pith. sign in

arxiv: 2605.04472 · v1 · submitted 2026-05-06 · 💻 cs.LG

Automated Formal Proofs of Combinatorial Identities via Wilf-Zeilberger Guidance and LLMs

Beibei Xiong , Hangyu Lv , Junqi Liu , Yisen Wang , Shaoshi Chen , Jianlin Wang , Zhengfeng Yang , Lihong Zhi This is my paper

Pith reviewed 2026-05-08 17:08 UTC · model grok-4.3

classification 💻 cs.LG
keywords Wilf-Zeilberger methodcombinatorial identitiesformal proofsLean 4neuro-symbolic AILLM-based proversautomated theorem provingrecurrence relations
0
0 comments X

The pith

The WZ-LLM framework combines Wilf-Zeilberger symbolic plans with LLM subgoal proving in Lean 4 to automate formal proofs of combinatorial identities.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper introduces a neuro-symbolic method that uses the Wilf-Zeilberger technique to construct auxiliary functions and recurrences for combinatorial identities, converts those into Lean 4 proof sketches, and then applies a trained LLM prover to complete the machine-checkable parts. Pure LLM provers often fail on these tasks because they require extended planning that leads to search explosion. By providing the WZ-derived structure as guidance, the system improves the ability to verify identities formally. Readers would care because successful automation could make formal verification of discrete mathematics results more routine and reliable.

Core claim

The paper claims that turning WZ proof plans into executable Lean sketches allows an LLM prover to discharge the subgoals, leading to improved performance on benchmarks of combinatorial identities compared to baselines that lack this guidance. It also describes training a dedicated WZ-Prover through a bootstrapping process verified in the Lean kernel.

What carries the argument

The WZ-LLM neuro-symbolic framework that translates Wilf-Zeilberger auxiliary functions and recurrence relations into Lean 4 sketches for LLM completion.

Load-bearing premise

That the Wilf-Zeilberger method can reliably produce useful executable Lean sketches for a broad range of combinatorial identities and that the LLM prover can consistently discharge the resulting subgoals without excessive search explosion.

What would settle it

Observing a large set of classic combinatorial identities where either WZ fails to yield valid sketches or the LLM cannot resolve the subgoals despite the guidance.

Figures

Figures reproduced from arXiv: 2605.04472 by Beibei Xiong, Hangyu Lv, Jianlin Wang, Junqi Liu, Lihong Zhi, Shaoshi Chen, Yisen Wang, Zhengfeng Yang.

Figure 1
Figure 1. Figure 1: Overview of WZ-LLM. (A) Symbolic decomposition. Given combinatorial-identity statements, the symbolic engine checks WZ applicability and, when applicable, constructs a WZ proof sketch that is decomposed into sub-goals. Statements that are not amenable to the WZ method are labeled as WZ-uncovered targets. All sub-goals and uncovered targets form a shared pool of proving tasks. (B) Model training. We cold-st… view at source ↗
Figure 2
Figure 2. Figure 2: Lean InfoView output of the wz prove tactic. The tactic automatically invokes symbolic computation and the trained prover to generate Lean proof suggestions, which can be accepted to fill in the complete formal proof. the complete proof is inserted into the Lean file, as shown in Listing 1, and verified by the kernel, yielding a fully checked end-to-end formal proof with minimal user intervention. 23 view at source ↗
read the original abstract

Automating formal proofs of combinatorial identities is challenging for LLM-based provers, as long-horizon proof planning is required and unconstrained search quickly explodes. Symbolic methods such as the Wilf-Zeilberger (WZ) method can achieve a mechanized proof of combinatorial identities by constructing special auxiliary functions and demonstrating that they satisfy specific recurrence relations. We propose WZ-LLM, a neuro-symbolic framework that turns WZ proof plans into executable proof sketches in Lean 4 and uses an LLM-based prover to discharge the resulting machine-checkable subgoals. We also train a dedicated WZ-Prover via a Lean-kernel-verified bootstrapping loop with expert-verified iteration, followed by DAPO-based refinement. Experiments show that WZ-LLM achieves a 34% proof success rate on LCI-Test (100 classic combinatorial identities), outperforming strong baselines such as DeepSeek-V3 and Goedel-Prover-V2, and delivering consistent gains on CombiBench and PutnamBench-Comb. These results indicate that our framework provides two complementary strengths: improved direct proving for identities beyond the scope of WZ, and substantially higher end-to-end success when WZ sketches guide a specialized prover.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper introduces WZ-LLM, a neuro-symbolic framework that uses the Wilf-Zeilberger method to produce executable Lean 4 proof sketches for combinatorial identities and employs a specialized LLM prover (trained via Lean-verified bootstrapping and DAPO refinement) to discharge the resulting subgoals. It reports a 34% success rate on the LCI-Test benchmark consisting of 100 classic identities, outperforming general-purpose baselines such as DeepSeek-V3 and Goedel-Prover-V2, along with consistent gains on CombiBench and PutnamBench-Comb.

Significance. If the performance improvements can be attributed to the WZ guidance rather than solely to specialized training, the work demonstrates a practical way to combine symbolic recurrence-based planning with neural search for long-horizon combinatorial proofs. The Lean-kernel-verified bootstrapping loop supplies a reproducible, machine-checked foundation for the prover component.

major comments (2)
  1. [Experimental results on LCI-Test] The LCI-Test evaluation compares WZ-LLM against general-purpose models but omits an ablation that runs the identical trained WZ-Prover without WZ-generated sketches. Because the prover itself is fine-tuned on combinatorial identities via bootstrapping, the reported 34% success rate and benchmark gains do not isolate whether the symbolic guidance contributes measurably beyond the fine-tuning alone; this directly affects the central neuro-symbolic claim.
  2. [Abstract and experimental section] The abstract and experimental description supply no information on controls for the baselines, implementation details of the prover, statistical tests for the reported gains, or failure-mode analysis. Without these, the concrete performance numbers cannot be fully verified or reproduced.
minor comments (1)
  1. [Method description] A short concrete example showing the conversion of one WZ plan into a Lean sketch would clarify the interface between the symbolic and neural components.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our work. We address each major comment below and commit to revisions that will strengthen the experimental validation and reproducibility of our results.

read point-by-point responses

  1. Referee: [Experimental results on LCI-Test] The LCI-Test evaluation compares WZ-LLM against general-purpose models but omits an ablation that runs the identical trained WZ-Prover without WZ-generated sketches. Because the prover itself is fine-tuned on combinatorial identities via bootstrapping, the reported 34% success rate and benchmark gains do not isolate whether the symbolic guidance contributes measurably beyond the fine-tuning alone; this directly affects the central neuro-symbolic claim.

    Authors: We agree that an ablation isolating the effect of WZ-generated sketches is crucial for substantiating the neuro-symbolic contribution. In the revised manuscript, we will add results from running the trained WZ-Prover directly on the LCI-Test identities without WZ sketches (e.g., using standard prompting for full proofs). This will allow direct comparison to the guided version and clarify the added value of the symbolic planning. We note that the current baselines are general-purpose models not specialized for combinatorics, while our WZ-Prover is tailored, but the ablation will further support the claim. revision: yes

  2. Referee: [Abstract and experimental section] The abstract and experimental description supply no information on controls for the baselines, implementation details of the prover, statistical tests for the reported gains, or failure-mode analysis. Without these, the concrete performance numbers cannot be fully verified or reproduced.

    Authors: We acknowledge the lack of these details in the current version. The revised experimental section will include: detailed descriptions of baseline controls and setups, full implementation details and hyperparameters for the WZ-Prover (including bootstrapping loop and DAPO refinement), statistical tests such as paired t-tests or bootstrap confidence intervals for the reported improvements, and a comprehensive failure-mode analysis breaking down unsuccessful cases (e.g., subgoal failures vs. sketch generation issues). We will also ensure the abstract highlights key methodological aspects for context. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation chain is self-contained

full rationale

The paper's core contribution is an empirical neuro-symbolic pipeline: the established Wilf-Zeilberger method supplies Lean sketches whose subgoals are discharged by an LLM prover (with a separately trained WZ-Prover obtained via kernel-verified bootstrapping and DAPO refinement). No equation, definition, or performance metric is shown to reduce by construction to its own inputs; the 34% success rate and benchmark gains are reported as measured outcomes on held-out test sets (LCI-Test, CombiBench, PutnamBench-Comb) rather than tautological re-statements of training data or self-citations. Standard LLM fine-tuning practices and an external symbolic method do not trigger any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework relies on the pre-existing Wilf-Zeilberger method as a domain assumption and standard practices in LLM training; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The Wilf-Zeilberger method can construct auxiliary functions satisfying recurrence relations that mechanize proofs of combinatorial identities.
    This is the core symbolic component invoked to generate the Lean proof sketches.

pith-pipeline@v0.9.0 · 5537 in / 1411 out tokens · 35190 ms · 2026-05-08T17:08:33.757297+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

  1. [1]

    local obligation

    doi: 10.1007/978-3-7091-0445-3. URL https: //doi.org/10.1007/978-3-7091-0445-3. Langley, P. Crafting papers on machine learning. In Langley, P. (ed.),Proceedings of the 17th International Conference on Machine Learning (ICML 2000), pp. 1207–1216, Stan- ford, CA, 2000. Morgan Kaufmann. Lin, Y ., Tang, S., Lyu, B., Yang, Z., Chung, J.-H., Zhao, H., Jiang, L...

    work page doi:10.1007/978-3-7091-0445-3 2000

  2. [2]

    The hypothesis1≤nimplies0< n, which lets us rewritenas(n−1) + 1viaNat.succ pred eq of pos

    work page

  3. [3]

    Therefore, once we rewritenas(n-1)+1, the goal reduces to a direct rewrite byNat.factorial succ

    Lean’s factorial recurrence is encoded as Nat.factorial succ: (m+1)! = (m+1) * m!. Therefore, once we rewritenas(n-1)+1, the goal reduces to a direct rewrite byNat.factorial succ

    work page

  4. [4]

    Proof plan

    No induction is needed: the proof is a one-step normalization that bridges the arithmetic identity n = (n-1)+1 with the definitional equation of factorial. Proof plan

    work page

  5. [5]

    Derive0 < nfrom the assumption1≤n

    work page

  6. [6]

    Rewritenas(n - 1) + 1usingNat.succ pred eq of pos

    work page

  7. [7]

    Replacen!by((n-1)+1)!and rewrite it usingNat.factorial succ

    work page

  8. [8]

    Complete Lean 4 Proof

    Finish by simplification (simp) and associativity/commutativity normalization (ringis unnecessary forNhere). Complete Lean 4 Proof. 1import Mathlib 2 3open Nat Finset BigOperators 4 5theorem factorial_succ_ge_one {n :N} (h : 1≤n) : 6n * (n - 1)! = n! := by 7have hn : 0 < n := Nat.pos_of_ne_zero (by 8-- 1≤n implies n̸=0 9exact Nat.ne_of_gt (Nat.lt_of_lt_of...

    work page 2026

  9. [9]

    This is a cleaner arithmetic target than reasoning directly about a casted subtraction inR

    From n > m we get m < n+ 1 in N immediately. This is a cleaner arithmetic target than reasoning directly about a casted subtraction inR

    work page

  10. [10]

    If we can show(m:R)<(n+ 1 :R), then (n−m+ 1 :R) = (n+ 1 :R)−(m:R)>0, hence it cannot be zero (discharged bylinarith)

    work page

  11. [11]

    The remaining linear arithmetic on naturals is solved byomega

    The bridge between the N-inequality and the R-inequality is handled by norm cast. The remaining linear arithmetic on naturals is solved byomega. Proof plan

    work page

  12. [12]

    Reduce the non-vanishing goal to a positivity/inequality statement: it suffices to prove (m:R)<(n+ 1 :R) , then conclude bylinarith

    work page

  13. [13]

    Usenorm castto turn(m:R)<(n+ 1 :R)into the natural-number inequalitym < n+ 1

    work page

  14. [14]

    Complete Lean 4 Proof

    Provem < n+ 1from the hypothesisn > musingomega. Complete Lean 4 Proof. 1import Mathlib 2 3open Real Nat Finset BigOperators Polynomial 4 5set_option maxHeartbeats 8000000000 6theorem hwz (n m :N) (htotalNumidx : n > m) : 7((↑n -↑m + 1) :R)̸=0 := by 8-- Reduce to a strict inequality inR, then finish by linear arithmetic. 9suffices h : (m :R) < (n + 1 :R) ...

    work page 2026

  15. [15]

    (n+1) := by 266rw [Finset.sum_range_add] 267simp only [range_one, sum_singleton, add_zero, add_left_inj, add_sub] 268congr 2 269apply Finset.sum_congr rfl 270intro k hidx 271simp only [mem_range] at hidx 272exact WZ k hidx 273_ = (G n n - G n 0) + F (n + 1) n - F n n + F (n + 1) (n+1) := by 274-- telescoping:Σ(G(k+1)-G(k)) = G(n)-G(0) 275congr 3 276apply ...

    work page