32 research outputs found
Definitional Functoriality for Dependent (Sub)Types
Dependently-typed proof assistant rely crucially on definitional equality,
which relates types and terms that are automatically identified in the
underlying type theory. This paper extends type theory with definitional
functor laws, equations satisfied propositionally by a large class of
container-like type constructors , equipped with a , such as lists or trees. Promoting these equations to
definitional ones strengthen the theory, enabling slicker proofs and more
automation for functorial type constructors. This extension is used to
modularly justify a structural form of coercive subtyping, propagating
subtyping through type formers in a map-like fashion. We show that the
resulting notion of coercive subtyping, thanks to the extra definitional
equations, is equivalent to a natural and implicit form of subsumptive
subtyping. The key result of decidability of type-checking in a dependent type
system with functor laws for lists has been entirely mechanized in Coq
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
A Fibrational Account of Local States
International audienceOne main challenge of the theory of computational effects is to understand how to combine various notions of effects in a meaningful way. Here, we study the particular case of the local state monad, which we would like to express as the result of combining together a family of global state monads parametrized by the number of available registers. To that purpose, we develop a notion of indexed monad inspired by the early work by Street, which refines and generalizes Power's recent notion of indexed Lawvere theory. One main achievement of the paper is to integrate the block structure necessary to encode allocation as part of the resulting notion of indexed state monad. We then explain how to recover the local state monad from the functorial data provided by our notion of indexed state monad. This reconstruction is based on the guiding idea that an algebra of the indexed state monad should be defined as a section of a 2-categorical notion of fibration associated to the indexed state monad by a Grothendieck construction
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
A Reasonably Gradual Type Theory
International audienceGradualizing 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 alternative to CastCIC called GRIP. GRIP is a reasonably gradual type theory that addresses the issues above, featuring internal precision and general exception handling. For consistent reasoning about gradual terms, GRIP features an impure sort of types inhabited by errors and unknown terms, and a pure sort of strict propositions. By adopting a novel interpretation of the unknown term that carefully accounts for universe levels, GRIP satisfies graduality for a large and well-defined class of terms, in addition to being normalizing and a conservative extension of CIC. 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
The Multiverse: Logical Modularity for Proof Assistants
Proof assistants play a dual role as programming languages and logical systems. As programming languages, proof assistants offer standard modularity mechanisms such as first-class functions, type polymorphism and modules. As logical systems, however, modularity is lacking, and understandably so: incompatible reasoning principles-such as univalence and uniqueness of identity proofs-can indirectly lead to logical inconsistency when used in a given development, even when they appear to be confined to different modules. The lack of logical modularity in proof assistants also hinders the adoption of richer programming constructs, such as effects. We propose the multiverse, a general type-theoretic approach to endow proof assistants with logical modularity. The multiverse consists of multiple universe hierarchies that statically describe the reasoning principles and effects available to define a term at a given type. We identify sufficient conditions for this structuring to modularly ensure that incompatible principles do not interfere, and to locally restrict the power of dependent elimination when necessary. This extensible approach generalizes the ad-hoc treatment of the sort of propositions in the Coq proof assistant. We illustrate the power of the multiverse by describing the inclusion of Coq-style propositions, the strict propositions of Gilbert et al., the exceptional type theory of Pédrot and Tabareau, and general axiomatic extensions of the logic
The Multiverse: Logical Modularity for Proof Assistants
Proof assistants play a dual role as programming languages and logical systems. As programming languages, proof assistants offer standard modularity mechanisms such as first-class functions, type polymorphism and modules. As logical systems, however, modularity is lacking, and understandably so: incompatible reasoning principles-such as univalence and uniqueness of identity proofs-can indirectly lead to logical inconsistency when used in a given development, even when they appear to be confined to different modules. The lack of logical modularity in proof assistants also hinders the adoption of richer programming constructs, such as effects. We propose the multiverse, a general type-theoretic approach to endow proof assistants with logical modularity. The multiverse consists of multiple universe hierarchies that statically describe the reasoning principles and effects available to define a term at a given type. We identify sufficient conditions for this structuring to modularly ensure that incompatible principles do not interfere, and to locally restrict the power of dependent elimination when necessary. This extensible approach generalizes the ad-hoc treatment of the sort of propositions in the Coq proof assistant. We illustrate the power of the multiverse by describing the inclusion of Coq-style propositions, the strict propositions of Gilbert et al., the exceptional type theory of Pédrot and Tabareau, and general axiomatic extensions of the logic
Dijkstra monads for all
This paper proposes a general semantic framework for verifying programs with arbitrary monadic side-effects using Dijkstra monads, which we define as monad-like structures indexed by a specification monad. We prove that any monad morphism between a computational monad and a specification monad gives rise to a Dijkstra monad, which provides great flexibility for obtaining Dijkstra monads tailored to the verification task at hand. We moreover show that a large variety of specification monads can be obtained by applying monad transformers to various base specification monads, including predicate transformers and Hoare-style pre- and postconditions. For defining correct monad transformers, we propose a language inspired by Moggi's monadic metalanguage that is parameterized by a dependent type theory. We also develop a notion of algebraic operations for Dijkstra monads, and start to investigate two ways of also accommodating effect handlers. We implement our framework in both Coq and F*, and illustrate that it supports a wide variety of verification styles for effects such as exceptions, nondeterminism, state, input-output, and general recursion