| import OAI.Combinatorics.InfiniteMatroid.TransfiniteConstruction |
| import OAI.Combinatorics.InfiniteMatroid.RankDescent |
| import OAI.Combinatorics.InfiniteMatroid.PackingCovering |
|
|
| namespace OAI |
|
|
| section |
|
|
| open Set Filter Matroid |
| namespace InfiniteMatroidCounterexample |
|
|
| theorem main : |
| Countable E ∧ Infinite E ∧ |
| ∃ M₀ M₁ : Matroid E, |
| M₀.E = Set.univ ∧ M₁.E = Set.univ ∧ |
| M₀.dual = M₀ ∧ M₁.dual = M₁ ∧ |
| (∀ I₀ I₁ : Set E, M₀.Indep I₀ → M₁.Indep I₁ → I₀ ∪ I₁ ≠ Set.univ) ∧ |
| ¬ HasPackingCovering M₀ M₁ := by |
| obtain ⟨R,hK⟩ := exists_concrete_rank |
| obtain ⟨S,hS⟩ := R.exists_basisSystem hK |
| let Q : Matroid E0 := S.matroid |
| have hE : Q.E = Set.univ := S.ground |
| have hQ : Q.dual = Q := S.self_dual |
| have hanti : ∀ ⦃X Y : Set D⦄, X ∈ R.ultra → Y ∈ R.ultra → X ⊆ Y → |
| R.rank Y ≤ R.rank X := |
| fun _ _ hX hY hXY ↦ R.antitone hX hY (R.ideal.modSubset_of_subset hXY) |
| have hb : ∀ B, Q.IsBase B → (Prod.mk · false) ⁻¹' B ∈ R.ultra → |
| ((Prod.mk · true) ⁻¹' B)ᶜ ∈ R.ultra ∧ |
| R.rank ((Prod.mk · true) ⁻¹' B)ᶜ < R.rank ((Prod.mk · false) ⁻¹' B) := by |
| intro B hB hl |
| have ha : R.Admissible B := hS B ((S.isBase_iff B).mp hB) |
| rcases ha with ha | ha |
| · exact ha.2 |
| · exact (ha.1 hl).elim |
| have hn : ∀ I J : Set E, (rayMatroid Q false).Indep I → |
| (rayMatroid Q true).Indep J → I ∪ J ≠ Set.univ := |
| no_independent_cover Q R.ultra R.rank hanti hb |
| refine ⟨inferInstance, inferInstance, rayMatroid Q false, rayMatroid Q true, |
| rayMatroid_ground Q hE false, rayMatroid_ground Q hE true, |
| rayMatroid_selfDual Q hE hQ false, rayMatroid_selfDual Q hE hQ true, hn, ?_⟩ |
| intro h |
| obtain ⟨I,J,hI,hJ,hIJ⟩ := independent_cover_of_packing_covering |
| (rayMatroid Q false) (rayMatroid Q true) |
| (rayMatroid_ground Q hE false) (rayMatroid_ground Q hE true) |
| (rayMatroid_selfDual Q hE hQ false) (rayMatroid_selfDual Q hE hQ true) h |
| exact hn I J hI hJ hIJ |
|
|
| end InfiniteMatroidCounterexample |
| end |
|
|
| end OAI |
|
|
