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ea76aac | 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 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
end OAI
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