Cut-elimination Theorem

The Cut-elimination Theorem is a fundamental result in proof theory stating that any logical proof can be transformed into a form that omits the "cut" rule—an inference step used to simplify proofs. In essence, it shows that proofs can be made more direct and transparent, relying only on basic, foundational steps, without relying on "shortcuts" through intermediate lemmas. This process enhances clarity, consistency, and understandability of logical systems, ensuring that complex proofs can be reconstructed without relying on the cut rule, thereby strengthening the structure and reliability of formal reasoning.