| import OAI.NumberTheory.Jacobsthal.Estimates.PositiveProgression |
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| namespace OAI |
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| namespace Erdos970 |
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| section |
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| namespace ErdosInverseTail |
| open ErdosInverseHits |
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| noncomputable def commonParentLength (Y p : ℕ) (Qplus : ℝ) : ℕ := |
| ⌊(Y : ℝ)/((p : ℝ)*Qplus)⌋₊ |
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| theorem commonParentLength_le_hits (Y p q b : ℕ) (hp : 0 < p) (hq : 0 < q) |
| (hb : b ≤ p*q) (Qplus : ℝ) (hqQ : (q : ℝ) ≤ Qplus) : |
| commonParentLength Y p Qplus ≤ hitLength Y (p*q) b := by |
| have hpR : (0 : ℝ) < p := by exact_mod_cast hp |
| have hqR : (0 : ℝ) < q := by exact_mod_cast hq |
| have hQ : 0 < Qplus := hqR.trans_le hqQ |
| have hA : 0 ≤ (Y : ℝ)/((p : ℝ)*Qplus) := by positivity |
| have hJ : (commonParentLength Y p Qplus : ℝ) ≤ (Y : ℝ)/((p : ℝ)*Qplus) := Nat.floor_le hA |
| have hden : (p : ℝ)*q ≤ (p : ℝ)*Qplus := mul_le_mul_of_nonneg_left hqQ hpR.le |
| have hJB : (commonParentLength Y p Qplus : ℝ) ≤ (Y : ℝ)/((p : ℝ)*q) := |
| hJ.trans (div_le_div_of_nonneg_left (Nat.cast_nonneg _) (mul_pos hpR hqR) hden) |
| have hnat : commonParentLength Y p Qplus*(p*q) ≤ Y := by |
| exact_mod_cast (le_div_iff₀ (mul_pos hpR hqR)).mp hJB |
| have hdiv : commonParentLength Y p Qplus ≤ Y/(p*q) := (Nat.le_div_iff_mul_le (mul_pos hp hq)).mpr hnat |
| exact hdiv.trans (Nat.div_le_div_right (by omega : Y ≤ Y+p*q-b)) |
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| theorem actual_cell_tail_length (Y p q b : ℕ) (hp : 0 < p) (hq : 0 < q) |
| (hb1 : 1 ≤ b) (hb : b ≤ p*q) (Qplus xi : ℝ) |
| (hxi : 0 ≤ xi) (hxi1 : xi ≤ 1) (hqQ : (q : ℝ) ≤ Qplus) (hQq : Qplus ≤ (1+xi)*(q : ℝ)) : |
| ((hitLength Y (p*q) b-commonParentLength Y p Qplus : ℕ) : ℝ) ≤ |
| xi*(commonParentLength Y p Qplus : ℝ)+3 := by |
| have hpR : (0 : ℝ) < p := by exact_mod_cast hp |
| have hqR : (0 : ℝ) < q := by exact_mod_cast hq |
| have hQ : 0 < Qplus := hqR.trans_le hqQ |
| have hA : 0 ≤ (Y : ℝ)/((p : ℝ)*Qplus) := by positivity |
| have hJlt : (Y : ℝ)/((p : ℝ)*Qplus) < (commonParentLength Y p Qplus : ℝ)+1 := Nat.lt_floor_add_one _ |
| have hratio : Qplus/(q : ℝ) ≤ 1+xi := (div_le_iff₀ hqR).mpr hQq |
| have hB : (Y : ℝ)/((p : ℝ)*q) ≤ (1+xi)*((Y : ℝ)/((p : ℝ)*Qplus)) := by |
| calc |
| _ = ((Y : ℝ)/((p : ℝ)*Qplus))*(Qplus/(q : ℝ)) := by field_simp |
| _ ≤ ((Y : ℝ)/((p : ℝ)*Qplus))*(1+xi) := mul_le_mul_of_nonneg_left hratio hA |
| _ = _ := by ring |
| have herr := hitLength_error Y (p*q) b (mul_pos hp hq) hb1 hb |
| have hN : (hitLength Y (p*q) b : ℝ) ≤ (Y : ℝ)/((p : ℝ)*q)+1 := by |
| have hh := (abs_le.mp herr).2 |
| push_cast at hh |
| linarith |
| have hJN := commonParentLength_le_hits Y p q b hp hq hb Qplus hqQ |
| rw [Nat.cast_sub hJN] |
| have hpay := mul_le_mul_of_nonneg_left hJlt.le (by linarith : 0 ≤ 1+xi) |
| nlinarith |
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| end ErdosInverseTail |
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| end |
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| section |
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| namespace ErdosInverseTail |
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| theorem common_length_from_log (Y p q u : ℕ) (hp : 0 < p) (hq : 0 < q) (hu : 0 < u) |
| (Qplus xi w aStar : ℝ) (hqQ : (q : ℝ) ≤ Qplus) (hQq : Qplus ≤ (1+xi)*(q : ℝ)) |
| (hxi : 0 ≤ xi) (hxi1 : xi ≤ 1) (hw : 1 < w) (ha : 0 < aStar) |
| (hlen : 3*aStar/4 ≤ Real.log ((Y : ℝ)/((p : ℝ)*q*u))/Real.log w) : |
| w^(3*aStar/4) ≤ 2*(commonParentLength Y p Qplus : ℝ)/(u : ℝ)+2 := by |
| have hpR : (0 : ℝ) < p := by exact_mod_cast hp |
| have hqR : (0 : ℝ) < q := by exact_mod_cast hq |
| have huR : (0 : ℝ) < u := by exact_mod_cast hu |
| have hu1 : (1 : ℝ) ≤ u := by exact_mod_cast hu |
| have hQ : 0 < Qplus := hqR.trans_le hqQ |
| have hw0 : 0 < w := by linarith |
| have hlogw : 0 < Real.log w := Real.log_pos hw |
| have hlog := (le_div_iff₀ hlogw).mp hlen |
| have hlogpos : 0 < Real.log ((Y : ℝ)/((p : ℝ)*q*u)) := by nlinarith |
| have hR1 : 1 < (Y : ℝ)/((p : ℝ)*q*u) := (Real.log_pos_iff (by positivity)).mp hlogpos |
| have hpow : w^(3*aStar/4) ≤ (Y : ℝ)/((p : ℝ)*q*u) := by |
| rw [Real.rpow_def_of_pos hw0] |
| calc |
| _ ≤ Real.exp (Real.log ((Y : ℝ)/((p : ℝ)*q*u))) := Real.exp_le_exp.mpr (by nlinarith [hlog]) |
| _ = _ := Real.exp_log (by linarith) |
| have hA : 0 ≤ (Y : ℝ)/((p : ℝ)*Qplus) := by positivity |
| have hJlt : (Y : ℝ)/((p : ℝ)*Qplus) < (commonParentLength Y p Qplus : ℝ)+1 := Nat.lt_floor_add_one _ |
| have hratio : Qplus/(q : ℝ) ≤ 1+xi := (div_le_iff₀ hqR).mpr hQq |
| have hRupper : (Y : ℝ)/((p : ℝ)*q*u) ≤ 2*((commonParentLength Y p Qplus : ℝ)+1)/(u : ℝ) := by |
| calc |
| _ = ((Y : ℝ)/((p : ℝ)*Qplus))*(Qplus/(q : ℝ))/(u : ℝ) := by field_simp |
| _ ≤ ((Y : ℝ)/((p : ℝ)*Qplus))*(1+xi)/(u : ℝ) := |
| div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left hratio hA) huR.le |
| _ ≤ ((commonParentLength Y p Qplus : ℝ)+1)*(1+xi)/(u : ℝ) := |
| div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_right hJlt.le (by linarith)) huR.le |
| _ ≤ 2*((commonParentLength Y p Qplus : ℝ)+1)/(u : ℝ) := by |
| apply div_le_div_of_nonneg_right _ huR.le |
| nlinarith [show (0 : ℝ) ≤ commonParentLength Y p Qplus from Nat.cast_nonneg _] |
| have htwo : (2 : ℝ)/(u : ℝ) ≤ 2 := (div_le_iff₀ huR).mpr (by linarith) |
| calc |
| _ ≤ (Y : ℝ)/((p : ℝ)*q*u) := hpow |
| _ ≤ 2*((commonParentLength Y p Qplus : ℝ)+1)/(u : ℝ) := hRupper |
| _ = 2*(commonParentLength Y p Qplus : ℝ)/(u : ℝ)+2/(u : ℝ) := by ring |
| _ ≤ _ := add_le_add le_rfl htwo |
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| end ErdosInverseTail |
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| end |
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| end Erdos970 |
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| end OAI |
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