import OAI.Analysis.HenonEmden.OriginBounds

namespace OAI

universe uE uE2

section Distribution

open Set Filter Topology MeasureTheory
open scoped ContDiff
open Set Filter Topology
open scoped RealInnerProductSpace ContDiff
open Set Filter Topology
open scoped ContDiff
open Set MeasureTheory MeasureTheory.Measure
open scoped ENNReal
open Set Filter Topology MeasureTheory
open Set Filter Topology
open MeasureTheory
open scoped Convolution
open Set Filter Topology
open MeasureTheory Set Filter Topology
open scoped ContDiff
noncomputable section


namespace HenonLaneEmden.WeakPDE
variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]

lemma cutoff_laplacian_integral_scale {u φ ξ : E → ℝ}
    (hφ : ContDiff ℝ 2 φ) (hξ : ContDiff ℝ 2 ξ) {R : ℝ} (hR : 0 < R) :
    (∫ x : E, u x*Laplacian.laplacian (fun y => φ y*ξ (R⁻¹ • y)) x) =
      R^(Module.finrank ℝ E)*(R⁻¹)^2*
        ∫ y : E, u (R • y)*Laplacian.laplacian (fun z => φ (R • z)*ξ z) y := by
  have hF : ContDiff ℝ 2 (fun z : E => φ (R • z)*ξ z) :=
    (hφ.comp (contDiff_const.smul contDiff_id)).mul hξ
  have hid : (fun y : E => φ y*ξ (R⁻¹ • y)) =
      (fun y : E => (fun z => φ (R • z)*ξ z) (R⁻¹ • y)) := by
    ext y
    simp [smul_smul,ne_of_gt hR]
  have hh (x : E) : Laplacian.laplacian (fun y => φ y*ξ (R⁻¹ • y)) (R • x) =
      (R⁻¹)^2*Laplacian.laplacian (fun z => φ (R • z)*ξ z) x := by
    rw [hid,laplacian_dilation hF]
    simp [smul_smul,ne_of_gt hR]
  have hc := Measure.integral_comp_smul_of_nonneg volume
    (fun x => u x*Laplacian.laplacian (fun y => φ y*ξ (R⁻¹ • y)) x) R (hR := hR.le)
  simp only [hh,smul_eq_mul] at hc
  have hC : (∫ x, u (R • x)*((R⁻¹)^2*Laplacian.laplacian (fun z => φ (R • z)*ξ z) x)) =
      (R⁻¹)^2*(∫ x, u (R • x)*Laplacian.laplacian (fun z => φ (R • z)*ξ z) x) := by
    simp_rw [← mul_assoc,mul_comm (u (R • _)) ((R⁻¹)^2),mul_assoc]
    exact integral_const_mul _ _
  rw [hC] at hc
  have hn : R^(Module.finrank ℝ E) ≠ 0 := pow_ne_zero _ (ne_of_gt hR)
  calc
    _ = R^Module.finrank ℝ E * ((R^Module.finrank ℝ E)⁻¹ *
      ∫ x, u x*Laplacian.laplacian (fun y => φ y*ξ (R⁻¹ • y)) x) := by field_simp
    _ = _ := by rw [← hc]; ring

lemma cutoff_laplacian_integral_tendsto_zero {u φ ξ : E → ℝ}
    (hn : 3 ≤ Module.finrank ℝ E) (hu : Continuous u)
    (hφ : ContDiff ℝ 2 φ) (hξ : ContDiff ℝ 2 ξ) (hξc : HasCompactSupport ξ) :
    Tendsto (fun R : ℝ => ∫ x : E, u x*Laplacian.laplacian (fun y => φ y*ξ (R⁻¹ • y)) x)
      (𝓝[>] 0) (𝓝 0) := by
  have heq : ∀ᶠ R : ℝ in 𝓝[>] 0, (∫ x : E, u x*Laplacian.laplacian (fun y => φ y*ξ (R⁻¹ • y)) x) =
      R^(Module.finrank ℝ E-2)*∫ y : E, u (R • y)*Laplacian.laplacian (fun z => φ (R • z)*ξ z) y := by
    filter_upwards [self_mem_nhdsWithin] with R hR
    rw [cutoff_laplacian_integral_scale hφ hξ hR]
    congr 1
    rw [inv_pow]
    have hp := pow_sub₀ R (ne_of_gt hR) (show 2 ≤ Module.finrank ℝ E by omega)
    exact hp.symm
  apply Tendsto.congr' (Filter.EventuallyEq.symm heq)
  have hpow : Tendsto (fun R : ℝ => R^(Module.finrank ℝ E-2)) (𝓝[>] 0) (𝓝 0) := by
    have := (continuous_pow (Module.finrank ℝ E-2) : Continuous (fun R : ℝ => R^(Module.finrank ℝ E-2))).tendsto 0
    simpa [show Module.finrank ℝ E-2 ≠ 0 by omega] using this.mono_left nhdsWithin_le_nhds
  simpa using hpow.mul ((continuous_cutoff_error hu hφ hξ hξc).tendsto 0 |>.mono_left nhdsWithin_le_nhds)
end HenonLaneEmden.WeakPDE

namespace HenonLaneEmden.WeakPDE
variable {E : Type uE2} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]

omit [MeasurableSpace E] [BorelSpace E] in
lemma bump_dilation_eventually_zero (ξ : ContDiffBump (0 : E)) {x : E} (hx : x ≠ 0) :
    (fun R : ℝ => ξ (R⁻¹ • x)) =ᶠ[𝓝[>] 0] 0 := by
  have he : 0 < ‖x‖ / ξ.rOut := div_pos (norm_pos_iff.mpr hx) ξ.rOut_pos
  filter_upwards [self_mem_nhdsWithin, (eventually_lt_nhds he).filter_mono nhdsWithin_le_nhds] with R hR hlt
  apply ξ.zero_of_le_dist
  rw [dist_zero_right,norm_smul,Real.norm_eq_abs,abs_of_pos (inv_pos.mpr hR),← div_eq_inv_mul]
  apply (le_div_iff₀ hR).mpr
  simpa only [mul_comm] using (le_of_lt ((lt_div_iff₀ ξ.rOut_pos).mp hlt))

lemma cutoff_source_integral_tendsto_zero [Nontrivial E] {g φ : E → ℝ}
    (hg : LocallyIntegrable g) (hφ : Continuous φ) (hφc : HasCompactSupport φ)
    (ξ : ContDiffBump (0 : E)) :
    Tendsto (fun R : ℝ => ∫ x : E, g x*(φ x*ξ (R⁻¹ • x))) (𝓝[>] 0) (𝓝 0) := by
  have hi : Integrable (fun x => g x*φ x) :=
    hg.integrable_smul_right_of_hasCompactSupport hφ hφc
  have hc (R : ℝ) : Continuous (fun x : E => ξ (R⁻¹ • x)) :=
    ξ.continuous.comp (continuous_const.smul continuous_id)
  have lim := tendsto_integral_filter_of_dominated_convergence
    (μ := volume) (l := 𝓝[>] (0 : ℝ)) (F := fun R x => g x*(φ x*ξ (R⁻¹ • x)))
    (f := fun _ => (0 : ℝ)) (fun x => ‖g x*φ x‖)
    (Eventually.of_forall fun R => by
      rw [show (fun x => g x*(φ x*ξ (R⁻¹ • x))) =
        ((fun x => g x*φ x)*(fun x => ξ (R⁻¹ • x))) by ext x; simp only [Pi.mul_apply]; ring]
      exact hi.aestronglyMeasurable.mul (hc R).aestronglyMeasurable)
    (Eventually.of_forall fun R => Eventually.of_forall fun x => by
      rw [← mul_assoc,norm_mul]
      exact mul_le_of_le_one_right (norm_nonneg _) (by
        rw [Real.norm_eq_abs,abs_of_nonneg ξ.nonneg]; exact ξ.le_one))
    hi.norm ?_
  · simpa using lim
  filter_upwards [ae_iff.mpr (show volume {x : E | ¬ x ≠ 0} = 0 by simp)] with x hx
  apply tendsto_const_nhds.congr'
  filter_upwards [bump_dilation_eventually_zero ξ hx] with R hR
  simp [hR]

omit [MeasurableSpace E] [BorelSpace E] in
lemma cutoff_test_off_origin {φ : E → ℝ} (hφ : ContDiff ℝ 2 φ)
    (hφc : HasCompactSupport φ) (ξ : ContDiffBump (0 : E)) {R : ℝ} (_hR : 0 < R) :
    ContDiff ℝ 2 (fun x => φ x-φ x*ξ (R⁻¹ • x)) ∧
    HasCompactSupport (fun x => φ x-φ x*ξ (R⁻¹ • x)) ∧
    0 ∉ tsupport (fun x => φ x-φ x*ξ (R⁻¹ • x)) := by
  refine ⟨hφ.sub (hφ.mul (ξ.contDiff.comp (contDiff_const.smul contDiff_id))),
    hφc.sub (hφc.mul_right), ?_⟩
  apply notMem_tsupport_iff_eventuallyEq.mpr
  have ht : Tendsto (fun x : E => R⁻¹ • x) (𝓝 0) (𝓝 0) := by
    have h : Continuous (fun x : E => R⁻¹ • x) := continuous_const.smul continuous_id
    have ht := h.tendsto 0
    change Tendsto (fun x : E => R⁻¹ • x) (𝓝 0) (𝓝 (R⁻¹ • (0:E))) at ht
    simpa only [smul_zero] using ht
  have he := ξ.eventuallyEq_one.comp_tendsto ht
  filter_upwards [he] with x hx
  change ξ (R⁻¹ • x) = 1 at hx
  simp [hx]

lemma distributional_extension (hn : 3 ≤ Module.finrank ℝ E) {u g : E → ℝ}
    (hu : Continuous u) (hg : LocallyIntegrable g)
    (heq : ∀ (φ : E → ℝ), ContDiff ℝ 2 φ → HasCompactSupport φ → 0 ∉ tsupport φ →
      (∫ x, u x*Laplacian.laplacian φ x) = -(∫ x, g x*φ x))
    {φ : E → ℝ} (hφ : ContDiff ℝ 2 φ) (hφc : HasCompactSupport φ) :
    (∫ x, u x*Laplacian.laplacian φ x) = -(∫ x, g x*φ x) := by
  have : Nontrivial E := (Module.finrank_pos_iff (R := ℝ)).mp (by omega)
  let ξ : ContDiffBump (0 : E) := ⟨1,2,by norm_num,by norm_num⟩
  have hi : Integrable (fun x => g x*φ x) :=
    hg.integrable_smul_right_of_hasCompactSupport hφ.continuous hφc
  have hui : Integrable (fun x => u x*Laplacian.laplacian φ x) :=
    integrable_mul_compact (compactSupport_laplacian hφc) hu.continuousOn (continuous_laplacian hφ).continuousOn
  have hevent : ∀ᶠ R : ℝ in 𝓝[>] 0,
      (∫ x, u x*Laplacian.laplacian φ x) -
        (∫ x, u x*Laplacian.laplacian (fun y => φ y*ξ (R⁻¹ • y)) x) =
        -(∫ x, g x*φ x) + (∫ x, g x*(φ x*ξ (R⁻¹ • x))) := by
    filter_upwards [self_mem_nhdsWithin] with R hR
    have hχ : ContDiff ℝ 2 (fun x => φ x*ξ (R⁻¹ • x)) :=
      hφ.mul (ξ.contDiff.comp (contDiff_const.smul contDiff_id))
    have hχc : HasCompactSupport (fun x => φ x*ξ (R⁻¹ • x)) := hφc.mul_right
    have hχi : Integrable (fun x => u x*Laplacian.laplacian (fun y => φ y*ξ (R⁻¹ • y)) x) :=
      integrable_mul_compact (compactSupport_laplacian hχc) hu.continuousOn (continuous_laplacian hχ).continuousOn
    have hgi : Integrable (fun x => g x*(φ x*ξ (R⁻¹ • x))) :=
      hg.integrable_smul_right_of_hasCompactSupport hχ.continuous hχc
    obtain ⟨hψ,hψc,hψ0⟩ := cutoff_test_off_origin hφ hφc ξ hR
    have HH := heq _ hψ hψc hψ0
    have hid (x : E) : Laplacian.laplacian (fun y => φ y-φ y*ξ (R⁻¹ • y)) x =
        Laplacian.laplacian φ x-Laplacian.laplacian (fun y => φ y*ξ (R⁻¹ • y)) x :=
      hφ.contDiffAt.laplacian_sub hχ.contDiffAt
    simp_rw [hid,mul_sub] at HH
    rw [integral_sub hui hχi,integral_sub hi hgi] at HH
    linarith
  have hlim1 := (tendsto_const_nhds (x := ∫ x, u x*Laplacian.laplacian φ x)).sub (cutoff_laplacian_integral_tendsto_zero hn hu hφ ξ.contDiff ξ.hasCompactSupport)
  have hlim2 := (tendsto_const_nhds (x := -(∫ x, g x*φ x))).add (cutoff_source_integral_tendsto_zero hg hφ.continuous hφc ξ)
  have HH := tendsto_nhds_unique hlim1 (hlim2.congr' (Filter.EventuallyEq.symm hevent))
  simpa only [sub_zero,add_zero] using HH
end HenonLaneEmden.WeakPDE

end
end Distribution

end OAI