lean/OAI/Geometry/IsometricImmersion/Calculus/FinitePrimitive.lean

import OAI.Geometry.IsometricImmersion.Energy.WeightPrimitive

namespace OAI

noncomputable section
open Set Filter MeasureTheory
open scoped ContDiff Topology Interval

namespace SmoothLocal.Flow
open SmoothLocal.ODE

def pairRectangle (R a b : ℝ) : Set (ℝ × ℝ) := Ioo (-R) R ×ˢ Ioo a b

def pairPrimitive (B : ℝ × ℝ → ℝ) (p : ℝ × ℝ) : ℝ :=
  ∫ t in 0..p.1, B (t, p.2)

def pairPartialY (B : ℝ × ℝ → ℝ) (p : ℝ × ℝ) : ℝ :=
  fderiv ℝ B p (0, 1)

def pairCompactFamily (B : ℝ × ℝ → ℝ) (r : ℝ) (y : ℝ) : IntervalFunctions r :=
  ContinuousMap.mkD (fun x : Icc (-r) r => B (x, y)) 0

theorem pairRectangle_isOpen (R a b : ℝ) : IsOpen (pairRectangle R a b) :=
  isOpen_Ioo.prod isOpen_Ioo

theorem exists_compact_time_interval {R x : ℝ} (hx : x ∈ Ioo (-R) R) :
    ∃ r : ℝ, 0 < r ∧ r < R ∧ x ∈ Ioo (-r) r := by
  have hxa : |x| < R := abs_lt.mpr hx
  refine ⟨(|x| + R) / 2, ?_, ?_, ?_⟩
  · nlinarith [abs_nonneg x]
  · linarith
  · apply abs_lt.mp
    linarith

theorem pairCompactFamily_joint_continuous {B : ℝ × ℝ → ℝ} {R a b r : ℝ}
    (hB : ContinuousOn B (pairRectangle R a b)) (hrR : r < R) :
    ContinuousOn (fun p : ℝ × Icc (-r) r => B (p.2, p.1))
      (Ioo a b ×ˢ (univ : Set (Icc (-r) r))) := by
  exact hB.comp (by fun_prop)
    (fun p hp => ⟨⟨(neg_lt_neg hrR).trans_le p.2.2.1,
      p.2.2.2.trans_lt hrR⟩, hp.1⟩)

theorem pairCompactFamily_continuousOn {B : ℝ × ℝ → ℝ} {R a b r : ℝ}
    (hB : ContinuousOn B (pairRectangle R a b)) (hrR : r < R) :
    ContinuousOn (pairCompactFamily B r) (Ioo a b) :=
  ContinuousMap.continuousOn_mkD_of_uncurry _ 0 (pairCompactFamily_joint_continuous hB hrR)

theorem pairCompactFamily_apply {B : ℝ × ℝ → ℝ} {R a b r y : ℝ}
    (hB : ContinuousOn B (pairRectangle R a b)) (hrR : r < R)
    (hy : y ∈ Ioo a b) (x : Icc (-r) r) : pairCompactFamily B r y x = B (x, y) := by
  have hc : Continuous (fun t : Icc (-r) r => B (t, y)) :=
    (pairCompactFamily_joint_continuous hB hrR).comp_continuous
      (continuous_const.prodMk continuous_id) (fun _ => ⟨hy, mem_univ _⟩)
  unfold pairCompactFamily
  rw [ContinuousMap.mkD_of_continuous hc]
  rfl

theorem pairPrimitive_eq_clamped {B : ℝ × ℝ → ℝ} {R a b r x y : ℝ}
    (hB : ContinuousOn B (pairRectangle R a b)) (hr : 0 < r) (hrR : r < R)
    (hx : x ∈ Ioo (-r) r) (hy : y ∈ Ioo a b) :
    pairPrimitive B (x, y) =
      intervalExtension r hr (primitiveCLM r hr (pairCompactFamily B r y)) x := by
  rw [intervalExtension_apply_of_mem r hr _ ⟨hx.1.le, hx.2.le⟩, primitiveCLM_apply]
  unfold pairPrimitive
  apply intervalIntegral.integral_congr
  intro t ht
  have ht' : t ∈ Icc (-r) r :=
    uIcc_subset_Icc ⟨by linarith, hr.le⟩ ⟨hx.1.le, hx.2.le⟩ ht
  rw [intervalExtension_apply_of_mem r hr _ ht', pairCompactFamily_apply hB hrR hy]

theorem pairPrimitive_continuousOn {B : ℝ × ℝ → ℝ} {R a b : ℝ}
    (hB : ContinuousOn B (pairRectangle R a b)) :
    ContinuousOn (pairPrimitive B) (pairRectangle R a b) := by
  intro p hp
  obtain ⟨r, hr, hrR, hpr⟩ := exists_compact_time_interval hp.1
  have hF := (primitiveCLM r hr).continuous.comp_continuousOn
    (pairCompactFamily_continuousOn hB hrR)
  have hc := intervalExtension_family_continuousOn r hr hF
  have hn : (univ : Set ℝ) ×ˢ Ioo a b ∈ 𝓝 p :=
    (isOpen_univ.prod isOpen_Ioo).mem_nhds ⟨mem_univ _, hp.2⟩
  have he : pairPrimitive B =ᶠ[𝓝 p]
      (fun q => intervalExtension r hr (primitiveCLM r hr (pairCompactFamily B r q.2)) q.1) := by
    filter_upwards [(pairRectangle_isOpen r a b).mem_nhds ⟨hpr, hp.2⟩] with q hq
    exact pairPrimitive_eq_clamped hB hr hrR hq.1 hq.2
  exact ((hc.continuousAt hn).congr_of_eventuallyEq he).continuousWithinAt

theorem pairPartialY_contDiffOn {B : ℝ × ℝ → ℝ} {R a b : ℝ} (n : ℕ)
    (hB : ContDiffOn ℝ (n + 1 : ℕ) B (pairRectangle R a b)) :
    ContDiffOn ℝ n (pairPartialY B) (pairRectangle R a b) := by
  have hd : ContDiffOn ℝ n (fderiv ℝ B) (pairRectangle R a b) :=
    hB.fderiv_of_isOpen (pairRectangle_isOpen R a b) (by simp)
  exact hd.clm_apply contDiffOn_const

theorem pairCompactFamily_hasDerivAt {B : ℝ × ℝ → ℝ} {R a b r y : ℝ}
    (hB : ContDiffOn ℝ 1 B (pairRectangle R a b)) (hrR : r < R) (hy : y ∈ Ioo a b) :
    HasDerivAt (pairCompactFamily B r) (pairCompactFamily (pairPartialY B) r y) y := by
  have hP : ContinuousOn (pairPartialY B) (pairRectangle R a b) :=
    (pairPartialY_contDiffOn 0 hB).continuousOn
  apply hasDerivAt_continuousMap_of_eval (pairCompactFamily_continuousOn hP hrR) _ hy
  intro u hu x
  have hx : (x : ℝ) ∈ Ioo (-R) R :=
    ⟨(neg_lt_neg hrR).trans_le x.2.1, x.2.2.trans_lt hrR⟩
  have hxu : ((x : ℝ), u) ∈ pairRectangle R a b := ⟨hx, hu⟩
  have hd := ((hB.contDiffAt ((pairRectangle_isOpen R a b).mem_nhds hxu)).differentiableAt
    (by norm_num)).hasFDerivAt
  have hs : HasDerivAt (fun v : ℝ => ((x : ℝ), v)) (0, 1) u :=
    (hasDerivAt_const u (x : ℝ)).prodMk (hasDerivAt_id u)
  have hpoint : HasDerivAt (fun v => B (x, v)) (pairPartialY B (x, u)) u :=
    hd.comp_hasDerivAt u hs
  rw [pairCompactFamily_apply hP hrR hu x]
  apply hpoint.congr_of_eventuallyEq
  filter_upwards [isOpen_Ioo.mem_nhds hu] with v hv
  exact pairCompactFamily_apply hB.continuousOn hrR hv x

theorem pairPrimitive_hasDerivAt_y {B : ℝ × ℝ → ℝ} {R a b x y : ℝ}
    (hB : ContDiffOn ℝ 1 B (pairRectangle R a b))
    (hx : x ∈ Ioo (-R) R) (hy : y ∈ Ioo a b) :
    HasDerivAt (fun v => pairPrimitive B (x, v)) (pairPrimitive (pairPartialY B) (x, y)) y := by
  obtain ⟨r, hr, hrR, hxr⟩ := exists_compact_time_interval hx
  have hP : ContinuousOn (pairPartialY B) (pairRectangle R a b) :=
    (pairPartialY_contDiffOn 0 hB).continuousOn
  have hFB := pairCompactFamily_hasDerivAt hB hrR hy
  have hPB := (primitiveCLM r hr).hasFDerivAt.comp_hasDerivAt y hFB
  have hd := intervalExtension_family_hasDerivAt r hr hPB x
  rw [← pairPrimitive_eq_clamped hP hr hrR hxr hy] at hd
  apply hd.congr_of_eventuallyEq
  filter_upwards [isOpen_Ioo.mem_nhds hy] with v hv
  exact pairPrimitive_eq_clamped hB.continuousOn hr hrR hxr hv

theorem pairPrimitive_hasDerivAt_x {B : ℝ × ℝ → ℝ} {R a b x y : ℝ}
    (hB : ContinuousOn B (pairRectangle R a b))
    (hx : x ∈ Ioo (-R) R) (hy : y ∈ Ioo a b) :
    HasDerivAt (fun t => pairPrimitive B (t, y)) (B (x, y)) x := by
  obtain ⟨r, hr, hrR, hxr⟩ := exists_compact_time_interval hx
  have hd := intervalExtension_primitive_hasDerivAt r hr (pairCompactFamily B r y) hxr
  rw [intervalExtension_apply_of_mem r hr _ ⟨hxr.1.le, hxr.2.le⟩,
    pairCompactFamily_apply hB hrR hy] at hd
  apply hd.congr_of_eventuallyEq
  filter_upwards [isOpen_Ioo.mem_nhds hxr] with t ht
  exact pairPrimitive_eq_clamped hB hr hrR ht hy

theorem pairPrimitive_contDiffOn_nat (n : ℕ) {B : ℝ × ℝ → ℝ} {R a b : ℝ}
    (hB : ContDiffOn ℝ n B (pairRectangle R a b)) :
    ContDiffOn ℝ n (pairPrimitive B) (pairRectangle R a b) := by
  induction n generalizing B with
  | zero =>
      simpa only [Nat.cast_zero, contDiffOn_zero] using pairPrimitive_continuousOn hB.continuousOn
  | succ n hn =>
      have hB1 : ContDiffOn ℝ 1 B (pairRectangle R a b) := hB.of_le (by simp)
      apply contDiffOn_succ_of_continuous_partials (pairRectangle_isOpen R a b) n
        (hB.of_le (by simp)) (hn (pairPartialY_contDiffOn n hB))
      · intro p hp
        exact pairPrimitive_hasDerivAt_x hB.continuousOn hp.1 hp.2
      · intro p hp
        exact pairPrimitive_hasDerivAt_y hB1 hp.1 hp.2

end SmoothLocal.Flow

end

end OAI