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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 53 54 55 56 57 58 59 60 61 62 63 64 | import OAI.Combinatorics.InfiniteMatroid.CorollaryModel
namespace OAI
namespace InfiniteMatroidCounterexample
open Set Matroid
lemma packingCovering_of_intersection {α : Type*} {M N : Matroid α}
(hE : N.E = M.E) (h : HasIntersection M N) : HasPackingCovering M N.dual := by
obtain ⟨J, J₀, J₁, hJM, hJN, hd, hJJ, hcl⟩ := h
have h0J : J₀ ⊆ J := hJJ ▸ subset_union_left
have h1J : J₁ ⊆ J := hJJ ▸ subset_union_right
have h0M := hJM.subset h0J
have h1M := hJM.subset h1J
have h0N := hJN.subset h0J
have h1N := hJN.subset h1J
let P := M.closure J₀
let C := M.E \ P
have hPE : P ⊆ M.E := M.closure_subset_ground J₀
have h0P : J₀ ⊆ P := M.subset_closure J₀
have h1C : J₁ ⊆ C := by
intro x hx
refine ⟨h1M.subset_ground hx, ?_⟩
intro hxP
have hx0 : x ∈ J₀ := (hJM.closure_inter_eq_self_of_subset h0J) ▸ ⟨hxP, h1J hx⟩
exact disjoint_left.mp hd hx0 hx
have hCcl : C ⊆ N.closure J₁ := by
intro x hx
have hxcl : x ∈ P ∪ N.closure J₁ := hcl.symm ▸ hx.1
exact hxcl.resolve_left hx.2
have h0basis : M.IsBasis J₀ P := h0M.isBasis_closure
have h1basis : N.IsBasis J₁ C := h1N.isBasis_of_subset_of_subset_closure h1C hCcl
have hJ0con : (N ／ C).Indep J₀ := by
apply h1basis.contract_indep_iff.mpr
refine ⟨hJJ ▸ hJN, ?_⟩
exact disjoint_sdiff_left.mono_right h0P
have hJ1con : (M ／ P).Indep J₁ := by
apply h0basis.contract_indep_iff.mpr
refine ⟨?_, disjoint_sdiff_right.mono_right h1C⟩
simpa only [union_comm J₁, hJJ] using hJM
have hNP : N.E \ C = P := by
rw [hE]
exact sdiff_sdiff_cancel_left hPE
have hdualS : (N.dual ↾ P).Spanning (P \ J₀) := by
have hc : (N.dual ↾ P).Coindep J₀ := by
change ((N.dual ↾ P).dual).Indep J₀
simpa only [Matroid.contract, Matroid.delete, Matroid.dual_ground, hNP] using hJ0con
exact hc.compl_spanning
have hN1 : (N ↾ C).Spanning J₁ :=
(N.restrict_spanning_iff h1C (by rw [hE]; exact sdiff_subset)).mpr hCcl
have hdualI : (contractOnto N.dual C).Indep (C \ J₁) := by
simpa only [contractOnto, Matroid.dual_dual, Matroid.restrict_ground_eq] using hN1.compl_coindep.indep
refine ⟨P, C, J₀, P \ J₀, J₁, C \ J₁,
disjoint_sdiff_right, union_sdiff_cancel hPE, ?_, h0P, sdiff_subset,
disjoint_sdiff_right, ?_, hdualS, h1C, sdiff_subset, ?_, hdualI, ?_⟩
· simpa only [Matroid.dual_ground] using hE
· exact (M.restrict_spanning_iff h0P hPE).mpr Subset.rfl
· exact hJ1con
· exact union_sdiff_cancel h1C
end InfiniteMatroidCounterexample
end OAI
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