| import Mathlib |
| import OAI.Computability.MinUncut.Model |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open MeasureTheory Set |
| namespace MinUncut.FiniteGaussian |
|
|
| lemma conditional_rounding_bound {Ω : Type*} [MeasurableSpace Ω] |
| {μ : Measure Ω} [IsProbabilityMeasure μ] {s : Set Ω} (hs : MeasurableSet s) |
| (hp : 0 < μ.real s) {F G : Ω → ℝ} (hF : Integrable F μ) (hG : Integrable G μ) |
| (hFrange : ∀ x, 0 ≤ F x ∧ F x ≤ 1) (hGrange : ∀ x, 0 ≤ G x ∧ G x ≤ 1) : |
| |(∫ x, F x ∂μ)-(∫ x in s, G x ∂μ)/μ.real s| ≤ |
| μ.real sᶜ+∫ x in s, |F x-G x| ∂μ := by |
| let p := μ.real s |
| let a := μ.real sᶜ |
| let B := ∫ x in sᶜ, F x ∂μ |
| let C := ∫ x in s, G x ∂μ |
| let A := ∫ x in s, F x-G x ∂μ |
| have ha : 0 ≤ a := measureReal_nonneg |
| have hpa : p+a=1 := by simpa only [p,a,probReal_univ] using measureReal_add_measureReal_compl (μ := μ) hs |
| have hB0 : 0 ≤ B := integral_nonneg (fun x => (hFrange x).1) |
| have hBa : B ≤ a := by |
| have hh := integral_mono hF.integrableOn (integrable_const (1:ℝ)) (fun x => (hFrange x).2) (μ := μ.restrict sᶜ) |
| simpa only [setIntegral_const,smul_eq_mul,mul_one] using hh |
| have hC0 : 0 ≤ C := integral_nonneg (fun x => (hGrange x).1) |
| have hCp : C ≤ p := by |
| have hh := integral_mono hG.integrableOn (integrable_const (1:ℝ)) (fun x => (hGrange x).2) (μ := μ.restrict s) |
| simpa only [setIntegral_const,smul_eq_mul,mul_one] using hh |
| have hratio : 0 ≤ a*C/p := by dsimp [p]; positivity |
| have hratiole : a*C/p ≤ a := by |
| apply (div_le_iff₀ hp).mpr |
| exact mul_le_mul_of_nonneg_left hCp ha |
| have hbound : |B-a*C/p| ≤ a := abs_le.mpr ⟨by linarith,by linarith⟩ |
| have he : (∫ x, F x ∂μ)-C/p=A+(B-a*C/p) := by |
| have hsplit := integral_add_compl hs hF |
| have hsub := integral_sub hF.integrableOn hG.integrableOn (μ := μ.restrict s) |
| change A=(∫ x in s, F x ∂μ)-C at hsub |
| have hx : C-C/p= -(a*C/p) := by |
| have hp0 : p ≠ 0 := ne_of_gt hp |
| apply (mul_right_cancel₀ hp0) |
| field_simp [hp0] |
| nlinarith |
| rw [← hsplit] |
| change (∫ x in s, F x ∂μ)+B-C/p=A+(B-a*C/p) |
| rw [hsub] |
| linear_combination hx |
| change |(∫ x, F x ∂μ)-C/p| ≤ a+_ |
| rw [he] |
| have hA : |A| ≤ ∫ x in s, |F x-G x| ∂μ := abs_integral_le_integral_abs |
| exact (abs_add_le _ _).trans (by linarith) |
|
|
| end MinUncut.FiniteGaussian |
|
|
| end |
| end |
|
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| end OAI |
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