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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 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 | import OAI.Probability.MatroidProphet.Density.Basic
namespace OAI
namespace MatroidProphet
open Set
variable {α : Type*} [Fintype α]
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']
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
end MatroidProphet
end OAI
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