| import OAI.NumberTheory.Jacobsthal.Primes.SmallEffectivePrimeSplit |
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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 ErdosVarianceCommon |
| open ErdosVarianceEffective ErdosInverseTail ErdosInverseHits |
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| noncomputable def commonLength (Y qf : ℕ) (R xi : ℝ) : ℕ := |
| commonParentLength Y qf ((1+xi)*R) |
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| theorem bin_upper_comparison (R xi : ℝ) (_hR : 0 < R) (hxi : 0 ≤ xi) |
| (p : ℕ) (hbin : R < (p : ℝ) ∧ (p : ℝ) ≤ (1+xi)*R) : |
| (p : ℝ) ≤ (1+xi)*R ∧ (1+xi)*R ≤ (1+xi)*(p : ℝ) := |
| ⟨hbin.2,mul_le_mul_of_nonneg_left hbin.1.le (by linarith)⟩ |
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| theorem commonLength_le_effective (Y : ℕ) (a : ℕ → ℕ) (r : ℚ) (qf : ℕ) (hq : Squarefree qf) |
| (p : ℕ) [NeZero p] (hHp : (effectiveModulus a r qf).Coprime p) |
| (R xi : ℝ) (hR : 0 < R) (hxi : 0 ≤ xi) |
| (hbin : R < (p : ℝ) ∧ (p : ℝ) ≤ (1+xi)*R) : |
| commonLength Y qf R xi ≤ hitLength Y (p*qf) (effectiveBase a r qf p hHp) := by |
| have hb := (effective_initial_bounds a r qf hq p hHp).2 |
| have hh := commonParentLength_le_hits Y qf p (effectiveBase a r qf p hHp) |
| (Nat.pos_of_ne_zero hq.ne_zero) (Nat.pos_of_ne_zero (NeZero.ne p)) (by simpa only [effectiveBase,Nat.mul_comm] using hb) |
| ((1+xi)*R) (bin_upper_comparison R xi hR hxi p hbin).1 |
| simpa only [commonLength,Nat.mul_comm] using hh |
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| theorem commonLength_tail_bound (Y : ℕ) (a : ℕ → ℕ) (r : ℚ) (qf : ℕ) (hq : Squarefree qf) |
| (p : ℕ) [NeZero p] (hHp : (effectiveModulus a r qf).Coprime p) |
| (R xi : ℝ) (hR : 0 < R) (hxi : 0 ≤ xi) (hxi1 : xi ≤ 1) |
| (hbin : R < (p : ℝ) ∧ (p : ℝ) ≤ (1+xi)*R) : |
| ((hitLength Y (p*qf) (effectiveBase a r qf p hHp)-commonLength Y qf R xi : ℕ) : ℝ) ≤ |
| xi*(commonLength Y qf R xi : ℝ)+3 := by |
| have hb := effective_initial_bounds a r qf hq p hHp |
| have hcomp := bin_upper_comparison R xi hR hxi p hbin |
| have hh := actual_cell_tail_length Y qf p (effectiveBase a r qf p hHp) |
| (Nat.pos_of_ne_zero hq.ne_zero) (Nat.pos_of_ne_zero (NeZero.ne p)) hb.1 |
| (by simpa only [effectiveBase,Nat.mul_comm] using hb.2) ((1+xi)*R) xi hxi hxi1 hcomp.1 hcomp.2 |
| simpa only [commonLength,Nat.mul_comm] using hh |
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| theorem commonLength_reference_bounds (Y qf p : ℕ) (hq : 0 < qf) (hp : 0 < p) |
| (R xi : ℝ) (hR : 0 < R) (hxi : 0 ≤ xi) (hxi1 : xi ≤ 1) |
| (hbin : R < (p : ℝ) ∧ (p : ℝ) ≤ (1+xi)*R) : |
| (commonLength Y qf R xi : ℝ) ≤ (Y : ℝ)/((p : ℝ)*qf) ∧ |
| (Y : ℝ)/((p : ℝ)*qf)-(commonLength Y qf R xi : ℝ) ≤ |
| xi*(commonLength Y qf R xi : ℝ)+2 := by |
| have hqR : (0 : ℝ) < qf := by exact_mod_cast hq |
| have hpR : (0 : ℝ) < p := by exact_mod_cast hp |
| have hU : 0 < (1+xi)*R := by positivity |
| have hratio : 0 ≤ (Y : ℝ)/((qf : ℝ)*((1+xi)*R)) := by positivity |
| have hfloor : (commonLength Y qf R xi : ℝ) ≤ (Y : ℝ)/((qf : ℝ)*((1+xi)*R)) := Nat.floor_le hratio |
| have hfloorHi : (Y : ℝ)/((qf : ℝ)*((1+xi)*R)) < (commonLength Y qf R xi : ℝ)+1 := Nat.lt_floor_add_one _ |
| have hcomp := bin_upper_comparison R xi hR hxi p hbin |
| have hdiv : (Y : ℝ)/((qf : ℝ)*((1+xi)*R)) ≤ (Y : ℝ)/((p : ℝ)*qf) := |
| div_le_div_of_nonneg_left (Nat.cast_nonneg _) (by positivity) (by nlinarith) |
| have hinfl : (Y : ℝ)/((p : ℝ)*qf) ≤ (1+xi)*((Y : ℝ)/((qf : ℝ)*((1+xi)*R))) := by |
| have hf : ((1+xi)*R)/(p : ℝ) ≤ 1+xi := (div_le_iff₀ hpR).mpr hcomp.2 |
| calc |
| _ = ((Y : ℝ)/((qf : ℝ)*((1+xi)*R)))*(((1+xi)*R)/(p : ℝ)) := by field_simp |
| _ ≤ ((Y : ℝ)/((qf : ℝ)*((1+xi)*R)))*(1+xi) := mul_le_mul_of_nonneg_left hf hratio |
| _ = _ := by ring |
| refine ⟨hfloor.trans hdiv,?_⟩ |
| have hh := mul_le_mul_of_nonneg_left hfloorHi.le (show 0 ≤ 1+xi by linarith) |
| nlinarith |
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| end ErdosVarianceCommon |
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| end |
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| end Erdos970 |
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| end OAI |
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