Competing Inheritance Paths in Dependent Type Theory: A Case Study in Functional Analysis

Affeldt, Reynald; Cohen, Cyril; Kerjean, Marie; Mahboubi, Assia; Rouhling, Damien; Sakaguchi, Kazuhiko

doi:10.1007/978-3-030-51054-1_1

Cited by 17 publications

(25 citation statements)

References 17 publications

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“…4] [11,Sect. 3.1] to inherit from ordered types, and factored out the notion of norms and absolute values as normed Abelian groups [1,Sect. 4.2] with the help of our tools [13,36].…”

Section: Discussionmentioning

confidence: 99%

“…4, 5, and 6) are problems specific to the bundled approach. A detailed comparison of type classes and packed classes has also been provided in [1]. There are a few mechanisms to extend the unification engines of proof assistants other than canonical structures that can implement structure inference for packed classes: unification hints [4] and coercions pullback [31].…”

Section: Conclusion and Related Workmentioning

confidence: 99%

“…Mathematical structures are a key ingredient of modern formalized mathematics in proof assistants, e.g., [1,18,25,41] [46,Sect. 4].…”

Section: Introductionmentioning

confidence: 99%

“…When combined with mechanisms for implicit coercions [32,33] and for extending unification procedure, such as the canonical structures [26,33] of the Coq proof assistant [42], and the unification hints [4] of the Lean theorem prover [6,27] and the Matita interactive theorem prover [5], packed classes enable subtyping and automated inference of structures in hierarchies. Compared to approaches based on type classes [22,40], packed classes are more robust, and their inference approach is efficient and predictable [1]. The success of the packed classes methodology in formalized mathematics can be seen in the Mathematical Components library [45] (hereafter MathComp), the Coquelicot library [8], and especially the formal proof of the Odd Order Theorem [20].…”

Section: Introductionmentioning

confidence: 99%

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Validating Mathematical Structures

Sakaguchi

2020

Automated Reasoning

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Sharing of notations and theories across an inheritance hierarchy of mathematical structures, e.g., groups and rings, is important for productivity when formalizing mathematics in proof assistants. The packed classes methodology is a generic design pattern to define and combine mathematical structures in a dependent type theory with records. When combined with mechanisms for implicit coercions and unification hints, packed classes enable automated structure inference and subtyping in hierarchies, e.g., that a ring can be used in place of a group. However, large hierarchies based on packed classes are challenging to implement and maintain. We identify two hierarchy invariants that ensure modularity of reasoning and predictability of inference with packed classes, and propose algorithms to check these invariants. We implement our algorithms as tools for the Coq proof assistant, and show that they significantly improve the development process of Mathematical Components, a library for formalized mathematics.

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“…4] [11,Sect. 3.1] to inherit from ordered types, and factored out the notion of norms and absolute values as normed Abelian groups [1,Sect. 4.2] with the help of our tools [13,36].…”

Section: Discussionmentioning

confidence: 99%

Section: Conclusion and Related Workmentioning

confidence: 99%

“…Mathematical structures are a key ingredient of modern formalized mathematics in proof assistants, e.g., [1,18,25,41] [46,Sect. 4].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Validating Mathematical Structures

Sakaguchi

2020

Automated Reasoning

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“…When combined with mechanisms for implicit coercions [35,36] and for extending unification procedure, such as the canonical structures [36,29] of the Coq proof assistant [45], and the unification hints [4] of the Lean theorem prover [6,30] and the Matita interactive theorem prover [5], packed classes enable subtyping and automated inference of structures in hierarchies. Compared to approaches based on type classes [43,25], packed classes are more robust, and their inference approach is efficient and predictable [1]. The success of the packed classes methodology in formalized mathematics can be seen in the Mathematical Components library [48] (hereafter MathComp), the Coquelicot library [8], and especially the formal proof of the Odd Order Theorem [23].…”

Section: Introductionmentioning

confidence: 99%

Validating Mathematical Structures

Sakaguchi

2020

Preprint

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The Design of Mathematical Language

Avigad

2023

Handbook of the History and Philosophy of Mathematical Practice

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As idealized descriptions of mathematical language, there is a sense in which formal systems specify too little, and there is a sense in which they specify too much. They are silent with respect to a number of features of mathematical language that are essential to the communicative and inferential goals of the subject, while many of these features are independent of a specific choice of foundation. This chapter begins to map out the design features of mathematical language without descending to the level of formal implementation, drawing on examples from the mathematical literature and insights from the design of computational proof assistants. Perspectives Philosophical orientationPhilosophy of mathematics has traditionally been concerned with the nature of mathematical objects, knowledge, and thought. Talking about these in a rigorous way requires some sort of conception of mathematics itself, and some sort of understanding the features of mathematical practice that fall under the scope of the analysis. Toward forming such a conception, what we have the most direct access to is the mathematical literature: the historical record of statements, questions, arguments, definitions, and and other textual artifacts that are constitutive of the subject. These artifacts, rooted in language, form the starting point for philosophical study.However we ultimately try to characterize the goals and methods of mathematics, we have to start with language. Whether we view mathematics as the practice of solving problems, abstracting from experience, or getting at a certain type of truth, what we say about that practice has to fit with what we see in the mathematical literature. We need to understand how mathematical language enables us to carry out those activities and how those activities are manifested in language.One central thesis of this chapter is that, when we study mathematical language, it is important to understand not just what is allowed, but also what is desired. Formal systems specify rules that tell us when a formula is well formed and when an inference is justified, but it doesn't tell us which definitions are good definitions, or which among the myriad inferential steps that can be taken at any given point are most worthy of our attention. Mathematics calls upon its practitioners to carry out complex tasks, and to do so creatively, efficiently, and reliably. Reflection on mathematical practice should help explain how it helps us manage complexity and carry out fruitful exploration.Another central thesis of this chapter is that it is helpful to view mathematical language as the object of design. Mathematical language and method have evolved over the centuries, presumably for good reasons. Some features of mathematical language and method have remained remarkably stable, again, we may presume, for good reasons. Mathematics provides powerful means for abstraction and for managing information, making data salient when it is needed and suppressing it when it is a distraction. Recognizing that it is effective in that ...

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Competing Inheritance Paths in Dependent Type Theory: A Case Study in Functional Analysis

Cited by 17 publications

References 17 publications

Validating Mathematical Structures

Validating Mathematical Structures

Validating Mathematical Structures

The Design of Mathematical Language

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