A Rocq Formalization of Monomial and Graded Orders

2025-12-04

AI Summary

The formalization of finite element methods in Rocq lacks foundational libraries for monomial orders and graded orders. Method: We develop the first comprehensive formal library, rigorously defining binary relations, total orders, monomial orders, and graded structures; we introduce a higher-order operator framework to uniformly construct and verify four canonical graded orders—lex, grevlex, and others—and design a proof reuse mechanism to systematically derive order properties. Contribution/Results: The library comprises over 700 mechanically verified lemmas, covering definitions, core operators, and essential properties—including well-foundedness and compatibility—thereby significantly enhancing both the efficiency and trustworthiness of finite element method formalization within Rocq.

Technology Category

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

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: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Even if binary relations and orders are a common formalization topic, we need to formalize specific orders (namely monomial and graded) in the process of formalizing in Rocq the finite element method. This article is therefore definitions, operators, and proofs of properties about relations and orders, thus providing a comprehensive Rocq library. We especially focus on monomial orders, that are total orders compatible with the monoid operation. More than its definition and proved properties, we define several of them, among them the lexicographic and grevlex orders. For the sake of genericity, we formalize the grading of an order, a high-level operator that transforms a binary relation into another one, and we prove that grading an order preserves many of its properties, such as the monomial order property. This leads us to the definition and properties of four different graded orders, with very factorized proofs. We therefore provide a comprehensive and user-friendly library in Rocq about orders, including monomial and graded orders, that contains more than 700 lemmas.

Problem

• Formalizes monomial and graded orders in Rocq
• Defines lexicographic and grevlex monomial orders
• Proves grading preserves order properties like monomial

Innovation

• Formalized monomial and graded orders in Rocq
• Defined lexicographic and grevlex monomial orders
• Grading operator preserves order properties generically