import Mathlib namespace OAI noncomputable section open Set MeasureTheory Manifold Bundle open scoped ContDiff Manifold ENNReal NNReal Topology open Set Filter open scoped Topology NNReal open Set Filter open scoped Topology open Set Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology open Set open scoped Topology open Set Filter Manifold Bundle ContinuousLinearMap open scoped Topology ContDiff Manifold Bundle open Set Filter ContinuousLinearMap InnerProductSpace open scoped Topology ContDiff open Set Filter ContinuousLinearMap open scoped Topology ContDiff open Set Filter ContinuousLinearMap open scoped Topology ContDiff open Set Filter ContinuousLinearMap open scoped Topology ContDiff open scoped NNReal open Set Filter ContinuousLinearMap open scoped Topology ContDiff open Set Filter ContinuousLinearMap open scoped Topology open MeasureTheory open scoped ContDiff ENNReal open Set Filter Manifold Bundle ContinuousLinearMap MeasureTheory open scoped Topology ContDiff Manifold Bundle ENNReal open Set Filter Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology Manifold open Set Filter Manifold Bundle ContinuousLinearMap open scoped Topology ContDiff Manifold Bundle open Set Filter Manifold Bundle open scoped Topology ContDiff Manifold Bundle open Set Filter Manifold Bundle open scoped Topology ContDiff Manifold Bundle open Set Filter Bundle open scoped Topology Bundle open scoped Topology open Function Manifold Set open Manifold Bundle open scoped Manifold Bundle open Set open Set Filter open scoped Topology ContDiff open Set Filter Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology open Set Filter Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology open Set Filter Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology open Set Filter Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology open Set Filter Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology open Set Filter Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology open Set Filter open scoped ContDiff Topology open Set Filter Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology open Set Filter open scoped ContDiff Topology open Set Filter Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology open Set Filter Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology open Set Filter open scoped ContDiff Topology open Set Filter Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology open Set Filter Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology open Set Filter Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology open Set Filter open scoped ContDiff Topology open Set Filter Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology open Set Filter Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology open Set Filter open scoped ContDiff Topology open Filter Set open scoped Topology open Set Filter Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology open Set Filter Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology open Set Filter Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology open Set Filter Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology open Set Filter Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology open Set Filter Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology open Set Filter Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology open Set Filter Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology open Set Filter Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology open Set Filter Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology open Set Filter Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology open Set Filter Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology open Set Filter Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology open Set Filter Manifold MeasureTheory Bundle open scoped ENNReal ContDiff Topology open Set Filter open scoped Topology open Set Filter open scoped Topology ContDiff 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 end end OAI