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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 | import Mathlib
namespace OAI
open Set Matroid
namespace InfiniteMatroidCounterexample
abbrev D := (m : ℕ) × ((Fin m → Bool) → Bool)
abbrev E0 := D × Bool
abbrev E := ℤ × D
def contractOnto {α : Type*} (M : Matroid α) (C : Set α) : Matroid α :=
(M.dual ↾ C).dual
def HasPackingCovering {α : Type*} (M₀ M₁ : Matroid α) : Prop :=
∃ P C S₀ S₁ I₀ I₁ : Set α,
Disjoint P C ∧ P ∪ C = M₀.E ∧ M₁.E = M₀.E ∧
S₀ ⊆ P ∧ S₁ ⊆ P ∧ Disjoint S₀ S₁ ∧
(M₀ ↾ P).Spanning S₀ ∧ (M₁ ↾ P).Spanning S₁ ∧
I₀ ⊆ C ∧ I₁ ⊆ C ∧
(contractOnto M₀ C).Indep I₀ ∧
(contractOnto M₁ C).Indep I₁ ∧ I₀ ∪ I₁ = C
end InfiniteMatroidCounterexample
end OAI
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