Search results for "Type theory"

Very Large Container Molecules

2005

Wellfounded Trees and Dependent Polynomial Functors

2004

We set out to study the consequences of the assumption of types of wellfounded trees in dependent type theories. We do so by in- vestigating the categorical notion of wellfounded tree introduced in [16]. Our main result shows that wellfounded trees allow us to define initial algebras for a wide class of endofunctors on locally cartesian closed cat- egories.

The identity type weak factorisation system

2008

We show that the classifying category C(T) of a dependent type theory T with axioms for identity types admits a non-trivial weak factorisation system. We provide an explicit characterisation of the elements of both the left class and the right class of the weak factorisation system. This characterisation is applied to relate identity types and the homotopy theory of groupoids.

Collection Principles in Dependent Type Theory

2002

We introduce logic-enriched intuitionistic type theories, that extend intuitionistic dependent type theories with primitive judgements to express logic. By adding type theoretic rules that correspond to the collection axiom schemes of the constructive set theory CZF we obtain a generalisation of the type theoretic interpretation of CZF. Suitable logic-enriched type theories allow also the study of reinterpretations of logic. We end the paper with an application to the double-negation interpretation.

The generalised type-theoretic interpretation of constructive set theory

2006

We present a generalisation of the type-theoretic interpretation of constructive set theory into Martin-Löf type theory. The generalisation involves replacing Martin-Löf type theory with a new type theory in which logic is treated as primitive instead of being formulated via the propositions-as-types representation. The original interpretation treated logic in Martin-Löf type theory via the propositions-as-types interpretation. The generalisation involves replacing Martin-Löf type theory with a new type theory in which logic is treated as primitive. The primitive treatment of logic in type theories allows us to study reinterpretations of logic, such as the double-negation translation.

Selective collection of glass containers in the Valencian Community (Spain)

2000

Spanish legislation has recently adopted the norms defined by a directive of the European Parliament and the Council of Ministers, 20 December 1994, concerning containers and container residues. The purpose of the directive is to harmonize national measures for the processing of these products in member states. Thus, contributing to prevent or reduce their impact upon the environment both in the different member states and in third countries. To this effect, the directive requires the compliance within a given period of a series of recycling and processing objectives applicable to all the packaging and container materials on the market, including glass products. The aims of the present stu…

Inductive types in homotopy type theory

2012

Homotopy type theory is an interpretation of Martin-L\"of's constructive type theory into abstract homotopy theory. There results a link between constructive mathematics and algebraic topology, providing topological semantics for intensional systems of type theory as well as a computational approach to algebraic topology via type theory-based proof assistants such as Coq. The present work investigates inductive types in this setting. Modified rules for inductive types, including types of well-founded trees, or W-types, are presented, and the basic homotopical semantics of such types are determined. Proofs of all results have been formally verified by the Coq proof assistant, and the proof s…

Finiteness in a Minimalist Foundation

2008

We analyze the concepts of finite set and finite subset from the perspective of a minimalist foundational theory which has recently been introduced by Maria Emilia Maietti and the second author. The main feature of that theory and, as a consequence, of our approach is compatibility with other foundational theories such as Zermelo-Fraenkel set theory, Martin-Lof's intuitionistic Type Theory, topos theory, Aczel's CZF, Coquand's Calculus of Constructions. This compatibility forces our arguments to be constructive in a strong sense: no use is made of powerful principles such as the axiom of choice, the power-set axiom, the law of the excluded middle.

On mathematical modelling of the solid-liquid mixtures transport in porous axial-symmetrical container with Henry and Langmuir sorption kinetics

2018

In this paper we study diffusion and convection filtration problem of one substance through the pores of a porous material which may absorb and immobilize some of the diffusing substances. As an example we consider round cylinder with filtration process in the axial direction. The cylinder is filled with sorbent i.e. absorbent material that passed through dirty water or liquid solutions. We can derive the system of two partial differential equations (PDEs), one expressing the rate of change of concentration of water in the pores of the sorbent and the other - the rate of change of concentration in the sorbent or kinetical equation for absorption. The approximation of corresponding initial b…

Selective recognition of small hydrogen bond acceptors by a calix[6]arene-based molecular container

2019

Selective molecular recognition is of primary importance for applications such as sensing and separation of chemicals. This work describes the host-guest and crystallisation properties of a penta-carbamated calix[6]arene designed as a molecular container with a H-donating recognition group directed towards the heart of the cavity. As demonstrated by NMR spectroscopy and X-ray diffraction studies, this macrocyclic receptor can selectively recognise small H-bond acceptors through one or two hydrogen bonds, the guests nesting inside the polyaromatic cavity surrounded by eleven bulky tert-butyl groups.