import OAI.Geometry.IsometricImmersion.Flows.FlowParameterContinuity
import Mathlib.Analysis.Calculus.Deriv.Slope

namespace OAI

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

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

theorem hasDerivAt_of_exact_secant {f g : ℝ → ℝ} {s : ℝ}
    (hg : ContinuousAt g s)
    (he : ∀ᶠ u in 𝓝 s, f u - f s = (u - s) * g u) :
    HasDerivAt f (g s) s := by
  rw [hasDerivAt_iff_tendsto_slope]
  have hslope : slope f s =ᶠ[𝓝[≠] s] g := by
    filter_upwards [he.filter_mono nhdsWithin_le_nhds, self_mem_nhdsWithin] with u hu hne
    have hne' : u ≠ s := by simpa only [mem_compl_iff, mem_singleton_iff] using hne
    rw [slope_def_field, hu]
    exact mul_div_cancel_left₀ (g u) (sub_ne_zero.mpr hne')
  exact (hg.mono_left nhdsWithin_le_nhds).congr' hslope.symm

variable {q : Coord → ℝ} {U : Set Coord} {Y : ℝ → ℝ → ℝ}

theorem flow_difference_formula
    (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U)
    (hY : ContinuousOn (uncurry Y) (Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2))
    (hrange : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
      Y s t ∈ Icc (-3 : ℝ) 3)
    (hstart : ∀ s ∈ Icc (-2 : ℝ) 2, Y s 0 = s)
    (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)
    {s0 s1 t : ℝ} (hs0 : s0 ∈ Icc (-2 : ℝ) 2) (hs1 : s1 ∈ Icc (-2 : ℝ) 2)
    (ht : t ∈ Ioo (-2 : ℝ) 2) :
    Y s1 t - Y s0 t = (s1 - s0) *
      Real.exp (-(∫ u in 0..t, flowAverageCoefficient q Y s0 s1 u)) := by
  have hcoeff := flowAverageCoefficient_continuous hq hU hSU hY hrange
  have ha : ContinuousOn (flowAverageCoefficient q Y s0 s1) (Icc (-2 : ℝ) 2) := by
    rw [continuousOn_iff_continuous_domRestrict]
    exact hcoeff.comp
      ((continuous_const (y := (⟨s0, hs0⟩ : CapInterval))).prodMk
        ((continuous_const (y := (⟨s1, hs1⟩ : CapInterval))).prodMk continuous_id))
  have hz (u : ℝ) (hu : u ∈ Ioo (-2 : ℝ) 2) :
      HasDerivAt (fun v => Y s1 v - Y s0 v)
        (-flowAverageCoefficient q Y s0 s1 u * (Y s1 u - Y s0 u)) u := by
    exact (flow_difference_hasDerivWithinAt hq hU hSU ⟨hu.1.le, hu.2.le⟩
      (hrange s0 hs0 u ⟨hu.1.le, hu.2.le⟩) (hrange s1 hs1 u ⟨hu.1.le, hu.2.le⟩)
      (hode s0 hs0 u ⟨hu.1.le, hu.2.le⟩) (hode s1 hs1 u ⟨hu.1.le, hu.2.le⟩)).hasDerivAt
        (Icc_mem_nhds hu.1 hu.2)
  have he := scalar_linear_evolution_formula (show (0 : ℝ) < 2 by norm_num)
    (ha.mono Ioo_subset_Icc_self) hz ht
  simpa only [hstart s0 hs0, hstart s1 hs1] using he

theorem cap_flow_hasDerivAt_initial
    (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U)
    (hY : ContinuousOn (uncurry Y) (Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2))
    (hrange : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
      Y s t ∈ Icc (-3 : ℝ) 3)
    (hstart : ∀ s ∈ Icc (-2 : ℝ) 2, Y s 0 = s)
    (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)
    {s t : ℝ} (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) :
    HasDerivAt (fun u => Y u t)
      (Real.exp (-(∫ r in 0..t, coordPartial 1 q (coordinatePoint r (Y s r))))) s := by
  let G (u : ℝ) := Real.exp (-(∫ r in 0..t, flowAverageCoefficient q Y s u r))
  have hG : ContinuousOn G (Icc (-2 : ℝ) 2) := by
    rw [continuousOn_iff_continuous_domRestrict]
    exact (((flowAverageIntegral_continuous hq hU hSU hY hrange
      ⟨ht.1.le, ht.2.le⟩).comp
        ((continuous_const (y := (⟨s, ⟨hs.1.le, hs.2.le⟩⟩ : CapInterval))).prodMk continuous_id)).neg).rexp
  have he : ∀ᶠ u in 𝓝 s, Y u t - Y s t = (u - s) * G u := by
    filter_upwards [Icc_mem_nhds hs.1 hs.2] with u hu
    exact flow_difference_formula hq hU hSU hY hrange hstart hode ⟨hs.1.le, hs.2.le⟩ hu ht
  have hd := hasDerivAt_of_exact_secant (hG.continuousAt (Icc_mem_nhds hs.1 hs.2)) he
  have hdiag : G s = Real.exp (-(∫ r in 0..t,
      coordPartial 1 q (coordinatePoint r (Y s r)))) := by
    simp only [G, flowAverageCoefficient, verticalAverageDerivative_diagonal]
  rw [hdiag] at hd
  exact hd

theorem cap_flow_initial_deriv_pos
    (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U)
    (hY : ContinuousOn (uncurry Y) (Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2))
    (hrange : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
      Y s t ∈ Icc (-3 : ℝ) 3)
    (hstart : ∀ s ∈ Icc (-2 : ℝ) 2, Y s 0 = s)
    (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)
    {s t : ℝ} (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) :
    0 < deriv (fun u => Y u t) s := by
  rw [(cap_flow_hasDerivAt_initial hq hU hSU hY hrange hstart hode hs ht).deriv]
  exact Real.exp_pos _

theorem exists_cap_flow_first_variation
    (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U)
    (hsmall : ∀ p ∈ modelSquare, |q p| ≤ (1 : ℝ) / 100) :
    ∃ Y : ℝ → ℝ → ℝ,
      ContinuousOn (uncurry Y) (Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2) ∧
      (∀ s ∈ Icc (-2 : ℝ) 2, ContDiffOn ℝ ∞ (Y s) (Icc (-2 : ℝ) 2)) ∧
      (∀ s ∈ Icc (-2 : ℝ) 2, Y s 0 = s) ∧
      (∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
        HasDerivWithinAt (Y s) (-q (coordinatePoint t (Y s t))) (Icc (-2 : ℝ) 2) t) ∧
      (∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
        |Y s t - s| ≤ (1 : ℝ) / 50) ∧
      (∀ s ∈ Ioo (-2 : ℝ) 2, ∀ t ∈ Ioo (-2 : ℝ) 2,
        HasDerivAt (fun u => Y u t)
          (Real.exp (-(∫ r in 0..t, coordPartial 1 q (coordinatePoint r (Y s r))))) s ∧
        0 < deriv (fun u => Y u t) s) := by
  obtain ⟨Y, hY, htime, hstart, hode, hdisp, hdisp', hrange, hLip⟩ :=
    exists_cap_flow hq hU hSU hsmall
  have hrange' (s : ℝ) (hs : s ∈ Icc (-2 : ℝ) 2) (t : ℝ)
      (ht : t ∈ Icc (-2 : ℝ) 2) : Y s t ∈ Icc (-3 : ℝ) 3 :=
    abs_le.mp ((hrange s hs t ht).1.trans (by norm_num))
  refine ⟨Y, hY, htime, hstart, hode, hdisp', ?_⟩
  intro s hs t ht
  exact ⟨cap_flow_hasDerivAt_initial hq hU hSU hY hrange' hstart hode hs ht,
    cap_flow_initial_deriv_pos hq hU hSU hY hrange' hstart hode hs ht⟩

end SmoothLocal.Flow

end

end OAI