Formalising inductive and coinductive containers

2024-09-04

arXiv.org

AI Summary

This paper addresses the challenge of formalizing container theory in Cubical Agda without assuming Uniqueness of Identity Proofs (UIP). It generalizes classical container functors from the category of sets to the category of types—i.e., the “wild” category—and systematically develops their inductive and coinductive semantics. Methodologically, it replaces extensional equality with path types native to Cubical Agda, enabling a unified treatment of bisimulation and coinductive equivalence, and constructively proves—within intensional type theory—that container functors preserve least and greatest fixed points. The main contributions are: (1) eliminating dependence on UIP by establishing path types as central to coinductive reasoning; (2) extending container theory to homotopy type theory; and (3) providing a more general and robust formal semantic foundation for generic programming and higher-order abstraction.

Technology Category

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

Application Category

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

Abstract

Containers capture the concept of strictly positive data types in programming. The original development of containers is done in the internal language of locally cartesian closed categories (LCCCs) with disjoint coproducts and W-types, and uniqueness of identity proofs (UIP) is implicitly assumed throughout. Although it is claimed that these developments can also be interpreted in extensional Martin-L""of type theory, this interpretation is not made explicit. In this paper, we present a formalisation of the results that 'containers preserve least and greatest fixed points' in Cubical Agda, thereby giving a formulation in intensional type theory. Our proofs do not make use of UIP and thereby generalise the original results from talking about container functors on Set to container functors on the wild category of types. Our main incentive for using Cubical Agda is that its path type restores the equivalence between bisimulation and coinductive equality. Thus, besides developing container theory in a more general setting, we also demonstrate the usefulness of Cubical Agda's path type to coinductive proofs.

Problem

• Formalizing inductive and coinductive containers in Cubical Agda
• Generalizing container theory without uniqueness of identity proofs
• Demonstrating Cubical Agda's path type for coinductive proofs

Innovation

• Formalizing containers in Cubical Agda
• Avoiding UIP for generalized results
• Using path type for coinductive equality