import OAI.NumberTheory.Jacobsthal.Estimates.MovingLocalReference

namespace OAI

namespace Erdos970
open scoped _root_.Erdos970

section

namespace NumberTheoryLean.JacobsthalSourceScale

open _root_.Filter Asymptotics
open scoped Topology

noncomputable def sourceW (L : ℝ) : ℝ := L/(Real.log L)^2
noncomputable def sourceB (L : ℝ) : ℝ := L/Real.log (sourceW L)

theorem log_square_le_sqrt_eventually : ∀ᶠ L : ℝ in atTop,(Real.log L)^2 ≤ Real.sqrt L := by
  have h := (isLittleO_log_rpow_rpow_atTop (2:ℝ) (s:=1/2) (by norm_num)).bound (by norm_num : (0:ℝ)<1)
  filter_upwards [h,eventually_gt_atTop (1:ℝ)] with L hL hpos
  have hr : 0 ≤ L^((1/2:ℝ)) := Real.rpow_nonneg (by linarith) _
  simpa only [Real.rpow_ofNat,Real.norm_eq_abs,abs_of_nonneg (sq_nonneg (Real.log L)),
    abs_of_nonneg hr,one_mul,← Real.sqrt_eq_rpow,abs_of_nonneg (Real.sqrt_nonneg L)] using hL

theorem sourceW_bounds_eventually : ∀ᶠ L : ℝ in atTop,
    1 < sourceW L ∧ Real.sqrt L ≤ sourceW L ∧ sourceW L ≤ L := by
  filter_upwards [log_square_le_sqrt_eventually,eventually_gt_atTop (1:ℝ),
    Real.tendsto_log_atTop.eventually (eventually_ge_atTop (1:ℝ))] with L hsmall hL hlog
  have hL0 : 0 < L := by linarith
  have hden : 0 < (Real.log L)^2 := sq_pos_of_pos (by linarith)
  have hsq : Real.sqrt L ≤ sourceW L := by
    apply (le_div_iff₀ hden).mpr
    have hm := mul_le_mul_of_nonneg_left hsmall (Real.sqrt_nonneg L)
    simpa only [Real.mul_self_sqrt hL0.le] using hm
  have hlow : 1 < Real.sqrt L := by
    simpa only [Real.sqrt_one] using Real.sqrt_lt_sqrt (by norm_num : (0:ℝ) ≤ 1) hL
  refine ⟨hlow.trans_le hsq,hsq,?_⟩
  exact div_le_self hL0.le (by nlinarith : 1 ≤ (Real.log L)^2)

theorem sourceW_tendsto : Tendsto sourceW atTop atTop := by
  apply tendsto_atTop.mpr
  intro R
  filter_upwards [sourceW_bounds_eventually,Real.tendsto_sqrt_atTop.eventually (eventually_ge_atTop R)] with L hW hR
  exact hR.trans hW.2.1

theorem source_scale_eventually : ∀ᶠ L : ℝ in atTop,
    1 < sourceW L ∧ 0 < sourceB L ∧ Real.sqrt L ≤ sourceB L ∧ sourceB L ≤ L ∧
      Real.log (sourceB L) ≤ 2*Real.log (sourceW L) := by
  filter_upwards [sourceW_bounds_eventually,log_square_le_sqrt_eventually,eventually_gt_atTop (1:ℝ),
    Real.tendsto_log_atTop.eventually (eventually_ge_atTop (1:ℝ)),
    sourceW_tendsto.eventually (eventually_ge_atTop (Real.exp 1))] with L hW hsmall hL hlog hWexp
  have hL0 : 0 < L := by linarith
  have hW0 : 0 < sourceW L := by linarith [hW.1]
  have hlogW : 1 ≤ Real.log (sourceW L) := by simpa only [Real.log_exp] using Real.log_le_log (Real.exp_pos 1) hWexp
  have hlogWL : Real.log (sourceW L) ≤ Real.log L := Real.log_le_log hW0 hW.2.2
  have hlogSmall : Real.log (sourceW L) ≤ Real.sqrt L := by
    have hh : Real.log L ≤ (Real.log L)^2 := by nlinarith
    exact hlogWL.trans (hh.trans hsmall)
  have hBpos : 0 < sourceB L := div_pos hL0 (by linarith)
  have hBsq : Real.sqrt L ≤ sourceB L := by
    apply (le_div_iff₀ (show 0 < Real.log (sourceW L) by linarith)).mpr
    have hm := mul_le_mul_of_nonneg_left hlogSmall (Real.sqrt_nonneg L)
    simpa only [Real.mul_self_sqrt hL0.le] using hm
  have hBL : sourceB L ≤ L := div_le_self hL0.le hlogW
  have hLWsq : L ≤ (sourceW L)^2 := by
    have hh := mul_self_le_mul_self (Real.sqrt_nonneg L) hW.2.1
    nlinarith [Real.sq_sqrt hL0.le]
  have hcomp := Real.log_le_log hBpos (hBL.trans hLWsq)
  refine ⟨hW.1,hBpos,hBsq,hBL,?_⟩
  simpa only [Real.log_pow,Nat.cast_ofNat] using hcomp

theorem sourceB_tendsto : Tendsto sourceB atTop atTop := by
  apply tendsto_atTop.mpr
  intro R
  filter_upwards [source_scale_eventually,Real.tendsto_sqrt_atTop.eventually (eventually_ge_atTop R)] with L h hR
  exact hR.trans h.2.2.1

theorem sourceW_pow_sourceB {L : ℝ} (hW : 1 < sourceW L) :
    (sourceW L)^(sourceB L)=Real.exp L := by
  rw [sourceB,Real.rpow_def_of_pos (by linarith : 0 < sourceW L)]
  congr 1
  field_simp [ne_of_gt (Real.log_pos hW)]

end NumberTheoryLean.JacobsthalSourceScale

end

end Erdos970

end OAI