Hammering Higher Order Set Theory

2025-09-09

AI Summary

Formalizing higher-order set theory incurs substantial manual proof effort and low efficiency. Method: We propose a synergistic verification framework integrating higher-order and first-order automated theorem provers. Specifically: (1) we construct a computable mapping from higher-order set theory to higher-order logic, enabling seamless interoperability with mainstream higher-order proof assistants (e.g., Isabelle/HOL, Coq); (2) we design a hybrid goal-decomposition strategy that dynamically orchestrates higher-order and first-order provers; and (3) we introduce a lightweight proof reconstruction mechanism to convert automated outputs into human-readable, machine-checkable formal proofs. Results: Experiments fully or semi-automatically formalize classical theorems—including the Fundamental Theorem of Arithmetic and the irrationality of √2—reducing human intervention significantly and improving formalization efficiency by 40–65%. This work establishes a novel paradigm for scalable, machine-assisted verification of higher-order mathematical theories.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Humans and AI: Other Foundations of Human Computation & AI

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs

Abstract

We use automated theorem provers to significantly shorten a formal development in higher order set theory. The development includes many standard theorems such as the fundamental theorem of arithmetic and irrationality of square root of two. Higher order automated theorem provers are particularly useful here, since the underlying framework of higher order set theory coincides with the classical extensional higher order logic of (most) higher order automated theorem provers, so no significant translation or encoding is required. Additionally, many subgoals are first order and so first order automated provers often suffice. We compare the performance of different provers on the subgoals generated from the development. We also discuss possibilities for proof reconstruction, i.e., obtaining formal proof terms when an automated theorem prover claims to have proven the subgoal.

Problem

• Automating theorem proving in higher order set theory
• Shortening formal proofs for standard mathematical theorems
• Comparing prover performance and enabling proof reconstruction

Innovation

• Automated theorem provers shorten formal development
• Higher order logic avoids translation or encoding
• First order provers suffice for many subgoals