l3lab/ntp-mathlib huggingface dataset:
private theorem aux₂ : 0 < u - ⌊ξ⌋ * v ∧ u - ⌊ξ⌋ * v < v := by
  obtain ⟨hcop, _, h⟩ := h
  obtain ⟨hv₀, hv₀'⟩ := aux₀ (zero_lt_two.trans_le hv)
  have hv₁ : 0 < 2 * v - 1 := by linarith only [hv]
  rw [← one_div, lt_div_iff (mul_pos hv₀ hv₀'), ← abs_of_pos (mul_pos hv₀ hv₀'), ← abs_mul, sub_mul,
    ← mul_assoc, ← mul_assoc, div_mul_cancel _ hv₀.ne', abs_sub_comm, abs_lt, lt_sub_iff_add_lt,
    sub_lt_iff_lt_add, mul_assoc] at h
  have hu₀ : 0 ≤ u - ⌊ξ⌋ * v := by
    -- Porting note: this abused the definitional equality `-1 + 1 = 0`
    -- refine' (zero_le_mul_right hv₁).mp ((lt_iff_add_one_le (-1 : ℤ) _).mp _)
    refine' (zero_le_mul_right hv₁).mp ?_
    rw [← sub_one_lt_iff, zero_sub]
    replace h := h.1
    rw [← lt_sub_iff_add_lt, ← mul_assoc, ← sub_mul] at h
    exact mod_cast
      h.trans_le
        ((mul_le_mul_right <| hv₀').mpr <|
          (sub_le_sub_iff_left (u : ℝ)).mpr ((mul_le_mul_right hv₀).mpr (floor_le ξ)))
  

Proof state:
case intro.intro.intro
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
hcop : IsCoprime u v
left✝ : v = 1 → -(1 / 2) < ξ - ↑u
h : -1 + ξ * (↑v * (2 * ↑v - 1)) < ↑u * (2 * ↑v - 1) ∧ ↑u * (2 * ↑v - 1) < 1 + ξ * (↑v * (2 * ↑v - 1))
hv₀ : 0 < ↑v
hv₀' : 0 < 2 * ↑v - 1
hv₁ : 0 < 2 * v - 1
hu₀ : 0 ≤ u - ⌊ξ⌋ * v
⊢ 0 < u - ⌊ξ⌋ * v ∧ u - ⌊ξ⌋ * v < v

Next tactic:
have hu₁ : u - ⌊ξ⌋ * v ≤ v := by
    refine' le_of_mul_le_mul_right (le_of_lt_add_one _) hv₁
    replace h := h.2
    rw [← sub_lt_iff_lt_add, ← mul_assoc, ← sub_mul, ← add_lt_add_iff_right (v * (2 * v - 1) : ℝ),
      add_comm (1 : ℝ)] at h
    have :=
      (mul_lt_mul_right <| hv₀').mpr
        ((sub_lt_sub_iff_left (u : ℝ)).mpr <|
          (mul_lt_mul_right hv₀).mpr <| sub_right_lt_of_lt_add <| lt_floor_add_one ξ)
    rw [sub_mul ξ, one_mul, ← sub_add, add_mul] at this
    exact mod_cast this.trans h
private theorem aux₂ : 0 < u - ⌊ξ⌋ * v ∧ u - ⌊ξ⌋ * v < v := by
  obtain ⟨hcop, _, h⟩ := h
  obtain ⟨hv₀, hv₀'⟩ := aux₀ (zero_lt_two.trans_le hv)
  have hv₁ : 0 < 2 * v - 1 := by linarith only [hv]
  rw [← one_div, lt_div_iff (mul_pos hv₀ hv₀'), ← abs_of_pos (mul_pos hv₀ hv₀'), ← abs_mul, sub_mul,
    ← mul_assoc, ← mul_assoc, div_mul_cancel _ hv₀.ne', abs_sub_comm, abs_lt, lt_sub_iff_add_lt,
    sub_lt_iff_lt_add, mul_assoc] at h
  have hu₀ : 0 ≤ u - ⌊ξ⌋ * v := by
    -- Porting note: this abused the definitional equality `-1 + 1 = 0`
    -- refine' (zero_le_mul_right hv₁).mp ((lt_iff_add_one_le (-1 : ℤ) _).mp _)
    refine' (zero_le_mul_right hv₁).mp ?_
    rw [← sub_one_lt_iff, zero_sub]
    replace h := h.1
    rw [← lt_sub_iff_add_lt, ← mul_assoc, ← sub_mul] at h
    exact mod_cast
      h.trans_le
        ((mul_le_mul_right <| hv₀').mpr <|
          (sub_le_sub_iff_left (u : ℝ)).mpr ((mul_le_mul_right hv₀).mpr (floor_le ξ)))
  have hu₁ : u - ⌊ξ⌋ * v ≤ v := by
    

Proof state:
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
hcop : IsCoprime u v
left✝ : v = 1 → -(1 / 2) < ξ - ↑u
h : -1 + ξ * (↑v * (2 * ↑v - 1)) < ↑u * (2 * ↑v - 1) ∧ ↑u * (2 * ↑v - 1) < 1 + ξ * (↑v * (2 * ↑v - 1))
hv₀ : 0 < ↑v
hv₀' : 0 < 2 * ↑v - 1
hv₁ : 0 < 2 * v - 1
hu₀ : 0 ≤ u - ⌊ξ⌋ * v
⊢ u - ⌊ξ⌋ * v ≤ v

Next tactic:
refine' le_of_mul_le_mul_right (le_of_lt_add_one _) hv₁
private theorem aux₂ : 0 < u - ⌊ξ⌋ * v ∧ u - ⌊ξ⌋ * v < v := by
  obtain ⟨hcop, _, h⟩ := h
  obtain ⟨hv₀, hv₀'⟩ := aux₀ (zero_lt_two.trans_le hv)
  have hv₁ : 0 < 2 * v - 1 := by linarith only [hv]
  rw [← one_div, lt_div_iff (mul_pos hv₀ hv₀'), ← abs_of_pos (mul_pos hv₀ hv₀'), ← abs_mul, sub_mul,
    ← mul_assoc, ← mul_assoc, div_mul_cancel _ hv₀.ne', abs_sub_comm, abs_lt, lt_sub_iff_add_lt,
    sub_lt_iff_lt_add, mul_assoc] at h
  have hu₀ : 0 ≤ u - ⌊ξ⌋ * v := by
    -- Porting note: this abused the definitional equality `-1 + 1 = 0`
    -- refine' (zero_le_mul_right hv₁).mp ((lt_iff_add_one_le (-1 : ℤ) _).mp _)
    refine' (zero_le_mul_right hv₁).mp ?_
    rw [← sub_one_lt_iff, zero_sub]
    replace h := h.1
    rw [← lt_sub_iff_add_lt, ← mul_assoc, ← sub_mul] at h
    exact mod_cast
      h.trans_le
        ((mul_le_mul_right <| hv₀').mpr <|
          (sub_le_sub_iff_left (u : ℝ)).mpr ((mul_le_mul_right hv₀).mpr (floor_le ξ)))
  have hu₁ : u - ⌊ξ⌋ * v ≤ v := by
    refine' le_of_mul_le_mul_right (le_of_lt_add_one _) hv₁
    

Proof state:
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
hcop : IsCoprime u v
left✝ : v = 1 → -(1 / 2) < ξ - ↑u
h : -1 + ξ * (↑v * (2 * ↑v - 1)) < ↑u * (2 * ↑v - 1) ∧ ↑u * (2 * ↑v - 1) < 1 + ξ * (↑v * (2 * ↑v - 1))
hv₀ : 0 < ↑v
hv₀' : 0 < 2 * ↑v - 1
hv₁ : 0 < 2 * v - 1
hu₀ : 0 ≤ u - ⌊ξ⌋ * v
⊢ (u - ⌊ξ⌋ * v) * (2 * v - 1) < v * (2 * v - 1) + 1

Next tactic:
replace h := h.2
private theorem aux₂ : 0 < u - ⌊ξ⌋ * v ∧ u - ⌊ξ⌋ * v < v := by
  obtain ⟨hcop, _, h⟩ := h
  obtain ⟨hv₀, hv₀'⟩ := aux₀ (zero_lt_two.trans_le hv)
  have hv₁ : 0 < 2 * v - 1 := by linarith only [hv]
  rw [← one_div, lt_div_iff (mul_pos hv₀ hv₀'), ← abs_of_pos (mul_pos hv₀ hv₀'), ← abs_mul, sub_mul,
    ← mul_assoc, ← mul_assoc, div_mul_cancel _ hv₀.ne', abs_sub_comm, abs_lt, lt_sub_iff_add_lt,
    sub_lt_iff_lt_add, mul_assoc] at h
  have hu₀ : 0 ≤ u - ⌊ξ⌋ * v := by
    -- Porting note: this abused the definitional equality `-1 + 1 = 0`
    -- refine' (zero_le_mul_right hv₁).mp ((lt_iff_add_one_le (-1 : ℤ) _).mp _)
    refine' (zero_le_mul_right hv₁).mp ?_
    rw [← sub_one_lt_iff, zero_sub]
    replace h := h.1
    rw [← lt_sub_iff_add_lt, ← mul_assoc, ← sub_mul] at h
    exact mod_cast
      h.trans_le
        ((mul_le_mul_right <| hv₀').mpr <|
          (sub_le_sub_iff_left (u : ℝ)).mpr ((mul_le_mul_right hv₀).mpr (floor_le ξ)))
  have hu₁ : u - ⌊ξ⌋ * v ≤ v := by
    refine' le_of_mul_le_mul_right (le_of_lt_add_one _) hv₁
    replace h := h.2
    

Proof state:
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
hcop : IsCoprime u v
left✝ : v = 1 → -(1 / 2) < ξ - ↑u
hv₀ : 0 < ↑v
hv₀' : 0 < 2 * ↑v - 1
hv₁ : 0 < 2 * v - 1
hu₀ : 0 ≤ u - ⌊ξ⌋ * v
h : ↑u * (2 * ↑v - 1) < 1 + ξ * (↑v * (2 * ↑v - 1))
⊢ (u - ⌊ξ⌋ * v) * (2 * v - 1) < v * (2 * v - 1) + 1

Next tactic:
rw [← sub_lt_iff_lt_add, ← mul_assoc, ← sub_mul, ← add_lt_add_iff_right (v * (2 * v - 1) : ℝ),
      add_comm (1 : ℝ)] at h
private theorem aux₂ : 0 < u - ⌊ξ⌋ * v ∧ u - ⌊ξ⌋ * v < v := by
  obtain ⟨hcop, _, h⟩ := h
  obtain ⟨hv₀, hv₀'⟩ := aux₀ (zero_lt_two.trans_le hv)
  have hv₁ : 0 < 2 * v - 1 := by linarith only [hv]
  rw [← one_div, lt_div_iff (mul_pos hv₀ hv₀'), ← abs_of_pos (mul_pos hv₀ hv₀'), ← abs_mul, sub_mul,
    ← mul_assoc, ← mul_assoc, div_mul_cancel _ hv₀.ne', abs_sub_comm, abs_lt, lt_sub_iff_add_lt,
    sub_lt_iff_lt_add, mul_assoc] at h
  have hu₀ : 0 ≤ u - ⌊ξ⌋ * v := by
    -- Porting note: this abused the definitional equality `-1 + 1 = 0`
    -- refine' (zero_le_mul_right hv₁).mp ((lt_iff_add_one_le (-1 : ℤ) _).mp _)
    refine' (zero_le_mul_right hv₁).mp ?_
    rw [← sub_one_lt_iff, zero_sub]
    replace h := h.1
    rw [← lt_sub_iff_add_lt, ← mul_assoc, ← sub_mul] at h
    exact mod_cast
      h.trans_le
        ((mul_le_mul_right <| hv₀').mpr <|
          (sub_le_sub_iff_left (u : ℝ)).mpr ((mul_le_mul_right hv₀).mpr (floor_le ξ)))
  have hu₁ : u - ⌊ξ⌋ * v ≤ v := by
    refine' le_of_mul_le_mul_right (le_of_lt_add_one _) hv₁
    replace h := h.2
    rw [← sub_lt_iff_lt_add, ← mul_assoc, ← sub_mul, ← add_lt_add_iff_right (v * (2 * v - 1) : ℝ),
      add_comm (1 : ℝ)] at h
    

Proof state:
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
hcop : IsCoprime u v
left✝ : v = 1 → -(1 / 2) < ξ - ↑u
hv₀ : 0 < ↑v
hv₀' : 0 < 2 * ↑v - 1
hv₁ : 0 < 2 * v - 1
hu₀ : 0 ≤ u - ⌊ξ⌋ * v
h : (↑u - ξ * ↑v) * (2 * ↑v - 1) + ↑v * (2 * ↑v - 1) < ↑v * (2 * ↑v - 1) + 1
⊢ (u - ⌊ξ⌋ * v) * (2 * v - 1) < v * (2 * v - 1) + 1

Next tactic:
have :=
      (mul_lt_mul_right <| hv₀').mpr
        ((sub_lt_sub_iff_left (u : ℝ)).mpr <|
          (mul_lt_mul_right hv₀).mpr <| sub_right_lt_of_lt_add <| lt_floor_add_one ξ)
private theorem aux₂ : 0 < u - ⌊ξ⌋ * v ∧ u - ⌊ξ⌋ * v < v := by
  obtain ⟨hcop, _, h⟩ := h
  obtain ⟨hv₀, hv₀'⟩ := aux₀ (zero_lt_two.trans_le hv)
  have hv₁ : 0 < 2 * v - 1 := by linarith only [hv]
  rw [← one_div, lt_div_iff (mul_pos hv₀ hv₀'), ← abs_of_pos (mul_pos hv₀ hv₀'), ← abs_mul, sub_mul,
    ← mul_assoc, ← mul_assoc, div_mul_cancel _ hv₀.ne', abs_sub_comm, abs_lt, lt_sub_iff_add_lt,
    sub_lt_iff_lt_add, mul_assoc] at h
  have hu₀ : 0 ≤ u - ⌊ξ⌋ * v := by
    -- Porting note: this abused the definitional equality `-1 + 1 = 0`
    -- refine' (zero_le_mul_right hv₁).mp ((lt_iff_add_one_le (-1 : ℤ) _).mp _)
    refine' (zero_le_mul_right hv₁).mp ?_
    rw [← sub_one_lt_iff, zero_sub]
    replace h := h.1
    rw [← lt_sub_iff_add_lt, ← mul_assoc, ← sub_mul] at h
    exact mod_cast
      h.trans_le
        ((mul_le_mul_right <| hv₀').mpr <|
          (sub_le_sub_iff_left (u : ℝ)).mpr ((mul_le_mul_right hv₀).mpr (floor_le ξ)))
  have hu₁ : u - ⌊ξ⌋ * v ≤ v := by
    refine' le_of_mul_le_mul_right (le_of_lt_add_one _) hv₁
    replace h := h.2
    rw [← sub_lt_iff_lt_add, ← mul_assoc, ← sub_mul, ← add_lt_add_iff_right (v * (2 * v - 1) : ℝ),
      add_comm (1 : ℝ)] at h
    have :=
      (mul_lt_mul_right <| hv₀').mpr
        ((sub_lt_sub_iff_left (u : ℝ)).mpr <|
          (mul_lt_mul_right hv₀).mpr <| sub_right_lt_of_lt_add <| lt_floor_add_one ξ)
    

Proof state:
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
hcop : IsCoprime u v
left✝ : v = 1 → -(1 / 2) < ξ - ↑u
hv₀ : 0 < ↑v
hv₀' : 0 < 2 * ↑v - 1
hv₁ : 0 < 2 * v - 1
hu₀ : 0 ≤ u - ⌊ξ⌋ * v
h : (↑u - ξ * ↑v) * (2 * ↑v - 1) + ↑v * (2 * ↑v - 1) < ↑v * (2 * ↑v - 1) + 1
this : (↑u - ↑⌊ξ⌋ * ↑v) * (2 * ↑v - 1) < (↑u - (ξ - 1) * ↑v) * (2 * ↑v - 1)
⊢ (u - ⌊ξ⌋ * v) * (2 * v - 1) < v * (2 * v - 1) + 1

Next tactic:
rw [sub_mul ξ, one_mul, ← sub_add, add_mul] at this
private theorem aux₂ : 0 < u - ⌊ξ⌋ * v ∧ u - ⌊ξ⌋ * v < v := by
  obtain ⟨hcop, _, h⟩ := h
  obtain ⟨hv₀, hv₀'⟩ := aux₀ (zero_lt_two.trans_le hv)
  have hv₁ : 0 < 2 * v - 1 := by linarith only [hv]
  rw [← one_div, lt_div_iff (mul_pos hv₀ hv₀'), ← abs_of_pos (mul_pos hv₀ hv₀'), ← abs_mul, sub_mul,
    ← mul_assoc, ← mul_assoc, div_mul_cancel _ hv₀.ne', abs_sub_comm, abs_lt, lt_sub_iff_add_lt,
    sub_lt_iff_lt_add, mul_assoc] at h
  have hu₀ : 0 ≤ u - ⌊ξ⌋ * v := by
    -- Porting note: this abused the definitional equality `-1 + 1 = 0`
    -- refine' (zero_le_mul_right hv₁).mp ((lt_iff_add_one_le (-1 : ℤ) _).mp _)
    refine' (zero_le_mul_right hv₁).mp ?_
    rw [← sub_one_lt_iff, zero_sub]
    replace h := h.1
    rw [← lt_sub_iff_add_lt, ← mul_assoc, ← sub_mul] at h
    exact mod_cast
      h.trans_le
        ((mul_le_mul_right <| hv₀').mpr <|
          (sub_le_sub_iff_left (u : ℝ)).mpr ((mul_le_mul_right hv₀).mpr (floor_le ξ)))
  have hu₁ : u - ⌊ξ⌋ * v ≤ v := by
    refine' le_of_mul_le_mul_right (le_of_lt_add_one _) hv₁
    replace h := h.2
    rw [← sub_lt_iff_lt_add, ← mul_assoc, ← sub_mul, ← add_lt_add_iff_right (v * (2 * v - 1) : ℝ),
      add_comm (1 : ℝ)] at h
    have :=
      (mul_lt_mul_right <| hv₀').mpr
        ((sub_lt_sub_iff_left (u : ℝ)).mpr <|
          (mul_lt_mul_right hv₀).mpr <| sub_right_lt_of_lt_add <| lt_floor_add_one ξ)
    rw [sub_mul ξ, one_mul, ← sub_add, add_mul] at this
    

Proof state:
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
hcop : IsCoprime u v
left✝ : v = 1 → -(1 / 2) < ξ - ↑u
hv₀ : 0 < ↑v
hv₀' : 0 < 2 * ↑v - 1
hv₁ : 0 < 2 * v - 1
hu₀ : 0 ≤ u - ⌊ξ⌋ * v
h : (↑u - ξ * ↑v) * (2 * ↑v - 1) + ↑v * (2 * ↑v - 1) < ↑v * (2 * ↑v - 1) + 1
this : (↑u - ↑⌊ξ⌋ * ↑v) * (2 * ↑v - 1) < (↑u - ξ * ↑v) * (2 * ↑v - 1) + ↑v * (2 * ↑v - 1)
⊢ (u - ⌊ξ⌋ * v) * (2 * v - 1) < v * (2 * v - 1) + 1

Next tactic:
exact mod_cast this.trans h
private theorem aux₂ : 0 < u - ⌊ξ⌋ * v ∧ u - ⌊ξ⌋ * v < v := by
  obtain ⟨hcop, _, h⟩ := h
  obtain ⟨hv₀, hv₀'⟩ := aux₀ (zero_lt_two.trans_le hv)
  have hv₁ : 0 < 2 * v - 1 := by linarith only [hv]
  rw [← one_div, lt_div_iff (mul_pos hv₀ hv₀'), ← abs_of_pos (mul_pos hv₀ hv₀'), ← abs_mul, sub_mul,
    ← mul_assoc, ← mul_assoc, div_mul_cancel _ hv₀.ne', abs_sub_comm, abs_lt, lt_sub_iff_add_lt,
    sub_lt_iff_lt_add, mul_assoc] at h
  have hu₀ : 0 ≤ u - ⌊ξ⌋ * v := by
    -- Porting note: this abused the definitional equality `-1 + 1 = 0`
    -- refine' (zero_le_mul_right hv₁).mp ((lt_iff_add_one_le (-1 : ℤ) _).mp _)
    refine' (zero_le_mul_right hv₁).mp ?_
    rw [← sub_one_lt_iff, zero_sub]
    replace h := h.1
    rw [← lt_sub_iff_add_lt, ← mul_assoc, ← sub_mul] at h
    exact mod_cast
      h.trans_le
        ((mul_le_mul_right <| hv₀').mpr <|
          (sub_le_sub_iff_left (u : ℝ)).mpr ((mul_le_mul_right hv₀).mpr (floor_le ξ)))
  have hu₁ : u - ⌊ξ⌋ * v ≤ v := by
    refine' le_of_mul_le_mul_right (le_of_lt_add_one _) hv₁
    replace h := h.2
    rw [← sub_lt_iff_lt_add, ← mul_assoc, ← sub_mul, ← add_lt_add_iff_right (v * (2 * v - 1) : ℝ),
      add_comm (1 : ℝ)] at h
    have :=
      (mul_lt_mul_right <| hv₀').mpr
        ((sub_lt_sub_iff_left (u : ℝ)).mpr <|
          (mul_lt_mul_right hv₀).mpr <| sub_right_lt_of_lt_add <| lt_floor_add_one ξ)
    rw [sub_mul ξ, one_mul, ← sub_add, add_mul] at this
    exact mod_cast this.trans h
  

Proof state:
case intro.intro.intro
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
hcop : IsCoprime u v
left✝ : v = 1 → -(1 / 2) < ξ - ↑u
h : -1 + ξ * (↑v * (2 * ↑v - 1)) < ↑u * (2 * ↑v - 1) ∧ ↑u * (2 * ↑v - 1) < 1 + ξ * (↑v * (2 * ↑v - 1))
hv₀ : 0 < ↑v
hv₀' : 0 < 2 * ↑v - 1
hv₁ : 0 < 2 * v - 1
hu₀ : 0 ≤ u - ⌊ξ⌋ * v
hu₁ : u - ⌊ξ⌋ * v ≤ v
⊢ 0 < u - ⌊ξ⌋ * v ∧ u - ⌊ξ⌋ * v < v

Next tactic:
have huv_cop : IsCoprime (u - ⌊ξ⌋ * v) v := by
    rwa [sub_eq_add_neg, ← neg_mul, IsCoprime.add_mul_right_left_iff]
private theorem aux₂ : 0 < u - ⌊ξ⌋ * v ∧ u - ⌊ξ⌋ * v < v := by
  obtain ⟨hcop, _, h⟩ := h
  obtain ⟨hv₀, hv₀'⟩ := aux₀ (zero_lt_two.trans_le hv)
  have hv₁ : 0 < 2 * v - 1 := by linarith only [hv]
  rw [← one_div, lt_div_iff (mul_pos hv₀ hv₀'), ← abs_of_pos (mul_pos hv₀ hv₀'), ← abs_mul, sub_mul,
    ← mul_assoc, ← mul_assoc, div_mul_cancel _ hv₀.ne', abs_sub_comm, abs_lt, lt_sub_iff_add_lt,
    sub_lt_iff_lt_add, mul_assoc] at h
  have hu₀ : 0 ≤ u - ⌊ξ⌋ * v := by
    -- Porting note: this abused the definitional equality `-1 + 1 = 0`
    -- refine' (zero_le_mul_right hv₁).mp ((lt_iff_add_one_le (-1 : ℤ) _).mp _)
    refine' (zero_le_mul_right hv₁).mp ?_
    rw [← sub_one_lt_iff, zero_sub]
    replace h := h.1
    rw [← lt_sub_iff_add_lt, ← mul_assoc, ← sub_mul] at h
    exact mod_cast
      h.trans_le
        ((mul_le_mul_right <| hv₀').mpr <|
          (sub_le_sub_iff_left (u : ℝ)).mpr ((mul_le_mul_right hv₀).mpr (floor_le ξ)))
  have hu₁ : u - ⌊ξ⌋ * v ≤ v := by
    refine' le_of_mul_le_mul_right (le_of_lt_add_one _) hv₁
    replace h := h.2
    rw [← sub_lt_iff_lt_add, ← mul_assoc, ← sub_mul, ← add_lt_add_iff_right (v * (2 * v - 1) : ℝ),
      add_comm (1 : ℝ)] at h
    have :=
      (mul_lt_mul_right <| hv₀').mpr
        ((sub_lt_sub_iff_left (u : ℝ)).mpr <|
          (mul_lt_mul_right hv₀).mpr <| sub_right_lt_of_lt_add <| lt_floor_add_one ξ)
    rw [sub_mul ξ, one_mul, ← sub_add, add_mul] at this
    exact mod_cast this.trans h
  have huv_cop : IsCoprime (u - ⌊ξ⌋ * v) v := by
    

Proof state:
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
hcop : IsCoprime u v
left✝ : v = 1 → -(1 / 2) < ξ - ↑u
h : -1 + ξ * (↑v * (2 * ↑v - 1)) < ↑u * (2 * ↑v - 1) ∧ ↑u * (2 * ↑v - 1) < 1 + ξ * (↑v * (2 * ↑v - 1))
hv₀ : 0 < ↑v
hv₀' : 0 < 2 * ↑v - 1
hv₁ : 0 < 2 * v - 1
hu₀ : 0 ≤ u - ⌊ξ⌋ * v
hu₁ : u - ⌊ξ⌋ * v ≤ v
⊢ IsCoprime (u - ⌊ξ⌋ * v) v

Next tactic:
rwa [sub_eq_add_neg, ← neg_mul, IsCoprime.add_mul_right_left_iff]
private theorem aux₂ : 0 < u - ⌊ξ⌋ * v ∧ u - ⌊ξ⌋ * v < v := by
  obtain ⟨hcop, _, h⟩ := h
  obtain ⟨hv₀, hv₀'⟩ := aux₀ (zero_lt_two.trans_le hv)
  have hv₁ : 0 < 2 * v - 1 := by linarith only [hv]
  rw [← one_div, lt_div_iff (mul_pos hv₀ hv₀'), ← abs_of_pos (mul_pos hv₀ hv₀'), ← abs_mul, sub_mul,
    ← mul_assoc, ← mul_assoc, div_mul_cancel _ hv₀.ne', abs_sub_comm, abs_lt, lt_sub_iff_add_lt,
    sub_lt_iff_lt_add, mul_assoc] at h
  have hu₀ : 0 ≤ u - ⌊ξ⌋ * v := by
    -- Porting note: this abused the definitional equality `-1 + 1 = 0`
    -- refine' (zero_le_mul_right hv₁).mp ((lt_iff_add_one_le (-1 : ℤ) _).mp _)
    refine' (zero_le_mul_right hv₁).mp ?_
    rw [← sub_one_lt_iff, zero_sub]
    replace h := h.1
    rw [← lt_sub_iff_add_lt, ← mul_assoc, ← sub_mul] at h
    exact mod_cast
      h.trans_le
        ((mul_le_mul_right <| hv₀').mpr <|
          (sub_le_sub_iff_left (u : ℝ)).mpr ((mul_le_mul_right hv₀).mpr (floor_le ξ)))
  have hu₁ : u - ⌊ξ⌋ * v ≤ v := by
    refine' le_of_mul_le_mul_right (le_of_lt_add_one _) hv₁
    replace h := h.2
    rw [← sub_lt_iff_lt_add, ← mul_assoc, ← sub_mul, ← add_lt_add_iff_right (v * (2 * v - 1) : ℝ),
      add_comm (1 : ℝ)] at h
    have :=
      (mul_lt_mul_right <| hv₀').mpr
        ((sub_lt_sub_iff_left (u : ℝ)).mpr <|
          (mul_lt_mul_right hv₀).mpr <| sub_right_lt_of_lt_add <| lt_floor_add_one ξ)
    rw [sub_mul ξ, one_mul, ← sub_add, add_mul] at this
    exact mod_cast this.trans h
  have huv_cop : IsCoprime (u - ⌊ξ⌋ * v) v := by
    rwa [sub_eq_add_neg, ← neg_mul, IsCoprime.add_mul_right_left_iff]
  

Proof state:
case intro.intro.intro
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
hcop : IsCoprime u v
left✝ : v = 1 → -(1 / 2) < ξ - ↑u
h : -1 + ξ * (↑v * (2 * ↑v - 1)) < ↑u * (2 * ↑v - 1) ∧ ↑u * (2 * ↑v - 1) < 1 + ξ * (↑v * (2 * ↑v - 1))
hv₀ : 0 < ↑v
hv₀' : 0 < 2 * ↑v - 1
hv₁ : 0 < 2 * v - 1
hu₀ : 0 ≤ u - ⌊ξ⌋ * v
hu₁ : u - ⌊ξ⌋ * v ≤ v
huv_cop : IsCoprime (u - ⌊ξ⌋ * v) v
⊢ 0 < u - ⌊ξ⌋ * v ∧ u - ⌊ξ⌋ * v < v

Next tactic:
refine' ⟨lt_of_le_of_ne' hu₀ fun hf => _, lt_of_le_of_ne hu₁ fun hf => _⟩
private theorem aux₂ : 0 < u - ⌊ξ⌋ * v ∧ u - ⌊ξ⌋ * v < v := by
  obtain ⟨hcop, _, h⟩ := h
  obtain ⟨hv₀, hv₀'⟩ := aux₀ (zero_lt_two.trans_le hv)
  have hv₁ : 0 < 2 * v - 1 := by linarith only [hv]
  rw [← one_div, lt_div_iff (mul_pos hv₀ hv₀'), ← abs_of_pos (mul_pos hv₀ hv₀'), ← abs_mul, sub_mul,
    ← mul_assoc, ← mul_assoc, div_mul_cancel _ hv₀.ne', abs_sub_comm, abs_lt, lt_sub_iff_add_lt,
    sub_lt_iff_lt_add, mul_assoc] at h
  have hu₀ : 0 ≤ u - ⌊ξ⌋ * v := by
    -- Porting note: this abused the definitional equality `-1 + 1 = 0`
    -- refine' (zero_le_mul_right hv₁).mp ((lt_iff_add_one_le (-1 : ℤ) _).mp _)
    refine' (zero_le_mul_right hv₁).mp ?_
    rw [← sub_one_lt_iff, zero_sub]
    replace h := h.1
    rw [← lt_sub_iff_add_lt, ← mul_assoc, ← sub_mul] at h
    exact mod_cast
      h.trans_le
        ((mul_le_mul_right <| hv₀').mpr <|
          (sub_le_sub_iff_left (u : ℝ)).mpr ((mul_le_mul_right hv₀).mpr (floor_le ξ)))
  have hu₁ : u - ⌊ξ⌋ * v ≤ v := by
    refine' le_of_mul_le_mul_right (le_of_lt_add_one _) hv₁
    replace h := h.2
    rw [← sub_lt_iff_lt_add, ← mul_assoc, ← sub_mul, ← add_lt_add_iff_right (v * (2 * v - 1) : ℝ),
      add_comm (1 : ℝ)] at h
    have :=
      (mul_lt_mul_right <| hv₀').mpr
        ((sub_lt_sub_iff_left (u : ℝ)).mpr <|
          (mul_lt_mul_right hv₀).mpr <| sub_right_lt_of_lt_add <| lt_floor_add_one ξ)
    rw [sub_mul ξ, one_mul, ← sub_add, add_mul] at this
    exact mod_cast this.trans h
  have huv_cop : IsCoprime (u - ⌊ξ⌋ * v) v := by
    rwa [sub_eq_add_neg, ← neg_mul, IsCoprime.add_mul_right_left_iff]
  refine' ⟨lt_of_le_of_ne' hu₀ fun hf => _, lt_of_le_of_ne hu₁ fun hf => _⟩ <;>
    · 

Proof state:
case intro.intro.intro.refine'_1
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
hcop : IsCoprime u v
left✝ : v = 1 → -(1 / 2) < ξ - ↑u
h : -1 + ξ * (↑v * (2 * ↑v - 1)) < ↑u * (2 * ↑v - 1) ∧ ↑u * (2 * ↑v - 1) < 1 + ξ * (↑v * (2 * ↑v - 1))
hv₀ : 0 < ↑v
hv₀' : 0 < 2 * ↑v - 1
hv₁ : 0 < 2 * v - 1
hu₀ : 0 ≤ u - ⌊ξ⌋ * v
hu₁ : u - ⌊ξ⌋ * v ≤ v
huv_cop : IsCoprime (u - ⌊ξ⌋ * v) v
hf : u - ⌊ξ⌋ * v = 0
⊢ False

Next tactic:
rw [hf] at huv_cop
private theorem aux₂ : 0 < u - ⌊ξ⌋ * v ∧ u - ⌊ξ⌋ * v < v := by
  obtain ⟨hcop, _, h⟩ := h
  obtain ⟨hv₀, hv₀'⟩ := aux₀ (zero_lt_two.trans_le hv)
  have hv₁ : 0 < 2 * v - 1 := by linarith only [hv]
  rw [← one_div, lt_div_iff (mul_pos hv₀ hv₀'), ← abs_of_pos (mul_pos hv₀ hv₀'), ← abs_mul, sub_mul,
    ← mul_assoc, ← mul_assoc, div_mul_cancel _ hv₀.ne', abs_sub_comm, abs_lt, lt_sub_iff_add_lt,
    sub_lt_iff_lt_add, mul_assoc] at h
  have hu₀ : 0 ≤ u - ⌊ξ⌋ * v := by
    -- Porting note: this abused the definitional equality `-1 + 1 = 0`
    -- refine' (zero_le_mul_right hv₁).mp ((lt_iff_add_one_le (-1 : ℤ) _).mp _)
    refine' (zero_le_mul_right hv₁).mp ?_
    rw [← sub_one_lt_iff, zero_sub]
    replace h := h.1
    rw [← lt_sub_iff_add_lt, ← mul_assoc, ← sub_mul] at h
    exact mod_cast
      h.trans_le
        ((mul_le_mul_right <| hv₀').mpr <|
          (sub_le_sub_iff_left (u : ℝ)).mpr ((mul_le_mul_right hv₀).mpr (floor_le ξ)))
  have hu₁ : u - ⌊ξ⌋ * v ≤ v := by
    refine' le_of_mul_le_mul_right (le_of_lt_add_one _) hv₁
    replace h := h.2
    rw [← sub_lt_iff_lt_add, ← mul_assoc, ← sub_mul, ← add_lt_add_iff_right (v * (2 * v - 1) : ℝ),
      add_comm (1 : ℝ)] at h
    have :=
      (mul_lt_mul_right <| hv₀').mpr
        ((sub_lt_sub_iff_left (u : ℝ)).mpr <|
          (mul_lt_mul_right hv₀).mpr <| sub_right_lt_of_lt_add <| lt_floor_add_one ξ)
    rw [sub_mul ξ, one_mul, ← sub_add, add_mul] at this
    exact mod_cast this.trans h
  have huv_cop : IsCoprime (u - ⌊ξ⌋ * v) v := by
    rwa [sub_eq_add_neg, ← neg_mul, IsCoprime.add_mul_right_left_iff]
  refine' ⟨lt_of_le_of_ne' hu₀ fun hf => _, lt_of_le_of_ne hu₁ fun hf => _⟩ <;>
    · rw [hf] at huv_cop
      

Proof state:
case intro.intro.intro.refine'_1
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
hcop : IsCoprime u v
left✝ : v = 1 → -(1 / 2) < ξ - ↑u
h : -1 + ξ * (↑v * (2 * ↑v - 1)) < ↑u * (2 * ↑v - 1) ∧ ↑u * (2 * ↑v - 1) < 1 + ξ * (↑v * (2 * ↑v - 1))
hv₀ : 0 < ↑v
hv₀' : 0 < 2 * ↑v - 1
hv₁ : 0 < 2 * v - 1
hu₀ : 0 ≤ u - ⌊ξ⌋ * v
hu₁ : u - ⌊ξ⌋ * v ≤ v
huv_cop : IsCoprime 0 v
hf : u - ⌊ξ⌋ * v = 0
⊢ False

Next tactic:
simp only [isCoprime_zero_left, isCoprime_self, isUnit_iff] at huv_cop
private theorem aux₂ : 0 < u - ⌊ξ⌋ * v ∧ u - ⌊ξ⌋ * v < v := by
  obtain ⟨hcop, _, h⟩ := h
  obtain ⟨hv₀, hv₀'⟩ := aux₀ (zero_lt_two.trans_le hv)
  have hv₁ : 0 < 2 * v - 1 := by linarith only [hv]
  rw [← one_div, lt_div_iff (mul_pos hv₀ hv₀'), ← abs_of_pos (mul_pos hv₀ hv₀'), ← abs_mul, sub_mul,
    ← mul_assoc, ← mul_assoc, div_mul_cancel _ hv₀.ne', abs_sub_comm, abs_lt, lt_sub_iff_add_lt,
    sub_lt_iff_lt_add, mul_assoc] at h
  have hu₀ : 0 ≤ u - ⌊ξ⌋ * v := by
    -- Porting note: this abused the definitional equality `-1 + 1 = 0`
    -- refine' (zero_le_mul_right hv₁).mp ((lt_iff_add_one_le (-1 : ℤ) _).mp _)
    refine' (zero_le_mul_right hv₁).mp ?_
    rw [← sub_one_lt_iff, zero_sub]
    replace h := h.1
    rw [← lt_sub_iff_add_lt, ← mul_assoc, ← sub_mul] at h
    exact mod_cast
      h.trans_le
        ((mul_le_mul_right <| hv₀').mpr <|
          (sub_le_sub_iff_left (u : ℝ)).mpr ((mul_le_mul_right hv₀).mpr (floor_le ξ)))
  have hu₁ : u - ⌊ξ⌋ * v ≤ v := by
    refine' le_of_mul_le_mul_right (le_of_lt_add_one _) hv₁
    replace h := h.2
    rw [← sub_lt_iff_lt_add, ← mul_assoc, ← sub_mul, ← add_lt_add_iff_right (v * (2 * v - 1) : ℝ),
      add_comm (1 : ℝ)] at h
    have :=
      (mul_lt_mul_right <| hv₀').mpr
        ((sub_lt_sub_iff_left (u : ℝ)).mpr <|
          (mul_lt_mul_right hv₀).mpr <| sub_right_lt_of_lt_add <| lt_floor_add_one ξ)
    rw [sub_mul ξ, one_mul, ← sub_add, add_mul] at this
    exact mod_cast this.trans h
  have huv_cop : IsCoprime (u - ⌊ξ⌋ * v) v := by
    rwa [sub_eq_add_neg, ← neg_mul, IsCoprime.add_mul_right_left_iff]
  refine' ⟨lt_of_le_of_ne' hu₀ fun hf => _, lt_of_le_of_ne hu₁ fun hf => _⟩ <;>
    · rw [hf] at huv_cop
      simp only [isCoprime_zero_left, isCoprime_self, isUnit_iff] at huv_cop
      

Proof state:
case intro.intro.intro.refine'_1
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
hcop : IsCoprime u v
left✝ : v = 1 → -(1 / 2) < ξ - ↑u
h : -1 + ξ * (↑v * (2 * ↑v - 1)) < ↑u * (2 * ↑v - 1) ∧ ↑u * (2 * ↑v - 1) < 1 + ξ * (↑v * (2 * ↑v - 1))
hv₀ : 0 < ↑v
hv₀' : 0 < 2 * ↑v - 1
hv₁ : 0 < 2 * v - 1
hu₀ : 0 ≤ u - ⌊ξ⌋ * v
hu₁ : u - ⌊ξ⌋ * v ≤ v
hf : u - ⌊ξ⌋ * v = 0
huv_cop : v = 1 ∨ v = -1
⊢ False

Next tactic:
cases' huv_cop with huv_cop huv_cop