| import OAI.Combinatorics.InfiniteMatroid.CoveringReduction |
| import OAI.Combinatorics.InfiniteMatroid.PairDuality |
| import OAI.Combinatorics.InfiniteMatroid.Main |
|
|
| namespace OAI |
|
|
| namespace InfiniteMatroidCounterexample |
| open Set Matroid |
|
|
| theorem separate_covering_packing_counterexamples : |
| (∃ (F : Set E) (M : Bool → Matroid E), |
| (∀ i, (M i).E = F) ∧ ¬ CoveringConjectureFor F M) ∧ |
| (∃ (F : Set E) (M : Bool → Matroid E), |
| (∀ i, (M i).E = F) ∧ ¬ PackingConjectureFor F M) := by |
| obtain ⟨_, _, M₀, M₁, hE₀, hE₁, _, _, _, hno⟩ := main |
| let M : Bool → Matroid E := fun i ↦ if i then M₁ else M₀ |
| have hg : ∀ i, (M i).E = univ := by |
| intro i |
| cases i <;> simp [M, hE₀, hE₁] |
| have hn : ¬ HasPackingCovering (M false) (M true) := hno |
| obtain ⟨C, N, hN, hnc, hcond⟩ := separate_covering_of_no_packingCovering hg hn |
| refine ⟨⟨C, N, hN, fun h ↦ hnc (h.mpr hcond)⟩, |
| ⟨C, fun i ↦ (N i).dual, ?_, ?_⟩⟩ |
| · simpa only [Matroid.dual_ground] using hN |
| · exact separate_packing_of_covering_counterexample hN hnc hcond |
|
|
| end InfiniteMatroidCounterexample |
|
|
| end OAI |
|
|
