Proofstep Generation in Formal Verification

• Proofstep generation is the process of constructing, predicting, or synthesizing the next logical step in a formal proof, integrating both symbolic and neural methods.
• It employs state-based techniques like finite-state models and explicit search to guide tactic selection and ensure verifiable, stepwise proof construction.
• Modern frameworks leverage autoregressive transformers and reinforcement learning to iteratively refine proof steps, reducing human intervention in formal verification.

Proofstep generation is the process of constructing, predicting, or synthesizing the next logical step within a formal proof, typically in an interactive theorem prover (ITP), SMT-based verifier, or automated proof assistant. Central to formal mathematics and program verification, proofstep generation entails, at each intermediate proof state, proposing the next tactic, rule application, or low-level proof action that advances toward discharging the overall proof obligation. Modern research spans symbolic, statistical, and hybrid approaches, drawing from deep learning, state-machine inference, fine-grained proof object generation, and closely coupled neuro-symbolic pipelines.

1. Formal Definition and Background

Let $s_t$ denote the proof state at step $t$, with $G_t$ the current goal, $\Gamma_t$ the local context of assumptions, and $\mathcal{E}$ the global environment of proven lemmas. A proofstep $p_t$ is typically a tuple $(\tau, \alpha)$, where $\tau$ is a tactic name in a tactic set $\mathbb{T}$, and $\alpha$ are argument choices (from $t$0). Proofstep generation is thus the modeling and selection of the conditional distribution

$t$1

and, for whole-proof synthesis, iterating maximally likely steps:

$t$2

In autoregressive or sequence-to-sequence modeling, the training objective is to maximize log-likelihood of each ground-truth step:

$t$3

Alternative frameworks instantiate $t$4 as a sequence of text tokens encoding a tactic or proof sentence, directly applicable to text-based assistance or to systems lacking a fixed tactic grammar (Li et al., 2024).

2. Symbolic and State-Based Approaches

Symbolic frameworks structure proofstep generation as explicit search or inference over abstract state machines, algorithmic rule sets, or verified calculus schemas:

• Sequent calculus and proof object generation: Systems such as SC-TPTP extend the TPTP format to encode proofs as sequences of sequent-calculus steps, with explicit grammar and a suite of 30 low-level inference rules, including structural, connective introduction/elimination, and equality rules. High-level steps (e.g., congruence closure, multi-substitution) can be automatically unfolded into primitive proof steps, supporting stepwise reconstruction and export (e.g., to Coq). Certified libraries check compliance and reconstruct unfolded steps, ensuring that each advanced inference is grounded in checkable, tool-independent atomic transitions (Cailler et al., 15 Jul 2025).
• Antiunification and parameterized proof objects: In matching logic or certified symbolic execution, proofstep generation can be parameterized via algorithms (e.g., Plotkin antiunification), where each intermediate algorithmic transformation corresponds to an explicit proof schema. Proof objects record fine-grained meta-logical justifications for each step, with line-by-line tracking and kernel-level verification (Arusoaie et al., 2021).
• Finite-state models and tactic sequences: Proofstep suggestion can be automated by inferring extended finite-state machines (EFSMs) from large proof corpora. Each EFSM state represents an abstract proof context, and transitions correspond to tactics guarded by learned constraints on parameters. The resulting EFSMs guide search or suggest the next step given the current context; transition traces, guards by decision-tree classifiers, and prefix-tree state merging yield compact, data-driven models for both step selection and proof search (Gransden et al., 2014).

3. Neural, Retrieval-Augmented, and RL-Driven Frameworks

Recent advances leverage deep learning—particularly LLMs—as the engine for proofstep generation, using various architectural strategies:

• Autoregressive Transformer models: Models such as LLMSTEP (Welleck et al., 2023), StepFun-Prover (Shang et al., 27 Jul 2025), and those surveyed in (Li et al., 2024), encode the current proof state (goal, context, local declarations) as text, and autoregressively predict the next valid tactic or proofstep. Fine-tuning on millions of (state, tactic) pairs harvested from large formalization libraries yields high single-step and end-to-end proof synthesis rates.
• Tool-integrated RL and iterative refinement: Correctness is enforced by tight coupling with the proof assistant. For instance, StepFun-Prover (Shang et al., 27 Jul 2025) interleaves model-generated Lean tactic blocks with REPL output ("no goals left" or error traces) and iteratively adapts the model policy using reinforcement learning. Binary rewards from the verifier drive policy improvement (e.g. GRPO); failures feed back as input to repair or propose alternative steps.
• Verifier feedback as reward and filter: In Theorem Prover as a Judge (TP-as-a-Judge) and RLTPF frameworks (Leang et al., 18 Feb 2025), external tool verification (e.g., Lean) is used to gate acceptance of candidate proofsteps. Iterative autoformalization repairs invalid steps, and binary verdicts from the checker supply reward signals for RL fine-tuning. This induces data-efficient pipelines capable of generating high-quality, verifiable synthetic proofs, and ensures all intermediate steps are machine-checkable.
• Retrieval-augmented generation: Proofstep generation benefits from retrieval of structurally or semantically related prior proofs. PROMISE (Ahn et al., 7 Apr 2026) indexes large corpora of intermediate proof states, embeddings, and tactic transitions; at each step, it retrieves structurally nearest neighbors—using AST embeddings and token overlaps—adapts their tactic templates via context-dependent argument filling, and steers iterative, structure-aware search. Stepwise (He et al., 20 Mar 2026) combines beam/tree search with neural ranking and symbolic model pruning; candidate steps are filtered and ranked in a best-first queue, and failed actions are repaired or pruned via counterexample checkers or premise repair.
• Hierarchical decomposition and RAG for high-complexity proofs: The TLAPS RAG method (Zhou, 6 Jan 2025) decomposes complex obligations into subgoals via hand-built templates or LLM-guided pattern-splitting; then, per sub-obligation, top-k similar proof steps are retrieved from a curated database, and the LLM adapts these to the new context. All candidate proof fragments are parsed and verified by the prover, forming an interactive RAG loop.

4. Data, Benchmarks, and Evaluation Metrics

Empirical assessment of proofstep generation hinges on large-scale datasets and well-defined metrics. Major benchmarks include:

• Lean: LeanDojo, MiniF2F, mathlib4-test, and derived synthetic datasets provide millions of (state, tactic) examples (Welleck et al., 2023, Shang et al., 27 Jul 2025).
• Coq: Gamepad (71K proofs), CoqGym (71K proofs), PRISM (repair examples), ListNat/Bool/Coqlib/Values (EFSM mining) (Gransden et al., 2014, Li et al., 2024).
• Isabelle: PISA (183K theorems, 2.16M proof steps), FVEL seL4 (29K theorems) (He et al., 20 Mar 2026).
• TLA+: Sentence-level proof-step corpora, used for TLAPS RAG (Zhou, 6 Jan 2025).
• WebMath, GSM8K: For mathematical reasoning with external symbolic filtering (Leang et al., 18 Feb 2025).

Metrics:

• Single-step accuracy: Fraction of times the ground-truth next step is in the model’s top-$t$5 predictions, $t$6.
• Proof success rate: Fraction of theorems fully discharged (within time or step limits) on test suites.
• Sample and compute efficiency: Steps per proof, mean proof time, effort saving (measured as fraction of proof lines generated or subgoal reductions).
• Verifier-passed steps: Fraction of generated proofsteps that pass parsing, typechecking, and full verification.

Empirical results show, for example, pass@1 rates of 70.0% for StepFun-Prover-32B on MiniF2F-test (Shang et al., 27 Jul 2025), and 77.6% seL4 theorem coverage for Stepwise (Mistral-7B) (He et al., 20 Mar 2026), substantially outperforming hammer-only or single-shot LLM approaches.

5. Integration with Proof Assistants and Formal Frameworks

Robust proofstep generation requires seamless integration with the target proof ecosystem:

• Lean: Tools like LLMSTEP expose the Lean 4 internal goal state and context, serialize it to text, and call an external LM server for tactic suggestions. Suggestions are parsed, executed, and classified as valid/complete/invalid, with feedback highlighted in the user interface (Welleck et al., 2023).
• Isabelle: Stepwise and PROMISE interact with an Isabelle REPL via Python/Scala bridges, instrumenting state extraction, tactic application, counterexample search, and session management (He et al., 20 Mar 2026, Ahn et al., 7 Apr 2026).
• Coq: SC-TPTP and associated libraries support round-trip proof object generation, low-level step reconstruction (including Coq lemma exports), and kernel-level checking (Cailler et al., 15 Jul 2025).
• Verus/Rust: SAFE uses Verus as an external SMT-backed Rust proof checker; proof-annotation steps (invariants, assertions) are synthesized and validated, with failed steps and error messages included in the model input during self-debug rounds (Chen et al., 2024).
• TLA+: RAG approaches index and retrieve TLAPS-script fragments, using subgoal decomposition and strict TLAPS verification as the filter (Zhou, 6 Jan 2025).

6. Open Challenges and Future Directions

Despite substantial progress, open problems remain:

• Long context and scaling: Proof states (especially with large contexts or many subgoals) can exceed LLM context windows; compositional, GNN-based, or retrieval-augmented encodings are being investigated (Li et al., 2024).
• Generalization and sample efficiency: Current models often overfit to observed patterns and may fail on out-of-distribution theorems. Data-centric strategies—bootstrapping with synthetic examples, self-evolution, expert iteration—are actively explored (Chen et al., 2024, Leang et al., 18 Feb 2025).
• Fine-grained verification and search orchestration: Integrating counterexample search, premise repair, tactic repair, and symbolic pruning into search loops sharpens success rates and reduces search space as evidenced by neuro-symbolic systems (He et al., 20 Mar 2026).
• Interplay with formal artifacts: Generating Coq or Lean proof terms (rather than scripts) and reconstructing high-level steps into atomic, kernel-verified inferences supports robust transfer between ATPs and ITPs (Cailler et al., 15 Jul 2025).
• Human-in-the-loop and provable correctness: Tight feedback loops, explicit suggestion classification (valid/complete/invalid), and interactive suggestion verification chart the path toward semi-automated, provably correct mathematical reasoning assistants (Welleck et al., 2023).

Ongoing research continues to hybridize retrieval, symbolic search, and RL—integrating Kreutzer-style beam search, AST and goal embeddings, automated decomposition, and RL-tuned verifier feedback—establishing proofstep generation as a vibrant interface between formal logic, automated reasoning, and machine learning.