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f89a930 | 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 | import Mathlib
namespace OAI
namespace Erdos970
section
open MeasureTheory Set
namespace ErdosContinuousBoundary
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)
end ErdosContinuousBoundary
end
end Erdos970
end OAI
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