import OAI.MeasureTheory.Falconer.Recursion.CriticalScalarPotential
import OAI.MeasureTheory.Falconer.Recursion.TruncatedScalarPotential

namespace OAI

/-! # Critical maximum-truncated potentials with logarithmic scale loss -/

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

theorem truncated_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/max (a x) (1/(2:ℝ)^(g+k)))^q) ∂μ) ≤
      ENNReal.ofReal ((2:ℝ)^q)*
        (ENNReal.ofReal K+(k:ℝ≥0∞)*ENNReal.ofReal (K*(2:ℝ)^q)) := by
  have hp (x : X) : (1/max (a x) (1/(2:ℝ)^(g+k)))^q ≤
      (2:ℝ)^q*(1/(a x+1/(2:ℝ)^(g+k)))^q := by
    have hax := ha0 x
    calc
      _ ≤ (2/(a x+1/(2:ℝ)^(g+k)))^q := Real.rpow_le_rpow (by positivity)
        (inverse_max_le_two_inverse_add (ha0 x) (by positivity)) hq
      _ = _ := by
        rw [show 2/(a x+1/(2:ℝ)^(g+k)) = 2*(1/(a x+1/(2:ℝ)^(g+k))) by ring,
          Real.mul_rpow (by norm_num) (by positivity)]
  calc
    _ ≤ ∫⁻ x in {x | a x < 1/(2:ℝ)^g},
        ENNReal.ofReal ((2:ℝ)^q)*ENNReal.ofReal ((1/(a x+1/(2:ℝ)^(g+k)))^q) ∂μ := by
      apply lintegral_mono
      intro x
      dsimp only
      rw [← ENNReal.ofReal_mul (by positivity)]
      exact ENNReal.ofReal_le_ofReal (hp x)
    _ = ENNReal.ofReal ((2:ℝ)^q)*(∫⁻ x in {x | a x < 1/(2:ℝ)^g},
        ENNReal.ofReal ((1/(a x+1/(2:ℝ)^(g+k)))^q) ∂μ) :=
      lintegral_const_mul' _ _ ENNReal.ofReal_ne_top
    _ ≤ _ := mul_le_mul' le_rfl
      (regularized_scalar_potential_critical μ a ha ha0 hq g k hsmall)

end Falconer

end OAI