
Consistency of the Predicative Calculus of Cumulative Inductive Constructions (pCuIC)
In order to avoid wellknow paradoxes associated with selfreferential d...
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Functional Pearl: The Distributive λCalculus
We introduce a simple extension of the λcalculus with pairs—called the ...
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Complete Bidirectional Typing for the Calculus of Inductive Constructions
This article presents a bidirectional type system for the Calculus of In...
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Touring the MetaCoq Project (Invited Paper)
Proof assistants are getting more widespread use in research and industr...
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Dependent Pearl: Normalization by realizability
For those of us who generally live in the world of syntax, semantic proo...
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Normalization by Evaluation for CallbyPushValue and Polarized LambdaCalculus
We observe that normalization by evaluation for simplytyped lambdacalc...
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On principal types and wellfoundedness of terms in ECC
When we investigate a type system, it is helpful if we can establish the...
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Gradualizing the Calculus of Inductive Constructions
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 nogo 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 dependentlytyped 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 dependentlytyped programming languages.
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