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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 | import OAI.Probability.MatroidProphet.Main
namespace OAI
namespace MatroidProphet
open Set
variable {α : Type*} [Fintype α]
/-- `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
end MatroidProphet
end OAI
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