PrimeNumberTheoremAnd

We can now obtain an upper bound on $E_\pi$ in terms of $E_\theta$:

Next, we bound the integral appearing in Sublemma \ref{fks2-eq-17}.

We obtain our final bound for converting bounds on $E_\theta$ to bounds on $E_\pi$. -/

/- The following lemmas are used for theorem_3. -/

-- Helper: admissible_bound is linear in A lemma admissible_bound_mul (A K B C R x : ℝ) : admissible_bound (A * K) B C R x = K * admissible_bound A B C R x := by simp [admissible_bound]; ring

/- Helper: the ratio log x / (x * admissible_bound A B C R x) equals R^B / A * g_bound 1 (1-B) (C/√R) x

We obtain our final bound for converting bounds on $E_\theta$ to bounds on $E_\pi$. -/

/- The following lemmas are used for theorem_3. -/

-- Helper: admissible_bound is linear in A lemma admissible_bound_mul (A K B C R x : ℝ) : admissible_bound (A * K) B C R x = K * admissible_bound A B C R x := by simp [admissible_bound]; ring

/- Helper: the ratio log x / (x * admissible_bound A B C R x) equals R^B / A * g_bound 1 (1-B) (C/√R) x -/ lemma ratio_eq_g {A B C R x : ℝ} (hR : R > 0) (hx : x > 0) (hlogx : log x > 0) : log x / (x * admissible_bound A B C R x) = R ^ B / A * g_bound 1 (1 - B) (C / sqrt R) x := by unfold admissible_bound g_bound; ring_nf; rw [ Real.mul_rpow ( by positivity ) ( by positivity ), Real.inv_rpow ( by positivity ) ] ; norm_num [ Real.rpow_sub, Real.rpow_neg, Real.sqrt_mul, hR.le, hx.le, hlogx.le ] ; ring_nf; rw [ Real.rpow_sub hlogx, Real.rpow_one ] ; norm_num [ Real.exp_neg ] ; ring_nf; next => norm_num left rw [show (log x * R⁻¹) ^ (1 / 2 : ℝ) = Real.sqrt (log x * R⁻¹) by rw [Real.sqrt_eq_rpow]] rw [Real.sqrt_mul, Real.sqrt_inv] <;> linarith

/- Helper: for x ≥ x₁, the ratio log x / (x * admissible_bound) is at most the value at x₁

We obtain our final bound for converting bounds on $E_\theta$ to bounds on $E_\pi$. -/

/- The following lemmas are used for theorem_3. -/

-- Helper: admissible_bound is linear in A lemma admissible_bound_mul (A K B C R x : ℝ) : admissible_bound (A * K) B C R x = K * admissible_bound A B C R x := by simp [admissible_bound]; ring

/- Helper: the ratio log x / (x * admissible_bound A B C R x) equals R^B / A * g_bound 1 (1-B) (C/√R) x -/ lemma ratio_eq_g {A B C R x : ℝ} (hR : R > 0) (hx : x > 0) (hlogx : log x > 0) : log x / (x * admissible_bound A B C R x) = R ^ B / A * g_bound 1 (1 - B) (C / sqrt R) x := by unfold admissible_bound g_bound; ring_nf; rw [ Real.mul_rpow ( by positivity ) ( by positivity ), Real.inv_rpow ( by positivity ) ] ; norm_num [ Real.rpow_sub, Real.rpow_neg, Real.sqrt_mul, hR.le, hx.le, hlogx.le ] ; ring_nf; rw [ Real.rpow_sub hlogx, Real.rpow_one ] ; norm_num [ Real.exp_neg ] ; ring_nf; next => norm_num left rw [show (log x * R⁻¹) ^ (1 / 2 : ℝ) = Real.sqrt (log x * R⁻¹) by rw [Real.sqrt_eq_rpow]] rw [Real.sqrt_mul, Real.sqrt_inv] <;> linarith

/- Helper: for x ≥ x₁, the ratio log x / (x * admissible_bound) is at most the value at x₁ -/ lemma ratio_mono {A B C R x₁ x : ℝ} (hB : B ≥ 1 + C ^ 2 / (16 * R)) (hR : R > 0) (hx1 : x₁ > 1) (hx : x ≥ x₁) (hA : A > 0) : log x / (x * admissible_bound A B C R x) ≤ log x₁ / (x₁ * admissible_bound A B C R x₁) := by have h_ratio_simplified : R ^ B / A * g_bound 1 (1 - B) (C / Real.sqrt R) x ≤ R ^ B / A * g_bound 1 (1 - B) (C / Real.sqrt R) x₁ := by have h_decreasing : ∀ x y : ℝ, 1 < x → x ≤ y → g_bound 1 (1 - B) (C / Real.sqrt R) x ≥ g_bound 1 (1 - B) (C / Real.sqrt R) y := by intros x y hx hy have h_deriv_neg : ∀ x : ℝ, 1 < x → deriv (g_bound 1 (1 - B) (C / Real.sqrt R)) x ≤ 0 := by intros x hx have h_deriv_neg : deriv (g_bound 1 (1 - B) (C / Real.sqrt R)) x = (-1 * Real.log x + (1 - B) + (C / (2 * Real.sqrt R)) * Real.sqrt (Real.log x)) * x ^ (-2 : ℝ) * (Real.log x) ^ ((1 - B) - 1) * Real.exp ((C / Real.sqrt R) * Real.sqrt (Real.log x)) := by field_simp; rw [ lemma_10_substep hx ] ; ring_nf ; norm_num [ Real.sqrt_ne_zero'.mpr hR, Real.sqrt_ne_zero'.mpr ( Real.log_pos hx ), Real.rpow_neg_one ] ; ring_nf; grind generalize_proofs at *; ( have h_quad_neg : ∀ u : ℝ, u > 0 → -u^2 + (C / (2 * Real.sqrt R)) * u + (1 - B) ≤ 0 := by intros u hu have h_quad_neg : -u^2 + (C / (2 * Real.sqrt R)) * u + (1 - B) ≤ 0 := by have h_discriminant : (C / (2 * Real.sqrt R))^2 - 4 * (-1) * (1 - B) ≤ 0 := by field_simp at *; ring_nf at *; nlinarith [ inv_mul_cancel₀ ( ne_of_gt hR ), inv_pos.mpr hR, Real.sqrt_nonneg R, Real.sq_sqrt hR.le, mul_inv_cancel₀ ( ne_of_gt ( Real.sqrt_pos.mpr hR ) ), Real.sqrt_nonneg ( R⁻¹ ), Real.sq_sqrt ( inv_nonneg.mpr hR.le ) ] ; linarith [ sq_nonneg ( C / ( 2 * Real.sqrt R ) - 2 * u ) ] generalize_proofs at *; (exact h_quad_neg) generalize_proofs at *; ( exact h_deriv_neg.symm ▸ mul_nonpos_of_nonpos_of_nonneg ( mul_nonpos_of_nonpos_of_nonneg ( mul_nonpos_of_nonpos_of_nonneg ( by have := h_quad_neg ( Real.sqrt ( Real.log x ) ) ( Real.sqrt_pos.mpr ( Real.log_pos hx ) ) ; rw [ Real.sq_sqrt ( Real.log_nonneg hx.le ) ] at this; linarith ) ( by positivity ) ) ( by exact Real.rpow_nonneg ( Real.log_nonneg hx.le ) _ ) ) ( by positivity ) ;)) generalize_proofs at *; ( by_contra h_contra; have := exists_deriv_eq_slope ( g_bound 1 ( 1 - B ) ( C / Real.sqrt R ) ) ( show x < y from hy.lt_of_ne ( by rintro rfl; linarith ) ) ; norm_num at this; exact absurd ( this ( by exact ContinuousOn.mono ( show ContinuousOn ( g_bound 1 ( 1 - B ) ( C / Real.sqrt R ) ) ( Set.Icc x y ) from by exact ContinuousOn.mul ( ContinuousOn.mul ( ContinuousOn.rpow continuousOn_id continuousOn_const <| by intro u hu; exact Or.inl <| by linarith [ hu.1 ] ) <| ContinuousOn.rpow ( Real.continuousOn_log.mono <| by exact fun u hu => ne_of_gt <| by linarith [ hu.1 ] ) continuousOn_const <| by intro u hu; exact Or.inl <| ne_of_gt <| Real.log_pos <| by linarith [ hu.1 ] ) <| ContinuousOn.rexp <| ContinuousOn.mul continuousOn_const <| ContinuousOn.sqrt <| Real.continuousOn_log.mono <| by exact fun u hu => ne_of_gt <| by linarith [ hu.1 ] ) <| Set.Icc_subset_Icc ( by linarith ) le_rfl ) <| by exact fun u hu => DifferentiableAt.differentiableWithinAt <| by exact DifferentiableAt.mul ( DifferentiableAt.mul ( DifferentiableAt.rpow ( differentiableAt_id ) ( differentiableAt_const _ ) <| by linarith [ hu.1 ] ) <| DifferentiableAt.rpow ( Real.differentiableAt_log <| by linarith [ hu.1 ] ) ( differentiableAt_const _ ) <| by exact ne_of_gt <| Real.log_pos <| by linarith [ hu.1 ] ) <| DifferentiableAt.exp <| DifferentiableAt.mul ( differentiableAt_const _ ) <| DifferentiableAt.sqrt ( Real.differentiableAt_log <| by linarith [ hu.1 ] ) <| by exact ne_of_gt <| Real.log_pos <| by linarith [ hu.1 ] ) ( by rintro ⟨ c, ⟨ hxc, hcy ⟩, hcd ⟩ ; rw [ eq_div_iff ] at hcd <;> nlinarith [ h_deriv_neg c <| by linarith ] ) ;); exact mul_le_mul_of_nonneg_left ( h_decreasing _ _ hx1 hx ) ( by positivity ); convert h_ratio_simplified using 1 <;> norm_num [ ratio_eq_g hR ( by linarith : 0 < x ) ( Real.log_pos ( by linarith ) ), ratio_eq_g hR ( by linarith : 0 < x₁ ) ( Real.log_pos ( by linarith ) ) ]

/- Helper: for B ≥ 3/2 and x ≥ x₁ ≥ x₀, the m(x₀,x)*(log x)^(1-B) factor simplifies

We obtain our final bound for converting bounds on $E_\theta$ to bounds on $E_\pi$. -/

/- The following lemmas are used for theorem_3. -/

-- Helper: admissible_bound is linear in A lemma admissible_bound_mul (A K B C R x : ℝ) : admissible_bound (A * K) B C R x = K * admissible_bound A B C R x := by simp [admissible_bound]; ring

/- Helper: the ratio log x / (x * admissible_bound A B C R x) equals R^B / A * g_bound 1 (1-B) (C/√R) x -/ lemma ratio_eq_g {A B C R x : ℝ} (hR : R > 0) (hx : x > 0) (hlogx : log x > 0) : log x / (x * admissible_bound A B C R x) = R ^ B / A * g_bound 1 (1 - B) (C / sqrt R) x := by unfold admissible_bound g_bound; ring_nf; rw [ Real.mul_rpow ( by positivity ) ( by positivity ), Real.inv_rpow ( by positivity ) ] ; norm_num [ Real.rpow_sub, Real.rpow_neg, Real.sqrt_mul, hR.le, hx.le, hlogx.le ] ; ring_nf; rw [ Real.rpow_sub hlogx, Real.rpow_one ] ; norm_num [ Real.exp_neg ] ; ring_nf; next => norm_num left rw [show (log x * R⁻¹) ^ (1 / 2 : ℝ) = Real.sqrt (log x * R⁻¹) by rw [Real.sqrt_eq_rpow]] rw [Real.sqrt_mul, Real.sqrt_inv] <;> linarith

/- Helper: for x ≥ x₁, the ratio log x / (x * admissible_bound) is at most the value at x₁ -/ lemma ratio_mono {A B C R x₁ x : ℝ} (hB : B ≥ 1 + C ^ 2 / (16 * R)) (hR : R > 0) (hx1 : x₁ > 1) (hx : x ≥ x₁) (hA : A > 0) : log x / (x * admissible_bound A B C R x) ≤ log x₁ / (x₁ * admissible_bound A B C R x₁) := by have h_ratio_simplified : R ^ B / A * g_bound 1 (1 - B) (C / Real.sqrt R) x ≤ R ^ B / A * g_bound 1 (1 - B) (C / Real.sqrt R) x₁ := by have h_decreasing : ∀ x y : ℝ, 1 < x → x ≤ y → g_bound 1 (1 - B) (C / Real.sqrt R) x ≥ g_bound 1 (1 - B) (C / Real.sqrt R) y := by intros x y hx hy have h_deriv_neg : ∀ x : ℝ, 1 < x → deriv (g_bound 1 (1 - B) (C / Real.sqrt R)) x ≤ 0 := by intros x hx have h_deriv_neg : deriv (g_bound 1 (1 - B) (C / Real.sqrt R)) x = (-1 * Real.log x + (1 - B) + (C / (2 * Real.sqrt R)) * Real.sqrt (Real.log x)) * x ^ (-2 : ℝ) * (Real.log x) ^ ((1 - B) - 1) * Real.exp ((C / Real.sqrt R) * Real.sqrt (Real.log x)) := by field_simp; rw [ lemma_10_substep hx ] ; ring_nf ; norm_num [ Real.sqrt_ne_zero'.mpr hR, Real.sqrt_ne_zero'.mpr ( Real.log_pos hx ), Real.rpow_neg_one ] ; ring_nf; grind generalize_proofs at *; ( have h_quad_neg : ∀ u : ℝ, u > 0 → -u^2 + (C / (2 * Real.sqrt R)) * u + (1 - B) ≤ 0 := by intros u hu have h_quad_neg : -u^2 + (C / (2 * Real.sqrt R)) * u + (1 - B) ≤ 0 := by have h_discriminant : (C / (2 * Real.sqrt R))^2 - 4 * (-1) * (1 - B) ≤ 0 := by field_simp at *; ring_nf at *; nlinarith [ inv_mul_cancel₀ ( ne_of_gt hR ), inv_pos.mpr hR, Real.sqrt_nonneg R, Real.sq_sqrt hR.le, mul_inv_cancel₀ ( ne_of_gt ( Real.sqrt_pos.mpr hR ) ), Real.sqrt_nonneg ( R⁻¹ ), Real.sq_sqrt ( inv_nonneg.mpr hR.le ) ] ; linarith [ sq_nonneg ( C / ( 2 * Real.sqrt R ) - 2 * u ) ] generalize_proofs at *; (exact h_quad_neg) generalize_proofs at *; ( exact h_deriv_neg.symm ▸ mul_nonpos_of_nonpos_of_nonneg ( mul_nonpos_of_nonpos_of_nonneg ( mul_nonpos_of_nonpos_of_nonneg ( by have := h_quad_neg ( Real.sqrt ( Real.log x ) ) ( Real.sqrt_pos.mpr ( Real.log_pos hx ) ) ; rw [ Real.sq_sqrt ( Real.log_nonneg hx.le ) ] at this; linarith ) ( by positivity ) ) ( by exact Real.rpow_nonneg ( Real.log_nonneg hx.le ) _ ) ) ( by positivity ) ;)) generalize_proofs at *; ( by_contra h_contra; have := exists_deriv_eq_slope ( g_bound 1 ( 1 - B ) ( C / Real.sqrt R ) ) ( show x < y from hy.lt_of_ne ( by rintro rfl; linarith ) ) ; norm_num at this; exact absurd ( this ( by exact ContinuousOn.mono ( show ContinuousOn ( g_bound 1 ( 1 - B ) ( C / Real.sqrt R ) ) ( Set.Icc x y ) from by exact ContinuousOn.mul ( ContinuousOn.mul ( ContinuousOn.rpow continuousOn_id continuousOn_const <| by intro u hu; exact Or.inl <| by linarith [ hu.1 ] ) <| ContinuousOn.rpow ( Real.continuousOn_log.mono <| by exact fun u hu => ne_of_gt <| by linarith [ hu.1 ] ) continuousOn_const <| by intro u hu; exact Or.inl <| ne_of_gt <| Real.log_pos <| by linarith [ hu.1 ] ) <| ContinuousOn.rexp <| ContinuousOn.mul continuousOn_const <| ContinuousOn.sqrt <| Real.continuousOn_log.mono <| by exact fun u hu => ne_of_gt <| by linarith [ hu.1 ] ) <| Set.Icc_subset_Icc ( by linarith ) le_rfl ) <| by exact fun u hu => DifferentiableAt.differentiableWithinAt <| by exact DifferentiableAt.mul ( DifferentiableAt.mul ( DifferentiableAt.rpow ( differentiableAt_id ) ( differentiableAt_const _ ) <| by linarith [ hu.1 ] ) <| DifferentiableAt.rpow ( Real.differentiableAt_log <| by linarith [ hu.1 ] ) ( differentiableAt_const _ ) <| by exact ne_of_gt <| Real.log_pos <| by linarith [ hu.1 ] ) <| DifferentiableAt.exp <| DifferentiableAt.mul ( differentiableAt_const _ ) <| DifferentiableAt.sqrt ( Real.differentiableAt_log <| by linarith [ hu.1 ] ) <| by exact ne_of_gt <| Real.log_pos <| by linarith [ hu.1 ] ) ( by rintro ⟨ c, ⟨ hxc, hcy ⟩, hcd ⟩ ; rw [ eq_div_iff ] at hcd <;> nlinarith [ h_deriv_neg c <| by linarith ] ) ;); exact mul_le_mul_of_nonneg_left ( h_decreasing _ _ hx1 hx ) ( by positivity ); convert h_ratio_simplified using 1 <;> norm_num [ ratio_eq_g hR ( by linarith : 0 < x ) ( Real.log_pos ( by linarith ) ), ratio_eq_g hR ( by linarith : 0 < x₁ ) ( Real.log_pos ( by linarith ) ) ]

/- Helper: for B ≥ 3/2 and x ≥ x₁ ≥ x₀, the m(x₀,x)*(log x)^(1-B) factor simplifies -/ lemma m_simplify {B x₀ x x₁ : ℝ} (hB : B ≥ 3 / 2) (hx₀ : x₀ > 1) (hx₁ : x₁ > 1) (hx₁x₀ : x₁ ≥ x₀) (hx : x ≥ x₁) (hlogx : log x > 0) : max ((log x₀) ^ ((2 * B - 3) / 2)) ((log x) ^ ((2 * B - 3) / 2)) * (log x) ^ (1 - B) ≤ 1 / sqrt (log x₁) := by rw [ max_def ]; split_ifs <;> norm_num [ Real.sqrt_eq_rpow, ← Real.rpow_add hlogx ]; · rw [ ← Real.rpow_neg ( Real.log_nonneg hx₁.le ) ] ; ring_nf ; norm_num [ hlogx ]; rw [ Real.rpow_le_rpow_iff_of_neg ] <;> linarith [ Real.log_pos hx₁, Real.log_le_log ( by linarith ) hx ]; · exact False.elim <| ‹¬log x₀ ^ ( ( 2 * B - 3 ) / 2 ) ≤ log x ^ ( ( 2 * B - 3 ) / 2 ) › <| Real.rpow_le_rpow ( Real.log_nonneg <| by linarith ) ( Real.log_le_log ( by linarith ) <| by linarith ) <| by linarith;

/- Helper: admissible_bound is positive under suitable conditions -/ lemma admissible_bound_pos {A B C R x : ℝ} (hA : A > 0) (hR : R > 0) (hlogx : log x > 0) : admissible_bound A B C R x > 0 := by exact mul_pos ( mul_pos hA ( Real.rpow_pos_of_pos ( div_pos hlogx hR ) _ ) ) ( Real.exp_pos _ )

/- Helper: Eθ is non-negative -/ lemma Eθ_nonneg (x : ℝ) (hx : x > 0) : Eθ x ≥ 0 := by exact div_nonneg ( abs_nonneg _ ) hx.le

/- Helper: δ is non-negative -/ lemma delta_nonneg (x₀ : ℝ) : δ x₀ ≥ 0 := by exact abs_nonneg _

/- Helper: from ratio_mono, derive the bound on log x / x in terms of admissible_bound -/ lemma logx_over_x_bound {A B C R x₁ x : ℝ} (hB : B ≥ 1 + C ^ 2 / (16 * R)) (hR : R > 0) (hA : A > 0) (hx1_gt1 : x₁ > 1) (hx : x ≥ x₁) : log x / x ≤ (log x₁ / (x₁ * admissible_bound A B C R x₁)) * admissible_bound A B C R x := by convert! mul_le_mul_of_nonneg_right ( ratio_mono hB hR hx1_gt1 hx hA ) ( admissible_bound_pos hA hR ( Real.log_pos <| show 1 < x by linarith ) |> le_of_lt ) using 1 ; ring_nf; norm_num [ ne_of_gt ( admissible_bound_pos hA hR ( Real.log_pos <| show 1 < x by linarith ) ) ]

/- PROBLEM Bound the delta term from eq_30.

PROVIDED SOLUTION Use logx_over_x_bound to get log x / x ≤ (log x₁ / (x₁ * admissible_bound A B C R x₁)) * admissible_bound A B C R x. Multiply both sides by (x₀ / log x₀) * δ x₀ (which is non-negative since δ is non-negative by delta_nonneg and x₀/log x₀ > 0 from hlogx₀). The LHS becomes (log x / x) * (x₀ / log x₀) * δ x₀. The RHS becomes (log x₁ / (x₁ * admissible_bound A B C R x₁)) * (x₀ / log x₀) * δ x₀ * admissible_bound A B C R x which equals (x₀ * log x₁) / (admissible_bound A B C R x₁ * x₁ * log x₀) * δ x₀ * admissible_bound A B C R x by algebraic rearrangement.

We obtain our final bound for converting bounds on $E_\theta$ to bounds on $E_\pi$. -/

/- The following lemmas are used for theorem_3. -/

-- Helper: admissible_bound is linear in A lemma admissible_bound_mul (A K B C R x : ℝ) : admissible_bound (A * K) B C R x = K * admissible_bound A B C R x := by simp [admissible_bound]; ring

/- Helper: the ratio log x / (x * admissible_bound A B C R x) equals R^B / A * g_bound 1 (1-B) (C/√R) x -/ lemma ratio_eq_g {A B C R x : ℝ} (hR : R > 0) (hx : x > 0) (hlogx : log x > 0) : log x / (x * admissible_bound A B C R x) = R ^ B / A * g_bound 1 (1 - B) (C / sqrt R) x := by unfold admissible_bound g_bound; ring_nf; rw [ Real.mul_rpow ( by positivity ) ( by positivity ), Real.inv_rpow ( by positivity ) ] ; norm_num [ Real.rpow_sub, Real.rpow_neg, Real.sqrt_mul, hR.le, hx.le, hlogx.le ] ; ring_nf; rw [ Real.rpow_sub hlogx, Real.rpow_one ] ; norm_num [ Real.exp_neg ] ; ring_nf; next => norm_num left rw [show (log x * R⁻¹) ^ (1 / 2 : ℝ) = Real.sqrt (log x * R⁻¹) by rw [Real.sqrt_eq_rpow]] rw [Real.sqrt_mul, Real.sqrt_inv] <;> linarith

/- Helper: for x ≥ x₁, the ratio log x / (x * admissible_bound) is at most the value at x₁ -/ lemma ratio_mono {A B C R x₁ x : ℝ} (hB : B ≥ 1 + C ^ 2 / (16 * R)) (hR : R > 0) (hx1 : x₁ > 1) (hx : x ≥ x₁) (hA : A > 0) : log x / (x * admissible_bound A B C R x) ≤ log x₁ / (x₁ * admissible_bound A B C R x₁) := by have h_ratio_simplified : R ^ B / A * g_bound 1 (1 - B) (C / Real.sqrt R) x ≤ R ^ B / A * g_bound 1 (1 - B) (C / Real.sqrt R) x₁ := by have h_decreasing : ∀ x y : ℝ, 1 < x → x ≤ y → g_bound 1 (1 - B) (C / Real.sqrt R) x ≥ g_bound 1 (1 - B) (C / Real.sqrt R) y := by intros x y hx hy have h_deriv_neg : ∀ x : ℝ, 1 < x → deriv (g_bound 1 (1 - B) (C / Real.sqrt R)) x ≤ 0 := by intros x hx have h_deriv_neg : deriv (g_bound 1 (1 - B) (C / Real.sqrt R)) x = (-1 * Real.log x + (1 - B) + (C / (2 * Real.sqrt R)) * Real.sqrt (Real.log x)) * x ^ (-2 : ℝ) * (Real.log x) ^ ((1 - B) - 1) * Real.exp ((C / Real.sqrt R) * Real.sqrt (Real.log x)) := by field_simp; rw [ lemma_10_substep hx ] ; ring_nf ; norm_num [ Real.sqrt_ne_zero'.mpr hR, Real.sqrt_ne_zero'.mpr ( Real.log_pos hx ), Real.rpow_neg_one ] ; ring_nf; grind generalize_proofs at *; ( have h_quad_neg : ∀ u : ℝ, u > 0 → -u^2 + (C / (2 * Real.sqrt R)) * u + (1 - B) ≤ 0 := by intros u hu have h_quad_neg : -u^2 + (C / (2 * Real.sqrt R)) * u + (1 - B) ≤ 0 := by have h_discriminant : (C / (2 * Real.sqrt R))^2 - 4 * (-1) * (1 - B) ≤ 0 := by field_simp at *; ring_nf at *; nlinarith [ inv_mul_cancel₀ ( ne_of_gt hR ), inv_pos.mpr hR, Real.sqrt_nonneg R, Real.sq_sqrt hR.le, mul_inv_cancel₀ ( ne_of_gt ( Real.sqrt_pos.mpr hR ) ), Real.sqrt_nonneg ( R⁻¹ ), Real.sq_sqrt ( inv_nonneg.mpr hR.le ) ] ; linarith [ sq_nonneg ( C / ( 2 * Real.sqrt R ) - 2 * u ) ] generalize_proofs at *; (exact h_quad_neg) generalize_proofs at *; ( exact h_deriv_neg.symm ▸ mul_nonpos_of_nonpos_of_nonneg ( mul_nonpos_of_nonpos_of_nonneg ( mul_nonpos_of_nonpos_of_nonneg ( by have := h_quad_neg ( Real.sqrt ( Real.log x ) ) ( Real.sqrt_pos.mpr ( Real.log_pos hx ) ) ; rw [ Real.sq_sqrt ( Real.log_nonneg hx.le ) ] at this; linarith ) ( by positivity ) ) ( by exact Real.rpow_nonneg ( Real.log_nonneg hx.le ) _ ) ) ( by positivity ) ;)) generalize_proofs at *; ( by_contra h_contra; have := exists_deriv_eq_slope ( g_bound 1 ( 1 - B ) ( C / Real.sqrt R ) ) ( show x < y from hy.lt_of_ne ( by rintro rfl; linarith ) ) ; norm_num at this; exact absurd ( this ( by exact ContinuousOn.mono ( show ContinuousOn ( g_bound 1 ( 1 - B ) ( C / Real.sqrt R ) ) ( Set.Icc x y ) from by exact ContinuousOn.mul ( ContinuousOn.mul ( ContinuousOn.rpow continuousOn_id continuousOn_const <| by intro u hu; exact Or.inl <| by linarith [ hu.1 ] ) <| ContinuousOn.rpow ( Real.continuousOn_log.mono <| by exact fun u hu => ne_of_gt <| by linarith [ hu.1 ] ) continuousOn_const <| by intro u hu; exact Or.inl <| ne_of_gt <| Real.log_pos <| by linarith [ hu.1 ] ) <| ContinuousOn.rexp <| ContinuousOn.mul continuousOn_const <| ContinuousOn.sqrt <| Real.continuousOn_log.mono <| by exact fun u hu => ne_of_gt <| by linarith [ hu.1 ] ) <| Set.Icc_subset_Icc ( by linarith ) le_rfl ) <| by exact fun u hu => DifferentiableAt.differentiableWithinAt <| by exact DifferentiableAt.mul ( DifferentiableAt.mul ( DifferentiableAt.rpow ( differentiableAt_id ) ( differentiableAt_const _ ) <| by linarith [ hu.1 ] ) <| DifferentiableAt.rpow ( Real.differentiableAt_log <| by linarith [ hu.1 ] ) ( differentiableAt_const _ ) <| by exact ne_of_gt <| Real.log_pos <| by linarith [ hu.1 ] ) <| DifferentiableAt.exp <| DifferentiableAt.mul ( differentiableAt_const _ ) <| DifferentiableAt.sqrt ( Real.differentiableAt_log <| by linarith [ hu.1 ] ) <| by exact ne_of_gt <| Real.log_pos <| by linarith [ hu.1 ] ) ( by rintro ⟨ c, ⟨ hxc, hcy ⟩, hcd ⟩ ; rw [ eq_div_iff ] at hcd <;> nlinarith [ h_deriv_neg c <| by linarith ] ) ;); exact mul_le_mul_of_nonneg_left ( h_decreasing _ _ hx1 hx ) ( by positivity ); convert h_ratio_simplified using 1 <;> norm_num [ ratio_eq_g hR ( by linarith : 0 < x ) ( Real.log_pos ( by linarith ) ), ratio_eq_g hR ( by linarith : 0 < x₁ ) ( Real.log_pos ( by linarith ) ) ]

/- Helper: for B ≥ 3/2 and x ≥ x₁ ≥ x₀, the m(x₀,x)*(log x)^(1-B) factor simplifies -/ lemma m_simplify {B x₀ x x₁ : ℝ} (hB : B ≥ 3 / 2) (hx₀ : x₀ > 1) (hx₁ : x₁ > 1) (hx₁x₀ : x₁ ≥ x₀) (hx : x ≥ x₁) (hlogx : log x > 0) : max ((log x₀) ^ ((2 * B - 3) / 2)) ((log x) ^ ((2 * B - 3) / 2)) * (log x) ^ (1 - B) ≤ 1 / sqrt (log x₁) := by rw [ max_def ]; split_ifs <;> norm_num [ Real.sqrt_eq_rpow, ← Real.rpow_add hlogx ]; · rw [ ← Real.rpow_neg ( Real.log_nonneg hx₁.le ) ] ; ring_nf ; norm_num [ hlogx ]; rw [ Real.rpow_le_rpow_iff_of_neg ] <;> linarith [ Real.log_pos hx₁, Real.log_le_log ( by linarith ) hx ]; · exact False.elim <| ‹¬log x₀ ^ ( ( 2 * B - 3 ) / 2 ) ≤ log x ^ ( ( 2 * B - 3 ) / 2 ) › <| Real.rpow_le_rpow ( Real.log_nonneg <| by linarith ) ( Real.log_le_log ( by linarith ) <| by linarith ) <| by linarith;

/- Helper: admissible_bound is positive under suitable conditions -/ lemma admissible_bound_pos {A B C R x : ℝ} (hA : A > 0) (hR : R > 0) (hlogx : log x > 0) : admissible_bound A B C R x > 0 := by exact mul_pos ( mul_pos hA ( Real.rpow_pos_of_pos ( div_pos hlogx hR ) _ ) ) ( Real.exp_pos _ )

/- Helper: Eθ is non-negative -/ lemma Eθ_nonneg (x : ℝ) (hx : x > 0) : Eθ x ≥ 0 := by exact div_nonneg ( abs_nonneg _ ) hx.le

/- Helper: δ is non-negative -/ lemma delta_nonneg (x₀ : ℝ) : δ x₀ ≥ 0 := by exact abs_nonneg _

/- Helper: from ratio_mono, derive the bound on log x / x in terms of admissible_bound -/ lemma logx_over_x_bound {A B C R x₁ x : ℝ} (hB : B ≥ 1 + C ^ 2 / (16 * R)) (hR : R > 0) (hA : A > 0) (hx1_gt1 : x₁ > 1) (hx : x ≥ x₁) : log x / x ≤ (log x₁ / (x₁ * admissible_bound A B C R x₁)) * admissible_bound A B C R x := by convert! mul_le_mul_of_nonneg_right ( ratio_mono hB hR hx1_gt1 hx hA ) ( admissible_bound_pos hA hR ( Real.log_pos <| show 1 < x by linarith ) |> le_of_lt ) using 1 ; ring_nf; norm_num [ ne_of_gt ( admissible_bound_pos hA hR ( Real.log_pos <| show 1 < x by linarith ) ) ]

/- PROBLEM Bound the delta term from eq_30.

PROVIDED SOLUTION Use logx_over_x_bound to get log x / x ≤ (log x₁ / (x₁ * admissible_bound A B C R x₁)) * admissible_bound A B C R x. Multiply both sides by (x₀ / log x₀) * δ x₀ (which is non-negative since δ is non-negative by delta_nonneg and x₀/log x₀ > 0 from hlogx₀). The LHS becomes (log x / x) * (x₀ / log x₀) * δ x₀. The RHS becomes (log x₁ / (x₁ * admissible_bound A B C R x₁)) * (x₀ / log x₀) * δ x₀ * admissible_bound A B C R x which equals (x₀ * log x₁) / (admissible_bound A B C R x₁ * x₁ * log x₀) * δ x₀ * admissible_bound A B C R x by algebraic rearrangement. -/ lemma delta_term_bound {A B C R x₀ x₁ x : ℝ} (hB : B ≥ 1 + C ^ 2 / (16 * R)) (hR : R > 0) (hA : A > 0) (hx1_gt1 : x₁ > 1) (hx : x ≥ x₁) (hx0_pos : x₀ > 0) (hlogx₀ : log x₀ > 0) : (log x / x) * (x₀ / log x₀) * δ x₀ ≤ (x₀ * log x₁) / (admissible_bound A B C R x₁ * x₁ * log x₀) * δ x₀ * admissible_bound A B C R x := by have h1 := logx_over_x_bound hB hR hA hx1_gt1 hx have h2 : x₀ / log x₀ ≥ 0 := div_nonneg hx0_pos.le hlogx₀.le have h3 := delta_nonneg x₀ calc (log x / x) * (x₀ / log x₀) * δ x₀ ≤ ((log x₁ / (x₁ * admissible_bound A B C R x₁)) * admissible_bound A B C R x) * (x₀ / log x₀) * δ x₀ := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right h1 h2) h3 _ = (x₀ * log x₁) / (admissible_bound A B C R x₁ * x₁ * log x₀) * δ x₀ * admissible_bound A B C R x := by ring

/- Helper: Dawson function is monotone decreasing for large arguments (after ~0.924) We use remark_after_corollary_11 which gives the existence of a maximum around 0.924 -/ lemma dawson_antitoneOn : ∃ x₀ : ℝ, x₀ < 1 ∧ StrictAntiOn dawson (Set.Ioi x₀) := by obtain ⟨ x₀, hx₀ ⟩ := remark_after_corollary_11; exact ⟨ x₀, by norm_num at hx₀; linarith, hx₀.2.2 ⟩

/- Helper: Eθ t = |Eθ t| since Eθ is non-negative -/ lemma Eθ_eq_abs {t : ℝ} (ht : t > 0) : Eθ t = |Eθ t| := by rw [ abs_of_nonneg ( Eθ_nonneg t ht ) ]

/- Algebraic identity: (log x / x) * (2A/R^B) * x * m * exp(-C*√(log x/R)) * D = 2 * m * (log x)^(1-B) * D * admissible_bound A B C R x

We obtain our final bound for converting bounds on $E_\theta$ to bounds on $E_\pi$. -/

/- The following lemmas are used for theorem_3. -/

-- Helper: admissible_bound is linear in A lemma admissible_bound_mul (A K B C R x : ℝ) : admissible_bound (A * K) B C R x = K * admissible_bound A B C R x := by simp [admissible_bound]; ring

/- Helper: the ratio log x / (x * admissible_bound A B C R x) equals R^B / A * g_bound 1 (1-B) (C/√R) x -/ lemma ratio_eq_g {A B C R x : ℝ} (hR : R > 0) (hx : x > 0) (hlogx : log x > 0) : log x / (x * admissible_bound A B C R x) = R ^ B / A * g_bound 1 (1 - B) (C / sqrt R) x := by unfold admissible_bound g_bound; ring_nf; rw [ Real.mul_rpow ( by positivity ) ( by positivity ), Real.inv_rpow ( by positivity ) ] ; norm_num [ Real.rpow_sub, Real.rpow_neg, Real.sqrt_mul, hR.le, hx.le, hlogx.le ] ; ring_nf; rw [ Real.rpow_sub hlogx, Real.rpow_one ] ; norm_num [ Real.exp_neg ] ; ring_nf; next => norm_num left rw [show (log x * R⁻¹) ^ (1 / 2 : ℝ) = Real.sqrt (log x * R⁻¹) by rw [Real.sqrt_eq_rpow]] rw [Real.sqrt_mul, Real.sqrt_inv] <;> linarith

/- Helper: for x ≥ x₁, the ratio log x / (x * admissible_bound) is at most the value at x₁ -/ lemma ratio_mono {A B C R x₁ x : ℝ} (hB : B ≥ 1 + C ^ 2 / (16 * R)) (hR : R > 0) (hx1 : x₁ > 1) (hx : x ≥ x₁) (hA : A > 0) : log x / (x * admissible_bound A B C R x) ≤ log x₁ / (x₁ * admissible_bound A B C R x₁) := by have h_ratio_simplified : R ^ B / A * g_bound 1 (1 - B) (C / Real.sqrt R) x ≤ R ^ B / A * g_bound 1 (1 - B) (C / Real.sqrt R) x₁ := by have h_decreasing : ∀ x y : ℝ, 1 < x → x ≤ y → g_bound 1 (1 - B) (C / Real.sqrt R) x ≥ g_bound 1 (1 - B) (C / Real.sqrt R) y := by intros x y hx hy have h_deriv_neg : ∀ x : ℝ, 1 < x → deriv (g_bound 1 (1 - B) (C / Real.sqrt R)) x ≤ 0 := by intros x hx have h_deriv_neg : deriv (g_bound 1 (1 - B) (C / Real.sqrt R)) x = (-1 * Real.log x + (1 - B) + (C / (2 * Real.sqrt R)) * Real.sqrt (Real.log x)) * x ^ (-2 : ℝ) * (Real.log x) ^ ((1 - B) - 1) * Real.exp ((C / Real.sqrt R) * Real.sqrt (Real.log x)) := by field_simp; rw [ lemma_10_substep hx ] ; ring_nf ; norm_num [ Real.sqrt_ne_zero'.mpr hR, Real.sqrt_ne_zero'.mpr ( Real.log_pos hx ), Real.rpow_neg_one ] ; ring_nf; grind generalize_proofs at *; ( have h_quad_neg : ∀ u : ℝ, u > 0 → -u^2 + (C / (2 * Real.sqrt R)) * u + (1 - B) ≤ 0 := by intros u hu have h_quad_neg : -u^2 + (C / (2 * Real.sqrt R)) * u + (1 - B) ≤ 0 := by have h_discriminant : (C / (2 * Real.sqrt R))^2 - 4 * (-1) * (1 - B) ≤ 0 := by field_simp at *; ring_nf at *; nlinarith [ inv_mul_cancel₀ ( ne_of_gt hR ), inv_pos.mpr hR, Real.sqrt_nonneg R, Real.sq_sqrt hR.le, mul_inv_cancel₀ ( ne_of_gt ( Real.sqrt_pos.mpr hR ) ), Real.sqrt_nonneg ( R⁻¹ ), Real.sq_sqrt ( inv_nonneg.mpr hR.le ) ] ; linarith [ sq_nonneg ( C / ( 2 * Real.sqrt R ) - 2 * u ) ] generalize_proofs at *; (exact h_quad_neg) generalize_proofs at *; ( exact h_deriv_neg.symm ▸ mul_nonpos_of_nonpos_of_nonneg ( mul_nonpos_of_nonpos_of_nonneg ( mul_nonpos_of_nonpos_of_nonneg ( by have := h_quad_neg ( Real.sqrt ( Real.log x ) ) ( Real.sqrt_pos.mpr ( Real.log_pos hx ) ) ; rw [ Real.sq_sqrt ( Real.log_nonneg hx.le ) ] at this; linarith ) ( by positivity ) ) ( by exact Real.rpow_nonneg ( Real.log_nonneg hx.le ) _ ) ) ( by positivity ) ;)) generalize_proofs at *; ( by_contra h_contra; have := exists_deriv_eq_slope ( g_bound 1 ( 1 - B ) ( C / Real.sqrt R ) ) ( show x < y from hy.lt_of_ne ( by rintro rfl; linarith ) ) ; norm_num at this; exact absurd ( this ( by exact ContinuousOn.mono ( show ContinuousOn ( g_bound 1 ( 1 - B ) ( C / Real.sqrt R ) ) ( Set.Icc x y ) from by exact ContinuousOn.mul ( ContinuousOn.mul ( ContinuousOn.rpow continuousOn_id continuousOn_const <| by intro u hu; exact Or.inl <| by linarith [ hu.1 ] ) <| ContinuousOn.rpow ( Real.continuousOn_log.mono <| by exact fun u hu => ne_of_gt <| by linarith [ hu.1 ] ) continuousOn_const <| by intro u hu; exact Or.inl <| ne_of_gt <| Real.log_pos <| by linarith [ hu.1 ] ) <| ContinuousOn.rexp <| ContinuousOn.mul continuousOn_const <| ContinuousOn.sqrt <| Real.continuousOn_log.mono <| by exact fun u hu => ne_of_gt <| by linarith [ hu.1 ] ) <| Set.Icc_subset_Icc ( by linarith ) le_rfl ) <| by exact fun u hu => DifferentiableAt.differentiableWithinAt <| by exact DifferentiableAt.mul ( DifferentiableAt.mul ( DifferentiableAt.rpow ( differentiableAt_id ) ( differentiableAt_const _ ) <| by linarith [ hu.1 ] ) <| DifferentiableAt.rpow ( Real.differentiableAt_log <| by linarith [ hu.1 ] ) ( differentiableAt_const _ ) <| by exact ne_of_gt <| Real.log_pos <| by linarith [ hu.1 ] ) <| DifferentiableAt.exp <| DifferentiableAt.mul ( differentiableAt_const _ ) <| DifferentiableAt.sqrt ( Real.differentiableAt_log <| by linarith [ hu.1 ] ) <| by exact ne_of_gt <| Real.log_pos <| by linarith [ hu.1 ] ) ( by rintro ⟨ c, ⟨ hxc, hcy ⟩, hcd ⟩ ; rw [ eq_div_iff ] at hcd <;> nlinarith [ h_deriv_neg c <| by linarith ] ) ;); exact mul_le_mul_of_nonneg_left ( h_decreasing _ _ hx1 hx ) ( by positivity ); convert h_ratio_simplified using 1 <;> norm_num [ ratio_eq_g hR ( by linarith : 0 < x ) ( Real.log_pos ( by linarith ) ), ratio_eq_g hR ( by linarith : 0 < x₁ ) ( Real.log_pos ( by linarith ) ) ]

/- Helper: for B ≥ 3/2 and x ≥ x₁ ≥ x₀, the m(x₀,x)*(log x)^(1-B) factor simplifies -/ lemma m_simplify {B x₀ x x₁ : ℝ} (hB : B ≥ 3 / 2) (hx₀ : x₀ > 1) (hx₁ : x₁ > 1) (hx₁x₀ : x₁ ≥ x₀) (hx : x ≥ x₁) (hlogx : log x > 0) : max ((log x₀) ^ ((2 * B - 3) / 2)) ((log x) ^ ((2 * B - 3) / 2)) * (log x) ^ (1 - B) ≤ 1 / sqrt (log x₁) := by rw [ max_def ]; split_ifs <;> norm_num [ Real.sqrt_eq_rpow, ← Real.rpow_add hlogx ]; · rw [ ← Real.rpow_neg ( Real.log_nonneg hx₁.le ) ] ; ring_nf ; norm_num [ hlogx ]; rw [ Real.rpow_le_rpow_iff_of_neg ] <;> linarith [ Real.log_pos hx₁, Real.log_le_log ( by linarith ) hx ]; · exact False.elim <| ‹¬log x₀ ^ ( ( 2 * B - 3 ) / 2 ) ≤ log x ^ ( ( 2 * B - 3 ) / 2 ) › <| Real.rpow_le_rpow ( Real.log_nonneg <| by linarith ) ( Real.log_le_log ( by linarith ) <| by linarith ) <| by linarith;

/- Helper: admissible_bound is positive under suitable conditions -/ lemma admissible_bound_pos {A B C R x : ℝ} (hA : A > 0) (hR : R > 0) (hlogx : log x > 0) : admissible_bound A B C R x > 0 := by exact mul_pos ( mul_pos hA ( Real.rpow_pos_of_pos ( div_pos hlogx hR ) _ ) ) ( Real.exp_pos _ )

/- Helper: Eθ is non-negative -/ lemma Eθ_nonneg (x : ℝ) (hx : x > 0) : Eθ x ≥ 0 := by exact div_nonneg ( abs_nonneg _ ) hx.le

/- Helper: δ is non-negative -/ lemma delta_nonneg (x₀ : ℝ) : δ x₀ ≥ 0 := by exact abs_nonneg _

/- Helper: from ratio_mono, derive the bound on log x / x in terms of admissible_bound -/ lemma logx_over_x_bound {A B C R x₁ x : ℝ} (hB : B ≥ 1 + C ^ 2 / (16 * R)) (hR : R > 0) (hA : A > 0) (hx1_gt1 : x₁ > 1) (hx : x ≥ x₁) : log x / x ≤ (log x₁ / (x₁ * admissible_bound A B C R x₁)) * admissible_bound A B C R x := by convert! mul_le_mul_of_nonneg_right ( ratio_mono hB hR hx1_gt1 hx hA ) ( admissible_bound_pos hA hR ( Real.log_pos <| show 1 < x by linarith ) |> le_of_lt ) using 1 ; ring_nf; norm_num [ ne_of_gt ( admissible_bound_pos hA hR ( Real.log_pos <| show 1 < x by linarith ) ) ]

/- PROBLEM Bound the delta term from eq_30.

PROVIDED SOLUTION Use logx_over_x_bound to get log x / x ≤ (log x₁ / (x₁ * admissible_bound A B C R x₁)) * admissible_bound A B C R x. Multiply both sides by (x₀ / log x₀) * δ x₀ (which is non-negative since δ is non-negative by delta_nonneg and x₀/log x₀ > 0 from hlogx₀). The LHS becomes (log x / x) * (x₀ / log x₀) * δ x₀. The RHS becomes (log x₁ / (x₁ * admissible_bound A B C R x₁)) * (x₀ / log x₀) * δ x₀ * admissible_bound A B C R x which equals (x₀ * log x₁) / (admissible_bound A B C R x₁ * x₁ * log x₀) * δ x₀ * admissible_bound A B C R x by algebraic rearrangement. -/ lemma delta_term_bound {A B C R x₀ x₁ x : ℝ} (hB : B ≥ 1 + C ^ 2 / (16 * R)) (hR : R > 0) (hA : A > 0) (hx1_gt1 : x₁ > 1) (hx : x ≥ x₁) (hx0_pos : x₀ > 0) (hlogx₀ : log x₀ > 0) : (log x / x) * (x₀ / log x₀) * δ x₀ ≤ (x₀ * log x₁) / (admissible_bound A B C R x₁ * x₁ * log x₀) * δ x₀ * admissible_bound A B C R x := by have h1 := logx_over_x_bound hB hR hA hx1_gt1 hx have h2 : x₀ / log x₀ ≥ 0 := div_nonneg hx0_pos.le hlogx₀.le have h3 := delta_nonneg x₀ calc (log x / x) * (x₀ / log x₀) * δ x₀ ≤ ((log x₁ / (x₁ * admissible_bound A B C R x₁)) * admissible_bound A B C R x) * (x₀ / log x₀) * δ x₀ := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right h1 h2) h3 _ = (x₀ * log x₁) / (admissible_bound A B C R x₁ * x₁ * log x₀) * δ x₀ * admissible_bound A B C R x := by ring

/- Helper: Dawson function is monotone decreasing for large arguments (after ~0.924) We use remark_after_corollary_11 which gives the existence of a maximum around 0.924 -/ lemma dawson_antitoneOn : ∃ x₀ : ℝ, x₀ < 1 ∧ StrictAntiOn dawson (Set.Ioi x₀) := by obtain ⟨ x₀, hx₀ ⟩ := remark_after_corollary_11; exact ⟨ x₀, by norm_num at hx₀; linarith, hx₀.2.2 ⟩

/- Helper: Eθ t = |Eθ t| since Eθ is non-negative -/ lemma Eθ_eq_abs {t : ℝ} (ht : t > 0) : Eθ t = |Eθ t| := by rw [ abs_of_nonneg ( Eθ_nonneg t ht ) ]

/- Algebraic identity: (log x / x) * (2A/R^B) * x * m * exp(-C*√(log x/R)) * D = 2 * m * (log x)^(1-B) * D * admissible_bound A B C R x -/ lemma integral_algebra {A B C R x m D : ℝ} (hR : R > 0) (hx : x > 1) : (log x / x) * ((2 * A) / R ^ B * x * m * exp (-C * √(log x / R)) * D) = 2 * m * (log x) ^ (1 - B) * D * admissible_bound A B C R x := by unfold admissible_bound; ring_nf; rw [ Real.rpow_sub ( by linarith [ Real.log_pos hx ] ), Real.rpow_one ] ; ring_nf; rw [ Real.mul_rpow ( by linarith [ Real.log_pos hx ] ) ( by positivity ), Real.inv_rpow ( by positivity ) ] ; ring_nf ; norm_num [ ne_of_gt ( zero_lt_one.trans hx ) ] ; rw [ mul_inv_cancel_right₀ ( ne_of_gt ( Real.rpow_pos_of_pos ( Real.log_pos hx ) _ ) ) ] have heq: Real.exp (-(C * (Real.log x * R⁻¹)^(1/2 : ℝ))) = Real.exp (-(C * Real.sqrt (Real.log x * R⁻¹))) := by congr; rw [Real.sqrt_eq_rpow] rw [heq] linarith

/- Dawson monotonicity for arguments ≥ 1 -/ lemma dawson_mono_ge_one {a b : ℝ} (ha : a ≥ 1) (hab : a ≤ b) : dawson b ≤ dawson a := by obtain ⟨ x₀, hx₀ ⟩ := dawson_antitoneOn; exact hx₀.2.le_iff_ge ( show x₀ < b by linarith ) ( show x₀ < a by linarith ) |>.2 ( by linarith )

/- Derive that √(log x₁) - C/(2√R) ≥ 1 from the hypothesis on x₁

We obtain our final bound for converting bounds on $E_\theta$ to bounds on $E_\pi$. -/

/- The following lemmas are used for theorem_3. -/

-- Helper: admissible_bound is linear in A lemma admissible_bound_mul (A K B C R x : ℝ) : admissible_bound (A * K) B C R x = K * admissible_bound A B C R x := by simp [admissible_bound]; ring

/- Helper: the ratio log x / (x * admissible_bound A B C R x) equals R^B / A * g_bound 1 (1-B) (C/√R) x -/ lemma ratio_eq_g {A B C R x : ℝ} (hR : R > 0) (hx : x > 0) (hlogx : log x > 0) : log x / (x * admissible_bound A B C R x) = R ^ B / A * g_bound 1 (1 - B) (C / sqrt R) x := by unfold admissible_bound g_bound; ring_nf; rw [ Real.mul_rpow ( by positivity ) ( by positivity ), Real.inv_rpow ( by positivity ) ] ; norm_num [ Real.rpow_sub, Real.rpow_neg, Real.sqrt_mul, hR.le, hx.le, hlogx.le ] ; ring_nf; rw [ Real.rpow_sub hlogx, Real.rpow_one ] ; norm_num [ Real.exp_neg ] ; ring_nf; next => norm_num left rw [show (log x * R⁻¹) ^ (1 / 2 : ℝ) = Real.sqrt (log x * R⁻¹) by rw [Real.sqrt_eq_rpow]] rw [Real.sqrt_mul, Real.sqrt_inv] <;> linarith

/- Helper: for x ≥ x₁, the ratio log x / (x * admissible_bound) is at most the value at x₁ -/ lemma ratio_mono {A B C R x₁ x : ℝ} (hB : B ≥ 1 + C ^ 2 / (16 * R)) (hR : R > 0) (hx1 : x₁ > 1) (hx : x ≥ x₁) (hA : A > 0) : log x / (x * admissible_bound A B C R x) ≤ log x₁ / (x₁ * admissible_bound A B C R x₁) := by have h_ratio_simplified : R ^ B / A * g_bound 1 (1 - B) (C / Real.sqrt R) x ≤ R ^ B / A * g_bound 1 (1 - B) (C / Real.sqrt R) x₁ := by have h_decreasing : ∀ x y : ℝ, 1 < x → x ≤ y → g_bound 1 (1 - B) (C / Real.sqrt R) x ≥ g_bound 1 (1 - B) (C / Real.sqrt R) y := by intros x y hx hy have h_deriv_neg : ∀ x : ℝ, 1 < x → deriv (g_bound 1 (1 - B) (C / Real.sqrt R)) x ≤ 0 := by intros x hx have h_deriv_neg : deriv (g_bound 1 (1 - B) (C / Real.sqrt R)) x = (-1 * Real.log x + (1 - B) + (C / (2 * Real.sqrt R)) * Real.sqrt (Real.log x)) * x ^ (-2 : ℝ) * (Real.log x) ^ ((1 - B) - 1) * Real.exp ((C / Real.sqrt R) * Real.sqrt (Real.log x)) := by field_simp; rw [ lemma_10_substep hx ] ; ring_nf ; norm_num [ Real.sqrt_ne_zero'.mpr hR, Real.sqrt_ne_zero'.mpr ( Real.log_pos hx ), Real.rpow_neg_one ] ; ring_nf; grind generalize_proofs at *; ( have h_quad_neg : ∀ u : ℝ, u > 0 → -u^2 + (C / (2 * Real.sqrt R)) * u + (1 - B) ≤ 0 := by intros u hu have h_quad_neg : -u^2 + (C / (2 * Real.sqrt R)) * u + (1 - B) ≤ 0 := by have h_discriminant : (C / (2 * Real.sqrt R))^2 - 4 * (-1) * (1 - B) ≤ 0 := by field_simp at *; ring_nf at *; nlinarith [ inv_mul_cancel₀ ( ne_of_gt hR ), inv_pos.mpr hR, Real.sqrt_nonneg R, Real.sq_sqrt hR.le, mul_inv_cancel₀ ( ne_of_gt ( Real.sqrt_pos.mpr hR ) ), Real.sqrt_nonneg ( R⁻¹ ), Real.sq_sqrt ( inv_nonneg.mpr hR.le ) ] ; linarith [ sq_nonneg ( C / ( 2 * Real.sqrt R ) - 2 * u ) ] generalize_proofs at *; (exact h_quad_neg) generalize_proofs at *; ( exact h_deriv_neg.symm ▸ mul_nonpos_of_nonpos_of_nonneg ( mul_nonpos_of_nonpos_of_nonneg ( mul_nonpos_of_nonpos_of_nonneg ( by have := h_quad_neg ( Real.sqrt ( Real.log x ) ) ( Real.sqrt_pos.mpr ( Real.log_pos hx ) ) ; rw [ Real.sq_sqrt ( Real.log_nonneg hx.le ) ] at this; linarith ) ( by positivity ) ) ( by exact Real.rpow_nonneg ( Real.log_nonneg hx.le ) _ ) ) ( by positivity ) ;)) generalize_proofs at *; ( by_contra h_contra; have := exists_deriv_eq_slope ( g_bound 1 ( 1 - B ) ( C / Real.sqrt R ) ) ( show x < y from hy.lt_of_ne ( by rintro rfl; linarith ) ) ; norm_num at this; exact absurd ( this ( by exact ContinuousOn.mono ( show ContinuousOn ( g_bound 1 ( 1 - B ) ( C / Real.sqrt R ) ) ( Set.Icc x y ) from by exact ContinuousOn.mul ( ContinuousOn.mul ( ContinuousOn.rpow continuousOn_id continuousOn_const <| by intro u hu; exact Or.inl <| by linarith [ hu.1 ] ) <| ContinuousOn.rpow ( Real.continuousOn_log.mono <| by exact fun u hu => ne_of_gt <| by linarith [ hu.1 ] ) continuousOn_const <| by intro u hu; exact Or.inl <| ne_of_gt <| Real.log_pos <| by linarith [ hu.1 ] ) <| ContinuousOn.rexp <| ContinuousOn.mul continuousOn_const <| ContinuousOn.sqrt <| Real.continuousOn_log.mono <| by exact fun u hu => ne_of_gt <| by linarith [ hu.1 ] ) <| Set.Icc_subset_Icc ( by linarith ) le_rfl ) <| by exact fun u hu => DifferentiableAt.differentiableWithinAt <| by exact DifferentiableAt.mul ( DifferentiableAt.mul ( DifferentiableAt.rpow ( differentiableAt_id ) ( differentiableAt_const _ ) <| by linarith [ hu.1 ] ) <| DifferentiableAt.rpow ( Real.differentiableAt_log <| by linarith [ hu.1 ] ) ( differentiableAt_const _ ) <| by exact ne_of_gt <| Real.log_pos <| by linarith [ hu.1 ] ) <| DifferentiableAt.exp <| DifferentiableAt.mul ( differentiableAt_const _ ) <| DifferentiableAt.sqrt ( Real.differentiableAt_log <| by linarith [ hu.1 ] ) <| by exact ne_of_gt <| Real.log_pos <| by linarith [ hu.1 ] ) ( by rintro ⟨ c, ⟨ hxc, hcy ⟩, hcd ⟩ ; rw [ eq_div_iff ] at hcd <;> nlinarith [ h_deriv_neg c <| by linarith ] ) ;); exact mul_le_mul_of_nonneg_left ( h_decreasing _ _ hx1 hx ) ( by positivity ); convert h_ratio_simplified using 1 <;> norm_num [ ratio_eq_g hR ( by linarith : 0 < x ) ( Real.log_pos ( by linarith ) ), ratio_eq_g hR ( by linarith : 0 < x₁ ) ( Real.log_pos ( by linarith ) ) ]

/- Helper: for B ≥ 3/2 and x ≥ x₁ ≥ x₀, the m(x₀,x)*(log x)^(1-B) factor simplifies -/ lemma m_simplify {B x₀ x x₁ : ℝ} (hB : B ≥ 3 / 2) (hx₀ : x₀ > 1) (hx₁ : x₁ > 1) (hx₁x₀ : x₁ ≥ x₀) (hx : x ≥ x₁) (hlogx : log x > 0) : max ((log x₀) ^ ((2 * B - 3) / 2)) ((log x) ^ ((2 * B - 3) / 2)) * (log x) ^ (1 - B) ≤ 1 / sqrt (log x₁) := by rw [ max_def ]; split_ifs <;> norm_num [ Real.sqrt_eq_rpow, ← Real.rpow_add hlogx ]; · rw [ ← Real.rpow_neg ( Real.log_nonneg hx₁.le ) ] ; ring_nf ; norm_num [ hlogx ]; rw [ Real.rpow_le_rpow_iff_of_neg ] <;> linarith [ Real.log_pos hx₁, Real.log_le_log ( by linarith ) hx ]; · exact False.elim <| ‹¬log x₀ ^ ( ( 2 * B - 3 ) / 2 ) ≤ log x ^ ( ( 2 * B - 3 ) / 2 ) › <| Real.rpow_le_rpow ( Real.log_nonneg <| by linarith ) ( Real.log_le_log ( by linarith ) <| by linarith ) <| by linarith;

/- Helper: admissible_bound is positive under suitable conditions -/ lemma admissible_bound_pos {A B C R x : ℝ} (hA : A > 0) (hR : R > 0) (hlogx : log x > 0) : admissible_bound A B C R x > 0 := by exact mul_pos ( mul_pos hA ( Real.rpow_pos_of_pos ( div_pos hlogx hR ) _ ) ) ( Real.exp_pos _ )

/- Helper: Eθ is non-negative -/ lemma Eθ_nonneg (x : ℝ) (hx : x > 0) : Eθ x ≥ 0 := by exact div_nonneg ( abs_nonneg _ ) hx.le

/- Helper: δ is non-negative -/ lemma delta_nonneg (x₀ : ℝ) : δ x₀ ≥ 0 := by exact abs_nonneg _

/- Helper: from ratio_mono, derive the bound on log x / x in terms of admissible_bound -/ lemma logx_over_x_bound {A B C R x₁ x : ℝ} (hB : B ≥ 1 + C ^ 2 / (16 * R)) (hR : R > 0) (hA : A > 0) (hx1_gt1 : x₁ > 1) (hx : x ≥ x₁) : log x / x ≤ (log x₁ / (x₁ * admissible_bound A B C R x₁)) * admissible_bound A B C R x := by convert! mul_le_mul_of_nonneg_right ( ratio_mono hB hR hx1_gt1 hx hA ) ( admissible_bound_pos hA hR ( Real.log_pos <| show 1 < x by linarith ) |> le_of_lt ) using 1 ; ring_nf; norm_num [ ne_of_gt ( admissible_bound_pos hA hR ( Real.log_pos <| show 1 < x by linarith ) ) ]

/- PROBLEM Bound the delta term from eq_30.

PROVIDED SOLUTION Use logx_over_x_bound to get log x / x ≤ (log x₁ / (x₁ * admissible_bound A B C R x₁)) * admissible_bound A B C R x. Multiply both sides by (x₀ / log x₀) * δ x₀ (which is non-negative since δ is non-negative by delta_nonneg and x₀/log x₀ > 0 from hlogx₀). The LHS becomes (log x / x) * (x₀ / log x₀) * δ x₀. The RHS becomes (log x₁ / (x₁ * admissible_bound A B C R x₁)) * (x₀ / log x₀) * δ x₀ * admissible_bound A B C R x which equals (x₀ * log x₁) / (admissible_bound A B C R x₁ * x₁ * log x₀) * δ x₀ * admissible_bound A B C R x by algebraic rearrangement. -/ lemma delta_term_bound {A B C R x₀ x₁ x : ℝ} (hB : B ≥ 1 + C ^ 2 / (16 * R)) (hR : R > 0) (hA : A > 0) (hx1_gt1 : x₁ > 1) (hx : x ≥ x₁) (hx0_pos : x₀ > 0) (hlogx₀ : log x₀ > 0) : (log x / x) * (x₀ / log x₀) * δ x₀ ≤ (x₀ * log x₁) / (admissible_bound A B C R x₁ * x₁ * log x₀) * δ x₀ * admissible_bound A B C R x := by have h1 := logx_over_x_bound hB hR hA hx1_gt1 hx have h2 : x₀ / log x₀ ≥ 0 := div_nonneg hx0_pos.le hlogx₀.le have h3 := delta_nonneg x₀ calc (log x / x) * (x₀ / log x₀) * δ x₀ ≤ ((log x₁ / (x₁ * admissible_bound A B C R x₁)) * admissible_bound A B C R x) * (x₀ / log x₀) * δ x₀ := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right h1 h2) h3 _ = (x₀ * log x₁) / (admissible_bound A B C R x₁ * x₁ * log x₀) * δ x₀ * admissible_bound A B C R x := by ring

/- Helper: Dawson function is monotone decreasing for large arguments (after ~0.924) We use remark_after_corollary_11 which gives the existence of a maximum around 0.924 -/ lemma dawson_antitoneOn : ∃ x₀ : ℝ, x₀ < 1 ∧ StrictAntiOn dawson (Set.Ioi x₀) := by obtain ⟨ x₀, hx₀ ⟩ := remark_after_corollary_11; exact ⟨ x₀, by norm_num at hx₀; linarith, hx₀.2.2 ⟩

/- Helper: Eθ t = |Eθ t| since Eθ is non-negative -/ lemma Eθ_eq_abs {t : ℝ} (ht : t > 0) : Eθ t = |Eθ t| := by rw [ abs_of_nonneg ( Eθ_nonneg t ht ) ]

/- Algebraic identity: (log x / x) * (2A/R^B) * x * m * exp(-C*√(log x/R)) * D = 2 * m * (log x)^(1-B) * D * admissible_bound A B C R x -/ lemma integral_algebra {A B C R x m D : ℝ} (hR : R > 0) (hx : x > 1) : (log x / x) * ((2 * A) / R ^ B * x * m * exp (-C * √(log x / R)) * D) = 2 * m * (log x) ^ (1 - B) * D * admissible_bound A B C R x := by unfold admissible_bound; ring_nf; rw [ Real.rpow_sub ( by linarith [ Real.log_pos hx ] ), Real.rpow_one ] ; ring_nf; rw [ Real.mul_rpow ( by linarith [ Real.log_pos hx ] ) ( by positivity ), Real.inv_rpow ( by positivity ) ] ; ring_nf ; norm_num [ ne_of_gt ( zero_lt_one.trans hx ) ] ; rw [ mul_inv_cancel_right₀ ( ne_of_gt ( Real.rpow_pos_of_pos ( Real.log_pos hx ) _ ) ) ] have heq: Real.exp (-(C * (Real.log x * R⁻¹)^(1/2 : ℝ))) = Real.exp (-(C * Real.sqrt (Real.log x * R⁻¹))) := by congr; rw [Real.sqrt_eq_rpow] rw [heq] linarith

/- Dawson monotonicity for arguments ≥ 1 -/ lemma dawson_mono_ge_one {a b : ℝ} (ha : a ≥ 1) (hab : a ≤ b) : dawson b ≤ dawson a := by obtain ⟨ x₀, hx₀ ⟩ := dawson_antitoneOn; exact hx₀.2.le_iff_ge ( show x₀ < b by linarith ) ( show x₀ < a by linarith ) |>.2 ( by linarith )

/- Derive that √(log x₁) - C/(2√R) ≥ 1 from the hypothesis on x₁ -/ lemma sqrt_log_minus_ge_one {C R x₁ : ℝ} (hx1 : x₁ ≥ exp ((1 + C / (2 * sqrt R)) ^ 2)) : √(log x₁) - C / (2 * √R) ≥ 1 := by -- Taking the natural logarithm of both sides of the inequality $x₁ \geq \exp((1 + C / (2 * \sqrt{R}))^2)$, we get $\log x₁ \geq (1 + C / (2 * \sqrt{R}))^2$. have h_log : Real.log x₁ ≥ (1 + C / (2 * Real.sqrt R)) ^ 2 := by simpa using Real.log_le_log ( by positivity ) hx1; exact le_tsub_of_add_le_right ( Real.le_sqrt_of_sq_le ( by linarith ) )

/- PROBLEM Helper: the integral term from eq_30 is bounded by the Dawson component of μ_asymp

Bound the integral term from eq_30.

PROVIDED SOLUTION Step 1: Since Eθ t ≥ 0, we have ∫ Eθ t / log t^2 ≤ ∫ |Eθ t| / log t^2. Step 2: Apply lemma_12 to bound the integral. Step 3: Multiply by log x / x and use integral_algebra to simplify: = 2 * m(x₀,x) * (log x)^(1-B) * dawson(√(log x) - C/(2√R)) * admissible_bound. Step 4: Apply m_simplify to get m * (log x)^(1-B) ≤ 1/√(log x₁). Step 5: Apply dawson_mono_ge_one (using sqrt_log_minus_ge_one for the ≥ 1 condition) to get dawson(√(log x)-C/(2√R)) ≤ dawson(√(log x₁)-C/(2√R)). Step 6: Combine: ≤ 2/√(log x₁) * dawson(√(log x₁)-C/(2√R)) * admissible_bound.

Bound for ψ(y) for large y.

Differentiate [ \frac{d}{dx} \left( \Li(x) - \frac{x}{\log(x)} \right) = \frac{1}{\log(x)} + \frac{1 - \log(x)}{(\log(x))^2} = \frac{1}{(\log(x))^2} ] to see that the difference is strictly increasing. Evaluating at $x = 6.58$ and applying the mean value theorem gives the announced result. -/) (latexEnv := "lemma") (discussion := 713)] theorem lemma_20_a : StrictMonoOn (fun x ↦ Li x - x / log x) (Set.Ioi 6.58) := by have hpos (x : ℝ) (hx : x ∈ Set.Ioi 6.58) := log_pos (by linarith [Set.mem_Ioi.mp hx]) |>.ne' apply strictMonoOn_of_deriv_pos (convex_Ioi _) · apply HasDerivAt.continuousOn (by apply hasDerivAt_Li) |>.sub fun_prop (disch := simp_all) · intro x hx rw [interior_Ioi, Set.mem_Ioi] at hx rw [deriv_fun_sub (hasDerivAt_Li hx).differentiableAt (by fun_prop (disch := simp_all)), deriv_fun_div differentiableAt_fun_id (differentiableAt_log (by linarith)) (hpos x hx)] simp [(hasDerivAt_Li hx).deriv, field, pow_two_pos_of_ne_zero, (hpos x hx), - sub_pos]

private lemma Li_ibp {x : ℝ} (hx : x > 2) : Li x - x / log x = -2 / log 2 + ∫ t in (2:ℝ)..x, 1 / (log t) ^ 2 := by have h_parts : ∀ a b : ℝ, 2 ≤ a → a < b → ∫ t in a..b, (1 : ℝ) / Real.log t = (b / Real.log b) - (a / Real.log a) + ∫ t in a..b, (1 : ℝ) / Real.log t ^ 2 := by intro a b _ _ rw [intervalIntegral.integral_eq_sub_of_hasDerivAt] rotate_right next => use fun x => x / Real.log x + ∫ t in a..x, 1 / Real.log t ^ 2 · norm_num; ring · intro x hx exact Ramanujan.hasDerivAt_li_antideriv (by linarith) (by cases Set.mem_uIcc.mp hx <;> linarith) · apply_rules [ContinuousOn.intervalIntegrable] exact continuousOn_of_forall_continuousAt fun t ht => ContinuousAt.div continuousAt_const (Real.continuousAt_log (by cases Set.mem_uIcc.mp ht <;> linarith)) (ne_of_gt (Real.log_pos (by cases Set.mem_uIcc.mp ht <;> linarith))) convert! congr_arg (fun y => y - x / Real.log x) (h_parts 2 x (by norm_num) hx) using 1 ring!

/-The following lemmas are used for the proof of lemma_20_b. The first lemma was originally used for the proof of theorem_6_3, but was moved up here to be used for lemma_20_b as well.

Now we can start estimating $E_\pi$. We make the following running hypotheses. Let $x_0 > 0$ be chosen such that $\pi(x_0)$ and $\theta(x_0)$ are computable, and let $x_1 \geq \max(x_0, 14)$. Let $\{b_i\}_{i=1}^N$ be a finite partition of $[\log x_0, \log x_1]$, with $b_1 = \log x_0$ and $b_N = \log x_1$, and suppose that $\varepsilon_{\theta,\mathrm{num}}$ gives numerical bounds for $x = \exp(b_i)$, for each $i=1,\dots,N$.

This is obtained by combining Sublemma \ref{fks2-eq-30} with the admissibility of $\varepsilon_{\theta,num}$ and Lemma \ref{fks2-lemma-19}. -/) (latexEnv := "sublemma") (discussion := 715)] theorem theorem_6_1 {x₀ x₁ : ℝ} (h : x₁ ≥ max x₀ 14) {N : ℕ} (b : Fin (N + 1) → ℝ) (hmono : Monotone b) (h_b_start : b 0 = log x₀) (h_b_end : b (Fin.last N) = log x₁) (εθ_num : ℝ → ℝ) (h_εθ_num : ∀ i : Fin (N+1), Eθ.numericalBound (exp (b i)) εθ_num) (x : ℝ) (hx : x ≥ x₁) (hx₀ : x₀ ≥ 2) : Eπ x ≤ εθ_num x₁ + (log x / x) * (x₀ / log x₀) * (Eπ x₀ + Eθ x₀) + (log x / x) * ∑ i ∈ Finset.Iio (Fin.last N), εθ_num (exp (b i)) * (Li (exp (b (i + 1))) - Li (exp (b i)) + exp (b i) / b i - exp (b (i + 1)) / b (i + 1)) + εθ_num x₁ * (log x / x) * ∫ t in x₁..x, 1 / (log t) ^ 2 := by have hx₀_le_x₁ : x₀ ≤ x₁ := le_trans (le_max_left _ _) h have hx₀_le_x : x₀ ≤ x := le_trans hx₀_le_x₁ hx have h30 := eq_30 hx₀_le_x hx₀ have hEθ_x_le : Eθ x ≤ εθ_num x₁ := by have := h_εθ_num (Fin.last N) rw [h_b_end, exp_log (by linarith [h])] at this exact this x hx have hδ_le : δ x₀ ≤ Eπ x₀ + Eθ x₀ := by rw [δ, Eπ, Eθ] refine (abs_sub_le _ 0 ).trans ? have hx₀_pos : 0 < x₀ := by positivity have hlogx₀_pos : 0 < log x₀ := Real.log_pos (by linarith) simp [abs_div, abs_of_pos hx₀_pos, abs_of_pos (by positivity : 0 < x₀ / log x₀)] have hx₁_ge2 : x₁ ≥ 2 := by linarith [h] have hsplit := intervalIntegral.integral_add_adjacent_intervals (Eθ_integrable hx₀ hx₀_le_x₁) (Eθ_integrable hx₁_ge2 hx) have hlogx_x_nonneg : 0 ≤ log x / x := div_nonneg (Real.log_nonneg (by linarith [hx, h])) (by linarith [hx, h]) have h_coeff_nonneg : 0 ≤ (log x / x) * (x₀ / log x₀) := by apply mul_nonneg hlogx_x_nonneg have : 0 < log x₀ := Real.log_pos (by linarith) exact div_nonneg (by linarith) this.le calc Eπ x ≤ Eθ x + (log x / x) * (x₀ / log x₀) * δ x₀ + (log x / x) * ∫ t in x₀..x, Eθ t / log t ^ 2 := h30 _ ≤ εθ_num x₁ + (log x / x) * (x₀ / log x₀) * (Eπ x₀ + Eθ x₀) + (log x / x) * ∫ t in x₀..x, Eθ t / log t ^ 2 := by gcongr _ ≤ εθ_num x₁ + (log x / x) * (x₀ / log x₀) * (Eπ x₀ + Eθ x₀) + (log x / x) * ((∫ t in x₀..x₁, Eθ t / log t ^ 2) + ∫ t in x₁..x, Eθ t / log t ^ 2) := by rw [hsplit] _ ≤ εθ_num x₁ + (log x / x) * (x₀ / log x₀) * (Eπ x₀ + Eθ x₀) + (log x / x) * ∑ i ∈ Finset.Iio (Fin.last N), εθ_num (exp (b i)) * (Li (exp (b (i + 1))) - Li (exp (b i)) + exp (b i) / b i - exp (b (i + 1)) / b (i + 1)) + εθ_num x₁ * (log x / x) * ∫ t in x₁..x, 1 / (log t) ^ 2 := by have h_bound_x1_x : ∫ t in x₁..x, Eθ t / log t ^ 2 ≤ εθ_num x₁ * ∫ t in x₁..x, 1 / (log t) ^ 2 := by have := h_εθ_num (Fin.last N) rw [h_b_end, exp_log (by linarith [h])] at this exact Eθ_bound_integral_last hx₁_ge2 hx εθ_num this have h_bound_x0_x1 := bound_x0_x1 hx₀ hx₀_le_x₁ b hmono h_b_start h_b_end εθ_num h_εθ_num nlinarith [h_bound_x0_x1, h_bound_x1_x, hlogx_x_nonneg]

/- The following 2 lemmas are used for theorem_6_3.