openAI-math-dataset / lean /OAI /Analysis /BiholderTransport /Regularity /ContDiffQuadraticUpper.lean
| 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 | |