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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 | import OAI.Probability.MatroidProphet.Density.Basic
import Mathlib.Algebra.Order.BigOperators.Group.Finset
namespace OAI
namespace MatroidProphet
open Set Finset
variable {α : Type*} [Fintype α]
noncomputable def conditionalRank (M : Matroid α) (S P : Set α) : ℕ :=
natRank M (S ∪ P) - natRank M P
lemma conditionalRank_add_base (M : Matroid α) (S P : Set α) :
conditionalRank M S P + natRank M P = natRank M (S ∪ P) := by
exact Nat.sub_add_cancel (natRank_mono M subset_union_right)
lemma conditionalRank_antitone (M : Matroid α) (S : Set α)
{P Q : Set α} (hPQ : P ⊆ Q) :
conditionalRank M S Q ≤ conditionalRank M S P := by
have hsub := natRank_submod M (S ∪ P) Q
have hi := natRank_mono M (show P ⊆ (S ∪ P) ∩ Q from
fun x hx => ⟨Or.inr hx, hPQ hx⟩)
have hu : (S ∪ P) ∪ Q = S ∪ Q := by
ext x
simp only [mem_union]
constructor
· rintro ((h | h) | h)
· exact Or.inl h
· exact Or.inr (hPQ h)
· exact Or.inr h
· rintro (h | h)
· exact Or.inl (Or.inl h)
· exact Or.inr h
rw [hu] at hsub
have hP := conditionalRank_add_base M S P
have hQ := conditionalRank_add_base M S Q
omega
lemma conditionalRank_exchange (M : Matroid α) (S J A : Set α) :
conditionalRank M S A + conditionalRank M J (A ∪ S) =
conditionalRank M J A + conditionalRank M S (A ∪ J) := by
have hSA := conditionalRank_add_base M S A
have hJA := conditionalRank_add_base M J A
have hJAS := conditionalRank_add_base M J (A ∪ S)
have hSAJ := conditionalRank_add_base M S (A ∪ J)
have hUS : S ∪ A = A ∪ S := union_comm _ _
have hUJ : J ∪ A = A ∪ J := union_comm _ _
have hU : J ∪ (A ∪ S) = S ∪ (A ∪ J) := by
ext x
simp only [mem_union]
tauto
rw [hUS] at hSA
rw [hUJ] at hJA
rw [hU] at hJAS
omega
omit [Fintype α] in
lemma conditionalRank_closure (M : Matroid α) (S P : Set α) :
conditionalRank M S (M.closure P) = conditionalRank M S P := by
simp only [conditionalRank, natRank, M.eRk_union_closure_right_eq,
M.eRk_closure_eq]
lemma conditionalRank_le_rank (M : Matroid α) (J A : Set α) :
conditionalRank M J A ≤ natRank M J := by
have h := conditionalRank_antitone M J (empty_subset A)
simpa [conditionalRank] using h
lemma rankCost_step (M : Matroid α) (J S A B X Y : Set α)
(hXA : X ⊆ A) (hAB : A ⊆ B)
(hB : B ⊆ M.closure (A ∪ J)) (hY : A ∪ S ⊆ Y) :
conditionalRank M S A - conditionalRank M S B ≤
conditionalRank M J X - conditionalRank M J Y := by
have hSJ := conditionalRank_antitone M S hB
rw [conditionalRank_closure] at hSJ
have hSA := conditionalRank_antitone M S hAB
have hJA := conditionalRank_antitone M J hXA
have hJY := conditionalRank_antitone M J hY
have heq := conditionalRank_exchange M S J A
omega
lemma sum_potential_drop_le (p : ℕ → ℕ) (m : ℕ)
(hp : ∀ k < m, p (k + 1) ≤ p k) :
(∑ k ∈ range m, (p k - p (k + 1))) ≤ p 0 := by
suffices h : (∑ k ∈ range m, (p k - p (k + 1))) + p m = p 0 by omega
induction m with
| zero => simp
| succ m ih =>
rw [sum_range_succ]
have hprev := ih (fun k hk => hp k (Nat.lt_trans hk (Nat.lt_succ_self m)))
have hlast := hp m (Nat.lt_succ_self m)
omega
lemma rankCost_chain (M : Matroid α) (J : Set α) (m : ℕ)
(X A B S : ℕ → Set α)
(hXA : ∀ k < m, X k ⊆ A k)
(hAB : ∀ k < m, A k ⊆ B k)
(hB : ∀ k < m, B k ⊆ M.closure (A k ∪ J))
(hY : ∀ k < m, A k ∪ S k ⊆ X (k + 1)) :
(∑ k ∈ range m, conditionalRank M (S k) (A k)) ≤
(∑ k ∈ range m, conditionalRank M (S k) (B k)) + natRank M J := by
have hmono : ∀ k < m, X k ⊆ X (k + 1) := by
intro k hk
exact (hXA k hk).trans (subset_union_left.trans (hY k hk))
have hp : ∀ k < m, conditionalRank M J (X (k + 1)) ≤
conditionalRank M J (X k) := fun k hk => conditionalRank_antitone M J (hmono k hk)
have hdrop := sum_potential_drop_le (fun k => conditionalRank M J (X k)) m hp
have hsum : (∑ k ∈ range m, conditionalRank M (S k) (A k)) ≤
∑ k ∈ range m, (conditionalRank M (S k) (B k) +
(conditionalRank M J (X k) - conditionalRank M J (X (k + 1)))) := by
apply sum_le_sum
intro k hk
have hk' := mem_range.1 hk
have hs := rankCost_step M J (S k) (A k) (B k) (X k) (X (k + 1))
(hXA k hk') (hAB k hk') (hB k hk') (hY k hk')
have ha := conditionalRank_antitone M (S k) (hAB k hk')
omega
rw [sum_add_distrib] at hsum
have hbound := conditionalRank_le_rank M J (X 0)
omega
end MatroidProphet
end OAI
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