import OAI.NumberTheory.TwoPoint.ShortIntervals.MRTMultiscaleEnergy

namespace OAI

/-! The explicit amplification order used between two successive MRT
prime scales. The integer ceiling satisfies the exact support inequalities
needed by the mixed cofactor moment. -/

namespace TwoPointCorrelations

open Finset MeasureTheory
open scoped Classical

noncomputable def mrtAmplificationOrder (Y u : ℝ) : ℕ :=
  ⌈Real.log u / Real.log Y⌉₊

lemma mrt_nat_pow_eq_exp {Y : ℝ} (hY : 0 < Y) (r : ℕ) :
    Y ^ r = Real.exp ((r : ℝ) * Real.log Y) := by
  rw [← Real.rpow_natCast, Real.rpow_def_of_pos hY]
  congr 1
  ring

theorem mrt_amplification_order_bounds {Y u : ℝ} (hY : 1 < Y) (hu : 1 ≤ u) :
    u ≤ Y ^ mrtAmplificationOrder Y u ∧
      Y ^ mrtAmplificationOrder Y u ≤ u * Y ∧
      (mrtAmplificationOrder Y u : ℝ) < Real.log u / Real.log Y + 1 := by
  have hY0 : 0 < Y := by linarith
  have hu0 : 0 < u := by linarith
  have hlogY : 0 < Real.log Y := Real.log_pos hY
  have hquot : 0 ≤ Real.log u / Real.log Y :=
    div_nonneg (Real.log_nonneg hu) hlogY.le
  have hlo : Real.log u / Real.log Y ≤ (mrtAmplificationOrder Y u : ℝ) :=
    Nat.le_ceil _
  have hhi : (mrtAmplificationOrder Y u : ℝ) < Real.log u / Real.log Y + 1 :=
    Nat.ceil_lt_add_one hquot
  refine ⟨?_, ?_, hhi⟩
  · calc
      u = Real.exp (Real.log u) := (Real.exp_log hu0).symm
      _ ≤ Real.exp ((mrtAmplificationOrder Y u : ℝ) * Real.log Y) :=
        Real.exp_le_exp.mpr ((div_le_iff₀ hlogY).mp hlo)
      _ = _ := (mrt_nat_pow_eq_exp hY0 _).symm
  · have hh : (mrtAmplificationOrder Y u : ℝ) * Real.log Y <
        Real.log u + Real.log Y := by
      have hh := (mul_lt_mul_of_pos_right hhi hlogY)
      rw [add_mul, div_mul_cancel₀ _ hlogY.ne', one_mul] at hh
      exact hh
    calc
      _ = Real.exp ((mrtAmplificationOrder Y u : ℝ) * Real.log Y) :=
        mrt_nat_pow_eq_exp hY0 _
      _ ≤ Real.exp (Real.log u + Real.log Y) := Real.exp_le_exp.mpr hh.le
      _ = u * Y := by rw [Real.exp_add, Real.exp_log hu0, Real.exp_log hY0]

theorem mrt_short_class_cofactor_energy_chosen (P Q : Finset ℕ)
    (hP : ∀ p ∈ P, p.Prime) {Y N : ℕ} (hY : 1 < Y) (hN : 0 < N)
    (hbin : P ⊆ Icc Y (2 * Y)) (a : ℕ → ℂ) (ha : ∀ p ∈ P, ‖a p‖ ≤ 1)
    (F : ℕ → ℂ) (hF : OneBounded F) {u : ℝ} (hu : 1 ≤ u)
    {T V : ℝ} (hT : 0 < T) (hV : 0 < V)
    {E : Set ℝ} (hE : E ⊆ Set.Ioc (-T) T)
    (hlarge : ∀ t ∈ E, V ≤ ‖mrtExponentialPolynomial P
      (fun p => a p / (p : ℂ)) (fun p => -Real.log (p : ℝ)) t‖) :
    (∫ t in E, ‖mrtCofactorPolynomial Q F N u t‖ ^ 2) ≤
      (16 * Real.exp 10 *
        (T / (N : ℝ) + (2 : ℝ) ^ (mrtAmplificationOrder Y u + 1) * Y) *
          ((mrtAmplificationOrder Y u).factorial : ℝ) ^ 2) /
        V ^ (2 * mrtAmplificationOrder Y u) := by
  obtain ⟨hlo, hhi, _⟩ := mrt_amplification_order_bounds
    (show (1 : ℝ) < Y by exact_mod_cast hY) hu
  exact mrt_short_class_cofactor_energy P Q hP (by omega) hN hbin a ha F hF hu _
    hlo hhi hT hV hE hlarge

end TwoPointCorrelations

end OAI