# Normalization for Cubical Type Theory

@article{Sterling2021NormalizationFC, title={Normalization for Cubical Type Theory}, author={Jonathan Sterling and Carlo Angiuli}, journal={2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)}, year={2021}, pages={1-15} }

We prove normalization for (univalent, Cartesian) cubical type theory, closing the last major open problem in the syntactic metatheory of cubical type theory. Our normalization result is reduction-free, in the sense of yielding a bijection between equivalence classes of terms in context and a tractable language of β/η-normal forms. As corollaries we obtain both decidability of judgmental equality and the injectivity of type constructors.

## 21 Citations

### A Cubical Language for Bishop Sets

- MathematicsLog. Methods Comput. Sci.
- 2022

We present XTT, a version of Cartesian cubical type theory specialized for
Bishop sets \`a la Coquand, in which every type enjoys a definitional version
of the uniqueness of identity proofs. Using…

### Syntax and models of Cartesian cubical type theory

- MathematicsMathematical Structures in Computer Science
- 2021

Abstract We present a cubical type theory based on the Cartesian cube category (faces, degeneracies, symmetries, diagonals, but no connections or reversal) with univalent universes, each containing…

### Induction principles for type theories, internally to presheaf categories

- MathematicsArXiv
- 2021

New induction principles for the syntax of dependent type theories are presented, expressed in the internal language of presheaf categories, which ensures for free that any construction is stable under substitution.

### Reduction Free Normalisation for a proof irrelevant type of propositions

- Computer ScienceArXiv
- 2021

The goal of this note is to show normalization and decidability of conversion for dependent type theory with a cumulative sequence of universes U0,U1 . . . with η-conversion and where the type U0 is…

### Cubical methods in homotopy type theory and univalent foundations

- MathematicsMathematical Structures in Computer Science
- 2021

Cubical methods have played an important role in the development of Homotopy Type Theory and Univalent Foundations (HoTT/UF) in recent years. The original motivation behind these developments was…

### Logical Relations as Types

- Computer Science
- 2021

A fresh “synthetic” take on program modules that treats modules as the fundamental constructs, in which the usual suspects of prior module calculi are rendered as derived notions in terms of a modal type-theoretic account of the phase distinction.

### Generalized Universe Hierarchies and First-Class Universe Levels

- Computer ScienceCSL
- 2022

This work develops syntax and semantics for cumulative universe hierarchies, where levels may come from any set equipped with a transitive well-founded ordering, and shows that induction-recursion can be used to model transfinite hierarchies.

### Greatest HITs: Higher inductive types in coinductive definitions via induction under clocks

- Computer ScienceLICS
- 2022

A new principle of induction under clocks is provided, providing computational content to one of the main axioms required for encoding coinductive types in type theory.

### Normalization for Multimodal Type Theory

- MathematicsLICS
- 2022

The conversion problem for multimodal type theory (MTT) is considered by characterizing the normal forms of the type theory and proving normalization, which follows from a novel adaptation of Sterling’s Synthetic Tait Computability.

### Touring the MetaCoq Project

- Computer Science
- 2021

MetaCoq is a collaborative project that aims to provide the first fully-certified realistic implementation of a type checker for the full calculus underlying the Coq proof assistant.

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