The mechanization of science illustrated by the Lean formalization of the multi-graded Proj construction

2025-09-18

AI Summary

The multigraded Proj construction lacks a complete formalization in interactive theorem provers, hindering mechanized verification in algebraic geometry. Method: Building on Lean 4 and mathlib, and grounded in dependent type theory, we present the first full formalization of the multigraded Proj construction—including its definition, categorical structure, functoriality from graded rings to graded schemes, and properties of projective morphisms. Contribution: We develop the first reusable, machine-checked library for multigraded Proj, formally verifying key results such as graded localization, open immersions, and the universal property of relative Proj. This foundation enables rigorous formalization of more advanced constructions in algebraic geometry—e.g., sheaves of graded modules and relative projective space—and advances the integration of generative science into formal mathematics.

Technology Category

• Machine Learning: Multimodal Learning
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We formalize the multi-graded Proj construction in Lean4, illustrating mechanized mathematics and formalization.

Problem

• Formalizing multi-graded Proj construction in Lean4
• Demonstrating mechanized mathematics through formalization
• Advancing automated theorem proving capabilities

Innovation

• Lean4 formalizes multi-graded Proj
• Mechanized mathematics implementation
• Automated theorem proving framework