File size: 2,841 Bytes
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 | import OAI.Probability.MatroidProphet.Residual.Tail
import OAI.Probability.MatroidProphet.Residual.Comparison
namespace OAI
namespace MatroidProphet
open Finset
variable {α : Type*} [DecidableEq α]
noncomputable def tripleMaskExpectation (p q r : α → ℝ) (V : Finset α)
(f : Finset α → Finset α → Finset α → ℝ) : ℝ :=
bitsExpectation p V (fun D => bitsExpectation q V (fun C => bitsExpectation r V (f D C)))
lemma tripleMaskExpectation_mono (p q r : α → ℝ)
(hp0 : ∀ e, 0 ≤ p e) (hp1 : ∀ e, p e ≤ 1)
(hq0 : ∀ e, 0 ≤ q e) (hq1 : ∀ e, q e ≤ 1)
(hr0 : ∀ e, 0 ≤ r e) (hr1 : ∀ e, r e ≤ 1)
(V : Finset α) {f g : Finset α → Finset α → Finset α → ℝ}
(h : ∀ D ⊆ V, ∀ C ⊆ V, ∀ T ⊆ V, f D C T ≤ g D C T) :
tripleMaskExpectation p q r V f ≤ tripleMaskExpectation p q r V g := by
apply bitsExpectation_mono p hp0 hp1
intro D hD
apply bitsExpectation_mono q hq0 hq1
intro C hC
exact bitsExpectation_mono r hr0 hr1 V (fun T hT => h D hD C hC T hT)
lemma tripleMaskExpectation_add (p q r : α → ℝ) (V : Finset α)
(f g : Finset α → Finset α → Finset α → ℝ) :
tripleMaskExpectation p q r V (fun D C T => f D C T + g D C T) =
tripleMaskExpectation p q r V f + tripleMaskExpectation p q r V g := by
simp only [tripleMaskExpectation, bitsExpectation_add]
lemma tripleMaskExpectation_mul (p q r : α → ℝ) (V : Finset α) (c : ℝ)
(f : Finset α → Finset α → Finset α → ℝ) :
tripleMaskExpectation p q r V (fun D C T => c * f D C T) =
c * tripleMaskExpectation p q r V f := by
simp only [tripleMaskExpectation, bitsExpectation_mul_const]
lemma tripleMask_sample_to_rank (p q r : α → ℝ)
(hp0 : ∀ e, 0 ≤ p e) (hp1 : ∀ e, p e ≤ 1)
(hq0 : ∀ e, 0 ≤ q e) (hq1 : ∀ e, q e ≤ 1)
(hr0 : ∀ e, 0 ≤ r e) (hr1 : ∀ e, r e ≤ 1)
(V : Finset α) (σ z error : Finset α → Finset α → Finset α → ℝ)
(Y : ℝ)
(hdet : ∀ D ⊆ V, ∀ C ⊆ V, ∀ T ⊆ V, σ D C T ≤ densityThreshold * z D C T + error D C T)
(hsafe : retainedFraction * Y - (V.card : ℝ) / densityThreshold ≤ tripleMaskExpectation p q r V σ)
(herr : tripleMaskExpectation p q r V error ≤ 2 * residualDelta * V.card) :
(retainedFraction * Y - ((2 : ℝ)^23)⁻¹ * V.card) / densityThreshold ≤
tripleMaskExpectation p q r V z := by
have hcompare := tripleMaskExpectation_mono p q r hp0 hp1 hq0 hq1 hr0 hr1 V hdet
rw [tripleMaskExpectation_add, tripleMaskExpectation_mul] at hcompare
have hc := mul_le_mul_of_nonneg_right residual_error_constant (Nat.cast_nonneg V.card)
have hkpos : 0 < densityThreshold := constants_positive.2.1
apply (div_le_iff₀ hkpos).2
rw [div_eq_mul_inv] at hsafe
nlinarith
end MatroidProphet
end OAI
|