[Paper Review] Discrete Jordan Curve Theorem: A proof formalized in Coq with hypermaps

Jean‐François Dufourd|ArXiv.org|Feb 20, 2008

Logic, programming, and type systems10 references3 citations

TL;DR

This paper presents a formalized proof of a discrete Jordan Curve Theorem using hypermaps and the Coq proof assistant. It introduces a novel hypermap model with open orbits to establish constructive planarity and connectivity criteria, proving that breaking a ring of faces in a planar hypermap increases the number of connected components by exactly one.

ABSTRACT

This paper presents a formalized proof of a discrete form of the Jordan Curve Theorem. It is based on a hypermap model of planar subdivisions, formal specifications and proofs assisted by the Coq system. Fundamental properties are proven by structural or noetherian induction: Genus Theorem, Euler's Formula, constructive planarity criteria. A notion of ring of faces is inductively defined and a Jordan Curve Theorem is stated and proven for any planar hypermap.

Motivation & Objective

• To formalize a discrete version of the Jordan Curve Theorem within a purely combinatorial framework using hypermaps.
• To develop a Coq-based formal specification of planar subdivisions that avoids geometric embeddings and relies on algebraic structures.
• To establish constructive criteria for planarity and connectivity using structural and noetherian induction on hypermap components.
• To define and reason about rings of faces in hypermaps as a foundation for proving the discrete JCT.
• To provide a complete, machine-checked proof of the discrete JCT, demonstrating that breaking a ring increases component count by one.

Proposed method

• Modeling planar subdivisions using hypermaps as a triple (D, α₀, α₁), where D is a finite set of darts and α₀, α₁ are permutations.
• Defining orbits via function compositions: α₀-orbit (edge), α₁-orbit (vertex), φ-orbit (face), and α₀,α₁-orbit (component).
• Using structural and noetherian induction to prove foundational results: the Genus Theorem, Euler’s Formula, and planarity criteria.
• Inductively defining rings of faces using four preconditions (pre_ring0 to pre_ring3) to ensure topological consistency.
• Proving the discrete JCT via induction on the ring structure, with key lemmas showing that breaking a ring link increases component count by one.
• Leveraging Coq’s interactive proof environment to verify all lemmas and theorems, with ~5,000 lines of Coq code and 400 formalized lemmas.

Experimental results

Research questions

• RQ1 Can a discrete Jordan Curve Theorem be formally proven using a purely combinatorial hypermap model without geometric embeddings?
• RQ2 How can constructive planarity and connectivity criteria be derived using structural and noetherian induction on hypermap components?
• RQ3 What inductive definition of a ring of faces ensures topological correctness and enables a formal proof of the discrete JCT?
• RQ4 How does breaking a ring link in a planar hypermap affect the number of connected components, and can this be formally verified?
• RQ5 To what extent can a Coq-based formal specification framework support the development of topological invariants in computational geometry?

Key findings

• The discrete Jordan Curve Theorem is formally proven in Coq: breaking a ring of faces in a planar hypermap increases the number of connected components by exactly one.
• The proof relies on a novel hypermap model with open orbits, enabling precise and constructive reasoning about connectivity and planarity.
• The framework includes 25 new definitions and 400 lemmas, with all dimension-0 results formally proven and dimension-1 results admitted due to symmetry.
• The hypermap model supports constructive planarity criteria, including the equivalence of the hypermap Jordan property to global planarity.
• The formalization is complete and machine-checked, forming a solid foundation for further work in 2D and 3D computational geometry and geometric modeling.
• The approach demonstrates the effectiveness of Coq in formalizing complex topological reasoning in discrete geometry, with practical implications for image processing and geometric modeling.