234 research outputs found
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
Type classes for efficient exact real arithmetic in Coq
Floating point operations are fast, but require continuous effort on the part
of the user in order to ensure that the results are correct. This burden can be
shifted away from the user by providing a library of exact analysis in which
the computer handles the error estimates. Previously, we [Krebbers/Spitters
2011] provided a fast implementation of the exact real numbers in the Coq proof
assistant. Our implementation improved on an earlier implementation by O'Connor
by using type classes to describe an abstract specification of the underlying
dense set from which the real numbers are built. In particular, we used dyadic
rationals built from Coq's machine integers to obtain a 100 times speed up of
the basic operations already. This article is a substantially expanded version
of [Krebbers/Spitters 2011] in which the implementation is extended in the
various ways. First, we implement and verify the sine and cosine function.
Secondly, we create an additional implementation of the dense set based on
Coq's fast rational numbers. Thirdly, we extend the hierarchy to capture order
on undecidable structures, while it was limited to decidable structures before.
This hierarchy, based on type classes, allows us to share theory on the
naturals, integers, rationals, dyadics, and reals in a convenient way. Finally,
we obtain another dramatic speed-up by avoiding evaluation of termination
proofs at runtime.Comment: arXiv admin note: text overlap with arXiv:1105.275
Coercive subtyping: Theory and implementation
International audienceCoercive subtyping is a useful and powerful framework of subtyping for type theories. The key idea of coercive subtyping is subtyping as abbreviation. In this paper, we give a new and adequate formulation of T[C], the system that extends a type theory T with coercive subtyping based on a set C of basic subtyping judgements, and show that coercive subtyping is a conservative extension and, in a more general sense, a definitional extension. We introduce an intermediate system, the star-calculus T[C]^@?, in which the positions that require coercion insertions are marked, and show that T[C]^@? is a conservative extension of T and that T[C]^@? is equivalent to T[C]. This makes clear what we mean by coercive subtyping being a conservative extension, on the one hand, and amends a technical problem that has led to a gap in the earlier conservativity proof, on the other. We also compare coercive subtyping with the 'ordinary' notion of subtyping - subsumptive subtyping, and show that the former is adequate for type theories with canonical objects while the latter is not. An improved implementation of coercive subtyping is done in the proof assistant Plastic
Practical Datatype Specializations with Phantom Types and Recursion Schemes
Datatype specialization is a form of subtyping that captures program
invariants on data structures that are expressed using the convenient and
intuitive datatype notation. Of particular interest are structural invariants
such as well-formedness. We investigate the use of phantom types for describing
datatype specializations. We show that it is possible to express
statically-checked specializations within the type system of Standard ML. We
also show that this can be done in a way that does not lose useful programming
facilities such as pattern matching in case expressions.Comment: 25 pages. Appeared in the Proc. of the 2005 ACM SIGPLAN Workshop on
M
Closed Type Families with Overlapping Equations
Open, type-level functions are a recent innovation in Haskell that move Haskell towards the expressiveness of dependent types, while retaining the look and feel of a practical programming language. This paper shows how to increase expressiveness still further, by adding closed type functions whose equations may overlap, and may have non-linear patterns over an open type universe. Although practically useful and simple to implement, these features go be- yond conventional dependent type theory in some respects, and have a subtle metatheory
Closed Type Families with Overlapping Equations
Open, type-level functions are a recent innovation in Haskell that move Haskell towards the expressiveness of dependent types, while retaining the look and feel of a practical programming language. This paper shows how to increase expressiveness still further, by adding closed type functions whose equations may overlap, and may have non-linear patterns over an open type universe. Although practically useful and simple to implement, these features go be- yond conventional dependent type theory in some respects, and have a subtle metatheory
Denotational Semantics for Subtyping Between Recursive Types
Inheritance in the form of subtyping is considered in the framework of a polymorphic type discipline with records, variants, and recursive types. We give a denotational semantics based on the paradigm that interprets subtyping as explicit coercion. The main technical result gives a coherent interpretation for a strong rule for deriving inheritances between recursive types
An Overabundance of Equality: Implementing Kind Equalities into Haskell
Haskell, as embodied by version 7.10.1 of the Glasgow Haskell Compiler (GHC), supports reasoning about equality among types, via generalized algebraic datatypes (GADTs) and type families. However, these features are not available among the kinds that clas- sify the types. Motivated by a concrete example of how kind equali- ties can help programmers today, this paper presents the challenges and solutions encountered in integrating kind equalities into GHC, an industrial-strength compiler. The challenges addressed here all surround the many notions of type equality that exist in GHC to- day, and in particular around GHC’s role mechanism. These many different relations on types complicate the theory considerably
An update of GHC supporting reasoning about kind equalities is a key part of this work
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