openAI-math-dataset / lean /OAI /Combinatorics /IndependentSets /Reduction /SliceDifferenceVariation.lean
| 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 | |