This repo is aiming to formalize the bread and butter of matroid theory in lean4. The goal is eventually to have is all included in the lean4 math library : https://github.com/leanprover-community/mathlib4
At the time of writing, the contents of this repo are being submitted to mathlib piece by piece. Currently already in mathlib are :
- The definition of a matroid in terms of the base axioms : Mathlib/Data/Matroid/Basic
- Construction of matroids in terms of independence axioms : Mathlib/Data/Matroid/IndepAxioms
- Matroid duality : Mathlib/Data/Matroid/Dual
- Restricting a matroid to a set, and the restriction order : Mathlib/Data/Matroid/Restrict
- Matroids with at most one base : Mathlib/Data/Matroid/Constructions
- Maps and comaps between matroids : Matroid/Map
- Matroid closure : Mathlib/Data/Matroid/Closure
- Matroid circuits : Mathlib/Data/Matroid/Circuit
- Cardinal-valued matroid rank : Mathlib/Data/Matroid/Rank/Cardinal
- Finite-rank sets : Mathlib/Data/Matroid/Rank/Finite
- Direct sums : Mathlib/Data/Matroid/Sum
Topics in this repo are listed below.
- Isomorphism : Matroid/Equiv
- Circuits and cocircuits : Matroid/IsCircuit
- Loops : Matroid/Loop
- The rank function : Matroid/Rank/*
- Minors : Matroid/Minor/Basic and Matroid/Minor/Iso
- Relative Rank : Matroid/Minor/Rank
- Flats, Hyperplanes, Covers : Matroid/Flat/*
- Truncations : Matroid/Constructions/Truncate
- Parallel elements : Matroid/Parallel
- Parallel and series extension : Matroid/Constructions/ParallelExtension
- Simplicity and simplification : Matroid/Simple
- Modular sets, matroids, pairs and families : Matroid/Modular/*
- Modular cuts, extensions and projections : Matroid/Extension
- Circuit-hyperplane relaxation : Matroid/Constructions/Relax
- Paving matroids : Matroid/Paving
- The matroid intersection theorem : Matroid/Intersection
- Uniform matroids (finite and infinite) : Matroid/Uniform
- Connectivity, local connectivity, skewness and connectivity between sets : Matroid/Connectivity/*
- Cryptomorphism for flats, circuits, closure and rank : Matroid/Axioms/*
- Quotients and the weak order : Matroid/Order/*
- Representability : Matroid/Representation/*
- Circuit/cocircuit intersections : Matroid/Binary/Crossing
- Tutte's excluded minor theorem for finitary binary matroids : Matroid/Binary/Representation
If you have any feature requests or thoughts, please contact me at apnelson@uwaterloo.ca - I am more than happy to see others make use of this code!