Definitional Functoriality for Dependent (Sub)Types - Extended Version

핵심 개념

This paper extends Martin-Löf Type Theory (MLTT) with definitional functor laws for type constructors like lists, trees, and dependent products, enabling slicker proofs and more automation for functorial type constructors. It also uses this functorial structure to modularly justify a structural form of coercive subtyping.

초록

The paper extends MLTT with definitional functor laws for type constructors like lists, trees, and dependent products. This allows type constructors to be treated as functors, with a map operation that satisfies the functor laws definitionally.
The key points are:

Type constructors like lists, trees, and dependent products are equipped with a map operation that satisfies the functor laws definitionally. This avoids the need to explicitly reason about the functor laws, enabling more automation and slicker proofs.

The definitional functor laws are used to modularly justify a structural form of coercive subtyping, where subtyping is propagated through type constructors. This is shown to be equivalent to a more implicit form of subsumptive subtyping.

A type theory called MLTTmap is developed that extends MLTT with the definitional functor laws. The metatheory of MLTTmap, including normalization, canonicity, and decidable type-checking, is mechanized in Coq for a representative fragment.

Two other type theories, MLTTcoe and MLTTsub, are introduced to study the relationship between explicit coercive subtyping and implicit subsumptive subtyping. Translations between these systems are established, leveraging the definitional functor laws.

The key technical contributions are the design of MLTTmap, the metatheoretic analysis, and the connections drawn between the different subtyping approaches.

통계

None.

인용구

None.

핵심 통찰 요약

Definitional Functoriality for Dependent (Sub)Types -- Extended version

by Théo... 게시일 arxiv.org 04-10-2024

https://arxiv.org/pdf/2310.14929.pdf

더 깊은 질문

What are some potential applications or use cases for the definitional functor laws introduced in this work, beyond the subtyping example

The definitional functor laws introduced in this work have various potential applications and use cases beyond the subtyping example. One key application is in the development of more efficient and automated proof assistants. By enforcing functor laws definitionally, the type theory becomes stronger, enabling slicker proofs and more automation for functorial type constructors. This can lead to faster and more reliable verification of complex mathematical and computational properties.
Another application is in the development of modular and reusable code. By promoting equations to definitional ones, the theory becomes more robust and coherent, allowing for easier composition and manipulation of data structures. This can be particularly useful in software development, where maintaining consistency and correctness of code is crucial.
Additionally, the definitional functor laws can enhance the expressivity and flexibility of dependent type systems. By explicitly capturing the functorial structure of type formers, it becomes easier to reason about and manipulate complex data structures and relationships between types. This can lead to more elegant and concise formalizations of mathematical concepts and properties.

How could the techniques developed in this paper be extended to handle more general inductive types, beyond the specific ones considered (lists, trees, etc.)

The techniques developed in this paper can be extended to handle more general inductive types beyond the specific ones considered, such as lists and trees. One approach to extending these techniques is to generalize the functor laws to apply to a wider range of inductive types with different structures and constructors. This would involve defining appropriate map operations for each new type former and ensuring that the functor laws hold definitionally for these types.
Another extension could involve incorporating more advanced features of dependent type theory, such as dependent pattern matching and indexed inductive types. By extending the logical relations and algorithmic conversion techniques to handle these advanced features, the framework developed in this work could be applied to a broader class of inductive types and data structures.
Furthermore, exploring the interaction between definitional functor laws and other aspects of type theory, such as coinductive types and higher inductive types, could lead to a more comprehensive and versatile framework for reasoning about complex data structures and properties in dependent type theory.

Are there any limitations or drawbacks to the definitional functor laws approach compared to alternative ways of handling functoriality in dependent type theory

While the definitional functor laws approach offers several advantages in terms of strengthening the equational theory and enabling more automation and modularity, there are some limitations and drawbacks compared to alternative ways of handling functoriality in dependent type theory.
One limitation is the potential complexity and overhead introduced by enforcing definitional functor laws for a wide range of type formers. This approach may require significant formalization and verification effort, especially when dealing with more complex inductive types and data structures.
Another drawback is the potential impact on the usability and readability of the type theory. Enforcing definitional functor laws may introduce additional complexity and constraints, making it harder for users to understand and reason about the code. This could lead to a steeper learning curve for developers and users of the type theory.
Additionally, the definitional functor laws approach may not be suitable for all scenarios or applications. In some cases, a more flexible and lightweight approach to handling functoriality, such as using propositional equalities or explicit proof obligations, may be more practical and efficient.
Overall, while definitional functor laws offer significant benefits in terms of strengthening the equational theory and enabling more automation, it is important to consider the trade-offs and limitations of this approach in the context of specific use cases and requirements.