| import OAI.Probability.MatroidProphet.Main |
|
|
| namespace OAI |
|
|
| /-! Literal `eq:raw-pattern-count` (transfer.tex:282 |
| exponential relaxation. The family is the actual fixed labeled residual family, |
| not a replacement cardinality expression. The empty-group case is included. -/ |
|
|
| namespace MatroidProphet |
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|
| open Set Finset Pivots |
|
|
| variable {α : Type*} [Fintype α] [LinearOrder α] {n : ℕ} |
|
|
| theorem integerResidualFamily_card_raw (M : Matroid α) (hE : M.E = Set.univ) |
| (d : α) (U : Finset α) (q : ℕ) (F : ℤ → Set α) (a : ℤ) (ε : Fin 2) |
| (K : ParityWindow a ε → Set α) (test : Fin n → α) : |
| (integerResidualFamily M U q F a ε K test).card ≤ |
| (binomialSum U.card q)^2 * (4^(2*q) * binomialSum n (2*q))^2 := by |
| exact card_residualFamily_le M d U q _ _ _ _ test |
| (by simp [hE]) (by simp [hE]) (by simp [hE]) |
|
|
| / |
| No lower bound on the group size or density threshold is needed. -/ |
| theorem labeledResidualFamily_card_raw (M : Matroid α) (hE : M.E = Set.univ) |
| (U : Finset α) (κ : ℕ) (F : ℤ → Set α) (a : ℤ) (ε : Fin 2) |
| (K : ParityWindow a ε → Set α) : |
| (labeledResidualFamily M U κ F a ε K).card ≤ |
| (binomialSum U.card (U.card / κ))^2 * |
| (4^(2*(U.card / κ)) * binomialSum U.card (2*(U.card / κ)))^2 := by |
| classical |
| by_cases hU : U.Nonempty |
| · obtain ⟨d, _⟩ := hU |
| exact Finset.card_image_le.trans |
| (integerResidualFamily_card_raw M hE d U (U.card / κ) F a ε K (groupLabel U)) |
| · have hU0 : U = ∅ := Finset.not_nonempty_iff_eq_empty.mp hU |
| subst U |
| have hsub : labeledResidualFamily M ∅ κ F a ε K ⊆ ({∅} : Finset (Finset α)) := by |
| intro R hR |
| have hR0 := labeledResidualFamily_subsets M ∅ κ F a ε K R hR |
| simpa only [Finset.mem_singleton] using Finset.subset_empty.mp hR0 |
| have hc := Finset.card_le_card hsub |
| have hzero : Finset.Iic (0 : ℕ) = {0} := by |
| ext k |
| simp only [Finset.mem_Iic, Finset.mem_singleton, Nat.le_zero] |
| simpa [binomialSum, hzero] using hc |
|
|
| end MatroidProphet |
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|
| end OAI |
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|
