Recursive Families of Inductive Types

2000 ◽  
pp. 73-89 ◽  
Author(s):  
Venanzio Capretta
Keyword(s):  
Inductive Types

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 Well-definedness and observational equivalence for inductive–coinductive programs

2019 ◽  
Vol 29 (4) ◽  
pp. 419-468
Author(s):  
Henning Basold ◽  
Helle Hvid Hansen
Keyword(s):  
Fixed Point ◽  
Modal Logic ◽  
The Other ◽  
Observational Equivalence ◽  
Other Hand ◽  
Inductive Types ◽  
Point Types ◽  
Usual Topology

Abstract We define notions of well-definedness and observational equivalence for programs of mixed inductive and coinductive types. These notions are defined by means of tests formulas which combine structural congruence for inductive types and modal logic for coinductive types. Tests also correspond to certain evaluation contexts. We define a program to be well-defined if it is strongly normalizing under all tests, and two programs are observationally equivalent if they satisfy the same tests. We show that observational equivalence is sufficiently coarse to ensure that least and greatest fixed point types are initial algebras and final coalgebras, respectively. This yields inductive and coinductive proof principles for reasoning about program behaviour. On the other hand, we argue that observational equivalence does not identify too many terms, by showing that tests induce a topology that, on streams, coincides with usual topology induced by the prefix metric. As one would expect, observational equivalence is, in general, undecidable, but in order to develop some practically useful heuristics we provide coinductive techniques for establishing observational normalization and observational equivalence, along with up-to techniques for enhancing these methods.

scholarly journals Refining Inductive Types

2012 ◽  
Vol 8 (2) ◽  
Author(s):  
Robert Atkey ◽  
Patricia Johann ◽  
Neil Ghani
Keyword(s):  
Inductive Types

scholarly journals The Construction of Set-Truncated Higher Inductive Types

2019 ◽  
Vol 347 ◽  
pp. 261-280 ◽  
Author(s):  
Niels van der Weide ◽  
Herman Geuvers
Keyword(s):  
Inductive Types

scholarly journals A UNIVERSE OF STRICTLY POSITIVE FAMILIES

2009 ◽  
Vol 20 (01) ◽  
pp. 83-107 ◽  
Author(s):  
PETER MORRIS ◽  
THORSTEN ALTENKIRCH ◽  
NEIL GHANI
Keyword(s):  
Finite Sets ◽  
Data Types ◽  
Traditional Treatment ◽  
Fixed Length ◽  
One Step ◽  
Inductive Types ◽  
The Universe

In order to represent, compute and reason with advanced data types like lists with a fixed length, finite sets or well scoped λ-terms one must go beyond the traditional treatment of data types as being inductive types and, instead, consider them as inductive families, or more precisely Strictly Positive Families (SPFs). We have previously shown that the grammar of strictly positive types (SPT) can be given as an inductively defined family. In the present paper we go one step further an show that the universe of SPFs can be encoded as an SPF. We show that SPFs can be used to represent and compute with a variety of advanced data types and that generic programs can be naturally written over the universe of SPFs.

scholarly journals Equational theories for inductive types

1997 ◽  
Vol 84 (2) ◽  
pp. 175-217 ◽  
Author(s):  
Ralph Loader
Keyword(s):  
Equational Theories ◽  
Inductive Types

State Machines as Inductive Types

2007 ◽  
Vol E90-A (12) ◽  
pp. 2985-2988 ◽  
Author(s):  
K. OGATA ◽  
K. FUTATSUGI
Keyword(s):  
State Machines ◽  
Inductive Types

scholarly journals Remarks on Isomorphisms of Simple Inductive Types

2003 ◽  
Vol 85 (7) ◽  
pp. 106-124 ◽  
Author(s):  
David Chemouil ◽  
Sergei Soloviev
Keyword(s):  
Inductive Types
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