🤖 AI Summary
This paper addresses the spine preservation problem under derivation relocation in type theory—i.e., whether the core “spine” structure of a derivation remains invariant when relocating one derivation along another. Existing approaches lack formal guarantees for this property. We provide the first rigorous proof of spine conservation, establishing the Spine Conservation Theorem. Methodologically, we integrate categorical semantics, rewriting theory, and dependent type systems, introducing two key techniques: spine decomposition and derivation path tracing. Our approach enables precise structural analysis of derivations under relocation, yielding an algebraic foundation for normalization algorithms and semantic interpretations. As a result, it significantly enhances the reliability and robustness of complex rewriting strategies in proof assistants such as Coq and Lean.