80,360 research outputs found
Elaboration in Dependent Type Theory
To be usable in practice, interactive theorem provers need to provide
convenient and efficient means of writing expressions, definitions, and proofs.
This involves inferring information that is often left implicit in an ordinary
mathematical text, and resolving ambiguities in mathematical expressions. We
refer to the process of passing from a quasi-formal and partially-specified
expression to a completely precise formal one as elaboration. We describe an
elaboration algorithm for dependent type theory that has been implemented in
the Lean theorem prover. Lean's elaborator supports higher-order unification,
type class inference, ad hoc overloading, insertion of coercions, the use of
tactics, and the computational reduction of terms. The interactions between
these components are subtle and complex, and the elaboration algorithm has been
carefully designed to balance efficiency and usability. We describe the central
design goals, and the means by which they are achieved
A Bi-Directional Refinement Algorithm for the Calculus of (Co)Inductive Constructions
The paper describes the refinement algorithm for the Calculus of
(Co)Inductive Constructions (CIC) implemented in the interactive theorem prover
Matita. The refinement algorithm is in charge of giving a meaning to the terms,
types and proof terms directly written by the user or generated by using
tactics, decision procedures or general automation. The terms are written in an
"external syntax" meant to be user friendly that allows omission of
information, untyped binders and a certain liberal use of user defined
sub-typing. The refiner modifies the terms to obtain related well typed terms
in the internal syntax understood by the kernel of the ITP. In particular, it
acts as a type inference algorithm when all the binders are untyped. The
proposed algorithm is bi-directional: given a term in external syntax and a
type expected for the term, it propagates as much typing information as
possible towards the leaves of the term. Traditional mono-directional
algorithms, instead, proceed in a bottom-up way by inferring the type of a
sub-term and comparing (unifying) it with the type expected by its context only
at the end. We propose some novel bi-directional rules for CIC that are
particularly effective. Among the benefits of bi-directionality we have better
error message reporting and better inference of dependent types. Moreover,
thanks to bi-directionality, the coercion system for sub-typing is more
effective and type inference generates simpler unification problems that are
more likely to be solved by the inherently incomplete higher order unification
algorithms implemented. Finally we introduce in the external syntax the notion
of vector of placeholders that enables to omit at once an arbitrary number of
arguments. Vectors of placeholders allow a trivial implementation of implicit
arguments and greatly simplify the implementation of primitive and simple
tactics
Consistency and Completeness of Rewriting in the Calculus of Constructions
Adding rewriting to a proof assistant based on the Curry-Howard isomorphism,
such as Coq, may greatly improve usability of the tool. Unfortunately adding an
arbitrary set of rewrite rules may render the underlying formal system
undecidable and inconsistent. While ways to ensure termination and confluence,
and hence decidability of type-checking, have already been studied to some
extent, logical consistency has got little attention so far. In this paper we
show that consistency is a consequence of canonicity, which in turn follows
from the assumption that all functions defined by rewrite rules are complete.
We provide a sound and terminating, but necessarily incomplete algorithm to
verify this property. The algorithm accepts all definitions that follow
dependent pattern matching schemes presented by Coquand and studied by McBride
in his PhD thesis. It also accepts many definitions by rewriting, containing
rules which depart from standard pattern matching.Comment: 20 page
Homotopy Type Theory in Lean
We discuss the homotopy type theory library in the Lean proof assistant. The
library is especially geared toward synthetic homotopy theory. Of particular
interest is the use of just a few primitive notions of higher inductive types,
namely quotients and truncations, and the use of cubical methods.Comment: 17 pages, accepted for ITP 201
Duality Anomaly Cancellation, Minimal String Unification and the Effective Low-Energy Lagrangian of 4-D Strings
We present a systematic study of the constraints coming from target-space
duality and the associated duality anomaly cancellations on orbifold-like 4-D
strings. A prominent role is played by the modular weights of the massless
fields. We present a general classification of all possible modular weights of
massless fields in Abelian orbifolds. We show that the cancellation of modular
anomalies strongly constrains the massless fermion content of the theory, in
close analogy with the standard ABJ anomalies. We emphasize the validity of
this approach not only for (2,2) orbifolds but for (0,2) models with and
without Wilson lines. As an application one can show that one cannot build a
or orbifold whose massless charged sector with respect
to the (level one) gauge group is that of the
minimal supersymmetric standard model, since any such model would necessarily
have duality anomalies. A general study of those constraints for Abelian
orbifolds is presented. Duality anomalies are also related to the computation
of string threshold corrections to gauge coupling constants. We present an
analysis of the possible relevance of those threshold corrections to the
computation of and for all Abelian orbifolds. Some
particular {\it minimal} scenarios, namely those based on all
orbifolds except Comment: 69 page
Axion Couplings and Effective Cut-Offs in Superstring Compactifications
We use the linear supermultiplet formalism of supergravity to study axion
couplings and chiral anomalies in the context of field-theoretical Lagrangians
describing orbifold compactifications beyond the classical approximation. By
matching amplitudes computed in the effective low energy theory with the
results of string loop calculations we determine the appropriate counterterm in
this effective theory that assures modular invariance to all loop order. We use
supersymmetry consistency constraints to identify the correct ultra-violet
cut-offs for the effective low energy theory. Our results have a simple
interpretation in terms of two-loop unification of gauge coupling constants at
the string scale.Comment: 25 page
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