161 research outputs found
On Irrelevance and Algorithmic Equality in Predicative Type Theory
Dependently typed programs contain an excessive amount of static terms which
are necessary to please the type checker but irrelevant for computation. To
separate static and dynamic code, several static analyses and type systems have
been put forward. We consider Pfenning's type theory with irrelevant
quantification which is compatible with a type-based notion of equality that
respects eta-laws. We extend Pfenning's theory to universes and large
eliminations and develop its meta-theory. Subject reduction, normalization and
consistency are obtained by a Kripke model over the typed equality judgement.
Finally, a type-directed equality algorithm is described whose completeness is
proven by a second Kripke model.Comment: 36 pages, superseds the FoSSaCS 2011 paper of the first author,
titled "Irrelevance in Type Theory with a Heterogeneous Equality Judgement
On Equivalence and Canonical Forms in the LF Type Theory
Decidability of definitional equality and conversion of terms into canonical
form play a central role in the meta-theory of a type-theoretic logical
framework. Most studies of definitional equality are based on a confluent,
strongly-normalizing notion of reduction. Coquand has considered a different
approach, directly proving the correctness of a practical equivalance algorithm
based on the shape of terms. Neither approach appears to scale well to richer
languages with unit types or subtyping, and neither directly addresses the
problem of conversion to canonical.
In this paper we present a new, type-directed equivalence algorithm for the
LF type theory that overcomes the weaknesses of previous approaches. The
algorithm is practical, scales to richer languages, and yields a new notion of
canonical form sufficient for adequate encodings of logical systems. The
algorithm is proved complete by a Kripke-style logical relations argument
similar to that suggested by Coquand. Crucially, both the algorithm itself and
the logical relations rely only on the shapes of types, ignoring dependencies
on terms.Comment: 41 page
Martin-L\"of \`a la Coq
We present an extensive mechanization of the meta-theory of Martin-L\"of Type
Theory (MLTT) in the Coq proof assistant. Our development builds on
pre-existing work in Agda to show not only the decidability of conversion, but
also the decidability of type checking, using an approach guided by
bidirectional type checking. From our proof of decidability, we obtain a
certified and executable type checker for a full-fledged version of MLTT with
support for , , , and identity types, and one
universe. Furthermore, our development does not rely on impredicativity,
induction-recursion or any axiom beyond MLTT with a schema for indexed
inductive types and a handful of predicative universes, narrowing the gap
between the object theory and the meta-theory to a mere difference in
universes. Finally, we explain our formalization choices, geared towards a
modular development relying on Coq's features, e.g. meta-programming facilities
provided by tactics and universe polymorphism
Encoding of Predicate Subtyping with Proof Irrelevance in the ??-Calculus Modulo Theory
The ??-calculus modulo theory is a logical framework in which various logics and type systems can be encoded, thus helping the cross-verification and interoperability of proof systems based on those logics and type systems. In this paper, we show how to encode predicate subtyping and proof irrelevance, two important features of the PVS proof assistant. We prove that this encoding is correct and that encoded proofs can be mechanically checked by Dedukti, a type checker for the ??-calculus modulo theory using rewriting
Coinductive Formal Reasoning in Exact Real Arithmetic
In this article we present a method for formally proving the correctness of
the lazy algorithms for computing homographic and quadratic transformations --
of which field operations are special cases-- on a representation of real
numbers by coinductive streams. The algorithms work on coinductive stream of
M\"{o}bius maps and form the basis of the Edalat--Potts exact real arithmetic.
We use the machinery of the Coq proof assistant for the coinductive types to
present the formalisation. The formalised algorithms are only partially
productive, i.e., they do not output provably infinite streams for all possible
inputs. We show how to deal with this partiality in the presence of syntactic
restrictions posed by the constructive type theory of Coq. Furthermore we show
that the type theoretic techniques that we develop are compatible with the
semantics of the algorithms as continuous maps on real numbers. The resulting
Coq formalisation is available for public download.Comment: 40 page
The Confluent Terminating Context-Free Substitutive Rewriting System for the lambda-Calculus with Surjective Pairing and Terminal Type
For the lambda-calculus with surjective pairing and terminal type, Curien and Di Cosmo, inspired by Knuth-Bendix completion, introduced a confluent rewriting system of the naive rewriting system. Their system is a confluent (CR) rewriting system stable under contexts. They left the strong normalization (SN) of their rewriting system open. By Girard\u27s reducibility method with restricting reducibility theorem, we prove SN of their rewriting, and SN of the extensions by polymorphism and (terminal types caused by parametric polymorphism). We extend their system by sum types and eta-like reductions, and prove the SN. We compare their system to type-directed expansions
A Two-Level Linear Dependent Type Theory
We present a type theory combining both linearity and dependency by
stratifying typing rules into a level for logics and a level for programs. The
distinction between logics and programs decouples their semantics, allowing the
type system to assume tight resource bounds. A natural notion of irrelevancy is
established where all proofs and types occurring inside programs are fully
erasable without compromising their operational behavior. Through a heap-based
operational semantics, we show that extracted programs always make
computational progress and run memory clean. Additionally, programs can be
freely reflected into the logical level for conducting deep proofs in the style
of standard dependent type theories. This enables one to write resource safe
programs and verify their correctness using a unified language
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