| import OAI.Combinatorics.InfiniteMatroid.Basic |
|
|
| namespace OAI |
|
|
| namespace InfiniteMatroidCounterexample |
| open Set Matroid |
|
|
| def IsUniform {α : Type*} (M : Matroid α) : Prop := |
| ∀ X ⊆ M.E, M.Indep X ∨ M.Spanning X |
|
|
| def IsPartitional {α : Type} (M : Matroid α) : Prop := |
| ∃ (ι : Type) (β : ι → Type) (N : ∀ i, Matroid (β i)) (e : (Σ i, β i) ≃ α), |
| (∀ i, IsUniform (N i)) ∧ M = (Matroid.sigma N).mapEquiv e |
|
|
| def HasIntersection {α : Type*} (M N : Matroid α) : Prop := |
| ∃ J J₀ J₁ : Set α, M.Indep J ∧ N.Indep J ∧ |
| Disjoint J₀ J₁ ∧ J₀ ∪ J₁ = J ∧ M.closure J₀ ∪ N.closure J₁ = M.E |
|
|
| def FamilyCovering {α ι : Type*} (F : Set α) (M : ι → Matroid α) : Prop := |
| ∃ I : ι → Set α, (∀ i, (M i).Indep (I i)) ∧ (⋃ i, I i) = F |
|
|
| def FamilyPacking {α ι : Type*} (M : ι → Matroid α) : Prop := |
| ∃ S : ι → Set α, (Pairwise fun i j ↦ Disjoint (S i) (S j)) ∧ |
| ∀ i, (M i).Spanning (S i) |
|
|
| def CoveringConjectureFor {α ι : Type*} (F : Set α) (M : ι → Matroid α) : Prop := |
| FamilyCovering F M ↔ ∀ Y ⊆ F, |
| FamilyPacking (fun i ↦ M i ↾ Y) → FamilyCovering Y (fun i ↦ M i ↾ Y) |
|
|
| def PackingConjectureFor {α ι : Type*} (F : Set α) (M : ι → Matroid α) : Prop := |
| FamilyPacking M ↔ ∀ Y ⊆ F, |
| FamilyCovering Y (fun i ↦ contractOnto (M i) Y) → |
| FamilyPacking (fun i ↦ contractOnto (M i) Y) |
|
|
| end InfiniteMatroidCounterexample |
|
|
| end OAI |
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|
