openAI-math-dataset / lean /OAI /Analysis /BiholderTransport /Regularity /ContDiffParametricSecant.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 | |
| 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 | |
| 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 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_parametric_secant_lower {E : Type*} [NormedAddCommGroup E] | |
| [NormedSpace ℝ E] {F : E×ℝ → ℝ} {x : E} | |
| (hF : ContDiffAt ℝ 1 F (x,0)) {K : ℝ} | |
| (hK : K < fderiv ℝ F (x,0) (0,1)) : | |
| ∀ᶠ q : E×ℝ in 𝓝 (x,0), 0≤q.2 → K*q.2≤F q-F (q.1,0) := by | |
| have hd : ContinuousAt (fun q : E×ℝ => fderiv ℝ F q (0,1)) (x,0) := | |
| (hF.continuousAt_fderiv (by norm_num)).clm_apply continuousAt_const | |
| have H : ∀ᶠ q : E×ℝ in 𝓝 (x,0), ContDiffAt ℝ 1 F q ∧ K<fderiv ℝ F q (0,1) := | |
| (hF.eventually (by norm_num)).and (hd.preimage_mem_nhds (Ioi_mem_nhds hK)) | |
| obtain ⟨U,hU,V,hV,hUV⟩ := mem_nhds_prod_iff.mp H | |
| obtain ⟨δ,hδ,hδV⟩ := Metric.mem_nhds_iff.mp hV | |
| have hnear : U ×ˢ Metric.ball (0:ℝ) δ ∈ 𝓝 (x,0) := by | |
| rw [nhds_prod_eq] | |
| exact Filter.prod_mem_prod hU (Metric.ball_mem_nhds (0:ℝ) hδ) | |
| filter_upwards [hnear] with q hq ht | |
| have hqt : q.2<δ := (le_abs_self q.2).trans_lt (by simpa only [Metric.mem_ball,Real.dist_eq,sub_zero] using hq.2) | |
| have Hj : ∀ r∈Icc 0 q.2, ContDiffAt ℝ 1 F (q.1,r) ∧ K<fderiv ℝ F (q.1,r) (0,1) := by | |
| intro r hr | |
| exact hUV ⟨hq.1,hδV (by simpa only [Metric.mem_ball,Real.dist_eq,sub_zero,abs_of_nonneg hr.1] using hr.2.trans_lt hqt)⟩ | |
| have hD : ∀ r∈Icc 0 q.2, HasDerivAt (fun r => F (q.1,r)) | |
| (fderiv ℝ F (q.1,r) (0,1)) r := by | |
| intro r hr | |
| exact ((Hj r hr).1.differentiableAt_one.hasFDerivAt).comp_hasDerivAt r | |
| ((hasDerivAt_const r q.1).prodMk (hasDerivAt_id r)) | |
| rcases ht.eq_or_lt with he|ht | |
| · have hqeq : q = (q.1, 0) := by ext <;> simp [←he] | |
| rw [hqeq] | |
| simp | |
| obtain ⟨r,hr,he⟩ := exists_hasDerivAt_eq_slope (fun r => F (q.1,r)) | |
| (fun r => fderiv ℝ F (q.1,r) (0,1)) ht | |
| (fun r hr => (hD r hr).continuousAt.continuousWithinAt) | |
| (fun r hr => hD r ⟨hr.1.le,hr.2.le⟩) | |
| have hk := (Hj r ⟨hr.1.le,hr.2.le⟩).2.le | |
| rw [he,sub_zero] at hk | |
| exact (le_div_iff₀ ht).mp hk | |
| end WeakMTWTransport | |
| end | |
| end OAI | |