Coercive subtyping in lambda-free logical frameworks

Linguistics and Philosophy, 2012

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Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs

In this article, we show how one can formalise in type theory mathematical objects, for which dependent types are usually deemed unavoidable, using only simple types. We outline a method to encode most of the terms of Lean's dependent type theory into the simple type theory of Isabelle/HOL. Taking advantage of Isabelle's automation, we illustrate our method with the formalisation in Isabelle/HOL of a mathematical notion developed in the 1980s: strict omega-categories.

Logical Methods in Computer Science, 2008

We present a type theory with some proof-irrelevance built into the conversion rule. We argue that this feature is useful when type theory is used as the logical formalism underlying a theorem prover. We also show a close relation with the subset types of the theory of PVS. We show that in these theories, because of the additional extentionality, the axiom of choice implies the decidability of equality, that is, almost classical logic. Finally we describe a simple set-theoretic semantics.

Theory and Applications of Models of Computation, 2017

In a dependent type theory satisfying the propositions as types correspondence together with the proofs-as-programs paradigm, the validity of the unique choice rule or even more of the choice rule says that the extraction of a computable witness from an existential statement under hypothesis can be performed within the same theory. Here we show that the unique choice rule, and hence the choice rule, are not valid both in Coquand's Calculus of Constructions with indexed sum types, list types and binary disjoint sums and in its predicative version implemented in the intensional level of the Minimalist Foundation. This means that in these theories the extraction of computational witnesses from existential statements must be performed in a more expressive proofs-as-programs theory.

Information and Computation, 2013

Coercive 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.

Oslo Studies in Language

This paper presents an analysis of coercion and related phenomena in the framework of Dependent Type Semantics (DTS). Using underspecified terms in DTS, we present an analysis of selectional restriction as presupposition; we then combine it with a type called ‘transfer frame’ to provide an analysis of coercion. Our analysis focuses on the fact that coercion is triggered not only by type mismatch between predicates and their arguments, but also by more general inference with contextual information. We show how the analysis can be extended to copredication of logical polysemy and complement coercion. Finally, we will suggest that this analysis can shed light on an aspect of complicity that is invoked in interpreting coercion and other meaning-shifting phenomena.

2008

Otter-lambda is a theorem-prover based on an untyped logic with lambda calculus, called lambda logic. Otter-lambda is built on Otter, so it uses resolution proof search, supplemented by demodulation and paramodulation for equality reasoning, but it also uses a new algorithm, lambda unification, to instantiate variables for functions or predicates. The idea of "implicit typing" is to "type" the function and predicate symbols by specifying the legal types of their arguments and return values. The hope is that if the axioms can be typed in this way then the consequences should be typeable too. This is true (with one restriction) in first-order logic. We show that by placing suitable restrictions on lambda unification, one can extend this theorem to lambda logic. All the interesting proofs obtained so far with Otter-lambda, except those explicitly involving untypeable axioms, are covered by this theorem. "Explicit typing" refers to the use of simple type-checking in addition to implicit typing.

Electronic Proceedings in Theoretical Computer Science, 2015

We present two extensions of the LF Constructive Type Theory featuring monadic locks. A lock is a monadic type construct that captures the effect of an external call to an oracle. Such calls are the basic tool for gluing together diverse Type Theories and proof development environments. The oracle can be invoked either to check that a constraint holds or to provide a suitable witness. The systems are presented in the canonical style developed by the CMU School. The first system, CLLF P , is the canonical version of the system LLF P , presented earlier by the authors. The second system, CLLF P? , features the possibility of invoking the oracle to obtain a witness satisfying a given constraint. We discuss encodings of Fitch-Prawitz Set theory, call-by-value λ-calculi, and systems of Light Linear Logic. Finally, we show how to use Fitch-Prawitz Set Theory to define a type system that types precisely the strongly normalizing terms.

Lecture Notes in Computer Science, 2007

A type-theoretic framework for formal reasoning with different logical foundations is introduced and studied. With logic-enriched type theories formulated in a logical framework, it allows various logical systems such as classical logic as well as intuitionistic logic to be used effectively alongside inductive data types and type universes. This provides an adequate basis for wider applications of type theory based theorem proving technology. Two notions of set are introduced in the framework and used in two case studies of classical reasoning: a predicative one in the formalisation of Weyl's predicative mathematics and an impredicative one in the verification of security protocols.

Lecture Notes in Computer Science, 2005

We extend classical first-order logic with subtyping by type predicates and type coercion. Type predicates assert that the value of a term belongs to a more special type than the signature guarantees, while type coercion allows using terms of a more general type where the signature calls for a more special one. These operations are important e.g. in the specification and verification of object-oriented programs. We present a tableau calculus for this logic and prove its completeness.