| import Mathlib |
| import OAI.Geometry.IntegralFillings.Differentiation.ErrorModulus |
|
|
| 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 Filter Set |
| open scoped Topology NNReal |
|
|
| namespace SharpIntegralFillings |
| open Metric Asymptotics |
|
|
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| lemma seminorm_sub_bound_of_sphere {p q : Seminorm ℝ E} {ε : ℝ} (_hε : 0 ≤ ε) |
| (H : ∀ v : E, ‖v‖ = 1 → |p v-q v| ≤ ε) (v : E) : |
| |p v-q v| ≤ ε*‖v‖ := by |
| by_cases hv : v = 0 |
| · simp [hv] |
| have hv0 : 0 < ‖v‖ := norm_pos_iff.mpr hv |
| let u := ‖v‖⁻¹ • v |
| have hu : ‖u‖ = 1 := by simp [u,norm_smul,hv0.ne'] |
| have he : ‖v‖ • u = v := by simp [u,hv0.ne'] |
| have hpv : p v = ‖v‖ * p u := by |
| calc p v = p (‖v‖ • u) := congrArg p he.symm |
| _ = ‖v‖ * p u := by rw [map_smul_eq_mul, Real.norm_of_nonneg (norm_nonneg v)] |
| have hqv : q v = ‖v‖ * q u := by |
| calc q v = q (‖v‖ • u) := congrArg q he.symm |
| _ = ‖v‖ * q u := by rw [map_smul_eq_mul, Real.norm_of_nonneg (norm_nonneg v)] |
| calc |p v-q v| = ‖v‖ * |p u-q u| := by |
| rw [hpv,hqv,←mul_sub,abs_mul,abs_of_pos hv0] |
| _ ≤ ‖v‖*ε := mul_le_mul_of_nonneg_left (H u hu) hv0.le |
| _ = ε*‖v‖ := mul_comm _ _ |
|
|
| variable [FiniteDimensional ℝ E] |
| lemma eventually_seminorm_bound_of_dense |
| {ι κ : Type*} {l : Filter ι} {p : ι → Seminorm ℝ E} {q : Seminorm ℝ E} |
| {K : ℝ≥0} (hp : ∀ i, LipschitzWith K (p i)) (hq : LipschitzWith K q) |
| {D : κ → E} (hD : DenseRange D) |
| (hc : ∀ j, Tendsto (fun i => p i (D j)) l (𝓝 (q (D j)))) |
| {ε : ℝ} (hε : 0 < ε) : |
| ∀ᶠ i in l, ∀ v : E, |p i v-q v| ≤ ε*‖v‖ := by |
| have hlim := tendsto_of_lipschitz_dense hp hq hD hc |
| have h := uniformly_tendsto_of_lipschitz hp hq (isCompact_sphere (0 : E) 1) |
| (fun v _ => hlim v) hε |
| filter_upwards [h] with i hi v |
| apply seminorm_sub_bound_of_sphere hε.le _ v |
| intro w hw |
| exact (hi w (by simpa only [mem_sphere,dist_zero_right] using hw)).le |
|
|
| variable {X : Type*} [MetricSpace X] |
|
|
| def HasTwoPointMetricDifferential (f : E → X) (p : Seminorm ℝ E) |
| (s : Set E) (x : E) : Prop := |
| (fun q : E × E => dist (f q.1) (f q.2) - p (q.1 - q.2)) =o[𝓝[s ×ˢ s] (x, x)] |
| (fun q : E × E => ‖q.1 - x‖ + ‖q.2 - x‖) |
|
|
| end SharpIntegralFillings |
|
|
| namespace SharpIntegralFillings.MetricDifferentiation |
| open Metric Asymptotics |
|
|
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] |
| {X : Type*} [MetricSpace X] [SeparableSpace X] [Nonempty X] |
| {f : E → X} {K : ℝ≥0} |
|
|
| lemma hasTwoPointMetricDifferential_of_uniformModulus |
| (hf : LipschitzWith K f) {s : Set E} {x : E} |
| (hc : ∀ j : ℕ, ContinuousWithinAt (fun y => metricSeminorm f y (denseSeq E j)) s x) |
| {r : ℕ → ℝ} (hr : ∀ n, 0 < r n) |
| (hu : TendstoUniformlyOn (fun n => metricErrorModulus f (r n)) (fun _ => 0) atTop s) : |
| HasTwoPointMetricDifferential f (metricSeminorm f x) s x := by |
| apply Asymptotics.IsLittleO.of_bound |
| intro ε hε |
| have hp := eventually_seminorm_bound_of_dense (fun y => metricSeminorm_lipschitz hf y) |
| (metricSeminorm_lipschitz hf x) (denseRange_denseSeq E) hc (half_pos hε) |
| obtain ⟨N,hN⟩ := (eventually_atTop.1 ((Metric.tendstoUniformlyOn_iff.1 hu) (ε/2) (half_pos hε))) |
| have hNN (y : E) (hy : y ∈ s) : metricErrorModulus f (r N) y < ε/2 := by |
| have h := hN N le_rfl y hy |
| simpa only [dist_zero_left,Real.norm_eq_abs,abs_of_nonneg (metricErrorModulus_nonneg hf _ _)] using h |
| have hfst : Tendsto (fun q : E × E => q.1) (𝓝[s ×ˢ s] (x,x)) (𝓝[s] x) := |
| continuous_fst.continuousWithinAt.tendsto_nhdsWithin (fun _ hq => hq.1) |
| have hnear : ∀ᶠ q : E × E in 𝓝[s ×ˢ s] (x,x), ‖q.2-q.1‖ < r N := by |
| have ht : Tendsto (fun q : E × E => ‖q.2-q.1‖) (𝓝[s ×ˢ s] (x,x)) (𝓝 0) := by |
| simpa using ((continuous_snd.sub continuous_fst).norm.continuousWithinAt.tendsto (s := s ×ˢ s) (x := (x,x))) |
| exact (tendsto_order.1 ht).2 _ (hr N) |
| filter_upwards [hfst.eventually hp,hnear,self_mem_nhdsWithin] with q hqp hqr hqs |
| have he₁ := metricErrorModulus_bound hf q.1 (q.2-q.1) hqr |
| have he₂ := hqp (q.2-q.1) |
| have hm := hNN q.1 hqs.1 |
| have ht : ‖q.2-q.1‖ ≤ ‖q.1-x‖+‖q.2-x‖ := by |
| calc ‖q.2-q.1‖ = ‖(q.2-x)-(q.1-x)‖ := by congr 1; abel |
| _ ≤ ‖q.2-x‖+‖q.1-x‖ := norm_sub_le _ _ |
| _ = _ := add_comm _ _ |
| have he : |dist (f q.1) (f q.2)-metricSeminorm f x (q.1-q.2)| ≤ |
| ε*(‖q.1-x‖+‖q.2-x‖) := by |
| have hpneg : metricSeminorm f x (q.1-q.2) = metricSeminorm f x (q.2-q.1) := by |
| rw [←neg_sub q.2 q.1,map_neg_eq_map] |
| rw [hpneg,dist_comm] |
| calc |dist (f q.2) (f q.1)-metricSeminorm f x (q.2-q.1)| ≤ |
| |dist (f q.2) (f q.1)-metricSeminorm f q.1 (q.2-q.1)| + |
| |metricSeminorm f q.1 (q.2-q.1)-metricSeminorm f x (q.2-q.1)| := abs_sub_le _ _ _ |
| _ ≤ metricErrorModulus f (r N) q.1 * ‖q.2-q.1‖ + ε/2*‖q.2-q.1‖ := by |
| apply add_le_add _ he₂ |
| simpa only [←add_sub_assoc,add_sub_cancel_left] using he₁ |
| _ ≤ ε*‖q.2-q.1‖ := by nlinarith [norm_nonneg (q.2-q.1)] |
| _ ≤ ε*(‖q.1-x‖+‖q.2-x‖) := mul_le_mul_of_nonneg_left ht hε.le |
| simpa only [Real.norm_eq_abs,abs_of_nonneg (add_nonneg (norm_nonneg _) (norm_nonneg _))] using he |
|
|
| end SharpIntegralFillings.MetricDifferentiation |
|
|
| open Set Filter MeasureTheory TopologicalSpace |
| open scoped Topology ENNReal |
|
|
| end |
|
|
| end OAI |
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|
