import OAI.NumberTheory.TwoPoint.Halasz.HalaszCollisionMoment
import OAI.NumberTheory.TwoPoint.Halasz.HalaszVinogradovProduct

namespace OAI

/-! Absorbing the contribution of repeated long coordinates by the
diagonal lower bound for the complete mean value. -/
namespace TwoPointCorrelations

lemma halasz_collision_algebra {k : ℕ} (hk : 0<k) {A B I C R : ℝ}
    (hA : 0<A) (hB : 0<B) (hC : 0≤C) (hR : C^2<R)
    (hdiag : R^k*B≤A) (hmoment : I^(2*k)≤A^(2*k-1)*B) : C*I<A := by
  by_contra hn
  have hAI : A≤C*I := le_of_not_gt hn
  have hpow := pow_le_pow_left₀ hA.le hAI (2*k)
  rw [mul_pow] at hpow
  have ha : A^(2*k) ≤ C^(2*k)*A^(2*k-1)*B := by
    calc
      A^(2*k) ≤ C^(2*k)*I^(2*k) := hpow
      _ ≤ C^(2*k)*(A^(2*k-1)*B) :=
        mul_le_mul_of_nonneg_left hmoment (pow_nonneg hC _)
      _ = _ := by ring
  have hb := halasz_collision_power_absorb (by omega : 1≤2*k) hA ha
  have hRpow : C^(2*k)<R^k := by
    rw [pow_mul]
    exact pow_lt_pow_left₀ hR (sq_nonneg C) (by omega)
  have hstrict : C^(2*k)*B<R^k*B := mul_lt_mul_of_pos_right hRpow hB
  linarith

theorem halasz_collision_moment_small {s k N : ℕ} (hs : 0<s) (hk : 2≤k)
    {C : ℝ} (hC : 0≤C) (hN : C^2<N) :
    C*halaszCollisionMoment s k N < (halaszVinogradovCount (s+k) k N:ℝ) := by
  have hN0 : 0<N := by exact_mod_cast lt_of_le_of_lt (sq_nonneg C) hN
  have hJs : 0<(halaszVinogradovCount s k N:ℝ) := by
    exact_mod_cast lt_of_lt_of_le (Nat.pow_pos hN0) (halasz_vinogradov_count_lower s k N)
  have hJ : 0<(halaszVinogradovCount (s+k) k N:ℝ) := by
    exact_mod_cast lt_of_lt_of_le (Nat.pow_pos hN0)
      (halasz_vinogradov_count_lower (s+k) k N)
  have hdiag : (N:ℝ)^k*(halaszVinogradovCount s k N:ℝ) ≤
      (halaszVinogradovCount (s+k) k N:ℝ) := by
    exact_mod_cast (by simpa only [Nat.add_comm k s] using
      halasz_vinogradov_diagonal_extension s k N)
  exact halasz_collision_algebra (by omega : 0<k) hJ hJs hC hN hdiag
    (halasz_collision_moment_bound hs hk N)

end TwoPointCorrelations

end OAI