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c270adf | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 | 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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