Positive Inductive-Recursive Definitions

Ghani, Neil; Forsberg, Fredrik Nordvall; Malatesta, L.

doi:10.2168/lmcs-11(1:13)2015

Cited by 5 publications

(3 citation statements)

References 18 publications

(20 reference statements)

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“…This is interesting future work, as it will enhance our understanding of induction-recursion from a syntactic perspective. Recent advances in understanding induction-recursion might aid this research [Ghani et al 2015].…”

Section: Resultsmentioning

confidence: 99%

Decidability of conversion for type theory in type theory

Abel

Öhman

Vezzosi

2017

Proc. ACM Program. Lang.

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Type theory should be able to handle its own meta-theory, both to justify its foundational claims and to obtain a verified implementation. At the core of a type checker for intensional type theory lies an algorithm to check equality of types, or in other words, to check whether two types are convertible. We have formalized in Agda a practical conversion checking algorithm for a dependent type theory with one universe à la Russell, natural numbers, and η-equality for Π types. We prove the algorithm correct via a Kripke logical relation parameterized by a suitable notion of equivalence of terms. We then instantiate the parameterized fundamental lemma twice: once to obtain canonicity and injectivity of type formers, and once again to prove the completeness of the algorithm. Our proof relies on inductive-recursive definitions, but not on the uniqueness of identity proofs. Thus, it is valid in variants of intensional Martin-Löf Type Theory as long as they support induction-recursion, for instance, Extensional, Observational, or Homotopy Type Theory. algorithm correctly implements equality of types is based on a Kripke logical relation and follows Abel and Scherer [2012] and Abel et al. [2016].Our novel contribution is a full formalization of the algorithm and its proof of soundness, completeness, and termination, in Martin-Löf Type Theory [Martin-Löf 1975] with intensional equality, plus some well-understood extensions such as induction-recursion [Dybjer 2000]. As a mechanization language, we use Agda itself in its latest version, using the language variant without Streicher's Axiom K [1993]. 1 Consequently, our proof is directly transferable to related foundations such as Homotopy Type Theory extended by induction-recursion.The Agda formalization is quite sizeable: around 10.000 lines of code (500.000 characters). Especially the proof of the fundamental theorem of logical relations is substantial (5.000 lines). The congruence rule for the recursor for natural numbers alone requires a lemma that stretches over more than 500 lines. It is not that the proof is mathematically deep, once the right definition of the logical relation and the right formulation of the fundamental theorem are in placeÐit is just that a formalization requires us to get all the technicalities right. In research articles with pen and paper proofs only, the proof of the fundamental theorem is often skipped or reduced to the single sentence łproof by induction on the typing and equality derivationsž. Yet checking that each case of the induction goes through would require a reviewer many hours of disciplined technical reasoning. Written out, the proof would stretch over many pages.In previous works [Abel et al. 2007;Abel and Scherer 2012;Harper and Pfenning 2005], two logical relations and two fundamental theorems are needed for the meta theory: one that entails soundness of the algorithmic equality, and one for completeness. While in pen-and-paper proofs we could get away with remarks like łproof analogousž, a formalization would require us to do ...

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Section: Resultsmentioning

confidence: 99%

Decidability of conversion for type theory in type theory

Abel

Öhman

Vezzosi

2017

Proc. ACM Program. Lang.

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“…These are sufficient to model transfinite hierarchies, but as we noted in Section 4.1, we do not know how to model strict inclusions with them. Variants of induction-recursion [9,10,11] are particularly flexible and powerful extensions to universes.…”

Section: Effects Of Choice Of Level Structurementioning

confidence: 99%

Generalized Universe Hierarchies and First-Class Universe Levels

Kovács¹

2021

Preprint

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In type theories, universe hierarchies are commonly used to increase the expressive power of the theory while avoiding inconsistencies arising from size issues. There are numerous ways to specify universe hierarchies, and theories may differ in details of cumulativity, choice of universe levels, specification of type formers and eliminators, and available internal operations on levels. In the current work, we aim to provide a framework which covers a large part of the design space. First, we develop syntax and semantics for cumulative universe hierarchies, where levels may come from any set equipped with a transitive well-founded ordering. In the semantics, we show that induction-recursion can be used to model transfinite hierarchies, and also support lifting operations on type codes which strictly preserve type formers. Then, we consider a setup where universe levels are first-class types and subject to arbitrary internal reasoning. This generalizes the bounded polymorphism features of Coq and at the same time the internal level computations in Agda.

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“…To inform the design and justify the reduction rules provided for CastCIC, we build a syntactic model of CastCIC by translation to CIC augmented with induction-recursion [Dybjer and Setzer 2003;Ghani et al 2015;Martin-Löf 1996] ( § 6.1). From a type theory point of view, what makes CastCIC peculiar is first of all the possibility of having errors (both "pessimistic" as err and "optimistic" as ?…”

Section: Realizing a Dependent Cast Calculus: Castcicmentioning

confidence: 99%

Gradualizing the Calculus of Inductive Constructions

Lennon-Bertrand,

Maillard,

Tabareau

et al. 2020

Preprint

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Acknowledging the ordeal of a fully formal development in a proof assistant such as Coq, we investigate gradual variations on the Calculus of Inductive Construction (CIC) for swifter prototyping with imprecise types and terms. We observe, with a no-go theorem, a crucial tradeoff between graduality and the key properties of normalization and closure of universes under dependent product that CIC enjoys. Beyond this Fire Triangle of Graduality, we explore the gradualization of CIC with three different compromises, each relaxing one edge of the Fire Triangle. We develop a parametrized presentation of Gradual CIC that encompasses all three variations, and develop their metatheory. We first present a bidirectional elaboration of Gradual CIC to a dependently-typed cast calculus, which elucidates the interrelation between typing, conversion, and the gradual guarantees. We use a syntactic model into CIC to inform the design of a safe, confluent reduction, and establish, when applicable, normalization. We also study the stronger notion of graduality as embeddingprojection pairs formulated by New and Ahmed, using appropriate semantic model constructions. This work informs and paves the way towards the development of malleable proof assistants and dependently-typed programming languages.

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Positive Inductive-Recursive Definitions

Cited by 5 publications

References 18 publications

Decidability of conversion for type theory in type theory

Decidability of conversion for type theory in type theory

Generalized Universe Hierarchies and First-Class Universe Levels

Gradualizing the Calculus of Inductive Constructions

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