scholarly journals Wittgenstein's Uniqueness Rule as an Elimination Rule of Inductive Types:

2021 ◽  
Vol 53 (2) ◽  
pp. 95-114
Author(s):  
Mitsuhiro Okada
Keyword(s):  
Elimination Rule ◽  
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.

Multiplicative conjunction and an algebraic meaning of contraction and weakening

1998 ◽  
Vol 63 (3) ◽  
pp. 831-859 ◽  
Author(s):  
A. Avron
Keyword(s):  
Classical Logic ◽  
Linear Logic ◽  
Substructural Logics ◽  
Interesting Property ◽  
Relevance Logic ◽  
Proper Subset ◽  
Algebraic Structures ◽  
Elimination Rule ◽  
First Case ◽  
Soundness And Completeness

AbstractWe show that the elimination rule for the multiplicative (or intensional) conjunction Λ is admissible in many important multiplicative substructural logics. These include LLm (the multiplicative fragment of Linear Logic) and RMIm (the system obtained from LLm by adding the contraction axiom and its converse, the mingle axiom.) An exception is Rm (the intensional fragment of the relevance logic R, which is LLm together with the contraction axiom). Let SLLm and SRm be, respectively, the systems which are obtained from LLm and Rm by adding this rule as a new rule of inference. The set of theorems of SRm is a proper extension of that of Rm, but a proper subset of the set of theorems of RMIm. Hence it still has the variable-sharing property. SRm has also the interesting property that classical logic has a strong translation into it. We next introduce general algebraic structures, called strong multiplicative structures, and prove strong soundness and completeness of SLLm relative to them. We show that in the framework of these structures, the addition of the weakening axiom to SLLm corresponds to the condition that there will be exactly one designated element, while the addition of the contraction axiom corresponds to the condition that there will be exactly one nondesignated element (in the first case we get the system BCKm, in the second - the system SRm). Various other systems in which multiplicative conjunction functions as a true conjunction are studied, together with their algebraic counterparts.

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.

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