Lookahead Proof-Strategy Selection: Evaluating Alternative Proof Approaches Before Committing to a Complete Derivation

Mathematical theorem proving often involves selecting an appropriate proof strategy before developing a complete derivation. A theorem may admit multiple proof approaches, including direct proof, proof by contradiction, mathematical induction, contrapositive reasoning, construction, case analysis, and the application of auxiliary lemmas. However, committing to one strategy too early may result in lengthy derivations, unnecessary intermediate steps, or proof attempts that eventually become difficult to complete. Lookahead Proof-Strategy Selection (LPSS) investigates whether evaluating alternative proof approaches through preliminary reasoning previews can provide useful information for selecting a proof strategy before committing computational or reasoning resources to a complete derivation. The proposed framework treats proof-strategy selection as a prospective decision problem. Rather than immediately expanding the first plausible approach, LPSS generates multiple candidate strategies, constructs limited previews of their potential derivations, estimates their expected usefulness, and selects a strategy according to a quality-, feasibility-, uncertainty-, and cost-aware decision policy. The framework consists of five principal components: candidate proof-strategy generation, preliminary derivation construction, downstream proof-quality prediction, uncertainty-aware strategy evaluation, and adaptive strategy selection. Candidate approaches are evaluated using several proposed dimensions, including logical validity, theorem coverage, expected derivation length, dependency complexity, formal verification feasibility, and computational cost. The framework also considers the possibility that a strategy with a promising initial derivation may encounter substantial difficulties at later stages. A mathematical formulation is introduced to represent the expected utility of alternative proof strategies. The formulation distinguishes between the apparent quality of a preliminary derivation and the estimated probability of completing a correct proof. It also incorporates the computational resources required to evaluate candidate approaches. This paper presents LPSS as a conceptual and testable framework for prospective mathematical reasoning. It does not claim experimental superiority over existing theorem-proving systems or establish that preliminary proof evaluation reliably predicts final proof success. Instead, it identifies a structured research direction for investigating whether limited reasoning previews can improve proof-strategy decisions, reduce unproductive derivation attempts, and support more efficient allocation of mathematical reasoning resources.