import OAI.MeasureTheory.Falconer.Graphs.GraphOriginalFubini
import OAI.MeasureTheory.Falconer.Graphs.GraphFourTransformExpansion

namespace OAI

namespace PlanarFalconer
open Set MeasureTheory

theorem sourceGraphFourTransform_original_spatial
    (ρ : Fin 4 → Measure Plane) [∀ i, IsFiniteMeasure (ρ i)]
    (X : Fin 4 → Set Plane) (M : Fin 4 → ℝ)
    (b : Fin 4 → Plane → UnitCircle → ℂ) (W : UnitCircle × UnitCircle → ℂ)
    (hb : ∀ i, Measurable (fun p : Plane × UnitCircle => b i p.1 p.2))
    (hW : Measurable W) (B : Fin 4 → ℝ) (hB : ∀ i, 0 ≤ B i)
    (hbound : ∀ i x ω, ‖b i x ω‖ ≤ B i) (hw : ∀ p, ‖W p‖ ≤ 1)
    {Ψ : ℝ → ℝ} (hΨ : Measurable Ψ)
    (hsupp : ∀ t, Ψ t ≠ 0 → t ∈ Ioo (1/4:ℝ) 4)
    {C : ℝ} (hC : 0 ≤ C) (hc : ∀ t, |Ψ t| ≤ C) {R : ℝ} (hR : 0 < R) :
    R⁻¹ • (∫ r : ℝ, ∫ u, ∫ v, Ψ (r/R) • sourceGraphFourTransform ρ X M b W r u v
      ∂sourceArcLength ∂sourceArcLength) =
      ((M 0*M 1)*(M 2*M 3))⁻¹ •
        (∫ x, R⁻¹ • (∫ r : ℝ, ∫ u, ∫ v, Ψ (r/R) • graphFourKernel b W x r u v
          ∂sourceArcLength ∂sourceArcLength)
          ∂Measure.pi (fun i => (ρ i).restrict (X i))) := by
  let μ := Measure.pi (fun i => (ρ i).restrict (X i))
  let c := ((M 0*M 1)*(M 2*M 3))⁻¹
  have he (r : ℝ) (u v : UnitCircle) :
      Ψ (r/R) • sourceGraphFourTransform ρ X M b W r u v =
        c • (∫ x, graphOriginalKernel b W Ψ R (r,(u,v)) x ∂μ) := by
    rw [sourceGraphFourTransform_endpoint_integral]
    change Ψ (r/R) • (c • (∫ x, graphFourKernel b W x r u v ∂μ)) = _
    rw [smul_comm]
    congr 1
    exact (integral_smul _ _).symm
  simp_rw [he, integral_smul]
  rw [smul_comm (R⁻¹) c]
  congr 1
  rw [graphOriginalKernel_spatial_swap μ b W hb hW B hB hbound hw hΨ hsupp hC hc hR]
  simp only [graphOriginalKernel, integral_smul, μ]

end PlanarFalconer

end OAI