import OAI.NumberTheory.PrimeGaps.RieszDecay

namespace OAI

namespace LargePrimeGaps

open Complex Real Set MeasureTheory Filter Topology

open Complex Filter Topology

open Complex Filter Topology MeasureTheory

open scoped FourierTransform

open Complex Real Set MeasureTheory Filter Topology

open Complex Real Set MeasureTheory Filter Topology

open Complex Real Set MeasureTheory Filter Topology

open scoped FourierTransform

open Complex Real Set MeasureTheory Filter Topology

open Complex Real Set MeasureTheory Filter Topology

open Complex Real Set MeasureTheory Filter Topology

lemma primeCharacterError_log_bound (A B : ℝ) (hA : 0<A) (hB : 0<B) :
    ∃ M : ℝ, 0<M ∧ ∀ (q : ℕ) [NeZero q] (χ : DirichletCharacter ℂ q), χ.IsPrimitive →
      ∀ x : ℝ, 2≤x → 1≤Real.log x → (Real.log x)^(2*A+2)≤x →
        (q:ℝ)≤(Real.log x)^B →
        ‖primeCharacterError x χ‖≤M*x*(Real.log x)^(-A) := by
  obtain ⟨D,hD,hb⟩ := primeWeighted_error_bound (B+2*A+2)
  refine ⟨15*D+8,by positivity,?_⟩
  intro q _ χ hp x hx hL hpower hq
  let L := Real.log x
  let K := A+2
  let a := B+2*A+2
  let Q := ((q:ℝ)+2)^(1/2:ℝ)
  let h := x/L^K
  have hx0 : 0<x := by linarith
  have hx1 : 1<x := by linarith
  have hL0 : 0<L := lt_of_lt_of_le zero_lt_one hL
  have hK : 0<K := by dsimp [K]; linarith
  have ha : 0<a := by dsimp [a]; linarith
  have hpK : 0<L^K := Real.rpow_pos_of_pos hL0 _
  have hh0 : 0<h := div_pos hx0 hpK
  have hK1 : 1≤L^K := Real.one_le_rpow hL hK.le
  have hhx : h≤x := (div_le_self hx0.le hK1)
  have hKh : L^K≤x := by
    apply (Real.rpow_le_rpow_of_exponent_le hL (show K≤2*A+2 by dsimp [K]; linarith)).trans
    exact hpower
  have hh1 : 1≤h := (le_div_iff₀ hpK).mpr (by simpa using hKh)
  have hyx : x≤x+h := by linarith
  have hy : 1<x+h := by linarith
  have hy2 : x+h≤2*x := by linarith
  have hLy : L≤Real.log (x+h) := Real.log_le_log hx0 hyx
  have hLy1 : 1≤Real.log (x+h) := hL.trans hLy
  have hLy2 : Real.log (x+h)≤2*L := by
    calc
      _ ≤ Real.log (x^2) := Real.log_le_log (by linarith) (by nlinarith)
      _ = _ := by rw [Real.log_pow]; rfl
  have hQ0 : 0≤Q := by dsimp [Q]; positivity
  have hQ : Q≤3*L^B := by
    have hq2 : 1≤(q:ℝ)+2 := by have := Nat.cast_nonneg (α:=ℝ) q; linarith
    have hqpow : Q≤(q:ℝ)+2 := by
      simpa only [Real.rpow_one] using
        Real.rpow_le_rpow_of_exponent_le hq2 (show (1/2:ℝ)≤1 by norm_num)
    have hLB : 1≤L^B := Real.one_le_rpow hL hB.le
    linarith
  let E (y : ℝ) := ‖weightedPrefix (primeCoefficients χ) y-characterMain χ*(y:ℂ)^2/2‖
  have hEx : E x≤D*Q*x^2*L^(-a)+x := hb q χ hp x hx1 hL
  have hEy : E (x+h)≤4*D*Q*x^2*L^(-a)+2*x := by
    calc
      _ ≤ D*Q*(x+h)^2*(Real.log (x+h))^(-a)+(x+h) := hb q χ hp (x+h) hy hLy1
      _ ≤ D*Q*(2*x)^2*L^(-a)+2*x := by
        apply add_le_add _ hy2
        apply mul_le_mul _ (Real.rpow_le_rpow_of_nonpos hL0 hLy (neg_nonpos.mpr ha.le)) (by positivity) (by positivity)
        gcongr
      _ = _ := by ring
  have hcoeff (n : ℕ) (hn : n∈Finset.Ioc ⌊x⌋₊ ⌊x+h⌋₊) :
      ‖primeCoefficients χ n‖≤2*L := by
    have hn0 : 0<(n:ℝ) := by exact_mod_cast (show 0<n by have := (Finset.mem_Ioc.mp hn).1; omega)
    have hnx : (n:ℝ)≤x+h := (Nat.le_floor_iff (by linarith : 0≤x+h)).mp (Finset.mem_Ioc.mp hn).2
    exact (primeCoefficients_norm_le χ n).trans ((Real.log_le_log hn0 hnx).trans hLy2)
  have heq : primeCharacterError x χ=
      (∑ n∈Finset.Ioc 0 ⌊x⌋₊, primeCoefficients χ n)-characterMain χ*(x:ℂ) := by
    classical
    simp only [primeCharacterError,primeCharacterPrefix,primeCoefficients,characterMain]
    split_ifs <;> simp
  rw [heq]
  have hu := prefix_unsmoothing hx0.le hh0 (show 0≤2*L by positivity)
    (primeCoefficients χ) (characterMain χ) hcoeff
  change _≤(E (x+h)+E x)/h+(h+1)*(2*L)+‖characterMain χ‖*h/2 at hu
  have hδ : ‖characterMain χ‖*h/2≤h := by
    have := mul_le_mul_of_nonneg_right (characterMain_norm_le χ) hh0.le
    linarith
  have htail : (h+1)*(2*L)≤4*h*L := by nlinarith
  have herr : (E (x+h)+E x)/h ≤ 15*D*x*L^(-A)+3*x*L^(-A) := by
    calc
      _ ≤ (5*D*Q*x^2*L^(-a)+3*x)/h := by
        apply div_le_div_of_nonneg_right _ hh0.le
        linarith
      _ ≤ (5*D*(3*L^B)*x^2*L^(-a)+3*x)/h := by gcongr
      _ = (15*D*L^B*x^2*L^(-a)+3*x)/h := by ring
      _ = 15*D*x*L^(-A)+3*L^K := by
        dsimp [h]
        rw [div_eq_mul_inv,inv_div]
        have hr : L^B*L^(-a)*L^K=L^(-A) := by
          rw [←Real.rpow_add hL0,←Real.rpow_add hL0]
          congr 1
          dsimp [a,K]
          ring
        field_simp
        linear_combination 15*D*x*hr
      _ ≤ _ := by
        have ht : L^K≤x*L^(-A) := by
          calc
            _ = L^(2*A+2)*L^(-A) := by rw [←Real.rpow_add hL0]; congr 1; dsimp [K]; ring
            _ ≤ _ := mul_le_mul_of_nonneg_right hpower (by positivity)
        nlinarith [ht]
  have hsmall : h≤x*L^(-A) := by
    dsimp [h]
    rw [div_eq_mul_inv,←Real.rpow_neg hL0.le]
    exact mul_le_mul_of_nonneg_left
      (Real.rpow_le_rpow_of_exponent_le hL (show -K≤-A by dsimp [K]; linarith)) hx0.le
  have htail' : 4*h*L≤4*x*L^(-A) := by
    calc
      _ = 4*x*L^(1-K) := by
        dsimp [h]
        rw [div_eq_mul_inv,Real.rpow_sub hL0,Real.rpow_one]
        ring
      _ ≤ _ := mul_le_mul_of_nonneg_left
        (Real.rpow_le_rpow_of_exponent_le hL (show 1-K≤-A by dsimp [K]; linarith)) (by positivity)
  calc
    _ ≤ (E (x+h)+E x)/h+(h+1)*(2*L)+‖characterMain χ‖*h/2 := hu
    _ ≤ (15*D*x*L^(-A)+3*x*L^(-A))+4*x*L^(-A)+x*L^(-A) :=
      add_le_add (add_le_add herr (htail.trans htail')) (hδ.trans hsmall)
    _ = _ := by ring

theorem siegelWalfisz : SiegelWalfisz := by
  intro A B hA hB
  obtain ⟨M,hM,hb⟩ := primeCharacterError_log_bound A B hA hB
  have hevent : ∀ᶠ x : ℝ in atTop, 2≤x ∧ 1≤Real.log x ∧ (Real.log x)^(2*A+2)≤x := by
    filter_upwards [eventually_ge_atTop (2:ℝ),eventually_ge_atTop (Real.exp 1),
      eventually_log_rpow_le_rpow (2*A+2) (by norm_num : (0:ℝ)<1)] with x hx hex hp
    refine ⟨hx,?_,by simpa only [Real.rpow_one] using hp⟩
    rw [←Real.log_exp (1:ℝ)]
    exact Real.log_le_log (Real.exp_pos _) hex
  obtain ⟨x₀,hx₀⟩ := eventually_atTop.mp hevent
  refine ⟨M,hM,max x₀ 2,by have := le_max_right x₀ (2:ℝ); linarith,?_⟩
  intro x hx d hd hdlog χ hχ
  let : NeZero d := ⟨ne_of_gt hd⟩
  obtain ⟨hx2,hlog,hpower⟩ := hx₀ x ((le_max_left _ _).trans hx)
  exact hb d χ hχ x hx2 hlog hpower hdlog

theorem bombieriVinogradov : BombieriVinogradov :=
  bombieriVinogradov_of_siegelWalfisz siegelWalfisz

theorem main : ∀ C : ℝ, 0 < C →
    ∃ c : ℝ, 0 < c ∧ ∃ N₀ : ℕ, ∀ N : ℕ, N₀ ≤ N →
      c * (N : ℝ) ≤ ((largeGapIndices C N).card : ℝ) :=
  index_density_of_bombieriVinogradov bombieriVinogradov

end LargePrimeGaps

end OAI