| import OAI.Combinatorics.InfiniteMatroid.Basic |
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| namespace OAI |
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| section |
| open Set Matroid |
| namespace InfiniteMatroidCounterexample |
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| lemma contraction_indep_compl_spanning {α : Type*} (M : Matroid α) (hM : M.dual = M) |
| {C I : Set α} (hI : (contractOnto M C).Indep I) : |
| (M ↾ C).Spanning (C \ I) := by |
| unfold contractOnto at hI |
| rw [hM] at hI |
| have h : (M ↾ C).Coindep I := hI |
| simpa only [Matroid.restrict_ground_eq] using h.compl_spanning |
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| lemma spanning_union_of_restrictions {α : Type*} (M : Matroid α) (hE : M.E = univ) |
| {P C S T : Set α} (hPC : P ∪ C = univ) (hSP : S ⊆ P) (hTC : T ⊆ C) |
| (hS : (M ↾ P).Spanning S) (hT : (M ↾ C).Spanning T) : M.Spanning (S ∪ T) := by |
| apply (M.spanning_iff_ground_subset_closure (by rw [hE]; exact subset_univ _)).mpr |
| have hPg : P ⊆ M.E := by rw [hE]; exact subset_univ _ |
| have hCg : C ⊆ M.E := by rw [hE]; exact subset_univ _ |
| have hp := (M.restrict_spanning_iff hSP hPg).mp hS |
| have hc := (M.restrict_spanning_iff hTC hCg).mp hT |
| rw [hE,←hPC] |
| exact union_subset (hp.trans (M.closure_mono subset_union_left)) |
| (hc.trans (M.closure_mono subset_union_right)) |
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| lemma indep_compl_of_selfDual_spanning {α : Type*} (M : Matroid α) (hE : M.E = univ) |
| (hM : M.dual = M) {A : Set α} (hA : M.Spanning A) : M.Indep Aᶜ := by |
| have h : M.dual.Indep (M.E \ A) := hA.compl_coindep |
| simpa only [hM,hE,← Set.compl_eq_univ_sdiff] using h |
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| /-- The mixed packing/covering reduction, not merely the two pure cases. -/ |
| lemma independent_cover_of_packing_covering {α : Type*} (M₀ M₁ : Matroid α) |
| (hE₀ : M₀.E = univ) (hE₁ : M₁.E = univ) (hM₀ : M₀.dual = M₀) (hM₁ : M₁.dual = M₁) |
| (h : HasPackingCovering M₀ M₁) : |
| ∃ J₀ J₁ : Set α, M₀.Indep J₀ ∧ M₁.Indep J₁ ∧ J₀ ∪ J₁ = univ := by |
| classical |
| obtain ⟨P,C,S₀,S₁,I₀,I₁,hdPC,hPC,_,hS₀P,hS₁P,hdS,hS₀,hS₁,_,_,hI₀,hI₁,hIC⟩ := h |
| have hPC' : P ∪ C = univ := hPC.trans hE₀ |
| have hT₀ := contraction_indep_compl_spanning M₀ hM₀ hI₀ |
| have hT₁ := contraction_indep_compl_spanning M₁ hM₁ hI₁ |
| have hA₀ := spanning_union_of_restrictions M₀ hE₀ hPC' hS₀P sdiff_subset hS₀ hT₀ |
| have hA₁ := spanning_union_of_restrictions M₁ hE₁ hPC' hS₁P sdiff_subset hS₁ hT₁ |
| have hd : Disjoint (S₀ ∪ (C \ I₀)) (S₁ ∪ (C \ I₁)) := by |
| apply Set.disjoint_left.mpr |
| intro x hx hy |
| rcases hx with hx|hx <;> rcases hy with hy|hy |
| · exact Set.disjoint_left.mp hdS hx hy |
| · exact Set.disjoint_left.mp hdPC (hS₀P hx) hy.1 |
| · exact Set.disjoint_left.mp hdPC (hS₁P hy) hx.1 |
| · have hm : x ∈ I₀ ∪ I₁ := hIC.symm ▸ hx.1 |
| exact hm.elim hx.2 hy.2 |
| refine ⟨_,_,indep_compl_of_selfDual_spanning M₀ hE₀ hM₀ hA₀, |
| indep_compl_of_selfDual_spanning M₁ hE₁ hM₁ hA₁,?_⟩ |
| rw [←Set.compl_inter,hd.inter_eq,Set.compl_empty] |
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| end InfiniteMatroidCounterexample |
| end |
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| end OAI |
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