papers/zoo-identity-chain/sections/08-formal-verification.tex

\section{Formal Verification}


\subsection{Lean 4 Proof Architecture}

The trust semantics of \IChain{} are formally verified in Lean 4 across three modules:

\textbf{Trust/Authority.lean} --- Proves that authority delegation is transitive but bounded. A DID with authority level $\ell$ can delegate to another DID at level $\leq \ell - 1$. This prevents authority inflation.

\begin{theorem}[Authority Non-Inflation]
For any chain of authority delegations $d_0 \to d_1 \to \cdots \to d_k$, the authority level is strictly decreasing:
\[
\text{auth}(d_0) > \text{auth}(d_1) > \cdots > \text{auth}(d_k)
\]
\end{theorem}

\textbf{Trust/Vouch.lean} --- Proves that the vouching system is Sybil-resistant under the assumption that at most $f < k/2$ of the vouching DIDs are controlled by a single entity. The proof establishes that forging a Level 1 credential requires controlling $\geq k/2 + 1$ independent DIDs.

\begin{theorem}[Vouch Sybil Resistance]
If an adversary controls at most $f$ DIDs where $f < \lfloor k/2 \rfloor + 1$, then the adversary cannot produce a valid vouch chain for a Level 1 credential.
\end{theorem}

\textbf{Trust/Revocation.lean} --- Proves that revocation is:
\begin{enumerate}
\item \textbf{Immediate}: A revoked credential fails verification in the next block.
\item \textbf{Irreversible}: Once revoked, a credential cannot be un-revoked (the accumulator update is one-way).
\item \textbf{Non-interfering}: Revoking one credential does not affect the validity of others.
\end{enumerate}

\subsection{Verification Coverage}

\begin{table}[h]
\centering
\begin{tabular}{llc}
\toprule
\textbf{Module} & \textbf{Property} & \textbf{Status} \\
\midrule
Trust/Authority.lean & Non-inflation & Verified \\
Trust/Authority.lean & Bounded delegation depth & Verified \\
Trust/Vouch.lean & Sybil resistance & Verified \\
Trust/Vouch.lean & Convergence (finite vouching) & Verified \\
Trust/Revocation.lean & Immediate revocation & Verified \\
Trust/Revocation.lean & Irreversibility & Verified \\
Trust/Revocation.lean & Non-interference & Verified \\
\bottomrule
\end{tabular}
\caption{Formal verification status}
\label{tab:verification}
\end{table}