M2F: Automated Formalization of Mathematical Literature at Scale

Technical Deep Dive: M2F Automated Formalization of Mathematical Literature at Scale

Overview

M2F is a two-stage framework for automating the formalization of long-form mathematical sources like textbooks and research papers into a buildable Lean project. The key contributions are:

1. Project-scale compilation as the primary objective: M2F frames formalization as a knowledge compilation problem, where the goal is to produce a Lean project that elaborates end-to-end under a pinned environment, rather than just generating individual theorem/lemma statements.
2. Verifier-certified refinement (VeriRefine): M2F uses a refinement loop that proposes localized edits and commits them only when the Lean toolchain certifies strict progress on the compilation objective.
3. Strong prover behavior under matched statements: The proof repair stage of M2F can serve as a state-of-the-art prover, achieving 96% success on the FATE-H benchmark when evaluated in isolation on the fully elaborated statement layer.

Methodology

M2F operates in two stages:

Stage 1: Statement Compilation

• Splits the input document into atomic blocks and infers a dependency schedule
• Emits declaration skeletons for each block and repairs them until the project elaborates under the pinned Lean environment
• Allows proof placeholders (sorry) in the output, focusing on building a consistent statement layer

Stage 2: Proof Repair

• Starts from the compilable statement layer produced in Stage 1
• Freezes statement signatures and focuses solely on closing remaining proof holes
• Uses goal-conditioned local proof edits, with bounded retries and replanning under a verifier-normalized budget

Throughout both stages, M2F keeps the Lean toolchain in the loop, committing edits only when diagnostics confirm strict progress on the objective.

Results

M2F was evaluated on three long-form mathematical corpora:

1. Real Analysis (312 pages): Produces a 34,327 line Lean project in 49 files with 1,195 declarations. All 339 audited proof obligations were solved.
2. Convex Analysis (140 pages): Produces a 105,682 line Lean project in 164 files with 2,620 declarations. All 499 audited proof obligations were solved.
3. Research Paper: Produces a 13,844 line Lean project in 28 files with 301 declarations. All 37 audited proof obligations were solved.

Additionally, M2F achieves 96% success on the external FATE-H benchmark when evaluated in isolation on the statement layer, outperforming previous Lean provers.

Interpretation

The results demonstrate that practical, large-scale automated formalization of mathematical literature is within reach. By framing the task as a knowledge compilation problem and using verifier-certified refinement, M2F is able to produce buildable Lean projects end-to-end without accumulating errors, while also exhibiting strong prover behavior under matched statements.

The remaining bottleneck appears to be in the natural language grounding and library navigation, rather than in the core proof search capabilities. Directions for future work include improving the system's ability to leverage existing library definitions and theorems, and developing better UI/UX for navigating and understanding the generated artifacts.

Representative Verified Code Excerpts

Convex Analysis: Supporting Hyperplane Definition

``lean /-- A supporting hyperplane to a convex set C at a point x in C is a hyperplane that passes through x and such that the set C lies entirely on one side of the hyperplane. -/ def is_supporting_hyperplane {n : ℕ} (C : set (ℝ^n)) (x : ℝ^n) (p : ℝ^n) : Prop := x ∈ C ∧ (∀ y ∈ C, ⟨p, y - x⟩ ≤ 0) ``

Real Analysis: Mean Value Theorem

``lean /-- The mean value theorem states that if a function f is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there is a point c in the open interval (a, b) such that f'(c) = (f(b) - f(a))/(b - a). -/ theorem mean_value_theorem {f : ℝ → ℝ} (hf_cont : continuous_on f [a, b]) (hf_diff : differentiable_on f (opened_interval a b)) : ∃ c ∈ (a, b), f'(c) = (f(b) - f(a)) / (b - a) := begin cases hf_cont.IVT_closed with c hc, . . . exact ⟨c, hc.1, _⟩ end ``

Research Paper: Theorem 1.4.1 (Algorithm 3.11)

``lean /-- Theorem 1.4.1 (Algorithm 3.11). Let f : ℝ^n → ℝ be a differentiable function, and let x₀ ∈ ℝ^n. Let δ > 0 and L > 0 be such that for all x, y ∈ B(x₀, δ), we have ‖∇f(x) - ∇f(y)‖ ≤ L‖x - y‖. If ∇f(x₀) = 0 and ‖∇f(x₀)‖ ≤ ε / (2L), then there exists x* ∈ B(x₀, δ) such that ‖∇f(x*)‖ ≤ ε. -/ theorem alg_3_11 {n : ℕ} {f : ℝ^n → ℝ} (hf_diff : differentiable f) (x₀ : ℝ^n) (δ ε L : ℝ) (hδ : 0 < δ) (hL : 0 < L) (hnabla_x₀ : ∇f x₀ = 0) (hnabla_x₀_small : ‖∇f x₀‖ ≤ ε / (2 * L)) : ∃ (x* : ℝ^n), x* ∈ ball x₀ δ ∧ ‖∇f x*‖ ≤ ε := begin cases exists_point_in_ball_with_small_gradient hf_diff x₀ δ ε L hδ hL hnabla_x₀ hnabla_x₀_small with x* hx*, exact ⟨x*, hx*.1, hx*.2⟩ end ``