Synthetic Theorem Generation

• Synthetic theorem generation is an automated process that constructs novel, formally valid theorems and proofs using symbolic, deductive, neural, or template-based methods.
• It leverages structured theory presentations, grammar-driven expansions, and type-checking kernels to ensure correctness and auditability of generated statements.
• Recent frameworks improve scalability and proof relevance, supporting applications in algebra, formal analysis, and neural theorem proving to expand formal libraries.

Synthetic theorem generation refers to the fully or partially automated construction of formally valid, novel theorems—both statements and proofs—often for the purpose of augmenting theorem-prover libraries, benchmarking automated reasoning systems, or generating training data for neural theorem provers. Across formal logic, algebra, combinatorics, and contemporary automated theorem proving, the challenge is to not only invent new conjectures but also to provide their formal validation, leveraging symbolic, deductive, combinatorial, neural, or template-based procedures. Recent research has advanced this field with generative frameworks offering provable correctness, scalability, auditability, and immediate utility for downstream proof synthesis and verification.

1. Foundational Principles and Formal Frameworks

Synthetic theorem generation is anchored in declarative logic and formal algebraic structures. At its foundation, a theory presentation is treated as a structured datatype—typically a triple $T = (S, F, E)$ where $S$ is a sort (carrier set), $F$ is a set of function symbols (with arities), and $E$ is a list of equational axioms. This paradigm allows higher-order "builders" to automatically derive signature types, term languages, product algebras, homomorphism record-types, and associated theorems from any base theory. Each construction is a mechanical traversal or expansion over the theory presentation, ensuring that all generated statements are type-correct by construction and validated by the type-checking kernel of the target proof assistant (such as Tog or Lean) (Carette et al., 2020).

In first-order logic (FOL) and combinatorial settings, synthetic theorems are often generated via controlled random walks over proof-state transitions, grammar-driven expansions, or symbolic template filling. In propositional and first-order contexts, clause space (conjunctive normal form) is used for resolution-based synthesis, while for algebraic or higher-order frameworks, terms are constructed by syntax-guided enumeration and induction principles.

2. Generative Methodologies: Programmatic, Deductive, and Neural

a. Declarative and Deductive Approaches

Declarative grammar frameworks (e.g., Unigram) express generation rules as $$R(\text{output type}, \text{input types}, \text{realizers}, \text{constraints})$$ and sample complete logical statements by recursive tree expansion. Constraints enforce context-sensitivity (e.g., avoiding unwanted nesting of logical operators) and permit auditability and extensibility (Sileo, 16 Jun 2024).

Deductive symbolic systems (e.g., TheSy) for bottom-up theory exploration avoid random testing and SMT-based filtering, instead combining term enumeration, symbolic observational equivalence (SOE), and one-step induction provers. This enables efficient pruning of redundant or trivial conjectures and supports compositional reasoning over user-defined recursive functions and algebraic datatypes (Singher et al., 2020).

b. Combinatorial and Template-Based Synthesis

Combinatorial generators systematically enumerate formula skeletons (binary trees for implicational logic, full n-ary constructions for richer logics), assign variable partitions for maximal coverage (Bell and Catalan number scaling), and can apply structural transformations to yield canonical classes of equiprovable formulas. The Mints transform and other formula transformers can simplify or harden test-cases, while minimal unsatisfiable cores are explicitly constructed via the Rectangular Standard Contradiction (RSC) approach (Xu et al., 6 Nov 2025, Tarau, 2019).

Template-based frameworks, notably for algebraic structures and logic, use polarity matrices and schema expansion to generate entire families of theorems, parameterized by sets of literals or terms. This class of generators provides rigorous correctness and nonredundancy guarantees, as shown by the RSC approach for minimal unsatisfiable clause sets (Xu et al., 6 Nov 2025).

c. Neural and RL-Guided Synthesis

Neural methods, such as MetaGen and forward-proposer–based neural saturation-guided provers, synthesize theorems by learning probabilistic models over sequences of proof steps, with GRU-based relevance and substitution networks guiding subtree and token-level sampling (Wang et al., 2020). Reinforcement learning, including RL from theorem-prover feedback (RLTPF), replaces human annotation with formal-validation signals, directly optimizing for provability and ranking competing proof attempts (Leang et al., 18 Feb 2025). Large-scale synthetic data pipelines feed neurosymbolic models for both proof-policy training and downstream LLM-based theorem proving (e.g., DeepSeek-Prover, LeanConjecturer, Alchemy, and proof-state exploration with adaptive beam scheduling) (Xin et al., 23 May 2024, Onda et al., 27 Jun 2025, Wu et al., 21 Oct 2024, Lai et al., 17 May 2025).

3. Application Domains and Synthesis Pipelines

Synthetic theorem generation is deployed across diverse mathematical and logical domains:

• Algebra: Automated mining of signatures, products, term algebras, and homomorphism records from declarative theory presentations, with modular expansion to hundreds of new record types per base theory, each certified correct (Carette et al., 2020).
• First-Order and Propositional Logic: Massive generation of FOL problems and neutral, entailment, or contradiction labels via declarative grammars, clause-based resolution walks, or Monte Carlo Tree Search with policy-value networks for Metamath (Sileo, 16 Jun 2024, Lin et al., 5 May 2024).
• Formal Analysis and Topology: LeanConjecturer produces thousands of novel, nontrivial conjectures via LLM-guided iterative synthesis, enabling reinforcement learning for domain-specific proof-policy optimization (Onda et al., 27 Jun 2025).
• Symbolic Mutation and Library Amplification: Alchemy mutates candidate Lean theorems by tactic-guided rewriting (rw) and implication application (apply), scaling Mathlib from 110k to >6M validated entries, pretraining LLMs for enhanced in-distribution and out-of-distribution performance (Wu et al., 21 Oct 2024).
• Combinatorial Logic: Efficient $O(N)$–space, $O(N)$–time generation of all normal-form linear lambda terms and their principal types for the implicational fragment of linear logic, producing billions of theorem/proof pairs (Tarau et al., 2020).
• Template-Based Contradictions: Systematic construction of minimally unsatisfiable templates for automated theorem generation in CNF, with guaranteed logical equivalence and provability (Xu et al., 6 Nov 2025).
• Automated Theory Exploration in Separation Logic: Synthesis of inductive lemmas (including unknown predicates and guarded relational constraints) via cyclic abduction and root-matching to augment entailment provers (Le, 2017).

4. Correctness, Auditability, and Practical Validation

Correctness in synthetic theorem generation is ensured either by construction (via type-checking kernels, Curry–Howard correspondence, or explicit inductive proofs), by formal verification in theorem provers (Lean, Tog, Mathlib), or by structural guarantees (minimal unsatisfiable sets, sound combinatorial transforms). Declarative approaches enable full traceability—every clause, rule, or tactic can be backtracked to its origin template or rule instantiation (Sileo, 16 Jun 2024, Xu et al., 6 Nov 2025). Iterative autoformalisation and refinement loops elevate execution rates by systematically repairing formalisation errors, with Lean-based verdicts giving robust feedback, leading to substantial improvements in downstream model accuracy with minimal human annotation (Leang et al., 18 Feb 2025).

Deduplication and decontamination of synthetic theorems—removing trivial, redundant, or near-duplicate statements—are critical for controlling dataset size and maintaining the utility and diversity of training corpora, as evidenced by Alchemy and other large-scale mutation frameworks (Wu et al., 21 Oct 2024).

5. Impact on Machine Learning, Automated Proving, and Library Expansion

The synthesis of large, structurally rich, and balanced theorem corpora has demonstrably improved the performance of neural theorem provers, LLM-based ATPs, and formal verification tools. Integrating millions of synthetic records yields absolute improvements of up to 4.7%–6% on in-distribution and out-of-distribution benchmarks (Leandojo, MiniF2F, FIMO) compared to pre-synthetic baselines (Wu et al., 21 Oct 2024, Xin et al., 23 May 2024). Empirical studies reveal strong transfer from purely synthetic training data to human-written benchmarks, with models trained solely on synthetic theorems outperforming those trained on limited curated corpora (Firoiu et al., 2021, Aygün et al., 2020, Wang et al., 2020).

Synthetic lemma generation (ATG) reduces proof search complexity, enabling proof of deeper or harder theorems by providing reusable intermediate results otherwise absent from initial libraries (Lin et al., 5 May 2024). Benchmarks that partition libraries by proof depth measure the average proof reduction and human-aligned precision, highlighting both the value and current limitations of model-based synthesis.

6. Limitations, Challenges, and Outlook

Current systematic generators are generally restricted to first-order equational theories, propositional or first-order logic, or symbolic algebra. Extending to higher-order functions, non-equational domains, coinduction, modal logics, and more sophisticated semantic domains requires nontrivial template engineering, richer deductive engines, and potentially neural-symbolic hybridization. Computational complexity (e.g., exponential clause scaling, search over deep or large theorems) limits brute-force approaches.

Precision for human-aligned theorem matches remains modest (<2%), with most generated statements novel but not coinciding with standard human-written lemmas (Lin et al., 5 May 2024). Neural-theorem usefulness metrics (e.g., downstream utility, reusability across problems) remain underexplored but are recognized as crucial next steps (Firoiu et al., 2021).

Emerging frameworks such as RSC, proof-state exploration with adaptive scheduling, modular self-play mutation, and RL-based feedback promise scalable, rigorous, and extensible synthetic generation, potentially enabling unprecedented library growth and automated mathematical discovery.

7. Future Directions and Research Significance

Advances in template-based contradiction, theory-exploration engines, and neural-guided generative schemes position synthetic theorem generation as a pivotal technology for formal mathematics, AI, and software verification. Current research aims to integrate domain-specific templates (e.g., algebraic axioms, topology operations), extend auditability and correctness guarantees to richer logical fragments, and combine deterministic template methods with neural generators for semantically meaningful conjecture synthesis (Xu et al., 6 Nov 2025, Onda et al., 27 Jun 2025). Released synthetic datasets and codebases provide a fertile ground for benchmarking, replication, and further method development.

By systematically elevating machine agents from mere verifiers to discoverers, synthetic theorem generation reshapes the capabilities and workflows of formal reasoning, offering scalable alternatives to costly manual curation and enabling data-rich, automated problem solving in logic and mathematics.