| import OAI.Probability.MatroidProphet.RankAccounting | |
| namespace OAI | |
| namespace MatroidProphet | |
| namespace Pivots | |
| open Set Finset | |
| variable {α : Type*} [Fintype α] | |
| def oldPrefix (b : ℕ → α) (k : ℕ) : Set α := {x | ∃ i < k, b i = x} | |
| @[simp] lemma oldPrefix_zero | |
| {α : Type u_1} [Fintype α] (b : ℕ → α) : oldPrefix b 0 = ∅ := by | |
| ext x | |
| simp [oldPrefix] | |
| lemma oldPrefix_succ | |
| {α : Type u_1} [Fintype α] (b : ℕ → α) (k : ℕ) : | |
| oldPrefix b (k + 1) = insert (b k) (oldPrefix b k) := by | |
| ext x | |
| constructor | |
| · rintro ⟨i, hi, rfl⟩ | |
| rcases Nat.lt_succ_iff_lt_or_eq.mp hi with hi | rfl | |
| · exact Or.inr ⟨i, hi, rfl⟩ | |
| · exact Or.inl rfl | |
| · rintro (rfl | ⟨i, hi, rfl⟩) | |
| · exact ⟨k, Nat.lt_succ_self k, rfl⟩ | |
| · exact ⟨i, Nat.lt_trans hi (Nat.lt_succ_self k), rfl⟩ | |
| lemma oldPrefix_mono | |
| {α : Type u_1} [Fintype α] (b : ℕ → α) : Monotone (oldPrefix b) := by | |
| intro j k hjk x hx | |
| obtain ⟨i, hi, rfl⟩ := hx | |
| exact ⟨i, lt_of_lt_of_le hi hjk, rfl⟩ | |
| lemma conditionalRank_singleton_of_notMem_closure (M : Matroid α) | |
| (P : Set α) {e : α} (he : e ∈ M.E) (hne : e ∉ M.closure P) : | |
| conditionalRank M {e} P = 1 := by | |
| unfold conditionalRank natRank | |
| rw [Set.singleton_union, M.eRk_insert_eq_add_one ⟨he, hne⟩, | |
| ENat.toNat_add (rank_ne_top M P) (by simp)] | |
| simp | |
| lemma conditionalRank_singleton_of_mem_closure | |
| {α : Type u_1} [Fintype α] (M : Matroid α) | |
| (P : Set α) {e : α} (he : e ∈ M.closure P) : | |
| conditionalRank M {e} P = 0 := by | |
| have h := congrArg (natRank M) (M.closure_insert_eq_of_mem_closure he) | |
| simp only [natRank_closure] at h | |
| simp only [conditionalRank, Set.singleton_union, h, Nat.sub_self] | |
| noncomputable def omittable (M : Matroid α) (b : ℕ → α) (A : Set α) (r : ℕ) : Finset ℕ := by | |
| classical | |
| exact (Finset.range r).filter (fun k => b k ∈ M.closure (oldPrefix b k ∪ A)) | |
| lemma omittable_drop (M : Matroid α) (b : ℕ → α) (A : Set α) (k : ℕ) | |
| (he : b k ∈ M.E) (hind : b k ∉ M.closure (oldPrefix b k)) | |
| (homit : b k ∈ M.closure (oldPrefix b k ∪ A)) : | |
| conditionalRank M A (oldPrefix b (k + 1)) + 1 = | |
| conditionalRank M A (oldPrefix b k) := by | |
| have hex := conditionalRank_exchange M {b k} A (oldPrefix b k) | |
| rw [conditionalRank_singleton_of_notMem_closure M _ he hind, | |
| conditionalRank_singleton_of_mem_closure M _ homit] at hex | |
| rw [oldPrefix_succ, Set.union_singleton] at * | |
| omega | |
| lemma card_omittable_le_rank (M : Matroid α) (b : ℕ → α) (A : Set α) (r : ℕ) | |
| (he : ∀ k < r, b k ∈ M.E) | |
| (hind : ∀ k < r, b k ∉ M.closure (oldPrefix b k)) : | |
| (omittable M b A r).card ≤ natRank M A := by | |
| classical | |
| let p := fun k => conditionalRank M A (oldPrefix b k) | |
| have hp : ∀ k < r, p (k + 1) ≤ p k := by | |
| intro k _ | |
| exact conditionalRank_antitone M A (oldPrefix_mono b (Nat.le_succ k)) | |
| have hdrop := sum_potential_drop_le p r hp | |
| have hsum : (omittable M b A r).card ≤ ∑ k ∈ range r, (p k - p (k + 1)) := by | |
| rw [Finset.card_eq_sum_ones] | |
| calc | |
| (∑ k ∈ omittable M b A r, 1) ≤ | |
| ∑ k ∈ omittable M b A r, (p k - p (k + 1)) := by | |
| apply Finset.sum_le_sum | |
| intro k hk | |
| have hk' := Finset.mem_filter.mp hk | |
| have hkr := Finset.mem_range.mp hk'.1 | |
| have hd := omittable_drop M b A k (he k hkr) (hind k hkr) hk'.2 | |
| change p (k + 1) + 1 = p k at hd | |
| omega | |
| _ ≤ ∑ k ∈ range r, (p k - p (k + 1)) := by | |
| apply Finset.sum_le_sum_of_subset_of_nonneg | |
| · exact Finset.filter_subset _ _ | |
| · intro _ _ _ | |
| exact Nat.zero_le _ | |
| have hzero : p 0 = natRank M A := by simp [p, conditionalRank] | |
| omega | |
| lemma notMem_closure_before_nonomittable | |
| {α : Type u_1} [Fintype α] (M : Matroid α) (b : ℕ → α) | |
| (A P : Set α) (r k : ℕ) (hk : k < r) | |
| (hkeep : k ∉ omittable M b A r) (hP : P ⊆ oldPrefix b k ∪ A) : | |
| b k ∉ M.closure P := by | |
| classical | |
| intro he | |
| apply hkeep | |
| exact Finset.mem_filter.mpr ⟨Finset.mem_range.mpr hk, | |
| M.closure_subset_closure hP he⟩ | |
| noncomputable def vulnerable (M : Matroid α) (b : ℕ → α) (A : Set α) (r : ℕ) : Finset ℕ := by | |
| classical | |
| exact (Finset.range r).filter (fun k => | |
| b k ∉ M.closure (oldPrefix b k) ∧ b k ∈ M.closure (oldPrefix b k ∪ A)) | |
| lemma card_vulnerable_le_rank (M : Matroid α) (b : ℕ → α) (A : Set α) (r : ℕ) | |
| (he : ∀ k < r, b k ∈ M.E) : | |
| (vulnerable M b A r).card ≤ natRank M A := by | |
| classical | |
| let p := fun k => conditionalRank M A (oldPrefix b k) | |
| have hp : ∀ k < r, p (k + 1) ≤ p k := by | |
| intro k _ | |
| exact conditionalRank_antitone M A (oldPrefix_mono b (Nat.le_succ k)) | |
| have hdrop := sum_potential_drop_le p r hp | |
| have hsum : (vulnerable M b A r).card ≤ ∑ k ∈ range r, (p k - p (k + 1)) := by | |
| rw [Finset.card_eq_sum_ones] | |
| calc | |
| (∑ k ∈ vulnerable M b A r, 1) ≤ | |
| ∑ k ∈ vulnerable M b A r, (p k - p (k + 1)) := by | |
| apply Finset.sum_le_sum | |
| intro k hk | |
| have hk' := Finset.mem_filter.mp hk | |
| have hkr := Finset.mem_range.mp hk'.1 | |
| have hd := omittable_drop M b A k (he k hkr) hk'.2.1 hk'.2.2 | |
| change p (k + 1) + 1 = p k at hd | |
| omega | |
| _ ≤ ∑ k ∈ range r, (p k - p (k + 1)) := by | |
| apply Finset.sum_le_sum_of_subset_of_nonneg | |
| · exact Finset.filter_subset _ _ | |
| · intro _ _ _ | |
| exact Nat.zero_le _ | |
| have hzero : p 0 = natRank M A := by simp [p, conditionalRank] | |
| omega | |
| end Pivots | |
| end MatroidProphet | |
| end OAI | |