File size: 2,417 Bytes
dd14c26 | 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 65 66 67 68 69 70 71 | import Mathlib.Combinatorics.Matroid.Closure
namespace OAI
namespace MatroidProphet
namespace Pivots
open Set
variable {α : Type*} {M : Matroid α} {I J : Set α} {e : α}
def support (M : Matroid α) (I : Set α) (e : α) : Set α :=
⋂₀ {J | J ⊆ I ∧ e ∈ M.closure J}
lemma support_subset (he : e ∈ M.closure I) : support M I e ⊆ I :=
Set.sInter_subset_of_mem ⟨Subset.rfl, he⟩
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⟩
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)
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)
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
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)
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)
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⟩
end Pivots
end MatroidProphet
end OAI
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