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 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