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110f4f3 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 | 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
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