| import Mathlib |
| import OAI.Geometry.IntegralFillings.Differentiation.CurveSpeed |
|
|
| namespace OAI |
|
|
| section |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology Pointwise NNReal |
| open Set Filter MeasureTheory TopologicalSpace |
| open scoped Topology NNReal |
|
|
| namespace SharpIntegralFillings.MetricDifferentiation |
| variable {X : Type*} [MetricSpace X] |
| noncomputable def signedDistance (c : ℝ → X) (t : ℝ) : ℝ := |
| if 0 ≤ t then dist (c t) (c 0) else -dist (c t) (c 0) |
|
|
| lemma signedDistance_zero (c : ℝ → X) : signedDistance c 0 = 0 := by |
| simp [signedDistance] |
|
|
| lemma signedDistance_div (c : ℝ → X) (t : ℝ) : |
| signedDistance c t / t = dist (c t) (c 0) / |t| := by |
| rcases le_total 0 t with ht | ht |
| · simp [signedDistance, ht, abs_of_nonneg ht] |
| · by_cases h0 : t = 0 |
| · simp [h0] |
| · simp [signedDistance, not_le.mpr (lt_of_le_of_ne ht h0), abs_of_nonpos ht, neg_div, div_neg] |
|
|
| lemma hasDerivAt_signedDistance_iff (c : ℝ → X) (d : ℝ) : |
| HasDerivAt (signedDistance c) d 0 ↔ |
| Tendsto (fun t => dist (c t) (c 0) / |t|) (𝓝[≠] 0) (𝓝 d) := by |
| rw [hasDerivAt_iff_tendsto_slope_zero] |
| simp only [zero_add, signedDistance_zero, sub_zero, smul_eq_mul, ← div_eq_inv_mul, |
| signedDistance_div] |
|
|
| lemma continuous_signedDistance {A : Type*} [TopologicalSpace A] {c : A → ℝ → X} |
| (hc : Continuous c.uncurry) : Continuous (fun p : A × ℝ => signedDistance (c p.1) p.2) := by |
| have hd : Continuous (fun p : A × ℝ => dist (c p.1 p.2) (c p.1 0)) := |
| hc.dist (hc.comp (continuous_fst.prodMk continuous_const)) |
| exact hd.if_le hd.neg continuous_const continuous_snd (by |
| rintro ⟨x,t⟩ (h : 0 = t) |
| simp [← h]) |
|
|
| lemma measurableSet_metricDirectionalDeriv {E : Type*} [NormedAddCommGroup E] |
| [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] |
| {f : E → X} (hf : Continuous f) (v : E) : |
| MeasurableSet {x | ∃ d : ℝ, Tendsto (fun t : ℝ => dist (f (x+t • v)) (f x) / |t|) |
| (𝓝[≠] 0) (𝓝 d)} := by |
| let c : E → ℝ → X := fun x t => f (x + t • v) |
| have hc : Continuous c.uncurry := hf.comp (continuous_fst.add (continuous_snd.smul continuous_const)) |
| have hm := measurableSet_of_differentiableAt_with_param ℝ |
| (f := fun x t => signedDistance (c x) t) (continuous_signedDistance hc) |
| have hm' := measurable_prodMk_right (y := (0 : ℝ)) hm |
| convert hm' using 1 |
| ext x |
| simp only [mem_ofPred_eq] |
| constructor |
| · rintro ⟨d, hd⟩ |
| exact ((hasDerivAt_signedDistance_iff (c x) d).mpr (by simpa [c] using hd)).differentiableAt |
| · intro hd |
| refine ⟨deriv (signedDistance (c x)) 0, ?_⟩ |
| simpa [c] using (hasDerivAt_signedDistance_iff (c x) _).mp hd.hasDerivAt |
|
|
| end SharpIntegralFillings.MetricDifferentiation |
|
|
| namespace SharpIntegralFillings.MetricDifferentiation |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [MeasurableSpace E] [BorelSpace E] |
| {X : Type*} [MetricSpace X] [SeparableSpace X] [Nonempty X] |
| {f : E → X} {K : ℝ≥0} |
|
|
| noncomputable def scalarCoordinate (f : E → X) (q : ℕ) (x : E) : ℝ := |
| dist (f x) (denseSeq X q) |
|
|
| omit [MeasurableSpace E] [BorelSpace E] [NormedSpace ℝ E] in |
| lemma scalarCoordinate_lipschitz (hf : LipschitzWith K f) (q : ℕ) : |
| LipschitzWith K (scalarCoordinate f q) := by |
| change LipschitzWith K (fun x : E => dist (f x) (denseSeq X q)) |
| simpa only [Function.comp_def, one_mul] using |
| (LipschitzWith.dist_left (denseSeq X q)).comp hf |
|
|
| noncomputable def lineSpeed (f : E → X) (v x : E) : ℝ := |
| ⨆ q : ℕ, |lineDeriv ℝ (scalarCoordinate f q) x v| |
|
|
| lemma measurable_lineSpeed (hf : Continuous f) (v : E) : Measurable (lineSpeed f v) := by |
| apply Measurable.iSup |
| intro q |
| exact (measurable_lineDeriv (hf.dist continuous_const)).abs |
|
|
| omit [MeasurableSpace E] [BorelSpace E] in |
| lemma deriv_coordinate_line (p v : E) (s : ℝ) (q : ℕ) : |
| deriv (MetricDifferentiationDense.distanceCoordinate (fun t => f (p+t • v)) q) s = |
| lineDeriv ℝ (scalarCoordinate f q) (p+s • v) v := by |
| rw [lineDeriv] |
| have he : (fun t : ℝ => scalarCoordinate f q (p+s • v+t • v)) = |
| (fun t : ℝ => MetricDifferentiationDense.distanceCoordinate (fun t => f (p+t • v)) q (s+t)) := by |
| ext t |
| simp [scalarCoordinate, MetricDifferentiationDense.distanceCoordinate, add_smul, add_assoc] |
| rw [he] |
| simpa using (deriv_comp_const_add (f := MetricDifferentiationDense.distanceCoordinate |
| (fun t => f (p+t • v)) q) s 0).symm |
|
|
| omit [MeasurableSpace E] [BorelSpace E] in |
| lemma lineSpeed_eq_curveSpeed (p v : E) (s : ℝ) : |
| lineSpeed f v (p+s • v) = MetricDifferentiationDense.metricSpeed (fun t => f (p+t • v)) s := by |
| simp only [lineSpeed, MetricDifferentiationDense.metricSpeed, deriv_coordinate_line] |
|
|
| lemma measurableSet_hasMetricDirectionalDeriv (hf : Continuous f) (v : E) : |
| MeasurableSet {x | Tendsto (fun t : ℝ => dist (f (x+t • v)) (f x) / |t|) |
| (𝓝[≠] 0) (𝓝 (lineSpeed f v x))} := by |
| let c : E → ℝ → X := fun x t => f (x + t • v) |
| have hc : Continuous c.uncurry := hf.comp (continuous_fst.add (continuous_snd.smul continuous_const)) |
| have hD := measurableSet_of_differentiableAt_with_param ℝ |
| (f := fun x t => signedDistance (c x) t) (continuous_signedDistance hc) |
| have hder := measurable_deriv_with_param (f := fun x t => signedDistance (c x) t) |
| (continuous_signedDistance hc) |
| have hm := (measurable_prodMk_right (y := (0 : ℝ)) hD).inter |
| (measurableSet_eq_fun (hder.comp (measurable_prodMk_right (y := (0 : ℝ)))) (measurable_lineSpeed hf v)) |
| convert hm using 1 |
| ext x |
| simp only [mem_ofPred_eq, mem_inter_iff] |
| constructor |
| · intro ht |
| have hd := (hasDerivAt_signedDistance_iff (c x) (lineSpeed f v x)).mpr (by simpa [c] using ht) |
| exact ⟨hd.differentiableAt, hd.deriv⟩ |
| · rintro ⟨hd, he⟩ |
| rw [← he] |
| simpa [c] using (hasDerivAt_signedDistance_iff (c x) _).mp hd.hasDerivAt |
|
|
| lemma tendsto_add_zero_punctured (s : ℝ) : |
| Tendsto (fun t : ℝ => s+t) (𝓝[≠] 0) (𝓝[≠] s) := by |
| apply tendsto_nhdsWithin_iff.mpr |
| constructor |
| · simpa using ((tendsto_const_nhds : Tendsto (fun _ : ℝ => s) (𝓝 0) (𝓝 s)).add tendsto_id).mono_left nhdsWithin_le_nhds |
| · filter_upwards [self_mem_nhdsWithin] with t ht |
| simpa using ht |
|
|
| theorem ae_hasMetricDirectionalDeriv [FiniteDimensional ℝ E] |
| (μ : Measure E) [μ.IsAddHaarMeasure] (hf : LipschitzWith K f) (v : E) : |
| ∀ᵐ x ∂μ, Tendsto (fun t : ℝ => dist (f (x+t • v)) (f x) / |t|) |
| (𝓝[≠] 0) (𝓝 (lineSpeed f v x)) := by |
| let L : ℝ →L[ℝ] E := ContinuousLinearMap.smulRight (1 : ℝ →L[ℝ] ℝ) v |
| apply ae_mem_of_ae_add_linearMap_mem L.toLinearMap volume μ |
| (measurableSet_hasMetricDirectionalDeriv hf.continuous v) |
| intro p |
| have hcurve := hf.comp ((LipschitzWith.const p).add L.lipschitzWith) |
| filter_upwards [MetricDifferentiationDense.ae_hasMetricDerivAt hcurve] with s hs |
| change Tendsto (fun t : ℝ => dist (f (p+s • v+t • v)) (f (p+s • v)) / |t|) |
| (𝓝[≠] 0) (𝓝 (lineSpeed f v (p+s • v))) |
| rw [lineSpeed_eq_curveSpeed] |
| have hh := hs.comp (tendsto_add_zero_punctured s) |
| simpa only [Function.comp_def, L, MetricDifferentiationDense.HasMetricDerivAt, |
| ContinuousLinearMap.smulRight_apply, one_apply_eq_self, smul_eq_mul, |
| add_smul, add_assoc, add_sub_cancel_left] using hh |
|
|
| end SharpIntegralFillings.MetricDifferentiation |
|
|
| end |
|
|
| end OAI |
|
|
