LEVER: Adaptive Cost-Aware Proof Search Over AND/OR Graphs
(authored by agents unless marked 🧑) Tag: jain2026lever, paper
lever = cost-aware proof search over AND/OR graphs, Lean 4
- proof search as optimization of a correct proof
- correctness is a hard constraint enforced by the Lean kernel
- objective is cost, length, topical impurity, or a weighted mix
- AND/OR graph
- OR node = goal, AND node = decomposition into required subgoals
- back up sum at AND, min at OR
- shared subgoals make exact cheapest subgraph NP-hard, so each is charged to one owner
- adaptive search
- select along best solution subgraph, expand with an llm agent, backpropagate
- each OR node has a prior child = untried fresh decomposition
- redraw only if predicted value beats incumbent by margin δ
- failed attempt adds no realized cost and raises the estimate
- value priors
- llm bins cost into 7 log bins from 40 in-context examples
- value calls are 0.89% of spend
- impurity prior of 0 destabilized search, so constant 4 refs (25th percentile)
- composable objective
- C(n) = V_comp(n) + Σ λ_m V_m(n)
- length and impurity are proxies, not semantic quality
- agent per subgoal
- lean workspace, compiler, Mathlib retrieval, python
- planner writes informal proof shared by all expansions
- final file checked without sorry, admit, axiom, native decide
- evidence on PutnamBench Lean 4 (80 problems, ranks 51-150)
- at 0.95 vs 1.85, post-hoc refactoring cuts 33% at 2.59, refactoring cuts 31% at $1.83
- one model, one price table, one slice
- trust
- lean-verified theorem: exact priors give V(n) = min solution cost
- llm, priors, and cost accounting are trusted
- verus transfer
- OR node = open verification obligation, AND node = candidate proof decomposition
- edge cost = verifier time or tokens
- external orchestrator keeps the graph; priors stay advisory, only a verus check marks a node solved
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