Datasets:

roseking
/

openAI-math-dataset

import OAI.Geometry.IsometricImmersion.Flows.FlowBounds

namespace OAI

noncomputable section
open Set Filter MeasureTheory Function
open scoped ContDiff Topology Interval

namespace SmoothLocal.Flow
open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted

def initialFlowDerivative (Y : ℝ → ℝ → ℝ) (p : ℝ × ℝ) : ℝ :=
  deriv (fun s => Y s p.1) p.2

section ActualFlow
variable {q : Coord → ℝ} {U : Set Coord} {Y : ℝ → ℝ → ℝ}
variable (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U)
variable (hY : ContinuousOn (uncurry Y) (Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2))
variable (hrange : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
  Y s t ∈ Icc (-3 : ℝ) 3)
variable (hstart : ∀ s ∈ Icc (-2 : ℝ) 2, Y s 0 = s)
variable (hode : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
  HasDerivWithinAt (Y s) (-q (coordinatePoint t (Y s t))) (Icc (-2 : ℝ) 2) t)
include hq hU hSU hY hrange hstart hode

theorem flowCoordinateMap_mem_modelSquare {p : ℝ × ℝ}
    (hp : p ∈ pairRectangle 2 (-2) 2) : flowCoordinateMap Y p ∈ modelSquare := by
  have hF := cap_flow_joint_contDiffOn hq hU hSU hY hrange hstart hode
  have hbound : Y p.2 p.1 ∈ Icc (-3 : ℝ) 3 := by
    apply isClosed_Icc.mem_of_tendsto
      (hF.contDiffAt ((pairRectangle_isOpen 2 (-2) 2).mem_nhds hp)).continuousAt
    filter_upwards [(pairRectangle_isOpen 2 (-2) 2).mem_nhds hp] with point hpoint
    exact hrange point.2 ⟨hpoint.2.1.le, hpoint.2.2.le⟩
      point.1 ⟨hpoint.1.1.le, hpoint.1.2.le⟩
  exact coordinatePoint_mem_modelSquare ⟨by linarith [hp.1.1], by linarith [hp.1.2]⟩
    hbound

theorem flowPullback_contDiffOn {C : Coord → ℝ} (hC : ContDiffOn ℝ ∞ C U) :
    ContDiffOn ℝ ∞ (fun p => C (flowCoordinateMap Y p)) (pairRectangle 2 (-2) 2) := by
  have hF := cap_flow_joint_contDiffOn hq hU hSU hY hrange hstart hode
  have hT : ContDiffOn ℝ ∞ (flowCoordinateMap Y) (pairRectangle 2 (-2) 2) :=
    (contDiffOn_fst.smul contDiffOn_const).add (hF.smul contDiffOn_const)
  exact hC.comp hT (fun p hp => hSU
    (flowCoordinateMap_mem_modelSquare hq hU hSU hY hrange hstart hode hp))

theorem initialFlowDerivative_eq_pairPartialY {p : ℝ × ℝ}
    (hp : p ∈ pairRectangle 2 (-2) 2) :
    initialFlowDerivative Y p = pairPartialY (fun v : ℝ × ℝ => Y v.2 v.1) p := by
  have hF := cap_flow_joint_contDiffOn hq hU hSU hY hrange hstart hode
  have hd := ((hF.contDiffAt ((pairRectangle_isOpen 2 (-2) 2).mem_nhds hp)).differentiableAt
    (by simp)).hasFDerivAt
  exact (hd.comp_hasDerivAt p.2 ((hasDerivAt_const p.2 p.1).prodMk (hasDerivAt_id p.2))).deriv

theorem initialFlowDerivative_contDiffOn :
    ContDiffOn ℝ ∞ (initialFlowDerivative Y) (pairRectangle 2 (-2) 2) := by
  have hF := cap_flow_joint_contDiffOn hq hU hSU hY hrange hstart hode
  have hd : ContDiffOn ℝ ∞ (pairPartialY (fun v : ℝ × ℝ => Y v.2 v.1))
      (pairRectangle 2 (-2) 2) :=
    (hF.fderiv_of_isOpen (pairRectangle_isOpen 2 (-2) 2) (by simp)).clm_apply contDiffOn_const
  exact hd.congr (fun p hp => initialFlowDerivative_eq_pairPartialY
    hq hU hSU hY hrange hstart hode hp)

theorem flowPullback_partial_y {C : Coord → ℝ} (hC : ContDiffOn ℝ ∞ C U)
    {p : ℝ × ℝ} (hp : p ∈ pairRectangle 2 (-2) 2) :
    pairPartialY (fun v => C (flowCoordinateMap Y v)) p =
      coordPartial 1 C (flowCoordinateMap Y p) * initialFlowDerivative Y p := by
  have hsY : HasDerivAt (fun s => Y s p.1) (initialFlowDerivative Y p) p.2 :=
    (cap_flow_hasDerivAt_initial hq hU hSU hY hrange hstart hode hp.2 hp.1).differentiableAt.hasDerivAt
  have hsT : HasDerivAt (fun s => flowCoordinateMap Y (p.1, s))
      (initialFlowDerivative Y p • Pi.single 1 (1 : ℝ)) p.2 := by
    convert
      ((hasDerivAt_const p.2 p.1).smul_const (Pi.single 0 (1 : ℝ) : Coord)).add
        (hsY.smul_const (Pi.single 1 (1 : ℝ) : Coord)) using 1 <;>
      first | rfl | simp
  have hmem := hSU (flowCoordinateMap_mem_modelSquare hq hU hSU hY hrange hstart hode hp)
  have hdC := ((hC.contDiffAt (hU.mem_nhds hmem)).differentiableAt (by simp)).hasFDerivAt
  have hchain : HasDerivAt (fun s => C (flowCoordinateMap Y (p.1, s)))
      (coordPartial 1 C (flowCoordinateMap Y p) * initialFlowDerivative Y p) p.2 := by
    convert hdC.comp_hasDerivAt p.2 hsT using 1 <;>
      first | rfl | simp only [map_smul, smul_eq_mul, coordPartial, mul_comm]
  have hB := flowPullback_contDiffOn hq hU hSU hY hrange hstart hode hC
  have hdB : HasFDerivAt (fun v : ℝ × ℝ => C (flowCoordinateMap Y v))
      (fderiv ℝ (fun v : ℝ × ℝ => C (flowCoordinateMap Y v)) p) p :=
    ((hB.contDiffAt ((pairRectangle_isOpen 2 (-2) 2).mem_nhds hp)).differentiableAt
      (by simp)).hasFDerivAt
  have hpair : HasDerivAt (fun s => C (flowCoordinateMap Y (p.1, s)))
      (pairPartialY (fun v => C (flowCoordinateMap Y v)) p) p.2 := by
    convert hdB.comp_hasDerivAt p.2
      ((hasDerivAt_const p.2 p.1).prodMk (hasDerivAt_id p.2)) using 1 <;> rfl
  exact hpair.unique hchain

theorem cap_flow_initial_second_hasDerivAt {s t : ℝ}
    (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) :
    HasDerivAt (fun u => deriv (fun v => Y v t) u)
      (-deriv (fun v => Y v t) s *
        (∫ r in 0..t, coordPartial 1 (coordPartial 1 q) (coordinatePoint r (Y s r)) *
          deriv (fun v => Y v r) s)) s := by
  let B (p : ℝ × ℝ) := coordPartial 1 q (flowCoordinateMap Y p)
  have hB : ContDiffOn ℝ ∞ B (pairRectangle 2 (-2) 2) :=
    flowPullback_contDiffOn hq hU hSU hY hrange hstart hode (partial_contDiffOn hq hU 1)
  have hI := pairPrimitive_hasDerivAt_y (hB.of_le (by simp)) ht hs
  have hdiag : pairPrimitive (pairPartialY B) (t, s) =
      ∫ r in 0..t, coordPartial 1 (coordPartial 1 q) (coordinatePoint r (Y s r)) *
        deriv (fun v => Y v r) s := by
    apply intervalIntegral.integral_congr
    intro r hr
    have hr' : r ∈ Ioo (-2 : ℝ) 2 := by
      have hz : (0 : ℝ) ∈ Ioo (-2 : ℝ) 2 := by norm_num
      exact ⟨(lt_min hz.1 ht.1).trans_le hr.1, hr.2.trans_lt (max_lt hz.2 ht.2)⟩
    exact flowPullback_partial_y hq hU hSU hY hrange hstart hode
      (partial_contDiffOn hq hU 1) ⟨hr', hs⟩
  have hV : HasDerivAt (fun u => Real.exp (-pairPrimitive B (t, u)))
      (Real.exp (-pairPrimitive B (t, s)) * -pairPrimitive (pairPartialY B) (t, s)) s := hI.neg.exp
  have he : (fun u => deriv (fun v => Y v t) u) =ᶠ[𝓝 s]
      (fun u => Real.exp (-pairPrimitive B (t, u))) := by
    filter_upwards [isOpen_Ioo.mem_nhds hs] with u hu
    exact (cap_flow_hasDerivAt_initial hq hU hSU hY hrange hstart hode hu ht).deriv
  have hVs : Real.exp (-pairPrimitive B (t, s)) = deriv (fun v => Y v t) s :=
    (cap_flow_hasDerivAt_initial hq hU hSU hY hrange hstart hode hs ht).deriv.symm
  rw [hdiag, hVs] at hV
  convert hV.congr_of_eventuallyEq he using 1
  first | rfl | ring

theorem cap_flow_initial_second_formula {s t : ℝ}
    (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) :
    deriv (fun u => deriv (fun v => Y v t) u) s =
      -deriv (fun v => Y v t) s *
        (∫ r in 0..t, coordPartial 1 (coordPartial 1 q) (coordinatePoint r (Y s r)) *
          deriv (fun v => Y v r) s) :=
  (cap_flow_initial_second_hasDerivAt hq hU hSU hY hrange hstart hode hs ht).deriv

theorem flow_initial_second_integral_bound {M : ℝ} (hM : 0 ≤ M)
    (hMq : ∀ p ∈ modelSquare, |coordPartial 1 q p| ≤ M)
    (hMqq : ∀ p ∈ modelSquare, |coordPartial 1 (coordPartial 1 q) p| ≤ M)
    {s t : ℝ} (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) :
    IntervalIntegrable (fun r => coordPartial 1 (coordPartial 1 q) (coordinatePoint r (Y s r)) *
      deriv (fun v => Y v r) s) volume 0 t ∧
    |∫ r in 0..t, coordPartial 1 (coordPartial 1 q) (coordinatePoint r (Y s r)) *
      deriv (fun v => Y v r) s| ≤ 2 * M * Real.exp (2 * M) := by
  let C (p : ℝ × ℝ) := coordPartial 1 (coordPartial 1 q) (flowCoordinateMap Y p) * initialFlowDerivative Y p
  have hC : ContDiffOn ℝ ∞ C (pairRectangle 2 (-2) 2) :=
    (flowPullback_contDiffOn hq hU hSU hY hrange hstart hode
      (partial_contDiffOn (partial_contDiffOn hq hU 1) hU 1)).mul
      (initialFlowDerivative_contDiffOn hq hU hSU hY hrange hstart hode)
  have hc : ContinuousOn (fun r => C (r, s)) (Ioo (-2 : ℝ) 2) :=
    hC.continuousOn.comp (f := fun r => (r, s)) (by fun_prop) (fun r hr => ⟨hr, hs⟩)
  have hseg : uIcc (0 : ℝ) t ⊆ Ioo (-2 : ℝ) 2 := by
    intro r hr
    have hz : (0 : ℝ) ∈ Ioo (-2 : ℝ) 2 := by norm_num
    exact ⟨(lt_min hz.1 ht.1).trans_le hr.1, hr.2.trans_lt (max_lt hz.2 ht.2)⟩
  refine ⟨(hc.mono hseg).intervalIntegrable, ?_⟩
  have hbound (r : ℝ) (hr : r ∈ uIcc (0 : ℝ) t) :
      ‖coordPartial 1 (coordPartial 1 q) (coordinatePoint r (Y s r)) * deriv (fun v => Y v r) s‖ ≤
        M * Real.exp (2 * M) := by
    have hr' := hseg hr
    have hys := cap_flow_initial_deriv_bounds hq hU hSU hY hrange hstart hode hM hMq hs hr'
    have hpos := cap_flow_initial_deriv_pos hq hU hSU hY hrange hstart hode hs hr'
    rw [Real.norm_eq_abs, abs_mul, abs_of_pos hpos]
    exact mul_le_mul (hMqq _ (flowCoordinateMap_mem_modelSquare (p := (r, s)) hq hU hSU hY hrange hstart hode
      ⟨hr', hs⟩)) hys.2 hpos.le hM
  have hi := intervalIntegral.norm_integral_le_of_norm_le_const
    (a := 0) (b := t) (C := M * Real.exp (2 * M))
    (fun r hr => hbound r (uIoc_subset_uIcc hr))
  have ht2 : |t| ≤ 2 := (abs_lt.mpr ht).le
  calc
    _ ≤ (M * Real.exp (2 * M)) * |t| := by simpa only [Real.norm_eq_abs, sub_zero] using hi
    _ ≤ (M * Real.exp (2 * M)) * 2 := mul_le_mul_of_nonneg_left ht2
      (mul_nonneg hM (Real.exp_pos _).le)
    _ = 2 * M * Real.exp (2 * M) := by ring

theorem cap_flow_initial_second_bound {M : ℝ} (hM : 0 ≤ M)
    (hMq : ∀ p ∈ modelSquare, |coordPartial 1 q p| ≤ M)
    (hMqq : ∀ p ∈ modelSquare, |coordPartial 1 (coordPartial 1 q) p| ≤ M)
    {s t : ℝ} (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) :
    |deriv (fun u => deriv (fun v => Y v t) u) s| ≤ 2 * M * Real.exp (4 * M) := by
  rw [cap_flow_initial_second_formula hq hU hSU hY hrange hstart hode hs ht, abs_mul,
    abs_neg, abs_of_pos (cap_flow_initial_deriv_pos hq hU hSU hY hrange hstart hode hs ht)]
  calc
    _ ≤ Real.exp (2 * M) * (2 * M * Real.exp (2 * M)) :=
      mul_le_mul (cap_flow_initial_deriv_bounds hq hU hSU hY hrange hstart hode hM hMq hs ht).2
        (flow_initial_second_integral_bound hq hU hSU hY hrange hstart hode hM hMq hMqq hs ht).2
        (abs_nonneg _) (Real.exp_pos _).le
    _ = (2 * M) * (Real.exp (2 * M) * Real.exp (2 * M)) := by ring
    _ = 2 * M * Real.exp (4 * M) := by
      rw [← Real.exp_add]
      congr 2
      ring

end ActualFlow
end SmoothLocal.Flow

end

end OAI

	import OAI.Geometry.IsometricImmersion.Flows.FlowBounds

	namespace OAI

	noncomputable section
	open Set Filter MeasureTheory Function
	open scoped ContDiff Topology Interval

	namespace SmoothLocal.Flow
	open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted

	def initialFlowDerivative (Y : ℝ → ℝ → ℝ) (p : ℝ × ℝ) : ℝ :=
	deriv (fun s => Y s p.1) p.2

	section ActualFlow
	variable {q : Coord → ℝ} {U : Set Coord} {Y : ℝ → ℝ → ℝ}
	variable (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U)
	variable (hY : ContinuousOn (uncurry Y) (Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2))
	variable (hrange : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
	Y s t ∈ Icc (-3 : ℝ) 3)
	variable (hstart : ∀ s ∈ Icc (-2 : ℝ) 2, Y s 0 = s)
	variable (hode : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
	HasDerivWithinAt (Y s) (-q (coordinatePoint t (Y s t))) (Icc (-2 : ℝ) 2) t)
	include hq hU hSU hY hrange hstart hode

	theorem flowCoordinateMap_mem_modelSquare {p : ℝ × ℝ}
	(hp : p ∈ pairRectangle 2 (-2) 2) : flowCoordinateMap Y p ∈ modelSquare := by
	have hF := cap_flow_joint_contDiffOn hq hU hSU hY hrange hstart hode
	have hbound : Y p.2 p.1 ∈ Icc (-3 : ℝ) 3 := by
	apply isClosed_Icc.mem_of_tendsto
	(hF.contDiffAt ((pairRectangle_isOpen 2 (-2) 2).mem_nhds hp)).continuousAt
	filter_upwards [(pairRectangle_isOpen 2 (-2) 2).mem_nhds hp] with point hpoint
	exact hrange point.2 ⟨hpoint.2.1.le, hpoint.2.2.le⟩
	point.1 ⟨hpoint.1.1.le, hpoint.1.2.le⟩
	exact coordinatePoint_mem_modelSquare ⟨by linarith [hp.1.1], by linarith [hp.1.2]⟩
	hbound

	theorem flowPullback_contDiffOn {C : Coord → ℝ} (hC : ContDiffOn ℝ ∞ C U) :
	ContDiffOn ℝ ∞ (fun p => C (flowCoordinateMap Y p)) (pairRectangle 2 (-2) 2) := by
	have hF := cap_flow_joint_contDiffOn hq hU hSU hY hrange hstart hode
	have hT : ContDiffOn ℝ ∞ (flowCoordinateMap Y) (pairRectangle 2 (-2) 2) :=
	(contDiffOn_fst.smul contDiffOn_const).add (hF.smul contDiffOn_const)
	exact hC.comp hT (fun p hp => hSU
	(flowCoordinateMap_mem_modelSquare hq hU hSU hY hrange hstart hode hp))

	theorem initialFlowDerivative_eq_pairPartialY {p : ℝ × ℝ}
	(hp : p ∈ pairRectangle 2 (-2) 2) :
	initialFlowDerivative Y p = pairPartialY (fun v : ℝ × ℝ => Y v.2 v.1) p := by
	have hF := cap_flow_joint_contDiffOn hq hU hSU hY hrange hstart hode
	have hd := ((hF.contDiffAt ((pairRectangle_isOpen 2 (-2) 2).mem_nhds hp)).differentiableAt
	(by simp)).hasFDerivAt
	exact (hd.comp_hasDerivAt p.2 ((hasDerivAt_const p.2 p.1).prodMk (hasDerivAt_id p.2))).deriv

	theorem initialFlowDerivative_contDiffOn :
	ContDiffOn ℝ ∞ (initialFlowDerivative Y) (pairRectangle 2 (-2) 2) := by
	have hF := cap_flow_joint_contDiffOn hq hU hSU hY hrange hstart hode
	have hd : ContDiffOn ℝ ∞ (pairPartialY (fun v : ℝ × ℝ => Y v.2 v.1))
	(pairRectangle 2 (-2) 2) :=
	(hF.fderiv_of_isOpen (pairRectangle_isOpen 2 (-2) 2) (by simp)).clm_apply contDiffOn_const
	exact hd.congr (fun p hp => initialFlowDerivative_eq_pairPartialY
	hq hU hSU hY hrange hstart hode hp)

	theorem flowPullback_partial_y {C : Coord → ℝ} (hC : ContDiffOn ℝ ∞ C U)
	{p : ℝ × ℝ} (hp : p ∈ pairRectangle 2 (-2) 2) :
	pairPartialY (fun v => C (flowCoordinateMap Y v)) p =
	coordPartial 1 C (flowCoordinateMap Y p) * initialFlowDerivative Y p := by
	have hsY : HasDerivAt (fun s => Y s p.1) (initialFlowDerivative Y p) p.2 :=
	(cap_flow_hasDerivAt_initial hq hU hSU hY hrange hstart hode hp.2 hp.1).differentiableAt.hasDerivAt
	have hsT : HasDerivAt (fun s => flowCoordinateMap Y (p.1, s))
	(initialFlowDerivative Y p • Pi.single 1 (1 : ℝ)) p.2 := by
	convert
	((hasDerivAt_const p.2 p.1).smul_const (Pi.single 0 (1 : ℝ) : Coord)).add
	(hsY.smul_const (Pi.single 1 (1 : ℝ) : Coord)) using 1 <;>
	first \| rfl \| simp
	have hmem := hSU (flowCoordinateMap_mem_modelSquare hq hU hSU hY hrange hstart hode hp)
	have hdC := ((hC.contDiffAt (hU.mem_nhds hmem)).differentiableAt (by simp)).hasFDerivAt
	have hchain : HasDerivAt (fun s => C (flowCoordinateMap Y (p.1, s)))
	(coordPartial 1 C (flowCoordinateMap Y p) * initialFlowDerivative Y p) p.2 := by
	convert hdC.comp_hasDerivAt p.2 hsT using 1 <;>
	first \| rfl \| simp only [map_smul, smul_eq_mul, coordPartial, mul_comm]
	have hB := flowPullback_contDiffOn hq hU hSU hY hrange hstart hode hC
	have hdB : HasFDerivAt (fun v : ℝ × ℝ => C (flowCoordinateMap Y v))
	(fderiv ℝ (fun v : ℝ × ℝ => C (flowCoordinateMap Y v)) p) p :=
	((hB.contDiffAt ((pairRectangle_isOpen 2 (-2) 2).mem_nhds hp)).differentiableAt
	(by simp)).hasFDerivAt
	have hpair : HasDerivAt (fun s => C (flowCoordinateMap Y (p.1, s)))
	(pairPartialY (fun v => C (flowCoordinateMap Y v)) p) p.2 := by
	convert hdB.comp_hasDerivAt p.2
	((hasDerivAt_const p.2 p.1).prodMk (hasDerivAt_id p.2)) using 1 <;> rfl
	exact hpair.unique hchain

	theorem cap_flow_initial_second_hasDerivAt {s t : ℝ}
	(hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) :
	HasDerivAt (fun u => deriv (fun v => Y v t) u)
	(-deriv (fun v => Y v t) s *
	(∫ r in 0..t, coordPartial 1 (coordPartial 1 q) (coordinatePoint r (Y s r)) *
	deriv (fun v => Y v r) s)) s := by
	let B (p : ℝ × ℝ) := coordPartial 1 q (flowCoordinateMap Y p)
	have hB : ContDiffOn ℝ ∞ B (pairRectangle 2 (-2) 2) :=
	flowPullback_contDiffOn hq hU hSU hY hrange hstart hode (partial_contDiffOn hq hU 1)
	have hI := pairPrimitive_hasDerivAt_y (hB.of_le (by simp)) ht hs
	have hdiag : pairPrimitive (pairPartialY B) (t, s) =
	∫ r in 0..t, coordPartial 1 (coordPartial 1 q) (coordinatePoint r (Y s r)) *
	deriv (fun v => Y v r) s := by
	apply intervalIntegral.integral_congr
	intro r hr
	have hr' : r ∈ Ioo (-2 : ℝ) 2 := by
	have hz : (0 : ℝ) ∈ Ioo (-2 : ℝ) 2 := by norm_num
	exact ⟨(lt_min hz.1 ht.1).trans_le hr.1, hr.2.trans_lt (max_lt hz.2 ht.2)⟩
	exact flowPullback_partial_y hq hU hSU hY hrange hstart hode
	(partial_contDiffOn hq hU 1) ⟨hr', hs⟩
	have hV : HasDerivAt (fun u => Real.exp (-pairPrimitive B (t, u)))
	(Real.exp (-pairPrimitive B (t, s)) * -pairPrimitive (pairPartialY B) (t, s)) s := hI.neg.exp
	have he : (fun u => deriv (fun v => Y v t) u) =ᶠ[𝓝 s]
	(fun u => Real.exp (-pairPrimitive B (t, u))) := by
	filter_upwards [isOpen_Ioo.mem_nhds hs] with u hu
	exact (cap_flow_hasDerivAt_initial hq hU hSU hY hrange hstart hode hu ht).deriv
	have hVs : Real.exp (-pairPrimitive B (t, s)) = deriv (fun v => Y v t) s :=
	(cap_flow_hasDerivAt_initial hq hU hSU hY hrange hstart hode hs ht).deriv.symm
	rw [hdiag, hVs] at hV
	convert hV.congr_of_eventuallyEq he using 1
	first \| rfl \| ring

	theorem cap_flow_initial_second_formula {s t : ℝ}
	(hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) :
	deriv (fun u => deriv (fun v => Y v t) u) s =
	-deriv (fun v => Y v t) s *
	(∫ r in 0..t, coordPartial 1 (coordPartial 1 q) (coordinatePoint r (Y s r)) *
	deriv (fun v => Y v r) s) :=
	(cap_flow_initial_second_hasDerivAt hq hU hSU hY hrange hstart hode hs ht).deriv

	theorem flow_initial_second_integral_bound {M : ℝ} (hM : 0 ≤ M)
	(hMq : ∀ p ∈ modelSquare, \|coordPartial 1 q p\| ≤ M)
	(hMqq : ∀ p ∈ modelSquare, \|coordPartial 1 (coordPartial 1 q) p\| ≤ M)
	{s t : ℝ} (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) :
	IntervalIntegrable (fun r => coordPartial 1 (coordPartial 1 q) (coordinatePoint r (Y s r)) *
	deriv (fun v => Y v r) s) volume 0 t ∧
	\|∫ r in 0..t, coordPartial 1 (coordPartial 1 q) (coordinatePoint r (Y s r)) *
	deriv (fun v => Y v r) s\| ≤ 2 * M * Real.exp (2 * M) := by
	let C (p : ℝ × ℝ) := coordPartial 1 (coordPartial 1 q) (flowCoordinateMap Y p) * initialFlowDerivative Y p
	have hC : ContDiffOn ℝ ∞ C (pairRectangle 2 (-2) 2) :=
	(flowPullback_contDiffOn hq hU hSU hY hrange hstart hode
	(partial_contDiffOn (partial_contDiffOn hq hU 1) hU 1)).mul
	(initialFlowDerivative_contDiffOn hq hU hSU hY hrange hstart hode)
	have hc : ContinuousOn (fun r => C (r, s)) (Ioo (-2 : ℝ) 2) :=
	hC.continuousOn.comp (f := fun r => (r, s)) (by fun_prop) (fun r hr => ⟨hr, hs⟩)
	have hseg : uIcc (0 : ℝ) t ⊆ Ioo (-2 : ℝ) 2 := by
	intro r hr
	have hz : (0 : ℝ) ∈ Ioo (-2 : ℝ) 2 := by norm_num
	exact ⟨(lt_min hz.1 ht.1).trans_le hr.1, hr.2.trans_lt (max_lt hz.2 ht.2)⟩
	refine ⟨(hc.mono hseg).intervalIntegrable, ?_⟩
	have hbound (r : ℝ) (hr : r ∈ uIcc (0 : ℝ) t) :
	‖coordPartial 1 (coordPartial 1 q) (coordinatePoint r (Y s r)) * deriv (fun v => Y v r) s‖ ≤
	M * Real.exp (2 * M) := by
	have hr' := hseg hr
	have hys := cap_flow_initial_deriv_bounds hq hU hSU hY hrange hstart hode hM hMq hs hr'
	have hpos := cap_flow_initial_deriv_pos hq hU hSU hY hrange hstart hode hs hr'
	rw [Real.norm_eq_abs, abs_mul, abs_of_pos hpos]
	exact mul_le_mul (hMqq _ (flowCoordinateMap_mem_modelSquare (p := (r, s)) hq hU hSU hY hrange hstart hode
	⟨hr', hs⟩)) hys.2 hpos.le hM
	have hi := intervalIntegral.norm_integral_le_of_norm_le_const
	(a := 0) (b := t) (C := M * Real.exp (2 * M))
	(fun r hr => hbound r (uIoc_subset_uIcc hr))
	have ht2 : \|t\| ≤ 2 := (abs_lt.mpr ht).le
	calc
	_ ≤ (M * Real.exp (2 * M)) * \|t\| := by simpa only [Real.norm_eq_abs, sub_zero] using hi
	_ ≤ (M * Real.exp (2 * M)) * 2 := mul_le_mul_of_nonneg_left ht2
	(mul_nonneg hM (Real.exp_pos _).le)
	_ = 2 * M * Real.exp (2 * M) := by ring

	theorem cap_flow_initial_second_bound {M : ℝ} (hM : 0 ≤ M)
	(hMq : ∀ p ∈ modelSquare, \|coordPartial 1 q p\| ≤ M)
	(hMqq : ∀ p ∈ modelSquare, \|coordPartial 1 (coordPartial 1 q) p\| ≤ M)
	{s t : ℝ} (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) :
	\|deriv (fun u => deriv (fun v => Y v t) u) s\| ≤ 2 * M * Real.exp (4 * M) := by
	rw [cap_flow_initial_second_formula hq hU hSU hY hrange hstart hode hs ht, abs_mul,
	abs_neg, abs_of_pos (cap_flow_initial_deriv_pos hq hU hSU hY hrange hstart hode hs ht)]
	calc
	_ ≤ Real.exp (2 * M) * (2 * M * Real.exp (2 * M)) :=
	mul_le_mul (cap_flow_initial_deriv_bounds hq hU hSU hY hrange hstart hode hM hMq hs ht).2
	(flow_initial_second_integral_bound hq hU hSU hY hrange hstart hode hM hMq hMqq hs ht).2
	(abs_nonneg _) (Real.exp_pos _).le
	_ = (2 * M) * (Real.exp (2 * M) * Real.exp (2 * M)) := by ring
	_ = 2 * M * Real.exp (4 * M) := by
	rw [← Real.exp_add]
	congr 2
	ring

	end ActualFlow
	end SmoothLocal.Flow

	end

	end OAI