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namespace OAI
namespace MatroidProphet
open Set Finset
variable {α : Type*} [Fintype α]
lemma natRank_le_ncard (M : Matroid α) (S : Set α) : natRank M S ≤ S.ncard := by
have h := ENat.toNat_le_toNat (M.eRk_le_encard S) (Set.toFinite S).encard_lt_top.ne
simpa only [natRank, Set.ncard_def] using h
lemma conditionalRank_mono_left (M : Matroid α) (P : Set α)
{S T : Set α} (hST : S ⊆ T) : conditionalRank M S P ≤ conditionalRank M T P := by
exact Nat.sub_le_sub_right (natRank_mono M (union_subset_union_left P hST)) _
lemma conditionalRank_spanned (M : Matroid α) (hE : M.E = Set.univ)
(S P H : Set α) (hspan : S ⊆ M.closure (P ∪ H)) :
conditionalRank M S P ≤ H.ncard := by
have hP : P ⊆ M.closure (P ∪ H) :=
subset_union_left.trans (M.subset_closure _ (by simp [hE]))
have hr := natRank_mono M (union_subset hspan hP)
rw [natRank_closure] at hr
have ht : conditionalRank M S P ≤ conditionalRank M H P := by
unfold conditionalRank
rw [union_comm H P]
exact Nat.sub_le_sub_right hr _
exact ht.trans ((conditionalRank_le_rank M H P).trans (natRank_le_ncard M H))
theorem layerGreedy_rank_lower (M : Matroid α) (hE : M.E = Set.univ)
(E : Finset α) (base : ℤ → Set α) (layer : α → ℤ) (time : α → ℕ)
(b : ℤ) (S K : Set α) (high : α → Prop) [DecidablePred high]
(hS : S \ base b ⊆ {e | e ∈ E ∧ layer e = b})
(hK : ∀ e ∈ layerGreedy M E base layer time, layer e = b → ¬ high e → e ∈ K) :
conditionalRank M S (base b ∪ K) ≤
((layerGreedy M E base layer time).filter (fun e => layer e = b ∧ high e)).card := by
classical
let H := (layerGreedy M E base layer time).filter (fun e => layer e = b ∧ high e)
have hspan : S ⊆ M.closure ((base b ∪ K) ∪ (H : Set α)) := by
intro e he
by_cases heF : e ∈ base b
· exact M.mem_closure_of_mem (Or.inl (Or.inl heF)) (by simp [hE])
have hel := hS ⟨he, heF⟩
have hs := layerGreedy_spans M E base layer time hE b hel
apply M.closure_subset_closure ?_ hs
rintro f (hf | ⟨hf, hb⟩)
· exact Or.inl (Or.inl hf)
· by_cases hh : high f
· exact Or.inr (mem_filter.mpr ⟨hf, hb, hh⟩)
· exact Or.inl (Or.inr (hK f hf hb hh))
have hc := conditionalRank_spanned M hE S (base b ∪ K) (H : Set α) hspan
simpa only [Set.ncard_coe_finset, H] using hc
theorem layerGreedy_sum_rank_lower [DecidableEq α]
(M : Matroid α) (hE : M.E = Set.univ)
(E : Finset α) (base : ℤ → Set α) (layer : α → ℤ) (time : α → ℕ)
(points : Finset ℤ) (S K : ℤ → Set α) (high : α → Prop) [DecidablePred high]
(hS : ∀ b ∈ points, S b \ base b ⊆ {e | e ∈ E ∧ layer e = b})
(hK : ∀ b ∈ points, ∀ e ∈ layerGreedy M E base layer time,
layer e = b → ¬ high e → e ∈ K b) :
(∑ b ∈ points, conditionalRank M (S b) (base b ∪ K b)) ≤
((layerGreedy M E base layer time).filter high).card := by
classical
let A := layerGreedy M E base layer time
calc
_ ≤ ∑ b ∈ points, (A.filter (fun e => layer e = b ∧ high e)).card := by
exact Finset.sum_le_sum fun b hb =>
layerGreedy_rank_lower M hE E base layer time b (S b) (K b) high (hS b hb) (hK b hb)
_ = ∑ b ∈ points, ((A.filter high).filter (fun e => layer e = b)).card := by
apply Finset.sum_congr rfl
intro b hb
congr 1
ext e
simp only [Finset.mem_filter]
tauto
_ = ((A.filter high).filter (fun e => layer e ∈ points)).card :=
Finset.sum_card_fiberwise_eq_card_filter (A.filter high) points layer
_ ≤ (A.filter high).card := Finset.card_filter_le _ _
end MatroidProphet
end OAI
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