| import Mathlib |
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| namespace OAI |
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| namespace Erdos970 |
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| section |
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| open MeasureTheory Set |
| namespace ErdosContinuousBoundary |
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| theorem integral_upper_difference_bound |
| (F G : ℝ → ℝ) (hF : Continuous F) (hG : Continuous G) |
| (a b D : ℝ) (ha : a ∈ Icc 1 2) (hD : 0 ≤ D) |
| (hFG : ∀ x, |F x-G x| ≤ D) (hGb : ∀ x, |G x| ≤ 1) : |
| |(∫ x in (1:ℝ)..a,F x)-(∫ x in (1:ℝ)..b,G x)| ≤ D+|a-b| := by |
| have hs := intervalIntegral.integral_sub (hF.intervalIntegrable (μ:=volume) 1 a) (hG.intervalIntegrable (μ:=volume) 1 a) |
| have hadd := intervalIntegral.integral_add_adjacent_intervals |
| (hG.intervalIntegrable (μ:=volume) 1 b) (hG.intervalIntegrable (μ:=volume) b a) |
| have he : (∫ x in (1:ℝ)..a,F x)-(∫ x in (1:ℝ)..b,G x) = |
| (∫ x in (1:ℝ)..a,F x-G x)+(∫ x in b..a,G x) := by |
| rw [hs] |
| linarith |
| have hfirst := intervalIntegral.norm_integral_le_of_norm_le_const (a:=1) (b:=a) |
| (f:=fun x => F x-G x) (C:=D) (fun x _ => by simpa only [Real.norm_eq_abs] using hFG x) |
| have hsecond := intervalIntegral.norm_integral_le_of_norm_le_const (a:=b) (b:=a) |
| (f:=G) (C:=1) (fun x _ => by simpa only [Real.norm_eq_abs] using hGb x) |
| rw [Real.norm_eq_abs,abs_of_nonneg (by linarith [ha.1] : 0 ≤ a-1)] at hfirst |
| rw [Real.norm_eq_abs,one_mul] at hsecond |
| have hfirst' : |∫ x in (1:ℝ)..a,F x-G x| ≤ D := by |
| apply hfirst.trans |
| nlinarith [ha.2] |
| rw [he] |
| exact (abs_add_le _ _).trans (add_le_add hfirst' hsecond) |
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| end ErdosContinuousBoundary |
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| end |
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| end Erdos970 |
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| end OAI |
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