| 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 |
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|
