\subsection{Soundness of Reduction}
\label{sec:soundness-of-reduction}

Let $\text{IN}_0$ be the IN translated from a neural network $\text{NN}$. Let $\text{IN}_n$ be
the IN after $n$ reduction steps. Then we need to prove:
$$
\forall n \in \mathbb{N} \quad \llbracket \text{IN}_n \rrbracket = \llbracket \text{NN} \rrbracket
$$

\paragraph{Base case: $n = 0$} By \textbf{\Cref{sec:soundness-of-translation}}, the initial
$\text{IN}_0$ is constructed such that its semantics $\llbracket \text{IN}_0 \rrbracket$ exactly
match the mathematical definition of the ONNX operators in $\text{NN}$.

\paragraph{Induction step: $n \to n + 1$} Assume $\llbracket \text{IN}_n \rrbracket = \llbracket \text{NN} \rrbracket$.
If $\text{IN}_n$ is in normal form, the proof is complete. Otherwise, there exists an active pair
$A \bowtie B$ that reduces $\text{IN}_n$ to $\text{IN}_{n+1}$. By \textbf{\Cref{sec:soundness-of-interaction-rules}},
the mathematical definition is preserved after any reduction step, it follows that:
$$
\llbracket \text{IN}_{n+1} \rrbracket = \llbracket \text{IN}_{n} \rrbracket
$$
By the inductive hypothesis:
$$
\llbracket \text{IN}_{n+1} \rrbracket = \llbracket \text{NN} \rrbracket
$$

By the principle of mathematical induction, $\text{IN}_n$ remains semantically equivalent to the
original $\text{NN}$ at every step of the reduction process. Additionally, since IN are confluent,
the reduced mathematical expression is unique regardless of order in which rules are applied.