The RedPRL Proof Assistant (Invited Paper)

Angiuli, Carlo; Cavallo, Evan; Hou, Kuen-Bang; Harper, Robert; Sterling, Jonathan

doi:10.4204/eptcs.274.1

Cited by 5 publications

(4 citation statements)

References 10 publications

(12 reference statements)

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“…Using this idea, Angiuli et al (2017bAngiuli et al ( , 2018b, Angiuli (2019) define a computational Cartesian cubical type theory with a univalent universe hierarchy as well as an extensional equality judgment internalized as an equality pre-type. This type theory was implemented by Angiuli et al (2018a) in the RedPRL proof assistant. 5 In this paper, we define a formal Cartesian cubical type theory that abstracts from the computational semantics, along with a machine-verified Cartesian cubical sets model.…”

Section: The De Morgan (Cchm) Modelmentioning

confidence: 99%

Syntax and models of Cartesian cubical type theory

Angiuli

Brunerie

Coquand

et al. 2021

Math. Struct. Comp. Sci.

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We present a cubical type theory based on the Cartesian cube category (faces, degeneracies, symmetries, diagonals, but no connections or reversal) with univalent universes, each containing Π, Σ, path, identity, natural number, boolean, suspension, and glue (equivalence extension) types. The type theory includes a syntactic description of a uniform Kan operation, along with judgmental equality rules defining the Kan operation on each type. The Kan operation uses both a different set of generating trivial cofibrations and a different set of generating cofibrations than the Cohen, Coquand, Huber, and Mörtberg (CCHM) model. Next, we describe a constructive model of this type theory in Cartesian cubical sets. We give a mechanized proof, using Agda as the internal language of cubical sets in the style introduced by Orton and Pitts, that glue, Π, Σ, path, identity, boolean, natural number, suspension types, and the universe itself are Kan in this model, and that the universe is univalent. An advantage of this formal approach is that our construction can also be interpreted in a range of other models, including cubical sets on the connections cube category and the De Morgan cube category, as used in the CCHM model, and bicubical sets, as used in directed type theory.

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Section: The De Morgan (Cchm) Modelmentioning

confidence: 99%

Syntax and models of Cartesian cubical type theory

Angiuli

Brunerie

Coquand

et al. 2021

Math. Struct. Comp. Sci.

Self Cite

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“…In other words, A ∧ B follows from S if each of A and B follow from S. Each of A and B follow from S if they can be refined using other rules. Refinement logics such as Nuprl (Constable et al, 1986) and Red-PRL (Angiuli et al, 2018) as well as other proof assistants following in the LCF tradition (Section 4.2) encourage this style of reasoning. Sterling and Harper (2017) contains an overview of proof refinement.…”

Section: S Pr a ∧ B By S Pr A S Pr Bmentioning

confidence: 99%

“…Refinement logics such as Nuprl (Constable et al, 1986) and Red-PRL (Angiuli et al, 2018) as well as other proof assistants following in the LCF tradition (Section 4.2) encourage this style of reasoning. Sterling and Harper (2017) contains an overview of proof refinement.…”

Section: Proof Design Principlesmentioning

confidence: 99%

QED at Large: A Survey of Engineering of Formally Verified Software

Ringer

Palmskog

Sergey

et al. 2019

FNT in Programming Languages

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Development of formal proofs of correctness of programs can increase actual and perceived reliability and facilitate better understanding of program specifications and their underlying assumptions. Tools supporting such development have been available for over 40 years, but have only recently seen wide practical use. Projects based on construction of machine-checked formal proofs are now reaching an unprecedented scale, comparable to large software projects, which leads to new challenges in proof development and maintenance. Despite its increasing importance, the field of proof engineering is seldom considered in its own right; related theories, techniques, and tools span many fields and venues. This survey of the literature presents a holistic understanding of proof engineering for program correctness, covering impact in practice, foundations, proof automation, proof organization, and practical proof development.

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“…This, or some variation of it, is essentially the underlying type theoretic setup in the various cubical systems that have been implemented in recent years. These include cubical (2013), cubicaltt (2015), yacctt (2018), RedPRL (2016) (Angiuli et al, 2018a), redtt (2018), cooltt (2020), mlang (2019), and Cubical Agda (Vezzosi et al, 2019). All of these systems build on different cubical type theories and have different standard cubical models (Angiuli et al 2021a(Angiuli et al , 2018bBezem et al 2014Bezem et al , 2019Cavallo and Harper 2019;Cohen et al 2018;Coquand et al 2018); however, the ideas underlying them are very similar, and one of the goals of this paper is to give sufficient background to understand and work with the various systems.…”

Section: Introductionmentioning

confidence: 99%

Cubical methods in homotopy type theory and univalent foundations

Mörtberg

2021

Math. Struct. Comp. Sci.

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Cubical methods have played an important role in the development of Homotopy Type Theory and Univalent Foundations (HoTT/UF) in recent years. The original motivation behind these developments was to give constructive meaning to Voevodsky’s univalence axiom, but they have since then led to a range of new results. Among the achievements of these methods is the design of new type theories and proof assistants with native support for notions from HoTT/UF, syntactic and semantic consistency results for HoTT/UF, as well as a variety of independence results and establishing that the univalence axiom does not increase the proof theoretic strength of type theory. This paper is based on lecture notes that were written for the 2019 Homotopy Type Theory Summer School at Carnegie Mellon University. The goal of these lectures was to give an introduction to cubical methods and provide sufficient background in order to make the current research in this very active area of HoTT/UF more accessible to newcomers. The focus of these notes is hence on both the syntactic and semantic aspects of these methods, in particular on cubical type theory and the various cubical set categories that give meaning to these theories.

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The RedPRL Proof Assistant (Invited Paper)

Cited by 5 publications

References 10 publications

Syntax and models of Cartesian cubical type theory

Syntax and models of Cartesian cubical type theory

QED at Large: A Survey of Engineering of Formally Verified Software

Cubical methods in homotopy type theory and univalent foundations

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