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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 | import OAI.Probability.MatroidProphet.Main
namespace OAI
/-! Literal `eq:raw-pattern-count` (transfer.tex:282--285), before the
exponential relaxation. The family is the actual fixed labeled residual family,
not a replacement cardinality expression. The empty-group case is included. -/
namespace MatroidProphet
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])
/-- Both subset choices and both marked-sequence counts are retained exactly.
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
end OAI
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