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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
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