Normalization for Cubical Type Theory

  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)},
  • Jonathan Sterling, C. Angiuli
  • Published 27 January 2021
  • Mathematics
  • 2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
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. 

Figures from this paper

A Cubical Language for Bishop Sets

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

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

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

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

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

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

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

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

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

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.



Canonicity for Cubical Type Theory

  • Simon Huber
  • Mathematics
    Journal of Automated Reasoning
  • 2018
This paper proves canonicity for cubical type theory: any natural number in a context build from only name variables is judgmentally equal to a numeral.

A Cubical Language for Bishop Sets

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

Cubical Syntax for Reflection-Free Extensional Equality

An algebraic canonicity theorem is established using a novel cubical extension of the logical families or categorical gluing argument inspired by Coquand and Shulman: every closed element of boolean type is derivably equal to either 'true' or 'false'.

Canonicity and normalisation for Dependent Type Theory

On Higher Inductive Types in Cubical Type Theory

A constructive semantics, expressed in a presheaf topos with suitable structure inspired by cubical sets, of some higher inductive types of spheres, torus, suspensions, truncations, and pushouts is described.

Cartesian Cubical Computational Type Theory: Constructive Reasoning with Paths and Equalities

A dependent type theory organized around a Cartesian notion of cubes, supporting both fibrant and non-fibrant types, and is the first two-level type theory to satisfy the canonicity property: all closed terms of boolean type evaluate to either true or false.

Cubical Type Theory: A Constructive Interpretation of the Univalence Axiom

A type theory in which it is possible to directly manipulate n-dimensional cubes based on an interpretation of dependenttype theory in a cubical set model that enables new ways to reason about identity types, for instance, function extensionality is directly provable in the system.

Internal Type Theory

We introduce categories with families as a new notion of model for a basic framework of dependent types. This notion is close to ordinary syntax and yet has a clean categorical description. We also

Type theory in type theory using quotient inductive types

The formalisation of type theory avoids referring to preterms or a typability relation but defines directly well typed objects by an inductive definition and uses the elimination principle to define the set-theoretic and logical predicate interpretation.

Normalization by evaluation for typed lambda calculus with coproducts

This method is based on the semantic technique known as "normalization by evaluation", and involves inverting the interpretation of the syntax in a suitable sheaf model and extracting an appropriate unique normal form from this.