Are Dependent Types in Set Theory Feasible?

TL;DR

This paper demonstrates a feasible approach to implementing dependent types within set theory by embedding them into first-order logic, enabling verified automated reasoning in a proof assistant.

Contribution

It introduces a novel embedding of dependent function types into set theory within a proof assistant, supporting verified automated reasoning with subtyping and equality.

Findings

Successful embedding of dependent types into set theory

Implementation of a proof-producing type-checking tactic

Automated reasoning verified from set-theoretic axioms

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.