LLM-Driven Formal Verification of Complex Theorems: Engineering Practices

As the reasoning capabilities of large models continue to improve, formal verification of complex mathematical theorems is shifting from long-term, expert-led efforts toward engineering practices built on human-machine collaboration, tool-chain support, and platform-level infrastructure.

Formal verification is not merely asking a model to “write a proof.” It requires a system to understand rigorous mathematical semantics, generate checkable proof steps, and use an independent verifier to confirm that every step follows logical rules. This article systematically discusses the methodology for using large models in formal verification of complex theorems, covering technical principles, engineering workflows, evaluation metrics, risk governance, and implementation recommendations.

1. Why Formal Verification Is a Key Testbed for Complex Reasoning

The core value of formal verification is to transform mathematical arguments—traditionally completed by human experts through experience, intuition, and long-chain reasoning—into a symbolic process that machines can check, reproduce, and audit. In traditional mathematical proofs, a conclusion often depends on multiple intermediate theorems, boundary conditions, induction assumptions, and algebraic transformations. Without strict verification tools, a proof may still fail because of hidden assumptions, skipped derivation steps, or missing boundary cases, even if it appears plausible.

Formal verification is also important in software engineering, protocol design, cryptography, financial risk control, and control systems. For example, critical software systems may need to prove that they cannot produce out-of-bounds access under any valid input; concurrent systems may need to prove the absence of deadlock or data races; security protocols may need to prove that confidentiality and integrity hold under a specified attack model. These tasks are not merely about writing code. They involve mathematical modeling of complex state spaces and proving properties in a machine-checkable way.

The introduction of large models can significantly change how traditional formal verification is performed. In the past, proof strategy generation, lemma decomposition, pattern matching, and failure-path analysis relied heavily on expert experience. Today, large models can act as strategy generators, knowledge retrievers, error interpreters, and proof-draft generators, forming a closed loop with formal verification tools. The model proposes potentially effective proof directions, the verifier determines whether they are valid, and the system iterates based on the feedback.

2. Key Technical Paths for LLM Participation in Formal Proofs

2.1 From Natural-Language Statements to Formal Goals

The first step in formalizing a complex theorem is to convert a natural-language mathematical statement into a rigorous formal expression. This process involves several difficulties: how mathematical objects should be defined, how variable ranges should be constrained, how implicit conditions should be made explicit, and how the theorem goal should be decomposed into verifiable subgoals.

Large models can play a role in “semantic alignment” at this stage. For example, given a natural-language statement about number theory, geometry, or algebraic structures, the model can extract key objects, relations, and constraints, and generate candidate formal definitions. However, this process cannot rely on a single model output. Structured validation mechanisms are required to check whether variables are correctly declared, types are consistent, definitions are self-consistent, and the target is equivalent to the original statement.

In engineering practice, a formal modeling layer is usually established, dividing the task into three types of objects:

• Definitions: describe mathematical objects and their properties;
• Axioms or known premises: clarify the initial assumptions accepted by the system;
• Theorem goals: define the final conclusions that need to be proven.

Only when these three types of objects are clear and accepted by the verifier can subsequent proof search be meaningful.

2.2 Modeling Proof States and Generating Strategies

Formal proof is not a one-shot generation of a final answer. It continuously searches for the next operation based on the current state. The current state can be abstracted as:

• the current goal to be proven;
• established assumptions;
• introduced lemmas;
• historical search paths;
• available proof strategies or rules.

The value of large models at this layer is to propose candidate strategies with a high probability of being effective. The model can decompose a complex goal into multiple subgoals, or draw analogies from existing proof patterns, generating strategies such as induction, contradiction, construction, and reduction. However, model-generated strategies must be confirmed by the verifier and cannot be treated directly as correct proofs.

A typical strategy-generation flow is as follows:

def proof_search(goal, model, verifier, max_depth=20):
    frontier = [Node(goal=goal, path=[])]
    while frontier:
        node = frontier.pop(0)
        for tactic in model.propose(node):
            candidate = apply(node, tactic)
            report = verifier.check(candidate)
            if not report.valid:
                continue
            if report.qed:
                return candidate.path
            if len(candidate.path) < max_depth:
                frontier.append(candidate)
    return None

This pseudocode represents a “generate-verify-expand” loop: the model proposes candidate tactics, the system applies the tactic to generate a new state, the verifier checks whether the state is valid, and if the goal is completed, the proof path is returned; otherwise, the search space continues to expand.

2.3 Retrieval Augmentation and Reuse of Historical Proofs

In complex theorem proving, many subproblems are not entirely new. They may have structural similarities to existing theorems, common lemmas, or historical proof fragments. Therefore, retrieval augmentation is an important means of improving efficiency.

The system can build a formal proof library that stores verified lemmas, definitions, proof patterns, and failure cases. When a large model faces a new goal, it can retrieve relevant patterns from the library, such as:

• whether there is a similar induction structure;
• whether there are reusable inequality techniques;
• whether there is similar handling of boundary conditions;
• whether there is a known theorem that can be reduced to.

The significance of retrieval augmentation is not only to improve speed, but more importantly to reduce search-space explosion. Formal proofs often contain many invalid branches. If the system relies entirely on free-form model generation, resource consumption can be very high. By guiding the model with historical proof patterns, the system can prioritize exploring high-value paths.

2.4 Symbolic Verification Loops and Error Feedback

One of the most common problems with large models is that they may generate proof steps that look reasonable but cannot be accepted by the machine. Therefore, a formal verification system must insist that the verifier is the final judge. Every step output by the model must enter the symbolic verification stage, where the verifier checks syntax, types, rule references, and logical dependencies.

When a strategy fails, the system should not simply discard the result. Instead, it should extract the failure reason, such as:

• type mismatch;
• goal not fully simplified;
• cited lemma invalid;
• induction hypothesis not satisfied;
• boundary condition missing;
• algebraic transformation applied in the wrong direction.

This failure information can be fed back to the model as context for the next round of reasoning. Through multiple rounds of feedback, the system can gradually converge toward a valid proof path.

3. A Complete Engineering Workflow for Formalizing Complex Theorems

Using complex theorems in number theory or algebra as an example, a complete formal verification process usually includes the following stages.

3.1 Problem Definition and Goal Decomposition

First, the original statement, applicable scope, preconditions, and final goal of the theorem should be clarified. For highly difficult statements, the system should not ask the model to complete the entire proof at once. Instead, the statement should be decomposed into several intermediate lemmas.

For example, a complex proposition can be decomposed into:

1. proving that the base structure satisfies closure;
2. proving that a key invariant remains unchanged during transformations;
3. proving that the target set is contained in an acceptable set;
4. proving that boundary conditions hold;
5. combining the lemmas to derive the final conclusion.

The quality of decomposition directly determines subsequent efficiency. If the decomposition is too coarse, the model may find it difficult to control the proof path. If it is too fine, the system may generate many low-value subtasks. In practice, a common approach is: human experts define the skeleton, the model fills in the details, and the verifier continuously checks the results.

3.2 Formal Knowledge Preparation

Before starting the proof, a mathematical foundation that can be accepted by the verifier must be prepared. This includes type definitions, algebraic structures, axiom systems, existing lemma libraries, and commonly used inference rules. The more complete this knowledge layer is, the less likely the model is to fall into ineffective search.

The knowledge-preparation stage should also establish a terminology mapping table that links natural-language concepts to formal objects. For example, concepts such as “the set of positive integers,” “the divisibility relation,” “congruence structures,” and “monotonic functions” all need clear definitions and checkable constraints. Otherwise, the model may produce semantic deviations, causing subsequent proofs to drift away from the original problem.

3.3 Parallel Candidate Generation and Pruning

Complex theorems often have multiple feasible proof paths. The system can launch multiple reasoning branches in parallel and let the model generate different strategy candidates, such as induction, contradiction, construction, reduction, or algebraic transformation.

After parallel generation, pruning is required. Common pruning criteria include:

• whether the strategy can be accepted by the verifier;
• whether it significantly narrows the goal;
• whether it depends on too many unproven assumptions;
• whether it introduces new boundary conditions;
• whether it causes search depth to become uncontrollable;
• whether it repeats known failed paths.

Through pruning, the system can concentrate limited resources on high-value proof branches and avoid getting trapped in paths that look complex but cannot converge.

3.4 Verification, Failure Attribution, and Replanning

A failure returned by the verifier does not mean the task is over. Instead, it provides an improvement signal. The system needs to analyze the cause of failure and decide whether to replan.

Common handling methods include:

• if there is a type mismatch, correct the definition or re-constrain variables;
• if the goal is not simplified enough, introduce additional lemmas or adjust the reduction direction;
• if a boundary condition is missing, add a range proof;
• if the strategy is too broad, narrow the search space;
• if multiple branches continue to fail, return to the previous layer and decompose the problem again.

This failure-attribution-driven replanning mechanism is key to enabling automated formal verification to handle highly difficult tasks.

3.5 Preservation and Reproducibility of Proof Artifacts

A successful proof should not be treated as a one-time result. It should be preserved as a reproducible asset. Proof artifacts include:

• formal definitions;
• lemma sets;
• proof scripts;
• verification logs;
• search paths;
• failure cases;
• human review records.

This information can be used for regression testing, benchmark construction, and model evaluation. For complex projects, the value of formal verification is not only to obtain a proof once, but also to build a continuously evolving proof knowledge system.

4. Evaluation: More Than Just Completing a Proof

When evaluating the ability of large models to participate in formal verification, a single metric such as “whether the proof was completed” is not enough. A more complete evaluation system should consider the following dimensions.

4.1 Verification Pass Rate

The verification pass rate is the most basic metric. It indicates how many proofs the system can independently generate that are accepted by the verifier. However, this metric must be combined with task-difficulty stratification. If only simple formalization tasks are evaluated, the results may be misleading.

4.2 Proof Length and Step Efficiency

Even when a proof is completed, path length can vary significantly. An overly long proof implies high search cost, high maintenance difficulty, and possibly redundant steps. Proof length, the number of key steps, and reduction depth can all be used as quality metrics.

4.3 Search Rounds and Resource Consumption

Automated proof usually requires multiple rounds of generation, verification, and backtracking. The system should track average search rounds, verifier invocation counts, inference resource consumption, and the proportion of failed retries. For engineering systems, resource consumption directly determines whether the approach is practical.

4.4 Human Intervention Rate

The human intervention rate reflects the degree of automation. Ideally, the model should request expert confirmation only at a small number of key nodes, rather than relying frequently on human repair. A high intervention rate indicates that the system is still at the auxiliary-draft stage.

4.5 Reproducibility and Stability

Whether the same task can be completed stably across different times, different inputs, and different random strategies is a core measure of engineering reliability. If model output is highly dependent on randomness, it becomes difficult to deploy the system in production.

4.6 Resistance to Invalid Proofs

The system must be able to identify and reject invalid proofs. Evaluation should include misleading error cases, such as derivations that appear reasonable but omit boundary conditions, circular definitions, invalid induction, and hidden assumptions, in order to test whether the verification loop is sufficiently strict.

5. Engineering Platform Support: Runnable, Observable, and Governable Formal Verification

Large-model-driven formal verification is not a single algorithm problem. It is an engineering system that typically requires task scheduling, inference services, verification execution, data storage, log auditing, and monitoring and alerting.

In cloud infrastructure environments, including Baidu AI Cloud, developers can use common capabilities such as object storage, container platforms, message queues, logging services, and monitoring and alerting to build verification pipelines. For example:

• use object storage to store formal definitions, proof scripts, failure cases, and historical versions;
• use container platforms to isolate different verification tasks and avoid interference;
• use message queues to decouple task generation, proof search, and result return;
• use logging services to record model outputs, verifier feedback, and human review traces;
• use monitoring and alerting to track resource consumption, failure rates, and abnormal tasks.

The significance of this platform-oriented design is to turn formal verification from a “single experiment” into a “continuously running production system.” When task volume grows, the system can scale verification nodes horizontally, reuse proof libraries, optimize search strategies, and track abnormal behavior.

5.1 Task Isolation and Secure Execution

Formal verification tasks often involve large amounts of symbolic computation and proof-script execution. To prevent abnormal scripts from affecting other tasks, the runtime environment should be isolated through containerization, and resource limits should be configured. For high-risk scenarios, access control, audit logs, and the principle of least privilege should also be enabled.

5.2 Version Management for Proof Artifacts

Mathematical proofs evolve continuously. Changes to definitions, axioms, or verifier rules may invalidate old proofs. Therefore, proof artifacts must be bound to version information. Version management helps answer questions such as:

• which definition version the current proof is based on;
• which lemma failed first;
• whether a failure was caused by a rule upgrade;
• whether the system can roll back to a stable version.

5.3 Building a Failure-Case Library

Failure cases are often more valuable than successful cases for engineering purposes. The system should automatically classify failure reasons, such as definition errors, strategy errors, overly deep search, verifier timeouts, and lemma dependency conflicts. A long-term failure-case library can help models learn more robust proof paths and help developers identify gaps in the knowledge base.

6. Main Risks and Governance Mechanisms

Although large models show potential in complex reasoning, formal verification scenarios impose very high reliability requirements. System design must take risks seriously.

6.1 Model Hallucination Risk

Models may generate intermediate steps that appear reasonable but are actually incorrect. The solution is not to trust the model’s self-description. Instead, all critical steps must be forced through verifier checks. The model can only act as a candidate generator, not as the final proof judge.

6.2 Verifier Vulnerability Risk

If the verifier itself has implementation defects, it may accept invalid proofs. Therefore, the verifier must undergo independent testing, boundary testing, and rule-consistency checks. For critical systems, multiple verifiers can be used for cross-checking, or human sampling review can be applied.

6.3 Benchmark Overfitting Risk

If a model is overly exposed to specific proof patterns during training or evaluation, it may perform well on fixed tasks but lack generalization ability. Real capability should be evaluated through dynamically generated tasks, hidden variant tasks, and cross-domain tasks.

6.4 Runaway Resource Consumption

Complex proof search may trigger a large number of parallel branches, causing resource waste. The system should set limits on maximum depth, maximum node count, timeouts, and priority scheduling to avoid infinite expansion.

6.5 Unclear Accountability Boundaries

In critical scenarios, formal verification results must be reviewed and signed off by humans. Models can improve efficiency, but they cannot replace the responsible party. Engineering workflows should preserve human review nodes, especially in scenarios involving safety, finance, healthcare, or critical infrastructure.

7. Practical Recommendations for Developers

For teams that want to build automated formal verification capabilities, the following suggestions can be useful.

7.1 Start with Verifiable Tasks

Do not begin by challenging highly difficult open-ended theorems. It is better to start with tasks that have clear structures, explicit goals, and mature verifier support, such as algebraic properties, set relations, simple invariants, or finite-state system properties. Build a complete closed loop on small tasks first, then gradually expand to complex theorems.

7.2 Adopt a “Model Generation + Tool Verification” Architecture

The system architecture should always follow the principle that the model is responsible for proposing possibilities and tools are responsible for confirming correctness. Any model output should be treated only as a candidate draft before it has passed symbolic verification.

7.3 Build a Formal Knowledge Base

The knowledge base should include definitions, lemmas, proof templates, failure cases, and human annotations. Without a knowledge base, models often have to rely only on general reasoning and may struggle with highly specialized tasks.

7.4 Design Explainable Search Logs

Every strategy choice, failure reason, branch pruning, and verification result should be recorded. Explainable logs are useful not only for debugging, but also for helping experts understand system behavior and decide when to intervene.

7.5 Use Multiple Models or Voting Strategies

A single model may produce bias. Different strategies or different inference configurations can be used in parallel to generate multiple candidate proofs, which are then filtered by the verifier. For critical steps, human review can be introduced.

7.6 Integrate Formal Verification into Continuous Integration

In software or algorithm-system development, formal tasks can be included as part of continuous verification. Whenever definitions, rules, or interfaces change, regression proofs can be triggered automatically to prevent system degradation.

8. Conclusion

The significance of large models for formal verification of complex theorems is not merely to let machines complete a difficult proof once. It is to transform long-chain reasoning tasks that originally depend on expert experience into a decomposable, searchable, verifiable, and reproducible engineering workflow.

In this workflow, large models provide strategy generation, knowledge organization, and failure explanation. Formal verifiers provide strict validation. Engineering platforms provide task scheduling, data governance, and runtime monitoring.

A truly mature automated proof system does not pursue a one-time impressive result. Instead, it builds a stable closed loop: it can correctly understand problems, generate candidate paths, strictly verify every step, accumulate failure experience, and continuously improve reliability through iteration. For developers and enterprises, the value of this direction is not limited to mathematical research itself. It also provides a technical path that can be applied to complex software verification, protocol analysis, risk control, and the design of high-reliability systems.