeys{totvalset}{chkp=C_{e+1}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \cap \setkeys{totvalset}{chkp=C_{e}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \right) }\right|^{\weightofset@Chkp}\\ &= \setkeys{weightofset}{chkp=C_{e+1}} \left|{ \setkeys{totvalset}{chkp=C_{e+1}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \setminus \setkeys{totvalset}{chkp=C_{e}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} }\right|^{\weightofset@Chkp} + \setkeys{weightofset}{chkp=C_{e+1}} \left|{ \setkeys{totvalset}{chkp=C_{e+1}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \cap \setkeys{totvalset}{chkp=C_{e}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} }\right|^{\weightofset@Chkp} \\ &\leq \setkeys{totvalsetweight}{chkp=C_e} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \epsilon + \setkeys{weightofset}{chkp=C_{e+1}} \left|{ \setkeys{totvalset}{chkp=C_{e+1}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \cap \setkeys{totvalset}{chkp=C_{e}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp}}\right|^{\weightofset@Chkp} && \ifthenelse{\equal{}{}}{ \text{--- By applying Property~\ref{prop:changing-balances:entry}} }{ \begin{minipage}[t]{} \begin{itemize}[label=---,nosep,left=0pt] \item By applying Property~\ref{prop:changing-balances:entry} \end{itemize} \end{minipage} } \\ &\leq \setkeys{totvalsetweight}{chkp=C_e} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \epsilon + \setkeys{totvalsetweight}{chkp=C_e} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}(1+\rho) && \ifthenelse{\equal{5cm}{}}{ \text{--- As \left( \setkeys{totvalset}{chkp=C_{e+1}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \cap \setkeys{totvalset}{chkp=C_{e}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp}\right) \subseteq \setkeys{totvalset}{chkp=C_{e}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} and then apply Property~\ref{prop:changing-balances:rewardr}} }{ \begin{minipage}[t]{5cm} \begin{itemize}[label=---,nosep,left=0pt] \item As \left( \setkeys{totvalset}{chkp=C_{e+1}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \cap \setkeys{totvalset}{chkp=C_{e}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp}\right) \subseteq \setkeys{totvalset}{chkp=C_{e}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} and then apply Property~\ref{prop:changing-balances:rewardr} \end{itemize} \end{minipage} } \\ &= \setkeys{totvalsetweight}{chkp=C_e} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}(1+\epsilon+ \rho) && \ifthenelse{\equal{}{}}{ \text{--- Simplification} }{ \begin{minipage}[t]{} \begin{itemize}[label=---,nosep,left=0pt] \item Simplification \end{itemize} \end{minipage} } \end{align}\] ◻

Proof. \[\begin{align} \setkeys{attsetweightfromblock}{from=b',to=s,val=v,when=t,chkp=C_2} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=} &= \setkeys{weightofset}{chkp=C_2} \left|{ \setkeys{attsetfromblock}{from=b',to=s,val=v,when=t,chkp=C_2} {\mathcal{H}}_{\attsetfromblock@From}^{\attsetfromblock@To,\attsetfromblock@Val,\attsetfromblock@When,\attsetfromblock@ValSetChkp} }\right|^{\weightofset@Chkp}\\ &\geq \setkeys{weightofset}{chkp=C_2} \left|{ \setkeys{attsetfromblock}{from=b',to=s,val=v,when=t,chkp=C_1} {\mathcal{H}}_{\attsetfromblock@From}^{\attsetfromblock@To,\attsetfromblock@Val,\attsetfromblock@When,\attsetfromblock@ValSetChkp} \cap \setkeys{attsetfromblock}{from=b',to=s,val=v,when=t,chkp=C_2} {\mathcal{H}}_{\attsetfromblock@From}^{\attsetfromblock@To,\attsetfromblock@Val,\attsetfromblock@When,\attsetfromblock@ValSetChkp} }\right|^{\weightofset@Chkp}\\ &\geq \setkeys{weightofset}{chkp=C_1} \left|{ \setkeys{attsetfromblock}{from=b',to=s,val=v,when=t,chkp=C_1} {\mathcal{H}}_{\attsetfromblock@From}^{\attsetfromblock@To,\attsetfromblock@Val,\attsetfromblock@When,\attsetfromblock@ValSetChkp} \cap \setkeys{attsetfromblock}{from=b',to=s,val=v,when=t,chkp=C_2} {\mathcal{H}}_{\attsetfromblock@From}^{\attsetfromblock@To,\attsetfromblock@Val,\attsetfromblock@When,\attsetfromblock@ValSetChkp} }\right|^{\weightofset@Chkp}(1-\pi) && \ifthenelse{\equal{5cm}{}}{ \text{--- By applying the fact that honest validators never get slashed and \Cref{prop:changing-balances:penr}.} }{ \begin{minipage}[t]{5cm} \begin{itemize}[label=---,nosep,left=0pt] \item By applying the fact that honest validators never get slashed and \Cref{prop:changing-balances:penr}. \end{itemize} \end{minipage} } \\ &= \setkeys{weightofset}{chkp=C_1} \left|{ \setkeys{attsetfromblock}{from=b',to=s,val=v,when=t,chkp=C_1} {\mathcal{H}}_{\attsetfromblock@From}^{\attsetfromblock@To,\attsetfromblock@Val,\attsetfromblock@When,\attsetfromblock@ValSetChkp} \setminus \left( \setkeys{attsetfromblock}{from=b',to=s,val=v,when=t,chkp=C_1} {\mathcal{H}}_{\attsetfromblock@From}^{\attsetfromblock@To,\attsetfromblock@Val,\attsetfromblock@When,\attsetfromblock@ValSetChkp} \setminus \setkeys{attsetfromblock}{from=b',to=s,val=v,when=t,chkp=C_2} {\mathcal{H}}_{\attsetfromblock@From}^{\attsetfromblock@To,\attsetfromblock@Val,\attsetfromblock@When,\attsetfromblock@ValSetChkp} \right)}\right|^{\weightofset@Chkp}(1-\pi)\\ &= \left( \setkeys{attsetweightfromblock}{from=b',to=s,val=v,when=t,chkp=C_1} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=} - \setkeys{weightofset}{chkp=C_1} \left|{ \setkeys{attsetfromblock}{from=b',to=s,val=v,when=t,chkp=C_1} {\mathcal{H}}_{\attsetfromblock@From}^{\attsetfromblock@To,\attsetfromblock@Val,\attsetfromblock@When,\attsetfromblock@ValSetChkp} \setminus \setkeys{attsetfromblock}{from=b',to=s,val=v,when=t,chkp=C_2} {\mathcal{H}}_{\attsetfromblock@From}^{\attsetfromblock@To,\attsetfromblock@Val,\attsetfromblock@When,\attsetfromblock@ValSetChkp} }\right|^{\weightofset@Chkp} \right) (1-\pi) \\ &= \left( \setkeys{attsetweightfromblock}{from=b',to=s,val=v,when=t,chkp=C_1} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=} - \setkeys{weightofset}{chkp=C_1} \left|{ \left( \setkeys{attsetfromblockunfiltered}{from=b',to=s,val=v,when=t} {\mathcal{H}}_{\attsetfromblockunfiltered@From}^{\attsetfromblockunfiltered@To,\attsetfromblockunfiltered@Val,\attsetfromblockunfiltered@When} \setkeys{attsetfromblockunfiltered}{from=,to=,val=,when=} \cap \setkeys{valsetfromblock}{from=b',to=s,chkp=C_1} {\mathcal{J}}_{\valsetfromblock@From}^{\valsetfromblock@To,\valsetfromblock@Chkp} \right) \setminus \left( \setkeys{attsetfromblockunfiltered}{from=b',to=s,val=v,when=t} {\mathcal{H}}_{\attsetfromblockunfiltered@From}^{\attsetfromblockunfiltered@To,\attsetfromblockunfiltered@Val,\attsetfromblockunfiltered@When} \setkeys{attsetfromblockunfiltered}{from=,to=,val=,when=} \cap \setkeys{valsetfromblock}{from=b',to=s,chkp=C_2} {\mathcal{J}}_{\valsetfromblock@From}^{\valsetfromblock@To,\valsetfromblock@Chkp} \right) }\right|^{\weightofset@Chkp} \right) (1-\pi) && \ifthenelse{\equal{}{}}{ \text{--- By definition} }{ \begin{minipage}[t]{} \begin{itemize}[label=---,nosep,left=0pt] \item By definition \end{itemize} \end{minipage} } \\ &= \left( \setkeys{attsetweightfromblock}{from=b',to=s,val=v,when=t,chkp=C_1} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=} - \setkeys{weightofset}{chkp=C_1} \left|{ \left( \setkeys{attsetfromblockunfiltered}{from=b',to=s,val=v,when=t} {\mathcal{H}}_{\attsetfromblockunfiltered@From}^{\attsetfromblockunfiltered@To,\attsetfromblockunfiltered@Val,\attsetfromblockunfiltered@When} \setkeys{attsetfromblockunfiltered}{from=,to=,val=,when=} \cap \setkeys{valsetfromblock}{from=b',to=s,chkp=C_1} {\mathcal{J}}_{\valsetfromblock@From}^{\valsetfromblock@To,\valsetfromblock@Chkp} \right) \setminus \setkeys{valsetfromblock}{from=b',to=s,chkp=C_2} {\mathcal{J}}_{\valsetfromblock@From}^{\valsetfromblock@To,\valsetfromblock@Chkp} }\right|^{\weightofset@Chkp} \right) (1-\pi) && \ifthenelse{\equal{5cm}{}}{ \text{--- As (A \cap B) \setminus (A \cap C) = (A \cap B) \setminus C} }{ \begin{minipage}[t]{5cm} \begin{itemize}[label=---,nosep,left=0pt] \item As (A \cap B) \setminus (A \cap C) = (A \cap B) \setminus C \end{itemize} \end{minipage} } \\ &\geq \left( \setkeys{attsetweightfromblock}{from=b',to=s,val=v,when=t,chkp=C_1} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=} - \setkeys{weightofset}{chkp=C_1} \left|{ \setkeys{valsetfromblock}{from=b',to=s,chkp=C_1} {\mathcal{J}}_{\valsetfromblock@From}^{\valsetfromblock@To,\valsetfromblock@Chkp} \setminus \setkeys{valsetfromblock}{from=b',to=s,chkp=C_2} {\mathcal{J}}_{\valsetfromblock@From}^{\valsetfromblock@To,\valsetfromblock@Chkp} }\right|^{\weightofset@Chkp} \right) (1-\pi) \\ &\geq \left( \setkeys{attsetweightfromblock}{from=b',to=s,val=v,when=t,chkp=C_1} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=} - \setkeys{totvalsetweight}{chkp=C_1} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \epsilon \right) (1-\pi) && \ifthenelse{\equal{}{}}{ \text{--- By \Cref{prop:changing-balances:exitr}.} }{ \begin{minipage}[t]{} \begin{itemize}[label=---,nosep,left=0pt] \item By \Cref{prop:changing-balances:exitr}. \end{itemize} \end{minipage} } \end{align}\]  ◻

Proof. We can follow the same reasoning applied in the proof of 1 to prove that \[\begin{align} \setkeys{indicatorfromblock}{from=b',to=\mathsf{slot}(t')-1,val=v',when=t',chkp=C} {{{\mathit{P}}}}_{\indicatorfromblock@From}^{\indicatorfromblock@To,\indicatorfromblock@Val,\indicatorfromblock@When,\indicatorfromblock@ValSetChkp} \setkeys{indicatorfromblock}{from=,to=,val=,when=,chkp=} \geq \frac{ \setkeys{attsetweightfromblock}{from=b',to=\mathsf{slot}(t)-1,val=v,when=t,chkp=C} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=\mathsf{slot}(t)-1,chkp=C} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} \end{align}\]

Then, by applying Lemma 23 and Property [prop:changing-balances:rewardr], we have that \[\begin{align} \frac{ \setkeys{attsetweightfromblock}{from=b',to=\mathsf{slot}(t)-1,val=v,when=t,chkp=C} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=\mathsf{slot}(t)-1,chkp=C} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} &\geq \frac{ \left( \setkeys{attsetweightfromblock}{from=b',to=\mathsf{slot}(t)-1,val=v,when=t,chkp=C_1} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=} - \setkeys{totvalsetweight}{chkp=C_1} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \epsilon\right)(1-\pi) }{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=\mathsf{slot}(t)-1,chkp=C_1} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}(1+\rho)} \end{align}\] which concludes the proof. ◻

Proof. First, we proceed as follows to work out a lower bound on \(\setkeys{indicatorfromblock}{from=b',to=s,val=v,when=t,chkp=C_1} {{{\mathit{P}}}}_{\indicatorfromblock@From}^{\indicatorfromblock@To,\indicatorfromblock@Val,\indicatorfromblock@When,\indicatorfromblock@ValSetChkp} \setkeys{indicatorfromblock}{from=,to=,val=,when=,chkp=}\).

Then we have, \[\begin{align} \frac{\left( \setkeys{attsetweightfromblock}{from=b',to=s,val=v,when=t,chkp=C_1} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=} - \epsilon \setkeys{totvalsetweight}{chkp=C_1} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}\right)(1-\pi)}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C_1} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}(1+\rho)} &\geq \frac{\left( \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C_1} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} \setkeys{indicatorfromblock}{from=b',to=s,val=v,when=t,chkp=C_1} {{{\mathit{P}}}}_{\indicatorfromblock@From}^{\indicatorfromblock@To,\indicatorfromblock@Val,\indicatorfromblock@When,\indicatorfromblock@ValSetChkp} \setkeys{indicatorfromblock}{from=,to=,val=,when=,chkp=}- \epsilon \setkeys{totvalsetweight}{chkp=C_1} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}\right)(1-\pi)}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C_1} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}(1+\rho)} && \ifthenelse{\equal{6cm}{}}{ \text{--- By definition of\setkeys{indicatorfromblock}{from=b',to=s,val=v,when=t,chkp=C_1} {{{\mathit{P}}}}_{\indicatorfromblock@From}^{\indicatorfromblock@To,\indicatorfromblock@Val,\indicatorfromblock@When,\indicatorfromblock@ValSetChkp} \setkeys{indicatorfromblock}{from=,to=,val=,when=,chkp=}} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item By definition of\setkeys{indicatorfromblock}{from=b',to=s,val=v,when=t,chkp=C_1} {{{\mathit{P}}}}_{\indicatorfromblock@From}^{\indicatorfromblock@To,\indicatorfromblock@Val,\indicatorfromblock@When,\indicatorfromblock@ValSetChkp} \setkeys{indicatorfromblock}{from=,to=,val=,when=,chkp=} \end{itemize} \end{minipage} } \\ &> \frac{1}{2(1-\beta)} \left( 1+ \frac{ \setkeys{boostweight}{chkp=C_2} W_p^{\boostweight@Chkp}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C_1} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} \right) && \ifthenelse{\equal{6cm}{}}{ \text{--- By applying\setkeys{indicatorfromblock}{from=b',to=s,val=v,when=t,chkp=C} {{{\mathit{P}}}}_{\indicatorfromblock@From}^{\indicatorfromblock@To,\indicatorfromblock@Val,\indicatorfromblock@When,\indicatorfromblock@ValSetChkp} \setkeys{indicatorfromblock}{from=,to=,val=,when=,chkp=} > \frac{1+\rho}{2(1-\pi)(1-\beta)} \left( 1+ \frac{ \setkeys{boostweight}{chkp=C_2} W_p^{\boostweight@Chkp} }{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C_1} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} \right) + \frac{\epsilon \setkeys{totvalsetweight}{chkp=C_1} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C_1} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} } }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item By applying\setkeys{indicatorfromblock}{from=b',to=s,val=v,when=t,chkp=C} {{{\mathit{P}}}}_{\indicatorfromblock@From}^{\indicatorfromblock@To,\indicatorfromblock@Val,\indicatorfromblock@When,\indicatorfromblock@ValSetChkp} \setkeys{indicatorfromblock}{from=,to=,val=,when=,chkp=} > \frac{1+\rho}{2(1-\pi)(1-\beta)} \left( 1+ \frac{ \setkeys{boostweight}{chkp=C_2} W_p^{\boostweight@Chkp} }{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C_1} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} \right) + \frac{\epsilon \setkeys{totvalsetweight}{chkp=C_1} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s,chkp=C_1} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}}\end{itemize} \end{minipage} } \end{align}\] ◻

Proof. We proceed by induction on \(t'\) under the condition that \(\forall v'' \in \setkeys{commfromslot}{from=,to=,val=,when=} \setkeys{commfromslot}{from=\mathsf{slot}(t),to=\mathsf{slot}(t')} \overline{\mathcal{J}}_{\commfromslot@From}^{\commfromslot@To \ifthenelse{\equal{\commfromslot@Val}{}}{}{,\commfromslot@Val} \ifthenelse{\equal{\commfromslot@When}{}}{}{,\commfromslot@When} },\, b \in \setkeys{filtered}{time=t',val=v''} \mathsf{filt}_{\mathsf{hfc}}^{\filtered@Time\ifthenelse{\equal{\filtered@Val}{}}{}{,\filtered@Val}}\).

Base case.

This is a strong induction quantified on \(t'\), so there is no need for a base case. Alternatively, we can take \(t' < t\) as base case for which the Lemma is vacuously true.

Inductive step: \(t' \geq t\).

Let \(s':= \mathsf{slot}(t')\), \(v'\) be any honest validator, \(C' := \setkeys{chkpattime}{val=v',time=t'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}\) and \(b'\) be any block such that \(b' \preceq b\). We assume that the Lemma holds for any time \(t''\) such that \(t'' < t'\) and we prove that it holds at time \(t'\) as well.

From condition [itm:lem:lmd-confirmed-changing-balances-part-1:cond-on-latest-j] we have that \[C' \succeq C \land \mathsf{epoch}(C') \leq \mathsf{epoch}(C)+1\]

Hence, given that by the inductive hypothesis all validators in \(\setkeys{commfromslot}{from=,to=,val=,when=} \setkeys{commfromslot}{from=\mathsf{slot}(t),to=\mathsf{slot}(t')-1} \overline{\mathcal{J}}_{\commfromslot@From}^{\commfromslot@To \ifthenelse{\equal{\commfromslot@Val}{}}{}{,\commfromslot@Val} \ifthenelse{\equal{\commfromslot@When}{}}{}{,\commfromslot@When} }\) ¹² GHOST vote for a descendant of \(b\), we can apply Lemma 24 to conclude that \[\frac{ \setkeys{attsetweightfromblock}{from=b',to=s'-1,val=v',when=t',chkp=C'} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s'-1,chkp=C'} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} \geq \frac{ \left( \setkeys{attsetweightfromblock}{from=b',to=\mathsf{slot}(t)-1,val=v,when=t,chkp={C}} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=} - \epsilon \setkeys{totvalsetweight}{chkp={C}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \right) (1-\pi) }{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=\mathsf{slot}(t)-1,chkp={C}} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}(1+\rho)}\].

From Lemma 25, we have that
\[\frac{ \left( \setkeys{attsetweightfromblock}{from=b',to=\mathsf{slot}(t)-1,val=v,when=t,chkp={C}} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=} - \epsilon \setkeys{totvalsetweight}{chkp={C}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \right) (1-\pi) }{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=\mathsf{slot}(t)-1,chkp={C}} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}(1+\rho)} > \frac{1}{2(1-\beta)} \left( 1+ \frac{ \setkeys{boostweight}{chkp={C}} W_p^{\boostweight@Chkp} (1+ \epsilon+ \rho) }{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=\mathsf{slot}(t)-1,chkp={C}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} \right)\]

Hence, \[\label{eqn:thm:lmd-confirmed-changing-balances:1} \frac{ \setkeys{attsetweightfromblock}{from=b',to=s'-1,val=v',when=t',chkp=C'} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s'-1,chkp=C'} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} \geq \frac{\left( \setkeys{attsetweightfromblock}{from=b',to=\mathsf{slot}(t)-1,val=v,when=t,chkp={C}} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=} - \epsilon \setkeys{totvalsetweight}{chkp={C}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}\right)(1-\pi)}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=\mathsf{slot}(t)-1,chkp={C}} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}(1+\rho)} > \frac{1}{2(1-\beta)} \left( 1+ \frac{ \setkeys{boostweight}{chkp=C'} W_p^{\boostweight@Chkp}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=\mathsf{slot}(t)-1,chkp={C}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} \right)\tag{1}\] as, by Lemma 22, \(\setkeys{boostweight}{chkp=C'} W_p^{\boostweight@Chkp} \leq \setkeys{boostweight}{chkp={C}} W_p^{\boostweight@Chkp} (1 + \epsilon+ \rho)\).

Now, let us proceed by cases to show that \(\setkeys{attsetweightfromblock}{from=b',to=s'-1,val=v',when=t',chkp=C'} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=} > \frac{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s'-1,chkp=C'} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} + \setkeys{boostweight}{chkp=C'} W_p^{\boostweight@Chkp}}{2}\).

Case 1.1: \(\setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s'-1,chkp=C'} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} \geq \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=\mathsf{slot}(t)-1,chkp={C}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}\).

In this case we have that \[\frac{ \setkeys{attsetweightfromblock}{from=b',to=s'-1,val=v',when=t',chkp=C'} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s'-1,chkp=C'} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} > \frac{1}{2(1-\beta)} \left( 1+ \frac{ \setkeys{boostweight}{chkp=C'} W_p^{\boostweight@Chkp}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=\mathsf{slot}(t)-1,chkp={C}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} \right) \geq \frac{1}{2(1-\beta)} \left( 1+ \frac{ \setkeys{boostweight}{chkp=C'} W_p^{\boostweight@Chkp} }{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s'-1,chkp=C'} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} \right)\]

Hence, we can apply Lemma 3 to conclude that \[\setkeys{attsetweightfromblock}{from=b',to=s'-1,val=v',when=t',chkp=C'} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=} > \frac{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s'-1,chkp=C'} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} + \setkeys{boostweight}{chkp=C'} W_p^{\boostweight@Chkp}}{2}\].

Case 1.2: \(\setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s'-1,chkp=C'} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} < \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=\mathsf{slot}(t)-1,chkp={C}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}\).

\[\begin{align} \setkeys{attsetweightfromblock}{from=b',to=s'-1,val=v',when=t',chkp=C'} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=} &\geq \left( \setkeys{attsetweightfromblock}{from=b',to=\mathsf{slot}(t')-1,val=v,when=t,chkp={C}} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=}- \epsilon \setkeys{totvalsetweight}{chkp={C}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}\right) (1-\pi) && \ifthenelse{\equal{}{}}{ \text{--- by Lemma~\ref{lem:bound-on-h-next-chkp}} }{ \begin{minipage}[t]{} \begin{itemize}[label=---,nosep,left=0pt] \item by Lemma~\ref{lem:bound-on-h-next-chkp} \end{itemize} \end{minipage} } \\ \\ &> \frac{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=\mathsf{slot}(t')-1,chkp={C}} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}(1+\rho)}{2(1-\beta)} \left( 1+ \frac{ \setkeys{boostweight}{chkp=C'} W_p^{\boostweight@Chkp} }{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=\mathsf{slot}(t')-1,chkp={C}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} \right) && \ifthenelse{\equal{}{}}{ \text{--- by applying \eqref{eqn:thm:lmd-confirmed-changing-balances:1}} }{ \begin{minipage}[t]{} \begin{itemize}[label=---,nosep,left=0pt] \item by applying \eqref{eqn:thm:lmd-confirmed-changing-balances:1} \end{itemize} \end{minipage} } \\ &\geq \frac{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=\mathsf{slot}(t')-1,chkp={C}} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}}{2(1-\beta)} \left( 1+ \frac{ \setkeys{boostweight}{chkp=C'} W_p^{\boostweight@Chkp} }{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=\mathsf{slot}(t')-1,chkp={C}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} \right) && \ifthenelse{\equal{}{}}{ \text{--- as \rho\geq 0} }{ \begin{minipage}[t]{} \begin{itemize}[label=---,nosep,left=0pt] \item as \rho\geq 0 \end{itemize} \end{minipage} } \\ &\geq \frac{(1-\beta) \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=\mathsf{slot}(t')-1,chkp={C}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}}{2(1-\beta)} \left( 1+ \frac{ \setkeys{boostweight}{chkp=C'} W_p^{\boostweight@Chkp} }{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=\mathsf{slot}(t')-1,chkp={C}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} \right) &&\text{ \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/avkmnpri.png}\label{wfoysamu}\end{figure} }\\ &\geq \frac{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s'-1,chkp={C}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} + \setkeys{boostweight}{chkp=C'} W_p^{\boostweight@Chkp}}{2} && \ifthenelse{\equal{}{}}{ \text{--- by simplifications and \mathsf{slot}(t')=s'} }{ \begin{minipage}[t]{} \begin{itemize}[label=---,nosep,left=0pt] \item by simplifications and \mathsf{slot}(t')=s' \end{itemize} \end{minipage} } \\ &\geq \frac{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s'-1,chkp=C'} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} + \setkeys{boostweight}{chkp=C'} W_p^{\boostweight@Chkp}}{2} && \ifthenelse{\equal{}{}}{ \text{--- as\setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s'-1,chkp=C'} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} \leq \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s'-1,chkp={C}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} }{ \begin{minipage}[t]{} \begin{itemize}[label=---,nosep,left=0pt] \item as\setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s'-1,chkp=C'} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} \leq \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s'-1,chkp={C}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} \end{itemize} \end{minipage} } \end{align}\tag{2}\]

Now we can apply 2 to conclude the proof.

 ◻

Proof. Let \(C' := \mathsf{C}(\mathsf{block}(C''),\mathsf{epoch}(C)+1)\). By definition of checkpoints, \(C \preceq C' \preceq C'' \land \mathsf{epoch}(C') = \mathsf{epoch}(C)+1\).

\[\label{eqn:lem:bound-on-exit-two-epoch-distance:1} \begin{align} \setkeys{weightofset}{chkp=C} \left|{ \setkeys{totvalset}{chkp=C} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \setminus \setkeys{totvalset}{chkp=C''} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} }\right|^{\weightofset@Chkp} &= \setkeys{weightofset}{chkp=C} \left|{ \setkeys{totvalset}{chkp=C} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \setminus \setkeys{totvalset}{chkp=C'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} }\right|^{\weightofset@Chkp} + \setkeys{weightofset}{chkp=C} \left|{\left( \setkeys{totvalset}{chkp=C} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \cap \setkeys{totvalset}{chkp=C'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \right)\setminus \setkeys{totvalset}{chkp=C''} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} }\right|^{\weightofset@Chkp}\\ \end{align}\tag{3}\]

\[\label{eqn:lem:bound-on-exit-two-epoch-distance:2} \begin{align} \setkeys{weightofset}{chkp=C} \left|{\left( \setkeys{totvalset}{chkp=C} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \cap \setkeys{totvalset}{chkp=C'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \right)\setminus \setkeys{totvalset}{chkp=C''} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} }\right|^{\weightofset@Chkp} &\leq \frac{ \setkeys{weightofset}{chkp=C'} \left|{\left( \setkeys{totvalset}{chkp=C} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \cap \setkeys{totvalset}{chkp=C'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \right)\setminus \setkeys{totvalset}{chkp=C''} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} }\right|^{\weightofset@Chkp}}{1-\pi} && \ifthenelse{\equal{}{}}{ \text{--- By Property~\ref{prop:changing-balances:penr}} }{ \begin{minipage}[t]{} \begin{itemize}[label=---,nosep,left=0pt] \item By Property~\ref{prop:changing-balances:penr} \end{itemize} \end{minipage} } \\ &\leq \frac{ \setkeys{weightofset}{chkp=C'} \left|{ \setkeys{totvalset}{chkp=C'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \setminus \setkeys{totvalset}{chkp=C''} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} }\right|^{\weightofset@Chkp}}{1-\pi}\\ &\leq \frac{\epsilon \setkeys{weightofset}{chkp=C'} \left|{ \setkeys{totvalset}{chkp=C'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} }\right|^{\weightofset@Chkp}}{1-\pi} && \ifthenelse{\equal{}{}}{ \text{--- By Property~\ref{prop:changing-balances:exitr}} }{ \begin{minipage}[t]{} \begin{itemize}[label=---,nosep,left=0pt] \item By Property~\ref{prop:changing-balances:exitr} \end{itemize} \end{minipage} } \\ &= \frac{\epsilon \left( \setkeys{weightofset}{chkp=C'} \left|{ \setkeys{totvalset}{chkp=C'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \setminus \setkeys{totvalset}{chkp=C} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp}}\right|^{\weightofset@Chkp} + \setkeys{weightofset}{chkp=C'} \left|{\left( \setkeys{totvalset}{chkp=C} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \cap \setkeys{totvalset}{chkp=C'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \right) }\right|^{\weightofset@Chkp} \right) }{1-\pi}\\ &\leq \frac{\epsilon \left( \epsilon \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} + \setkeys{weightofset}{chkp=C'} \left|{\left( \setkeys{totvalset}{chkp=C} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \cap \setkeys{totvalset}{chkp=C'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \right) }\right|^{\weightofset@Chkp} \right) }{1-\pi} && \ifthenelse{\equal{}{}}{ \text{--- By Property~\ref{prop:changing-balances:entry}} }{ \begin{minipage}[t]{} \begin{itemize}[label=---,nosep,left=0pt] \item By Property~\ref{prop:changing-balances:entry} \end{itemize} \end{minipage} } \\ &\leq \frac{\epsilon \left( \epsilon \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} + \setkeys{weightofset}{chkp=C} \left|{\left( \setkeys{totvalset}{chkp=C} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \cap \setkeys{totvalset}{chkp=C'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \right) }\right|^{\weightofset@Chkp} (1+\rho) \right) }{1-\pi} && \ifthenelse{\equal{}{}}{ \text{--- By Property~\ref{prop:changing-balances:rewardr}} }{ \begin{minipage}[t]{} \begin{itemize}[label=---,nosep,left=0pt] \item By Property~\ref{prop:changing-balances:rewardr} \end{itemize} \end{minipage} } \\ &= \frac{\epsilon \left( \epsilon \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} + \setkeys{weightofset}{chkp=C} \left|{ \setkeys{totvalset}{chkp=C} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \setminus \left( \setkeys{totvalset}{chkp=C} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \setminus \setkeys{totvalset}{chkp=C'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \right) }\right|^{\weightofset@Chkp} (1+\rho) \right) }{1-\pi}\\ &= \frac{\epsilon \left( \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}(1+\epsilon+\rho) - \setkeys{weightofset}{chkp=C'} \left|{ \setkeys{totvalset}{chkp=C} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \setminus \setkeys{totvalset}{chkp=C} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} }\right|^{\weightofset@Chkp} (1+\rho) \right) }{1-\pi}\\ \end{align}\tag{4}\]

By combining 3 and 4 , we obtain \[\begin{align} \setkeys{weightofset}{chkp=C} \left|{ \setkeys{totvalset}{chkp=C} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \setminus \setkeys{totvalset}{chkp=C''} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} }\right|^{\weightofset@Chkp} &\leq \frac{ \setkeys{weightofset}{chkp=C} \left|{ \setkeys{totvalset}{chkp=C} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \setminus \setkeys{totvalset}{chkp=C'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} }\right|^{\weightofset@Chkp}\left(1 -\pi- (1+\rho)\epsilon\right) + \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}(1+\epsilon+\rho)\epsilon }{1-\pi} \\ &\leq \frac{ \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}\epsilon\left(1 -\pi- (1+\rho)\epsilon\right) + \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}(1+\epsilon+\rho)\epsilon }{1-\pi} && \ifthenelse{\equal{}{}}{ \text{--- By Properties~\ref{prop:changing-balances:exitr} and \ref{prop:changing-balances:cond-on-penr-exitr-rewardr}} }{ \begin{minipage}[t]{} \begin{itemize}[label=---,nosep,left=0pt] \item By Properties~\ref{prop:changing-balances:exitr} and \ref{prop:changing-balances:cond-on-penr-exitr-rewardr} \end{itemize} \end{minipage} } \\ &= \frac{ \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}\epsilon\left(2 + \rho(1-\epsilon) - \pi\right) }{1-\pi} \end{align}\] ◻

Proof. Let \(s:= \mathsf{slot}(t)\), \(v\) be any honest validator and \(b'\) be any block such that \(b' \preceq b\). First, we can proceed as follows to show that \(\setkeys{attsetweightfromblock}{from=b',to=s-1,val=v,when=t,chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=} \geq \frac{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s'-1,chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} + \setkeys{boostweight}{chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} W_p^{\boostweight@Chkp}}{2}\).

\[\begin{align} \setkeys{attsetweightfromblock}{from=b',to=s-1,val=v,when=t,chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {\addweight{\mathcal{H}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=} &\geq \setkeys{weightofset}{chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \left|{ \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=e} {\hat{\overline{\mathcal{J}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } }\right|^{\weightofset@Chkp} && \ifthenelse{\equal{}{}}{ \text{--- Due to condition \ref{itm:lem:lmd-confirmed-changing-balances-part-2:epoch} \ref{itm:lem:lmd-confirmed-changing-balances-part-2:vote-desc} \ref{itm:lem:lmd-confirmed-changing-balances-part-2:t-ge-e}.} }{ \begin{minipage}[t]{} \begin{itemize}[label=---,nosep,left=0pt] \item Due to condition \ref{itm:lem:lmd-confirmed-changing-balances-part-2:epoch} \ref{itm:lem:lmd-confirmed-changing-balances-part-2:vote-desc} \ref{itm:lem:lmd-confirmed-changing-balances-part-2:t-ge-e}. \end{itemize} \end{minipage} } \\ &\geq \setkeys{weightofset}{chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \left|{ \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch={\mathsf{epoch}( \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}})}} {\hat{\overline{\mathcal{J}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } \cap \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=e} {\hat{\overline{\mathcal{J}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } }\right|^{\weightofset@Chkp}\\ &= \setkeys{weightofset}{chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \left|{ \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch={\mathsf{epoch}( \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}})}} {\hat{\overline{\mathcal{J}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } \setminus \left( \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch={\mathsf{epoch}( \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}})}} {\hat{\overline{\mathcal{J}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } \setminus \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=e} {\hat{\overline{\mathcal{J}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } \right)}\right|^{\weightofset@Chkp}\\ &= \setkeys{commweightatepoch}{epoch=,val=,when=,chkp=} \setkeys{commweightatepoch}{epoch={\mathsf{epoch}( \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}})},chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \hat{\overline{\addweight{\mathcal{J}}}}{}^{\commweightatepoch@Epoch, \ifthenelse{\equal{\commweightatepoch@Val}{}}{}{\commweightatepoch@Val,} \ifthenelse{\equal{\commweightatepoch@When}{}}{}{\commweightatepoch@When,} \commweightatepoch@Chkp } - \setkeys{weightofset}{chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \left|{ \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch={\mathsf{epoch}( \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}})}} {\hat{\overline{\mathcal{J}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } \setminus \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=e} {\hat{\overline{\mathcal{J}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } }\right|^{\weightofset@Chkp}\\ &= \setkeys{totvalsetweight}{chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {\addweight{\mathcal{J}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} - \setkeys{weightofset}{chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \left|{ \setkeys{totvalset}{chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {\mathcal{J}}_{\mathsf{t}}^{\totvalset@Chkp} \setminus \setkeys{totvalset}{chkp={\mathsf{epoch}(\mathsf{block}( \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}),e)}} {\mathcal{J}}_{\mathsf{t}}^{\totvalset@Chkp}}\right|^{\weightofset@Chkp} &&\text{ \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/qxchyvzf.png}\label{fybkuocs}\end{figure} }\\ &\geq \setkeys{totvalsetweight}{chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {\addweight{\mathcal{J}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} - \setkeys{totvalsetweight}{chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}\frac{ \epsilon\left(2 + \rho(1-\epsilon) - \pi\right) }{1-\pi} && \ifthenelse{\equal{}{}}{ \text{--- From \Cref{lem:bound-on-exit-two-epoch-distance}.} }{ \begin{minipage}[t]{} \begin{itemize}[label=---,nosep,left=0pt] \item From \Cref{lem:bound-on-exit-two-epoch-distance}. \end{itemize} \end{minipage} } \\ &\geq \setkeys{totvalsetweight}{chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} (1-\beta) - \setkeys{totvalsetweight}{chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}\frac{ \epsilon\left(2 + \rho(1-\epsilon) - \pi\right) }{1-\pi} && \ifthenelse{\equal{7cm}{}}{ \text{--- As, due to \Cref{assum:beta} and \Cref{prop:wt-we-link},\setkeys{totvalsetweight}{chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {\addweight{\mathcal{J}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \geq \setkeys{totvalsetweight}{chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} (1-\beta).} }{ \begin{minipage}[t]{7cm} \begin{itemize}[label=---,nosep,left=0pt] \item As, due to \Cref{assum:beta} and \Cref{prop:wt-we-link},\setkeys{totvalsetweight}{chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {\addweight{\mathcal{J}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \geq \setkeys{totvalsetweight}{chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} (1-\beta). \end{itemize} \end{minipage} } \\ &= \setkeys{totvalsetweight}{chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \left( 1 - \beta - \frac{ \epsilon\left(2 + \rho(1-\epsilon) - \pi\right) }{1-\pi} \right)\\ &> \setkeys{totvalsetweight}{chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \frac{1}{2}\left(1+\frac{p}{{E}}\right) && \ifthenelse{\equal{7cm}{}}{ \text{--- As Property~\ref{prop:changing-balances:cond-on-p-exitr-beta} and Assumption~\ref{assum:ffg-assumptions:beta} imply that \beta < \frac{1}{3}=\frac{1}{2}-\frac{1}{6}\leq\frac{1}{2} - \frac{p}{2{E}} - \frac{2\epsilon}{1-\pi} - \frac{\epsilon(\rho(1-\epsilon) - \pi)}{1 - \pi} which in turn implies that \left( 1 - \beta - \frac{ \epsilon\left(2 + \rho(1-\epsilon) - \pi\right) }{1-\pi} \right) > \frac{1}{2}\left(1+\frac{p}{{E}}\right) } }{ \begin{minipage}[t]{7cm} \begin{itemize}[label=---,nosep,left=0pt] \item As Property~\ref{prop:changing-balances:cond-on-p-exitr-beta} and Assumption~\ref{assum:ffg-assumptions:beta} imply that \beta < \frac{1}{3}=\frac{1}{2}-\frac{1}{6}\leq\frac{1}{2} - \frac{p}{2{E}} - \frac{2\epsilon}{1-\pi} - \frac{\epsilon(\rho(1-\epsilon) - \pi)}{1 - \pi} which in turn implies that \left( 1 - \beta - \frac{ \epsilon\left(2 + \rho(1-\epsilon) - \pi\right) }{1-\pi} \right) > \frac{1}{2}\left(1+\frac{p}{{E}}\right) \end{itemize} \end{minipage} } \\ &= \frac{ \setkeys{totvalsetweight}{chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} + \setkeys{boostweight}{chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} W_p^{\boostweight@Chkp}}{2} && \ifthenelse{\equal{}{}}{ \text{--- By simplifications and definition of\setkeys{boostweight}{chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} W_p^{\boostweight@Chkp}.} }{ \begin{minipage}[t]{} \begin{itemize}[label=---,nosep,left=0pt] \item By simplifications and definition of\setkeys{boostweight}{chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} W_p^{\boostweight@Chkp}. \end{itemize} \end{minipage} } \\ &\geq \frac{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s'-1,chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} + \setkeys{boostweight}{chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} W_p^{\boostweight@Chkp}}{2} && \ifthenelse{\equal{}{}}{ \text{--- As, by definition,\setkeys{totvalsetweight}{chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \geq \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s'-1,chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} .} }{ \begin{minipage}[t]{} \begin{itemize}[label=---,nosep,left=0pt] \item As, by definition,\setkeys{totvalsetweight}{chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \geq \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=s'-1,chkp={ \setkeys{chkpattime}{val=v,time=t} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} . \end{itemize} \end{minipage} } \end{align}\tag{5}\]

Then, we can apply 2 to conclude the proof. ◻

Proof. We proceed by induction on \(t'\) under the condition that \(\forall v'' \in \setkeys{commfromslot}{from=,to=,val=,when=} \setkeys{commfromslot}{from=\mathsf{slot}(t),to=\mathsf{slot}(t')} \overline{\mathcal{J}}_{\commfromslot@From}^{\commfromslot@To \ifthenelse{\equal{\commfromslot@Val}{}}{}{,\commfromslot@Val} \ifthenelse{\equal{\commfromslot@When}{}}{}{,\commfromslot@When} },\, b \in \setkeys{filtered}{time=t',val=v''} \mathsf{filt}_{\mathsf{hfc}}^{\filtered@Time\ifthenelse{\equal{\filtered@Val}{}}{}{,\filtered@Val}}\).

Base case.

This is a strong induction quantified over \(t'\), so there is no need for a base case. Alternatively, we can take \(t' < t\) as base case for which the Theorem is vacuously true.

Inductive step: \(t' \geq t\).

Let \(s':= \mathsf{slot}(t')\), \(v'\) be any honest validator and \(b'\) be any block such that \(b' \preceq b\). We assume that the Lemma holds for any time \(t''\) such that \(t'' < t'\) and we prove that it holds at time \(t'\) as well.

We distinguish between two cases.

Case 1: \(\mathsf{epoch}(t') \leq \mathsf{epoch}(t)+1\).

Due to Lemma 5, we know that by time \(t'\), \(b\) is in the view of validator \(v'\). Because of this, conditions [itm:thm:lmd-confirmed-changing-balances:cond-on-j] and [itm:thm:lmd-confirmed-changing-balances:cond-on-latest-j] in the Lemma’s statement, and [prop:gasper-basic:highest-justified-not-from-current-epoch], we can conclude that \[\setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}} \succeq C \land \mathsf{epoch}( \setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}) \leq \mathsf{epoch}(C)+1\]

Hence, we can apply Lemma 26 to conclude the proof for this case.

Case 2: \(\mathsf{epoch}(t') > \mathsf{epoch}(t)+1\).

Let \(e := \min(\mathsf{epoch}( \setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}})+2,\mathsf{epoch}(t')-1)\). Observe that conditions [itm:thm:lmd-confirmed-changing-balances-prev-epoch:cond-on-j] and [itm:thm:lmd-confirmed-changing-balances-prev-epoch:cond-on-latest-j], and [prop:gasper-basic:highest-justified-not-from-current-epoch] imply that \(epoch( \setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}) \in [\mathsf{epoch}(t)-1,\mathsf{epoch}(t')-1]\). From the inductive hypothesis, we also know that \(b\) has been canonical for any validator during any epoch in the set \([\mathsf{epoch}(t)+1,\mathsf{epoch}(t')-1]\). Hence, \(b\) has been canonical in the view any honest validator during the entire epoch \(e\). By definition, \(\mathsf{epoch}( \setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}})\leq e \leq \mathsf{epoch}( \setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}})+2\). Also, from the Lemma’s conditions, we have that \(\mathsf{epoch}(b) <e\). This allows us to apply Lemma 28 to conclude the proof for this case.

 ◻

Proof. We proceed by induction on \(t'\) under the condition that \(\forall v'' \in \setkeys{commfromslot}{from=,to=,val=,when=} \setkeys{commfromslot}{from=\mathsf{slot}(t),to=\mathsf{slot}(t')} \overline{\mathcal{J}}_{\commfromslot@From}^{\commfromslot@To \ifthenelse{\equal{\commfromslot@Val}{}}{}{,\commfromslot@Val} \ifthenelse{\equal{\commfromslot@When}{}}{}{,\commfromslot@When} },\, b \in \setkeys{filtered}{time=t',val=v''} \mathsf{filt}_{\mathsf{hfc}}^{\filtered@Time\ifthenelse{\equal{\filtered@Val}{}}{}{,\filtered@Val}}\).

Base case.

This is a strong induction quantified over \(t'\), so there is no need for a base case. Alternatively, we can take \(t' < t\) as base case for which the Theorem is vacuously true.

Inductive step: \(t' \geq t\).

Let \(s':= \mathsf{slot}(t')\), \(v'\) be any honest validator and \(b'\) be any block such that \(b' \preceq b\). We assume that the Lemma holds for any time \(t''\) such that \(t'' < t'\) and we prove that it holds at time \(t'\) as well.

We distinguish between two cases.

Case 1: \(\mathsf{epoch}(t') = \mathsf{epoch}(t)\).

Due to Lemma 5, we know that by time \(t'\), \(b\) is in the view of validator \(v'\). Because of this, conditions [itm:thm:lmd-confirmed-changing-balances-prev-epoch:cond-on-j] and [itm:thm:lmd-confirmed-changing-balances-prev-epoch:cond-on-latest-j], and Property [prop:gasper-basic:highest-justified-not-from-current-epoch] imply that \[\setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}} \succeq C \land \mathsf{epoch}( \setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}) \leq \mathsf{epoch}(C)+1\]

Hence, we can apply Lemma 26 to conclude the proof for this case.

Case 2: \(\mathsf{epoch}(t') > \mathsf{epoch}(t)\).

Let \(e := \min(\mathsf{epoch}( \setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}})+2,\mathsf{epoch}(t')-1)\). Observe that conditions [itm:thm:lmd-confirmed-changing-balances-prev-epoch:cond-on-j] and [itm:thm:lmd-confirmed-changing-balances-prev-epoch:cond-on-latest-j], and [prop:gasper-basic:highest-justified-not-from-current-epoch] imply that \(epoch( \setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}) \in [\mathsf{epoch}(t)-2,\mathsf{epoch}(t')-1]\). Note that \(t = \mathsf{st}(\mathsf{first\_slot}(\mathsf{epoch}(t)))\) implies \(\mathsf{slot}(t) = \mathsf{first\_slot}(\mathsf{epoch}(t))\) which further implies that \(b\) has been canonical for any validator during any epoch in the set \([\mathsf{epoch}(t),\mathsf{epoch}(t')-1]\). Hence, \(b\) has been canonical in the view any honest validator during the entire epoch \(e\). By definition, \(\mathsf{epoch}( \setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}})\leq e \leq \mathsf{epoch}( \setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}})+2\). Also, from the Lemma’s conditions, we have that \(\mathsf{epoch}(b) <e\). This allows us to apply Lemma 28 to conclude the proof for this case.

 ◻

Proof. Direct consequence of [thm:lmd-confirmed-changing-balances,thm:lmd-confirmed-changing-balances-prev-epoch]. ◻

Proof. Let \(C_b := \mathsf{C}(b,e)\) and \(\mathit{VS}_b := \setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b,time=e} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )\), \(t'\) be any time such that \(t' \geq \mathsf{st}(\mathsf{epoch}(b)+1)\), and \(b'\) be any block such that \(b' \succeq \mathsf{C}(b,e)\).

We can now proceed as follows to prove the Lemma. \[\begin{align} \setkeys{weightofset}{chkp=b'} \left|{ \setkeys{ffgvalsetallsentraw}{source=\mathit{VS}_b,target={C_b},time=t'} \overset{T}{\mathcal{F}}{}_{\ffgvalsetallsentraw@Source \to \ffgvalsetallsentraw@Target}^{\ffgvalsetallsentraw@Time}\setminus \setkeys{slashedset}{chkp=b'} \mathcal{D}^{\slashedset@Chkp}}\right|^{\weightofset@Chkp} &\geq \setkeys{weightofset}{chkp=b'} \left|{ \left( \setkeys{ffgvalsettoslot}{to=s-1,source=\mathit{VS}_b,target=C_b,time=t,val=v} \mathcal{F}_{\ffgvalsettoslot@Source \to \ffgvalsettoslot@Target}^{\ffgvalsettoslot@To,\ffgvalsettoslot@Val,\ffgvalsettoslot@Time} \sqcup \setkeys{commfromslot}{from=,to=,val=,when=} \setkeys{commfromslot}{from=s',to=\mathsf{last\_slot}(e)} \overline{\mathcal{J}}_{\commfromslot@From}^{\commfromslot@To \ifthenelse{\equal{\commfromslot@Val}{}}{}{,\commfromslot@Val} \ifthenelse{\equal{\commfromslot@When}{}}{}{,\commfromslot@When} } \right) \setminus \setkeys{slashedset}{chkp=b'} \mathcal{D}^{\slashedset@Chkp} }\right|^{\weightofset@Chkp} && \ifthenelse{\equal{6cm}{}}{ \text{--- Given that t' \geq \mathsf{st}(e+1), by time t' every honest validator in slots [s,\mathsf{last\_slot}(e)] has \textsf{GHOST} voted for a block b'' \succeq b, which, by Property~\ref{prop:gasper-basic:ldm-vote-for-b-is-ffg-vote-for-cb} equates to an FFG vote for\setkeys{ffgvote}{from=\mathit{VS}_b, to=C_b} {\ffgvote@From}\to{\ffgvote@To} . To this, we add the validators whose \textsf{GHOST} votes have already been received at time t. } }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item Given that t' \geq \mathsf{st}(e+1), by time t' every honest validator in slots [s,\mathsf{last\_slot}(e)] has \textsf{GHOST} voted for a block b'' \succeq b, which, by Property~\ref{prop:gasper-basic:ldm-vote-for-b-is-ffg-vote-for-cb} equates to an FFG vote for\setkeys{ffgvote}{from=\mathit{VS}_b, to=C_b} {\ffgvote@From}\to{\ffgvote@To} . To this, we add the validators whose \textsf{GHOST} votes have already been received at time t. \end{itemize} \end{minipage} } \\ &\geq \setkeys{weightofset}{chkp={C_b}} \left|{ \left( \left( \setkeys{ffgvalsettoslot}{to=s-1,source=\mathit{VS}_b,target=C_b,time=t,val=v} \mathcal{F}_{\ffgvalsettoslot@Source \to \ffgvalsettoslot@Target}^{\ffgvalsettoslot@To,\ffgvalsettoslot@Val,\ffgvalsettoslot@Time} \sqcup \setkeys{commfromslot}{from=,to=,val=,when=} \setkeys{commfromslot}{from=s',to=\mathsf{last\_slot}(e)} \overline{\mathcal{J}}_{\commfromslot@From}^{\commfromslot@To \ifthenelse{\equal{\commfromslot@Val}{}}{}{,\commfromslot@Val} \ifthenelse{\equal{\commfromslot@When}{}}{}{,\commfromslot@When} } \right) \setminus \setkeys{slashedset}{chkp=b'} \mathcal{D}^{\slashedset@Chkp} \right) \cap \setkeys{totvalset}{chkp=b'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} }\right|^{\weightofset@Chkp} (1-\pi) && \ifthenelse{\equal{6cm}{}}{ \text{--- By Property~\ref{prop:changing-balances:penr}.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item By Property~\ref{prop:changing-balances:penr}. \end{itemize} \end{minipage} } \\ &= \setkeys{weightofset}{chkp={C_b}} \left|{ \left( \left( \setkeys{ffgvalsettoslot}{to=s-1,source=\mathit{VS}_b,target=C_b,time=t,val=v} \mathcal{F}_{\ffgvalsettoslot@Source \to \ffgvalsettoslot@Target}^{\ffgvalsettoslot@To,\ffgvalsettoslot@Val,\ffgvalsettoslot@Time} \sqcup \setkeys{commfromslot}{from=,to=,val=,when=} \setkeys{commfromslot}{from=s',to=\mathsf{last\_slot}(e)} \overline{\mathcal{J}}_{\commfromslot@From}^{\commfromslot@To \ifthenelse{\equal{\commfromslot@Val}{}}{}{,\commfromslot@Val} \ifthenelse{\equal{\commfromslot@When}{}}{}{,\commfromslot@When} } \right) \cap \setkeys{totvalset}{chkp=b'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \right) \setminus \setkeys{slashedset}{chkp=b'} \mathcal{D}^{\slashedset@Chkp} }\right|^{\weightofset@Chkp} (1-\pi) && \ifthenelse{\equal{6cm}{}}{ \text{--- As (A \setminus B) \cap C = (A \cap C)\setminus B.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item As (A \setminus B) \cap C = (A \cap C)\setminus B. \end{itemize} \end{minipage} } \\ &\geq \left( \setkeys{weightofset}{chkp={C_b}} \left|{ \left( \setkeys{ffgvalsettoslot}{to=s-1,source=\mathit{VS}_b,target=C_b,time=t,val=v} \mathcal{F}_{\ffgvalsettoslot@Source \to \ffgvalsettoslot@Target}^{\ffgvalsettoslot@To,\ffgvalsettoslot@Val,\ffgvalsettoslot@Time} \sqcup \setkeys{commfromslot}{from=,to=,val=,when=} \setkeys{commfromslot}{from=s',to=\mathsf{last\_slot}(e)} \overline{\mathcal{J}}_{\commfromslot@From}^{\commfromslot@To \ifthenelse{\equal{\commfromslot@Val}{}}{}{,\commfromslot@Val} \ifthenelse{\equal{\commfromslot@When}{}}{}{,\commfromslot@When} } \right) \cap \setkeys{totvalset}{chkp=b'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} }\right|^{\weightofset@Chkp} - \setkeys{weightofset}{chkp={C_b}} \left|{ \setkeys{slashedset}{chkp=b'} \mathcal{D}^{\slashedset@Chkp}}\right|^{\weightofset@Chkp} \right) (1-\pi) \\ &= \begin{aligned}[t] &\left( \setkeys{weightofset}{chkp={C_b}} \left|{ \setkeys{ffgvalsettoslot}{to=s-1,source=\mathit{VS}_b,target=C_b,time=t,val=v} \mathcal{F}_{\ffgvalsettoslot@Source \to \ffgvalsettoslot@Target}^{\ffgvalsettoslot@To,\ffgvalsettoslot@Val,\ffgvalsettoslot@Time} \sqcup \setkeys{commfromslot}{from=,to=,val=,when=} \setkeys{commfromslot}{from=s',to=\mathsf{last\_slot}(e)} \overline{\mathcal{J}}_{\commfromslot@From}^{\commfromslot@To \ifthenelse{\equal{\commfromslot@Val}{}}{}{,\commfromslot@Val} \ifthenelse{\equal{\commfromslot@When}{}}{}{,\commfromslot@When} } }\right|^{\weightofset@Chkp}\right.\\ &\left.\quad- \setkeys{weightofset}{chkp={C_b}} \left|{ \left( \setkeys{ffgvalsettoslot}{to=s-1,source=\mathit{VS}_b,target=C_b,time=t,val=v} \mathcal{F}_{\ffgvalsettoslot@Source \to \ffgvalsettoslot@Target}^{\ffgvalsettoslot@To,\ffgvalsettoslot@Val,\ffgvalsettoslot@Time} \sqcup \setkeys{commfromslot}{from=,to=,val=,when=} \setkeys{commfromslot}{from=s',to=\mathsf{last\_slot}(e)} \overline{\mathcal{J}}_{\commfromslot@From}^{\commfromslot@To \ifthenelse{\equal{\commfromslot@Val}{}}{}{,\commfromslot@Val} \ifthenelse{\equal{\commfromslot@When}{}}{}{,\commfromslot@When} }\right) \setminus \setkeys{totvalset}{chkp=b'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} }\right|^{\weightofset@Chkp} - \setkeys{weightofset}{chkp={C_b}} \left|{ \setkeys{slashedset}{chkp=b'} \mathcal{D}^{\slashedset@Chkp}}\right|^{\weightofset@Chkp} \right) (1-\pi) \end{aligned} && \ifthenelse{\equal{6cm}{}}{ \text{--- As A \cap B = A \setminus (A \setminus B).} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item As A \cap B = A \setminus (A \setminus B). \end{itemize} \end{minipage} } \\ &\geq \begin{align}[t] &\left( \setkeys{weightofset}{chkp={C_b}} \left|{ \setkeys{ffgvalsettoslot}{to=s-1,source=\mathit{VS}_b,target=C_b,time=t,val=v} \mathcal{F}_{\ffgvalsettoslot@Source \to \ffgvalsettoslot@Target}^{\ffgvalsettoslot@To,\ffgvalsettoslot@Val,\ffgvalsettoslot@Time} \sqcup \setkeys{commfromslot}{from=,to=,val=,when=} \setkeys{commfromslot}{from=s',to=\mathsf{last\_slot}(e)} \overline{\mathcal{J}}_{\commfromslot@From}^{\commfromslot@To \ifthenelse{\equal{\commfromslot@Val}{}}{}{,\commfromslot@Val} \ifthenelse{\equal{\commfromslot@When}{}}{}{,\commfromslot@When} } }\right|^{\weightofset@Chkp}\right.\\ &\left.\quad- \setkeys{weightofset}{chkp={C_b}} \left|{ \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=e} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } \setminus \setkeys{totvalset}{chkp=b'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} }\right|^{\weightofset@Chkp} - \setkeys{weightofset}{chkp={C_b}} \left|{ \setkeys{slashedset}{chkp=b'} \mathcal{D}^{\slashedset@Chkp}}\right|^{\weightofset@Chkp} \right) (1-\pi) \end{align} && \ifthenelse{\equal{6cm}{}}{ \text{--- As \left( \setkeys{ffgvalsettoslot}{to=s-1,source=\mathit{VS}_b,target=C_b,time=t,val=v} \mathcal{F}_{\ffgvalsettoslot@Source \to \ffgvalsettoslot@Target}^{\ffgvalsettoslot@To,\ffgvalsettoslot@Val,\ffgvalsettoslot@Time} \sqcup \setkeys{commfromslot}{from=,to=,val=,when=} \setkeys{commfromslot}{from=s',to=\mathsf{last\_slot}(e)} \overline{\mathcal{J}}_{\commfromslot@From}^{\commfromslot@To \ifthenelse{\equal{\commfromslot@Val}{}}{}{,\commfromslot@Val} \ifthenelse{\equal{\commfromslot@When}{}}{}{,\commfromslot@When} }\right)\subseteq \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=e} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } .} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item As \left( \setkeys{ffgvalsettoslot}{to=s-1,source=\mathit{VS}_b,target=C_b,time=t,val=v} \mathcal{F}_{\ffgvalsettoslot@Source \to \ffgvalsettoslot@Target}^{\ffgvalsettoslot@To,\ffgvalsettoslot@Val,\ffgvalsettoslot@Time} \sqcup \setkeys{commfromslot}{from=,to=,val=,when=} \setkeys{commfromslot}{from=s',to=\mathsf{last\_slot}(e)} \overline{\mathcal{J}}_{\commfromslot@From}^{\commfromslot@To \ifthenelse{\equal{\commfromslot@Val}{}}{}{,\commfromslot@Val} \ifthenelse{\equal{\commfromslot@When}{}}{}{,\commfromslot@When} }\right)\subseteq \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=e} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } . \end{itemize} \end{minipage} } \\ &= \begin{align}[t] &\left( \setkeys{weightofset}{chkp={C_b}} \left|{ \setkeys{ffgvalsettoslot}{to=s-1,source=\mathit{VS}_b,target=C_b,time=t,val=v} \mathcal{F}_{\ffgvalsettoslot@Source \to \ffgvalsettoslot@Target}^{\ffgvalsettoslot@To,\ffgvalsettoslot@Val,\ffgvalsettoslot@Time} \sqcup \setkeys{commfromslot}{from=,to=,val=,when=} \setkeys{commfromslot}{from=s',to=\mathsf{last\_slot}(e)} \overline{\mathcal{J}}_{\commfromslot@From}^{\commfromslot@To \ifthenelse{\equal{\commfromslot@Val}{}}{}{,\commfromslot@Val} \ifthenelse{\equal{\commfromslot@When}{}}{}{,\commfromslot@When} } }\right|^{\weightofset@Chkp}\right.\\ &\left.\quad- \setkeys{weightofset}{chkp={C_b}} \left|{ \setkeys{totvalset}{chkp=C_b} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \setminus \setkeys{totvalset}{chkp=b'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} }\right|^{\weightofset@Chkp} - \setkeys{weightofset}{chkp={C_b}} \left|{ \setkeys{slashedset}{chkp=b'} \mathcal{D}^{\slashedset@Chkp}}\right|^{\weightofset@Chkp} \right) (1-\pi) \end{align} && \ifthenelse{\equal{6cm}{}}{ \text{--- As, due to Condition~\ref{itm:lem:sufficient-condition-for-justification-changing-balances:totvalset-commepoch} of the Lemma's statement and \Cref{prop:wt-we-link},\setkeys{totvalset}{chkp=C_b} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} = \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=e} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } .} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item As, due to Condition~\ref{itm:lem:sufficient-condition-for-justification-changing-balances:totvalset-commepoch} of the Lemma's statement and \Cref{prop:wt-we-link},\setkeys{totvalset}{chkp=C_b} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} = \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=e} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } . \end{itemize} \end{minipage} } \\ &\geq \begin{aligned}[t] &\left( \setkeys{ffgvalsettoslotweight}{to=s-1,source=\mathit{VS}_b,target=C_b,time=t,val=v,weight chkp=C_b} F_{\ffgvalsettoslotweight@Source \to \ffgvalsettoslotweight@Target}^{\ffgvalsettoslotweight@To, \ffgvalsettoslotweight@Val,\ffgvalsettoslotweight@Time,\ffgvalsettoslotweight@WeightChkp} + \setkeys{commweightfromslot}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromslot}{from=s',to=\mathsf{last\_slot}(e),chkp=C_b} \overline{{\addweight{\mathcal{J}}}}_{\commweightfromslot@From}^{\commweightfromslot@To, \ifthenelse{\equal{\commweightfromslot@Val}{}}{}{\commweightfromslot@Val,} \ifthenelse{\equal{\commweightfromslot@When}{}}{}{\commweightfromslot@When,} \commweightfromslot@ValsetChkp} \right.\\ &\left.\quad- \setkeys{weightofset}{chkp={C_b}} \left|{ \setkeys{totvalset}{chkp=C_b} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \setminus \setkeys{totvalset}{chkp=b'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} }\right|^{\weightofset@Chkp} - \setkeys{weightofset}{chkp={C_b}} \left|{ \setkeys{slashedset}{chkp=b'} \mathcal{D}^{\slashedset@Chkp}}\right|^{\weightofset@Chkp} \right) (1-\pi) \end{aligned} \\ &\geq \begin{align}[t] &\left( \setkeys{totvalsetweight}{chkp={\mathsf{C}(b,e)}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \left(\frac{2}{3}\frac{1+\rho-\epsilon\rho}{1-\pi} +\epsilon\right)\right.\\ &\quad + \min\left( W_e, \beta \setkeys{totvalsetweight}{chkp=C_b} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \right)\\ &\quad-(1-\beta) \setkeys{valsetweightfromslot}{from=s',to=\mathsf{last\_slot}(e),chkp=C_b} \overline{{\addweight{\mathcal{W}}}}_{\valsetweightfromslot@From}^{\valsetweightfromslot@To,\valsetweightfromslot@ValsetChkp\ifthenelse{\equal{\valsetweightfromslot@WeightChkp}{}}{}{,\valsetweightfromslot@WeightChkp}} + \setkeys{commweightfromslot}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromslot}{from=s',to=\mathsf{last\_slot}(e),chkp=C_b} \overline{{\addweight{\mathcal{J}}}}_{\commweightfromslot@From}^{\commweightfromslot@To, \ifthenelse{\equal{\commweightfromslot@Val}{}}{}{\commweightfromslot@Val,} \ifthenelse{\equal{\commweightfromslot@When}{}}{}{\commweightfromslot@When,} \commweightfromslot@ValsetChkp} \\ &\left.\quad- \setkeys{weightofset}{chkp={C_b}} \left|{ \setkeys{totvalset}{chkp=C_b} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \setminus \setkeys{totvalset}{chkp=b'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} }\right|^{\weightofset@Chkp} - \setkeys{weightofset}{chkp={C_b}} \left|{ \setkeys{slashedset}{chkp=b'} \mathcal{D}^{\slashedset@Chkp}}\right|^{\weightofset@Chkp} \right) (1-\pi) \end{align} && \ifthenelse{\equal{6cm}{}}{ \text{--- By applying Condition~\ref{itm:lem:sufficient-condition-for-justification-changing-balances:f-cond} of the Lemma's statement.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item By applying Condition~\ref{itm:lem:sufficient-condition-for-justification-changing-balances:f-cond} of the Lemma's statement. \end{itemize} \end{minipage} } \\ &\geq \begin{align}[t] &\left( \setkeys{totvalsetweight}{chkp={\mathsf{C}(b,e)}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \left(\frac{2}{3}\frac{1+\rho-\epsilon\rho}{1-\pi} +\epsilon\right)\right.\\ \\ &\left.\quad- \setkeys{weightofset}{chkp={C_b}} \left|{ \setkeys{totvalset}{chkp=C_b} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \setminus \setkeys{totvalset}{chkp=b'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} }\right|^{\weightofset@Chkp} \right) (1-\pi) \end{align} && \ifthenelse{\equal{6cm}{}}{ \text{--- By \Cref{assum:beta},\setkeys{commweightfromslot}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromslot}{from=s',to=\mathsf{last\_slot}(e),chkp=C_b} \overline{{\addweight{\mathcal{J}}}}_{\commweightfromslot@From}^{\commweightfromslot@To, \ifthenelse{\equal{\commweightfromslot@Val}{}}{}{\commweightfromslot@Val,} \ifthenelse{\equal{\commweightfromslot@When}{}}{}{\commweightfromslot@When,} \commweightfromslot@ValsetChkp}\geq(1-\beta) \setkeys{valsetweightfromslot}{from=s',to=\mathsf{last\_slot}(e),chkp=C_b} \overline{{\addweight{\mathcal{W}}}}_{\valsetweightfromslot@From}^{\valsetweightfromslot@To,\valsetweightfromslot@ValsetChkp\ifthenelse{\equal{\valsetweightfromslot@WeightChkp}{}}{}{,\valsetweightfromslot@WeightChkp}}, and, by \Cref{assum:ffg-assumptions:max-slashing} and the fact that honest validators never get slashed, \min\left( W_e, \beta \setkeys{totvalsetweight}{chkp=C_b} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \right)\geq \setkeys{weightofset}{chkp={C_b}} \left|{ \setkeys{slashedset}{chkp=b'} \mathcal{D}^{\slashedset@Chkp}}\right|^{\weightofset@Chkp}.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item By \Cref{assum:beta},\setkeys{commweightfromslot}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromslot}{from=s',to=\mathsf{last\_slot}(e),chkp=C_b} \overline{{\addweight{\mathcal{J}}}}_{\commweightfromslot@From}^{\commweightfromslot@To, \ifthenelse{\equal{\commweightfromslot@Val}{}}{}{\commweightfromslot@Val,} \ifthenelse{\equal{\commweightfromslot@When}{}}{}{\commweightfromslot@When,} \commweightfromslot@ValsetChkp}\geq(1-\beta) \setkeys{valsetweightfromslot}{from=s',to=\mathsf{last\_slot}(e),chkp=C_b} \overline{{\addweight{\mathcal{W}}}}_{\valsetweightfromslot@From}^{\valsetweightfromslot@To,\valsetweightfromslot@ValsetChkp\ifthenelse{\equal{\valsetweightfromslot@WeightChkp}{}}{}{,\valsetweightfromslot@WeightChkp}}, and, by \Cref{assum:ffg-assumptions:max-slashing} and the fact that honest validators never get slashed, \min\left( W_e, \beta \setkeys{totvalsetweight}{chkp=C_b} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \right)\geq \setkeys{weightofset}{chkp={C_b}} \left|{ \setkeys{slashedset}{chkp=b'} \mathcal{D}^{\slashedset@Chkp}}\right|^{\weightofset@Chkp}. \end{itemize} \end{minipage} } \\ &= \begin{aligned}[t] & \setkeys{totvalsetweight}{chkp=C_b} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \left(\frac{2}{3}(1+\rho-\epsilon\rho) +\epsilon(1-\pi)\right)\\ &- \setkeys{weightofset}{chkp={C_b}} \left|{ \setkeys{totvalset}{chkp=C_b} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \setminus \setkeys{totvalset}{chkp=b'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} }\right|^{\weightofset@Chkp} (1-\pi) \end{aligned} && \ifthenelse{\equal{6cm}{}}{ \text{--- Simplification.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item Simplification. \end{itemize} \end{minipage} } \\ &= \begin{align}[t] & \setkeys{totvalsetweight}{chkp=C_b} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \left(\frac{2}{3}(1+\rho) +\epsilon\left(\frac{2}{3}-\frac{2}{3}+1-\pi-\frac{2}{3}\rho\right)\right)\\ &- \setkeys{weightofset}{chkp={C_b}} \left|{ \setkeys{totvalset}{chkp=C_b} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \setminus \setkeys{totvalset}{chkp=b'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} }\right|^{\weightofset@Chkp} (1-\pi) \end{align} && \ifthenelse{\equal{6cm}{}}{ \text{--- Terms manipulation.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item Terms manipulation. \end{itemize} \end{minipage} } \\ &= \begin{align}[t] & \setkeys{totvalsetweight}{chkp=C_b} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \left(\frac{2}{3}(1+\rho+\epsilon) +\left(1-\pi-\frac{2}{3}(1+\rho)\right)\right)\\ &- \setkeys{weightofset}{chkp={C_b}} \left|{ \setkeys{totvalset}{chkp=C_b} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \setminus \setkeys{totvalset}{chkp=b'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} }\right|^{\weightofset@Chkp} (1-\pi) \end{align} && \ifthenelse{\equal{6cm}{}}{ \text{--- Simplification.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item Simplification. \end{itemize} \end{minipage} } \\ &= \begin{aligned}[t] & \setkeys{totvalsetweight}{chkp=C_b} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \left(\frac{2}{3}(1+\rho+\epsilon)\right.\\ &\qquad\;\;\;\left.+ \setkeys{weightofset}{chkp={C_b}} \left|{ \setkeys{totvalset}{chkp=C_b} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \setminus \setkeys{totvalset}{chkp=b'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} }\right|^{\weightofset@Chkp}\left(1-\pi-\frac{2}{3}(1+\rho)\right)\right)\\ &- \setkeys{weightofset}{chkp={C_b}} \left|{ \setkeys{totvalset}{chkp=C_b} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \setminus \setkeys{totvalset}{chkp=b'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} }\right|^{\weightofset@Chkp} (1-\pi) \end{aligned} && \ifthenelse{\equal{6cm}{}}{ \text{--- Due to Properties~\ref{prop:changing-balances:exitr}, \ref{prop:changing-balances:cond-on-exitr} and~\ref{prop:changing-balances:cond-on-penr-exitr-rewardr}.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item Due to Properties~\ref{prop:changing-balances:exitr}, \ref{prop:changing-balances:cond-on-exitr} and~\ref{prop:changing-balances:cond-on-penr-exitr-rewardr}. \end{itemize} \end{minipage} } \\ &= \setkeys{totvalsetweight}{chkp=C_b} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \frac{2}{3}(1+\rho+\epsilon) - \setkeys{weightofset}{chkp={C_b}} \left|{ \setkeys{totvalset}{chkp=C_b} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \setminus \setkeys{totvalset}{chkp=b'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} }\right|^{\weightofset@Chkp} \frac{2}{3}(1+\rho) && \ifthenelse{\equal{6cm}{}}{ \text{--- Simplification.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item Simplification. \end{itemize} \end{minipage} } \\ &= \frac{2}{3} \left(\left( \setkeys{totvalsetweight}{chkp=C_b} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} - \setkeys{weightofset}{chkp={C_b}} \left|{ \setkeys{totvalset}{chkp=C_b} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \setminus \setkeys{totvalset}{chkp=b'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} }\right|^{\weightofset@Chkp}\right) (1+\rho) + \setkeys{totvalsetweight}{chkp=C_b} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \epsilon\right) && \ifthenelse{\equal{6cm}{}}{ \text{--- Simplification.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item Simplification. \end{itemize} \end{minipage} } \\ &= \frac{2}{3} \left( \setkeys{weightofset}{chkp={C_b}} \left|{ \setkeys{totvalset}{chkp=C_b} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \cap \setkeys{totvalset}{chkp=b'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} }\right|^{\weightofset@Chkp} (1+\rho) + \setkeys{totvalsetweight}{chkp=C_b} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \epsilon\right) && \ifthenelse{\equal{6cm}{}}{ \text{--- As A \setminus (A \setminus B) = A \cap B.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item As A \setminus (A \setminus B) = A \cap B. \end{itemize} \end{minipage} } \\ &\geq \frac{2}{3} \left( \setkeys{weightofset}{chkp={b'}} \left|{ \setkeys{totvalset}{chkp=C_b} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \cap \setkeys{totvalset}{chkp=b'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} }\right|^{\weightofset@Chkp} + \setkeys{weightofset}{chkp={b'}} \left|{ \setkeys{totvalset}{chkp=b'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \setminus \setkeys{totvalset}{chkp=C_b} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} }\right|^{\weightofset@Chkp} \right) && \ifthenelse{\equal{6cm}{}}{ \text{--- By applying Properties~\ref{prop:changing-balances:entry} and~\ref{prop:changing-balances:rewardr}.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item By applying Properties~\ref{prop:changing-balances:entry} and~\ref{prop:changing-balances:rewardr}. \end{itemize} \end{minipage} } \\ &= \frac{2}{3} \setkeys{totvalsetweight}{chkp=b'} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} && \ifthenelse{\equal{6cm}{}}{ \text{--- Simplification.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item Simplification. \end{itemize} \end{minipage} } \end{align}\]  ◻

Proof. Below we show that by

1. referring to 7 rather than 5

2. dropping 1

3. replacing

   1. 9 with with 11

   2. 10 with 31

   3. 13 with 32

the proofs of [ lem:no-filtered-out-if-no-conflicting-checkpoints, lem:ffg-safety-from-sir, lem:no-filt-gj-prev-epoch, lem:no-filt-guj-two-epochs-ago, lem:base-case-current-epoch-for-safety-of-confirmation-ffg, lem:base-case-prev-epoch-for-safety-of-confirmation-ffg, lem:ffg-no-chaching-safety, lem:ffg-overall-safety ]{reference-type=“ref+Label” reference=” lem:no-filtered-out-if-no-conflicting-checkpoints, lem:ffg-safety-from-sir, lem:no-filt-gj-prev-epoch, lem:no-filt-guj-two-epochs-ago, lem:base-case-current-epoch-for-safety-of-confirmation-ffg, lem:base-case-prev-epoch-for-safety-of-confirmation-ffg, lem:ffg-no-chaching-safety, lem:ffg-overall-safety “} still holds.

11.

Not affected by the change.

12.

10 is used twice in the proof of this Lemma. Given that [def:general-induction-conditions:is-lmd-confirmed] is replaced with [def:general-induction-conditions:is-lmd-confirmed], it is then possible to apply 31 in place of 10 both times. Also, 13 is not used in the proof of this Lemma.

[lem:no-filt-gj-prev-epoch,lem:no-filt-guj-two-epochs-ago].

Unaffected.

16.

10 is used twice in the proof of this Lemma. In both cases we have that \(\mathsf{epoch}(t) = \mathsf{epoch}(b) \land \mathsf{epoch}(\mathsf{GJ}(b)) = \mathsf{epoch}(t)-1\). Hence, in both cases it is possible to apply 31.

13 is also used twice in the proof of this Lemma. Note that by following the reasoning outlined in the proof of 16, it is easy to show that for any time \(t'\) such that \(\mathsf{epoch}(t') \in \{\mathsf{epoch}(t),\mathsf{epoch}(t)+1\}\), \(\mathsf{C}(b) \succeq \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}\). Hence, in both cases, it is possible to apply 32 in place of 13 to reach the same conclusion.

The above also implies that [def:general-induction-conditions:is-lmd-confirmed] is satisfied.

17.

10 is used once in the proof of this Lemma. In the proof of this Lemma have that \(\mathsf{epoch}(b) < \mathsf{epoch}(t) \land \mathsf{epoch}(\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b',time=t} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )) \geq \mathsf{epoch}(t)-2 \land t = \mathsf{st}(\mathsf{epoch}(t))\). Hence, it is possible to apply 31.

13 is also used once in the proof of this Lemma. Note that by following the reasoning outlined in the proof of 16, it is easy to show that for any time \(t'\) such that \(\mathsf{epoch}(t') = \mathsf{epoch}(t)\), \(\mathsf{C}(b) \succeq \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}\). Hence, it is possible to apply 32 in place of 13 to reach the same conclusion.

The above also implies that [def:general-induction-conditions:is-lmd-confirmed] is satisfied.

[lem:ffg-no-chaching-safety,lem:ffg-overall-safety].

Unaffected.

 ◻

Proof. \[\begin{align} \setkeys{totvalsetweight}{chkp=C_{e+1}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} &\geq \setkeys{weightofset}{chkp=C_{e+1}} \left|{ \setkeys{totvalset}{chkp=C_{e+1}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \cap \setkeys{totvalset}{chkp=C_{e}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp}}\right|^{\weightofset@Chkp} \\ &\geq \setkeys{weightofset}{chkp=C_{e}} \left|{ \setkeys{totvalset}{chkp=C_{e+1}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \cap \setkeys{totvalset}{chkp=C_{e}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp}}\right|^{\weightofset@Chkp}(1-\sigma) && \ifthenelse{\equal{5cm}{}}{ \text{--- By Properties~\ref{prop:changing-balances:penr}, \ref{prop:changing-balances:slashing} and~\ref{prop:changing-balances:slashpenr-higher-than-penr}.} }{ \begin{minipage}[t]{5cm} \begin{itemize}[label=---,nosep,left=0pt] \item By Properties~\ref{prop:changing-balances:penr}, \ref{prop:changing-balances:slashing} and~\ref{prop:changing-balances:slashpenr-higher-than-penr}. \end{itemize} \end{minipage} } \\ &= \setkeys{weightofset}{chkp=C_{e}} \left|{ \setkeys{totvalset}{chkp=C_{e}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \setminus ( \setkeys{totvalset}{chkp=C_{e}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \setminus \setkeys{totvalset}{chkp=C_{e+1}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp})}\right|^{\weightofset@Chkp}(1-\pi) \\ &\geq \setkeys{totvalsetweight}{chkp=C_{e}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}(1-\epsilon)(1-\sigma) && \ifthenelse{\equal{5cm}{}}{ \text{--- By \Cref{prop:changing-balances:exitr}.} }{ \begin{minipage}[t]{5cm} \begin{itemize}[label=---,nosep,left=0pt] \item By \Cref{prop:changing-balances:exitr}. \end{itemize} \end{minipage} } \end{align}\]  ◻

Proof. This proof is very similar to the proof of 21. The proof is by induction on \(t'\geq t\). We assume that the Lemma is satisfied for all times \(t_p<t'\) such that \(\mathsf{slot}(t')-1<\mathsf{slot}(t_p)\), and we show that the the Lemma also holds at any time \(t'\) as well. We proceed directly with the inductive argument as this is a total induction and therefore it does not necessitate of an analysis of the base case.

Given that we assume \(\setkeys{varforvalattime}{val=v} \ifthenelse {\equal{\varforvalattime@Time} {}} {{\mathrm{isConfirmed}}_{\varforvalattime@Val}} {{\mathrm{isConfirmed}}_{\varforvalattime@Val}^{\varforvalattime@Time}} \setkeys{varforvalattime}{time=,val=}(b,t_p)\), we know that there exists a block \(b_{t_p}\succeq b\) and a slot \(s_{t_p} \in [\mathsf{first\_slot}(\mathsf{epoch}(t_p)-1)+1,\mathsf{slot}(t_p)]\) such that \(\setkeys{varforvalattime}{val=v} \ifthenelse {\equal{\varforvalattime@Time} {}} {{\mathrm{isConfirmedNoCaching}}_{\varforvalattime@Val}} {{\mathrm{isConfirmedNoCaching}}_{\varforvalattime@Val}^{\varforvalattime@Time}} \setkeys{varforvalattime}{time=,val=}(b_{t_p},\mathsf{st}(s_{t_p}))\).

Now, let \(b' := \setkeys{varforvalattime}{val=v} \ifthenelse {\equal{\varforvalattime@Time} {}} {{\mathrm{highestConfirmedSinceEpoch}}_{\varforvalattime@Val}} {{\mathrm{highestConfirmedSinceEpoch}}_{\varforvalattime@Val}^{\varforvalattime@Time}} \setkeys{varforvalattime}{time=,val=}(\mathsf{epoch}(t')-1,t')\). Then, there exists a slot \(s' \in [\mathsf{first\_slot}(\mathsf{epoch}(t')-1)+1,\mathsf{slot}(t')]\) such that \(\setkeys{varforvalattime}{val=v} \ifthenelse {\equal{\varforvalattime@Time} {}} {{\mathrm{isConfirmedNoCaching}}_{\varforvalattime@Val}} {{\mathrm{isConfirmedNoCaching}}_{\varforvalattime@Val}^{\varforvalattime@Time}} \setkeys{varforvalattime}{time=,val=}(b',\mathsf{st}(s'))\).

Given that \(t_p \geq t\) and \(\mathsf{st}(\mathsf{epoch}(t)-1) \geq \mathbb{GST}\), we can apply 33 to conclude that both \(b_{t_p}\) is canonical in the view of any honest validator starting from \(\mathsf{st}(s_{t_p})\), and that both \(b'\) and \(b'\) are canonical at time \(\mathsf{st}(\mathsf{slot}(t'))\).

Let us now proceed by cases keeping in mind that, by definition, \(\mathsf{epoch}(t') \in \{\mathsf{epoch}(t_p),\mathsf{epoch}(t_p)+1\}\).

Case 1: \(s_{t_p} \in [\mathsf{first\_slot}(\mathsf{epoch}(t')-1)+1,\mathsf{slot}(t')]\).

This implies that \(\mathsf{slot}(b')\geq\mathsf{slot}(b_{t_p})\). Given that both blocks are canonical for any honest validator at time \(\mathsf{st}(\mathsf{slot}(t'))\), we can conclude that \(b\preceq b_{t_p}\preceq b'\).

Case 2: \(s_{t_p} \notin [\mathsf{first\_slot}(\mathsf{epoch}(t')-1)+1,\mathsf{slot}(t')]\).

By following the same reasoning used in the proof of the same case on 21, we can show that there exists a checkpoint \(C\) such that

1. \(C \succeq b_{t_p}\)

2. \(\mathsf{epoch}(C) = \mathsf{epoch}(t_p) = \mathsf{epoch}(t')-1\)

3. by time \(\mathsf{st}(\mathsf{epoch}(t_p)+1) = \mathsf{st}(\mathsf{epoch}(t'))\), \(\displaystyle \frac{ \setkeys{weightofset}{chkp=C} \left|{ \setkeys{ffgvalsetraw}{source={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{chkp=C} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}, target=C,time=\mathsf{st}(\mathsf{epoch}(t')),val=v} \mathcal{F}_{\ffgvalsetraw@Source \to \ffgvalsetraw@Target}^{\ffgvalsetraw@Val,\ffgvalsetraw@Time}\cap\mathcal{J}}\right|^{\weightofset@Chkp}}{ \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{J}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}} > \mathsf{honFFGratio}(\beta)\),
   where \(\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{chkp=C} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )=\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=\mathsf{block}(C),time=\mathsf{epoch}(C)} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )\)

4. by time \(\mathsf{st}(\mathsf{last\_slot}(\mathsf{epoch}(t_p))) = \mathsf{st}(\mathsf{last\_slot}(\mathsf{epoch}(t')-1))\) the view of validator \(v\) includes a block \(b''\) such that \(b''\succeq C \land \mathsf{epoch}(b'') < \mathsf{epoch}(t') \land \mathsf{C}(b_{t_p},\mathsf{epoch}(t_p)-1) \in \mathsf{AU}(b'')\).

Now we want to show that are satisfied for \(\setkeys{varforvalattime}{val=v} \ifthenelse {\equal{\varforvalattime@Time} {}} {{\mathrm{isConfirmedNoChaching}}_{\varforvalattime@Val}} {{\mathrm{isConfirmedNoChaching}}_{\varforvalattime@Val}^{\varforvalattime@Time}} \setkeys{varforvalattime}{time=,val=}(\mathsf{block}(C),\mathsf{st}(\mathsf{slot}(t')))\) which, as shown in the proof of 21, is sufficient to conclude the proof.

Lines  and [ln:ffg-changing-balances:exists] to [ln:ffg-changing-balances:prev-epoch-vs-check].

Same as the proof for lines  and [ln:ffg:exists] to [ln:ffg:prev-epoch-vs-check] in 21.

[ln:ffg-changing-balances:call-will-chkp-be-justified-prev-epoch]

Note that \(C = (\mathsf{block}(C),\mathsf{epoch}(t')-1)\). Then proceed as follows. \[\begin{align} \setkeys{weightofset}{chkp=C} \left|{ \setkeys{ffgvalsetraw}{source=, target=C,time=\mathsf{st}(\mathsf{epoch}(t')),val=v} \mathcal{F}_{\ffgvalsetraw@Source \to \ffgvalsetraw@Target}^{\ffgvalsetraw@Val,\ffgvalsetraw@Time}}\right|^{\weightofset@Chkp} &> \mathsf{honFFGratioVar}(\beta) \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{J}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} && \ifthenelse{\equal{7cm}{}}{ \text{--- By applying condition 3 above.} }{ \begin{minipage}[t]{7cm} \begin{itemize}[label=---,nosep,left=0pt] \item By applying condition 3 above. \end{itemize} \end{minipage} } \\ &\geq \mathsf{honFFGratioVar}(\beta)(1-\beta) \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} && \ifthenelse{\equal{7cm}{}}{ \text{--- As\setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{J}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}\geq(1-\beta) \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}.} }{ \begin{minipage}[t]{7cm} \begin{itemize}[label=---,nosep,left=0pt] \item As\setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{J}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}\geq(1-\beta) \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}. \end{itemize} \end{minipage} } \\ &\geq \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}\left(\frac{2}{3}\left(\frac{1+\rho-\epsilon\rho}{1-\pi}\right)+\epsilon\right) + \beta \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} && \ifthenelse{\equal{7cm}{}}{ \text{--- By expanding \mathsf{honFFGratioVar}(\beta).} }{ \begin{minipage}[t]{7cm} \begin{itemize}[label=---,nosep,left=0pt] \item By expanding \mathsf{honFFGratioVar}(\beta). \end{itemize} \end{minipage} } \\ &\geq \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}\left(\frac{2}{3}\left(\frac{1+\rho-\epsilon\rho}{1-\pi}\right)+\epsilon\right) + \min\left( W_e, \beta \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \right) \\ &= \begin{aligned}[t] & \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}\left(\frac{2}{3}\left(\frac{1+\rho-\epsilon\rho}{1-\pi}\right)+\epsilon\right) + \min\left( W_e, \beta \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \right)\\ &- (1-\beta) \setkeys{commweightfromslot}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromslot}{from=\mathsf{slot}(t'),to=\mathsf{last\_slot}(\mathsf{epoch}(C)),chkp=C} \overline{{\addweight{\mathcal{W}}}}_{\commweightfromslot@From}^{\commweightfromslot@To, \ifthenelse{\equal{\commweightfromslot@Val}{}}{}{\commweightfromslot@Val,} \ifthenelse{\equal{\commweightfromslot@When}{}}{}{\commweightfromslot@When,} \commweightfromslot@ValsetChkp} \end{aligned} && \ifthenelse{\equal{7cm}{}}{ \text{--- As, given that \mathsf{slot}(t')>\mathsf{last\_slot}(\mathsf{epoch}(b)),\setkeys{commweightfromslot}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromslot}{from=\mathsf{slot}(t'),to=\mathsf{last\_slot}(\mathsf{epoch}(C)),chkp=C} \overline{{\addweight{\mathcal{W}}}}_{\commweightfromslot@From}^{\commweightfromslot@To, \ifthenelse{\equal{\commweightfromslot@Val}{}}{}{\commweightfromslot@Val,} \ifthenelse{\equal{\commweightfromslot@When}{}}{}{\commweightfromslot@When,} \commweightfromslot@ValsetChkp} =0.} }{ \begin{minipage}[t]{7cm} \begin{itemize}[label=---,nosep,left=0pt] \item As, given that \mathsf{slot}(t')>\mathsf{last\_slot}(\mathsf{epoch}(b)),\setkeys{commweightfromslot}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromslot}{from=\mathsf{slot}(t'),to=\mathsf{last\_slot}(\mathsf{epoch}(C)),chkp=C} \overline{{\addweight{\mathcal{W}}}}_{\commweightfromslot@From}^{\commweightfromslot@To, \ifthenelse{\equal{\commweightfromslot@Val}{}}{}{\commweightfromslot@Val,} \ifthenelse{\equal{\commweightfromslot@When}{}}{}{\commweightfromslot@When,} \commweightfromslot@ValsetChkp} =0. \end{itemize} \end{minipage} } \end{align}\]

[ln:ffg-changing-balances:is-lmd-confimed-no-caching-prev-epoch].

To reduce that size of the expressions below, let \(e':= \mathsf{epoch}(t')\), \(e^{\mathit{tp}} := \mathsf{epoch}(t_p)\) and \(C_{\mathsf{vs}}:=\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{blck=b'',time=t'} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )\). Note also that the definition of \(\setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}\) (3), the conclusion reached in the case above, 32 and [prop:gasper-basic:highest-justified-not-from-current-epoch] imply that \(\setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}\preceq C\) and \(C_{\mathsf{vs}}\preceq C\).

Then, we can proceed as follows.

\[\begin{align} & \setkeys{indicatorfromblock}{from=b''',to=\mathsf{slot}(t')-1,val=v,when=t',chkp={C_{\mathsf{vs}}}} {{{\mathit{Q}}}}_{\indicatorfromblock@From}^{\indicatorfromblock@To,\indicatorfromblock@Val,\indicatorfromblock@When,\indicatorfromblock@ValSetChkp} \setkeys{indicatorfromblock}{from=,to=,val=,when=,chkp=}\\ &= \frac{ \setkeys{attsetweightfromblock}{from=b''',to=\mathsf{slot}(t')-1,val=v,when=t',chkp={C_{\mathsf{vs}}}} {\addweight{\mathcal{S}}}_{\attsetweightfromblock@From}^{\attsetweightfromblock@To,\attsetweightfromblock@Val,\attsetweightfromblock@When,\attsetweightfromblock@ValSetChkp} \setkeys{attsetweightfromblock}{from=,to=,val=,when=,chkp=}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b''',to=\mathsf{slot}(t')-1,chkp={C_{\mathsf{vs}}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} \\ &\geq \frac{ \setkeys{weightofset}{chkp={C_{\mathsf{vs}}}} \left|{ \setkeys{ffgvalsetraw}{source={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{chkp=C} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}, target=C,time=\mathsf{st}(e'),val=v} \mathcal{F}_{\ffgvalsetraw@Source \to \ffgvalsetraw@Target}^{\ffgvalsetraw@Val,\ffgvalsetraw@Time}\cap\mathcal{J}}\right|^{\weightofset@Chkp}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b''',to=\mathsf{slot}(t')-1,chkp={C_{\mathsf{vs}}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} && \ifthenelse{\equal{6cm}{}}{ \text{--- By \Cref{prop:gasper-basic:ffg-vote-for-c-ghost-vote-for-b-succ-c}.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item By \Cref{prop:gasper-basic:ffg-vote-for-c-ghost-vote-for-b-succ-c}. \end{itemize} \end{minipage} } \\ &\geq \frac{ \setkeys{weightofset}{chkp={C_{\mathsf{vs}}}} \left|{ \setkeys{ffgvalsetraw}{source={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{chkp=C} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}, target=C,time=\mathsf{st}(e'),val=v} \mathcal{F}_{\ffgvalsetraw@Source \to \ffgvalsetraw@Target}^{\ffgvalsetraw@Val,\ffgvalsetraw@Time}\cap\mathcal{J}}\right|^{\weightofset@Chkp}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b''',to=\mathsf{slot}(t')-1,chkp={C_{\mathsf{vs}}}} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} (1-\beta) && \ifthenelse{\equal{6cm}{}}{ \text{--- As, by Assumption~\ref{assum:beta},\setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b''',to=s',chkp={C_{\mathsf{vs}}}} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} \geq \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b''',to=s',chkp={C_{\mathsf{vs}}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}(1-\beta).} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item As, by Assumption~\ref{assum:beta},\setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b''',to=s',chkp={C_{\mathsf{vs}}}} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp} \geq \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b''',to=s',chkp={C_{\mathsf{vs}}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}(1-\beta). \end{itemize} \end{minipage} } \\ &= \frac{ \setkeys{weightofset}{chkp={C_{\mathsf{vs}}}} \left|{ \setkeys{ffgvalsetraw}{source={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{chkp=C} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}, target=C,time=\mathsf{st}(e'),val=v} \mathcal{F}_{\ffgvalsetraw@Source \to \ffgvalsetraw@Target}^{\ffgvalsetraw@Val,\ffgvalsetraw@Time}\cap\mathcal{J}}\right|^{\weightofset@Chkp}}{ \setkeys{commweightatepoch}{epoch=,val=,when=,chkp=} \setkeys{commweightatepoch}{epoch=e^{\mathit{tp}},chkp=C_{\mathsf{vs}}} \hat{\overline{\addweight{\mathcal{J}}}}{}^{\commweightatepoch@Epoch, \ifthenelse{\equal{\commweightatepoch@Val}{}}{}{\commweightatepoch@Val,} \ifthenelse{\equal{\commweightatepoch@When}{}}{}{\commweightatepoch@When,} \commweightatepoch@Chkp } } (1-\beta) && \ifthenelse{\equal{6cm}{}}{ \text{--- Given that {\mathsf{ps}^{+1}}(b''')\leq{\mathsf{ps}^{+1}}(b'')<\mathsf{first\_slot}(\mathsf{epoch}(t_p))\leq\mathsf{last\_slot}(\mathsf{epoch}(t_p))\leq\mathsf{slot}(t')-1,\setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=\mathsf{epoch}(t_p)} {\hat{\overline{\mathcal{J}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } = \setkeys{commfromblock}{from=,to=,val=,when=} \setkeys{commfromblock}{from=b''',to=\mathsf{slot}(t')-1} {\mathcal{J}}_{\commfromblock@From}^{\commfromblock@To \ifthenelse{\equal{\commfromblock@Val}{}}{}{,\commfromblock@Val} \ifthenelse{\equal{\commfromblock@When}{}}{}{,\commfromblock@When} }} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item Given that {\mathsf{ps}^{+1}}(b''')\leq{\mathsf{ps}^{+1}}(b'')<\mathsf{first\_slot}(\mathsf{epoch}(t_p))\leq\mathsf{last\_slot}(\mathsf{epoch}(t_p))\leq\mathsf{slot}(t')-1,\setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=\mathsf{epoch}(t_p)} {\hat{\overline{\mathcal{J}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } = \setkeys{commfromblock}{from=,to=,val=,when=} \setkeys{commfromblock}{from=b''',to=\mathsf{slot}(t')-1} {\mathcal{J}}_{\commfromblock@From}^{\commfromblock@To \ifthenelse{\equal{\commfromblock@Val}{}}{}{,\commfromblock@Val} \ifthenelse{\equal{\commfromblock@When}{}}{}{,\commfromblock@When} } \end{itemize} \end{minipage} } \\ &\geq \frac{ \setkeys{weightofset}{chkp={C_{\mathsf{vs}}}} \left|{ \setkeys{ffgvalsetraw}{source={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{chkp=C} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}, target=C,time=\mathsf{st}(e'),val=v} \mathcal{F}_{\ffgvalsetraw@Source \to \ffgvalsetraw@Target}^{\ffgvalsetraw@Val,\ffgvalsetraw@Time}\cap\mathcal{J}}\right|^{\weightofset@Chkp}}{ \setkeys{weightofset}{chkp=C_{\mathsf{vs}}} \left|{ \setkeys{totvalset}{chkp=C} {\mathcal{J}}_{\mathsf{t}}^{\totvalset@Chkp}}\right|^{\weightofset@Chkp}} (1-\beta) && \ifthenelse{\equal{6cm}{}}{ \text{--- As, due to \mathsf{epoch}(C)=e^{\mathit{tp}} and \Cref{prop:wt-we-link},\setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=e^{\mathit{tp}}} {\hat{\overline{\mathcal{J}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } = \setkeys{totvalset}{chkp=C} {\mathcal{J}}_{\mathsf{t}}^{\totvalset@Chkp}.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item As, due to \mathsf{epoch}(C)=e^{\mathit{tp}} and \Cref{prop:wt-we-link},\setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=e^{\mathit{tp}}} {\hat{\overline{\mathcal{J}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } = \setkeys{totvalset}{chkp=C} {\mathcal{J}}_{\mathsf{t}}^{\totvalset@Chkp}. \end{itemize} \end{minipage} } \\ &\geq \frac{ \setkeys{weightofset}{chkp={C_{\mathsf{vs}}}} \left|{ \setkeys{ffgvalsetraw}{source={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{chkp=C} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}, target=C,time=\mathsf{st}(e'),val=v} \mathcal{F}_{\ffgvalsetraw@Source \to \ffgvalsetraw@Target}^{\ffgvalsetraw@Val,\ffgvalsetraw@Time}\cap\mathcal{J}}\right|^{\weightofset@Chkp}}{ \setkeys{totvalsetweight}{chkp=C_{\mathsf{vs}}} {\addweight{\mathcal{J}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}} (1-\beta) && \ifthenelse{\equal{6cm}{}}{ \text{--- As, by definition,\setkeys{weightofset}{chkp=C_{\mathsf{vs}}} \left|{ \setkeys{totvalset}{chkp=C} {\mathcal{J}}_{\mathsf{t}}^{\totvalset@Chkp}}\right|^{\weightofset@Chkp}\leq \setkeys{totvalsetweight}{chkp=C_{\mathsf{vs}}} {\addweight{\mathcal{J}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item As, by definition,\setkeys{weightofset}{chkp=C_{\mathsf{vs}}} \left|{ \setkeys{totvalset}{chkp=C} {\mathcal{J}}_{\mathsf{t}}^{\totvalset@Chkp}}\right|^{\weightofset@Chkp}\leq \setkeys{totvalsetweight}{chkp=C_{\mathsf{vs}}} {\addweight{\mathcal{J}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}. \end{itemize} \end{minipage} } \\ &\geq \frac{ \setkeys{weightofset}{chkp={C_{\mathsf{vs}}}} \left|{ \setkeys{ffgvalsetraw}{source={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{chkp=C} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}, target=C,time=\mathsf{st}(e'),val=v} \mathcal{F}_{\ffgvalsetraw@Source \to \ffgvalsetraw@Target}^{\ffgvalsetraw@Val,\ffgvalsetraw@Time}\cap\mathcal{J}}\right|^{\weightofset@Chkp}(1-\pi)(1-\epsilon)}{ \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{J}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}} (1-\beta) && \ifthenelse{\equal{6cm}{}}{ \text{--- As, due to \Cref{lem:lower-bound-tot-next-chkp},\\ \setkeys{totvalsetweight}{chkp=C_{\mathsf{vs}}} {\addweight{\mathcal{J}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}(1-\pi)(1-\epsilon)\leq \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{J}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item As, due to \Cref{lem:lower-bound-tot-next-chkp},\\ \setkeys{totvalsetweight}{chkp=C_{\mathsf{vs}}} {\addweight{\mathcal{J}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}(1-\pi)(1-\epsilon)\leq \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{J}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}. \end{itemize} \end{minipage} } \\ &\geq \frac{ \setkeys{weightofset}{chkp=C} \left|{ \setkeys{ffgvalsetraw}{source={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{chkp=C} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}, target=C,time=\mathsf{st}(e'),val=v} \mathcal{F}_{\ffgvalsetraw@Source \to \ffgvalsetraw@Target}^{\ffgvalsetraw@Val,\ffgvalsetraw@Time}\cap\mathcal{J}\cap \setkeys{totvalset}{chkp=C_{\mathsf{vs}}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp}}\right|^{\weightofset@Chkp}(1-\pi)(1-\epsilon)}{(1+\rho) \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{J}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}} (1-\beta) && \ifthenelse{\equal{6cm}{}}{ \text{--- As, for a given set \mathcal{X},\\ \begin{aligned}[t] & \setkeys{weightofset}{chkp=C_{\mathsf{vs}}} \left|{\mathcal{X}}\right|^{\weightofset@Chkp}\\ &\geq \setkeys{weightofset}{chkp=C_{\mathsf{vs}}} \left|{\mathcal{X}\cap \setkeys{totvalset}{chkp=C} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp}\cap \setkeys{totvalset}{chkp=C_{\mathsf{vs}}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp}}\right|^{\weightofset@Chkp}\\ &\geq \frac{ \setkeys{weightofset}{chkp=C} \left|{\mathcal{X}\cap \setkeys{totvalset}{chkp=C} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp}\cap \setkeys{totvalset}{chkp=C_{\mathsf{vs}}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp}}\right|^{\weightofset@Chkp}}{1+\rho} && \ifthenelse{\equal{1cm}{}}{ \text{--- by \Cref{prop:changing-balances:rewardr}} }{ \begin{minipage}[t]{1cm} \begin{itemize}[label=---,nosep,left=0pt] \item by \Cref{prop:changing-balances:rewardr} \end{itemize} \end{minipage} } \\ &=\frac{ \setkeys{weightofset}{chkp=C} \left|{\mathcal{X}\cap \setkeys{totvalset}{chkp=C_{\mathsf{vs}}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp}}\right|^{\weightofset@Chkp}}{1+\rho} \end{aligned}.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item As, for a given set \mathcal{X},\\ \begin{align}[t] & \setkeys{weightofset}{chkp=C_{\mathsf{vs}}} \left|{\mathcal{X}}\right|^{\weightofset@Chkp}\\ &\geq \setkeys{weightofset}{chkp=C_{\mathsf{vs}}} \left|{\mathcal{X}\cap \setkeys{totvalset}{chkp=C} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp}\cap \setkeys{totvalset}{chkp=C_{\mathsf{vs}}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp}}\right|^{\weightofset@Chkp}\\ &\geq \frac{ \setkeys{weightofset}{chkp=C} \left|{\mathcal{X}\cap \setkeys{totvalset}{chkp=C} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp}\cap \setkeys{totvalset}{chkp=C_{\mathsf{vs}}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp}}\right|^{\weightofset@Chkp}}{1+\rho} && \ifthenelse{\equal{1cm}{}}{ \text{--- by \Cref{prop:changing-balances:rewardr}} }{ \begin{minipage}[t]{1cm} \begin{itemize}[label=---,nosep,left=0pt] \item by \Cref{prop:changing-balances:rewardr} \end{itemize} \end{minipage} } \\ &=\frac{ \setkeys{weightofset}{chkp=C} \left|{\mathcal{X}\cap \setkeys{totvalset}{chkp=C_{\mathsf{vs}}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp}}\right|^{\weightofset@Chkp}}{1+\rho} \end{align}. \end{itemize} \end{minipage} } \\ &\geq \frac{\left( \setkeys{weightofset}{chkp=C} \left|{ \setkeys{ffgvalsetraw}{source={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{chkp=C} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}, target=C,time=\mathsf{st}(e'),val=v} \mathcal{F}_{\ffgvalsetraw@Source \to \ffgvalsetraw@Target}^{\ffgvalsetraw@Val,\ffgvalsetraw@Time}\cap\mathcal{J}}\right|^{\weightofset@Chkp}-\epsilon \setkeys{totvalset}{chkp=C_{\mathsf{vs}}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp}\right)(1-\pi)(1-\epsilon)}{(1+\rho) \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{J}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}} (1-\beta) \\ &\geq \frac{ \setkeys{weightofset}{chkp=C} \left|{ \setkeys{ffgvalsetraw}{source={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{chkp=C} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}, target=C,time=\mathsf{st}(e'),val=v} \mathcal{F}_{\ffgvalsetraw@Source \to \ffgvalsetraw@Target}^{\ffgvalsetraw@Val,\ffgvalsetraw@Time}\cap\mathcal{J}}\right|^{\weightofset@Chkp}-\epsilon\frac{ \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}}{(1-\epsilon)(1-\sigma) }}{(1+\rho) \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{J}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}} (1-\epsilon)(1-\pi)(1-\beta) && \ifthenelse{\equal{6cm}{}}{ \text{--- By \Cref{lem:lower-bound-tot-next-chkp}.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item By \Cref{lem:lower-bound-tot-next-chkp}. \end{itemize} \end{minipage} } \\ &= \begin{align}[t] &\frac{(1-\epsilon)(1-\pi)}{1+\rho}\\ &\left(\frac{ \setkeys{weightofset}{chkp=C} \left|{ \setkeys{ffgvalsetraw}{source={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{chkp=C} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}, target=C,time=\mathsf{st}(e'),val=v} \mathcal{F}_{\ffgvalsetraw@Source \to \ffgvalsetraw@Target}^{\ffgvalsetraw@Val,\ffgvalsetraw@Time}\cap\mathcal{J}}\right|^{\weightofset@Chkp}}{ \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{J}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}} (1-\beta) - \frac{\epsilon \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}(1-\beta)}{((1-\epsilon)(1-\sigma) ) \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{J}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}} \right) \end{align} \\ \\ &\geq \begin{aligned}[t] &\frac{(1-\epsilon)(1-\pi)}{1+\rho}\\ &\left(\frac{ \setkeys{weightofset}{chkp=C} \left|{ \setkeys{ffgvalsetraw}{source={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{chkp=C} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}, target=C,time=\mathsf{st}(e'),val=v} \mathcal{F}_{\ffgvalsetraw@Source \to \ffgvalsetraw@Target}^{\ffgvalsetraw@Val,\ffgvalsetraw@Time}\cap\mathcal{J}}\right|^{\weightofset@Chkp}}{ \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{J}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}} (1-\beta) - \frac{\epsilon}{(1-\epsilon)(1-\sigma) } \right) \end{aligned} && \ifthenelse{\equal{6cm}{}}{ \text{--- As, due to\setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}\preceq C, \mathsf{epoch}(C)=\mathsf{epoch}(t_p) and \Cref{prop:wt-we-link},\setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{J}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \geq (1-\beta) \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item As, due to\setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}\preceq C, \mathsf{epoch}(C)=\mathsf{epoch}(t_p) and \Cref{prop:wt-we-link},\setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{J}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \geq (1-\beta) \setkeys{totvalsetweight}{chkp=C} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}. \end{itemize} \end{minipage} } \\ &\geq \begin{align}[t] &\frac{(1-\epsilon)(1-\pi)}{1+\rho}\\ &\left(\frac{2}{3}\left(\frac{1+\rho-\epsilon\rho}{1-\pi}\right)+\epsilon+\beta - \frac{\epsilon}{(1-\epsilon)(1-\sigma) } \right) \end{align} && \ifthenelse{\equal{6cm}{}}{ \text{--- As, by point iii above, \frac{ \setkeys{weightofset}{chkp=C} \left|{ \setkeys{ffgvalsetraw}{source={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{chkp=C} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}, target=C,time=\mathsf{st}(e'),val=v} \mathcal{F}_{\ffgvalsetraw@Source \to \ffgvalsetraw@Target}^{\ffgvalsetraw@Val,\ffgvalsetraw@Time}\cap\mathcal{J}}\right|^{\weightofset@Chkp}}{ \setkeys{totvalset}{chkp=C} {\mathcal{J}}_{\mathsf{t}}^{\totvalset@Chkp}}>\mathsf{honFFGratioVar}(\beta).} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item As, by point iii above, \frac{ \setkeys{weightofset}{chkp=C} \left|{ \setkeys{ffgvalsetraw}{source={\setkeys{votsource}{time=,blck=,chkp=} \setkeys{votsource}{chkp=C} \mathsf{vs}( \ifthenelse{\equal{\votsource@Chkp}{}}{ \votsource@Blk,\votsource@Time }{ \votsource@Chkp} )}, target=C,time=\mathsf{st}(e'),val=v} \mathcal{F}_{\ffgvalsetraw@Source \to \ffgvalsetraw@Target}^{\ffgvalsetraw@Val,\ffgvalsetraw@Time}\cap\mathcal{J}}\right|^{\weightofset@Chkp}}{ \setkeys{totvalset}{chkp=C} {\mathcal{J}}_{\mathsf{t}}^{\totvalset@Chkp}}>\mathsf{honFFGratioVar}(\beta). \end{itemize} \end{minipage} } \\ &= \begin{align}[t] &\frac{(1-\epsilon)(1-\pi)}{1+\rho}\\ &\left(\frac{2}{3}\left(\frac{1+\rho-\epsilon\rho}{1-\pi}\right)+\epsilon+\beta - \frac{\epsilon}{(1-\epsilon)(1-\sigma) } \right) \\ &-\frac{\epsilon}{1-\epsilon} -\beta -\frac{1+\rho}{2(1-\pi)} +\frac{\epsilon}{1-\epsilon} +\beta +\frac{1+\rho}{2(1-\pi)} \end{align} \\ &> \begin{aligned}[t] &\frac{p(1+\rho)(1+\rho+\epsilon)}{{E}(1-\pi)(1-\epsilon)} +\frac{\epsilon}{1-\epsilon} +\beta +\frac{1+\rho}{2(1-\pi)} \end{aligned} && \ifthenelse{\equal{6cm}{}}{ \text{--- Due to \Cref{prop:ffg-stronger-justification-changing-balances:p-e}.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item Due to \Cref{prop:ffg-stronger-justification-changing-balances:p-e}. \end{itemize} \end{minipage} } \\ &= \frac{1+\rho}{2(1-\pi)}\left(1+ \frac{p(1+\rho+\epsilon)}{{E}(1-\epsilon)}\right) +\frac{\epsilon}{1-\epsilon} +\beta \\ &= \frac{1+\rho}{2(1-\pi)} \left(1+ \frac{ \setkeys{boostweight}{chkp={C_{\mathsf{vs}}}} W_p^{\boostweight@Chkp}(1+\rho+\epsilon)}{ \setkeys{totvalsetweight}{chkp={C_{\mathsf{vs}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}(1-\epsilon)} \right) +\frac{ \setkeys{totvalsetweight}{chkp={C_{\mathsf{vs}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}\epsilon}{ \setkeys{totvalsetweight}{chkp={C_{\mathsf{vs}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}(1-\epsilon)} +\beta && \ifthenelse{\equal{6cm}{}}{ \text{--- As, by definition,\setkeys{boostweight}{chkp={C_{\mathsf{vs}}}} W_p^{\boostweight@Chkp} = \setkeys{totvalsetweight}{chkp={C_{\mathsf{vs}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \frac{p}{{E}} .} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item As, by definition,\setkeys{boostweight}{chkp={C_{\mathsf{vs}}}} W_p^{\boostweight@Chkp} = \setkeys{totvalsetweight}{chkp={C_{\mathsf{vs}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \frac{p}{{E}} . \end{itemize} \end{minipage} } \\ &\geq \frac{1+\rho}{2(1-\pi)} \left(1+ \frac{ \setkeys{boostweight}{chkp={C_{\mathsf{vs}}}} W_p^{\boostweight@Chkp}(1+\rho+\epsilon)}{ \setkeys{weightofset}{chkp={C_{\mathsf{vs}}}} \left|{ \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=e^{tp}} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } }\right|^{\weightofset@Chkp}} \right) +\frac{ \setkeys{totvalsetweight}{chkp={C_{\mathsf{vs}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}\epsilon}{ \setkeys{weightofset}{chkp={C_{\mathsf{vs}}}} \left|{ \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=e^{tp}} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } }\right|^{\weightofset@Chkp}} +\beta && \ifthenelse{\equal{6cm}{}}{ \text{--- \begin{align}[t] & \setkeys{totvalsetweight}{chkp={C_{\mathsf{vs}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}(1-\epsilon)\\ &= \setkeys{totvalsetweight}{chkp={C_{\mathsf{vs}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}- \setkeys{totvalsetweight}{chkp={C_{\mathsf{vs}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}\epsilon \\ &\geq \setkeys{totvalsetweight}{chkp={C_{\mathsf{vs}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}- \setkeys{weightofset}{chkp={C_{\mathsf{vs}}}} \left|{ \setkeys{totvalset}{chkp={C_{\mathsf{vs}}}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \setminus \setkeys{totvalset}{chkp=C} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp}}\right|^{\weightofset@Chkp} && \ifthenelse{\equal{2cm}{}}{ \text{--- By \Cref{prop:changing-balances:exitr}.} }{ \begin{minipage}[t]{2cm} \begin{itemize}[label=---,nosep,left=0pt] \item By \Cref{prop:changing-balances:exitr}. \end{itemize} \end{minipage} } \\ &= \setkeys{weightofset}{chkp={C_{\mathsf{vs}}}} \left|{ \setkeys{totvalset}{chkp={C_{\mathsf{vs}}}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \cap \setkeys{totvalset}{chkp=C} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp}}\right|^{\weightofset@Chkp} \\ &\geq \setkeys{weightofset}{chkp={C_{\mathsf{vs}}}} \left|{ \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=e^{tp}} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } }\right|^{\weightofset@Chkp} && \ifthenelse{\equal{2cm}{}}{ \text{--- Due\setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}\preceq C, \mathsf{epoch}(C)=\mathsf{epoch}(t_p) and \Cref{prop:wt-we-link},\setkeys{totvalset}{chkp=C} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} = \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=\mathsf{epoch}(t_p)} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } .} }{ \begin{minipage}[t]{2cm} \begin{itemize}[label=---,nosep,left=0pt] \item Due\setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}\preceq C, \mathsf{epoch}(C)=\mathsf{epoch}(t_p) and \Cref{prop:wt-we-link},\setkeys{totvalset}{chkp=C} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} = \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=\mathsf{epoch}(t_p)} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } . \end{itemize} \end{minipage} } \end{align}} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item \begin{align}[t] & \setkeys{totvalsetweight}{chkp={C_{\mathsf{vs}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}(1-\epsilon)\\ &= \setkeys{totvalsetweight}{chkp={C_{\mathsf{vs}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}- \setkeys{totvalsetweight}{chkp={C_{\mathsf{vs}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}\epsilon \\ &\geq \setkeys{totvalsetweight}{chkp={C_{\mathsf{vs}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}- \setkeys{weightofset}{chkp={C_{\mathsf{vs}}}} \left|{ \setkeys{totvalset}{chkp={C_{\mathsf{vs}}}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \setminus \setkeys{totvalset}{chkp=C} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp}}\right|^{\weightofset@Chkp} && \ifthenelse{\equal{2cm}{}}{ \text{--- By \Cref{prop:changing-balances:exitr}.} }{ \begin{minipage}[t]{2cm} \begin{itemize}[label=---,nosep,left=0pt] \item By \Cref{prop:changing-balances:exitr}. \end{itemize} \end{minipage} } \\ &= \setkeys{weightofset}{chkp={C_{\mathsf{vs}}}} \left|{ \setkeys{totvalset}{chkp={C_{\mathsf{vs}}}} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} \cap \setkeys{totvalset}{chkp=C} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp}}\right|^{\weightofset@Chkp} \\ &\geq \setkeys{weightofset}{chkp={C_{\mathsf{vs}}}} \left|{ \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=e^{tp}} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } }\right|^{\weightofset@Chkp} && \ifthenelse{\equal{2cm}{}}{ \text{--- Due\setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}\preceq C, \mathsf{epoch}(C)=\mathsf{epoch}(t_p) and \Cref{prop:wt-we-link},\setkeys{totvalset}{chkp=C} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} = \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=\mathsf{epoch}(t_p)} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } .} }{ \begin{minipage}[t]{2cm} \begin{itemize}[label=---,nosep,left=0pt] \item Due\setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}\preceq C, \mathsf{epoch}(C)=\mathsf{epoch}(t_p) and \Cref{prop:wt-we-link},\setkeys{totvalset}{chkp=C} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp} = \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=\mathsf{epoch}(t_p)} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } . \end{itemize} \end{minipage} } \end{align} \end{itemize} \end{minipage} } \\ &= \frac{1+\rho}{2(1-\pi)} \left(1+ \frac{ \setkeys{boostweight}{chkp={C_{\mathsf{vs}}}} W_p^{\boostweight@Chkp}(1+\rho+\epsilon)}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b''',to=\mathsf{slot}(t')-1,chkp={C_{\mathsf{vs}}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} \right) +\frac{ \setkeys{totvalsetweight}{chkp={C_{\mathsf{vs}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}\epsilon}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b''',to=\mathsf{slot}(t')-1,chkp={C_{\mathsf{vs}}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} +\beta && \ifthenelse{\equal{6cm}{}}{ \text{--- Given that {\mathsf{ps}^{+1}}(b''')\leq{\mathsf{ps}^{+1}}(b'')<\mathsf{first\_slot}(\mathsf{epoch}(t_p))\leq\mathsf{last\_slot}(\mathsf{epoch}(t_p))\leq\mathsf{slot}(t')-1,\setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=\mathsf{epoch}(t_p)} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } \subseteq \setkeys{commfromblock}{from=,to=,val=,when=} \setkeys{commfromblock}{from=b''',to=\mathsf{slot}(t')-1} {\mathcal{W}}_{\commfromblock@From}^{\commfromblock@To \ifthenelse{\equal{\commfromblock@Val}{}}{}{,\commfromblock@Val} \ifthenelse{\equal{\commfromblock@When}{}}{}{,\commfromblock@When} }.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item Given that {\mathsf{ps}^{+1}}(b''')\leq{\mathsf{ps}^{+1}}(b'')<\mathsf{first\_slot}(\mathsf{epoch}(t_p))\leq\mathsf{last\_slot}(\mathsf{epoch}(t_p))\leq\mathsf{slot}(t')-1,\setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=\mathsf{epoch}(t_p)} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } \subseteq \setkeys{commfromblock}{from=,to=,val=,when=} \setkeys{commfromblock}{from=b''',to=\mathsf{slot}(t')-1} {\mathcal{W}}_{\commfromblock@From}^{\commfromblock@To \ifthenelse{\equal{\commfromblock@Val}{}}{}{,\commfromblock@Val} \ifthenelse{\equal{\commfromblock@When}{}}{}{,\commfromblock@When} }. \end{itemize} \end{minipage} } \end{align}\]

 ◻

Proof. Note that 7 implies 5. Hence, we can apply [lem:ffg-motonocity-changing-balances,lem:ffg-overall-safety-changing-balances] to conclude the proof. ◻

Proof. Let \(\mathcal{X}^t := \bigcup_{C \succeq b \land epoch(C) = e} \setkeys{ffgvalsettoslot}{to=\mathsf{slot}(t)-1,source=,target=C,time=t,val=v} \mathcal{F}_{\ffgvalsettoslot@Source \to \ffgvalsettoslot@Target}^{\ffgvalsettoslot@To,\ffgvalsettoslot@Val,\ffgvalsettoslot@Time}\) and \(\mathcal{X}^{\mathsf{st}(e+1)} := \mathcal{X}^t \sqcup \setkeys{commfromslot}{from=,to=,val=,when=} \setkeys{commfromslot}{from=\mathsf{slot}(t),to=\mathsf{last\_slot}(e)} \overline{\mathcal{J}}_{\commfromslot@From}^{\commfromslot@To \ifthenelse{\equal{\commfromslot@Val}{}}{}{,\commfromslot@Val} \ifthenelse{\equal{\commfromslot@When}{}}{}{,\commfromslot@When} }\). Now proceed as follows.

\[\begin{align} \setkeys{weightofset}{chkp={ \setkeys{finattime}{time=t,val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}}} \left|{ \setkeys{ffgvalsetallsentraw}{target=C,time=\mathsf{st}(e+1)} \overset{T}{\mathcal{F}}{}_{\ffgvalsetallsentraw@Source \to \ffgvalsetallsentraw@Target}^{\ffgvalsetallsentraw@Time}}\right|^{\weightofset@Chkp} &\geq \setkeys{weightofset}{chkp={ \setkeys{finattime}{time=t,val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}}} \left|{\mathcal{X}^{\mathsf{st}(e+1)}}\right|^{\weightofset@Chkp} && \ifthenelse{\equal{5cm}{}}{ \text{--- Condition 3 of the Lemma's statement and \Cref{prop:gasper-basic:ldm-vote-for-b-is-ffg-vote-for-cb} imply that \mathcal{X}^{\mathsf{st}(e+1)} \subseteq \setkeys{ffgvalsetallsentraw}{target=C,time=\mathsf{st}(e+1)} \overset{T}{\mathcal{F}}{}_{\ffgvalsetallsentraw@Source \to \ffgvalsetallsentraw@Target}^{\ffgvalsetallsentraw@Time}.} }{ \begin{minipage}[t]{5cm} \begin{itemize}[label=---,nosep,left=0pt] \item Condition 3 of the Lemma's statement and \Cref{prop:gasper-basic:ldm-vote-for-b-is-ffg-vote-for-cb} imply that \mathcal{X}^{\mathsf{st}(e+1)} \subseteq \setkeys{ffgvalsetallsentraw}{target=C,time=\mathsf{st}(e+1)} \overset{T}{\mathcal{F}}{}_{\ffgvalsetallsentraw@Source \to \ffgvalsetallsentraw@Target}^{\ffgvalsetallsentraw@Time}. \end{itemize} \end{minipage} } \\ &= \setkeys{weightofset}{chkp={ \setkeys{finattime}{time=t,val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}}} \left|{\mathcal{X}^{t}}\right|^{\weightofset@Chkp} + \setkeys{commweightfromslot}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromslot}{from=\mathsf{slot}(t),to=\mathsf{last\_slot}(e),chkp={ \setkeys{finattime}{time=t,val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}}} \overline{{\addweight{\mathcal{J}}}}_{\commweightfromslot@From}^{\commweightfromslot@To, \ifthenelse{\equal{\commweightfromslot@Val}{}}{}{\commweightfromslot@Val,} \ifthenelse{\equal{\commweightfromslot@When}{}}{}{\commweightfromslot@When,} \commweightfromslot@ValsetChkp} \\ &\geq \setkeys{weightofset}{chkp={ \setkeys{finattime}{time=t,val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}}} \left|{\mathcal{X}^{t}}\right|^{\weightofset@Chkp} + (1-\beta) \setkeys{commweightfromslot}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromslot}{from=\mathsf{slot}(t),to=\mathsf{last\_slot}(e),chkp={ \setkeys{finattime}{time=t,val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}}} \overline{{\addweight{\mathcal{W}}}}_{\commweightfromslot@From}^{\commweightfromslot@To, \ifthenelse{\equal{\commweightfromslot@Val}{}}{}{\commweightfromslot@Val,} \ifthenelse{\equal{\commweightfromslot@When}{}}{}{\commweightfromslot@When,} \commweightfromslot@ValsetChkp} && \ifthenelse{\equal{5cm}{}}{ \text{--- By \Cref{assum:beta}.} }{ \begin{minipage}[t]{5cm} \begin{itemize}[label=---,nosep,left=0pt] \item By \Cref{assum:beta}. \end{itemize} \end{minipage} } \\ &\geq \frac{2}{3}\mathit{f}^{\beta,e-\mathsf{epoch}( \setkeys{finattime}{time=t,val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}})} \setkeys{totvalsetweight}{chkp={ \setkeys{finattime}{time=t,val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} && \ifthenelse{\equal{5cm}{}}{ \text{--- By applying condition 5 of the Lemma's statement.} }{ \begin{minipage}[t]{5cm} \begin{itemize}[label=---,nosep,left=0pt] \item By applying condition 5 of the Lemma's statement. \end{itemize} \end{minipage} } \end{align}\] We can now apply [prop:xxx] to conclude the proof. ◻

Proof. First, we proceed by induction on \(t' \geq \mathsf{st}(\mathsf{slot}(t))\) to show that all of the following inductive conditions hold

1. there exists no justified checkpoint for an epoch in \([\mathsf{epoch}(\mathsf{C}(b)), \mathsf{epoch}(t')]\) conflicting with \(b\).

2. for any honest validator \(v''\) and time \(t''\) such that \(\mathsf{st}(\mathsf{slot}(t)) \leq t'' < t'\)

   1. \(b \in \setkeys{filtered}{val=v'',time=t''} \mathsf{filt}_{\mathsf{hfc}}^{\filtered@Time\ifthenelse{\equal{\filtered@Val}{}}{}{,\filtered@Val}}\)

   2. \(\setkeys{chkpattime}{time=t'',val=v''} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}} \succeq \mathsf{C}(b)\)

Let \(v'\) be any honest validator. In particular, by abuse of notation, in the below, we allow \(v'\) to refer to different honest validators every time that it is used. This is to avoid instantiating many different variables for honest validators.

Base Case: \(\mathsf{epoch}(t')= \mathsf{epoch}(t)-1\).

In this case, we just need to prove inductive hypothesis [itm:lem:ffg-no-extra-assum-old-epoch-safety:cond1] as inductive hypothesis [itm:lem:ffg-no-extra-assum-old-epoch-safety:cond2] is vacuously true. Due to [ln:alg:ffg-no-extra-assum:old-epoch-exists] [ln:alg:ffg-no-extra-assum:old-epoch-cb-in-aub-prime], and Property [prop:gasper-basic:only-one-justified-per-epoch], we can conclude that no checkpoint for epoch \(\mathsf{epoch}(\mathsf{C}(b))\) conflicting with \(b\) could ever be justified. From this, due to [ln:alg:ffg-no-extra-assum:old-epoch-prec-fin] [ln:alg:ffg-no-extra-assum:old-epoch-ffg-check] [ln:alg:ffg-no-extra-assum:old-epoch-forall-e], we can apply 36 to conclude that no checkpoint for epochs in \([\mathsf{epoch}(\mathsf{C}(b)),\mathsf{epoch}(t)]\) conflicting with \(b\) could ever be justified proving inductive hypothesis [itm:lem:ffg-no-extra-assum-old-epoch-safety:cond1].

Inductive Step: \(\mathsf{epoch}(t') \geq \mathsf{epoch}(t)\).

Let \(b'\) be any block \(b' \succeq b\) that satisfies lines [ln:alg:ffg-no-extra-assum:old-epoch-b-prime-succ-b] to [ln:alg:ffg-no-extra-assum:old-epoch-b-prime-less-than-t]. Given that \(b' \in \setkeys{viewattime}{val=v,time=\mathsf{st}(\mathsf{slot}(t)-1)} \mathcal{V}^{\viewattime@Val,\viewattime@Time}\) and that we assume \(\mathsf{st}(\mathsf{slot}(t)-1)\geq\mathbb{GST}\), \(b'\) is in the view of any honest validator at time \(t'\). This implies that \(\mathsf{epoch}( \setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}})\geq\mathsf{epoch}(\mathsf{C}(b))\). Because of the above and inductive hypothesis [itm:lem:ffg-no-extra-assum-old-epoch-safety:cond1], we can apply 11 to conclude that \(b \in \setkeys{filtered}{time=t',val=v'} \mathsf{filt}_{\mathsf{hfc}}^{\filtered@Time\ifthenelse{\equal{\filtered@Val}{}}{}{,\filtered@Val}}\), i.e., \(b\) does not get filtered out by any honest validator at time time \(t'\) which proves inductive hypothesis [itm:lem:ffg-no-extra-assum-old-epoch-safety:cond21] for \(t'\).

Also, inductive hypothesis [itm:lem:ffg-no-extra-assum-old-epoch-safety:cond1], \(\mathsf{epoch}( \setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}})\geq\mathsf{epoch}(\mathsf{C}(b))\), Property [prop:gasper-basic:highest-justified-not-from-current-epoch] and the definition of \(\setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}\) (Definition 3) imply that \(\setkeys{chkpattime}{time=t',val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}} \succeq C(b)\) which proves inductive condition [itm:lem:ffg-no-extra-assum-old-epoch-safety:cond22] for \(t'\).

Hence, we are left with having to prove hypothesis [itm:lem:ffg-no-extra-assum-old-epoch-safety:cond1] for epoch \(\mathsf{epoch}(t')\). To do so we proceed by cases.

Case \(\mathsf{epoch}(t') > \mathsf{epoch}(t)\).

Given that we have proven above that inductive hypothesis [itm:lem:ffg-no-extra-assum-old-epoch-safety:cond2] holds at time \(t'\), due to line [ln:alg:ffg-no-extra-assum:lmd-safe], we can apply Lemma 10 to conclude that \(b\) is always canonical in the view of all honest validators at any time during epoch \(\mathsf{epoch}(t')\). Properties [prop:gasper-basic:ldm-vote-for-b-is-ffg-vote-for-cb] and [prop:gasper-basic:no-conflicting-if-all-honest-votes-in-support-of-b] immediately imply that no checkpoint conflicting with \(b\) can be justified in epoch \(\mathsf{epoch}(t')\), which concludes the proof for the inductive hypothesis [itm:lem:ffg-no-extra-assum-old-epoch-safety:cond1].

Case \(\mathsf{epoch}(t') = \mathsf{epoch}(t)\).

The proof for this case is already given in the proof of the base case.

Given that we have just established that inductive condition [itm:lem:ffg-no-extra-assum-old-epoch-safety:cond2] holds for any time \(t' \geq \mathsf{st}(\mathsf{slot}(t))\), due to line [ln:alg:ffg-no-extra-assum:lmd-safe] we can apply Lemma 10 to complete the proof. ◻

Proof. If \(\mathsf{epoch}(b) \geq \mathsf{epoch}(t)-1\), then proof of 18 suffices. If \(\mathsf{epoch}(b) < \mathsf{epoch}(t)-1\), then we can apply 37. ◻

Proof. Let us proceed by cases.

Case 1: \(b \preceq \setkeys{finattime}{time=t,val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}\).

Let \(v'\) be any honest validator. Given that \(\mathsf{st}(\mathsf{epoch}(t)-1)\geq \mathbb{GST}\), Properties [prop:gasper-basic:finalization] and [prop:gasper-basic:just-succ-finalization] imply that \(b \preceq \setkeys{finattime}{time=\mathsf{st}(\mathsf{slot}(t)+1),val=v'} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}\preceq \setkeys{chkpattime}{time=\mathsf{st}(\mathsf{slot}(t)+1),val=v'} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}\). Hence, given the definition of LMD-GHOST-HFC(3), \(b\) is canonical in the view of any honest validator at time \(\mathsf{st}(\mathsf{slot}(t))\) and thereafter.

Case 2: \(b \npreceq \setkeys{finattime}{time=t,val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}\).

Same as the proof of Lemma 19 by replacing 18 with 38.

 ◻

Proof. First, we want to prove that 8 implies \(\beta < \frac{1}{4}\left(1-\frac{p}{{E}(1-\beta)}\right)\). To do so we proceed as follows. \[\begin{align} \frac{1}{4}\left(1-\frac{p}{{E}(1-\beta)}\right) &> \frac{1}{4} \left(1- \frac{p}{\frac{{E}}{8}\left(3+\sqrt{9+ 16\frac{p}{{E}}}\right)}\right) && \ifthenelse{\equal{5cm}{}}{ \text{--- By applying \Cref{assum:ffg-no-extra-assum:beta} to 1-\beta.} }{ \begin{minipage}[t]{5cm} \begin{itemize}[label=---,nosep,left=0pt] \item By applying \Cref{assum:ffg-no-extra-assum:beta} to 1-\beta. \end{itemize} \end{minipage} } \\ &= \frac{1}{4} \left(1- \frac{p\left(3-\sqrt{9+ 16\frac{p}{{E}}}\right)}{\frac{{E}}{8}\left(9-{\frac{9{E}+ 16p}{{E}}}\right)}\right) \\ &= \frac{1}{4} \left(1+ \frac{3-\sqrt{9+ 16\frac{p}{{E}}}}{2}\right) \\ &= \frac{1}{8}\left(5 - \sqrt{9+ 16\frac{p}{{E}}}\right) \\ &> \beta \end{align}\]

Now, let \(t'\) be any time \(t' \geq \mathsf{st}(e+1)\). Given that, as described in Section 2.2.2, honest validators always GHOST vote for the block returned by the fork-choice function executed at the time of voting, then any honest validator in the committees of epoch \(e\) GHOST votes in support of \(b\). Note that as per 1, honest validators only GHOST vote in support of blocks that are from previous slots. Therefore, \(\mathsf{slot}(b) < \mathsf{st}(e)\leq \mathsf{epoch}(t')-1\). Hence, we can proceed as follows.

\[\begin{align} \setkeys{indicatorfromblock}{from=b',to=\mathsf{slot}(t')-1,val=v,when=t',chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {{{\mathit{Q}}}}_{\indicatorfromblock@From}^{\indicatorfromblock@To,\indicatorfromblock@Val,\indicatorfromblock@When,\indicatorfromblock@ValSetChkp} \setkeys{indicatorfromblock}{from=,to=,val=,when=,chkp=} &= \frac{ \setkeys{attsetfromblock}{from=b',to=\mathsf{slot}(t')-1,val=v,when=t',chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {\mathcal{S}}_{\attsetfromblock@From}^{\attsetfromblock@To,\attsetfromblock@Val,\attsetfromblock@When,\attsetfromblock@ValSetChkp} }{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=\mathsf{slot}(t')-1,chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} \\ &\geq \frac{ \setkeys{weightofset}{chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \left|{ \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=e} {\hat{\overline{\mathcal{J}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } }\right|^{\weightofset@Chkp}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=\mathsf{slot}(t')-1,chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} && \ifthenelse{\equal{6cm}{}}{ \text{--- As, all honest validators \textsf{GHOST} vote in support of b' during epoch e.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item As, all honest validators \textsf{GHOST} vote in support of b' during epoch e. \end{itemize} \end{minipage} } \\ &= \frac{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=\mathsf{slot}(t')-1,chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \addweight{\mathcal{J}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=\mathsf{slot}(t')-1,chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}} && \ifthenelse{\equal{6cm}{}}{ \text{--- As, given \Cref{assum:no-change-to-the-validator-set} and that \mathsf{slot}(b)<\mathsf{epoch}(t')-1,\setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=e} {\hat{\overline{\mathcal{J}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} }includes all of the honest validators in any possible committee.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item As, given \Cref{assum:no-change-to-the-validator-set} and that \mathsf{slot}(b)<\mathsf{epoch}(t')-1,\setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=e} {\hat{\overline{\mathcal{J}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} }includes all of the honest validators in any possible committee. \end{itemize} \end{minipage} } \\ &\geq (1-\beta) && \ifthenelse{\equal{6cm}{}}{ \text{--- By applying Assumption~\ref{assum:beta}.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item By applying Assumption~\ref{assum:beta}. \end{itemize} \end{minipage} } \\ &= (1-2\beta+\beta) \\ &> \frac{1}{2} \left( 1+ \frac{p}{{E}(1-\beta)} \right) + \beta && \ifthenelse{\equal{6cm}{}}{ \text{--- By applying the condition \beta < \frac{1}{4}\left(1-\frac{p}{{E}(1-\beta)}\right), from Assumption~\ref{assum:beta-lmd-monotonicity}, to 2\beta.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item By applying the condition \beta < \frac{1}{4}\left(1-\frac{p}{{E}(1-\beta)}\right), from Assumption~\ref{assum:beta-lmd-monotonicity}, to 2\beta. \end{itemize} \end{minipage} } \\ &= \frac{1}{2} \left(1+ \frac{ \setkeys{boostweight}{chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} W_p^{\boostweight@Chkp}}{ \setkeys{totvalsetweight}{chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}(1-\beta)}\right)+\beta && \ifthenelse{\equal{6cm}{}}{ \text{--- As, by definition,\setkeys{boostweight}{chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} W_p^{\boostweight@Chkp} = \setkeys{totvalsetweight}{chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \frac{p}{{E}} .} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item As, by definition,\setkeys{boostweight}{chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} W_p^{\boostweight@Chkp} = \setkeys{totvalsetweight}{chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} \frac{p}{{E}} . \end{itemize} \end{minipage} } \\ &\geq \frac{1}{2}\left(1+\frac{ \setkeys{boostweight}{chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} W_p^{\boostweight@Chkp}}{ \setkeys{weightofset}{chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \left|{ \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=e} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } }\right|^{\weightofset@Chkp}}\right)+\beta && \ifthenelse{\equal{6cm}{}}{ \text{--- As, by \Cref{assum:beta,assum:no-change-to-the-validator-set},\setkeys{weightofset}{chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \left|{ \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=e} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } }\right|^{\weightofset@Chkp}\geq \setkeys{totvalsetweight}{chkp={b_{\texttt{gen}}}} {\addweight{\mathcal{J}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}\geq (1-\beta) \setkeys{totvalsetweight}{chkp={b_{\texttt{gen}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}\geq (1-\beta) \setkeys{totvalsetweight}{chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item As, by \Cref{assum:beta,assum:no-change-to-the-validator-set},\setkeys{weightofset}{chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \left|{ \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=e} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } }\right|^{\weightofset@Chkp}\geq \setkeys{totvalsetweight}{chkp={b_{\texttt{gen}}}} {\addweight{\mathcal{J}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}\geq (1-\beta) \setkeys{totvalsetweight}{chkp={b_{\texttt{gen}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}\geq (1-\beta) \setkeys{totvalsetweight}{chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp}. \end{itemize} \end{minipage} } \\ &= \frac{1}{2}\left(1+\frac{ \setkeys{boostweight}{chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} W_p^{\boostweight@Chkp}}{ \setkeys{commweightfromblock}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromblock}{from=b',to=\mathsf{slot}(t')-1,chkp={ \setkeys{chkpattime}{time=t',val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}}} \addweight{\mathcal{W}}_{\commweightfromblock@From}^{\commweightfromblock@To, \ifthenelse{\equal{\commweightfromblock@Val}{}}{}{\commweightfromblock@Val,} \ifthenelse{\equal{\commweightfromblock@When}{}}{}{\commweightfromblock@When,} \commweightfromblock@ValSetChkp}}\right)+\beta && \ifthenelse{\equal{6cm}{}}{ \text{--- As \mathsf{slot}(b) < \mathsf{st}(e)\leq \mathsf{epoch}(t')-1.} }{ \begin{minipage}[t]{6cm} \begin{itemize}[label=---,nosep,left=0pt] \item As \mathsf{slot}(b) < \mathsf{st}(e)\leq \mathsf{epoch}(t')-1. \end{itemize} \end{minipage} } \end{align}\] ◻

Proof. Let \(\mathcal{X}:= \bigcup_{C \succeq b \land epoch(C) = e} \setkeys{ffgvalsettoslot}{to=\mathsf{slot}(t)-1,source=,target=C,time=t,val=v} \mathcal{F}_{\ffgvalsettoslot@Source \to \ffgvalsettoslot@Target}^{\ffgvalsettoslot@To,\ffgvalsettoslot@Val,\ffgvalsettoslot@Time}\). Also, by [prop:wt-we-link-2], let \(C'\) be the checkpoint such that \(C' \succeq \setkeys{finattime}{val=v,time=t} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}} \land \mathsf{epoch}(C') = e \land \setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=e} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } = \setkeys{totvalset}{chkp=C'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp}\). \[\begin{align} \setkeys{weightofset}{chkp={{ \setkeys{finattime}{time=t,val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}}}} \left|{\mathcal{X}}\right|^{\weightofset@Chkp} + (1-\beta) \setkeys{commweightfromslot}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromslot}{from=\mathsf{slot}(t),to=\mathsf{last\_slot}(e),chkp={{ \setkeys{finattime}{time=t,val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}}}} \overline{{\addweight{\mathcal{W}}}}_{\commweightfromslot@From}^{\commweightfromslot@To, \ifthenelse{\equal{\commweightfromslot@Val}{}}{}{\commweightfromslot@Val,} \ifthenelse{\equal{\commweightfromslot@When}{}}{}{\commweightfromslot@When,} \commweightfromslot@ValsetChkp} &\geq \setkeys{commweightfromslot}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromslot}{from=\mathsf{first\_slot}(e),to=\mathsf{slot}(t)-1,chkp={{ \setkeys{finattime}{time=t,val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}}}} \overline{{\addweight{\mathcal{J}}}}_{\commweightfromslot@From}^{\commweightfromslot@To, \ifthenelse{\equal{\commweightfromslot@Val}{}}{}{\commweightfromslot@Val,} \ifthenelse{\equal{\commweightfromslot@When}{}}{}{\commweightfromslot@When,} \commweightfromslot@ValsetChkp} + (1-\beta) \setkeys{commweightfromslot}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromslot}{from=\mathsf{slot}(t),to=\mathsf{last\_slot}(e),chkp={ \setkeys{finattime}{time=t,val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}}} \overline{{\addweight{\mathcal{W}}}}_{\commweightfromslot@From}^{\commweightfromslot@To, \ifthenelse{\equal{\commweightfromslot@Val}{}}{}{\commweightfromslot@Val,} \ifthenelse{\equal{\commweightfromslot@When}{}}{}{\commweightfromslot@When,} \commweightfromslot@ValsetChkp} && \ifthenelse{\equal{5cm}{}}{ \text{--- \Cref{prop:gasper-basic:ldm-vote-for-b-is-ffg-vote-for-cb} and condition 1 of the Lemma's statement imply that\setkeys{commweightfromslot}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromslot}{from=\mathsf{first\_slot}(e),to=\mathsf{slot}(t)-1,chkp={\mathsf{C}(b)}} \overline{{\addweight{\mathcal{J}}}}_{\commweightfromslot@From}^{\commweightfromslot@To, \ifthenelse{\equal{\commweightfromslot@Val}{}}{}{\commweightfromslot@Val,} \ifthenelse{\equal{\commweightfromslot@When}{}}{}{\commweightfromslot@When,} \commweightfromslot@ValsetChkp}\subseteq \mathcal{X}.} }{ \begin{minipage}[t]{5cm} \begin{itemize}[label=---,nosep,left=0pt] \item \Cref{prop:gasper-basic:ldm-vote-for-b-is-ffg-vote-for-cb} and condition 1 of the Lemma's statement imply that\setkeys{commweightfromslot}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromslot}{from=\mathsf{first\_slot}(e),to=\mathsf{slot}(t)-1,chkp={\mathsf{C}(b)}} \overline{{\addweight{\mathcal{J}}}}_{\commweightfromslot@From}^{\commweightfromslot@To, \ifthenelse{\equal{\commweightfromslot@Val}{}}{}{\commweightfromslot@Val,} \ifthenelse{\equal{\commweightfromslot@When}{}}{}{\commweightfromslot@When,} \commweightfromslot@ValsetChkp}\subseteq \mathcal{X}. \end{itemize} \end{minipage} } \\ &= \setkeys{commweightfromslot}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromslot}{from=\mathsf{first\_slot}(e),to=\mathsf{slot}(t)-1,chkp={{ \setkeys{finattime}{time=t,val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}}}} \overline{{\addweight{\mathcal{J}}}}_{\commweightfromslot@From}^{\commweightfromslot@To, \ifthenelse{\equal{\commweightfromslot@Val}{}}{}{\commweightfromslot@Val,} \ifthenelse{\equal{\commweightfromslot@When}{}}{}{\commweightfromslot@When,} \commweightfromslot@ValsetChkp} + (1-\beta) \setkeys{commweightfromslot}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromslot}{from=\mathsf{slot}(t),to=\mathsf{last\_slot}(e),chkp={{ \setkeys{finattime}{time=t,val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}}}} \overline{{\addweight{\mathcal{W}}}}_{\commweightfromslot@From}^{\commweightfromslot@To, \ifthenelse{\equal{\commweightfromslot@Val}{}}{}{\commweightfromslot@Val,} \ifthenelse{\equal{\commweightfromslot@When}{}}{}{\commweightfromslot@When,} \commweightfromslot@ValsetChkp} && \ifthenelse{\equal{5cm}{}}{ \text{--- By \Cref{assum:no-change-to-the-validator-set}.} }{ \begin{minipage}[t]{5cm} \begin{itemize}[label=---,nosep,left=0pt] \item By \Cref{assum:no-change-to-the-validator-set}. \end{itemize} \end{minipage} } \\ &\geq (1-\beta) \setkeys{commweightfromslot}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromslot}{from=\mathsf{first\_slot}(e),to=\mathsf{slot}(t)-1,chkp={{ \setkeys{finattime}{time=t,val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}}}} \overline{{\addweight{\mathcal{W}}}}_{\commweightfromslot@From}^{\commweightfromslot@To, \ifthenelse{\equal{\commweightfromslot@Val}{}}{}{\commweightfromslot@Val,} \ifthenelse{\equal{\commweightfromslot@When}{}}{}{\commweightfromslot@When,} \commweightfromslot@ValsetChkp} + (1-\beta) \setkeys{commweightfromslot}{from=,to=,chkp=,when=,val=} \setkeys{commweightfromslot}{from=\mathsf{slot}(t),to=\mathsf{last\_slot}(e),chkp={{ \setkeys{finattime}{time=t,val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}}}} \overline{{\addweight{\mathcal{W}}}}_{\commweightfromslot@From}^{\commweightfromslot@To, \ifthenelse{\equal{\commweightfromslot@Val}{}}{}{\commweightfromslot@Val,} \ifthenelse{\equal{\commweightfromslot@When}{}}{}{\commweightfromslot@When,} \commweightfromslot@ValsetChkp} && \ifthenelse{\equal{5cm}{}}{ \text{--- By \Cref{assum:beta}.} }{ \begin{minipage}[t]{5cm} \begin{itemize}[label=---,nosep,left=0pt] \item By \Cref{assum:beta}. \end{itemize} \end{minipage} } \\ &\geq (1-\beta) \setkeys{commweightatepoch}{epoch=,val=,when=,chkp=} \setkeys{commweightatepoch}{epoch=e,chkp={ \setkeys{finattime}{time=t,val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}}} \hat{\overline{\addweight{\mathcal{W}}}}{}^{\commweightatepoch@Epoch, \ifthenelse{\equal{\commweightatepoch@Val}{}}{}{\commweightatepoch@Val,} \ifthenelse{\equal{\commweightatepoch@When}{}}{}{\commweightatepoch@When,} \commweightatepoch@Chkp } \\ &\geq (1-\beta) \setkeys{totvalsetweight}{chkp=C'} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} && \ifthenelse{\equal{5cm}{}}{ \text{--- As above we have established that\setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=e} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } = \setkeys{totvalset}{chkp=C'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp}.} }{ \begin{minipage}[t]{5cm} \begin{itemize}[label=---,nosep,left=0pt] \item As above we have established that\setkeys{commatepoch}{epoch=,val=,when=} \setkeys{commatepoch}{epoch=e} {\hat{\overline{\mathcal{W}}}}{}^{\commatepoch@Epoch \ifthenelse{\equal{\commatepoch@Val}{}}{}{,\commatepoch@Val} \ifthenelse{\equal{\commatepoch@When}{}}{}{,\commatepoch@When} } = \setkeys{totvalset}{chkp=C'} {\mathcal{W}}_{\mathsf{t}}^{\totvalset@Chkp}. \end{itemize} \end{minipage} } \\ &\geq \frac{2}{3}\mathit{f}^{\beta,e-\mathsf{epoch}( \setkeys{finattime}{time=t,val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}})} \setkeys{totvalsetweight}{chkp={ \setkeys{finattime}{time=t,val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}}} {\addweight{\mathcal{W}}}_{\mathsf{t}}^{\totvalsetweight@ValsetChkp} && \ifthenelse{\equal{5cm}{}}{ \text{--- By \Cref{prop:xxx}.} }{ \begin{minipage}[t]{5cm} \begin{itemize}[label=---,nosep,left=0pt] \item By \Cref{prop:xxx}. \end{itemize} \end{minipage} } \end{align}\]  ◻

Proof. Let us proceed by cases.

Case 1: \(b \preceq \setkeys{finattime}{time=t,val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}\).

Given that \(\mathsf{st}(\mathsf{epoch}(t)-1)\geq \mathbb{GST}\), Property [prop:gasper-basic:finalization] implies that \(b \preceq \setkeys{finattime}{time=t',val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}\). Hence, \(\mathrm{isConfirmed}_v(b, t')\).

Case 2: \(b \npreceq \setkeys{finattime}{time=t,val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}} \land b \preceq \setkeys{finattime}{time=t',val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}\).

Obvious.

Case 3: \(b \npreceq \setkeys{finattime}{time=t,val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}} \land b \npreceq \setkeys{finattime}{time=t',val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}\).

The condition \(\mathrm{isConfirmed}_v(b,t)\) implies that there exists a slot \(s \in [\mathsf{first\_slot}(epoch(t)-1)+1,\mathsf{slot}(t)]\) such that \(\setkeys{varforvalattime}{val=v} \ifthenelse {\equal{\varforvalattime@Time} {}} {{\mathrm{isConfirmedNoCaching}}_{\varforvalattime@Val}} {{\mathrm{isConfirmedNoCaching}}_{\varforvalattime@Val}^{\varforvalattime@Time}} \setkeys{varforvalattime}{time=,val=}(b,\mathsf{st}(s))\). Given that \(\mathsf{st}(s-1)\geq\mathsf{st}(\mathsf{epoch}(t)-1)\geq\mathbb{GST}\), 38 implies that \(b\) is canonical in the view of any honest validator from time \(\mathsf{st}(s)\) and thereafter.

Now, let \(b' := \setkeys{varforvalattime}{val=v} \ifthenelse {\equal{\varforvalattime@Time} {}} {{\mathrm{highestConfirmedSinceEpoch}}_{\varforvalattime@Val}} {{\mathrm{highestConfirmedSinceEpoch}}_{\varforvalattime@Val}^{\varforvalattime@Time}} \setkeys{varforvalattime}{time=,val=}(\mathsf{epoch}(t')-1,t')\). Then, there exists a slot \(s' \in [\mathsf{first\_slot}(\mathsf{epoch}(t')-1)+1,\mathsf{slot}(t')]\) such that \(\setkeys{varforvalattime}{val=v} \ifthenelse {\equal{\varforvalattime@Time} {}} {{\mathrm{isConfirmedNoCaching}}_{\varforvalattime@Val}} {{\mathrm{isConfirmedNoCaching}}_{\varforvalattime@Val}^{\varforvalattime@Time}} \setkeys{varforvalattime}{time=,val=}(b',\mathsf{st}(s'))\). Thanks to 38, this also implies that \(b'\) is canonical for any honest validator at time \(\mathsf{st}(\mathsf{slot}(t'))\).

We can now proceed by cases.

Case 3.1: \(s \in [\mathsf{first\_slot}(\mathsf{epoch}(t')-1)+1,\mathsf{slot}(t')]\).

This implies that \(\mathsf{slot}(b')\geq\mathsf{slot}(b)\). Given that \(b'\) is also canonical at time \(t'\), we can conclude that \(b\preceq b'\).

Case 3.2: \(s \notin [\mathsf{first\_slot}(\mathsf{epoch}(t')-1)+1,\mathsf{slot}(t')]\).

This case implies that \(\mathsf{st}(s)\leq\mathsf{st}(\mathsf{epoch}(t')-1)\). Also, given that \(\mathsf{slot}(b)<s\), this further implies that \(\mathsf{epoch}(b) < \mathsf{epoch}(t')-1\). Hence, given that \(b\) is canonical in the view of any honest validator from time \(\mathsf{st}(s)\) and thereafter, this further implies that \(b\) has been canonical in the view of any honest validator for the entirety of any epoch \(e\) such that \(\min(\mathsf{epoch}(t)+1,\mathsf{epoch}(t')-1)\leq e \leq \mathsf{epoch}(t')-1\).

Now we show that \(\setkeys{varforvalattime}{val=v} \ifthenelse {\equal{\varforvalattime@Time} {}} {{\mathrm{isConfirmedNoChaching}}_{\varforvalattime@Val}} {{\mathrm{isConfirmedNoChaching}}_{\varforvalattime@Val}^{\varforvalattime@Time}} \setkeys{varforvalattime}{time=,val=}(b,\mathsf{st}(\mathsf{slot}(t')))\) is True. Given that \(\mathsf{epoch}(b) < \mathsf{epoch}(t')\), this amounts to proving that are satisfied.

[ln:alg:ffg-no-extra-assum:lmd-safe].

Given that reasoning above, we can apply 40 to conclude that \(\setkeys{varforvalattime}{val=v} \ifthenelse {\equal{\varforvalattime@Time} {}} {{\mathrm{isLMDGHOSTSafe}}_{\varforvalattime@Val}} {{\mathrm{isLMDGHOSTSafe}}_{\varforvalattime@Val}^{\varforvalattime@Time}} \setkeys{varforvalattime}{time=,val=}(b,\mathsf{C}(b),\mathsf{st}(\mathsf{slot}(t')))\).

.

Let us proceed by cases.

Case 1: \(\mathsf{epoch}(b) = \mathsf{epoch}(t)\).

As established above, \(\mathsf{epoch}(t')\geq\mathsf{epoch}(b)+2=\mathsf{epoch}(t)+2\). Given that \(b\) is canonical in the view of any honest validator during the entire epoch \(\mathsf{epoch}(t)+1\) and that \(\mathsf{st}(\mathsf{epoch}(t)+1)\geq\mathbb{GST}\), line [ln:alg:ffg-no-extra-assum:will-chkp-be-justified], 13 and [assum:ffg-no-extra-assum:justified-checkpoint-next-epoch] prove this case.

Case 2: \(\mathsf{epoch}(b) = \mathsf{epoch}(t)-1\).

Obvious as this case implies that were already satisfied at time \(t\) and the view of any validator is monotonically increasing with respect to time.

[ln:alg:ffg-no-extra-assum:old-epoch-prec-fin].

Given that \(b\) is canonical at time \(\mathsf{st}(\mathsf{slot}(t'))\), the definition of LMD-GHOST-HFC(3) implies that \((b \preceq \setkeys{chkpattime}{time=\mathsf{st}(\mathsf{slot}(t')),val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}) \lor (\exists b' \succeq b,\; b' \succeq \setkeys{finattime}{time=\mathsf{st}(\mathsf{slot}(t')),val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}})\).

Case 1: \(b \preceq \setkeys{chkpattime}{time=\mathsf{st}(\mathsf{slot}(t')),val=v} \mathsf{GJ}^{\chkpattime@Time\ifthenelse{\equal{\chkpattime@Val}{}}{}{,\chkpattime@Val}}\).

[prop:gasper-basic:just-succ-finalization] implies that \(b \preceq \setkeys{finattime}{time=\mathsf{st}(\mathsf{slot}(t')),val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}} \lor b \succeq \setkeys{finattime}{time=\mathsf{st}(\mathsf{slot}(t')),val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}\). If \(b \preceq \setkeys{finattime}{time=\mathsf{st}(\mathsf{slot}(t')),val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}\), then, given that \(t' \geq \mathsf{st}(\mathsf{slot}(t'))\), by [prop:gasper-basic:finalization], \(b \preceq \setkeys{finattime}{time=t',val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}\) which contradicts the case 3’s assumption \(b \npreceq \setkeys{finattime}{time=t',val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}\). Hence, \(b \succeq \setkeys{finattime}{time=\mathsf{st}(\mathsf{slot}(t')),val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}\).

Case 2: \(\exists b' \succeq b,\; b' \succeq \setkeys{finattime}{time=\mathsf{st}(\mathsf{slot}(t')),val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}\).

This case implies that \(b \preceq \setkeys{finattime}{time=\mathsf{st}(\mathsf{slot}(t')),val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}} \lor b \succeq \setkeys{finattime}{time=\mathsf{st}(\mathsf{slot}(t')),val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}\). As established in the case above, \(b \preceq \setkeys{finattime}{time=\mathsf{st}(\mathsf{slot}(t')),val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}\) leads to a contradiction. Hence, \(b \succeq \setkeys{finattime}{time=\mathsf{st}(\mathsf{slot}(t')),val=v} \mathsf{GF}^{\finattime@Time\ifthenelse{\equal{\finattime@Val}{}}{}{,\finattime@Val}}\).

.

Note that above we have established that \(\mathsf{epoch}(b)+1\leq\mathsf{epoch}(t')-1\). Given that \(b\) is canonical in the view of any honest validator from time \(\mathsf{st}(\mathsf{epoch}(b)+1)\leq\mathsf{st}(\mathsf{epoch}(t')-1)\) until \(\mathsf{st}(\mathsf{slot}(t'))\), thanks to [prop:gasper-basic:ldm-vote-for-b-is-ffg-vote-for-cb] we can apply 41 to prove that these lines are satisfied.

From \(\setkeys{varforvalattime}{val=v} \ifthenelse {\equal{\varforvalattime@Time} {}} {{\mathrm{isConfirmedNoChaching}}_{\varforvalattime@Val}} {{\mathrm{isConfirmedNoChaching}}_{\varforvalattime@Val}^{\varforvalattime@Time}} \setkeys{varforvalattime}{time=,val=}(b,\mathsf{st}(\mathsf{slot}(t')))\), we can conclude that \(\mathsf{slot}(b')\geq\mathsf{slot}(b)\). Then, given that \(b'\) is also canonical at time \(t'\), we can conclude that \(b\preceq b'\).

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Proof. Direct consequence of [lem:ffg-no-extra-assum-safety,lem:ffg-motonocity-no-extra-assum]. ◻