| import Mathlib.Combinatorics.Matroid.Closure |
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| namespace OAI |
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| namespace MatroidProphet |
| namespace Pivots |
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| open Set |
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| variable {α : Type*} {M : Matroid α} {I J : Set α} {e : α} |
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| def support (M : Matroid α) (I : Set α) (e : α) : Set α := |
| ⋂₀ {J | J ⊆ I ∧ e ∈ M.closure J} |
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| lemma support_subset (he : e ∈ M.closure I) : support M I e ⊆ I := |
| Set.sInter_subset_of_mem ⟨Subset.rfl, he⟩ |
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| lemma support_subset_of_mem_closure (hJI : J ⊆ I) (he : e ∈ M.closure J) : |
| support M I e ⊆ J := |
| Set.sInter_subset_of_mem ⟨hJI, he⟩ |
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| lemma mem_closure_support (hI : M.Indep I) (he : e ∈ M.closure I) : |
| e ∈ M.closure (support M I e) := by |
| rw [support, hI.closure_sInter_eq_biInter_closure_of_forall_subset |
| (Js := {J | J ⊆ I ∧ e ∈ M.closure J}) |
| ⟨I, Subset.rfl, he⟩ (fun _ hJ => hJ.1)] |
| exact Set.mem_iInter₂.mpr (fun _ hJ => hJ.2) |
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| lemma mem_closure_iff_support_subset (hI : M.Indep I) |
| (he : e ∈ M.closure I) (hJI : J ⊆ I) : |
| e ∈ M.closure J ↔ support M I e ⊆ J := by |
| constructor |
| · exact support_subset_of_mem_closure hJI |
| · intro h |
| exact M.closure_subset_closure h (mem_closure_support hI he) |
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| lemma support_nonempty (hI : M.Indep I) (he : e ∈ M.closure I) |
| (hne : e ∉ M.closure ∅) : (support M I e).Nonempty := by |
| by_contra h |
| have hem := mem_closure_support hI he |
| rw [Set.not_nonempty_iff_eq_empty.mp h] at hem |
| exact hne hem |
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| lemma support_unique (hI : M.Indep I) (he : e ∈ M.closure I) |
| (hJI : J ⊆ I) (hJe : e ∈ M.closure J) |
| (hmin : ∀ K ⊆ I, e ∈ M.closure K → J ⊆ K) : J = support M I e := by |
| exact Set.Subset.antisymm |
| (hmin _ (support_subset he) (mem_closure_support hI he)) |
| (support_subset_of_mem_closure hJI hJe) |
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| lemma old_redundant_after_insertions (he : e ∈ M.closure I) (A : Set α) : |
| M.closure (insert e (I ∪ A)) = M.closure (I ∪ A) := by |
| exact M.closure_insert_eq_of_mem_closure |
| (M.closure_subset_closure Set.subset_union_left he) |
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| lemma mem_closure_prefix_iff (hI : M.Indep I) (he : e ∈ M.closure I) |
| (time : α → ℕ) (k : ℕ) : |
| e ∈ M.closure {x | x ∈ I ∧ time x ≤ k} ↔ |
| ∀ x ∈ support M I e, time x ≤ k := by |
| rw [mem_closure_iff_support_subset hI he (fun _ hx => hx.1)] |
| constructor |
| · intro h x hx |
| exact (h hx).2 |
| · intro h x hx |
| exact ⟨support_subset he hx, h x hx⟩ |
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| end Pivots |
| end MatroidProphet |
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| end OAI |
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