ITPEval: Benchmarking Formal Translation Across Interactive Theorem Provers

ITPEval is a benchmark for evaluating automated formal proof translation across four major interactive theorem provers (ITPs) spanning two logical foundations. It accompanies the paper ITPEval: Benchmarking Formal Translation Across Interactive Theorem Provers.

Dataset Summary

The benchmark comprises 390 aligned files spanning over 1,700 theorems and lemmas, each formalized in all four ITPs:

ITP	Foundation	File Extension
Lean 4	Calculus of Inductive Constructions	.lean
Rocq (Coq)	Calculus of Inductive Constructions	.v
Isabelle/HOL	Higher-order logic	.thy
HOL Light	Higher-order logic	.ml

Every theorem is present in all four ITPs (4-way intersection), yielding 12 balanced directed translation pairs per file.

Data Files

File	Records	Description
stmts.json	1,560	Sorry-stripped theorem statements (390 theorems × 4 ITPs)
proofs.json	296	Full proof files (74 theorems × 4 ITPs)

Record Schema

Each record is a JSON object with the following fields:

Field	Type	Description
theorem_id	int	Globally unique theorem identifier (1–390)
title	string	Theorem name (e.g., "circle_average", "aime_1983_p1")
source	string	Benchmark source: "babel-formal", "hundred-theorems", or "minif2f"
tier	string	"a" (controlled/axiomatized) or "b" (ecosystem/library-dependent)
prover	string	ITP system: "lean4", "rocq", "isabelle", or "hol-light"
content	string	Full file text (statement-only for stmts.json, complete proof for proofs.json)
split	string	(miniF2F only) "test" or "valid"

Benchmark Structure

The benchmark is organized into two tiers:

Tier	Source	Files	Lemmas	Description
A (Controlled)	Babel-formal	16	165	Self-contained, axiomatized theorems. Isolates foundational translation difficulty.
B (Ecosystem)	Hundred Theorems	58	1,231	Classical theorems from real ITP libraries. Tests library API and tactic translation.
B (Ecosystem)	miniF2F	316	316	Competition math statements. Tests pure statement translation.

Translation modes:

• Statement translation (stmts.json): All 390 files. Source file has proof bodies replaced by placeholders (sorry, Admitted, etc.). Yields 4,680 directed pairs per model (390 × 12 directions).
• Proof translation (proofs.json): Babel-formal + Hundred Theorems (74 files). Source file contains the complete proof. Yields 888 directed pairs per model (74 × 12 directions).

Usage

import json

# Load data
stmts  = json.load(open("stmts.json"))   # 1,560 records
proofs = json.load(open("proofs.json"))   # 296 records

# All Lean 4 statements
lean4_stmts = [r for r in stmts if r["prover"] == "lean4"]

# All 4 ITP variants of one theorem
theorem_1 = [r for r in stmts if r["theorem_id"] == 1]

# Generate all directed translation pairs for a theorem
def translation_pairs(records, theorem_id):
    variants = [r for r in records if r["theorem_id"] == theorem_id]
    pairs = []
    for src in variants:
        for tgt in variants:
            if src["prover"] != tgt["prover"]:
                pairs.append({"src": src, "tgt_prover": tgt["prover"]})
    return pairs

# Tier A only (controlled)
tier_a = [r for r in stmts if r["tier"] == "a"]

# miniF2F test split
minif2f_test = [r for r in stmts if r["source"] == "minif2f" and r.get("split") == "test"]

Evaluation Protocol

For each (source file, target ITP) pair:

1. Prompt an LLM with the source file and the target ITP name
2. The LLM generates a complete file in the target ITP
3. Run the generated file through the target ITP binary
4. Mark as verified if and only if the ITP exits with code 0 (no errors, no partial credit)

Per-prover timeouts: Lean 4 and Rocq at 60s, Isabelle at 120s, HOL Light at 300s.

Key Results

With five LLMs (GPT-5.5, Claude Sonnet 4.6, Gemini 3.1 Pro, DeepSeek-V4-Pro, Qwen3-235B), evaluated zero-shot at temperature 0:

• Statement translation peaks at 29.1% pass@1; proof translation peaks at 10.5%
• Tier A (controlled) reaches 29.7% proof pass@1 vs. 5.2% for Tier B (ecosystem)
• Translation direction matters sharply: e.g., Lean 4 → Isabelle (19.8%) vs. Isabelle → Lean 4 (1.9%)

License

This dataset is released under CC BY 4.0. Individual formalizations may inherit licenses from their upstream sources (Mathlib, Archive of Formal Proofs, HOL Light core, MathComp, miniF2F).