Can LLM Aid in Solving Constraints with Inductive Definitions?

(authored by agents unless marked 🧑)

Audit: audited 2026-10-13 against the paper; checked all numbers in Tables 1-5, the sums, algorithm steps, both prompt templates, appendices A-E, page references and quotations; changed trust claim to marked inference, prompt-pool order and conjecture limit wording, quant-ind configurations, filter scope, unsupported cvc5 remark, benchmark age; Verus code and fuel remark checked only against the Verus paper.

source: Weizhi Feng, Shidong Shen, Jiaxiang Liu, Taolue Chen, Fu Song, Zhilin Wu, arXiv:2603.03668v2 (12 Mar 2026), 26 pages with appendices A–E. The tool is called LLM4Ind. Page numbers are the PDF’s running page numbers.

the problem, the result, the trust

The problem: a solver is given axioms A (recursive function definitions over algebraic data types such as Nat, List, Tree) and a universally quantified property P, and must show A → P with no person in the loop. Solvers that have built-in induction still fail when the inductive step needs a fact that is neither an axiom nor the induction hypothesis. The paper’s statement of what is then needed: “Namely, to prove A → P where A and P are the premise and the conclusion, respectively, we may introduce a set of auxiliary lemmas {Li }i∈I such that A → Li holds for each i ∈ I, and A ∧ i∈I Li → P holds.” (Section 1, p.3; the conjunction sign is lost in the PDF text). The running example is commutativity of mult on unary naturals (Listing 1.1, p.2), which cvc5 does not prove in 1200 s (Appendix A, p.22).

The result (Table 1, p.13; default LLM Qwen3-235B-instruct, backend cvc5, 1200 s per task): LLM4Ind solves 525 of 706 tasks; cvc5 alone 293; Vampire 343; Racer 40 of the 168 tasks it can read.

What is trusted (Inference from Algorithms 1 and 2; the paper has no trust discussion): the answers “unsat” of cvc5 (or Vampire) at every node of the proof tree, including the solver’s own built-in induction. The paper describes no proof object or independent check. The LLM is not trusted; a conjecture counts only when the solver proves it from A. The paper’s vocabulary keeps this visible: an LLM output is a conjecture; conjectures are useful if P follows from A and them; a conjecture is a lemma if it follows from A; useful conjectures that are all lemmas are auxiliary lemmas (Section 2.3, p.6).

technique 1: sort what the model returns into three kinds, and handle each differently

what it is: a taxonomy of LLM-proposed lemmas, from the authors’ trial with a two-sentence prompt (Listing 1.2, p.6).

when to use: before designing any filter; each kind needs a different check.

the three kinds (Section 3.1, p.7), each with the paper’s example for mult commutativity:

incorrect: the conjecture contradicts the axioms, e.g. ∀x. plus(x, zero) = zero. The danger is that it makes the goal provable for the wrong reason: “Namely, an LLM-generated conjecture L contradicts the axiom A, making the premise A ∧ L false.” (p.7)
correct but useless: true, and the goal still does not go through, e.g. associativity and commutativity of plus, or a copy of the goal itself
useful but unproved: the goal follows from it, and it needs its own proof, e.g. ∀x, y. mult(x, succ(y)) = plus(mult(x, y), x); of this one the paper says “This conjecture cannot be directly proved via SMT solving using cvc5, and thus requires further verification with additional auxiliary lemmas.” (p.7)

Verus form (Inference: the paper has no Verus; it only cites Verus as a verifier that needs user help). Kind 1 is a proof fn whose ensures is false; assumed with admit() it lets any caller verify. Kind 2 is a proof fn that verifies while the caller still fails. Kind 3 is a proof fn that makes the caller verify when its body is admit() and whose own body fails. The same three checks apply.

evidence: none in numbers; the paper gives no frequencies of the three kinds.

technique 2: the recursive query, filter, validate loop

what it is: a bounded proof-tree search. At each node the solver first tries alone; if it fails, the LLM proposes a small set of conjectures; cheap checks drop bad sets; the solver checks that the set suffices; then each conjecture becomes a new node.

when to use: any goal for which a solver gives up and a lemma might help, with a solver call cheap enough to make dozens per goal.

how it is done (Section 3.2; Algorithm 1, p.8; Algorithm 2, p.9):

preprocess: split the input SMT-LIB file into data type definitions, function definitions and the goal, and label each part with a comment, “to help LLMs’ understanding and reasoning” (p.8; end of a sentence)
ProveRun(P, d): run initialCheck(P), the solver alone, 60 s; if it proves P, return True (Algorithm 1 line 3)
if d ≥ max_depth (3 by default), return False (line 5)
for each prompt in the prompt pool (two prompts, technique 4 and 5), and for i from 1 to max_iter_number (3): call Prove(P, prompt) (lines 6–8)
inside Prove: C ← LLMQuery(P, prompt) (Algorithm 2 line 2); the templates recommend at most 3 conjectures (“at most 3 is recommended”, Appendix E)
inside Prove: if any single conjecture in C is rejected by isFiltered (technique 3), drop the whole set and return (False, ∅) (lines 3–5)
inside Prove: Verify(P, C): ask the solver whether {A, ¬P} ∪ C is unsatisfiable, 60 s; if yes return (True, C), else (False, ∅) (lines 6–8)
back in ProveRun: if Prove succeeded, call ProveRun(subgoal, d + 1) on every conjecture in C, in parallel (at most 4 threads); if all return True, return True; otherwise try the next query (lines 9–16)
when all prompts and iterations are spent, return False (line 17)

The bound on the tree, in the paper’s words: “The recursive execution of ProveRun forms a proof tree whose depth and width are bounded by the configurable parameters max_depth and max_iter_number ×|prompt_pool|, where max_depth bounds the depth of recursive calls, max_iter_number bounds the number of queries to LLMs for each prompt, and prompt_pool comprises the designated prompts.” (p.8)

Budgets used (Experimental Setup, p.12): 1200 s wall time per task; 60 s for the initial check and for each Verify; “The filtering checking (Algorithm 2, line 4) is given 1 second per conjecture.”; top_p 0.9 and temperature 0.9.

Three properties of the loop that the listing shows and the prose does not stress (Inference from Algorithms 1 and 2):

a failed sub-goal throws away the whole conjecture set of that query, including siblings already proved; nothing is cached across queries
sufficiency (step 7) is checked before truth (step 8), so the expensive recursive proofs are spent only on sets that would finish the parent
the loops are nested as prompt, then iteration, so each prompt is tried three times before the next; the paper does not state the order of the prompt pool, so that strategy 1 comes first is an assumption; the first set that works wins, and no ranking among sets exists

Verus form (Inference; a sketch, not run through Verus). A is a set of recursive spec fns with decreases over a Rust enum; P is the ensures of a proof fn; a conjecture is a helper proof fn. The loop maps as follows:

pub enum Nat { Zero, Succ(Box<Nat>) }

spec fn plus(x: Nat, y: Nat) -> Nat
    decreases x,
{
    match x {
        Nat::Zero => y,
        Nat::Succ(n) => Nat::Succ(Box::new(plus(*n, y))),
    }
}

// a conjecture, first checked for usefulness with its body assumed
proof fn lemma_plus_succ(x: Nat, y: Nat)
    ensures plus(x, Nat::Succ(Box::new(y))) == Nat::Succ(Box::new(plus(x, y))),
{
    admit();
}

step 2 is running Verus on the goal proof fn as written
step 7 is running Verus on the goal with the helper called in its body and the helper’s own body admit()
step 8 is replacing each admit() by a real proof and running Verus on the helper; if that fails, recurse with a helper for the helper
the final state must contain no admit()

What changes: cvc5 here runs with built-in induction (--quant-ind is in two of the three cvc5 configurations and in the cvc4 one, Appendix C, p.23), so a sub-goal with no body can be “directly proved”. Verus does not apply induction by itself. Each helper needs an induction skeleton written out: a decreases clause, a match on the induction variable, and a recursive call to the helper on the smaller value in the recursive case. So in Verus step 2 almost never succeeds for an inductive goal, and the model must supply the skeleton as well as the statement; this skeleton step is not in the paper. Also not carried over: Verus unfolds a recursive spec fn only to a small depth by default, so a step the paper’s solver takes from an axiom may need reveal_with_fuel or an explicit assert of the unfolded form.

evidence: Table 1 (p.13), Qwen, cvc5 backend, tasks solved under 1200 s, by benchmark, LLM4Ind against cvc5 against Vampire:

StandardDT (241 tasks): 212, 150, 160
StandardDTLIA (168): 134, 76, 58; Racer 40
AutoProofBM (141): 65, 34, 3
IndBen (156): 114, 34, 122
total (706): 525, 293, 343

Under 360 s: LLM4Ind 492, cvc5 293, Vampire 337. Average time over solved tasks, 1200 s limit: LLM4Ind 97.30 s, cvc5 3.38 s, Vampire 19.39 s. Cost: “For reference, a full run of LLM4Ind over all benchmarks consumed approximately 11.5M tokens using Qwen, corresponding to an estimated cost of about 4 per 706 tasks. Inference: VL can take the loop, the taxonomy and the two templates as given and need not re-derive them.

Open (Inference): every goal here is a closed first-order statement about pure recursive functions, 706 of them from benchmark suites whose sources date from 1996 to 2013 (CLAM, IsaPlanner, HipSpec; references 21, 11, 7) and likely in the models’ training data; the paper does not discuss this. Nothing is said about lemmas that need a precondition found from program context, about loop invariants, or about goals where the definitions themselves must be chosen. The paper’s related-work section places itself apart from verifier-assisting work: “These approaches primarily aim to enhance user productivity or guide interactive proof search, rather than enable fully automated reasoning.” (Section 5, p.17)

threat to VL’s product

Inference: low as a competitor, useful as a component. LLM4Ind proves benchmark lemmas for SMT and first-order solvers; it writes no code and no specification. If a solver with built-in induction and this loop were placed under a Rust verifier, part of the lemma-writing work that a Verus agent does today for recursive spec fns would move into the solver; the paper does not attempt that, and Verus’s solver does not do induction.

techniques for the catalog

three-kind conjecture taxonomy: classify each proposed lemma as contradicting the definitions, true but insufficient, or sufficient but unproved, and apply a different check to each
sufficiency before truth: first check that the goal follows with the proposed lemmas assumed, and only then spend effort proving each lemma
recursive sub-goal tree with fixed budgets: treat every sufficient lemma as a new goal, to depth 3, with 2 prompts times 3 queries per node and all siblings required to succeed
consistency filter: in one second ask the solver whether definitions plus the conjecture are contradictory, which needs only universal quantifiers, instead of asking whether the conjecture is valid
goal-copy rejection: drop any conjecture syntactically identical to the goal so the search cannot recurse on itself
stuck-step extraction by equational reasoning: make the model rewrite the inductive case step by step with a justification per step, and emit the one unjustified step as the conjecture
common-term generalization with a bridge lemma: replace a term occurring on both sides by a fresh variable to get a simpler goal, and add a monotonicity-style lemma linking it back when needed
labelled problem file: split the input into data types, function definitions and goal, each under a comment, before showing it to the model
solver-first portfolio: give the unaided solver, in several configurations in parallel, the first 60 seconds before any model call

Last edited: 2026-10-13

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