A Minimalist Proof Language for Neural Theorem Proving over Isabelle/HOL

TL;DR

This paper introduces Minilang, a minimalist proof language designed to better align informal reasoning with mechanized proofs, improving neural theorem proving performance over Isabelle/HOL.

Contribution

It proposes Minilang, a simplified proof language with only 10 operations, and demonstrates its effectiveness in enhancing LLM-based theorem proving accuracy.

Findings

Minilang improves pass@1 success rate by up to 20/29 percentage points.

Achieves a pass@1 rate of 69.1%, surpassing prior work.

Pass@8 rate reaches 79.2%, exceeding state-of-the-art benchmarks.

Abstract

Neural Theorem Proving (NTP) employs LLMs to automate formal proofs in proof assistants. While LLMs have achieved relatively remarkable success in informal reasoning tasks using natural languages, the transition to mechanized formal theorem proving presents persistent challenges. Mechanized proof languages often contain many syntactic constructs and diverse, specialized proof tactics, which facilitate expert use but have no direct counterpart in informal mathematical proofs. These prover-specific idioms represent an additional burden for LLM-based NTPs that might be otherwise successful in generating informal proofs. Seeking to bridge this gap between formal proof construction and informal reasoning, in order to better facilitate NTP, this work approaches these challenges from a language design perspective. We look at common reasoning patterns in informal proofs and in existing mechanized proofs, and design Minilang -- a minimalist proof language that captures these reasoning patterns. In contrast to proof languages (informal and formal) that often feature a large collection of operations with unclear semantic boundaries, Minilang is deliberately kept minimalist -- its core design comprises only 10 operations, each with clear semantic distinctions. We further develop a rule-based translator from Isabelle's language (Isar) to Minilang, translating ~340K existing proofs with an ~85% success rate. Using this translated corpus, we finetune two LLMs to compare machine learning performance on Minilang versus the original Isar. Experiments show Minilang benefits the two LLMs by improving the pass@1 success rate on the PISA benchmark by up to 20/29 percentage points in comparison to the Isar-based LLMs w/wo Sledgehammer. The pass@1 rate reaches 69.1%, exceeding the prior work Baldur's pass@64 (65.7%); the pass@8 rate reaches 79.2%, exceeding the SOTA on PISA (71.0%) achieved by Magnushammer.