| import OAI.Probability.MatroidProphet.Density.Basic |
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| namespace OAI |
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| namespace MatroidProphet |
| open Set |
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| variable {α : Type*} [Fintype α] |
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| private lemma densityPath_prefix (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (hκ : 0 < κ) (D : Set α) (P : ℕ → Set α) (hP : Monotone P) (N : ℕ) : |
| ∃ J : ℕ → Set α, Monotone J ∧ (∀ k, J k ⊆ D) ∧ |
| (∀ k ≤ N, M.closure (P k ∪ J k) = densityExpansion M hE κ D (P k)) ∧ |
| κ * (J N).ncard ≤ (D ∩ densityExpansion M hE κ D (P N)).ncard ∧ |
| (∀ k, N ≤ k → J k = J N) := by |
| induction N with |
| | zero => |
| have hPQ : M.closure (P 0) ⊆ densityExpansion M hE κ D (P 0) := by |
| exact (M.closure_subset_closure (densityExpansion_extensive M hE κ D (P 0))).trans_eq |
| (densityExpansion_flat M hE κ D (P 0)).closure |
| obtain ⟨K, hK, hgen, hcard⟩ := densityExpansion_generator_step M hE κ hκ D (P 0) |
| (M.closure (P 0)) (Matroid.isFlat_closure (M := M) _) |
| (M.subset_closure _ (by simp [hE])) hPQ |
| refine ⟨fun _ => K, fun _ _ _ => Subset.rfl, ?_, ?_, ?_, ?_⟩ |
| · exact fun _ _ he => (hK he).1.1 |
| · intro k hk |
| have hk0 : k = 0 := by omega |
| subst k |
| simpa only [M.closure_union_closure_left_eq] using hgen |
| · dsimp only |
| omega |
| · exact fun _ _ => rfl |
| | succ N ih => |
| obtain ⟨J, hJmono, hJD, hJgen, hJcard, hJsat⟩ := ih |
| let Q := densityExpansion M hE κ D (P N) |
| let Q' := densityExpansion M hE κ D (P (N + 1)) |
| let R := M.closure (P (N + 1) ∪ Q) |
| have hQQ' : Q ⊆ Q' := densityExpansion_mono M hE κ D (hP (Nat.le_succ N)) |
| have hRflat : M.IsFlat R := Matroid.isFlat_closure (M := M) _ |
| have hPR : P (N + 1) ⊆ R := |
| subset_union_left.trans (M.subset_closure _ (by simp [hE])) |
| have hQR : Q ⊆ R := subset_union_right.trans (M.subset_closure _ (by simp [hE])) |
| have hRQ' : R ⊆ Q' := by |
| exact (M.closure_subset_closure |
| (union_subset (densityExpansion_extensive M hE κ D _) hQQ')).trans_eq |
| (densityExpansion_flat M hE κ D _).closure |
| obtain ⟨K, hK, hKgen, hKcard⟩ := |
| densityExpansion_generator_step M hE κ hκ D (P (N + 1)) R hRflat hPR hRQ' |
| have hJNQ : J N ⊆ Q := by |
| change J N ⊆ densityExpansion M hE κ D (P N) |
| rw [← hJgen N le_rfl] |
| exact subset_union_right.trans (M.subset_closure _ (by simp [hE])) |
| have hJNR : J N ⊆ R := hJNQ.trans hQR |
| have hdisj : Disjoint (J N) K := by |
| rw [Set.disjoint_left] |
| intro e heJ heK |
| exact (hK heK).2 (hJNR heJ) |
| have hRgen : M.closure (P (N + 1) ∪ J N) = R := by |
| apply Subset.antisymm |
| · exact M.closure_subset_closure (union_subset_union Subset.rfl hJNQ) |
| · apply Matroid.closure_subset_closure_of_subset_closure |
| apply union_subset |
| · exact subset_union_left.trans (M.subset_closure _ (by simp [hE])) |
| · change densityExpansion M hE κ D (P N) ⊆ _ |
| rw [← hJgen N le_rfl] |
| exact M.closure_subset_closure (union_subset_union (hP (Nat.le_succ N)) Subset.rfl) |
| have hnewgen : M.closure (P (N + 1) ∪ (J N ∪ K)) = Q' := by |
| calc |
| M.closure (P (N + 1) ∪ (J N ∪ K)) = |
| M.closure ((P (N + 1) ∪ J N) ∪ K) := by rw [union_assoc] |
| _ = M.closure (M.closure (P (N + 1) ∪ J N) ∪ K) := |
| (M.closure_union_closure_left_eq _ _).symm |
| _ = M.closure (R ∪ K) := by rw [hRgen] |
| _ = Q' := hKgen |
| have hnewcard : κ * (J N ∪ K).ncard ≤ (D ∩ Q').ncard := by |
| rw [Set.ncard_union_eq hdisj, Nat.mul_add] |
| have hm := Set.ncard_mono (inter_subset_inter_right D hQR) |
| change κ * K.ncard + (D ∩ R).ncard ≤ (D ∩ Q').ncard at hKcard |
| change κ * (J N).ncard ≤ (D ∩ Q).ncard at hJcard |
| omega |
| let J' : ℕ → Set α := fun k => if k ≤ N then J k else J N ∪ K |
| have hJ'last : J' (N + 1) = J N ∪ K := by simp [J'] |
| refine ⟨J', ?_, ?_, ?_, ?_, ?_⟩ |
| · intro i j hij |
| by_cases hi : i ≤ N |
| · by_cases hj : j ≤ N |
| · simpa [J', hi, hj] using hJmono hij |
| · change (if i ≤ N then J i else J N ∪ K) ⊆ _ |
| rw [ite_eq_left hi] |
| simpa only [J', ite_eq_right hj] using (hJmono hi).trans subset_union_left |
| · have hj : ¬j ≤ N := by omega |
| simp only [J', ite_eq_right hi, ite_eq_right hj] |
| exact Subset.rfl |
| · intro k e he |
| by_cases hk : k ≤ N |
| · exact hJD k (by simpa [J', hk] using he) |
| · have he' : e ∈ J N ∪ K := by simpa [J', hk] using he |
| rcases he' with heJ | heK |
| · exact hJD N heJ |
| · exact (hK heK).1.1 |
| · intro k hk |
| by_cases hk' : k ≤ N |
| · simpa [J', hk'] using hJgen k hk' |
| · have heq : k = N + 1 := by omega |
| subst k |
| rw [hJ'last] |
| exact hnewgen |
| · rw [hJ'last] |
| exact hnewcard |
| · intro k hk |
| have hk' : ¬ k ≤ N := by omega |
| rw [hJ'last] |
| simp [J', hk'] |
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| lemma densityPath_generators (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (hκ : 0 < κ) (D : Set α) (P : ℕ → Set α) (hP : Monotone P) (N : ℕ) : |
| ∃ Jtotal : Set α, Jtotal ⊆ D ∧ κ * Jtotal.ncard ≤ D.ncard ∧ |
| ∃ J : ℕ → Set α, Monotone J ∧ (∀ k ≤ N, J k ⊆ Jtotal) ∧ |
| (∀ k ≤ N, M.closure (P k ∪ J k) = densityExpansion M hE κ D (P k)) := by |
| obtain ⟨J, hmono, hD, hgen, hcard, _⟩ := densityPath_prefix M hE κ hκ D P hP N |
| refine ⟨J N, hD N, ?_, J, hmono, ?_, hgen⟩ |
| · exact hcard.trans (Set.ncard_mono inter_subset_left) |
| · exact fun k hk => hmono hk |
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| end MatroidProphet |
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| end OAI |
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