| import OAI.Probability.MatroidProphet.Main |
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| namespace OAI |
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| namespace MatroidProphet |
| open Set |
| variable {α : Type*} [Fintype α] |
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|
| /-- `lem:density`, including the convention of closing a nonflat argument. -/ |
| def DensityExpansionContract (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D : Set α) : Prop := |
| Monotone (densityExpansion M hE κ D) ∧ |
| ∀ P : Set α, |
| IsDensityMax M κ D P (densityExpansion M hE κ D P) ∧ |
| (∀ Q, IsDensityMax M κ D P Q → Q ⊆ densityExpansion M hE κ D P) ∧ |
| P ⊆ densityExpansion M hE κ D P ∧ |
| densityExpansion M hE κ D (M.closure P) = densityExpansion M hE κ D P ∧ |
| ∀ Z : Set α, |
| (D ∩ (M.closure (densityExpansion M hE κ D P ∪ Z) \ |
| densityExpansion M hE κ D P)).ncard ≤ |
| κ * conditionalRank M Z (densityExpansion M hE κ D P) |
|
|
| /-- `lem:paths`: both paths, all integer endpoints, and the true least birth. -/ |
| def PathPropertiesContract (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C : ℕ → Set α) : Prop := |
| ∀ h : ℕ, |
| Monotone (nominalPath M hE κ D C h) ∧ |
| Monotone (guardedPath M hE κ D C (h + 1)) ∧ |
| (∀ k : ℤ, |
| guardedPath M hE κ D C h k ⊆ nominalPath M hE κ D C h k ∧ |
| nominalPath M hE κ D C h k ⊆ guardedPath M hE κ D C (h + 1) k ∧ |
| guardedPath M hE κ D C (h + 1) k ⊆ nominalPath M hE κ D C h (k + 1)) ∧ |
| (∀ k : ℤ, 0 ≤ k → |
| nominalPath M hE κ D C h k = univ ∧ |
| guardedPath M hE κ D C (h + 1) k = univ) ∧ |
| (∀ k : ℤ, k < activation h → nominalPath M hE κ D C h k = M.closure ∅) ∧ |
| (∀ k : ℤ, k < activation h - 1 → |
| guardedPath M hE κ D C (h + 1) k = M.closure ∅) ∧ |
| ∀ e : α, e ∉ M.closure ∅ → |
| IsLeast {k : ℤ | e ∈ nominalPath M hE κ D C h k} |
| (nominalBirth M hE κ D C h e) ∧ |
| ((nominalBirth M hE κ D C h e = activation h ∧ |
| e ∈ densityExpansion M hE κ (D h) (M.closure ∅)) ∨ |
| (activation h + 2 ≤ nominalBirth M hE κ D C h e ∧ |
| nominalBirth M hE κ D C h e ≤ 0)) |
|
|
| /-- `lem:generators`: one budget and one increasing family for the entire ℤ path. -/ |
| def GeneratorBudgetContract (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C : ℕ → Set α) : Prop := |
| ∀ h : ℕ, |
| (∃ G : Set α, G ⊆ D h ∧ κ * G.ncard ≤ (D h).ncard ∧ |
| ∃ J : ℤ → Set α, Monotone J ∧ (∀ k, J k ⊆ G) ∧ |
| ∀ k : ℤ, nominalPath M hE κ D C h k = |
| M.closure (guardedPath M hE κ D C h k ∪ J k)) ∧ |
| κ * natRank M (densityExpansion M hE κ (D h) (M.closure ∅)) ≤ (D h).ncard |
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| end MatroidProphet |
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| end OAI |
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