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59d1c80 | 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 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 | 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
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