37 research outputs found
On the strength of proof-irrelevant type theories
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.Comment: 20 pages, Logical Methods in Computer Science, Long version of IJCAR
2006 pape
Proof-irrelevant model of CC with predicative induction and judgmental equality
We present a set-theoretic, proof-irrelevant model for Calculus of
Constructions (CC) with predicative induction and judgmental equality in
Zermelo-Fraenkel set theory with an axiom for countably many inaccessible
cardinals. We use Aczel's trace encoding which is universally defined for any
function type, regardless of being impredicative. Direct and concrete
interpretations of simultaneous induction and mutually recursive functions are
also provided by extending Dybjer's interpretations on the basis of Aczel's
rule sets. Our model can be regarded as a higher-order generalization of the
truth-table methods. We provide a relatively simple consistency proof of type
theory, which can be used as the basis for a theorem prover
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
CoqMTU: a higher-order type theory with a predicative hierarchy of universes parametrized by a decidable first-order theory
International audienceWe study a complex type theory, a Calculus of Inductive Constructions with a predicative hierarchy of universes and a first-order theory T built in its conversion relation. The theory T is specified abstractly, by a set of constructors, a set of defined symbols, axioms expressing that constructors are free and defined symbols completely defined, and a generic elimination principle relying on crucial properties of first-order structures satisfying the axioms. We first show that CoqMTU enjoys all basic meta-theoretical properties of such calculi, confluence, subject reduction and strong normalization when restricted to weak-elimination, implying the decidability of type-checking in this case as well as consistency. The case of strong elimination is left open
The Definitional Side of the Forcing
International audienceThis paper studies forcing translations of proofs in dependent type theory, through the Curry-Howard correspondence. Based on a call-by-push-value decomposition, we synthesize two simply-typed translations: i) one call-by-value, corresponding to the translation derived from the presheaf construction as studied in a previous paper ; ii) one call-by-name, whose intuitions already appear in Kriv-ine and Miquel's work. Focusing on the call-by-name translation, we adapt it to the dependent case and prove that it is compatible with the definitional equality of our system, thus avoiding coherence problems. This allows us to use any category as forcing conditions , which is out of reach with the call-by-value translation. Our construction also exploits the notion of storage operators in order to interpret dependent elimination for inductive types. This is a novel example of a dependent theory with side-effects, clarifying how dependent elimination for inductive types must be restricted in a non-pure setting. Being implemented as a Coq plugin, this work gives the possibility to formalize easily consistency results, for instance the consistency of the negation of Voevodsky's univalence axiom
A Proof of Strong Normalization for the Theory of Constructions Using a Kripke-Like Interpretation
We give a proof that all terms that type-check in the theory of contructions are strongly normalizing (under Ăź-reduction). The main novelty of this proof is that it uses a Kripke-like interpretation of the types and kinds, and that it does not use infinite contexts. We explore some consequences of strong normalization, consistency and decidability of typechecking. We also show that our proof yields another proof of strong normalization for LF (under Ăź-reduction), using the reducibility method
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Modal dependent type theory and dependent right adjoints
In recent years we have seen several new models of dependent type theory
extended with some form of modal necessity operator, including nominal type
theory, guarded and clocked type theory, and spatial and cohesive type theory.
In this paper we study modal dependent type theory: dependent type theory with
an operator satisfying (a dependent version of) the K-axiom of modal logic. We
investigate both semantics and syntax. For the semantics, we introduce
categories with families with a dependent right adjoint (CwDRA) and show that
the examples above can be presented as such. Indeed, we show that any finite
limit category with an adjunction of endofunctors gives rise to a CwDRA via the
local universe construction. For the syntax, we introduce a dependently typed
extension of Fitch-style modal lambda-calculus, show that it can be interpreted
in any CwDRA, and build a term model. We extend the syntax and semantics with
universes
Modal dependent type theory and dependent right adjoints
In recent years we have seen several new models of dependent type theory
extended with some form of modal necessity operator, including nominal type
theory, guarded and clocked type theory, and spatial and cohesive type theory.
In this paper we study modal dependent type theory: dependent type theory with
an operator satisfying (a dependent version of) the K-axiom of modal logic. We
investigate both semantics and syntax. For the semantics, we introduce
categories with families with a dependent right adjoint (CwDRA) and show that
the examples above can be presented as such. Indeed, we show that any finite
limit category with an adjunction of endofunctors gives rise to a CwDRA via the
local universe construction. For the syntax, we introduce a dependently typed
extension of Fitch-style modal lambda-calculus, show that it can be interpreted
in any CwDRA, and build a term model. We extend the syntax and semantics with
universes