scholarly journals Constructing Inductive-Inductive Types in Cubical Type Theory

2019 ◽  
pp. 295-312
Author(s):  
Jasper Hugunin
Keyword(s):  
Type Theory ◽  
Inductive Types ◽  
Cubical Type Theory

Cubical Agda: A dependently typed programming language with univalence and higher inductive types

2021 ◽  
Vol 31 ◽  
Author(s):  
ANDREA VEZZOSI ◽  
ANDERS MÖRTBERG ◽  
ANDREAS ABEL
Keyword(s):  
Programming Language ◽  
Type Theory ◽  
Type Checking ◽  
Dependent Type Theory ◽  
Wide Range ◽  
General Schema ◽  
Direct Definition ◽  
Inductive Types ◽  
Definition Of ◽  
Cubical Type Theory

Abstract Proof assistants based on dependent type theory provide expressive languages for both programming and proving within the same system. However, all of the major implementations lack powerful extensionality principles for reasoning about equality, such as function and propositional extensionality. These principles are typically added axiomatically which disrupts the constructive properties of these systems. Cubical type theory provides a solution by giving computational meaning to Homotopy Type Theory and Univalent Foundations, in particular to the univalence axiom and higher inductive types (HITs). This paper describes an extension of the dependently typed functional programming language Agda with cubical primitives, making it into a full-blown proof assistant with native support for univalence and a general schema of HITs. These new primitives allow the direct definition of function and propositional extensionality as well as quotient types, all with computational content. Additionally, thanks also to copatterns, bisimilarity is equivalent to equality for coinductive types. The adoption of cubical type theory extends Agda with support for a wide range of extensionality principles, without sacrificing type checking and constructivity.

scholarly journals Internal Parametricity for Cubical Type Theory

2021 ◽  
Vol Volume 17, Issue 4 ◽  
Author(s):  
Evan Cavallo ◽  
Robert Harper
Keyword(s):  
Type Theory ◽  
Formal Interface ◽  
Inductive Types ◽  
Cubical Type Theory

We define a computational type theory combining the contentful equality structure of cartesian cubical type theory with internal parametricity primitives. The combined theory supports both univalence and its relational equivalent, which we call relativity. We demonstrate the use of the theory by analyzing polymorphic functions between higher inductive types, observe how cubical equality regularizes parametric type theory, and examine the similarities and discrepancies between cubical and parametric type theory, which are closely related. We also abstract a formal interface to the computational interpretation and show that this also has a presheaf model.

scholarly journals Homotopy Type Theory: A Synthetic Approach to Higher Equalities

2018 ◽  
Author(s):  
Michael Shulman
Keyword(s):  
Set Theory ◽  
Homotopy Type ◽  
Type Theory ◽  
Synthetic Approach ◽  
Homotopy Type Theory ◽  
Inductive Types ◽  
Basic Ideas ◽  
Univalent Foundations ◽  
Modern Mathematics

Homotopy type theory and univalent foundations (HoTT/UF) is a new foundation of mathematics, based not on set theory but on “infinity-groupoids”, which consist of collections of objects, ways in which two objects can be equal, ways in which those ways-to-be-equal can be equal, ad infinitum. Though apparently complicated, such structures are increasingly important in mathematics. Philosophically, they are an inevitable result of the notion that whenever we form a collection of things, we must simultaneously consider when two of those things are the same. The “synthetic” nature of HoTT/UF enables a much simpler description of infinity groupoids than is available in set theory, thereby aligning with modern mathematics while placing “equality” back in the foundations of logic. This chapter will introduce the basic ideas of HoTT/UF for a philosophical audience, including Voevodsky’s univalence axiom and higher inductive types.

scholarly journals Guarded Cubical Type Theory

2018 ◽  
Vol 63 (2) ◽  
pp. 211-253 ◽  
Author(s):  
Lars Birkedal ◽  
Aleš Bizjak ◽  
Ranald Clouston ◽  
Hans Bugge Grathwohl ◽  
Bas Spitters ◽  
...  
Keyword(s):  
Type Theory ◽  
Cubical Type Theory

Algebra of programming in Agda: Dependent types for relational program derivation

2009 ◽  
Vol 19 (5) ◽  
pp. 545-579 ◽  
Author(s):  
SHIN-CHENG MU ◽  
HSIANG-SHANG KO ◽  
PATRIK JANSSON
Keyword(s):  
Programming Language ◽  
Type Theory ◽  
Type System ◽  
Dependent Types ◽  
Dependent Type Theory ◽  
Program Derivation ◽  
Inductive Types ◽  
Algebra Of Programming ◽  
Dependent Type

AbstractRelational program derivation is the technique of stepwise refining a relational specification to a program by algebraic rules. The program thus obtained is correct by construction. Meanwhile, dependent type theory is rich enough to express various correctness properties to be verified by the type checker. We have developed a library, AoPA (Algebra of Programming in Agda), to encode relational derivations in the dependently typed programming language Agda. A program is coupled with an algebraic derivation whose correctness is guaranteed by the type system. Two non-trivial examples are presented: an optimisation problem and a derivation of quicksort in which well-founded recursion is used to model terminating hylomorphisms in a language with inductive types.

A realizability interpretation of Martin-Löf’s type Theory

1998 ◽  
Author(s):  
Catarina Coquand
Keyword(s):  
Type Theory ◽  
The Other ◽  
Dependent Types ◽  
Powerful Method ◽  
Natural Numbers ◽  
Simple Argument ◽  
Calculus Of Constructions ◽  
Polymorphic Type ◽  
Inductive Types ◽  
System F

The notion of computability predicate developed by Tail gives a powerful method for proving normalization for various λ-calculi. There are actually two versions of this method: a typed and an untyped approach. In the former approach we define a computability predicate over well-typed terms. This technique was developed by Tail (1967) in order to prove normalization for Gödel’s system T. His method was extended by Girard (1971) for his System F and by Martin-Löf (1971) for his type theory. The second approach is similar to Kleene’s realizability interpretation (Kleene 1945), but in this approach formulas are realized by (not necessarily well-typed) λ-terms rather than Gödel numbers. This technique was developed by Tail (1975) where he uses this method to obtain a simplified proof of normalization for Girard’s system F. The method was later rediscovered independently by several others, for instance Mitchell (1986). There are two ways of defining typed λ-calculus: we have either equality as conversion (on raw terms) or equality as judgments. It is more difficult to show in the theory with equality as judgments that every well-typed term has a normal form of the same type. We can find different approaches to how to solve this problem in Altenkirch (1993), Coquand (1991), Goguen (1994) and Martin-Löf (1975). There are several papers addressing normalization for calculi with equality as untyped conversion; two relevant papers showing normalization are Altenkirch’s (1994) proof for system F using the untyped method and Werner’s (1994) proof for the calculus of constructions with inductive types using the other method. Another difference is the predicative and impredicative formulations of λ-calculi where it is more intricate to prove normalization for the impredicative ones. The aim of this paper is to give a simple argument of normalization for a theory with “real” dependent types, i.e. a theory in which we can define types by recursion on the natural numbers (like Nn). For this we choose to study the simplest theory that has such types, namely the N, Π, U-fragment of Martin-Löf’s polymorphic type theory (Martin-Löf 1972) which contains the natural numbers, dependent functions and one universe. This theory has equality as conversion and is predicative.

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