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
  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; ring
  

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
help₁ : ∀ {a b c : ℝ}, a ≠ 0 → b ≠ 0 → c ≠ 0 → |a⁻¹ - b / c| = |(a - c / b) * (b / c / a)|
⊢ |(fract ξ)⁻¹ - ↑v / ↑u'| < (↑u' * (2 * ↑u' - 1))⁻¹

Next tactic:
have help₂ :
    ∀ {a b c d : ℝ}, a ≠ 0 → b ≠ 0 → c ≠ 0 → d ≠ 0 → (b * c)⁻¹ * (b / d / a) = (d * c * a)⁻¹ := by
    intros; 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 intros; rw [abs_sub_comm]; congr 1; field_simp; ring
  have help₂ :
    ∀ {a b c d : ℝ}, a ≠ 0 → b ≠ 0 → c ≠ 0 → d ≠ 0 → (b * c)⁻¹ * (b / d / a) = (d * c * a)⁻¹ := 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
help₁ : ∀ {a b c : ℝ}, a ≠ 0 → b ≠ 0 → c ≠ 0 → |a⁻¹ - b / c| = |(a - c / b) * (b / c / a)|
⊢ ∀ {a b c d : ℝ}, a ≠ 0 → b ≠ 0 → c ≠ 0 → d ≠ 0 → (b * c)⁻¹ * (b / d / a) = (d * c * a)⁻¹

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; rw [abs_sub_comm]; congr 1; field_simp; ring
  have help₂ :
    ∀ {a b c d : ℝ}, a ≠ 0 → b ≠ 0 → c ≠ 0 → d ≠ 0 → (b * c)⁻¹ * (b / d / a) = (d * c * a)⁻¹ := 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
help₁ : ∀ {a b c : ℝ}, a ≠ 0 → b ≠ 0 → c ≠ 0 → |a⁻¹ - b / c| = |(a - c / b) * (b / c / a)|
a✝⁴ b✝ c✝ d✝ : ℝ
a✝³ : a✝⁴ ≠ 0
a✝² : b✝ ≠ 0
a✝¹ : c✝ ≠ 0
a✝ : d✝ ≠ 0
⊢ (b✝ * c✝)⁻¹ * (b✝ / d✝ / a✝⁴) = (d✝ * c✝ * a✝⁴)⁻¹

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; ring
  have help₂ :
    ∀ {a b c d : ℝ}, a ≠ 0 → b ≠ 0 → c ≠ 0 → d ≠ 0 → (b * c)⁻¹ * (b / d / a) = (d * c * a)⁻¹ := by
    intros; 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 : |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
help₁ : ∀ {a b c : ℝ}, a ≠ 0 → b ≠ 0 → c ≠ 0 → |a⁻¹ - b / c| = |(a - c / b) * (b / c / a)|
a✝⁴ b✝ c✝ d✝ : ℝ
a✝³ : a✝⁴ ≠ 0
a✝² : b✝ ≠ 0
a✝¹ : c✝ ≠ 0
a✝ : d✝ ≠ 0
⊢ b✝ * (d✝ * c✝ * a✝⁴) = b✝ * c✝ * (d✝ * a✝⁴)

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
    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; ring
  have help₂ :
    ∀ {a b c d : ℝ}, a ≠ 0 → b ≠ 0 → c ≠ 0 → d ≠ 0 → (b * c)⁻¹ * (b / d / a) = (d * c * a)⁻¹ := by
    intros; field_simp; ring
  

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
help₁ : ∀ {a b c : ℝ}, a ≠ 0 → b ≠ 0 → c ≠ 0 → |a⁻¹ - b / c| = |(a - c / b) * (b / c / a)|
help₂ : ∀ {a b c d : ℝ}, a ≠ 0 → b ≠ 0 → c ≠ 0 → d ≠ 0 → (b * c)⁻¹ * (b / d / a) = (d * c * a)⁻¹
⊢ |(fract ξ)⁻¹ - ↑v / ↑u'| < (↑u' * (2 * ↑u' - 1))⁻¹

Next tactic:
calc
    |(fract ξ)⁻¹ - v / u'| = |(fract ξ - u' / v) * (v / u' / fract ξ)| :=
      help₁ hξ₀.ne' Hv.ne' Hu.ne'
    _ = |fract ξ - u' / v| * (v / u' / fract ξ) := by rw [abs_mul, abs_of_pos H₁, abs_sub_comm]
    _ < ((v : ℝ) * (2 * v - 1))⁻¹ * (v / u' / fract ξ) := ((mul_lt_mul_right H₁).mpr h')
    _ = (u' * (2 * v - 1) * fract ξ)⁻¹ := (help₂ hξ₀.ne' Hv.ne' Hv'.ne' Hu.ne')
    _ ≤ ((u' : ℝ) * (2 * u' - 1))⁻¹ := by
      rwa [inv_le_inv (mul_pos (mul_pos Hu Hv') hξ₀) <| mul_pos Hu Hu', mul_assoc,
        mul_le_mul_left Hu]
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; ring
  have help₂ :
    ∀ {a b c d : ℝ}, a ≠ 0 → b ≠ 0 → c ≠ 0 → d ≠ 0 → (b * c)⁻¹ * (b / d / a) = (d * c * a)⁻¹ := by
    intros; field_simp; ring
  calc
    |(fract ξ)⁻¹ - v / u'| = |(fract ξ - u' / v) * (v / u' / fract ξ)| :=
      help₁ hξ₀.ne' Hv.ne' Hu.ne'
    _ = |fract ξ - u' / v| * (v / u' / 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
H : 2 * ↑u' - 1 ≤ (2 * ↑v - 1) * fract ξ
help₁ : ∀ {a b c : ℝ}, a ≠ 0 → b ≠ 0 → c ≠ 0 → |a⁻¹ - b / c| = |(a - c / b) * (b / c / a)|
help₂ : ∀ {a b c d : ℝ}, a ≠ 0 → b ≠ 0 → c ≠ 0 → d ≠ 0 → (b * c)⁻¹ * (b / d / a) = (d * c * a)⁻¹
⊢ |(fract ξ - ↑u' / ↑v) * (↑v / ↑u' / fract ξ)| = |fract ξ - ↑u' / ↑v| * (↑v / ↑u' / fract ξ)

Next tactic:
rw [abs_mul, abs_of_pos H₁, 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]; congr 1; field_simp; ring
  have help₂ :
    ∀ {a b c d : ℝ}, a ≠ 0 → b ≠ 0 → c ≠ 0 → d ≠ 0 → (b * c)⁻¹ * (b / d / a) = (d * c * a)⁻¹ := by
    intros; field_simp; ring
  calc
    |(fract ξ)⁻¹ - v / u'| = |(fract ξ - u' / v) * (v / u' / fract ξ)| :=
      help₁ hξ₀.ne' Hv.ne' Hu.ne'
    _ = |fract ξ - u' / v| * (v / u' / fract ξ) := by rw [abs_mul, abs_of_pos H₁, abs_sub_comm]
    _ < ((v : ℝ) * (2 * v - 1))⁻¹ * (v / u' / fract ξ) := ((mul_lt_mul_right H₁).mpr h')
    _ = (u' * (2 * v - 1) * fract ξ)⁻¹ := (help₂ hξ₀.ne' Hv.ne' Hv'.ne' Hu.ne')
    _ ≤ ((u' : ℝ) * (2 * u' - 1))⁻¹ := 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
help₁ : ∀ {a b c : ℝ}, a ≠ 0 → b ≠ 0 → c ≠ 0 → |a⁻¹ - b / c| = |(a - c / b) * (b / c / a)|
help₂ : ∀ {a b c d : ℝ}, a ≠ 0 → b ≠ 0 → c ≠ 0 → d ≠ 0 → (b * c)⁻¹ * (b / d / a) = (d * c * a)⁻¹
⊢ (↑u' * (2 * ↑v - 1) * fract ξ)⁻¹ ≤ (↑u' * (2 * ↑u' - 1))⁻¹

Next tactic:
rwa [inv_le_inv (mul_pos (mul_pos Hu Hv') hξ₀) <| mul_pos Hu Hu', mul_assoc,
        mul_le_mul_left Hu]
private theorem invariant : ContfracLegendre.Ass (fract ξ)⁻¹ v (u - ⌊ξ⌋ * v) := by
  

Proof state:
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
h : ContfracLegendre.Ass ξ u v
⊢ ContfracLegendre.Ass (fract ξ)⁻¹ v (u - ⌊ξ⌋ * v)

Next tactic:
refine' ⟨_, fun huv => _, mod_cast aux₃ hv h⟩
private theorem invariant : ContfracLegendre.Ass (fract ξ)⁻¹ v (u - ⌊ξ⌋ * v) := by
  refine' ⟨_, fun huv => _, mod_cast aux₃ hv h⟩
  · 

Proof state:
case refine'_1
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
h : ContfracLegendre.Ass ξ u v
⊢ IsCoprime v (u - ⌊ξ⌋ * v)

Next tactic:
rw [sub_eq_add_neg, ← neg_mul, isCoprime_comm, IsCoprime.add_mul_right_left_iff]
private theorem invariant : ContfracLegendre.Ass (fract ξ)⁻¹ v (u - ⌊ξ⌋ * v) := by
  refine' ⟨_, fun huv => _, mod_cast aux₃ hv h⟩
  · rw [sub_eq_add_neg, ← neg_mul, isCoprime_comm, IsCoprime.add_mul_right_left_iff]
    

Proof state:
case refine'_1
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
h : ContfracLegendre.Ass ξ u v
⊢ IsCoprime u v

Next tactic:
exact h.1
private theorem invariant : ContfracLegendre.Ass (fract ξ)⁻¹ v (u - ⌊ξ⌋ * v) := by
  refine' ⟨_, fun huv => _, mod_cast aux₃ hv h⟩
  · rw [sub_eq_add_neg, ← neg_mul, isCoprime_comm, IsCoprime.add_mul_right_left_iff]
    exact h.1
  · 

Proof state:
case refine'_2
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
h : ContfracLegendre.Ass ξ u v
huv : u - ⌊ξ⌋ * v = 1
⊢ -(1 / 2) < (fract ξ)⁻¹ - ↑v

Next tactic:
obtain hv₀' := (aux₀ (zero_lt_two.trans_le hv)).2
private theorem invariant : ContfracLegendre.Ass (fract ξ)⁻¹ v (u - ⌊ξ⌋ * v) := by
  refine' ⟨_, fun huv => _, mod_cast aux₃ hv h⟩
  · rw [sub_eq_add_neg, ← neg_mul, isCoprime_comm, IsCoprime.add_mul_right_left_iff]
    exact h.1
  · obtain hv₀' := (aux₀ (zero_lt_two.trans_le hv)).2
    

Proof state:
case refine'_2
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
h : ContfracLegendre.Ass ξ u v
huv : u - ⌊ξ⌋ * v = 1
hv₀' : 0 < 2 * ↑v - 1
⊢ -(1 / 2) < (fract ξ)⁻¹ - ↑v

Next tactic:
have Hv : (v * (2 * v - 1) : ℝ)⁻¹ + (v : ℝ)⁻¹ = 2 / (2 * v - 1) := by
      field_simp; ring
private theorem invariant : ContfracLegendre.Ass (fract ξ)⁻¹ v (u - ⌊ξ⌋ * v) := by
  refine' ⟨_, fun huv => _, mod_cast aux₃ hv h⟩
  · rw [sub_eq_add_neg, ← neg_mul, isCoprime_comm, IsCoprime.add_mul_right_left_iff]
    exact h.1
  · obtain hv₀' := (aux₀ (zero_lt_two.trans_le hv)).2
    have Hv : (v * (2 * v - 1) : ℝ)⁻¹ + (v : ℝ)⁻¹ = 2 / (2 * v - 1) := by
      

Proof state:
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
h : ContfracLegendre.Ass ξ u v
huv : u - ⌊ξ⌋ * v = 1
hv₀' : 0 < 2 * ↑v - 1
⊢ (↑v * (2 * ↑v - 1))⁻¹ + (↑v)⁻¹ = 2 / (2 * ↑v - 1)

Next tactic:
field_simp
private theorem invariant : ContfracLegendre.Ass (fract ξ)⁻¹ v (u - ⌊ξ⌋ * v) := by
  refine' ⟨_, fun huv => _, mod_cast aux₃ hv h⟩
  · rw [sub_eq_add_neg, ← neg_mul, isCoprime_comm, IsCoprime.add_mul_right_left_iff]
    exact h.1
  · obtain hv₀' := (aux₀ (zero_lt_two.trans_le hv)).2
    have Hv : (v * (2 * v - 1) : ℝ)⁻¹ + (v : ℝ)⁻¹ = 2 / (2 * v - 1) := by
      field_simp; 

Proof state:
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
h : ContfracLegendre.Ass ξ u v
huv : u - ⌊ξ⌋ * v = 1
hv₀' : 0 < 2 * ↑v - 1
⊢ (↑v + ↑v * (2 * ↑v - 1)) * (2 * ↑v - 1) = 2 * (↑v * (2 * ↑v - 1) * ↑v)

Next tactic:
ring
private theorem invariant : ContfracLegendre.Ass (fract ξ)⁻¹ v (u - ⌊ξ⌋ * v) := by
  refine' ⟨_, fun huv => _, mod_cast aux₃ hv h⟩
  · rw [sub_eq_add_neg, ← neg_mul, isCoprime_comm, IsCoprime.add_mul_right_left_iff]
    exact h.1
  · obtain hv₀' := (aux₀ (zero_lt_two.trans_le hv)).2
    have Hv : (v * (2 * v - 1) : ℝ)⁻¹ + (v : ℝ)⁻¹ = 2 / (2 * v - 1) := by
      field_simp; ring
    

Proof state:
case refine'_2
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
h : ContfracLegendre.Ass ξ u v
huv : u - ⌊ξ⌋ * v = 1
hv₀' : 0 < 2 * ↑v - 1
Hv : (↑v * (2 * ↑v - 1))⁻¹ + (↑v)⁻¹ = 2 / (2 * ↑v - 1)
⊢ -(1 / 2) < (fract ξ)⁻¹ - ↑v

Next tactic:
have Huv : (u / v : ℝ) = ⌊ξ⌋ + (v : ℝ)⁻¹ := by
      rw [sub_eq_iff_eq_add'.mp huv]; field_simp
private theorem invariant : ContfracLegendre.Ass (fract ξ)⁻¹ v (u - ⌊ξ⌋ * v) := by
  refine' ⟨_, fun huv => _, mod_cast aux₃ hv h⟩
  · rw [sub_eq_add_neg, ← neg_mul, isCoprime_comm, IsCoprime.add_mul_right_left_iff]
    exact h.1
  · obtain hv₀' := (aux₀ (zero_lt_two.trans_le hv)).2
    have Hv : (v * (2 * v - 1) : ℝ)⁻¹ + (v : ℝ)⁻¹ = 2 / (2 * v - 1) := by
      field_simp; ring
    have Huv : (u / v : ℝ) = ⌊ξ⌋ + (v : ℝ)⁻¹ := by
      

Proof state:
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
h : ContfracLegendre.Ass ξ u v
huv : u - ⌊ξ⌋ * v = 1
hv₀' : 0 < 2 * ↑v - 1
Hv : (↑v * (2 * ↑v - 1))⁻¹ + (↑v)⁻¹ = 2 / (2 * ↑v - 1)
⊢ ↑u / ↑v = ↑⌊ξ⌋ + (↑v)⁻¹

Next tactic:
rw [sub_eq_iff_eq_add'.mp huv]
private theorem invariant : ContfracLegendre.Ass (fract ξ)⁻¹ v (u - ⌊ξ⌋ * v) := by
  refine' ⟨_, fun huv => _, mod_cast aux₃ hv h⟩
  · rw [sub_eq_add_neg, ← neg_mul, isCoprime_comm, IsCoprime.add_mul_right_left_iff]
    exact h.1
  · obtain hv₀' := (aux₀ (zero_lt_two.trans_le hv)).2
    have Hv : (v * (2 * v - 1) : ℝ)⁻¹ + (v : ℝ)⁻¹ = 2 / (2 * v - 1) := by
      field_simp; ring
    have Huv : (u / v : ℝ) = ⌊ξ⌋ + (v : ℝ)⁻¹ := by
      rw [sub_eq_iff_eq_add'.mp huv]; 

Proof state:
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
h : ContfracLegendre.Ass ξ u v
huv : u - ⌊ξ⌋ * v = 1
hv₀' : 0 < 2 * ↑v - 1
Hv : (↑v * (2 * ↑v - 1))⁻¹ + (↑v)⁻¹ = 2 / (2 * ↑v - 1)
⊢ ↑(⌊ξ⌋ * v + 1) / ↑v = ↑⌊ξ⌋ + (↑v)⁻¹

Next tactic:
field_simp
private theorem invariant : ContfracLegendre.Ass (fract ξ)⁻¹ v (u - ⌊ξ⌋ * v) := by
  refine' ⟨_, fun huv => _, mod_cast aux₃ hv h⟩
  · rw [sub_eq_add_neg, ← neg_mul, isCoprime_comm, IsCoprime.add_mul_right_left_iff]
    exact h.1
  · obtain hv₀' := (aux₀ (zero_lt_two.trans_le hv)).2
    have Hv : (v * (2 * v - 1) : ℝ)⁻¹ + (v : ℝ)⁻¹ = 2 / (2 * v - 1) := by
      field_simp; ring
    have Huv : (u / v : ℝ) = ⌊ξ⌋ + (v : ℝ)⁻¹ := by
      rw [sub_eq_iff_eq_add'.mp huv]; field_simp
    

Proof state:
case refine'_2
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
h : ContfracLegendre.Ass ξ u v
huv : u - ⌊ξ⌋ * v = 1
hv₀' : 0 < 2 * ↑v - 1
Hv : (↑v * (2 * ↑v - 1))⁻¹ + (↑v)⁻¹ = 2 / (2 * ↑v - 1)
Huv : ↑u / ↑v = ↑⌊ξ⌋ + (↑v)⁻¹
⊢ -(1 / 2) < (fract ξ)⁻¹ - ↑v

Next tactic:
have h' := (abs_sub_lt_iff.mp h.2.2).1
private theorem invariant : ContfracLegendre.Ass (fract ξ)⁻¹ v (u - ⌊ξ⌋ * v) := by
  refine' ⟨_, fun huv => _, mod_cast aux₃ hv h⟩
  · rw [sub_eq_add_neg, ← neg_mul, isCoprime_comm, IsCoprime.add_mul_right_left_iff]
    exact h.1
  · obtain hv₀' := (aux₀ (zero_lt_two.trans_le hv)).2
    have Hv : (v * (2 * v - 1) : ℝ)⁻¹ + (v : ℝ)⁻¹ = 2 / (2 * v - 1) := by
      field_simp; ring
    have Huv : (u / v : ℝ) = ⌊ξ⌋ + (v : ℝ)⁻¹ := by
      rw [sub_eq_iff_eq_add'.mp huv]; field_simp
    have h' := (abs_sub_lt_iff.mp h.2.2).1
    

Proof state:
case refine'_2
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
h : ContfracLegendre.Ass ξ u v
huv : u - ⌊ξ⌋ * v = 1
hv₀' : 0 < 2 * ↑v - 1
Hv : (↑v * (2 * ↑v - 1))⁻¹ + (↑v)⁻¹ = 2 / (2 * ↑v - 1)
Huv : ↑u / ↑v = ↑⌊ξ⌋ + (↑v)⁻¹
h' : ξ - ↑u / ↑v < (↑v * (2 * ↑v - 1))⁻¹
⊢ -(1 / 2) < (fract ξ)⁻¹ - ↑v

Next tactic:
rw [Huv, ← sub_sub, sub_lt_iff_lt_add, self_sub_floor, Hv] at h'
private theorem invariant : ContfracLegendre.Ass (fract ξ)⁻¹ v (u - ⌊ξ⌋ * v) := by
  refine' ⟨_, fun huv => _, mod_cast aux₃ hv h⟩
  · rw [sub_eq_add_neg, ← neg_mul, isCoprime_comm, IsCoprime.add_mul_right_left_iff]
    exact h.1
  · obtain hv₀' := (aux₀ (zero_lt_two.trans_le hv)).2
    have Hv : (v * (2 * v - 1) : ℝ)⁻¹ + (v : ℝ)⁻¹ = 2 / (2 * v - 1) := by
      field_simp; ring
    have Huv : (u / v : ℝ) = ⌊ξ⌋ + (v : ℝ)⁻¹ := by
      rw [sub_eq_iff_eq_add'.mp huv]; field_simp
    have h' := (abs_sub_lt_iff.mp h.2.2).1
    rw [Huv, ← sub_sub, sub_lt_iff_lt_add, self_sub_floor, Hv] at h'
    

Proof state:
case refine'_2
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
h : ContfracLegendre.Ass ξ u v
huv : u - ⌊ξ⌋ * v = 1
hv₀' : 0 < 2 * ↑v - 1
Hv : (↑v * (2 * ↑v - 1))⁻¹ + (↑v)⁻¹ = 2 / (2 * ↑v - 1)
Huv : ↑u / ↑v = ↑⌊ξ⌋ + (↑v)⁻¹
h' : fract ξ < 2 / (2 * ↑v - 1)
⊢ -(1 / 2) < (fract ξ)⁻¹ - ↑v

Next tactic:
rwa [lt_sub_iff_add_lt', (by ring : (v : ℝ) + -(1 / 2) = (2 * v - 1) / 2),
      lt_inv (div_pos hv₀' zero_lt_two) (aux₁ hv h), inv_div]
private theorem invariant : ContfracLegendre.Ass (fract ξ)⁻¹ v (u - ⌊ξ⌋ * v) := by
  refine' ⟨_, fun huv => _, mod_cast aux₃ hv h⟩
  · rw [sub_eq_add_neg, ← neg_mul, isCoprime_comm, IsCoprime.add_mul_right_left_iff]
    exact h.1
  · obtain hv₀' := (aux₀ (zero_lt_two.trans_le hv)).2
    have Hv : (v * (2 * v - 1) : ℝ)⁻¹ + (v : ℝ)⁻¹ = 2 / (2 * v - 1) := by
      field_simp; ring
    have Huv : (u / v : ℝ) = ⌊ξ⌋ + (v : ℝ)⁻¹ := by
      rw [sub_eq_iff_eq_add'.mp huv]; field_simp
    have h' := (abs_sub_lt_iff.mp h.2.2).1
    rw [Huv, ← sub_sub, sub_lt_iff_lt_add, self_sub_floor, Hv] at h'
    rwa [lt_sub_iff_add_lt', (by 

Proof state:
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
h : ContfracLegendre.Ass ξ u v
huv : u - ⌊ξ⌋ * v = 1
hv₀' : 0 < 2 * ↑v - 1
Hv : (↑v * (2 * ↑v - 1))⁻¹ + (↑v)⁻¹ = 2 / (2 * ↑v - 1)
Huv : ↑u / ↑v = ↑⌊ξ⌋ + (↑v)⁻¹
h' : fract ξ < 2 / (2 * ↑v - 1)
⊢ ↑v + -(1 / 2) = (2 * ↑v - 1) / 2

Next tactic:
ring