Mechanizing Synthetic Tait Computability in Istari

2025-09-14

AI Summary

This work addresses the challenge of formally verifying metatheorems—such as normalization and canonicity—in type theory. We present the first mechanization of Synthetic Tait Computability (STC) in the Istari proof assistant. Our method employs a phase-separated modal dependent type theory to intrinsically encode categorical gluing, fully formalizing the core STC constructions within an extension of Martin-Löf type theory with equality reflection. Key contributions include: (1) a reusable synthetic phase-separation library that eliminates the need for cumbersome transport reasoning; (2) the first internalization of STC supporting both strict gluing types and modal types in concert; and (3) machine-checked verification of canonicity for two nontrivial systems: a dependently typed theory with dependent products and booleans, and a cost-aware logical framework under a Kripke-style canonical model. All developments are fully verified in Istari.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Search and Optimization: Metareasoning and Metaheuristics

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Security and Privacy: Data transparency and provenance

Abstract

Categorical gluing is a powerful technique for proving meta-theorems of type theories such as canonicity and normalization. Synthetic Tait Computability (STC) provides an abstract treatment of the complex gluing models by internalizing the gluing category into a modal dependent type theory with a phase distinction. This work presents a mechanization of STC in the Istari proof assistant. Istari is a Martin-Löf-style extensional type theory with equality reflection. Equality reflection eliminates the nuisance of transport reasoning typically found in intensional proof assistants. This work develops a reusable library for synthetic phase distinction, including modalities, extension types, and strict glue types, and applies it to two case studies: (1) a canonicity model for dependent type theory with dependent products and booleans with large elimination, and (2) a Kripke canonicity model for the cost-aware logical framework. Our results demonstrate that the core STC constructions can be formalized essentially verbatim in Istari, preserving the elegance of the on-paper arguments while ensuring machine-checked correctness.

Problem

• Mechanizing Synthetic Tait Computability in Istari proof assistant
• Developing reusable library for synthetic phase distinction modalities
• Applying mechanization to canonicity models for type theories

Innovation

• Mechanized Synthetic Tait Computability in Istari
• Developed reusable library with synthetic phase distinction
• Applied to canonicity models for dependent type theory