| import OAI.Combinatorics.InfiniteMatroid.CorollaryModel |
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| namespace OAI |
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| namespace InfiniteMatroidCounterexample |
| open Set Matroid |
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| 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] |
| 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 |
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| end InfiniteMatroidCounterexample |
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| end OAI |
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