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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 | 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
namespace WeakMTWTransport
lemma contDiffAt_quadratic_upper {f : ℝ → ℝ} (hf : ContDiffAt ℝ 2 f 0)
(hf0 : f 0=0) (hd0 : deriv f 0=0) :
∃ eps C : ℝ, 0<eps ∧ 0<C ∧ ∀ s ∈ Ioo 0 eps, f s≤C*s^2 := by
obtain ⟨K,U,hU,hLip⟩ := (hf.derivWithin (m := 1) (by norm_num)).exists_lipschitzOnWith
have hnear : ∀ᶠ s : ℝ in 𝓝 0, DifferentiableAt ℝ f s :=
(hf.eventually (by norm_num)).mono (fun _ h => h.differentiableAt (by norm_num))
obtain ⟨eps,heps,hball⟩ := Metric.mem_nhds_iff.mp (inter_mem hU hnear)
refine ⟨eps,(K:ℝ)+1,heps,by positivity,?_⟩
intro s hs
have hsub : Icc 0 s ⊆ Metric.ball (0:ℝ) eps := by
intro r hr
simpa only [Metric.mem_ball,Real.dist_eq,sub_zero,abs_of_nonneg hr.1] using hr.2.trans_lt hs.2
have hzero : (0:ℝ) ∈ U := (hball (Metric.mem_ball_self heps)).1
have hbound : ∀ r ∈ Icc 0 s, ‖deriv f r‖ ≤ (K:ℝ)*s := by
intro r hr
have H := hLip.dist_le_mul r (hball (hsub hr)).1 0 hzero
have H' : ‖deriv f r‖ ≤ (K:ℝ)*r := by
simpa only [hd0,dist_zero_right,Real.norm_eq_abs,abs_of_nonneg hr.1] using H
exact H'.trans (mul_le_mul_of_nonneg_left hr.2 K.coe_nonneg)
have H := Convex.norm_image_sub_le_of_norm_deriv_le
(fun r hr => (hball (hsub hr)).2) hbound (convex_Icc (0:ℝ) s)
(left_mem_Icc.mpr hs.1.le) (right_mem_Icc.mpr hs.1.le)
have H' : |f s|≤(K:ℝ)*s^2 := by
simpa only [hf0,sub_zero,Real.norm_eq_abs,abs_of_pos hs.1,pow_two,mul_assoc] using H
exact (le_abs_self (f s)).trans (H'.trans (by nlinarith [sq_nonneg s]))
end WeakMTWTransport
end
end OAI
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