Sharing Proofs with Predicative Theories Through Universe Polymorphic Elaboration

Core Concepts

The author presents a transformation for sharing proofs between systems by translating them from impredicative to predicative logics, focusing on universe polymorphism. This approach allows for the translation of many proofs that do not use impredicativity in an essential way.

Abstract

The content discusses the challenges of sharing formal proofs between proof assistants with different logics. It introduces a transformation method using universe polymorphism to translate proofs from impredicative to predicative systems. The proposed translation aims to make proofs available across various proof assistants, emphasizing the importance of handling impredicativity effectively. The implementation of this transformation is detailed, showcasing its partial algorithm and constraint-postponement strategy. The tool Predicativize is highlighted as a semi-automatic solution for translating mathematical developments efficiently. Additionally, the content explores the interoperability of proof assistants and logical frameworks like Dedukti, emphasizing the significance of defining logics uniformly within a logical framework. It also delves into the challenges posed by impredicativity in proof interoperability and discusses how universe polymorphism can address these challenges effectively.

Stats

During the elaboration, we need to solve unification problems in the equational theory of universe levels.
The proposed translation allows one to translate many proofs that do not use impredicativity in an essential way.
The tool Predicativize was used to translate Matita's arithmetic library to Agda.
Universo replaces each sort by a fresh metavariable representing a natural number.
Universe polymorphism is represented directly with the use of the framework's function type.

Quotes

"Proof assistants allow interactively writing correct proofs and verifying safety-critical software."
"One challenge is sharing proofs between systems with incompatible logics."
"The proposed translation tackles the problem by elaborating terms into well-typed ones."
"Universe polymorphism enables using terms at multiple universes during translation."
"The tool Universo helps in translating libraries efficiently."

Key Insights Distilled From

Sharing proofs with predicative theories through universe polymorphic elaboration

by Thia... at arxiv.org 03-06-2024

https://arxiv.org/pdf/2308.15465.pdf

Deeper Inquiries

How does universe polymorphism enhance interoperability among different proof assistants?

Universe polymorphism enhances interoperability among different proof assistants by allowing for the translation of proofs across systems with varying levels of universes. By abstracting over universe levels, a single proof can be written in a way that is general enough to be applicable at multiple universe levels. This flexibility enables proofs to be shared and reused more easily between proof assistants that may have different underlying theories or structures.

What are potential limitations or drawbacks of using universe polymorphism in proof transformations?

One potential limitation of using universe polymorphism in proof transformations is the complexity it introduces. Managing and handling multiple universes can make the transformation process more intricate and challenging, especially when dealing with constraints related to unification problems at different levels. Additionally, ensuring soundness and correctness becomes more crucial as the system becomes more intricate due to the presence of multiple universes.

How can addressing impredicativity effectively impact the field of formal proof sharing?

Addressing impredicativity effectively impacts the field of formal proof sharing by enabling proofs from impredicative systems to be translated into predicative ones. By developing techniques like predicativization algorithms, proofs that rely on impredicative principles can still be shared across various platforms without losing their integrity or validity. This approach broadens the scope of formal proof sharing by making it possible to bridge gaps between systems with differing logical foundations, ultimately fostering greater collaboration and knowledge exchange within the research community.