| import OAI.NumberTheory.Jacobsthal.Estimates.MovingLocalReference |
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| namespace OAI |
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| namespace Erdos970 |
| open scoped _root_.Erdos970 |
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| section |
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| namespace NumberTheoryLean.JacobsthalSourceScale |
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| open _root_.Filter Asymptotics |
| open scoped Topology |
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| noncomputable def sourceW (L : ℝ) : ℝ := L/(Real.log L)^2 |
| noncomputable def sourceB (L : ℝ) : ℝ := L/Real.log (sourceW L) |
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| 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 |
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| 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) |
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| 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 |
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| 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 |
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| 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 |
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| 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)] |
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| end NumberTheoryLean.JacobsthalSourceScale |
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| end |
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| end Erdos970 |
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| end OAI |
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