| import Mathlib | |
| import OAI.Geometry.IntegralFillings.Currents.Basic | |
| import OAI.Geometry.IntegralFillings.Slicing.SmoothCutoff | |
| namespace OAI | |
| section | |
| open Set MeasureTheory Measure Filter Module | |
| open Set Filter MeasureTheory Measure ContinuousLinearMap | |
| open scoped Topology Convolution NNReal | |
| open Set Filter MeasureTheory Measure Metric | |
| open scoped Topology ContDiff | |
| open Set Filter Metric | |
| open Filter Set | |
| open Set Filter MeasureTheory TopologicalSpace | |
| open scoped Topology ENNReal | |
| open Set MeasureTheory | |
| open scoped RealInnerProductSpace | |
| open Matrix | |
| open scoped RealInnerProductSpace MatrixOrder | |
| open Set Filter MeasureTheory | |
| open scoped Topology ENNReal NNReal | |
| open MeasureTheory Filter Set Metric | |
| open scoped Topology Pointwise NNReal | |
| open Set MeasureTheory Filter | |
| open scoped Topology NNReal | |
| namespace SharpIntegralFillings.SmoothCutoff | |
| open Set Filter | |
| open scoped Topology NNReal | |
| noncomputable def profile (a : ℝ≥0) (t x : ℝ) : ℝ := | |
| Real.smoothTransition ((a:ℝ) * (x-t)) | |
| noncomputable def dprofile (a : ℝ≥0) (t x : ℝ) : ℝ := | |
| (a:ℝ) * deriv Real.smoothTransition ((a:ℝ) * (x-t)) | |
| lemma affine_lipschitz (a : ℝ≥0) (t : ℝ) : | |
| LipschitzWith a (fun x : ℝ => (a:ℝ)*(x-t)) := by | |
| apply LipschitzWith.of_dist_le_mul | |
| intro x y | |
| simp only [Real.dist_eq,←mul_sub,sub_sub_sub_cancel_right,abs_mul,abs_of_nonneg a.coe_nonneg] | |
| exact le_rfl | |
| lemma profile_hasDerivAt (a : ℝ≥0) (t x : ℝ) : | |
| HasDerivAt (profile a t) (dprofile a t x) x := by | |
| have hd := ((@Real.smoothTransition.contDiff 1).differentiable (by norm_num) ((a:ℝ)*(x-t))).hasDerivAt | |
| change HasDerivAt (fun y => Real.smoothTransition ((a:ℝ)*(y-t))) | |
| ((a:ℝ)*deriv Real.smoothTransition ((a:ℝ)*(x-t))) x | |
| simpa only [Function.comp_def,id_eq,mul_one,mul_comm] using | |
| hd.comp x (((hasDerivAt_id x).sub_const t).const_mul (a:ℝ)) | |
| lemma profile_bounds (a : ℝ≥0) (t x : ℝ) : | |
| 0 ≤ profile a t x ∧ profile a t x ≤ 1 := | |
| ⟨Real.smoothTransition.nonneg _,Real.smoothTransition.le_one _⟩ | |
| lemma profile_zero (a : ℝ≥0) {t x : ℝ} (hx : x ≤ t) : profile a t x = 0 := | |
| Real.smoothTransition.zero_of_nonpos (mul_nonpos_of_nonneg_of_nonpos a.coe_nonneg (sub_nonpos.mpr hx)) | |
| lemma dprofile_zero {a : ℝ≥0} (ha : 0 < a) {t x : ℝ} | |
| (hx : x ∉ Icc t (t+(a:ℝ)⁻¹)) : dprofile a t x = 0 := by | |
| have ha' : 0 < (a:ℝ) := ha | |
| have hnot : (a:ℝ)*(x-t) ∉ Icc 0 1 := by | |
| intro h | |
| apply hx | |
| refine ⟨?_,?_⟩ | |
| · have : 0 ≤ x-t := (mul_nonneg_iff_of_pos_left ha').mp h.1 | |
| linarith | |
| · have : x-t ≤ (a:ℝ)⁻¹ := by | |
| rw [inv_eq_one_div] | |
| apply (le_div_iff₀ ha').mpr | |
| simpa [mul_comm] using h.2 | |
| linarith | |
| simp [dprofile,deriv_transition_eq_zero hnot] | |
| lemma exists_profile_bounds : ∃ C : ℝ≥0, ∀ (a : ℝ≥0) (t : ℝ), | |
| LipschitzWith (C*a) (profile a t) ∧ | |
| LipschitzWith (a*(C*a)) (dprofile a t) ∧ | |
| ∀ x, |dprofile a t x| ≤ (C*a:ℝ≥0) := by | |
| obtain ⟨C,hC,hD,hb⟩ := exists_lipschitz_transition | |
| refine ⟨C,fun a t => ⟨hC.comp (affine_lipschitz a t),?_,?_⟩⟩ | |
| · change LipschitzWith (a*(C*a)) (fun x => (a:ℝ)*deriv Real.smoothTransition ((a:ℝ)*(x-t))) | |
| simpa only [sub_zero,Function.comp_def] using | |
| (affine_lipschitz a 0).comp (hD.comp (affine_lipschitz a t)) | |
| · intro x | |
| simp only [dprofile,abs_mul,abs_of_nonneg a.coe_nonneg,NNReal.coe_mul] | |
| calc | |
| (a:ℝ) * |deriv Real.smoothTransition ((a:ℝ)*(x-t))| ≤ (a:ℝ)*C := | |
| mul_le_mul_of_nonneg_left (hb _) a.coe_nonneg | |
| _ = (C:ℝ)*a := mul_comm _ _ | |
| lemma profile_sequence_tendsto (t x : ℝ) : | |
| Tendsto (fun n : ℕ => profile ((n:ℝ≥0)+1) t x) atTop | |
| (𝓝 ((Ioi t).indicator (fun _ => (1:ℝ)) x)) := by | |
| by_cases hx : t < x | |
| · rw [indicator_of_mem (show x ∈ Ioi t from hx)] | |
| have hlim : Tendsto (fun n : ℕ => ((n:ℝ)+1)*(x-t)) atTop atTop := | |
| (tendsto_atTop_add_const_right atTop 1 tendsto_natCast_atTop_atTop).atTop_mul_const (sub_pos.mpr hx) | |
| have heq : (fun n : ℕ => profile ((n:ℝ≥0)+1) t x) =ᶠ[atTop] fun _ => (1:ℝ) := by | |
| filter_upwards [hlim.eventually (eventually_ge_atTop 1)] with n hn | |
| apply Real.smoothTransition.one_of_one_le | |
| simpa only [NNReal.coe_add,NNReal.coe_natCast,NNReal.coe_one] using hn | |
| exact tendsto_const_nhds.congr' heq.symm | |
| · rw [indicator_of_notMem (show x ∉ Ioi t from hx)] | |
| simpa only [profile_zero _ (le_of_not_gt hx)] using | |
| (tendsto_const_nhds : Tendsto (fun _ : ℕ => (0:ℝ)) atTop (𝓝 0)) | |
| end SharpIntegralFillings.SmoothCutoff | |
| open Set MeasureTheory Filter | |
| open scoped ENNReal NNReal Topology | |
| end | |
| end OAI | |