scholarly journals Constructive sheaf models of type theory

2021 ◽  
pp. 1-24
Author(s):  
Thierry Coquand ◽  
Fabian Ruch ◽  
Christian Sattler
Keyword(s):  
Type Theory ◽  
Grothendieck Topology ◽  
Constructive Models ◽  
Constructive Version ◽  
Base Category

Abstract We provide a constructive version of the notion of sheaf models of univalent type theory. We start by relativizing existing constructive models of univalent type theory to presheaves over a base category. Any Grothendieck topology of the base category then gives rise to a family of left-exact modalities, and we recover a model of type theory by localizing the presheaf model with respect to this family of left-exact modalities. We provide then some examples.

Hyperfinite type structures

1999 ◽  
Vol 64 (3) ◽  
pp. 1216-1242 ◽  
Author(s):  
Dag Normann ◽  
Erik Palmgren ◽  
Viggo Stoltenberg-Hansen
Keyword(s):  
Mathematical Model ◽  
Type Theory ◽  
Nonstandard Analysis ◽  
A Priori ◽  
Domain Theory ◽  
Transfer Principle ◽  
Finitely Based ◽  
Constructive Version ◽  
Continuous Functionals ◽  
Original Concept

The notion of a hyperfinite set comes from nonstandard analysis. Such a set has the internal cardinality of a nonstandard natural number. By a transfer principle such sets share many properties of finite sets. Here we apply this notion to give a hyperfinite model of the Kleene-Kreisel continuous functionals. We also extend the method to provide a hyperfinite characterisation of certain transfinite type structures, thus, through the work of Waagbø [14], constructing a hyperfinite model for Martin-Löf type theory.This kind of application is not new. Normann [6] gave a characterisation of the Kleene-Kreisel continuous functionals using ‘hyperfinitary’ functionals. The novelty here is that we use a constructive version of hyperfinite functionals and also generalise the method to transfinite types. Many of the results of this paper are constructive, though not the characterisation theorems themselves.Our characterisation of the Kleene-Kreisel continuous functionals is a supplement to a number of previous characterisations of topological and recursion-theoretical nature, see [6] for a brief survey. Altogether these characterisations show that the original concept of Kleene and Kreisel forms the correct mathematical model of the idea of finitely based functions of finite types.There is, however, no a priori reason to believe that there is a canonical way to extend the continuous functionals to cover transfinite objects of transfinite type used in, e.g., type theory. Our characterisation of Waagbø's model indicates that the model is natural, not only seen from domain theory but from a higher perspective. Normann and Waagbø (unpublished) have subsequently obtained a limit-space characterisation that further supports this view.

A general formulation of simultaneous inductive-recursive definitions in type theory

2000 ◽  
Vol 65 (2) ◽  
pp. 525-549 ◽  
Author(s):  
Peter Dybjer
Keyword(s):  
Type Theory ◽  
General Notion ◽  
Recursive Definition ◽  
Restricted Version ◽  
Inductive Definitions ◽  
Simultaneous Induction ◽  
Constructive Version ◽  
Small Set ◽  
Recursive Definitions ◽  
The Way

AbstractThe first example of a simultaneous inductive-recursive definition in intuitionistic type theory is Martin-Löfs universe à la Tarski. A set U0of codes for small sets is generated inductively at the same time as a function T0, which maps a code to the corresponding small set, is defined by recursion on the way the elements of U0are generated.In this paper we argue that there is an underlyinggeneralnotion of simultaneous inductive-recursive definition which is implicit in Martin-Löf's intuitionistic type theory. We extend previously given schematic formulations of inductive definitions in type theory to encompass a general notion of simultaneous induction-recursion. This enables us to give a unified treatment of several interesting constructions including various universe constructions by Palmgren, Griffor, Rathjen, and Setzer and a constructive version of Aczel's Frege structures. Consistency of a restricted version of the extension is shown by constructing a realisability model in the style of Allen.

Type Theory and Formal Proof

2009 ◽  
Author(s):  
Rob Nederpelt ◽  
Herman Geuvers
Keyword(s):  
Type Theory ◽  
Formal Proof

Computational Models for Multilayered Composite Shells with Application to Tires

1996 ◽  
Vol 24 (1) ◽  
pp. 11-38 ◽  
Author(s):  
G. M. Kulikov
Keyword(s):  
Type Theory ◽  
Computational Models ◽  
Geometrical Nonlinearity ◽  
Higher Order ◽  
Stress Strain ◽  
Strain Fields ◽  
First Order ◽  
Timoshenko Type ◽  
Joint Influence ◽  
Discrete Layer

Abstract This paper focuses on four tire computational models based on two-dimensional shear deformation theories, namely, the first-order Timoshenko-type theory, the higher-order Timoshenko-type theory, the first-order discrete-layer theory, and the higher-order discrete-layer theory. The joint influence of anisotropy, geometrical nonlinearity, and laminated material response on the tire stress-strain fields is examined. The comparative analysis of stresses and strains of the cord-rubber tire on the basis of these four shell computational models is given. Results show that neglecting the effect of anisotropy leads to an incorrect description of the stress-strain fields even in bias-ply tires.

Campus Social Climate Correlates of Environmental Type Dimensions

NASPA Journal ◽  
2004 ◽  
Vol 41 (4) ◽  
Author(s):  
Daniel W. Salter ◽  
Reynol Junco ◽  
Summer D. Irvin
Keyword(s):  
Work Environment ◽  
Personal Growth ◽  
Type Theory ◽  
General Purpose ◽  
Myers Briggs Type Indicator ◽  
Environment Scale ◽  
Type Indicator ◽  
Group Environment ◽  
System Maintenance ◽  
Environment Type

To address the ability of the Salter Environment Type Assessment (SETA) to measure different kinds of campus environments, data from three studies of the SETA with the Work Environment Scale, Group Environment Scale, and University Residence Environment Scale were reexamined (n = 534). Relationship dimension scales were very consistent with extraversion and feeling from environmental type theory. System maintenance and systems change scales were associated with judging and perception on the SETA, respectively. Results from the SETA and personal growth dimension scales were mixed. Based on this analysis, the SETA may serve as a general purpose environmental assessment for use with the Myers-Briggs Type Indicator.

Coq plugin for the Reasonably Exceptional Type Theory

2019 ◽  
Author(s):  
Pierre-Marie P�drot ◽  
Nicolas Tabareau ◽  
Hans Jacob Fehrmann ◽  
�ric Tanter
Keyword(s):  
Type Theory
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