Are Dependent Types in Set Theory Feasible?
This work addresses the feasibility of dependent types in set theory for automated theorem proving, but it is incremental as it builds on existing types-as-sets paradigms.
The paper tackles the problem of embedding dependent types into set theory by presenting a mechanized embedding in the Lisa proof assistant, enabling automated and verified reasoning for dependent types from set-theoretic axioms.
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.