openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Profile /RadialAverageUniform.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Profile.RadialInnerDerivative | |
| import Mathlib.Topology.MetricSpace.UniformConvergence | |
| namespace OAI | |
| /-! Uniform continuity of the regular radial derivative integral, including the origin. -/ | |
| open Set Filter MeasureTheory | |
| namespace DefocusingNLS | |
| theorem radialAverage_sub_on (R r : ℝ) (hr : r ∈ Icc 0 R) (f g : ℝ → ℝ) | |
| (hf : ContinuousOn f (Icc 0 R)) (hg : ContinuousOn g (Icc 0 R)) : | |
| radialAverage (fun t => f t-g t) r=radialAverage f r-radialAverage g r := by | |
| have hm : MapsTo (fun s => r*s) (Icc (0 : ℝ) 1) (Icc 0 R) := by | |
| intro s hs | |
| exact ⟨mul_nonneg hr.1 hs.1,(mul_le_of_le_one_right hr.1 hs.2).trans hr.2⟩ | |
| have hfc : ContinuousOn (fun s => f (r*s)*s^11) (Icc (0 : ℝ) 1) := | |
| (hf.comp (continuous_const.mul continuous_id).continuousOn hm).mul | |
| (continuous_id.pow 11).continuousOn | |
| have hgc : ContinuousOn (fun s => g (r*s)*s^11) (Icc (0 : ℝ) 1) := | |
| (hg.comp (continuous_const.mul continuous_id).continuousOn hm).mul | |
| (continuous_id.pow 11).continuousOn | |
| unfold radialAverage | |
| rw [← intervalIntegral.integral_sub | |
| (ContinuousOn.intervalIntegrable_of_Icc (by norm_num) hfc) | |
| (ContinuousOn.intervalIntegrable_of_Icc (by norm_num) hgc)] | |
| apply intervalIntegral.integral_congr | |
| intro s _ | |
| ring | |
| theorem radial_derivative_integral_uniform (R : ℝ) (hR : 0 ≤ R) | |
| (f : ℕ → ℝ → ℝ) (g : ℝ → ℝ) | |
| (hf : ∀ i, ContinuousOn (f i) (Icc 0 R)) (hg : ContinuousOn g (Icc 0 R)) | |
| (ht : TendstoUniformlyOn f g atTop (Icc 0 R)) : | |
| TendstoUniformlyOn (fun i r => r*radialAverage (f i) r) | |
| (fun r => r*radialAverage g r) atTop (Icc 0 R) := by | |
| rw [Metric.tendstoUniformlyOn_iff] | |
| intro ε hε | |
| let δ := ε/(R+1) | |
| have hδ : 0 < δ := div_pos hε (by linarith) | |
| have hδε : δ*(R+1)=ε := div_mul_cancel₀ _ (by linarith) | |
| filter_upwards [(Metric.tendstoUniformlyOn_iff.mp ht) δ hδ] with i hi r hr | |
| have ha := radialAverage_norm_le (fun t => f i t-g t) R δ r hr.1 hr.2 (by | |
| intro t ht | |
| have hh := (hi t ht).le | |
| simpa only [dist_comm,dist_eq_norm] using hh) | |
| rw [radialAverage_sub_on R r hr (f i) g (hf i) hg] at ha | |
| rw [dist_comm,dist_eq_norm,← mul_sub,norm_mul,Real.norm_eq_abs,abs_of_nonneg hr.1] | |
| have hh := mul_le_mul_of_nonneg_left ha hr.1 | |
| have hrδ := mul_le_mul_of_nonneg_right hr.2 (show 0 ≤ δ/12 by positivity) | |
| nlinarith | |
| end DefocusingNLS | |
| end OAI | |