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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.Waves
namespace OAI
namespace InfiniteMatroidCounterexample
open Set Matroid
lemma packingCovering_of_wave_covering {α : Type*} {M : Bool → Matroid α} {F : Set α}
(hE : ∀ i, (M i).E = F) {W : Set α × (Bool → Set α)}
(hW : Wave M F W) (hC : FamilyCovering (F \ W.1) (fun i ↦ M i ／ W.1)) :
HasPackingCovering (M false) (M true) := by
obtain ⟨I, hI, hcov⟩ := hC
have hfull : I false ∪ I true = F \ W.1 := by
rw [← hcov]
ext x
simp only [mem_union, mem_iUnion]
exact ⟨fun h ↦ h.elim (fun hx ↦ ⟨false, hx⟩) (fun hx ↦ ⟨true, hx⟩),
fun ⟨i, hi⟩ ↦ by cases i <;> simp_all⟩
have hsub : ∀ i, I i ⊆ F \ W.1 := by
intro i
simpa [hE] using (hI i).subset_ground
have hcont : ∀ i, (contractOnto (M i) (F \ W.1)).Indep (I i) := by
intro i
rw [← hE i, contractOnto_ground_sdiff]
exact hI i
refine ⟨W.1, F \ W.1, W.2 false, W.2 true, I false, I true,
disjoint_sdiff_right, ?_, (hE true).trans (hE false).symm,
hW.2.1 false, hW.2.1 true, hW.2.2.1 Bool.false_ne_true, ?_, ?_,
hsub false, hsub true, hcont false, hcont true, hfull⟩
· rw [hE false]
exact union_sdiff_cancel hW.1
· exact ((M false).restrict_spanning_iff (hW.2.1 false)
(by rw [hE]; exact hW.1)).mpr (hW.2.2.2 false)
· exact ((M true).restrict_spanning_iff (hW.2.1 true)
(by rw [hE]; exact hW.1)).mpr (hW.2.2.2 true)
theorem separate_covering_of_no_packingCovering {α : Type*} {M : Bool → Matroid α}
{F : Set α} (hE : ∀ i, (M i).E = F)
(hno : ¬ HasPackingCovering (M false) (M true)) :
∃ (C : Set α) (N : Bool → Matroid α),
(∀ i, (N i).E = C) ∧ ¬ FamilyCovering C N ∧
(∀ Y ⊆ C, FamilyPacking (fun i ↦ N i ↾ Y) →
FamilyCovering Y (fun i ↦ N i ↾ Y)) := by
obtain ⟨W, hW⟩ := exists_maximal_wave M F
refine ⟨F \ W.1, fun i ↦ M i ／ W.1, ?_, ?_, residual_covering_condition hE hW⟩
· simp [hE]
· exact fun h ↦ hno (packingCovering_of_wave_covering hE hW.1 h)
end InfiniteMatroidCounterexample
end OAI
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