l3lab/ntp-mathlib huggingface dataset:
/-- For two points on the same vertical line, the distance is equal to the distance between the
logarithms of their imaginary parts. -/
nonrec theorem dist_of_re_eq (h : z.re = w.re) : dist z w = dist (log z.im) (log w.im) := by
  have h₀ : 0 < z.im / w.im := div_pos z.im_pos w.im_pos
  rw [dist_eq_iff_dist_coe_center_eq, Real.dist_eq, ← abs_sinh, ← log_div z.im_ne_zero w.im_ne_zero,
    sinh_log h₀, dist_of_re_eq, coe_im, coe_im, center_im, cosh_abs, cosh_log h₀, inv_div] <;>
  [skip; exact h]
  

Proof state:
z w : ℍ
r R : ℝ
h : re z = re w
h₀ : 0 < im z / im w
⊢ dist (im z) (im w * ((im z / im w + im w / im z) / 2)) = im w * |(im z / im w - im w / im z) / 2|

Next tactic:
nth_rw 4 [← abs_of_pos w.im_pos]
/-- For two points on the same vertical line, the distance is equal to the distance between the
logarithms of their imaginary parts. -/
nonrec theorem dist_of_re_eq (h : z.re = w.re) : dist z w = dist (log z.im) (log w.im) := by
  have h₀ : 0 < z.im / w.im := div_pos z.im_pos w.im_pos
  rw [dist_eq_iff_dist_coe_center_eq, Real.dist_eq, ← abs_sinh, ← log_div z.im_ne_zero w.im_ne_zero,
    sinh_log h₀, dist_of_re_eq, coe_im, coe_im, center_im, cosh_abs, cosh_log h₀, inv_div] <;>
  [skip; exact h]
  nth_rw 4 [← abs_of_pos w.im_pos]
  

Proof state:
z w : ℍ
r R : ℝ
h : re z = re w
h₀ : 0 < im z / im w
⊢ dist (im z) (im w * ((im z / im w + im w / im z) / 2)) = |im w| * |(im z / im w - im w / im z) / 2|

Next tactic:
simp only [← _root_.abs_mul, coe_im, Real.dist_eq]
/-- For two points on the same vertical line, the distance is equal to the distance between the
logarithms of their imaginary parts. -/
nonrec theorem dist_of_re_eq (h : z.re = w.re) : dist z w = dist (log z.im) (log w.im) := by
  have h₀ : 0 < z.im / w.im := div_pos z.im_pos w.im_pos
  rw [dist_eq_iff_dist_coe_center_eq, Real.dist_eq, ← abs_sinh, ← log_div z.im_ne_zero w.im_ne_zero,
    sinh_log h₀, dist_of_re_eq, coe_im, coe_im, center_im, cosh_abs, cosh_log h₀, inv_div] <;>
  [skip; exact h]
  nth_rw 4 [← abs_of_pos w.im_pos]
  simp only [← _root_.abs_mul, coe_im, Real.dist_eq]
  

Proof state:
z w : ℍ
r R : ℝ
h : re z = re w
h₀ : 0 < im z / im w
⊢ |im z - im w * ((im z / im w + im w / im z) / 2)| = |im w * ((im z / im w - im w / im z) / 2)|

Next tactic:
congr 1
/-- For two points on the same vertical line, the distance is equal to the distance between the
logarithms of their imaginary parts. -/
nonrec theorem dist_of_re_eq (h : z.re = w.re) : dist z w = dist (log z.im) (log w.im) := by
  have h₀ : 0 < z.im / w.im := div_pos z.im_pos w.im_pos
  rw [dist_eq_iff_dist_coe_center_eq, Real.dist_eq, ← abs_sinh, ← log_div z.im_ne_zero w.im_ne_zero,
    sinh_log h₀, dist_of_re_eq, coe_im, coe_im, center_im, cosh_abs, cosh_log h₀, inv_div] <;>
  [skip; exact h]
  nth_rw 4 [← abs_of_pos w.im_pos]
  simp only [← _root_.abs_mul, coe_im, Real.dist_eq]
  congr 1
  

Proof state:
case e_a
z w : ℍ
r R : ℝ
h : re z = re w
h₀ : 0 < im z / im w
⊢ im z - im w * ((im z / im w + im w / im z) / 2) = im w * ((im z / im w - im w / im z) / 2)

Next tactic:
field_simp [z.im_pos, w.im_pos, z.im_ne_zero, w.im_ne_zero]
/-- For two points on the same vertical line, the distance is equal to the distance between the
logarithms of their imaginary parts. -/
nonrec theorem dist_of_re_eq (h : z.re = w.re) : dist z w = dist (log z.im) (log w.im) := by
  have h₀ : 0 < z.im / w.im := div_pos z.im_pos w.im_pos
  rw [dist_eq_iff_dist_coe_center_eq, Real.dist_eq, ← abs_sinh, ← log_div z.im_ne_zero w.im_ne_zero,
    sinh_log h₀, dist_of_re_eq, coe_im, coe_im, center_im, cosh_abs, cosh_log h₀, inv_div] <;>
  [skip; exact h]
  nth_rw 4 [← abs_of_pos w.im_pos]
  simp only [← _root_.abs_mul, coe_im, Real.dist_eq]
  congr 1
  field_simp [z.im_pos, w.im_pos, z.im_ne_zero, w.im_ne_zero]
  

Proof state:
case e_a
z w : ℍ
r R : ℝ
h : re z = re w
h₀ : 0 < im z / im w
⊢ im z * (im w * im z * 2) - im w * (im z * im z + im w * im w) = im w * (im z * im z - im w * im w)

Next tactic:
ring
/-- Hyperbolic distance between two points is greater than or equal to the distance between the
logarithms of their imaginary parts. -/
theorem dist_log_im_le (z w : ℍ) : dist (log z.im) (log w.im) ≤ dist z w :=
  calc
    dist (log z.im) (log w.im) = dist (mk ⟨0, z.im⟩ z.im_pos) (mk ⟨0, w.im⟩ w.im_pos) :=
      Eq.symm <| dist_of_re_eq rfl
    _ ≤ dist z w := mul_le_mul_of_nonneg_left (arsinh_le_arsinh.2 <|
      div_le_div_of_le (mul_nonneg zero_le_two (sqrt_nonneg _)) <| by
        

Proof state:
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
⊢ dist ↑(mk { re := 0, im := im z } (_ : 0 < im z)) ↑(mk { re := 0, im := im w } (_ : 0 < im w)) ≤ dist ↑z ↑w

Next tactic:
simpa [sqrt_sq_eq_abs] using Complex.abs_im_le_abs (z - w)
theorem im_le_im_mul_exp_dist (z w : ℍ) : z.im ≤ w.im * Real.exp (dist z w) := by
  

Proof state:
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
⊢ im z ≤ im w * rexp (dist z w)

Next tactic:
rw [← div_le_iff' w.im_pos, ← exp_log z.im_pos, ← exp_log w.im_pos, ← Real.exp_sub, exp_le_exp]
theorem im_le_im_mul_exp_dist (z w : ℍ) : z.im ≤ w.im * Real.exp (dist z w) := by
  rw [← div_le_iff' w.im_pos, ← exp_log z.im_pos, ← exp_log w.im_pos, ← Real.exp_sub, exp_le_exp]
  

Proof state:
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
⊢ log (im z) - log (im w) ≤ dist z w

Next tactic:
exact (le_abs_self _).trans (dist_log_im_le z w)
/-- An upper estimate on the complex distance between two points in terms of the hyperbolic distance
and the imaginary part of one of the points. -/
theorem dist_coe_le (z w : ℍ) : dist (z : ℂ) w ≤ w.im * (Real.exp (dist z w) - 1) :=
  calc
    dist (z : ℂ) w ≤ dist (z : ℂ) (w.center (dist z w)) + dist (w : ℂ) (w.center (dist z w)) :=
      dist_triangle_right _ _ _
    _ = w.im * (Real.exp (dist z w) - 1) := by
      

Proof state:
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
⊢ dist ↑z ↑(center w (dist z w)) + dist ↑w ↑(center w (dist z w)) = im w * (rexp (dist z w) - 1)

Next tactic:
rw [dist_center_dist, dist_self_center, ← mul_add, ← add_sub_assoc, Real.sinh_add_cosh]
/-- An upper estimate on the complex distance between two points in terms of the hyperbolic distance
and the imaginary part of one of the points. -/
theorem le_dist_coe (z w : ℍ) : w.im * (1 - Real.exp (-dist z w)) ≤ dist (z : ℂ) w :=
  calc
    w.im * (1 - Real.exp (-dist z w)) =
        dist (z : ℂ) (w.center (dist z w)) - dist (w : ℂ) (w.center (dist z w)) := by
      

Proof state:
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
⊢ im w * (1 - rexp (-dist z w)) = dist ↑z ↑(center w (dist z w)) - dist ↑w ↑(center w (dist z w))

Next tactic:
rw [dist_center_dist, dist_self_center, ← Real.cosh_sub_sinh]
/-- An upper estimate on the complex distance between two points in terms of the hyperbolic distance
and the imaginary part of one of the points. -/
theorem le_dist_coe (z w : ℍ) : w.im * (1 - Real.exp (-dist z w)) ≤ dist (z : ℂ) w :=
  calc
    w.im * (1 - Real.exp (-dist z w)) =
        dist (z : ℂ) (w.center (dist z w)) - dist (w : ℂ) (w.center (dist z w)) := by
      rw [dist_center_dist, dist_self_center, ← Real.cosh_sub_sinh]; 

Proof state:
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
⊢ im w * (1 - (Real.cosh (dist z w) - Real.sinh (dist z w))) =
    im w * Real.sinh (dist z w) - im w * (Real.cosh (dist z w) - 1)

Next tactic:
ring
/-- The hyperbolic metric on the upper half plane. We ensure that the projection to
`TopologicalSpace` is definitionally equal to the subtype topology. -/
instance : MetricSpace ℍ :=
  metricSpaceAux.replaceTopology <| by
    

Proof state:
z w : ℍ
r R : ℝ
⊢ instTopologicalSpaceUpperHalfPlane = UniformSpace.toTopologicalSpace

Next tactic:
refine' le_antisymm (continuous_id_iff_le.1 _) _
/-- The hyperbolic metric on the upper half plane. We ensure that the projection to
`TopologicalSpace` is definitionally equal to the subtype topology. -/
instance : MetricSpace ℍ :=
  metricSpaceAux.replaceTopology <| by
    refine' le_antisymm (continuous_id_iff_le.1 _) _
    · 

Proof state:
case refine'_1
z w : ℍ
r R : ℝ
⊢ Continuous id

Next tactic:
refine' (@continuous_iff_continuous_dist ℍ ℍ metricSpaceAux.toPseudoMetricSpace _ _).2 _
/-- The hyperbolic metric on the upper half plane. We ensure that the projection to
`TopologicalSpace` is definitionally equal to the subtype topology. -/
instance : MetricSpace ℍ :=
  metricSpaceAux.replaceTopology <| by
    refine' le_antisymm (continuous_id_iff_le.1 _) _
    · refine' (@continuous_iff_continuous_dist ℍ ℍ metricSpaceAux.toPseudoMetricSpace _ _).2 _
      

Proof state:
case refine'_1
z w : ℍ
r R : ℝ
⊢ Continuous fun x => dist (id x.1) (id x.2)

Next tactic:
have : ∀ x : ℍ × ℍ, 2 * Real.sqrt (x.1.im * x.2.im) ≠ 0 := fun x =>
        mul_ne_zero two_ne_zero (Real.sqrt_pos.2 <| mul_pos x.1.im_pos x.2.im_pos).ne'
/-- The hyperbolic metric on the upper half plane. We ensure that the projection to
`TopologicalSpace` is definitionally equal to the subtype topology. -/
instance : MetricSpace ℍ :=
  metricSpaceAux.replaceTopology <| by
    refine' le_antisymm (continuous_id_iff_le.1 _) _
    · refine' (@continuous_iff_continuous_dist ℍ ℍ metricSpaceAux.toPseudoMetricSpace _ _).2 _
      have : ∀ x : ℍ × ℍ, 2 * Real.sqrt (x.1.im * x.2.im) ≠ 0 := fun x =>
        mul_ne_zero two_ne_zero (Real.sqrt_pos.2 <| mul_pos x.1.im_pos x.2.im_pos).ne'
      -- `continuity` fails to apply `Continuous.div`
      

Proof state:
case refine'_1
z w : ℍ
r R : ℝ
this : ∀ (x : ℍ × ℍ), 2 * sqrt (im x.1 * im x.2) ≠ 0
⊢ Continuous fun x => dist (id x.1) (id x.2)

Next tactic:
apply_rules [Continuous.div, Continuous.mul, continuous_const, Continuous.arsinh,
        Continuous.dist, continuous_coe.comp, continuous_fst, continuous_snd,
        Real.continuous_sqrt.comp, continuous_im.comp]
/-- The hyperbolic metric on the upper half plane. We ensure that the projection to
`TopologicalSpace` is definitionally equal to the subtype topology. -/
instance : MetricSpace ℍ :=
  metricSpaceAux.replaceTopology <| by
    refine' le_antisymm (continuous_id_iff_le.1 _) _
    · refine' (@continuous_iff_continuous_dist ℍ ℍ metricSpaceAux.toPseudoMetricSpace _ _).2 _
      have : ∀ x : ℍ × ℍ, 2 * Real.sqrt (x.1.im * x.2.im) ≠ 0 := fun x =>
        mul_ne_zero two_ne_zero (Real.sqrt_pos.2 <| mul_pos x.1.im_pos x.2.im_pos).ne'
      -- `continuity` fails to apply `Continuous.div`
      apply_rules [Continuous.div, Continuous.mul, continuous_const, Continuous.arsinh,
        Continuous.dist, continuous_coe.comp, continuous_fst, continuous_snd,
        Real.continuous_sqrt.comp, continuous_im.comp]
    · 

Proof state:
case refine'_2
z w : ℍ
r R : ℝ
⊢ UniformSpace.toTopologicalSpace ≤ instTopologicalSpaceUpperHalfPlane

Next tactic:
letI : MetricSpace ℍ := metricSpaceAux
/-- The hyperbolic metric on the upper half plane. We ensure that the projection to
`TopologicalSpace` is definitionally equal to the subtype topology. -/
instance : MetricSpace ℍ :=
  metricSpaceAux.replaceTopology <| by
    refine' le_antisymm (continuous_id_iff_le.1 _) _
    · refine' (@continuous_iff_continuous_dist ℍ ℍ metricSpaceAux.toPseudoMetricSpace _ _).2 _
      have : ∀ x : ℍ × ℍ, 2 * Real.sqrt (x.1.im * x.2.im) ≠ 0 := fun x =>
        mul_ne_zero two_ne_zero (Real.sqrt_pos.2 <| mul_pos x.1.im_pos x.2.im_pos).ne'
      -- `continuity` fails to apply `Continuous.div`
      apply_rules [Continuous.div, Continuous.mul, continuous_const, Continuous.arsinh,
        Continuous.dist, continuous_coe.comp, continuous_fst, continuous_snd,
        Real.continuous_sqrt.comp, continuous_im.comp]
    · letI : MetricSpace ℍ := metricSpaceAux
      

Proof state:
case refine'_2
z w : ℍ
r R : ℝ
this : MetricSpace ℍ := metricSpaceAux
⊢ UniformSpace.toTopologicalSpace ≤ instTopologicalSpaceUpperHalfPlane

Next tactic:
refine' le_of_nhds_le_nhds fun z => _
/-- The hyperbolic metric on the upper half plane. We ensure that the projection to
`TopologicalSpace` is definitionally equal to the subtype topology. -/
instance : MetricSpace ℍ :=
  metricSpaceAux.replaceTopology <| by
    refine' le_antisymm (continuous_id_iff_le.1 _) _
    · refine' (@continuous_iff_continuous_dist ℍ ℍ metricSpaceAux.toPseudoMetricSpace _ _).2 _
      have : ∀ x : ℍ × ℍ, 2 * Real.sqrt (x.1.im * x.2.im) ≠ 0 := fun x =>
        mul_ne_zero two_ne_zero (Real.sqrt_pos.2 <| mul_pos x.1.im_pos x.2.im_pos).ne'
      -- `continuity` fails to apply `Continuous.div`
      apply_rules [Continuous.div, Continuous.mul, continuous_const, Continuous.arsinh,
        Continuous.dist, continuous_coe.comp, continuous_fst, continuous_snd,
        Real.continuous_sqrt.comp, continuous_im.comp]
    · letI : MetricSpace ℍ := metricSpaceAux
      refine' le_of_nhds_le_nhds fun z => _
      

Proof state:
case refine'_2
z✝ w : ℍ
r R : ℝ
this : MetricSpace ℍ := metricSpaceAux
z : ℍ
⊢ 𝓝 z ≤ 𝓝 z

Next tactic:
rw [nhds_induced]
/-- The hyperbolic metric on the upper half plane. We ensure that the projection to
`TopologicalSpace` is definitionally equal to the subtype topology. -/
instance : MetricSpace ℍ :=
  metricSpaceAux.replaceTopology <| by
    refine' le_antisymm (continuous_id_iff_le.1 _) _
    · refine' (@continuous_iff_continuous_dist ℍ ℍ metricSpaceAux.toPseudoMetricSpace _ _).2 _
      have : ∀ x : ℍ × ℍ, 2 * Real.sqrt (x.1.im * x.2.im) ≠ 0 := fun x =>
        mul_ne_zero two_ne_zero (Real.sqrt_pos.2 <| mul_pos x.1.im_pos x.2.im_pos).ne'
      -- `continuity` fails to apply `Continuous.div`
      apply_rules [Continuous.div, Continuous.mul, continuous_const, Continuous.arsinh,
        Continuous.dist, continuous_coe.comp, continuous_fst, continuous_snd,
        Real.continuous_sqrt.comp, continuous_im.comp]
    · letI : MetricSpace ℍ := metricSpaceAux
      refine' le_of_nhds_le_nhds fun z => _
      rw [nhds_induced]
      

Proof state:
case refine'_2
z✝ w : ℍ
r R : ℝ
this : MetricSpace ℍ := metricSpaceAux
z : ℍ
⊢ 𝓝 z ≤ comap Subtype.val (𝓝 ↑z)

Next tactic:
refine' (nhds_basis_ball.le_basis_iff (nhds_basis_ball.comap _)).2 fun R hR => _
/-- The hyperbolic metric on the upper half plane. We ensure that the projection to
`TopologicalSpace` is definitionally equal to the subtype topology. -/
instance : MetricSpace ℍ :=
  metricSpaceAux.replaceTopology <| by
    refine' le_antisymm (continuous_id_iff_le.1 _) _
    · refine' (@continuous_iff_continuous_dist ℍ ℍ metricSpaceAux.toPseudoMetricSpace _ _).2 _
      have : ∀ x : ℍ × ℍ, 2 * Real.sqrt (x.1.im * x.2.im) ≠ 0 := fun x =>
        mul_ne_zero two_ne_zero (Real.sqrt_pos.2 <| mul_pos x.1.im_pos x.2.im_pos).ne'
      -- `continuity` fails to apply `Continuous.div`
      apply_rules [Continuous.div, Continuous.mul, continuous_const, Continuous.arsinh,
        Continuous.dist, continuous_coe.comp, continuous_fst, continuous_snd,
        Real.continuous_sqrt.comp, continuous_im.comp]
    · letI : MetricSpace ℍ := metricSpaceAux
      refine' le_of_nhds_le_nhds fun z => _
      rw [nhds_induced]
      refine' (nhds_basis_ball.le_basis_iff (nhds_basis_ball.comap _)).2 fun R hR => _
      

Proof state:
case refine'_2
z✝ w : ℍ
r R✝ : ℝ
this : MetricSpace ℍ := metricSpaceAux
z : ℍ
R : ℝ
hR : 0 < R
⊢ ∃ i, 0 < i ∧ ball z i ⊆ Subtype.val ⁻¹' ball (↑z) R

Next tactic:
have h₁ : 1 < R / im z + 1 := lt_add_of_pos_left _ (div_pos hR z.im_pos)
/-- The hyperbolic metric on the upper half plane. We ensure that the projection to
`TopologicalSpace` is definitionally equal to the subtype topology. -/
instance : MetricSpace ℍ :=
  metricSpaceAux.replaceTopology <| by
    refine' le_antisymm (continuous_id_iff_le.1 _) _
    · refine' (@continuous_iff_continuous_dist ℍ ℍ metricSpaceAux.toPseudoMetricSpace _ _).2 _
      have : ∀ x : ℍ × ℍ, 2 * Real.sqrt (x.1.im * x.2.im) ≠ 0 := fun x =>
        mul_ne_zero two_ne_zero (Real.sqrt_pos.2 <| mul_pos x.1.im_pos x.2.im_pos).ne'
      -- `continuity` fails to apply `Continuous.div`
      apply_rules [Continuous.div, Continuous.mul, continuous_const, Continuous.arsinh,
        Continuous.dist, continuous_coe.comp, continuous_fst, continuous_snd,
        Real.continuous_sqrt.comp, continuous_im.comp]
    · letI : MetricSpace ℍ := metricSpaceAux
      refine' le_of_nhds_le_nhds fun z => _
      rw [nhds_induced]
      refine' (nhds_basis_ball.le_basis_iff (nhds_basis_ball.comap _)).2 fun R hR => _
      have h₁ : 1 < R / im z + 1 := lt_add_of_pos_left _ (div_pos hR z.im_pos)
      

Proof state:
case refine'_2
z✝ w : ℍ
r R✝ : ℝ
this : MetricSpace ℍ := metricSpaceAux
z : ℍ
R : ℝ
hR : 0 < R
h₁ : 1 < R / im z + 1
⊢ ∃ i, 0 < i ∧ ball z i ⊆ Subtype.val ⁻¹' ball (↑z) R

Next tactic:
have h₀ : 0 < R / im z + 1 := one_pos.trans h₁
/-- The hyperbolic metric on the upper half plane. We ensure that the projection to
`TopologicalSpace` is definitionally equal to the subtype topology. -/
instance : MetricSpace ℍ :=
  metricSpaceAux.replaceTopology <| by
    refine' le_antisymm (continuous_id_iff_le.1 _) _
    · refine' (@continuous_iff_continuous_dist ℍ ℍ metricSpaceAux.toPseudoMetricSpace _ _).2 _
      have : ∀ x : ℍ × ℍ, 2 * Real.sqrt (x.1.im * x.2.im) ≠ 0 := fun x =>
        mul_ne_zero two_ne_zero (Real.sqrt_pos.2 <| mul_pos x.1.im_pos x.2.im_pos).ne'
      -- `continuity` fails to apply `Continuous.div`
      apply_rules [Continuous.div, Continuous.mul, continuous_const, Continuous.arsinh,
        Continuous.dist, continuous_coe.comp, continuous_fst, continuous_snd,
        Real.continuous_sqrt.comp, continuous_im.comp]
    · letI : MetricSpace ℍ := metricSpaceAux
      refine' le_of_nhds_le_nhds fun z => _
      rw [nhds_induced]
      refine' (nhds_basis_ball.le_basis_iff (nhds_basis_ball.comap _)).2 fun R hR => _
      have h₁ : 1 < R / im z + 1 := lt_add_of_pos_left _ (div_pos hR z.im_pos)
      have h₀ : 0 < R / im z + 1 := one_pos.trans h₁
      

Proof state:
case refine'_2
z✝ w : ℍ
r R✝ : ℝ
this : MetricSpace ℍ := metricSpaceAux
z : ℍ
R : ℝ
hR : 0 < R
h₁ : 1 < R / im z + 1
h₀ : 0 < R / im z + 1
⊢ ∃ i, 0 < i ∧ ball z i ⊆ Subtype.val ⁻¹' ball (↑z) R

Next tactic:
refine' ⟨log (R / im z + 1), Real.log_pos h₁, _⟩
/-- The hyperbolic metric on the upper half plane. We ensure that the projection to
`TopologicalSpace` is definitionally equal to the subtype topology. -/
instance : MetricSpace ℍ :=
  metricSpaceAux.replaceTopology <| by
    refine' le_antisymm (continuous_id_iff_le.1 _) _
    · refine' (@continuous_iff_continuous_dist ℍ ℍ metricSpaceAux.toPseudoMetricSpace _ _).2 _
      have : ∀ x : ℍ × ℍ, 2 * Real.sqrt (x.1.im * x.2.im) ≠ 0 := fun x =>
        mul_ne_zero two_ne_zero (Real.sqrt_pos.2 <| mul_pos x.1.im_pos x.2.im_pos).ne'
      -- `continuity` fails to apply `Continuous.div`
      apply_rules [Continuous.div, Continuous.mul, continuous_const, Continuous.arsinh,
        Continuous.dist, continuous_coe.comp, continuous_fst, continuous_snd,
        Real.continuous_sqrt.comp, continuous_im.comp]
    · letI : MetricSpace ℍ := metricSpaceAux
      refine' le_of_nhds_le_nhds fun z => _
      rw [nhds_induced]
      refine' (nhds_basis_ball.le_basis_iff (nhds_basis_ball.comap _)).2 fun R hR => _
      have h₁ : 1 < R / im z + 1 := lt_add_of_pos_left _ (div_pos hR z.im_pos)
      have h₀ : 0 < R / im z + 1 := one_pos.trans h₁
      refine' ⟨log (R / im z + 1), Real.log_pos h₁, _⟩
      

Proof state:
case refine'_2
z✝ w : ℍ
r R✝ : ℝ
this : MetricSpace ℍ := metricSpaceAux
z : ℍ
R : ℝ
hR : 0 < R
h₁ : 1 < R / im z + 1
h₀ : 0 < R / im z + 1
⊢ ball z (log (R / im z + 1)) ⊆ Subtype.val ⁻¹' ball (↑z) R

Next tactic:
refine' fun w hw => (dist_coe_le w z).trans_lt _
/-- The hyperbolic metric on the upper half plane. We ensure that the projection to
`TopologicalSpace` is definitionally equal to the subtype topology. -/
instance : MetricSpace ℍ :=
  metricSpaceAux.replaceTopology <| by
    refine' le_antisymm (continuous_id_iff_le.1 _) _
    · refine' (@continuous_iff_continuous_dist ℍ ℍ metricSpaceAux.toPseudoMetricSpace _ _).2 _
      have : ∀ x : ℍ × ℍ, 2 * Real.sqrt (x.1.im * x.2.im) ≠ 0 := fun x =>
        mul_ne_zero two_ne_zero (Real.sqrt_pos.2 <| mul_pos x.1.im_pos x.2.im_pos).ne'
      -- `continuity` fails to apply `Continuous.div`
      apply_rules [Continuous.div, Continuous.mul, continuous_const, Continuous.arsinh,
        Continuous.dist, continuous_coe.comp, continuous_fst, continuous_snd,
        Real.continuous_sqrt.comp, continuous_im.comp]
    · letI : MetricSpace ℍ := metricSpaceAux
      refine' le_of_nhds_le_nhds fun z => _
      rw [nhds_induced]
      refine' (nhds_basis_ball.le_basis_iff (nhds_basis_ball.comap _)).2 fun R hR => _
      have h₁ : 1 < R / im z + 1 := lt_add_of_pos_left _ (div_pos hR z.im_pos)
      have h₀ : 0 < R / im z + 1 := one_pos.trans h₁
      refine' ⟨log (R / im z + 1), Real.log_pos h₁, _⟩
      refine' fun w hw => (dist_coe_le w z).trans_lt _
      

Proof state:
case refine'_2
z✝ w✝ : ℍ
r R✝ : ℝ
this : MetricSpace ℍ := metricSpaceAux
z : ℍ
R : ℝ
hR : 0 < R
h₁ : 1 < R / im z + 1
h₀ : 0 < R / im z + 1
w : ℍ
hw : w ∈ ball z (log (R / im z + 1))
⊢ im z * (rexp (dist w z) - 1) < R

Next tactic:
rwa [← lt_div_iff' z.im_pos, sub_lt_iff_lt_add, ← Real.lt_log_iff_exp_lt h₀]
theorem im_pos_of_dist_center_le {z : ℍ} {r : ℝ} {w : ℂ}
    (h : dist w (center z r) ≤ z.im * Real.sinh r) : 0 < w.im :=
  calc
    0 < z.im * (Real.cosh r - Real.sinh r) := mul_pos z.im_pos (sub_pos.2 <| sinh_lt_cosh _)
    _ = (z.center r).im - z.im * Real.sinh r := (mul_sub _ _ _)
    _ ≤ (z.center r).im - dist (z.center r : ℂ) w := (sub_le_sub_left (by 

Proof state:
z✝ w✝ : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
w : ℂ
h : dist w ↑(center z r) ≤ im z * Real.sinh r
⊢ dist (↑(center z r)) w ≤ im z * Real.sinh r

Next tactic:
rwa [dist_comm]
theorem image_coe_closedBall (z : ℍ) (r : ℝ) :
    ((↑) : ℍ → ℂ) '' closedBall (α := ℍ) z r = closedBall ↑(z.center r) (z.im * Real.sinh r) := by
  

Proof state:
z✝ w : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
⊢ UpperHalfPlane.coe '' closedBall z r = closedBall (↑(center z r)) (im z * Real.sinh r)

Next tactic:
ext w
theorem image_coe_closedBall (z : ℍ) (r : ℝ) :
    ((↑) : ℍ → ℂ) '' closedBall (α := ℍ) z r = closedBall ↑(z.center r) (z.im * Real.sinh r) := by
  ext w; 

Proof state:
case h
z✝ w✝ : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
w : ℂ
⊢ w ∈ UpperHalfPlane.coe '' closedBall z r ↔ w ∈ closedBall (↑(center z r)) (im z * Real.sinh r)

Next tactic:
constructor
theorem image_coe_closedBall (z : ℍ) (r : ℝ) :
    ((↑) : ℍ → ℂ) '' closedBall (α := ℍ) z r = closedBall ↑(z.center r) (z.im * Real.sinh r) := by
  ext w; constructor
  · 

Proof state:
case h.mp
z✝ w✝ : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
w : ℂ
⊢ w ∈ UpperHalfPlane.coe '' closedBall z r → w ∈ closedBall (↑(center z r)) (im z * Real.sinh r)

Next tactic:
rintro ⟨w, hw, rfl⟩
theorem image_coe_closedBall (z : ℍ) (r : ℝ) :
    ((↑) : ℍ → ℂ) '' closedBall (α := ℍ) z r = closedBall ↑(z.center r) (z.im * Real.sinh r) := by
  ext w; constructor
  · rintro ⟨w, hw, rfl⟩
    

Proof state:
case h.mp.intro.intro
z✝ w✝ : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
w : ℍ
hw : w ∈ closedBall z r
⊢ ↑w ∈ closedBall (↑(center z r)) (im z * Real.sinh r)

Next tactic:
exact dist_le_iff_dist_coe_center_le.1 hw
theorem image_coe_closedBall (z : ℍ) (r : ℝ) :
    ((↑) : ℍ → ℂ) '' closedBall (α := ℍ) z r = closedBall ↑(z.center r) (z.im * Real.sinh r) := by
  ext w; constructor
  · rintro ⟨w, hw, rfl⟩
    exact dist_le_iff_dist_coe_center_le.1 hw
  · 

Proof state:
case h.mpr
z✝ w✝ : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
w : ℂ
⊢ w ∈ closedBall (↑(center z r)) (im z * Real.sinh r) → w ∈ UpperHalfPlane.coe '' closedBall z r

Next tactic:
intro hw
theorem image_coe_closedBall (z : ℍ) (r : ℝ) :
    ((↑) : ℍ → ℂ) '' closedBall (α := ℍ) z r = closedBall ↑(z.center r) (z.im * Real.sinh r) := by
  ext w; constructor
  · rintro ⟨w, hw, rfl⟩
    exact dist_le_iff_dist_coe_center_le.1 hw
  · intro hw
    

Proof state:
case h.mpr
z✝ w✝ : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
w : ℂ
hw : w ∈ closedBall (↑(center z r)) (im z * Real.sinh r)
⊢ w ∈ UpperHalfPlane.coe '' closedBall z r

Next tactic:
lift w to ℍ using im_pos_of_dist_center_le hw
theorem image_coe_closedBall (z : ℍ) (r : ℝ) :
    ((↑) : ℍ → ℂ) '' closedBall (α := ℍ) z r = closedBall ↑(z.center r) (z.im * Real.sinh r) := by
  ext w; constructor
  · rintro ⟨w, hw, rfl⟩
    exact dist_le_iff_dist_coe_center_le.1 hw
  · intro hw
    lift w to ℍ using im_pos_of_dist_center_le hw
    

Proof state:
case h.mpr.intro
z✝ w✝ : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
w : ℍ
hw : ↑w ∈ closedBall (↑(center z r)) (im z * Real.sinh r)
⊢ ↑w ∈ UpperHalfPlane.coe '' closedBall z r

Next tactic:
exact mem_image_of_mem _ (dist_le_iff_dist_coe_center_le.2 hw)