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c270adf | 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 | import Mathlib
namespace OAI
namespace LargeIndependentSets.CubeGradient
open MeasureTheory Set
open scoped NNReal
theorem slice_difference_le_variation {L : ℝ≥0} {f : ℝ → ℝ}
(hf : LipschitzWith L f) {a b x y : ℝ}
(hax : a ≤ x) (hxy : x ≤ y) (hyb : y ≤ b) :
|f y - f x| ≤ ∫ t in a..b, |deriv f t| := by
have hac : AbsolutelyContinuousOnInterval f a b :=
hf.lipschitzOnWith.absolutelyContinuousOnInterval
have hxyac : AbsolutelyContinuousOnInterval f x y :=
hf.lipschitzOnWith.absolutelyContinuousOnInterval
rw [← hxyac.integral_deriv_eq_sub]
exact (intervalIntegral.abs_integral_le_integral_abs hxy).trans
(intervalIntegral.integral_mono_interval hax hxy hyb
(Filter.Eventually.of_forall (fun t => abs_nonneg _)) hac.intervalIntegrable_deriv.abs)
theorem bit_block_variation {L : ℝ≥0} {f : ℝ → ℝ}
(hf : LipschitzWith L f) (a : ℝ) {h : ℝ} (hh : 0 ≤ h) :
(∫ t in a..a+h, |f (t+h) - f t|) ≤
h * ∫ t in a..a+2*h, |deriv f t| := by
have hfc := hf.continuous
have hc : Continuous (fun t => |f (t+h) - f t|) := by fun_prop
have hle : (∫ t in a..a+h, |f (t+h) - f t|) ≤
∫ _t in a..a+h, ∫ u in a..a+2*h, |deriv f u| := by
apply intervalIntegral.integral_mono_on (by linarith)
(hc.intervalIntegrable _ _) (continuous_const.intervalIntegrable _ _)
intro t ht
exact slice_difference_le_variation hf ht.1 (by linarith) (by linarith [ht.2])
convert hle using 1
simp [intervalIntegral.integral_const, smul_eq_mul]
end LargeIndependentSets.CubeGradient
end OAI
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