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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 39 40 41 42 43 44 45 46 47 48 49 50 51 52 | import OAI.Combinatorics.InfiniteMatroid.TransfiniteConstruction
import OAI.Combinatorics.InfiniteMatroid.RankDescent
import OAI.Combinatorics.InfiniteMatroid.PackingCovering
namespace OAI
section
open Set Filter Matroid
namespace InfiniteMatroidCounterexample
theorem main :
Countable E ∧ Infinite E ∧
∃ M₀ M₁ : Matroid E,
M₀.E = Set.univ ∧ M₁.E = Set.univ ∧
M₀.dual = M₀ ∧ M₁.dual = M₁ ∧
(∀ I₀ I₁ : Set E, M₀.Indep I₀ → M₁.Indep I₁ → I₀ ∪ I₁ ≠ Set.univ) ∧
¬ HasPackingCovering M₀ M₁ := by
obtain ⟨R,hK⟩ := exists_concrete_rank
obtain ⟨S,hS⟩ := R.exists_basisSystem hK
let Q : Matroid E0 := S.matroid
have hE : Q.E = Set.univ := S.ground
have hQ : Q.dual = Q := S.self_dual
have hanti : ∀ ⦃X Y : Set D⦄, X ∈ R.ultra → Y ∈ R.ultra → X ⊆ Y →
R.rank Y ≤ R.rank X :=
fun _ _ hX hY hXY ↦ R.antitone hX hY (R.ideal.modSubset_of_subset hXY)
have hb : ∀ B, Q.IsBase B → (Prod.mk · false) ⁻¹' B ∈ R.ultra →
((Prod.mk · true) ⁻¹' B)ᶜ ∈ R.ultra ∧
R.rank ((Prod.mk · true) ⁻¹' B)ᶜ < R.rank ((Prod.mk · false) ⁻¹' B) := by
intro B hB hl
have ha : R.Admissible B := hS B ((S.isBase_iff B).mp hB)
rcases ha with ha | ha
· exact ha.2
· exact (ha.1 hl).elim
have hn : ∀ I J : Set E, (rayMatroid Q false).Indep I →
(rayMatroid Q true).Indep J → I ∪ J ≠ Set.univ :=
no_independent_cover Q R.ultra R.rank hanti hb
refine ⟨inferInstance, inferInstance, rayMatroid Q false, rayMatroid Q true,
rayMatroid_ground Q hE false, rayMatroid_ground Q hE true,
rayMatroid_selfDual Q hE hQ false, rayMatroid_selfDual Q hE hQ true, hn, ?_⟩
intro h
obtain ⟨I,J,hI,hJ,hIJ⟩ := independent_cover_of_packing_covering
(rayMatroid Q false) (rayMatroid Q true)
(rayMatroid_ground Q hE false) (rayMatroid_ground Q hE true)
(rayMatroid_selfDual Q hE hQ false) (rayMatroid_selfDual Q hE hQ true) h
exact hn I J hI hJ hIJ
end InfiniteMatroidCounterexample
end
end OAI
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