193 research outputs found
Expressing Ecumenical Systems in the ??-Calculus Modulo Theory
Systems in which classical and intuitionistic logics coexist are called ecumenical. Such a system allows for interoperability and hybridization between classical and constructive propositions and proofs. We study Ecumenical STT, a theory expressed in the logical framework of the ??-calculus modulo theory. We prove soudness and conservativity of four subtheories of Ecumenical STT with respect to constructive and classical predicate logic and simple type theory. We also prove the weak normalization of well-typed terms and thus the consistency of Ecumenical STT
Staged Compilation with Two-Level Type Theory
The aim of staged compilation is to enable metaprogramming in a way such that
we have guarantees about the well-formedness of code output, and we can also
mix together object-level and meta-level code in a concise and convenient
manner. In this work, we observe that two-level type theory (2LTT), a system
originally devised for the purpose of developing synthetic homotopy theory,
also serves as a system for staged compilation with dependent types. 2LTT has
numerous good properties for this use case: it has a concise specification,
well-behaved model theory, and it supports a wide range of language features
both at the object and the meta level. First, we give an overview of 2LTT's
features and applications in staging. Then, we present a staging algorithm and
prove its correctness. Our algorithm is "staging-by-evaluation", analogously to
the technique of normalization-by-evaluation, in that staging is given by the
evaluation of 2LTT syntax in a semantic domain. The staging algorithm together
with its correctness constitutes a proof of strong conservativity of 2LLT over
the object theory. To our knowledge, this is the first description of staged
compilation which supports full dependent types and unrestricted staging for
types
Arithmetical conservation results
In this paper we present a proof of Goodman's Theorem, a classical result in
the metamathematics of constructivism, which states that the addition of the
axiom of choice to Heyting arithmetic in finite types does not increase the
collection of provable arithmetical sentences. Our proof relies on several
ideas from earlier proofs by other authors, but adds some new ones as well. In
particular, we show how a recent paper by Jaap van Oosten can be used to
simplify a key step in the proof. We have also included an interesting
corollary for classical systems pointed out to us by Ulrich Kohlenbach
Definitional Extension in Type Theory
When we extend a type system, the relation between the original system and its extension is an important issue we want to know. Conservative extension is a traditional relation we study with. But in some cases, like coercive subtyping, it is not strong enough to capture all the properties, more powerful relation between the systems is required. We bring the idea definitional extension from mathematical logic into type theory. In this paper, we study the notion of definitional extension for type theories and explicate its use, both informally and formally, in the context of coercive subtyping
Well posedness of Lagrangian flows and continuity equations in metric measure spaces
We establish, in a rather general setting, an analogue of DiPerna-Lions
theory on well-posedness of flows of ODE's associated to Sobolev vector fields.
Key results are a well-posedness result for the continuity equation associated
to suitably defined Sobolev vector fields, via a commutator estimate, and an
abstract superposition principle in (possibly extended) metric measure spaces,
via an embedding into .
When specialized to the setting of Euclidean or infinite dimensional (e.g.
Gaussian) spaces, large parts of previously known results are recovered at
once. Moreover, the class of metric measure spaces object
of extensive recent research fits into our framework. Therefore we provide, for
the first time, well-posedness results for ODE's under low regularity
assumptions on the velocity and in a nonsmooth context.Comment: Slightly expanded some remarks on the technical assumption (7.11);
Journal reference inserte
A Reasonably Gradual Type Theory
Gradualizing the Calculus of Inductive Constructions (CIC) involves dealing
with subtle tensions between normalization, graduality, and conservativity with
respect to CIC. Recently, GCIC has been proposed as a parametrized gradual type
theory that admits three variants, each sacrificing one of these properties.
For devising a gradual proof assistant based on CIC, normalization and
conservativity with respect to CIC are key, but the tension with graduality
needs to be addressed. Additionally, several challenges remain: (1) The
presence of two wildcard terms at any type-the error and unknown terms-enables
trivial proofs of any theorem, jeopardizing the use of a gradual type theory in
a proof assistant; (2) Supporting general indexed inductive families, most
prominently equality, is an open problem; (3) Theoretical accounts of gradual
typing and graduality so far do not support handling type mismatches detected
during reduction; (4) Precision and graduality are external notions not
amenable to reasoning within a gradual type theory. All these issues manifest
primally in CastCIC, the cast calculus used to define GCIC. In this work, we
present an extension of CastCIC called GRIP. GRIP is a reasonably gradual type
theory that addresses the issues above, featuring internal precision and
general exception handling. GRIP features an impure (gradual) sort of types
inhabited by errors and unknown terms, and a pure (non-gradual) sort of strict
propositions for consistent reasoning about gradual terms. Internal precision
supports reasoning about graduality within GRIP itself, for instance to
characterize gradual exception-handling terms, and supports gradual subset
types. We develop the metatheory of GRIP using a model formalized in Coq, and
provide a prototype implementation of GRIP in Agda.Comment: 27pages + 2pages bibliograph
Synthetic Undecidability and Incompleteness of First-Order Axiom Systems in Coq
We mechanise the undecidability of various frst-order axiom systems in Coq, employing
the synthetic approach to computability underlying the growing Coq Library of Undecidability Proofs. Concretely, we cover both semantic and deductive entailment in fragments
of Peano arithmetic (PA) as well as ZF and related fnitary set theories, with their undecidability established by many-one reductions from solvability of Diophantine equations, i.e.
Hilbertâs tenth problem (H10), and the Post correspondence problem (PCP), respectively.
In the synthetic setting based on the computability of all functions defnable in a constructive foundation, such as Coqâs type theory, it sufces to defne these reductions as metalevel functions with no need for further encoding in a formalised model of computation.
The concrete cases of PA and the considered set theories are supplemented by a general
synthetic theory of undecidable axiomatisations, focusing on well-known connections to
consistency and incompleteness. Specifcally, our reductions rely on the existence of standard models, necessitating additional assumptions in the case of full ZF, and all axiomatic
extensions still justifed by such standard models are shown incomplete. As a by-product of
the undecidability of set theories formulated using only membership and no equality symbol, we obtain the undecidability of frst-order logic with a single binary relation
Internal Parametricity for 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, and we give an account of the identity extension lemma for internal parametricity
Proof-irrelevance out of excluded-middle and choice in the calculus of constructions
We present a short and direct syntactic proof of the fact that adding the axiom of choice and the principle of excluded-middle to Coquand-Huet's Calculus of Constructions gives proof-irrelevanc
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