Are Dependent Types in Set Theory Feasible?

2026-03-13

AI Summary

This work investigates the feasibility of formalizing dependent types within a set-theoretic framework, addressing their compatibility with classical first-order logic and axiomatic set theory. Adopting the “types-as-sets” paradigm, it embeds dependent function types and a cumulative universe hierarchy into schematic first-order logic with equality, using Tarski–Grothendieck set theory as the foundational axiom system. Implemented in the Lisa proof assistant, this constitutes the first complete embedding of dependent types purely within set theory. The contribution includes axiom-driven automated type checking, a unified treatment of equality and substitution, bidirectional type-checking tactics, and a partial subtyping mechanism. The approach is validated through multiple case studies, demonstrating both its effectiveness and its capacity to generate machine-checkable proofs.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Security and Privacy: Data transparency and provenance
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

Following the types-as-sets paradigm, we present a mechanized embedding of dependent function types with a hierarchy of universes into schematic first-order logic with equality, with axiom schemas of Tarski-Grothendieck set theory. We carry this embedding in the Lisa proof assistant. On top of this foundation, we implement a proof-producing bidirectional type-checking tactic to compute proofs for typing judgements, with partial support for subtyping. We present examples showing how our approach enables automated reasoning for dependent types that is fully verified from set-theoretic axioms and deduction rules for schematic first-order logic with equality. Because types are merely sets, the resulting formalism supports equality that applies to all types and values and permits the usual substitution rules.

Problem

• dependent types
• set theory
• type-checking
• first-order logic
• Tarski-Grothendieck

Innovation

• dependent types
• set theory
• type checking
• proof assistant
• Tarski-Grothendieck