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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 | import OAI.Combinatorics.InfiniteMatroid.Basic
namespace OAI
section
open Set Matroid
namespace InfiniteMatroidCounterexample
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
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))
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
/-- 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]
end InfiniteMatroidCounterexample
end
end OAI
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