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Visored: A Controlled-Natural-Language Prover for LLM-Generated Mathematics
Published 16 Jun 2026 in cs.PL and cs.AI | (2606.17581v1)
Abstract: We present a dependent-type-based prover designed around the way LLMs (and humans) tend to write mathematics, complementing existing systems such as Lean and Rocq. Its core design choices are a surface that imitates mathematical natural language and a rule-driven automation layer that closes the routine steps a textbook would omit, so that an accepted proof can be re-emitted as a checked Lean file. Early experiments suggest that, even without any prover-specific training data, LLMs can learn to use it effectively on the miniF2F benchmark. Lean output excerpts: https://github.com/xiyuzhai-husky-lang/visored/
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What is this paper about?
This paper introduces Visored, a new kind of math “proof checker” designed to work well with both people and AI LLMs. Instead of forcing you to write in a programming-like proof language, Visored lets you write math in clear, controlled English inside LaTeX (the way many math papers are written). It then turns that writing into a precise, machine-checkable proof and can even export it to Lean (a popular proof assistant) for extra verification.
Think of Visored as a bridge between how humans naturally write math and how computers need math written to check it perfectly.
What questions does the paper try to answer?
The authors focus on a big problem: AI and humans can write convincing-looking math, but how do we reliably check that it’s correct?
In simpler terms, they ask:
- Can we make a system that reads math written in a natural, textbook-like style and checks it exactly?
- Can this system help LLMs turn their informal math into formal, verified proofs more easily?
- Can it split problems into small, understandable steps and give helpful, pinpoint feedback when something goes wrong?
How does Visored work? (Methods explained simply)
Imagine turning a math paragraph into a set of precise instructions a computer can trust. Visored does this in stages:
- Controlled Natural Language (CNL): You write math using a limited, well-defined set of sentence patterns like “Let …”, “Assume …”, “Then …” inside LaTeX. This keeps the language clear enough for a computer to understand while still feeling like normal math.
- Parsing the text: Visored reads both the prose and the formulas, building a structured “syntax tree” (like a diagram showing how the sentence and symbols fit together).
- Adding types and meanings: It assigns “types” to things (like “this is a real number” or “this is a set”). This is called a dependently-typed system. In everyday terms, types are safety labels that make sure expressions make sense. For example:
- 1/x is only allowed if x ≠ 0.
- √x (or |x|) requires x ≥ 0.
- log x requires x > 0.
- Visored automatically checks these safety conditions while it reads your proof.
- Solver for “obvious” steps: When you write “Then …” (the conclusion of a step), Visored uses a rule-based engine to fill in routine steps math texts often skip (like simple algebra rearrangements or standard identities). The rules are cost-bounded, which means the system tries a controlled amount of reasoning and stops if it can’t conclude, instead of getting stuck.
- Clear feedback: If Visored can’t verify a step, it doesn’t just say “fail.” It points to the exact line and part of the text that caused the problem. That makes fixing mistakes much easier.
- Optional Lean export: If a proof passes, Visored can convert it to Lean code. Lean then re-checks the proof with its tiny, very trustworthy core. This is like having two independent judges agree your solution is solid.
A helpful analogy:
- CNL is like writing with a controlled vocabulary.
- The typed intermediate representation is like a detailed recipe the computer can follow precisely.
- The solver is a careful helper who fills in the easy algebra and logic steps.
- Lean is a strict final examiner that double-checks everything.
What did the authors find, and why does it matter?
The authors tested Visored on a well-known math benchmark called miniF2F (a collection of competition-style problems from middle school level up to Olympiad). They used a single AI agent to draft proofs in Visored’s CNL and let Visored check them. They also improved Visored’s rule library whenever they found routine steps it didn’t yet know.
- Result: On one subset (the “valid” split, 244 problems), the AI+Visored loop produced accepted, checked proofs for 222 problems (about 91%).
- Where it struggled: Most unsolved problems were hard Olympiad problems needing advanced techniques or rules that weren’t implemented yet (like certain inequalities, trigonometric sums, or finite fields). Importantly, the failures were usually due to missing features in Visored, not the AI being unable to reason once those tools are available.
- A trade-off they found: Converting to Lean made very long files (much longer than hand-written Lean proofs). It works, but it’s verbose and slow to recompile. This made them decide to stop extending this prototype and instead build a cleaner next version with what they learned.
Why this matters:
- It shows that an AI can write in math-like English and get a lot of problems fully checked, even without special training just for Visored.
- The system gives precise error messages, which is extremely helpful for teaching, debugging, and improving AI behavior.
What could this change in the future?
The authors see several uses:
- Real-time math checker: A math assistant or tutor could accept only proofs Visored verifies, reducing “hallucinated” (incorrect) reasoning.
- Autoformalization: Turning human- or AI-written math into formal proofs becomes easier, helping research groups organize and trust AI-generated ideas.
- Better training signals for AI: Because Visored checks steps one by one and pinpoints failures, it can give dense, detailed feedback for reinforcement learning instead of a vague “pass/fail” at the end.
- Data generation: You can produce a large, high-quality dataset of formally checked, natural-language-style proofs—great for training future models.
Current limitations (and they’re honest about them):
- Scope: The prototype covers basic algebra, arithmetic, sets, elementary number theory, and some real analysis. Advanced topics and heavy contest techniques need more rules.
- Lean output verbosity: The exported Lean proofs are very long and could be improved.
- Diagnostics: When a step fails, the system tells you where, but it could say more about “almost” matches to guide fixes even better.
- Engineering maturity: It’s research code and a starting point, not a polished final product.
In short: Visored is a promising blueprint for making AI-written math more trustworthy by keeping the human-friendly feel (natural language) while enforcing the computer’s need for precision. It doesn’t try to discover deep proofs all by itself; instead, it makes checking and organizing proofs more reliable and scalable. That can help students, teachers, and researchers alike.
Knowledge Gaps
Knowledge gaps, limitations, and open questions
Below is a single, focused list of concrete gaps and open questions that remain unresolved and could guide future work:
- Soundness of the trusted base: Visored returns its own verdict without routing through a small, auditable kernel; there is no formal soundness proof of the solver, rule interpreter, or TCC (well-definedness) machinery. What minimal trusted core and proofs are needed to meet the de Bruijn criterion?
- Rule soundness and maintenance: The rule database is extended ad hoc during development; there is no principled process to verify rule soundness, manage versions, or prevent regressions (e.g., the documented existential regression). How should rules be curated, validated, and regression-tested at scale?
- Upward semantic parsing ambiguity: The LaTeX→Visored lift is ill-posed and relies on an LLM prior, but there is no principled search over alternative parses or probabilistic disambiguation. Can beam search over candidate Visored parses with type-aware scoring reduce misparses, and how to measure misparse rates?
- Diagnostic granularity: When the solver fails (budget exhausted), Visored reports only the unclosed goal; it does not expose near-miss rules, failed side-conditions, or best partial matches. What interfaces and instrumentation would make diagnostics actionable (e.g., counterexample hints, almost-fired rule traces)?
- Completeness and scalability of “Miracle” search: The cost-bounded, nondeterministic solver has no completeness guarantees and no characterization of trade-offs between budget, latency, and success rate. How should budgets be tuned, and can adaptive search or portfolio strategies improve closure without blowups?
- Coverage of mathematical domains: Multiple capabilities are missing or partial (as evidenced by the 22 unsolved miniF2F-valid problems), including:
- Inequality machinery (AM–GM, rearrangement, descent techniques).
- Trigonometric identities and sums; nonlinear trig equation solving.
- Finite fields and modular arithmetic types (Z/nZ), divisibility chains.
- Limit/fixed-point reasoning and set fixed-point problems.
- Handling disjunctions in conclusions, absolute-value systems, constructive existence.
- What minimal rule sets, decision procedures, and types should be added first to close these gaps?
- Domain-constraint propagation: Well-definedness checks currently cover only a subset (e.g., 1/x, sqrt, log). How should domain obligations propagate through additional constructs (piecewise, inverse functions, compositions) and interact with conditionals and quantifiers?
- Number-system/coercion decisions: Although Visored makes casts explicit, the LLM still chooses N vs Z vs Q vs R with potential drift. What automated heuristics or constraint solvers can infer consistent number systems and casts, and how often do current choices induce unprovable goals?
- CNL surface coverage: The accepted CNL lacks standard idioms (e.g., “WLOG,” “for sufficiently large n,” “passing to a subsequence,” tactic-like hints). Which sentence templates and shorthands should be prioritized, and how should their semantics be specified to avoid ambiguity?
- Tactic-hint mode: There is no mechanism for user/LLM-supplied hints (“by induction,” “apply Cauchy–Schwarz,” “rewrite using lemma X”) to steer the solver. How can hint annotations be designed to remain declarative while meaningfully guiding search?
- Interoperability beyond Lean: UVL MIR is intended to be target-neutral, but only Lean emission is exercised and is extremely verbose. What is required to target Coq/Isabelle/HOL, and how to ensure logical alignment (e.g., classical vs constructive, type hierarchies)?
- Lean emission size and performance: Emitted Lean proofs are often 250×–25,000× longer than hand-written, with heavy compile costs. Which emission strategies (tactic selection, lemma packaging, proof term compression, modularization) most reduce size and compile time without sacrificing robustness?
- Independent, held-out evaluation: The reported 91% is a coverage figure on the development split with rules added during solving; there is no held-out test evaluation or comparison under matched protocols. What is the true generalization rate under fixed rules and bounded retries, across datasets?
- Reproducibility across LLMs and prompts: Results rely on a single coding agent with a bespoke skill; the sensitivity to model choice, prompting, and retry budgets is unknown. How does performance vary across models, seeds, and agent scaffolds?
- Inference-time verifier mode: The intended deployment (real-time CNL checking) is not evaluated for latency, throughput, and user experience. What end-to-end timings (parse→solve→emit) are achievable, and how do diagnostics impact interactive workflows?
- Dense RL reward utility: Per-step verdicts are proposed as a dense reward, but no training experiments or ablations validate their effectiveness vs. compile-level signals. Does RL with Visored rewards improve autoformalization or proof planning, and what reward shaping is needed?
- CNL data generation quality: The proposed proof-corpus generator (LLM→CNL→Visored filter) lacks assessments of diversity, deduplication, bias, and transfer. How to measure and ensure data quality, and what is the downstream benefit for reasoning models?
- Hybrid ATP integration: Visored uses a custom rule-driven solver; there is no study of hybridizing with SMT/ATP (e.g., congruence closure, linear arithmetic, Vampire/E) per obligation. Which obligations benefit most from external ATPs, and how to keep per-step diagnostics informative?
- Backtracking over earlier commitments: Failures caused by upstream mis-decisions (e.g., wrong subtype or cast) are only locally reported. Can the system backtrack and reconsider earlier semantic choices automatically, guided by solver feedback?
- Quantitative solver profiling: There is no breakdown of which rule families close which obligations, nor ablations showing their contributions. Which domains and rules deliver the largest marginal gains, and where are the long tails?
- Existential and constructive content: Handling of existence claims (witness search, constructive vs classical reasoning) is fragile (noted regression). What mechanisms (witness synthesis tactics, Skolemization, search over candidate terms) most reliably discharge existentials?
- Specification (.lpcsv) management: Extending the CNL via spec files raises questions about consistency, conflicts, and backward compatibility. What tooling (linting, CI tests, versioning, diff-based semantic checks) is needed to safely evolve the specs?
- Security and robustness: The parser/solver has not been tested against adversarial or malformed CNL/LaTeX inputs (e.g., injections, conflicting declarations). What hardening and sandboxing are needed for deployment in open LLM loops?
- Standardized traces: Intermediate states are exposed as human-readable traces, not a machine-readable schema. What minimal, stable trace format would support tooling (analytics, RL logging, dataset release) without coupling to internal code?
Practical Applications
Immediate Applications
The following use cases can be deployed now by leveraging Visored’s current capabilities (CNL LaTeX surface, typed IR, rule-driven solver, localized diagnostics, optional Lean emission) within its present math scope (arithmetic, basic algebra, sets, elementary number theory, parts of real analysis).
- Math-verifying chatbots and tutors with localized feedback (education, software)
- What: Embed Visored as an inference-time verifier so LLMs respond with only checked steps or pinpointed diagnostics.
- Tools/workflow: LLM → CNL-in-LaTeX draft → Visored check → user sees accepted output or error location; optional Lean file for audit.
- Assumptions/dependencies: Content kept within current scope; configured .lpcsv spec for course notation; latency budget acceptable for solver runs.
- Autoformalization loop for problem sets and lecture notes (academia, education)
- What: Convert informal LaTeX proofs for competition/intro courses into Visored-checked CNL with optional Lean re-verification.
- Tools/workflow: Overleaf/VScode extension → write CNL → Visored diagnostics → iterate → export Lean for archiving/interop.
- Assumptions/dependencies: Human-in-the-loop handles out-of-vocabulary constructs; Lean emission verbosity tolerable for archival (not rapid iteration).
- Auto-grading of proof-based assignments with actionable errors (education)
- What: Grade proofs via accept/fail plus precise failure spans, improving feedback versus binary formal-compilation errors.
- Tools/workflow: LMS plugin runs Visored on student CNL → per-step pass/fail → rubric mapping; aggregate analytics per cohort.
- Assumptions/dependencies: Assignments align with supported rules; institutional acceptance of CNL format.
- Continuous verification of mathematical claims in technical docs (software engineering, finance, energy)
- What: CI linter for Markdown/LaTeX sections asserting model invariants, feasibility conditions, or algebraic transformations.
- Tools/workflow: Docs include example environments; CI runs Visored and fails builds on unproved “Then …” goals.
- Assumptions/dependencies: Claims expressed in supported math; authors add guards (e.g., x ≠ 0) that Visored enforces at type-check time.
- Dense-reward training signal for math LLMs (AI research, software)
- What: Use per-step success/failure vectors from Visored as reinforcement learning rewards to train verifiers/solvers.
- Tools/workflow: Rollout → CNL generation → Visored stepwise verdicts → RL loss; store traces for supervised fine-tuning.
- Assumptions/dependencies: Stable API exposing MIR/diagnostics; compute budget; curriculum restricted to Visored’s current coverage.
- Synthetic, formally checked CNL proof corpora (AI research, data)
- What: Generate unbounded, domain-aligned training data by prompting LLMs and filtering with Visored.
- Tools/workflow: Prompt bank → batch proof generation → Visored filter → deduplicate → publish datasets.
- Assumptions/dependencies: Licensing of prompts/outputs; careful spec tuning to mimic target genres (competition, textbook).
- Lean-compatible normalization of existing solutions (academia, open source)
- What: Convert curated CNL solutions into Lean for reuse in Mathlib-style repositories while retaining human-readable source.
- Tools/workflow: CNL → Visored accept → UVL MIR → Lean file → lake build check; attach the Lean alongside the CNL in repos.
- Assumptions/dependencies: Heavy Lean verbosity acceptable post-acceptance; tactic closure works on small obligations.
- Safer LLM-driven analytics and modeling (finance, healthcare, enterprise software)
- What: Require Visored-checked derivations for risk, pricing, or statistical transformations in model docs, reducing hallucinated math.
- Tools/workflow: Documentation policy: all derivations must pass Visored; auditors examine Lean outputs for critical claims.
- Assumptions/dependencies: Claims framed within arithmetic/algebra; governance approval of toolchain; security sandboxing for CI.
- Competition-math practice assistants (education, consumer)
- What: Drill miniF2F-style problems with immediate, localized proof feedback and solver-closed routine steps.
- Tools/workflow: Front-end selects problems → user/LLM proposes proof → Visored feedback → hinting with “Then …” obligations.
- Assumptions/dependencies: Focus on non-IMO categories where coverage is near saturation; curated rule sets for target contests.
- Mathematical well-definedness linting (software, ML research)
- What: Catch 1/0, log of nonpositive arguments, and missing guards automatically in research notes and code comments.
- Tools/workflow: Pre-commit hook scans math snippets and runs Visored’s typecheck-time obligations.
- Assumptions/dependencies: Authors use designated environments; current well-definedness repertoire sufficient for the domain.
Long-Term Applications
These require further rule coverage, richer CNL idioms, stronger solver tactics, performance/UX work, or domain types beyond current scope (e.g., Z/nZ, limits, inequalities, advanced analysis).
- End-to-end autoformalization of research papers and publishing pipelines (academia, publishing)
- What: Journals accept Visored-checked CNL supplements with auto-generated Lean for kernel-level verification.
- Tools/workflow: Authoring in CNL LaTeX → Visored accept → Lean artifact archived → reviewer tooling highlights obligations and rules used.
- Assumptions/dependencies: Expanded CNL (“WLOG”, “sufficiently large n”, subsequences), broader libraries (limits, measure theory), streamlined Lean emission.
- Domain-vertical formal assurance for safety-critical systems (robotics, aerospace, energy, medical devices)
- What: Prove control stability, optimization feasibility, or safety invariants in natural language backed by formal kernels.
- Tools/workflow: Domain-specific specs and rule packs (Lyapunov, KKT, convexity, units/dimensions) → Visored check → regulator-facing artifacts.
- Assumptions/dependencies: New kinds/types (manifolds, matrices, dimensions), integration with domain solvers (SMT/CAS), auditor acceptance.
- Verified modeling and governance in regulated industries (finance, healthcare, public sector policy)
- What: Procurement and compliance standards mandate formally checked mathematical claims in AI/quant models and impact assessments.
- Tools/workflow: Policy templates with “example” sections for every asserted property; Visored + Lean artifacts attached to submissions.
- Assumptions/dependencies: Standards bodies codify acceptable kernels (de Bruijn criterion), open audits of toolchains, staff upskilling for CNL.
- Proof notebooks and interactive textbooks (education, software)
- What: Jupyter/Overleaf-like environments where natural-language proofs are live-checked and collapsible into formal steps.
- Tools/workflow: Rich CNL widgets, tactic-hint mode (“by induction”, “apply AM–GM”), incremental solver with provenance graphs.
- Assumptions/dependencies: UI/IDE integrations; solver guidance API; broader math idioms and pedagogical patterns.
- Training frontier formal-reasoning models at scale (AI research)
- What: Use Visored’s per-step rewards and infinite CNL corpora to train next-gen provers/verifiers that generalize beyond current benchmarks.
- Tools/workflow: Large-scale RLHF/RLAIF with vector rewards; self-play between conjecturing and proving; curriculum across spec variants.
- Assumptions/dependencies: Significant compute; robust deduplication/quality control; community-agreed eval suites.
- Cross-lingual mathematical CNL (education, global access)
- What: CNL specs localized to multiple languages while mapping to the same typed IR and solver, widening access to formal feedback.
- Tools/workflow: GF/grammar-based frontends tied to Visored IR; bilingual datasets; teacher tools for multi-language classes.
- Assumptions/dependencies: Carefully designed grammars to avoid ambiguity; cultural/notation variants encoded in specs.
- Industrial MLOps with verified math claims (software, ML engineering)
- What: Enforce Visored checks on convergence bounds, fairness constraints, and optimization steps in experiment reports and PRs.
- Tools/workflow: CI gates on CNL proofs for theoretical sections; dashboards showing which claims are kernel-verified.
- Assumptions/dependencies: New rule packs for statistics/probability/learning theory; integration with notebooks and experiment trackers.
- Cryptography and protocol proofs in readable CNL (security, software)
- What: Write security reductions and algebraic properties in controlled English backed by formal verification for broader reviewability.
- Tools/workflow: Crypto-specific rules/types (groups, fields, Z/nZ, lattices) → Visored → Lean/Isabelle targets for final proof objects.
- Assumptions/dependencies: Extensive algebraic libraries, probabilistic reasoning, and asymptotic notations.
- Government engineering approvals with formal annexes (public works, infrastructure)
- What: Require formally checked derivations for structural formulas, safety margins, and material models as annexes to bids/reports.
- Tools/workflow: Standardized CNL templates; traceable rule usage; cross-tool export for archival.
- Assumptions/dependencies: Policy uptake; training and certification; long-term tool maintenance and open standards.
- Low-latency, interactive tactic-guided solving (software, UX)
- What: Users/LLMs steer the solver via hints (“rewrite using lemma X”), enabling harder steps without leaving the CNL surface.
- Tools/workflow: Hint channels in the MIR; learnable heuristics for rule selection; improved diagnostics (“which rule almost fired”).
- Assumptions/dependencies: New solver modes; richer diagnostics; benchmarking on hard instances (e.g., IMO-like problems).
Notes on Feasibility and Dependencies
- Scope coverage is the main limiter today: extending to inequalities, limits, trigonometry, finite fields, and advanced analysis requires targeted rule libraries and types.
- Lean emission is functional but verbose; large-scale or real-time workflows benefit from keeping Visored as the primary verifier and using Lean only for archival/audit until emission is optimized.
- Adoption hinges on lightweight, editable specs (.lpcsv) tuned to house styles; governance and policy use cases depend on standardization and auditable kernels.
- LLMs need no prover-specific pretraining to be effective at the current scope, but stronger performance and broader coverage will benefit from fine-tuning on Visored-generated CNL corpora.
Glossary
- AM-GM inequality: The arithmetic mean–geometric mean inequality; a foundational inequality relating averages of nonnegative numbers. "General AM-GM inequality is not in the rule library."
- autoformalization: The process of mapping informal mathematical text into machine-checkable formal proofs. "The task we want to solve, the dashed LaTeX -> Lean edge, is autoformalization."
- automated theorem proving (ATP): Fully automated systems that search for proofs in logic, often via SAT/SMT or resolution methods. "Traditional automated theorem proving (ATP), namely SAT/SMT solvers and resolution provers, targets fully automated proof discovery, which is NP-hard or undecidable."
- CNL (controlled natural language): A restricted, precisely defined subset of natural language designed to be unambiguous and machine-interpretable. "Visored targets a different problem: with the LLM supplying the proof outline in controlled natural language (CNL), the solver only has to close the small obvious steps that mathematicians routinely omit, a much more tractable task ..."
- coercions: Implicit conversions between types (e.g., from integers to reals) handled by a type system or prover. "type systems, coercions, library naming, and partial-function side conditions must all be resolved before the kernel accepts the proof;"
- cost-bounded solver: A proof search component that limits exploration by a fixed cost or budget to ensure tractable checking. "Then ... " obligation is decided by a cost-bounded rule-driven solver (Appendix E)."
- de Bruijn criterion: The requirement that proof checkers rely on a small, trustworthy kernel to validate proofs independently of how they were produced. "One line of work removes the human from the loop with a small, auditable kernel that returns a definite yes/no on a candidate proof, independent of the model that produced it (the de Bruijn criterion [31, 30])."
- dependently typed: A type system feature where types can depend on values, enabling more expressive specifications and proofs. "The input is elaborated into a dependently-typed IR (Appendix C), and each "Then ... " obligation is decided by a cost-bounded rule-driven solver (Appendix E)."
- elaboration: The stage that refines parsed syntax into a typed, semantically rich form and generates proof obligations (including well-definedness checks). "Elaboration also enforces well-definedness: writing 1/x demands x # 0, Vx demands x ≥ 0, log x demands x > 0."
- Finset: A Lean/Mathlib construct for finite sets, often used for finite sums and combinatorics. "The CNL is three lines; the Lean side does explicit induction, Finset . sum_range_succ, and omega for truncated N subtraction."
- first-order logic: A formal logical system with quantification over individuals but not over predicates/functions. "Naproche translates ForTheL into first-order logic."
- ForTheL: A controlled natural language (“Formal Theory Language”) designed for writing readable, formalizable mathematics. "Isabelle/Naproche [24] is the modern reference point: a ForTheL CNL frontend whose obligations are discharged by automated theorem provers via TPTP."
- GFLean: A framework that uses the Grammatical Framework (GF) to parse a subset of ForTheL and emit Lean code. "GFLean [29] uses the Grammatical Framework as the parsing layer and emits Lean from a subset of ForTheL ("Simplified ForTheL")."
- interactive theorem provers (ITP): Proof assistants where users guide proof development interactively under a small kernel (e.g., Lean, Coq/Rocq). "On the ATP side, Visored needs a solver strong enough ... On the ITP side, producing the dependent-type-checked proofs that interactive theorem provers (ITP) such as Lean and Rocq require is hard, ..."
- intermediate representation (IR): A structured, machine-checkable representation of proofs/programs used between parsing and final output. "Visored is a semantically rich intermediate representation (IR) in which every decision that has no surface trace in the LaTeX, namely subtype, library lemma, edge-case convention, and cast, is made explicit."
- kind: A meta-level classification (often above types) used in type systems for organizing types and enabling dependent typing. "Names get resolved against the current scope ..., and each subexpression is assigned a type and a kind."
- kernel: The small, trusted core of a theorem prover that verifies proof terms and ensures soundness. "One line of work removes the human from the loop with a small, auditable kernel that returns a definite yes/no on a candidate proof, independent of the model that produced it ..."
- lake build: The Lean build tool command that compiles and checks Lean projects. "The formatter itself does no proof search, but the emitted Lean code calls Lean-side tactics ... which lake build then closes."
- Lean: A modern interactive theorem prover and programming language with dependent types. "so that an accepted proof can be re- emitted as a checked Lean file."
- Lean transpilation: Translating proofs from another representation into Lean code for kernel validation and ecosystem tooling. "The dominant sink was Lean transpilation: we spent several months on it, largely because we had not yet found the right way to organize the emitted proofs."
- Mathlib: The core mathematical library for Lean, containing thousands of definitions and theorems. "Mathlib, the Coq stdlib, and Isabelle developments together cover only a small fraction of the field."
- mid-level IR (MIR): An intermediate representation between high-level syntax and low-level proof terms, suited for solver interaction. "MIR (mid-level IR). The sem AST (a typed abstract syntax tree) is lowered to Visored MIR, ..."
- miniF2F: A benchmark suite for formalized mathematics, including competition-style problems. "Whole-proof theorem provers now drive miniF2F [50] pass rates close to saturation (Appendix A), ..."
- Miracle budget: The capped search budget used by Visored’s non-deterministic solver runtime (“Miracle”) to attempt closing goals. "When the Miracle budget is exhausted without closing a goal, the diagnostic reports the unclosed goal and source location but does not yet report which rule almost fired or how close the best candidate was."
- Naproche: A controlled natural language system and toolchain for writing and checking mathematical texts. "It shares the controlled-natural- language surface of Mizar and Naproche [38, 24], ..."
- non-linear arithmetic: Reasoning involving polynomial/nonlinear constraints rather than purely linear arithmetic. "and a non-linear arithmetic discharge that needs v1312 = 131 as an explicit hint."
- resolution provers: Automated theorem provers based on the resolution rule in first-order logic. "Traditional automated theorem proving (ATP), namely SAT/SMT solvers and resolution provers, targets fully automated proof discovery, which is NP-hard or undecidable."
- Rocq: The modern name for the Coq proof assistant; used alongside Lean as a target or comparison system. "complementing existing systems such as Lean and Rocq."
- SAT/SMT solvers: Engines that decide satisfiability of propositional (SAT) or richer logical (SMT) formulas, often used in ATP. "Traditional automated theorem proving (ATP), namely SAT/SMT solvers and resolution provers, targets fully automated proof discovery, which is NP-hard or undecidable."
- semantic parsing: Mapping surface-level LaTeX/natural language into a structured semantic representation (Visored). "The proposed route is the composition of the upward semantic parsing arrow (inverting the LaTeX projection) with the deterministic Visored -> Lean projection."
- silent semantic drift: A subtle change in meaning introduced during translation that still compiles but alters the intended statement. "but as a silent semantic drift: a statement that compiles yet no longer says what the prose claimed, or a goal that looks plausible but is unprovable for a reason several rewriting steps upstream."
- spec files (.lpcsv): External configuration files that define accepted sentence templates and LaTeX commands for the CNL parser. "The set of recognized sentence templates, LaTeX commands, and LaTeX environments lives in . lpcsv configuration files, so extending the surface language is a matter of adding entries (Appendix B), not modifying compiler code."
- subtype: A subset of a type defined by a predicate/constraint, used to model refined domains (e.g., nonzero reals). "Visored is a semantically rich intermediate representation (IR) in which every decision that has no surface trace in the LaTeX, namely subtype, library lemma, edge-case convention, and cast, is made explicit."
- tactic: A proof-step procedure used in interactive theorem provers to transform goals (e.g., simp, ring, omega). "the emitted Lean code calls Lean-side tactics - obvious, simp, assumption, custom rewrite tactics - on small per-step obligations, which lake build then closes."
- typed abstract syntax tree (AST): A tree representation of parsed terms where each node carries type information. "The sem AST (a typed abstract syntax tree) is lowered to Visored MIR, ..."
- UnknownCommand error: A structured error raised when an undeclared LaTeX command is encountered. "A LaTeX command the spec does not declare (the user writes \ foobar) is rejected at this stage with a structured UnknownCommand error pointing at the offending token, ..."
- UVL MIR: A target-neutral intermediate representation used for emitting Lean (or other provers) from Visored. "A separate stage further translates MIR plus the recorded derivation steps to UVL MIR, a target-neutral IR from which a Lean formatter emits proof text."
- Vampire: A state-of-the-art first-order automated theorem prover used by systems like Naproche. "Naproche calls out to general-purpose ATPs (E, Vampire) for each obligation."
- Vieta jumping: A classical number-theoretic technique frequently used in Olympiad inequalities and Diophantine problems. "Vieta jumping; a famously hard olympiad problem."
- well-definedness: Conditions ensuring expressions are meaningful (e.g., no division by zero), enforced during type checking. "Elaboration also enforces well-definedness: writing 1/x demands x # 0, Vx demands x ≥ 0, log x demands x > 0."
- WLOG: Abbreviation for “without loss of generality,” a standard proof simplification convention. "implicit side conditions ("for x sufficiently large", "WLOG x = 0"), ..."
- Z/nZ: Integers modulo n (finite ring/field), used in number theory and algebra. "Z/nZ constructor unimplemented."
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