import OAI.Geometry.IsometricImmersion.Flows.FlowJointSmooth
import Mathlib.Analysis.Calculus.Deriv.MeanValue

namespace OAI

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

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

theorem exists_cap_vertical_derivative_bound {q : Coord → ℝ} {U : Set Coord}
    (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U) :
    ∃ M : ℝ, 0 ≤ M ∧ ∀ p ∈ modelSquare, |coordPartial 1 q p| ≤ M := by
  obtain ⟨M, hM⟩ := modelSquare_isCompact.exists_bound_of_continuousOn
    ((partial_contDiffOn hq hU 1).continuousOn.mono hSU)
  refine ⟨max M 0, le_max_right _ _, ?_⟩
  intro p hp
  have hh : |coordPartial 1 q p| ≤ M := by simpa only [Real.norm_eq_abs] using hM p hp
  exact hh.trans (le_max_left _ _)

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

theorem flow_initial_integral_bound
    (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)
    {M : ℝ} (hM : 0 ≤ M) (hMq : ∀ p ∈ modelSquare, |coordPartial 1 q p| ≤ M)
    {s t : ℝ} (hs : s ∈ Icc (-2 : ℝ) 2) (ht : t ∈ Icc (-2 : ℝ) 2) :
    IntervalIntegrable (fun r => coordPartial 1 q (coordinatePoint r (Y s r))) volume 0 t ∧
      |∫ r in 0..t, coordPartial 1 q (coordinatePoint r (Y s r))| ≤ 2 * M := by
  have hmem (u : ℝ) (hu : u ∈ Icc (-2 : ℝ) 2) :
      coordinatePoint u (Y s u) ∈ modelSquare :=
    coordinatePoint_mem_modelSquare ⟨by linarith [hu.1], by linarith [hu.2]⟩ (hrange s hs u hu)
  have hy : ContinuousOn (Y s) (Icc (-2 : ℝ) 2) :=
    hY.comp (f := fun u : ℝ => (s, u)) (by fun_prop) (fun u hu => ⟨hs, hu⟩)
  have hmap : ContinuousOn (fun u => coordinatePoint u (Y s u)) (Icc (-2 : ℝ) 2) :=
    (continuousOn_id.smul continuousOn_const).add (hy.smul continuousOn_const)
  have hc : ContinuousOn (fun u => coordPartial 1 q (coordinatePoint u (Y s u)))
      (Icc (-2 : ℝ) 2) :=
    (partial_contDiffOn hq hU 1).continuousOn.comp hmap (fun u hu => hSU (hmem u hu))
  have hseg : uIcc (0 : ℝ) t ⊆ Icc (-2 : ℝ) 2 :=
    uIcc_subset_Icc ⟨by norm_num, by norm_num⟩ ht
  refine ⟨(hc.mono hseg).intervalIntegrable, ?_⟩
  have hh := intervalIntegral.norm_integral_le_of_norm_le_const (a := 0) (b := t) (C := M)
    (f := fun u => coordPartial 1 q (coordinatePoint u (Y s u)))
    (fun u hu => by
      rw [Real.norm_eq_abs]
      exact hMq _ (hmem u (hseg (uIoc_subset_uIcc hu))))
  have ht2 : |t| ≤ 2 := abs_le.mpr ht
  calc
    _ ≤ M * |t| := by simpa only [Real.norm_eq_abs, sub_zero] using hh
    _ ≤ M * 2 := mul_le_mul_of_nonneg_left ht2 hM
    _ = 2 * M := mul_comm _ _

theorem cap_flow_initial_deriv_bounds
    (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)
    {M : ℝ} (hM : 0 ≤ M) (hMq : ∀ p ∈ modelSquare, |coordPartial 1 q p| ≤ M)
    {s t : ℝ} (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) :
    Real.exp (-2 * M) ≤ deriv (fun u => Y u t) s ∧
      deriv (fun u => Y u t) s ≤ Real.exp (2 * M) := by
  have hI := (flow_initial_integral_bound hq hU hSU hY hrange hM hMq
    ⟨hs.1.le, hs.2.le⟩ ⟨ht.1.le, ht.2.le⟩).2
  rw [(cap_flow_hasDerivAt_initial hq hU hSU hY hrange hstart hode hs ht).deriv]
  constructor
  · apply Real.exp_monotone
    linarith [(abs_le.mp hI).2]
  · apply Real.exp_monotone
    linarith [(abs_le.mp hI).1]

def triangularFlow (Y : ℝ → ℝ → ℝ) (p : ℝ × ℝ) : ℝ × ℝ := (p.1, Y p.2 p.1)

theorem cap_flow_strictMonoOn_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)
    {t : ℝ} (ht : t ∈ Ioo (-2 : ℝ) 2) :
    StrictMonoOn (fun s => Y s t) (Ioo (-2 : ℝ) 2) := by
  have hc : ContinuousOn (fun s => Y s t) (Ioo (-2 : ℝ) 2) :=
    HasDerivAt.continuousOn
      (fun s hs => cap_flow_hasDerivAt_initial hq hU hSU hY hrange hstart hode hs ht)
  apply strictMonoOn_of_deriv_pos (convex_Ioo (-2 : ℝ) 2) hc
  intro s hs
  apply cap_flow_initial_deriv_pos hq hU hSU hY hrange hstart hode _ ht
  simpa only [interior_Ioo] using hs

theorem triangularFlow_injOn
    (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) :
    InjOn (triangularFlow Y) (pairRectangle 2 (-2) 2) := by
  intro p hp z hz he
  have ht : p.1 = z.1 := by
    simpa only [triangularFlow] using congrArg (fun v : ℝ × ℝ => v.1) he
  have hy : Y p.2 p.1 = Y z.2 z.1 := by
    simpa only [triangularFlow] using congrArg (fun v : ℝ × ℝ => v.2) he
  rw [ht] at hy
  exact Prod.ext ht
    ((cap_flow_strictMonoOn_initial hq hU hSU hY hrange hstart hode hz.1).injOn hp.2 hz.2 hy)

end SmoothLocal.Flow

end

end OAI