import OAI.NumberTheory.TwoPointCorrelations.HalaszSmoothBounds

namespace OAI

/-! A fixed-power change of the prime cutoff changes the squared
pretentious distance by an absolute constant. -/

namespace TwoPointCorrelations

open Finset Complex
open scoped ComplexConjugate

lemma halasz_prime_cutoff_subset {n N : ℕ} (hn : n ≤ N) : primesUpTo n ⊆ primesUpTo N := by
  intro p hp
  obtain ⟨hp, hprime⟩ := mem_filter.mp hp
  exact mem_filter.mpr ⟨mem_range.mpr (by have := mem_range.mp hp; omega), hprime⟩

theorem halasz_distance_cutoff_loss : ∃ K : ℝ, 0 ≤ K ∧
    ∀ (f : ℕ → ℂ), OneBounded f → ∀ (n N : ℕ), 2 ≤ n → n ≤ N → N ≤ n ^ 3 →
    ∀ t : ℝ, squaredDistance f (mrtArchimedeanTwist t) N ≤
      squaredDistance f (mrtArchimedeanTwist t) n + K := by
  obtain ⟨C, hC⟩ := primeReciprocalInput
  let K := 2 * (Real.log 3 + 2 * |C|)
  have hK : 0 ≤ K := by dsimp [K]; positivity
  refine ⟨K, hK, ?_⟩
  intro f hf n N hn hnN hNcube t
  have hn0 : (0 : ℝ) < n := by exact_mod_cast (by omega : 0 < n)
  have hN0 : (0 : ℝ) < N := by exact_mod_cast (by omega : 0 < N)
  have hln : 0 < Real.log (n : ℝ) := Real.log_pos (by exact_mod_cast (by omega : 1 < n))
  have hlN : 0 < Real.log (N : ℝ) := Real.log_pos (by exact_mod_cast (by omega : 1 < N))
  have hlNn : Real.log (N : ℝ) ≤ 3 * Real.log (n : ℝ) := by
    have h := Real.log_le_log hN0 (show (N : ℝ) ≤ (n : ℝ) ^ 3 by exact_mod_cast hNcube)
    simpa only [Real.log_pow, Nat.cast_ofNat] using h
  have hll : Real.log (Real.log (N : ℝ)) - Real.log (Real.log (n : ℝ)) ≤ Real.log 3 := by
    have h := Real.log_le_log hlN hlNn
    rw [Real.log_mul (by norm_num) hln.ne'] at h
    linarith
  have hmN := hC (N : ℝ) (by exact_mod_cast (show 2 ≤ N by omega))
  have hmn := hC (n : ℝ) (by exact_mod_cast hn)
  rw [mrt_sievePrimesUpTo_nat] at hmN hmn
  have hmass : (∑ p ∈ primesUpTo N, (1 : ℝ) / p) -
      (∑ p ∈ primesUpTo n, (1 : ℝ) / p) ≤ Real.log 3 + 2 * |C| := by
    have ha := (abs_le.mp hmN).2
    have hb := (abs_le.mp hmn).1
    linarith [le_abs_self C]
  let P := primesUpTo N \ primesUpTo n
  have hsub := halasz_prime_cutoff_subset hnN
  have hterm (p : ℕ) (hp : p ∈ P) :
      (1 - (f p * conj (mrtArchimedeanTwist t p)).re) / (p : ℝ) ≤ 2 * (1 / (p : ℝ)) := by
    have hprime : p.Prime := (mem_filter.mp (mem_sdiff.mp hp).1).2
    have hnrm : ‖f p * conj (mrtArchimedeanTwist t p)‖ ≤ 1 := by
      simpa only [norm_mul, norm_conj, mrtArchimedeanTwist_norm, mul_one] using hf p hprime.pos
    have hre := (neg_le_abs _).trans ((Complex.abs_re_le_norm _).trans hnrm)
    have hnum : 1 - (f p * conj (mrtArchimedeanTwist t p)).re ≤ 2 := by linarith
    simpa only [mul_one_div] using div_le_div_of_nonneg_right hnum (Nat.cast_nonneg p)
  have hs := sum_le_sum (fun p hp => hterm p hp)
  rw [← mul_sum] at hs
  have heDist := sum_sdiff hsub (f := fun p : ℕ =>
    (1 - (f p * conj (mrtArchimedeanTwist t p)).re) / (p : ℝ))
  have heMass := sum_sdiff hsub (f := fun p : ℕ => (1 : ℝ) / p)
  change (∑ p ∈ P, (1 - (f p * conj (mrtArchimedeanTwist t p)).re) / (p : ℝ)) +
    squaredDistance f (mrtArchimedeanTwist t) n = squaredDistance f (mrtArchimedeanTwist t) N at heDist
  change (∑ p ∈ P, (1 : ℝ) / p) + (∑ p ∈ primesUpTo n, (1 : ℝ) / p) =
    (∑ p ∈ primesUpTo N, (1 : ℝ) / p) at heMass
  dsimp [K]
  linarith

end TwoPointCorrelations

end OAI