Interactive Theorem Proving

Interactive theorem proving is the proof-assistant mode where a human or AI works inside a formal system and the checker validates each proof step or tactic. In 137. 对洪乐潼的4小时访谈：AI for Math、把数学变成Lean、数学天书中的证明、直觉、被创造与被发现的, Hong Letong / 洪乐潼 distinguishes this from older automatic theorem proving and says current AI systems often replace more of the human interaction loop while still relying on a formal checker.

Key Claims

• The formal checker gives AI Verification a stronger signal than ordinary natural-language reasoning.
• The hard work includes translating informal goals into formal statements, choosing tactics, managing proof state, and using existing library material.
• Lean Theorem Prover and Mathlib make interactive theorem proving usable for modern mathematics, but library gaps and slow verification remain bottlenecks.
• AI theorem proving can produce correct but long proofs, so proof quality and proof elegance remain separate evaluation dimensions.

Connections

• Axiom Prover, AI For Math, and AI Mathematician — AI systems and goals built around theorem proving.
• Auto-Formalization and Formal Specification — upstream translation and precise-statement requirements.
• Formal Verification and AI Coding Verification — software and systems verification branch.