Derivates for Containers in Univalent Foundations

2025-12-19

AI Summary

Prior work on container derivatives was restricted to set-like (0-truncated) types, limiting applicability in homotopy type theory. Method: We generalize the derivative operation for containers to arbitrary (non-truncated) types—both shapes and positions—within the Univalent Foundations framework; construct a derivative structure satisfying universal properties in the cartesian morphism category; and fully formalize all results in Cubical Agda. Contributions: We establish fundamental derivative laws for constant, sum, and product containers; prove the chain rule holds but global invertibility fails, revealing a deep connection between non-invertibility and non-classical logic; derive an explicit rule for the derivative of least fixed points; and precisely characterize necessary and sufficient conditions for its invertibility. This work provides a rigorous homotopy-consistent foundation and formal toolkit for modeling one-hole contexts and traversing recursive data structures.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Data transparency and provenance
• Semantics and Knowledge: Provenance, trust, security and privacy, and ethical issues in managing semantic data
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Containers conveniently represent a wide class of inductive data types. Their derivatives compute representations of types of one-hole contexts, useful for implementing tree-traversal algorithms. In the category of containers and cartesian morphisms, derivatives of discrete containers (whose positions have decidable equality) satisfy a universal property. Working in Univalent Foundations, we extend the derivative operation to untruncated containers (whose shapes and positions are arbitrary types). We prove that this derivative, defined in terms of a set of isolated positions, satisfies an appropriate universal property in the wild category of untruncated containers and cartesian morphisms, as well as basic laws with respect to constants, sums and products. A chain rule exists, but is in general non-invertible. In fact, a globally invertible chain rule is inconsistent in the presence of non-set types, and equivalent to a classical principle when restricted to set-truncated containers. We derive a rule for derivatives of smallest fixed points from the chain rule, and characterize its invertibility. All of our results are formalized in Cubical Agda.

Problem

• Extends derivative operation to untruncated containers in Univalent Foundations
• Proves universal property and basic laws for derivatives of arbitrary types
• Analyzes chain rule invertibility and derives fixed-point derivative rules

Innovation

• Extended derivative operation to untruncated containers
• Proved universal property in wild category of containers
• Formalized results in Cubical Agda with chain rule analysis