The metatheory of UTT

Goguen, Healfdene

doi:10.1007/3-540-60579-7_4

Cited by 5 publications

(7 citation statements)

References 12 publications

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“…This is a decreasing diagram The careful reader could object that our rewriting relations work modulo = α on terms, rather than on = α -equivalence classes of terms, and therefore we should have explicit = α -steps, with a minimal label as required in the corresponding extension of van Oostrom's theorem to rewriting modulo [10]. Renaming of bound variables being so specific an equivalence, we allow ourselves this little glitch, as is customary.…”

Section: Decreasing Diagramsmentioning

confidence: 99%

“…In dependent type theories, confluence is needed to prove type preservation and termination, making all three properties interdependent if termination is used in the confluence proof. This circularity can be possibly broken in two ways: by proving all properties together within a single induction [9]; or by proving confluence on untyped terms first, and then successively type preservation, confluence on typed terms, and strong normalization. We develop the latter way here, focusing on untyped confluence.…”

Section: Introductionmentioning

confidence: 99%

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Confluence in Non-Left-Linear Untyped Higher-Order Rewrite Theories

Férey

Jouannaud

2021

23rd International Symposium on Principles and Practice of Declarative Programming

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We develop techniques based on van Oostrom's decreasing diagrams that reduce confluence proofs to the checking of critical pairs for higher-order rewrite rules extending beta-reduction on pure lambda-terms. We show that confluence is preserved for a large subset of terms that contains all pure lambda terms. Our results are applied to famous Klop's examples of non-confluent behaviours in presence of convergent rewrite rules and to fragments of various encodings, in a dependent type theory with rewrite rules, of the Calculus of Constructions with polymorphic universes.

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Section: Decreasing Diagramsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Confluence in Non-Left-Linear Untyped Higher-Order Rewrite Theories

Férey

Jouannaud

2021

23rd International Symposium on Principles and Practice of Declarative Programming

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“…Type theories thus specified in LF are intensional type theories such as Martin-Löf's intensional type theory [15] and the Unifying Theory of dependent Types (UTT) [10]. Intensional type theories have nice meta-theoretic properties including Church-Rosser, Subject Reduction and Strong Normalisation (see Goguen's thesis on the meta-theory of UTT [4,5]).…”

Section: The Logical Framework Lfmentioning

confidence: 99%

“…A dependent record type is a type of labelled tuples. For instance, ⟨n : N at, v : V ect(n)⟩ is the dependent record type with objects (called records) such as ⟨n = 2, v = [5,6]⟩, where dependency has to be respected: [5,6] must be of type V ect (2).…”

Section: A Formulation Of Dependent Record Typesmentioning

confidence: 99%

Dependent record types revisited

Zhang

2009

Proceedings of the 1st Workshop on Modules and Libraries for Proof Assistants

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Dependently-typed records have been studied in type theory in several previous research attempts, with applications to the study of module mechanisms for both programming and proof languages. Recently, the author has proposed an improved formulation of dependent record types in the context of studying manifest fields of module types. In this paper, we study this formulation in more details by considering universes of record types and some application examples. In particular, we show that record types provide a more powerful mechanism (than record kinds) in expressing module types and additional useful means (as compared with Σ-types) in applications.

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“…a partial formal proof in COQ [2]. There is a paper proof of DTC for Luo's Extended Calculus of Constructions with weak elimination [3], an extension for the case of strong elimination restricted to a single predicative universe [4], and a paper proof of consistency which includes strong elimination together with arbitrarily many predicative universes [5]. There are paper proofs of DTC for various versions of CIC modulo a theory T [6]- [8], and a formal proof of the last version.…”

Section: Introductionmentioning

confidence: 99%

CoQMTU: A Higher-Order Type Theory with a Predicative Hierarchy of Universes Parametrized by a Decidable First-Order Theory

Barras

Jouannaud²,

Strub

et al. 2011

2011 IEEE 26th Annual Symposium on Logic in Computer Science

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We 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.

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The metatheory of UTT

Cited by 5 publications

References 12 publications

Confluence in Non-Left-Linear Untyped Higher-Order Rewrite Theories

Confluence in Non-Left-Linear Untyped Higher-Order Rewrite Theories

Dependent record types revisited

CoQMTU: A Higher-Order Type Theory with a Predicative Hierarchy of Universes Parametrized by a Decidable First-Order Theory

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