Automated Theorem Proving

• Automated theorem proving is the algorithmic process to generate and verify formal proofs using logical inference and computational search strategies.
• ATP systems employ bottom-up, top-down, evolutionary, reinforcement learning, and neural methods to discover human-readable and efficient proofs.
• Integration with proof assistants and large formalized libraries enables ATP to tackle complex mathematical theories and enhance formal reasoning.

Automated theorem proving (ATP) refers to the algorithmic generation and formal verification of mathematical proofs, often through a combination of search, deduction, and, more recently, learning-guided algorithmic techniques. ATP is foundational both to mathematical logic and the development of interactive proof assistants, and has recently emerged as a major testbed for artificial intelligence due to its requirements for symbolic reasoning, search in combinatorially large spaces, and formal correctness.

1. Core Principles and Approaches

Automated theorem proving operates across formal logic systems (e.g., first-order logic, higher-order logic, intuitionistic logic, dependent type theory) and encompasses a wide variety of technical methodologies. At the foundational level, ATP constructs proof objects by applying valid inference rules—such as resolution, tableau, or sequent calculus steps—to axioms and hypotheses until the target theorem is derived. The process can be fully automated ("push-button"), interactively guided, or hybrid. Success in ATP is defined not only by finding any proof, but often by discovering human-readable or short proofs, discovering alternative proofs, or translating informal mathematical arguments into formalized, machine-checkable objects.

Key ATP paradigms include:

• Bottom-up approaches that build proofs from primitive inference rules upwards from axioms (e.g., resolution, superposition, saturation, classical tableau)
• Top-down approaches that emulate human-level proof planning by proposing and semantically checking intermediate concepts before attempting formal justification (Larson et al., 2023)
• Proof search as a Markov Decision Process, casting inference as sequential decision-making (Kusumoto et al., 2018)
• Type-directed inhabitation and synthesis, where the problem is recast as the construction of terms with specified types in dependent or simply-typed lambda calculi (Norman et al., 8 Apr 2025, Armstrong et al., 2011)

The Curry–Howard correspondence underpins much of proof assistant design, equating proofs with programs and propositions with types, thereby allowing computational machinery to explore and check the validity of proposed term/proof objects (Yang et al., 2016).

2. ATP in Large Formalized Mathematics and Proof Assistants

Integration with large libraries (e.g., Mizar Mathematical Library, Mathlib for Lean, set.mm for Metamath) and with proof assistants (Coq, Lean, Isabelle) has driven significant advances in ATP.

Proof assistant integration:

• Proof assistants act as proof verifiers, rigorously type-checking scripts composed of tactic applications (e.g., "Proof. intros. ... Qed." in Coq) (Yang et al., 2016).
• ATP systems may generate candidate proof scripts as sequences of tactics, which are batch-tested in the assistant; success is detected by the complete discharge of all goals (Yang et al., 2016).
• Heuristics, ML classifiers, or LLMs provide ranking or direct prediction of tactics, premise selection, or proof steps (see below).

Large theory ATP:

• Key developments such as the Mizar/MPTP pipeline enable translation of tens of thousands of definitions and theorems into ATP-targetable problems in TPTP syntax (first-order or higher-order) (Urban et al., 2012).
• AI/ATP systems (e.g., MaLARea, MaLeCoP) iteratively combine ML-based premise selection, ATP proof attempts, and semantic model-checking to guide search in massive theory spaces (Urban et al., 2012).
• Machine learning-based premise selection, learned from prior proofs' feature vectors (symbol occurrence, term patterns, semantic clause evaluation), is crucial for tractable proof search in corpora with 10^4–10^5 possible axioms, boosting success rates from below 40% to over 60% in benchmarks such as MPTP2078 (Urban et al., 2012).

3. ATP Algorithms: Evolutionary, Reinforcement Learning, and Neural Approaches

Recent years have seen the advent of ATP systems driven by data-driven and learning-based methods:

• Evolutionary algorithms (EA): Proof scripts are represented as variable-length integer sequences corresponding to tactics. Populations of candidate proofs are generated, evaluated by a proof assistant for fitness (progress towards goal closure), and evolved by selection, recombination, and mutation. Complete, human-readable proofs for theorems that are out of reach for default proof automation have been synthesized in this manner (Yang et al., 2016).
• Deep reinforcement learning: Proof search is modeled as an MDP where states are sets of open sequents, actions are inference rule applications, and reward is 1 upon completion. Policy and value functions are approximated by graph neural networks trained on large, augmented datasets of synthetic sequents, yielding provers that outperform human-engineered tactics such as Coq's tauto in intuitionistic logic (Kusumoto et al., 2018).
• Transformer-based LLMs: Stepwise tactic generation, frame selection, and subgoal closure are guided by (often fine-tuned) generative models capable of emitting tactic scripts or lemma statements. Systems like GPT-f have discovered previously unknown, shorter proofs in established libraries such as Metamath, contributing directly to community-accepted formalizations (Polu et al., 2020).
• Multi-agent and recursive decomposition architectures: Recent systems combine LLMs specialized for various subtasks—formalization, syntactic and semantic checking, proof generation, theorem retrieval, proof sketch decomposition—coordinated in a recursive, tree-structured workflow. Proofs are decomposed into subgoals, attacked in parallel or recursively, and then reconstructed into complete formal proofs (Davis, 16 Dec 2025).
• Partial Label Learning (PLL) for ATP: Sequential policy models in ATP search can be viewed through the lens of multilabel learning, with sets of correct derivations serving as "partial labels" and specialized loss functions (NLL, meritocratic, Libra) improving learning from multiple alternative proofs (Zombori et al., 4 Jul 2025).

Recent advances demonstrate that large-scale synthetic data generation (e.g., via proof-state exploration or reinforcement-learning-driven proof path construction) is critical for training performant LLM-based ATPs, achieving state-of-the-art pass rates on standardized benchmarks (Lai et al., 17 May 2025, Xiong et al., 25 Feb 2025).

4. Specialization: Domains, Planning, and Geometry

ATP adapts to specialized mathematical or applied domains using tailored encodings and algorithms:

• Plane geometry: Graph-theoretic ATP frameworks (e.g., GraATP) encode geometric quantities as vertices, inference rules as labeled edges in a DAG, and proofs as dependency-ordered traversals culminating in the target conclusion. Soundness is guaranteed by restricting to correct geometric inference rules; completeness depends on the rule set's expressiveness (Mahmud et al., 2014).
• Abstract algebra and rings: ATP by automated planning encodes axioms and inference steps of, e.g., commutative rings, as actions in a PDDL planning domain. Proof discovery reduces to plan search by off-the-shelf planners (e.g., LAMA, PRP), yielding explicit action sequences corresponding to proof steps (Petrov et al., 2023).
• Information theory: By formalizing existential information inequalities (EII), ATP reduces achievability and converse problems in network information theory to existential quantifier statements over linear constraints on entropic vectors. Proof search becomes candidate substitution enumeration followed by LP feasibility checking, validated and pruned by polyhedral techniques (Li, 2021).
• Type-theory inhabitation: Synthesis of terms inhabiting dependent types (proofs-as-programs) is realized by bounded exhaustive search with explicit substitution, conversion checking, and entropy-guided refinement; this enables production of short, structurally natural proof terms (Norman et al., 8 Apr 2025, Armstrong et al., 2011).

5. Evaluation, Datasets, and Benchmarks

ATP progress is measured via proof discovery or pass rates on standardized formal mathematics corpora and challenging competitions:

• Mizar/MPTP, TPTP, miniF2F, Mathlib, FIMO: Large-scale benchmarks measure both coverage (fraction of theorems proved) and efficiency (proof length, time, number of proof search steps). For example, ATPs coupled with ML-guided premise selection reprove over 61% of Mizar theorems in the standard MPTP2078 challenge (Urban et al., 2012). Multi-agent LLM systems achieve 90.4% pass@32 on miniF2F, with recursive decomposition closing additional hard instances (Davis, 16 Dec 2025).
• IMO-level and combinatorial identities: Recent work formalizes International Mathematical Olympiad problems into benchmarks (FIMO, LeanComb), with advanced ATP systems such as Aristotle achieving gold-medal-equivalent performance on modern IMO sets (Achim et al., 1 Oct 2025, Liu et al., 2023, Xiong et al., 25 Feb 2025).
• Synthetic data and transfer: Provers trained solely on synthetic theorems (e.g., generated by forward resolution or RL-driven tactic tree search) often outperform those trained only on human-curated theorem sets, and transfer success to human-authored problems is demonstrated (Aygün et al., 2020, Lai et al., 17 May 2025).
• Proof reconstruction and minimization: Automated systems are increasingly able to synthesize and verify compact, human-comprehensible proofs, and to reconstruct externally-derived proofs robustly (e.g., via micro-step replay in dependently-typed frameworks) (Armstrong et al., 2011).

6. Limitations, Challenges, and Future Directions

Despite notable successes, automated theorem proving faces several persistent challenges:

• Scalability: The exponential growth of search spaces (e.g., in tactic composition or term inhabitation) requires increasingly sophisticated guidance, hierarchical decomposition, or data-efficient learning (Yang et al., 2016, Norman et al., 8 Apr 2025).
• Proof search heuristics: Classic heuristics for clause or tactic selection remain crucial at every scale, and integrating or learning such heuristics is ongoing work (Aygün et al., 2020, Lai et al., 17 May 2025).
• Formal–informal gap: Automated translation of informal mathematical arguments or natural language proofs remains brittle; autoformalization and hybrid pipelines with error feedback are active research topics (Liu et al., 2023, Davis, 16 Dec 2025, Achim et al., 1 Oct 2025).
• Completeness and soundness: Most ATPs are sound by construction (by restricting to verified rule sets and tactics), but no general completeness guarantees exist for arbitrary mathematical domains or problem classes within practical computational budgets (Mahmud et al., 2014, Petrov et al., 2023).
• Robust proof term extraction: For externally-discovered proofs (e.g., by ATPs not coupled to the assistant kernel), reconstruction into program terms or formal objects is sometimes nontrivial, especially with non-constructive or Skolemized proofs (Brown et al., 10 Sep 2025, Armstrong et al., 2011).
• Higher-order and domain transfer: While HOL ATPs and cross-domain pipelines see success in set theory and analysis (Brown et al., 10 Sep 2025), adapting ATP techniques to new mathematical domains (geometry, combinatorics, analysis) requires domain-specific encoding and sometimes major advances in symbolic reasoning and search.

Future directions include:

• Fully end-to-end neural/hybrid ATPs with joint formal/informal reasoning (Achim et al., 1 Oct 2025, Davis, 16 Dec 2025)
• Unified formal and informal mathematical libraries and benchmarks at multi-contest or research-expert level (Liu et al., 2023, Xiong et al., 25 Feb 2025)
• Advances in autoformalization, concept invention, and hybrid top-down/bottom-up proof architectures (Larson et al., 2023)
• Integration of reinforcement learning, proof-state decomposition, and plan search across broader logics (Kusumoto et al., 2018, Petrov et al., 2023)

Automated theorem proving is now recognized as both an engineering and scientific challenge at the intersection of logic, computation, artificial intelligence, and mathematical practice. ATP systems continue to push the frontier of what can be formally verified and discovered by machine (Urban et al., 2012, Yang et al., 2016, Kusumoto et al., 2018, Polu et al., 2020, Achim et al., 1 Oct 2025).