l3lab/ntp-mathlib huggingface dataset:
private theorem aux₃ :
    |(fract ξ)⁻¹ - v / (u - ⌊ξ⌋ * v)| < (((u : ℝ) - ⌊ξ⌋ * v) * (2 * (u - ⌊ξ⌋ * v) - 1))⁻¹ := by
  obtain ⟨hu₀, huv⟩ := aux₂ hv h
  have hξ₀ := aux₁ hv h
  set u' := u - ⌊ξ⌋ * v with hu'
  have hu'ℝ : (u' : ℝ) = u - ⌊ξ⌋ * v := mod_cast hu'
  rw [← hu'ℝ]
  replace hu'ℝ := (eq_sub_iff_add_eq.mp hu'ℝ).symm
  obtain ⟨Hu, Hu'⟩ := aux₀ hu₀
  obtain ⟨Hv, Hv'⟩ := aux₀ (zero_lt_two.trans_le hv)
  have H₁ := div_pos (div_pos Hv Hu) hξ₀
  replace h := h.2.2
  have h' : |fract ξ - u' / v| < ((v : ℝ) * (2 * v - 1))⁻¹ := by
    rwa [hu'ℝ, add_div, mul_div_cancel _ Hv.ne', ← sub_sub, sub_right_comm] at h
  

Proof state:
case intro.intro.intro
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
hξ₀ : 0 < fract ξ
u' : ℤ := u - ⌊ξ⌋ * v
hu₀ : 0 < u'
huv : u' < v
hu' : u' = u - ⌊ξ⌋ * v
hu'ℝ : ↑u = ↑u' + ↑⌊ξ⌋ * ↑v
Hu : 0 < ↑u'
Hu' : 0 < 2 * ↑u' - 1
Hv : 0 < ↑v
Hv' : 0 < 2 * ↑v - 1
H₁ : 0 < ↑v / ↑u' / fract ξ
h : |ξ - ↑u / ↑v| < (↑v * (2 * ↑v - 1))⁻¹
h' : |fract ξ - ↑u' / ↑v| < (↑v * (2 * ↑v - 1))⁻¹
⊢ |(fract ξ)⁻¹ - ↑v / ↑u'| < (↑u' * (2 * ↑u' - 1))⁻¹

Next tactic:
have H : (2 * u' - 1 : ℝ) ≤ (2 * v - 1) * fract ξ := by
    replace h := (abs_lt.mp h).1
    have : (2 * (v : ℝ) - 1) * (-((v : ℝ) * (2 * v - 1))⁻¹ + u' / v) = 2 * u' - (1 + u') / v := by
      field_simp; ring
    rw [hu'ℝ, add_div, mul_div_cancel _ Hv.ne', ← sub_sub, sub_right_comm, self_sub_floor,
      lt_sub_iff_add_lt, ← mul_lt_mul_left Hv', this] at h
    refine' LE.le.trans _ h.le
    rw [sub_le_sub_iff_left, div_le_one Hv, add_comm]
    exact mod_cast huv
private theorem aux₃ :
    |(fract ξ)⁻¹ - v / (u - ⌊ξ⌋ * v)| < (((u : ℝ) - ⌊ξ⌋ * v) * (2 * (u - ⌊ξ⌋ * v) - 1))⁻¹ := by
  obtain ⟨hu₀, huv⟩ := aux₂ hv h
  have hξ₀ := aux₁ hv h
  set u' := u - ⌊ξ⌋ * v with hu'
  have hu'ℝ : (u' : ℝ) = u - ⌊ξ⌋ * v := mod_cast hu'
  rw [← hu'ℝ]
  replace hu'ℝ := (eq_sub_iff_add_eq.mp hu'ℝ).symm
  obtain ⟨Hu, Hu'⟩ := aux₀ hu₀
  obtain ⟨Hv, Hv'⟩ := aux₀ (zero_lt_two.trans_le hv)
  have H₁ := div_pos (div_pos Hv Hu) hξ₀
  replace h := h.2.2
  have h' : |fract ξ - u' / v| < ((v : ℝ) * (2 * v - 1))⁻¹ := by
    rwa [hu'ℝ, add_div, mul_div_cancel _ Hv.ne', ← sub_sub, sub_right_comm] at h
  have H : (2 * u' - 1 : ℝ) ≤ (2 * v - 1) * fract ξ := by
    

Proof state:
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
hξ₀ : 0 < fract ξ
u' : ℤ := u - ⌊ξ⌋ * v
hu₀ : 0 < u'
huv : u' < v
hu' : u' = u - ⌊ξ⌋ * v
hu'ℝ : ↑u = ↑u' + ↑⌊ξ⌋ * ↑v
Hu : 0 < ↑u'
Hu' : 0 < 2 * ↑u' - 1
Hv : 0 < ↑v
Hv' : 0 < 2 * ↑v - 1
H₁ : 0 < ↑v / ↑u' / fract ξ
h : |ξ - ↑u / ↑v| < (↑v * (2 * ↑v - 1))⁻¹
h' : |fract ξ - ↑u' / ↑v| < (↑v * (2 * ↑v - 1))⁻¹
⊢ 2 * ↑u' - 1 ≤ (2 * ↑v - 1) * fract ξ

Next tactic:
replace h := (abs_lt.mp h).1
private theorem aux₃ :
    |(fract ξ)⁻¹ - v / (u - ⌊ξ⌋ * v)| < (((u : ℝ) - ⌊ξ⌋ * v) * (2 * (u - ⌊ξ⌋ * v) - 1))⁻¹ := by
  obtain ⟨hu₀, huv⟩ := aux₂ hv h
  have hξ₀ := aux₁ hv h
  set u' := u - ⌊ξ⌋ * v with hu'
  have hu'ℝ : (u' : ℝ) = u - ⌊ξ⌋ * v := mod_cast hu'
  rw [← hu'ℝ]
  replace hu'ℝ := (eq_sub_iff_add_eq.mp hu'ℝ).symm
  obtain ⟨Hu, Hu'⟩ := aux₀ hu₀
  obtain ⟨Hv, Hv'⟩ := aux₀ (zero_lt_two.trans_le hv)
  have H₁ := div_pos (div_pos Hv Hu) hξ₀
  replace h := h.2.2
  have h' : |fract ξ - u' / v| < ((v : ℝ) * (2 * v - 1))⁻¹ := by
    rwa [hu'ℝ, add_div, mul_div_cancel _ Hv.ne', ← sub_sub, sub_right_comm] at h
  have H : (2 * u' - 1 : ℝ) ≤ (2 * v - 1) * fract ξ := by
    replace h := (abs_lt.mp h).1
    

Proof state:
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
hξ₀ : 0 < fract ξ
u' : ℤ := u - ⌊ξ⌋ * v
hu₀ : 0 < u'
huv : u' < v
hu' : u' = u - ⌊ξ⌋ * v
hu'ℝ : ↑u = ↑u' + ↑⌊ξ⌋ * ↑v
Hu : 0 < ↑u'
Hu' : 0 < 2 * ↑u' - 1
Hv : 0 < ↑v
Hv' : 0 < 2 * ↑v - 1
H₁ : 0 < ↑v / ↑u' / fract ξ
h' : |fract ξ - ↑u' / ↑v| < (↑v * (2 * ↑v - 1))⁻¹
h : -(↑v * (2 * ↑v - 1))⁻¹ < ξ - ↑u / ↑v
⊢ 2 * ↑u' - 1 ≤ (2 * ↑v - 1) * fract ξ

Next tactic:
have : (2 * (v : ℝ) - 1) * (-((v : ℝ) * (2 * v - 1))⁻¹ + u' / v) = 2 * u' - (1 + u') / v := by
      field_simp; ring
private theorem aux₃ :
    |(fract ξ)⁻¹ - v / (u - ⌊ξ⌋ * v)| < (((u : ℝ) - ⌊ξ⌋ * v) * (2 * (u - ⌊ξ⌋ * v) - 1))⁻¹ := by
  obtain ⟨hu₀, huv⟩ := aux₂ hv h
  have hξ₀ := aux₁ hv h
  set u' := u - ⌊ξ⌋ * v with hu'
  have hu'ℝ : (u' : ℝ) = u - ⌊ξ⌋ * v := mod_cast hu'
  rw [← hu'ℝ]
  replace hu'ℝ := (eq_sub_iff_add_eq.mp hu'ℝ).symm
  obtain ⟨Hu, Hu'⟩ := aux₀ hu₀
  obtain ⟨Hv, Hv'⟩ := aux₀ (zero_lt_two.trans_le hv)
  have H₁ := div_pos (div_pos Hv Hu) hξ₀
  replace h := h.2.2
  have h' : |fract ξ - u' / v| < ((v : ℝ) * (2 * v - 1))⁻¹ := by
    rwa [hu'ℝ, add_div, mul_div_cancel _ Hv.ne', ← sub_sub, sub_right_comm] at h
  have H : (2 * u' - 1 : ℝ) ≤ (2 * v - 1) * fract ξ := by
    replace h := (abs_lt.mp h).1
    have : (2 * (v : ℝ) - 1) * (-((v : ℝ) * (2 * v - 1))⁻¹ + u' / v) = 2 * u' - (1 + u') / v := by
      

Proof state:
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
hξ₀ : 0 < fract ξ
u' : ℤ := u - ⌊ξ⌋ * v
hu₀ : 0 < u'
huv : u' < v
hu' : u' = u - ⌊ξ⌋ * v
hu'ℝ : ↑u = ↑u' + ↑⌊ξ⌋ * ↑v
Hu : 0 < ↑u'
Hu' : 0 < 2 * ↑u' - 1
Hv : 0 < ↑v
Hv' : 0 < 2 * ↑v - 1
H₁ : 0 < ↑v / ↑u' / fract ξ
h' : |fract ξ - ↑u' / ↑v| < (↑v * (2 * ↑v - 1))⁻¹
h : -(↑v * (2 * ↑v - 1))⁻¹ < ξ - ↑u / ↑v
⊢ (2 * ↑v - 1) * (-(↑v * (2 * ↑v - 1))⁻¹ + ↑u' / ↑v) = 2 * ↑u' - (1 + ↑u') / ↑v

Next tactic:
field_simp
private theorem aux₃ :
    |(fract ξ)⁻¹ - v / (u - ⌊ξ⌋ * v)| < (((u : ℝ) - ⌊ξ⌋ * v) * (2 * (u - ⌊ξ⌋ * v) - 1))⁻¹ := by
  obtain ⟨hu₀, huv⟩ := aux₂ hv h
  have hξ₀ := aux₁ hv h
  set u' := u - ⌊ξ⌋ * v with hu'
  have hu'ℝ : (u' : ℝ) = u - ⌊ξ⌋ * v := mod_cast hu'
  rw [← hu'ℝ]
  replace hu'ℝ := (eq_sub_iff_add_eq.mp hu'ℝ).symm
  obtain ⟨Hu, Hu'⟩ := aux₀ hu₀
  obtain ⟨Hv, Hv'⟩ := aux₀ (zero_lt_two.trans_le hv)
  have H₁ := div_pos (div_pos Hv Hu) hξ₀
  replace h := h.2.2
  have h' : |fract ξ - u' / v| < ((v : ℝ) * (2 * v - 1))⁻¹ := by
    rwa [hu'ℝ, add_div, mul_div_cancel _ Hv.ne', ← sub_sub, sub_right_comm] at h
  have H : (2 * u' - 1 : ℝ) ≤ (2 * v - 1) * fract ξ := by
    replace h := (abs_lt.mp h).1
    have : (2 * (v : ℝ) - 1) * (-((v : ℝ) * (2 * v - 1))⁻¹ + u' / v) = 2 * u' - (1 + u') / v := by
      field_simp; 

Proof state:
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
hξ₀ : 0 < fract ξ
u' : ℤ := u - ⌊ξ⌋ * v
hu₀ : 0 < u'
huv : u' < v
hu' : u' = u - ⌊ξ⌋ * v
hu'ℝ : ↑u = ↑u' + ↑⌊ξ⌋ * ↑v
Hu : 0 < ↑u'
Hu' : 0 < 2 * ↑u' - 1
Hv : 0 < ↑v
Hv' : 0 < 2 * ↑v - 1
H₁ : 0 < ↑v / ↑u' / fract ξ
h' : |fract ξ - ↑u' / ↑v| < (↑v * (2 * ↑v - 1))⁻¹
h : -(↑v * (2 * ↑v - 1))⁻¹ < ξ - ↑u / ↑v
⊢ (2 * ↑v - 1) * (-↑v + (↑u - ↑⌊ξ⌋ * ↑v) * (↑v * (2 * ↑v - 1))) * ↑v =
    (2 * (↑u - ↑⌊ξ⌋ * ↑v) * ↑v - (1 + (↑u - ↑⌊ξ⌋ * ↑v))) * (↑v * (2 * ↑v - 1) * ↑v)

Next tactic:
ring
private theorem aux₃ :
    |(fract ξ)⁻¹ - v / (u - ⌊ξ⌋ * v)| < (((u : ℝ) - ⌊ξ⌋ * v) * (2 * (u - ⌊ξ⌋ * v) - 1))⁻¹ := by
  obtain ⟨hu₀, huv⟩ := aux₂ hv h
  have hξ₀ := aux₁ hv h
  set u' := u - ⌊ξ⌋ * v with hu'
  have hu'ℝ : (u' : ℝ) = u - ⌊ξ⌋ * v := mod_cast hu'
  rw [← hu'ℝ]
  replace hu'ℝ := (eq_sub_iff_add_eq.mp hu'ℝ).symm
  obtain ⟨Hu, Hu'⟩ := aux₀ hu₀
  obtain ⟨Hv, Hv'⟩ := aux₀ (zero_lt_two.trans_le hv)
  have H₁ := div_pos (div_pos Hv Hu) hξ₀
  replace h := h.2.2
  have h' : |fract ξ - u' / v| < ((v : ℝ) * (2 * v - 1))⁻¹ := by
    rwa [hu'ℝ, add_div, mul_div_cancel _ Hv.ne', ← sub_sub, sub_right_comm] at h
  have H : (2 * u' - 1 : ℝ) ≤ (2 * v - 1) * fract ξ := by
    replace h := (abs_lt.mp h).1
    have : (2 * (v : ℝ) - 1) * (-((v : ℝ) * (2 * v - 1))⁻¹ + u' / v) = 2 * u' - (1 + u') / v := by
      field_simp; ring
    

Proof state:
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
hξ₀ : 0 < fract ξ
u' : ℤ := u - ⌊ξ⌋ * v
hu₀ : 0 < u'
huv : u' < v
hu' : u' = u - ⌊ξ⌋ * v
hu'ℝ : ↑u = ↑u' + ↑⌊ξ⌋ * ↑v
Hu : 0 < ↑u'
Hu' : 0 < 2 * ↑u' - 1
Hv : 0 < ↑v
Hv' : 0 < 2 * ↑v - 1
H₁ : 0 < ↑v / ↑u' / fract ξ
h' : |fract ξ - ↑u' / ↑v| < (↑v * (2 * ↑v - 1))⁻¹
h : -(↑v * (2 * ↑v - 1))⁻¹ < ξ - ↑u / ↑v
this : (2 * ↑v - 1) * (-(↑v * (2 * ↑v - 1))⁻¹ + ↑u' / ↑v) = 2 * ↑u' - (1 + ↑u') / ↑v
⊢ 2 * ↑u' - 1 ≤ (2 * ↑v - 1) * fract ξ

Next tactic:
rw [hu'ℝ, add_div, mul_div_cancel _ Hv.ne', ← sub_sub, sub_right_comm, self_sub_floor,
      lt_sub_iff_add_lt, ← mul_lt_mul_left Hv', this] at h
private theorem aux₃ :
    |(fract ξ)⁻¹ - v / (u - ⌊ξ⌋ * v)| < (((u : ℝ) - ⌊ξ⌋ * v) * (2 * (u - ⌊ξ⌋ * v) - 1))⁻¹ := by
  obtain ⟨hu₀, huv⟩ := aux₂ hv h
  have hξ₀ := aux₁ hv h
  set u' := u - ⌊ξ⌋ * v with hu'
  have hu'ℝ : (u' : ℝ) = u - ⌊ξ⌋ * v := mod_cast hu'
  rw [← hu'ℝ]
  replace hu'ℝ := (eq_sub_iff_add_eq.mp hu'ℝ).symm
  obtain ⟨Hu, Hu'⟩ := aux₀ hu₀
  obtain ⟨Hv, Hv'⟩ := aux₀ (zero_lt_two.trans_le hv)
  have H₁ := div_pos (div_pos Hv Hu) hξ₀
  replace h := h.2.2
  have h' : |fract ξ - u' / v| < ((v : ℝ) * (2 * v - 1))⁻¹ := by
    rwa [hu'ℝ, add_div, mul_div_cancel _ Hv.ne', ← sub_sub, sub_right_comm] at h
  have H : (2 * u' - 1 : ℝ) ≤ (2 * v - 1) * fract ξ := by
    replace h := (abs_lt.mp h).1
    have : (2 * (v : ℝ) - 1) * (-((v : ℝ) * (2 * v - 1))⁻¹ + u' / v) = 2 * u' - (1 + u') / v := by
      field_simp; ring
    rw [hu'ℝ, add_div, mul_div_cancel _ Hv.ne', ← sub_sub, sub_right_comm, self_sub_floor,
      lt_sub_iff_add_lt, ← mul_lt_mul_left Hv', this] at h
    

Proof state:
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
hξ₀ : 0 < fract ξ
u' : ℤ := u - ⌊ξ⌋ * v
hu₀ : 0 < u'
huv : u' < v
hu' : u' = u - ⌊ξ⌋ * v
hu'ℝ : ↑u = ↑u' + ↑⌊ξ⌋ * ↑v
Hu : 0 < ↑u'
Hu' : 0 < 2 * ↑u' - 1
Hv : 0 < ↑v
Hv' : 0 < 2 * ↑v - 1
H₁ : 0 < ↑v / ↑u' / fract ξ
h' : |fract ξ - ↑u' / ↑v| < (↑v * (2 * ↑v - 1))⁻¹
h : 2 * ↑u' - (1 + ↑u') / ↑v < (2 * ↑v - 1) * fract ξ
this : (2 * ↑v - 1) * (-(↑v * (2 * ↑v - 1))⁻¹ + ↑u' / ↑v) = 2 * ↑u' - (1 + ↑u') / ↑v
⊢ 2 * ↑u' - 1 ≤ (2 * ↑v - 1) * fract ξ

Next tactic:
refine' LE.le.trans _ h.le
private theorem aux₃ :
    |(fract ξ)⁻¹ - v / (u - ⌊ξ⌋ * v)| < (((u : ℝ) - ⌊ξ⌋ * v) * (2 * (u - ⌊ξ⌋ * v) - 1))⁻¹ := by
  obtain ⟨hu₀, huv⟩ := aux₂ hv h
  have hξ₀ := aux₁ hv h
  set u' := u - ⌊ξ⌋ * v with hu'
  have hu'ℝ : (u' : ℝ) = u - ⌊ξ⌋ * v := mod_cast hu'
  rw [← hu'ℝ]
  replace hu'ℝ := (eq_sub_iff_add_eq.mp hu'ℝ).symm
  obtain ⟨Hu, Hu'⟩ := aux₀ hu₀
  obtain ⟨Hv, Hv'⟩ := aux₀ (zero_lt_two.trans_le hv)
  have H₁ := div_pos (div_pos Hv Hu) hξ₀
  replace h := h.2.2
  have h' : |fract ξ - u' / v| < ((v : ℝ) * (2 * v - 1))⁻¹ := by
    rwa [hu'ℝ, add_div, mul_div_cancel _ Hv.ne', ← sub_sub, sub_right_comm] at h
  have H : (2 * u' - 1 : ℝ) ≤ (2 * v - 1) * fract ξ := by
    replace h := (abs_lt.mp h).1
    have : (2 * (v : ℝ) - 1) * (-((v : ℝ) * (2 * v - 1))⁻¹ + u' / v) = 2 * u' - (1 + u') / v := by
      field_simp; ring
    rw [hu'ℝ, add_div, mul_div_cancel _ Hv.ne', ← sub_sub, sub_right_comm, self_sub_floor,
      lt_sub_iff_add_lt, ← mul_lt_mul_left Hv', this] at h
    refine' LE.le.trans _ h.le
    

Proof state:
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
hξ₀ : 0 < fract ξ
u' : ℤ := u - ⌊ξ⌋ * v
hu₀ : 0 < u'
huv : u' < v
hu' : u' = u - ⌊ξ⌋ * v
hu'ℝ : ↑u = ↑u' + ↑⌊ξ⌋ * ↑v
Hu : 0 < ↑u'
Hu' : 0 < 2 * ↑u' - 1
Hv : 0 < ↑v
Hv' : 0 < 2 * ↑v - 1
H₁ : 0 < ↑v / ↑u' / fract ξ
h' : |fract ξ - ↑u' / ↑v| < (↑v * (2 * ↑v - 1))⁻¹
h : 2 * ↑u' - (1 + ↑u') / ↑v < (2 * ↑v - 1) * fract ξ
this : (2 * ↑v - 1) * (-(↑v * (2 * ↑v - 1))⁻¹ + ↑u' / ↑v) = 2 * ↑u' - (1 + ↑u') / ↑v
⊢ 2 * ↑u' - 1 ≤ 2 * ↑u' - (1 + ↑u') / ↑v

Next tactic:
rw [sub_le_sub_iff_left, div_le_one Hv, add_comm]
private theorem aux₃ :
    |(fract ξ)⁻¹ - v / (u - ⌊ξ⌋ * v)| < (((u : ℝ) - ⌊ξ⌋ * v) * (2 * (u - ⌊ξ⌋ * v) - 1))⁻¹ := by
  obtain ⟨hu₀, huv⟩ := aux₂ hv h
  have hξ₀ := aux₁ hv h
  set u' := u - ⌊ξ⌋ * v with hu'
  have hu'ℝ : (u' : ℝ) = u - ⌊ξ⌋ * v := mod_cast hu'
  rw [← hu'ℝ]
  replace hu'ℝ := (eq_sub_iff_add_eq.mp hu'ℝ).symm
  obtain ⟨Hu, Hu'⟩ := aux₀ hu₀
  obtain ⟨Hv, Hv'⟩ := aux₀ (zero_lt_two.trans_le hv)
  have H₁ := div_pos (div_pos Hv Hu) hξ₀
  replace h := h.2.2
  have h' : |fract ξ - u' / v| < ((v : ℝ) * (2 * v - 1))⁻¹ := by
    rwa [hu'ℝ, add_div, mul_div_cancel _ Hv.ne', ← sub_sub, sub_right_comm] at h
  have H : (2 * u' - 1 : ℝ) ≤ (2 * v - 1) * fract ξ := by
    replace h := (abs_lt.mp h).1
    have : (2 * (v : ℝ) - 1) * (-((v : ℝ) * (2 * v - 1))⁻¹ + u' / v) = 2 * u' - (1 + u') / v := by
      field_simp; ring
    rw [hu'ℝ, add_div, mul_div_cancel _ Hv.ne', ← sub_sub, sub_right_comm, self_sub_floor,
      lt_sub_iff_add_lt, ← mul_lt_mul_left Hv', this] at h
    refine' LE.le.trans _ h.le
    rw [sub_le_sub_iff_left, div_le_one Hv, add_comm]
    

Proof state:
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
hξ₀ : 0 < fract ξ
u' : ℤ := u - ⌊ξ⌋ * v
hu₀ : 0 < u'
huv : u' < v
hu' : u' = u - ⌊ξ⌋ * v
hu'ℝ : ↑u = ↑u' + ↑⌊ξ⌋ * ↑v
Hu : 0 < ↑u'
Hu' : 0 < 2 * ↑u' - 1
Hv : 0 < ↑v
Hv' : 0 < 2 * ↑v - 1
H₁ : 0 < ↑v / ↑u' / fract ξ
h' : |fract ξ - ↑u' / ↑v| < (↑v * (2 * ↑v - 1))⁻¹
h : 2 * ↑u' - (1 + ↑u') / ↑v < (2 * ↑v - 1) * fract ξ
this : (2 * ↑v - 1) * (-(↑v * (2 * ↑v - 1))⁻¹ + ↑u' / ↑v) = 2 * ↑u' - (1 + ↑u') / ↑v
⊢ ↑u' + 1 ≤ ↑v

Next tactic:
exact mod_cast huv
private theorem aux₃ :
    |(fract ξ)⁻¹ - v / (u - ⌊ξ⌋ * v)| < (((u : ℝ) - ⌊ξ⌋ * v) * (2 * (u - ⌊ξ⌋ * v) - 1))⁻¹ := by
  obtain ⟨hu₀, huv⟩ := aux₂ hv h
  have hξ₀ := aux₁ hv h
  set u' := u - ⌊ξ⌋ * v with hu'
  have hu'ℝ : (u' : ℝ) = u - ⌊ξ⌋ * v := mod_cast hu'
  rw [← hu'ℝ]
  replace hu'ℝ := (eq_sub_iff_add_eq.mp hu'ℝ).symm
  obtain ⟨Hu, Hu'⟩ := aux₀ hu₀
  obtain ⟨Hv, Hv'⟩ := aux₀ (zero_lt_two.trans_le hv)
  have H₁ := div_pos (div_pos Hv Hu) hξ₀
  replace h := h.2.2
  have h' : |fract ξ - u' / v| < ((v : ℝ) * (2 * v - 1))⁻¹ := by
    rwa [hu'ℝ, add_div, mul_div_cancel _ Hv.ne', ← sub_sub, sub_right_comm] at h
  have H : (2 * u' - 1 : ℝ) ≤ (2 * v - 1) * fract ξ := by
    replace h := (abs_lt.mp h).1
    have : (2 * (v : ℝ) - 1) * (-((v : ℝ) * (2 * v - 1))⁻¹ + u' / v) = 2 * u' - (1 + u') / v := by
      field_simp; ring
    rw [hu'ℝ, add_div, mul_div_cancel _ Hv.ne', ← sub_sub, sub_right_comm, self_sub_floor,
      lt_sub_iff_add_lt, ← mul_lt_mul_left Hv', this] at h
    refine' LE.le.trans _ h.le
    rw [sub_le_sub_iff_left, div_le_one Hv, add_comm]
    exact mod_cast huv
  

Proof state:
case intro.intro.intro
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
hξ₀ : 0 < fract ξ
u' : ℤ := u - ⌊ξ⌋ * v
hu₀ : 0 < u'
huv : u' < v
hu' : u' = u - ⌊ξ⌋ * v
hu'ℝ : ↑u = ↑u' + ↑⌊ξ⌋ * ↑v
Hu : 0 < ↑u'
Hu' : 0 < 2 * ↑u' - 1
Hv : 0 < ↑v
Hv' : 0 < 2 * ↑v - 1
H₁ : 0 < ↑v / ↑u' / fract ξ
h : |ξ - ↑u / ↑v| < (↑v * (2 * ↑v - 1))⁻¹
h' : |fract ξ - ↑u' / ↑v| < (↑v * (2 * ↑v - 1))⁻¹
H : 2 * ↑u' - 1 ≤ (2 * ↑v - 1) * fract ξ
⊢ |(fract ξ)⁻¹ - ↑v / ↑u'| < (↑u' * (2 * ↑u' - 1))⁻¹

Next tactic:
have help₁ : ∀ {a b c : ℝ}, a ≠ 0 → b ≠ 0 → c ≠ 0 → |a⁻¹ - b / c| = |(a - c / b) * (b / c / a)| :=
    by intros; rw [abs_sub_comm]; congr 1; field_simp; ring
private theorem aux₃ :
    |(fract ξ)⁻¹ - v / (u - ⌊ξ⌋ * v)| < (((u : ℝ) - ⌊ξ⌋ * v) * (2 * (u - ⌊ξ⌋ * v) - 1))⁻¹ := by
  obtain ⟨hu₀, huv⟩ := aux₂ hv h
  have hξ₀ := aux₁ hv h
  set u' := u - ⌊ξ⌋ * v with hu'
  have hu'ℝ : (u' : ℝ) = u - ⌊ξ⌋ * v := mod_cast hu'
  rw [← hu'ℝ]
  replace hu'ℝ := (eq_sub_iff_add_eq.mp hu'ℝ).symm
  obtain ⟨Hu, Hu'⟩ := aux₀ hu₀
  obtain ⟨Hv, Hv'⟩ := aux₀ (zero_lt_two.trans_le hv)
  have H₁ := div_pos (div_pos Hv Hu) hξ₀
  replace h := h.2.2
  have h' : |fract ξ - u' / v| < ((v : ℝ) * (2 * v - 1))⁻¹ := by
    rwa [hu'ℝ, add_div, mul_div_cancel _ Hv.ne', ← sub_sub, sub_right_comm] at h
  have H : (2 * u' - 1 : ℝ) ≤ (2 * v - 1) * fract ξ := by
    replace h := (abs_lt.mp h).1
    have : (2 * (v : ℝ) - 1) * (-((v : ℝ) * (2 * v - 1))⁻¹ + u' / v) = 2 * u' - (1 + u') / v := by
      field_simp; ring
    rw [hu'ℝ, add_div, mul_div_cancel _ Hv.ne', ← sub_sub, sub_right_comm, self_sub_floor,
      lt_sub_iff_add_lt, ← mul_lt_mul_left Hv', this] at h
    refine' LE.le.trans _ h.le
    rw [sub_le_sub_iff_left, div_le_one Hv, add_comm]
    exact mod_cast huv
  have help₁ : ∀ {a b c : ℝ}, a ≠ 0 → b ≠ 0 → c ≠ 0 → |a⁻¹ - b / c| = |(a - c / b) * (b / c / a)| :=
    by 

Proof state:
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
hξ₀ : 0 < fract ξ
u' : ℤ := u - ⌊ξ⌋ * v
hu₀ : 0 < u'
huv : u' < v
hu' : u' = u - ⌊ξ⌋ * v
hu'ℝ : ↑u = ↑u' + ↑⌊ξ⌋ * ↑v
Hu : 0 < ↑u'
Hu' : 0 < 2 * ↑u' - 1
Hv : 0 < ↑v
Hv' : 0 < 2 * ↑v - 1
H₁ : 0 < ↑v / ↑u' / fract ξ
h : |ξ - ↑u / ↑v| < (↑v * (2 * ↑v - 1))⁻¹
h' : |fract ξ - ↑u' / ↑v| < (↑v * (2 * ↑v - 1))⁻¹
H : 2 * ↑u' - 1 ≤ (2 * ↑v - 1) * fract ξ
⊢ ∀ {a b c : ℝ}, a ≠ 0 → b ≠ 0 → c ≠ 0 → |a⁻¹ - b / c| = |(a - c / b) * (b / c / a)|

Next tactic:
intros
private theorem aux₃ :
    |(fract ξ)⁻¹ - v / (u - ⌊ξ⌋ * v)| < (((u : ℝ) - ⌊ξ⌋ * v) * (2 * (u - ⌊ξ⌋ * v) - 1))⁻¹ := by
  obtain ⟨hu₀, huv⟩ := aux₂ hv h
  have hξ₀ := aux₁ hv h
  set u' := u - ⌊ξ⌋ * v with hu'
  have hu'ℝ : (u' : ℝ) = u - ⌊ξ⌋ * v := mod_cast hu'
  rw [← hu'ℝ]
  replace hu'ℝ := (eq_sub_iff_add_eq.mp hu'ℝ).symm
  obtain ⟨Hu, Hu'⟩ := aux₀ hu₀
  obtain ⟨Hv, Hv'⟩ := aux₀ (zero_lt_two.trans_le hv)
  have H₁ := div_pos (div_pos Hv Hu) hξ₀
  replace h := h.2.2
  have h' : |fract ξ - u' / v| < ((v : ℝ) * (2 * v - 1))⁻¹ := by
    rwa [hu'ℝ, add_div, mul_div_cancel _ Hv.ne', ← sub_sub, sub_right_comm] at h
  have H : (2 * u' - 1 : ℝ) ≤ (2 * v - 1) * fract ξ := by
    replace h := (abs_lt.mp h).1
    have : (2 * (v : ℝ) - 1) * (-((v : ℝ) * (2 * v - 1))⁻¹ + u' / v) = 2 * u' - (1 + u') / v := by
      field_simp; ring
    rw [hu'ℝ, add_div, mul_div_cancel _ Hv.ne', ← sub_sub, sub_right_comm, self_sub_floor,
      lt_sub_iff_add_lt, ← mul_lt_mul_left Hv', this] at h
    refine' LE.le.trans _ h.le
    rw [sub_le_sub_iff_left, div_le_one Hv, add_comm]
    exact mod_cast huv
  have help₁ : ∀ {a b c : ℝ}, a ≠ 0 → b ≠ 0 → c ≠ 0 → |a⁻¹ - b / c| = |(a - c / b) * (b / c / a)| :=
    by intros; 

Proof state:
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
hξ₀ : 0 < fract ξ
u' : ℤ := u - ⌊ξ⌋ * v
hu₀ : 0 < u'
huv : u' < v
hu' : u' = u - ⌊ξ⌋ * v
hu'ℝ : ↑u = ↑u' + ↑⌊ξ⌋ * ↑v
Hu : 0 < ↑u'
Hu' : 0 < 2 * ↑u' - 1
Hv : 0 < ↑v
Hv' : 0 < 2 * ↑v - 1
H₁ : 0 < ↑v / ↑u' / fract ξ
h : |ξ - ↑u / ↑v| < (↑v * (2 * ↑v - 1))⁻¹
h' : |fract ξ - ↑u' / ↑v| < (↑v * (2 * ↑v - 1))⁻¹
H : 2 * ↑u' - 1 ≤ (2 * ↑v - 1) * fract ξ
a✝³ b✝ c✝ : ℝ
a✝² : a✝³ ≠ 0
a✝¹ : b✝ ≠ 0
a✝ : c✝ ≠ 0
⊢ |a✝³⁻¹ - b✝ / c✝| = |(a✝³ - c✝ / b✝) * (b✝ / c✝ / a✝³)|

Next tactic:
rw [abs_sub_comm]
private theorem aux₃ :
    |(fract ξ)⁻¹ - v / (u - ⌊ξ⌋ * v)| < (((u : ℝ) - ⌊ξ⌋ * v) * (2 * (u - ⌊ξ⌋ * v) - 1))⁻¹ := by
  obtain ⟨hu₀, huv⟩ := aux₂ hv h
  have hξ₀ := aux₁ hv h
  set u' := u - ⌊ξ⌋ * v with hu'
  have hu'ℝ : (u' : ℝ) = u - ⌊ξ⌋ * v := mod_cast hu'
  rw [← hu'ℝ]
  replace hu'ℝ := (eq_sub_iff_add_eq.mp hu'ℝ).symm
  obtain ⟨Hu, Hu'⟩ := aux₀ hu₀
  obtain ⟨Hv, Hv'⟩ := aux₀ (zero_lt_two.trans_le hv)
  have H₁ := div_pos (div_pos Hv Hu) hξ₀
  replace h := h.2.2
  have h' : |fract ξ - u' / v| < ((v : ℝ) * (2 * v - 1))⁻¹ := by
    rwa [hu'ℝ, add_div, mul_div_cancel _ Hv.ne', ← sub_sub, sub_right_comm] at h
  have H : (2 * u' - 1 : ℝ) ≤ (2 * v - 1) * fract ξ := by
    replace h := (abs_lt.mp h).1
    have : (2 * (v : ℝ) - 1) * (-((v : ℝ) * (2 * v - 1))⁻¹ + u' / v) = 2 * u' - (1 + u') / v := by
      field_simp; ring
    rw [hu'ℝ, add_div, mul_div_cancel _ Hv.ne', ← sub_sub, sub_right_comm, self_sub_floor,
      lt_sub_iff_add_lt, ← mul_lt_mul_left Hv', this] at h
    refine' LE.le.trans _ h.le
    rw [sub_le_sub_iff_left, div_le_one Hv, add_comm]
    exact mod_cast huv
  have help₁ : ∀ {a b c : ℝ}, a ≠ 0 → b ≠ 0 → c ≠ 0 → |a⁻¹ - b / c| = |(a - c / b) * (b / c / a)| :=
    by intros; rw [abs_sub_comm]; 

Proof state:
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
hξ₀ : 0 < fract ξ
u' : ℤ := u - ⌊ξ⌋ * v
hu₀ : 0 < u'
huv : u' < v
hu' : u' = u - ⌊ξ⌋ * v
hu'ℝ : ↑u = ↑u' + ↑⌊ξ⌋ * ↑v
Hu : 0 < ↑u'
Hu' : 0 < 2 * ↑u' - 1
Hv : 0 < ↑v
Hv' : 0 < 2 * ↑v - 1
H₁ : 0 < ↑v / ↑u' / fract ξ
h : |ξ - ↑u / ↑v| < (↑v * (2 * ↑v - 1))⁻¹
h' : |fract ξ - ↑u' / ↑v| < (↑v * (2 * ↑v - 1))⁻¹
H : 2 * ↑u' - 1 ≤ (2 * ↑v - 1) * fract ξ
a✝³ b✝ c✝ : ℝ
a✝² : a✝³ ≠ 0
a✝¹ : b✝ ≠ 0
a✝ : c✝ ≠ 0
⊢ |b✝ / c✝ - a✝³⁻¹| = |(a✝³ - c✝ / b✝) * (b✝ / c✝ / a✝³)|

Next tactic:
congr 1
private theorem aux₃ :
    |(fract ξ)⁻¹ - v / (u - ⌊ξ⌋ * v)| < (((u : ℝ) - ⌊ξ⌋ * v) * (2 * (u - ⌊ξ⌋ * v) - 1))⁻¹ := by
  obtain ⟨hu₀, huv⟩ := aux₂ hv h
  have hξ₀ := aux₁ hv h
  set u' := u - ⌊ξ⌋ * v with hu'
  have hu'ℝ : (u' : ℝ) = u - ⌊ξ⌋ * v := mod_cast hu'
  rw [← hu'ℝ]
  replace hu'ℝ := (eq_sub_iff_add_eq.mp hu'ℝ).symm
  obtain ⟨Hu, Hu'⟩ := aux₀ hu₀
  obtain ⟨Hv, Hv'⟩ := aux₀ (zero_lt_two.trans_le hv)
  have H₁ := div_pos (div_pos Hv Hu) hξ₀
  replace h := h.2.2
  have h' : |fract ξ - u' / v| < ((v : ℝ) * (2 * v - 1))⁻¹ := by
    rwa [hu'ℝ, add_div, mul_div_cancel _ Hv.ne', ← sub_sub, sub_right_comm] at h
  have H : (2 * u' - 1 : ℝ) ≤ (2 * v - 1) * fract ξ := by
    replace h := (abs_lt.mp h).1
    have : (2 * (v : ℝ) - 1) * (-((v : ℝ) * (2 * v - 1))⁻¹ + u' / v) = 2 * u' - (1 + u') / v := by
      field_simp; ring
    rw [hu'ℝ, add_div, mul_div_cancel _ Hv.ne', ← sub_sub, sub_right_comm, self_sub_floor,
      lt_sub_iff_add_lt, ← mul_lt_mul_left Hv', this] at h
    refine' LE.le.trans _ h.le
    rw [sub_le_sub_iff_left, div_le_one Hv, add_comm]
    exact mod_cast huv
  have help₁ : ∀ {a b c : ℝ}, a ≠ 0 → b ≠ 0 → c ≠ 0 → |a⁻¹ - b / c| = |(a - c / b) * (b / c / a)| :=
    by intros; rw [abs_sub_comm]; congr 1; 

Proof state:
case e_a
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
hξ₀ : 0 < fract ξ
u' : ℤ := u - ⌊ξ⌋ * v
hu₀ : 0 < u'
huv : u' < v
hu' : u' = u - ⌊ξ⌋ * v
hu'ℝ : ↑u = ↑u' + ↑⌊ξ⌋ * ↑v
Hu : 0 < ↑u'
Hu' : 0 < 2 * ↑u' - 1
Hv : 0 < ↑v
Hv' : 0 < 2 * ↑v - 1
H₁ : 0 < ↑v / ↑u' / fract ξ
h : |ξ - ↑u / ↑v| < (↑v * (2 * ↑v - 1))⁻¹
h' : |fract ξ - ↑u' / ↑v| < (↑v * (2 * ↑v - 1))⁻¹
H : 2 * ↑u' - 1 ≤ (2 * ↑v - 1) * fract ξ
a✝³ b✝ c✝ : ℝ
a✝² : a✝³ ≠ 0
a✝¹ : b✝ ≠ 0
a✝ : c✝ ≠ 0
⊢ b✝ / c✝ - a✝³⁻¹ = (a✝³ - c✝ / b✝) * (b✝ / c✝ / a✝³)

Next tactic:
field_simp
private theorem aux₃ :
    |(fract ξ)⁻¹ - v / (u - ⌊ξ⌋ * v)| < (((u : ℝ) - ⌊ξ⌋ * v) * (2 * (u - ⌊ξ⌋ * v) - 1))⁻¹ := by
  obtain ⟨hu₀, huv⟩ := aux₂ hv h
  have hξ₀ := aux₁ hv h
  set u' := u - ⌊ξ⌋ * v with hu'
  have hu'ℝ : (u' : ℝ) = u - ⌊ξ⌋ * v := mod_cast hu'
  rw [← hu'ℝ]
  replace hu'ℝ := (eq_sub_iff_add_eq.mp hu'ℝ).symm
  obtain ⟨Hu, Hu'⟩ := aux₀ hu₀
  obtain ⟨Hv, Hv'⟩ := aux₀ (zero_lt_two.trans_le hv)
  have H₁ := div_pos (div_pos Hv Hu) hξ₀
  replace h := h.2.2
  have h' : |fract ξ - u' / v| < ((v : ℝ) * (2 * v - 1))⁻¹ := by
    rwa [hu'ℝ, add_div, mul_div_cancel _ Hv.ne', ← sub_sub, sub_right_comm] at h
  have H : (2 * u' - 1 : ℝ) ≤ (2 * v - 1) * fract ξ := by
    replace h := (abs_lt.mp h).1
    have : (2 * (v : ℝ) - 1) * (-((v : ℝ) * (2 * v - 1))⁻¹ + u' / v) = 2 * u' - (1 + u') / v := by
      field_simp; ring
    rw [hu'ℝ, add_div, mul_div_cancel _ Hv.ne', ← sub_sub, sub_right_comm, self_sub_floor,
      lt_sub_iff_add_lt, ← mul_lt_mul_left Hv', this] at h
    refine' LE.le.trans _ h.le
    rw [sub_le_sub_iff_left, div_le_one Hv, add_comm]
    exact mod_cast huv
  have help₁ : ∀ {a b c : ℝ}, a ≠ 0 → b ≠ 0 → c ≠ 0 → |a⁻¹ - b / c| = |(a - c / b) * (b / c / a)| :=
    by intros; rw [abs_sub_comm]; congr 1; field_simp; 

Proof state:
case e_a
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
hξ₀ : 0 < fract ξ
u' : ℤ := u - ⌊ξ⌋ * v
hu₀ : 0 < u'
huv : u' < v
hu' : u' = u - ⌊ξ⌋ * v
hu'ℝ : ↑u = ↑u' + ↑⌊ξ⌋ * ↑v
Hu : 0 < ↑u'
Hu' : 0 < 2 * ↑u' - 1
Hv : 0 < ↑v
Hv' : 0 < 2 * ↑v - 1
H₁ : 0 < ↑v / ↑u' / fract ξ
h : |ξ - ↑u / ↑v| < (↑v * (2 * ↑v - 1))⁻¹
h' : |fract ξ - ↑u' / ↑v| < (↑v * (2 * ↑v - 1))⁻¹
H : 2 * ↑u' - 1 ≤ (2 * ↑v - 1) * fract ξ
a✝³ b✝ c✝ : ℝ
a✝² : a✝³ ≠ 0
a✝¹ : b✝ ≠ 0
a✝ : c✝ ≠ 0
⊢ b✝ * a✝³ = a✝³ * b✝

Next tactic:
ring