import OAI.MeasureTheory.Falconer.Recursion.CriticalPotential
import Mathlib.MeasureTheory.Integral.Lebesgue.Map

namespace OAI

/-! # Critical dyadic inverse potentials of a nonnegative statistic -/

namespace Falconer
open Set Metric MeasureTheory
open scoped ENNReal
variable {X : Type*} [MeasurableSpace X]

theorem regularized_scalar_potential_critical (μ : Measure X) (a : X → ℝ)
    (ha : Measurable a) (ha0 : ∀ x, 0 ≤ a x)
    {K q : ℝ} (hq : 0 ≤ q) (g k : ℕ)
    (hsmall : ∀ j : ℕ, j ≤ g+k →
      μ {x | a x < 1/(2:ℝ)^j} ≤ ENNReal.ofReal (K*(1/(2:ℝ)^j)^q)) :
    (∫⁻ x in {x | a x < 1/(2:ℝ)^g},
      ENNReal.ofReal ((1/(a x+1/(2:ℝ)^(g+k)))^q) ∂μ) ≤
      ENNReal.ofReal K+(k:ℝ≥0∞)*ENNReal.ofReal (K*(2:ℝ)^q) := by
  have hpre (j : ℕ) : a ⁻¹' ball (0:ℝ) (1/(2:ℝ)^j) = {x | a x < 1/(2:ℝ)^j} := by
    ext x
    simp only [mem_preimage,mem_ball,Real.dist_eq,sub_zero,abs_of_nonneg (ha0 x),mem_ofPred_eq]
  have hball (j : ℕ) (hj : j ≤ g+k) :
      (Measure.map a μ) (ball (0:ℝ) (1/(2:ℝ)^j)) ≤
        ENNReal.ofReal (K*(1/(2:ℝ)^j)^q) := by
    rw [Measure.map_apply ha measurableSet_ball,hpre]
    exact hsmall j hj
  have hh := regularized_potential_critical (Measure.map a μ) (0:ℝ) hq g k hball
  have hf : Measurable (fun z : ℝ =>
      ENNReal.ofReal ((1/(dist 0 z+1/(2:ℝ)^(g+k)))^q)) := by fun_prop
  have he : (∫⁻ z in ball (0:ℝ) (1/(2:ℝ)^g),
      ENNReal.ofReal ((1/(dist 0 z+1/(2:ℝ)^(g+k)))^q) ∂Measure.map a μ) =
      ∫⁻ x in {x | a x < 1/(2:ℝ)^g},
        ENNReal.ofReal ((1/(a x+1/(2:ℝ)^(g+k)))^q) ∂μ := by
    rw [← lintegral_indicator measurableSet_ball,lintegral_map
      (hf.indicator measurableSet_ball) ha,
      ← lintegral_indicator (measurableSet_lt ha measurable_const)]
    apply lintegral_congr
    intro x
    by_cases hx : a x < 1/(2:ℝ)^g
    · have hz : a x ∈ ball (0:ℝ) (1/(2:ℝ)^g) := by
        change x ∈ a ⁻¹' ball (0:ℝ) (1/(2:ℝ)^g)
        rw [hpre]
        exact hx
      rw [indicator_of_mem hz,indicator_of_mem (show x ∈ {x | a x < 1/(2:ℝ)^g} from hx)]
      simp only [Real.dist_eq,zero_sub,abs_neg,abs_of_nonneg (ha0 x)]
    · have hz : a x ∉ ball (0:ℝ) (1/(2:ℝ)^g) := by
        change x ∉ a ⁻¹' ball (0:ℝ) (1/(2:ℝ)^g)
        rw [hpre]
        exact hx
      rw [indicator_of_notMem hz,indicator_of_notMem (show x ∉ {x | a x < 1/(2:ℝ)^g} from hx)]
  rw [he] at hh
  exact hh

end Falconer

end OAI