| import OAI.Combinatorics.InfiniteMatroid.CorollaryModel |
| import OAI.Combinatorics.InfiniteMatroid.Matroid |
| import OAI.Combinatorics.InfiniteMatroid.DoubleRay |
|
|
| namespace OAI |
|
|
| namespace InfiniteMatroidCounterexample |
| open Set Matroid |
|
|
| lemma BasisSystem.isUniform {α : Type*} (S : BasisSystem α) : IsUniform S.matroid := by |
| intro X hX |
| obtain ⟨I, hI⟩ := S.matroid.exists_isBasis X |
| by_cases hb : S.matroid.IsBase I |
| · exact Or.inr (hb.spanning_of_superset hI.subset hX) |
| · left |
| have hXI : X ⊆ I := fun e he ↦ hI.mem_of_insert_indep he |
| (S.insert_indep_of_not_base hI.indep hb e) |
| exact (Subset.antisymm hXI hI.subset) ▸ hI.indep |
|
|
| lemma mapEquiv_trans {α β γ : Type*} (M : Matroid α) (e : α ≃ β) (f : β ≃ γ) : |
| (M.mapEquiv e).mapEquiv f = M.mapEquiv (e.trans f) := by |
| apply Matroid.ext_indep |
| · simp only [Matroid.mapEquiv_ground_eq, image_image, Equiv.trans_apply] |
| rfl |
| · intro I _ |
| simp only [Matroid.mapEquiv_indep_iff, image_image] |
| rfl |
|
|
| lemma rayMatroid_isPartitional {α : Type} (Q : Matroid (α × Bool)) |
| (hQ : IsUniform Q) (p : Bool) : IsPartitional (rayMatroid Q p) := by |
| refine ⟨Vertex p, fun _ ↦ α × Bool, fun _ ↦ Q, |
| (Equiv.sigmaEquivProd (Vertex p) (α × Bool)).trans (componentEquiv α p), |
| fun _ ↦ hQ, ?_⟩ |
| exact mapEquiv_trans _ _ _ |
|
|
| end InfiniteMatroidCounterexample |
|
|
| end OAI |
|
|
