import OAI.Geometry.IsometricImmersion.Flows.FlowParameter

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 continuous_intervalIntegral_of_joint
    {X : Type*} [TopologicalSpace X] [FirstCountableTopology X] [LocallyCompactSpace X]
    {a b : ℝ} (hab : a ≤ b) {f : X → ℝ → ℝ}
    (hf : Continuous (fun p : X × Icc a b => f p.1 p.2)) :
    Continuous (fun x => ∫ u in a..b, f x u) := by
  let g : X → ℝ → ℝ := fun x u => f x (projIcc a b hab u)
  have hg : Continuous (uncurry g) :=
    hf.comp (continuous_fst.prodMk ((continuous_projIcc (h := hab)).comp continuous_snd))
  have hc : Continuous (fun x => ∫ u in Icc a b, g x u) :=
    continuous_parametric_integral_of_continuous hg isCompact_Icc
  apply hc.congr
  intro x
  calc
    (∫ u in Icc a b, g x u) = ∫ u in a..b, g x u := by
      rw [intervalIntegral.integral_of_le hab, integral_Icc_eq_integral_Ioc]
    _ = ∫ u in a..b, f x u := by
      apply intervalIntegral.integral_congr
      intro u hu
      have hu' : u ∈ Icc a b := by simpa only [uIcc_of_le hab] using hu
      simp only [g, projIcc_of_mem hab hu']

abbrev CapInterval := Icc (-2 : ℝ) 2
abbrev VerticalInterval := Icc (-3 : ℝ) 3

theorem verticalAverageDerivative_continuous {q : Coord → ℝ} {U : Set Coord}
    (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U) :
    Continuous (fun p : CapInterval × (VerticalInterval × VerticalInterval) =>
      verticalAverageDerivative q p.1 p.2.1 p.2.2) := by
  apply continuous_intervalIntegral_of_joint (show (0 : ℝ) ≤ 1 by norm_num)
  have hmap : Continuous
      (fun p : (CapInterval × (VerticalInterval × VerticalInterval)) × Icc (0 : ℝ) 1 =>
        coordinatePoint p.1.1 ((p.1.2.1 : ℝ) + p.2 * ((p.1.2.2 : ℝ) - p.1.2.1))) := by
    unfold coordinatePoint
    fun_prop
  exact (partial_contDiffOn hq hU 1).continuousOn.comp_continuous hmap
    (fun p => hSU (verticalSegment_mem_modelSquare p.1.1.2 p.1.2.1.2 p.1.2.2.2 p.2.2))

def flowAverageCoefficient (q : Coord → ℝ) (Y : ℝ → ℝ → ℝ) (s0 s1 t : ℝ) : ℝ :=
  verticalAverageDerivative q t (Y s0 t) (Y s1 t)

theorem flowAverageCoefficient_continuous {q : Coord → ℝ} {U : Set Coord}
    (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U)
    {Y : ℝ → ℝ → ℝ}
    (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) :
    Continuous (fun p : CapInterval × (CapInterval × CapInterval) =>
      flowAverageCoefficient q Y p.1 p.2.1 p.2.2) := by
  have hy0 : Continuous (fun p : CapInterval × (CapInterval × CapInterval) =>
      Y p.1 p.2.2) :=
    hY.comp_continuous
      (f := fun p : CapInterval × (CapInterval × CapInterval) => ((p.1 : ℝ), (p.2.2 : ℝ)))
      (by fun_prop) (fun p => ⟨p.1.2, p.2.2.2⟩)
  have hy1 : Continuous (fun p : CapInterval × (CapInterval × CapInterval) =>
      Y p.2.1 p.2.2) :=
    hY.comp_continuous
      (f := fun p : CapInterval × (CapInterval × CapInterval) => ((p.2.1 : ℝ), (p.2.2 : ℝ)))
      (by fun_prop) (fun p => ⟨p.2.1.2, p.2.2.2⟩)
  have hmap : Continuous
      (fun p : CapInterval × (CapInterval × CapInterval) =>
        (p.2.2, ((⟨Y p.1 p.2.2, hrange _ p.1.2 _ p.2.2.2⟩,
          ⟨Y p.2.1 p.2.2, hrange _ p.2.1.2 _ p.2.2.2⟩) :
          VerticalInterval × VerticalInterval))) :=
    (continuous_snd.snd).prodMk
      ((hy0.subtype_mk _).prodMk (hy1.subtype_mk _))
  exact (verticalAverageDerivative_continuous hq hU hSU).comp hmap

theorem flowAverageIntegral_continuous {q : Coord → ℝ} {U : Set Coord}
    (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U)
    {Y : ℝ → ℝ → ℝ}
    (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) {t : ℝ} (ht : t ∈ Icc (-2 : ℝ) 2) :
    Continuous (fun p : CapInterval × CapInterval =>
      ∫ u in 0..t, flowAverageCoefficient q Y p.1 p.2 u) := by
  have hcoeff := flowAverageCoefficient_continuous hq hU hSU hY hrange
  have haux (a b : ℝ) (hab : a ≤ b) (ha : -2 ≤ a) (hb : b ≤ 2) :
      Continuous (fun p : CapInterval × CapInterval =>
        ∫ u in a..b, flowAverageCoefficient q Y p.1 p.2 u) := by
    apply continuous_intervalIntegral_of_joint hab
    have hmap : Continuous
        (fun p : (CapInterval × CapInterval) × Icc a b =>
          (p.1.1, (p.1.2, (⟨p.2, ⟨ha.trans p.2.2.1, p.2.2.2.trans hb⟩⟩ : CapInterval)))) := by
      fun_prop
    exact hcoeff.comp hmap
  by_cases ht0 : 0 ≤ t
  · exact haux 0 t ht0 (by norm_num) ht.2
  · have hc := (haux t 0 (le_of_not_ge ht0) ht.1 (by norm_num)).neg
    apply hc.congr
    intro p
    exact (intervalIntegral.integral_symm _ _).symm

end SmoothLocal.Flow

end

end OAI