| import Mathlib |
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| namespace OAI |
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| noncomputable section |
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| open Set MeasureTheory Manifold Bundle |
| open scoped ContDiff Manifold ENNReal NNReal Topology |
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| open Set Filter |
| open scoped Topology NNReal |
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| open Set Filter |
| open scoped Topology |
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| open Set Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set |
| open scoped Topology |
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| open Set Filter Manifold Bundle ContinuousLinearMap |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter ContinuousLinearMap InnerProductSpace |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
| open scoped NNReal |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology |
| open MeasureTheory |
| open scoped ContDiff ENNReal |
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| open Set Filter Manifold Bundle ContinuousLinearMap MeasureTheory |
| open scoped Topology ContDiff Manifold Bundle ENNReal |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology Manifold |
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| open Set Filter Manifold Bundle ContinuousLinearMap |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter Bundle |
| open scoped Topology Bundle |
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| open scoped Topology |
| open Function Manifold Set |
| open Manifold Bundle |
| open scoped Manifold Bundle |
| open Set |
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| open Set Filter |
| open scoped Topology ContDiff |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter |
| open scoped ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter |
| open scoped ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter |
| open scoped ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter |
| open scoped ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter |
| open scoped ContDiff Topology |
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| open Filter Set |
| open scoped Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter |
| open scoped Topology |
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| open Set Filter |
| open scoped Topology ContDiff |
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| namespace WeakMTWTransport |
| lemma contDiffAt_second_order_remainder |
| {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| {f : E → ℝ} (hf : ContDiffAt ℝ 2 f 0) : |
| (fun h => f h-f 0-fderiv ℝ f 0 h-fderiv ℝ (fderiv ℝ f) 0 h h/2) |
| =o[𝓝 0] (fun h : E => ‖h‖^2) := by |
| let D := fderiv ℝ f 0 |
| let B := fderiv ℝ (fderiv ℝ f) 0 |
| have hs : ∀ v w, B v w=B w v := fun v w => (hf.isSymmSndFDerivAt (by norm_num)).eq v w |
| let R : E → ℝ := fun h => f h-f 0-D h-B h h/2 |
| let R' : E → E →L[ℝ] ℝ := fun h => fderiv ℝ f h-D-B h |
| have hB : HasFDerivAt (fderiv ℝ f) B 0 := |
| (hf.fderiv_right (m := 1) (by norm_num)).differentiableAt (by norm_num) |>.hasFDerivAt |
| obtain ⟨r,hr,hrdiff⟩ := Metric.mem_nhds_iff.mp ((hf.eventually (by norm_num)).mono fun h hh => |
| hh.differentiableAt (by norm_num)) |
| have hder : ∀ h ∈ Metric.ball (0:E) r, HasFDerivWithinAt R (R' h) (Metric.ball (0:E) r) h := by |
| intro h hh |
| have hquad : HasFDerivAt (fun z : E => B z z/2) (B h) h := by |
| have H := (B.hasFDerivAt_of_bilinear (hasFDerivAt_id h) (hasFDerivAt_id h)).const_smul (1/2:ℝ) |
| have heq : (fun z : E => B z z/2) = ((1/2:ℝ) • fun z : E => B z z) := by ext z; simp only [Pi.smul_apply,smul_eq_mul]; ring |
| rw [heq] |
| apply H.congr_fderiv |
| ext v |
| simp only [smul_apply,add_apply,ContinuousLinearMap.precompR_apply, |
| ContinuousLinearMap.precompL_apply,ContinuousLinearMap.compL_apply,ContinuousLinearMap.comp_apply, |
| ContinuousLinearMap.id_apply,smul_eq_mul,id_eq] |
| rw [hs v h] |
| ring |
| exact (((hrdiff hh).hasFDerivAt.sub_const (f 0)).sub D.hasFDerivAt |>.sub hquad).hasFDerivWithinAt |
| have ho : R' =o[𝓝[Metric.ball (0:E) r] (0:E)] fun h => ‖h-(0:E)‖^1 := by |
| simpa only [R',D,B,sub_zero,pow_one] using hB.isLittleO.norm_right.mono nhdsWithin_le_nhds |
| have H := (convex_ball (0:E) r).isLittleO_pow_succ (Metric.mem_ball_self hr) hder ho |
| rw [nhdsWithin_eq_nhds.mpr (Metric.ball_mem_nhds _ hr)] at H |
| simpa only [R,D,B,map_zero,zero_apply,zero_div,sub_self,sub_zero] using H |
| end WeakMTWTransport |
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| end |
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| end OAI |
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