l3lab/ntp-mathlib huggingface dataset:
theorem sinh_half_dist_add_dist (a b c : ℍ) : sinh ((dist a b + dist b c) / 2) =
    (dist (a : ℂ) b * dist (c : ℂ) (conj ↑b) + dist (b : ℂ) c * dist (a : ℂ) (conj ↑b)) /
      (2 * sqrt (a.im * c.im) * dist (b : ℂ) (conj ↑b)) := by
  simp only [add_div _ _ (2 : ℝ), sinh_add, sinh_half_dist, cosh_half_dist, div_mul_div_comm]
  rw [← add_div, Complex.dist_self_conj, coe_im, abs_of_pos b.im_pos, mul_comm (dist (b : ℂ) _),
    dist_comm (b : ℂ), Complex.dist_conj_comm, mul_mul_mul_comm, mul_mul_mul_comm _ _ _ b.im]
  congr 2
  rw [sqrt_mul, sqrt_mul, sqrt_mul, mul_comm (sqrt a.im), mul_mul_mul_comm, mul_self_sqrt,
      mul_comm] <;> 

Proof state:
case e_a.e_a
z w : ℍ
r R : ℝ
a b c : ℍ
⊢ 0 ≤ im b

Next tactic:
exact (im_pos _).le
theorem sinh_half_dist_add_dist (a b c : ℍ) : sinh ((dist a b + dist b c) / 2) =
    (dist (a : ℂ) b * dist (c : ℂ) (conj ↑b) + dist (b : ℂ) c * dist (a : ℂ) (conj ↑b)) /
      (2 * sqrt (a.im * c.im) * dist (b : ℂ) (conj ↑b)) := by
  simp only [add_div _ _ (2 : ℝ), sinh_add, sinh_half_dist, cosh_half_dist, div_mul_div_comm]
  rw [← add_div, Complex.dist_self_conj, coe_im, abs_of_pos b.im_pos, mul_comm (dist (b : ℂ) _),
    dist_comm (b : ℂ), Complex.dist_conj_comm, mul_mul_mul_comm, mul_mul_mul_comm _ _ _ b.im]
  congr 2
  rw [sqrt_mul, sqrt_mul, sqrt_mul, mul_comm (sqrt a.im), mul_mul_mul_comm, mul_self_sqrt,
      mul_comm] <;> 

Proof state:
case e_a.e_a.hx
z w : ℍ
r R : ℝ
a b c : ℍ
⊢ 0 ≤ im a

Next tactic:
exact (im_pos _).le
theorem sinh_half_dist_add_dist (a b c : ℍ) : sinh ((dist a b + dist b c) / 2) =
    (dist (a : ℂ) b * dist (c : ℂ) (conj ↑b) + dist (b : ℂ) c * dist (a : ℂ) (conj ↑b)) /
      (2 * sqrt (a.im * c.im) * dist (b : ℂ) (conj ↑b)) := by
  simp only [add_div _ _ (2 : ℝ), sinh_add, sinh_half_dist, cosh_half_dist, div_mul_div_comm]
  rw [← add_div, Complex.dist_self_conj, coe_im, abs_of_pos b.im_pos, mul_comm (dist (b : ℂ) _),
    dist_comm (b : ℂ), Complex.dist_conj_comm, mul_mul_mul_comm, mul_mul_mul_comm _ _ _ b.im]
  congr 2
  rw [sqrt_mul, sqrt_mul, sqrt_mul, mul_comm (sqrt a.im), mul_mul_mul_comm, mul_self_sqrt,
      mul_comm] <;> 

Proof state:
case e_a.e_a.hx
z w : ℍ
r R : ℝ
a b c : ℍ
⊢ 0 ≤ im b

Next tactic:
exact (im_pos _).le
protected theorem dist_comm (z w : ℍ) : dist z w = dist w z := by
  

Proof state:
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
⊢ dist z w = dist w z

Next tactic:
simp only [dist_eq, dist_comm (z : ℂ), mul_comm]
theorem dist_le_iff_le_sinh :
    dist z w ≤ r ↔ dist (z : ℂ) w / (2 * sqrt (z.im * w.im)) ≤ sinh (r / 2) := by
  

Proof state:
z w : ℍ
r R : ℝ
⊢ dist z w ≤ r ↔ dist ↑z ↑w / (2 * sqrt (im z * im w)) ≤ sinh (r / 2)

Next tactic:
rw [← div_le_div_right (zero_lt_two' ℝ), ← sinh_le_sinh, sinh_half_dist]
theorem dist_eq_iff_eq_sinh :
    dist z w = r ↔ dist (z : ℂ) w / (2 * sqrt (z.im * w.im)) = sinh (r / 2) := by
  

Proof state:
z w : ℍ
r R : ℝ
⊢ dist z w = r ↔ dist ↑z ↑w / (2 * sqrt (im z * im w)) = sinh (r / 2)

Next tactic:
rw [← div_left_inj' (two_ne_zero' ℝ), ← sinh_inj, sinh_half_dist]
theorem dist_eq_iff_eq_sq_sinh (hr : 0 ≤ r) :
    dist z w = r ↔ dist (z : ℂ) w ^ 2 / (4 * z.im * w.im) = sinh (r / 2) ^ 2 := by
  

Proof state:
z w : ℍ
r R : ℝ
hr : 0 ≤ r
⊢ dist z w = r ↔ dist ↑z ↑w ^ 2 / (4 * im z * im w) = sinh (r / 2) ^ 2

Next tactic:
rw [dist_eq_iff_eq_sinh, ← sq_eq_sq, div_pow, mul_pow, sq_sqrt, mul_assoc]
theorem dist_eq_iff_eq_sq_sinh (hr : 0 ≤ r) :
    dist z w = r ↔ dist (z : ℂ) w ^ 2 / (4 * z.im * w.im) = sinh (r / 2) ^ 2 := by
  rw [dist_eq_iff_eq_sinh, ← sq_eq_sq, div_pow, mul_pow, sq_sqrt, mul_assoc]
  · 

Proof state:
z w : ℍ
r R : ℝ
hr : 0 ≤ r
⊢ dist ↑z ↑w ^ 2 / (2 ^ 2 * (im z * im w)) = sinh (r / 2) ^ 2 ↔ dist ↑z ↑w ^ 2 / (4 * (im z * im w)) = sinh (r / 2) ^ 2

Next tactic:
norm_num
theorem dist_eq_iff_eq_sq_sinh (hr : 0 ≤ r) :
    dist z w = r ↔ dist (z : ℂ) w ^ 2 / (4 * z.im * w.im) = sinh (r / 2) ^ 2 := by
  rw [dist_eq_iff_eq_sinh, ← sq_eq_sq, div_pow, mul_pow, sq_sqrt, mul_assoc]
  · norm_num
  · 

Proof state:
z w : ℍ
r R : ℝ
hr : 0 ≤ r
⊢ 0 ≤ im z * im w

Next tactic:
exact (mul_pos z.im_pos w.im_pos).le
theorem dist_eq_iff_eq_sq_sinh (hr : 0 ≤ r) :
    dist z w = r ↔ dist (z : ℂ) w ^ 2 / (4 * z.im * w.im) = sinh (r / 2) ^ 2 := by
  rw [dist_eq_iff_eq_sinh, ← sq_eq_sq, div_pow, mul_pow, sq_sqrt, mul_assoc]
  · norm_num
  · exact (mul_pos z.im_pos w.im_pos).le
  · 

Proof state:
case ha
z w : ℍ
r R : ℝ
hr : 0 ≤ r
⊢ 0 ≤ dist ↑z ↑w / (2 * sqrt (im z * im w))

Next tactic:
exact div_nonneg dist_nonneg (mul_nonneg zero_le_two <| sqrt_nonneg _)
theorem dist_eq_iff_eq_sq_sinh (hr : 0 ≤ r) :
    dist z w = r ↔ dist (z : ℂ) w ^ 2 / (4 * z.im * w.im) = sinh (r / 2) ^ 2 := by
  rw [dist_eq_iff_eq_sinh, ← sq_eq_sq, div_pow, mul_pow, sq_sqrt, mul_assoc]
  · norm_num
  · exact (mul_pos z.im_pos w.im_pos).le
  · exact div_nonneg dist_nonneg (mul_nonneg zero_le_two <| sqrt_nonneg _)
  · 

Proof state:
case hb
z w : ℍ
r R : ℝ
hr : 0 ≤ r
⊢ 0 ≤ sinh (r / 2)

Next tactic:
exact sinh_nonneg_iff.2 (div_nonneg hr zero_le_two)
protected theorem dist_triangle (a b c : ℍ) : dist a c ≤ dist a b + dist b c := by
  

Proof state:
z w : ℍ
r R : ℝ
a b c : ℍ
⊢ dist a c ≤ dist a b + dist b c

Next tactic:
rw [dist_le_iff_le_sinh, sinh_half_dist_add_dist, div_mul_eq_div_div _ _ (dist _ _), le_div_iff,
    div_mul_eq_mul_div]
protected theorem dist_triangle (a b c : ℍ) : dist a c ≤ dist a b + dist b c := by
  rw [dist_le_iff_le_sinh, sinh_half_dist_add_dist, div_mul_eq_div_div _ _ (dist _ _), le_div_iff,
    div_mul_eq_mul_div]
  · 

Proof state:
z w : ℍ
r R : ℝ
a b c : ℍ
⊢ dist ↑a ↑c * dist (↑b) ((starRingEnd ℂ) ↑b) / (2 * sqrt (im a * im c)) ≤
    (dist ↑a ↑b * dist (↑c) ((starRingEnd ℂ) ↑b) + dist ↑b ↑c * dist (↑a) ((starRingEnd ℂ) ↑b)) /
      (2 * sqrt (im a * im c))

Next tactic:
exact div_le_div_of_le (mul_nonneg zero_le_two (sqrt_nonneg _))
      (EuclideanGeometry.mul_dist_le_mul_dist_add_mul_dist (a : ℂ) b c (conj (b : ℂ)))
protected theorem dist_triangle (a b c : ℍ) : dist a c ≤ dist a b + dist b c := by
  rw [dist_le_iff_le_sinh, sinh_half_dist_add_dist, div_mul_eq_div_div _ _ (dist _ _), le_div_iff,
    div_mul_eq_mul_div]
  · exact div_le_div_of_le (mul_nonneg zero_le_two (sqrt_nonneg _))
      (EuclideanGeometry.mul_dist_le_mul_dist_add_mul_dist (a : ℂ) b c (conj (b : ℂ)))
  · 

Proof state:
z w : ℍ
r R : ℝ
a b c : ℍ
⊢ 0 < dist (↑b) ((starRingEnd ℂ) ↑b)

Next tactic:
rw [dist_comm, dist_pos, Ne.def, Complex.conj_eq_iff_im]
protected theorem dist_triangle (a b c : ℍ) : dist a c ≤ dist a b + dist b c := by
  rw [dist_le_iff_le_sinh, sinh_half_dist_add_dist, div_mul_eq_div_div _ _ (dist _ _), le_div_iff,
    div_mul_eq_mul_div]
  · exact div_le_div_of_le (mul_nonneg zero_le_two (sqrt_nonneg _))
      (EuclideanGeometry.mul_dist_le_mul_dist_add_mul_dist (a : ℂ) b c (conj (b : ℂ)))
  · rw [dist_comm, dist_pos, Ne.def, Complex.conj_eq_iff_im]
    

Proof state:
z w : ℍ
r R : ℝ
a b c : ℍ
⊢ ¬(↑b).im = 0

Next tactic:
exact b.im_ne_zero
theorem dist_le_dist_coe_div_sqrt (z w : ℍ) : dist z w ≤ dist (z : ℂ) w / sqrt (z.im * w.im) := by
  

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

Next tactic:
rw [dist_le_iff_le_sinh, ← div_mul_eq_div_div_swap, self_le_sinh_iff]
theorem dist_le_dist_coe_div_sqrt (z w : ℍ) : dist z w ≤ dist (z : ℂ) w / sqrt (z.im * w.im) := by
  rw [dist_le_iff_le_sinh, ← div_mul_eq_div_div_swap, self_le_sinh_iff]
  

Proof state:
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
⊢ 0 ≤ dist ↑z ↑w / (2 * sqrt (im z * im w))

Next tactic:
exact div_nonneg dist_nonneg (mul_nonneg zero_le_two (sqrt_nonneg _))
/-- An auxiliary `MetricSpace` instance on the upper half-plane. This instance has bad projection
to `TopologicalSpace`. We replace it later. -/
def metricSpaceAux : MetricSpace ℍ where
  dist := dist
  dist_self z := by 

Proof state:
z✝ w : ℍ
r R : ℝ
z : ℍ
⊢ dist z z = 0

Next tactic:
rw [dist_eq, dist_self, zero_div, arsinh_zero, mul_zero]
/-- An auxiliary `MetricSpace` instance on the upper half-plane. This instance has bad projection
to `TopologicalSpace`. We replace it later. -/
def metricSpaceAux : MetricSpace ℍ where
  dist := dist
  dist_self z := by rw [dist_eq, dist_self, zero_div, arsinh_zero, mul_zero]
  dist_comm := UpperHalfPlane.dist_comm
  dist_triangle := UpperHalfPlane.dist_triangle
  eq_of_dist_eq_zero {z w} h := by
    simpa [dist_eq, Real.sqrt_eq_zero', (mul_pos z.im_pos w.im_pos).not_le, ext_iff] using h
  edist_dist _ _ := by 

Proof state:
z w : ℍ
r R : ℝ
x✝¹ x✝ : ℍ
⊢ (fun x y =>
        ↑{
            val :=
              2 *
                arsinh
                  ((let src := Complex.abs;
                        {
                          toAddGroupSeminorm :=
                            { toFun := src.toFun, map_zero' := Complex.instNormedAddCommGroupComplex.proof_1,
                              add_le' :=
                                (_ :
                                  ∀ (x y : ℂ),
                                    MulHom.toFun src.toMulHom (x + y) ≤
                                      MulHom.toFun src.toMulHom x + MulHom.toFun src.toMulHom y),
                              neg' := Complex.instNormedAddCommGroupComplex.proof_2 },
                          eq_zero_of_map_eq_zero' := Complex.instNormedAddCommGroupComplex.proof_3 }).toAddGroupSeminorm
                      (↑x - ↑y) /
                    (2 * sqrt (im x * im y))),
            property :=
              (_ :
                0 ≤
                  2 *
                    arsinh
                      ((let src := Complex.abs;
                            {
                              toAddGroupSeminorm :=
                                { toFun := src.toFun, map_zero' := Complex.instNormedAddCommGroupComplex.proof_1,
                                  add_le' :=
                                    (_ :
                                      ∀ (x y : ℂ),
                                        MulHom.toFun src.toMulHom (x + y) ≤
                                          MulHom.toFun src.toMulHom x + MulHom.toFun src.toMulHom y),
                                  neg' := Complex.instNormedAddCommGroupComplex.proof_2 },
                              eq_zero_of_map_eq_zero' :=
                                Complex.instNormedAddCommGroupComplex.proof_3 }).toAddGroupSeminorm
                          (↑x - ↑y) /
                        (2 * sqrt (im x * im y)))) })
      x✝¹ x✝ =
    ENNReal.ofReal (dist x✝¹ x✝)

Next tactic:
exact ENNReal.coe_nnreal_eq _
/-- An auxiliary `MetricSpace` instance on the upper half-plane. This instance has bad projection
to `TopologicalSpace`. We replace it later. -/
def metricSpaceAux : MetricSpace ℍ where
  dist := dist
  dist_self z := by rw [dist_eq, dist_self, zero_div, arsinh_zero, mul_zero]
  dist_comm := UpperHalfPlane.dist_comm
  dist_triangle := UpperHalfPlane.dist_triangle
  eq_of_dist_eq_zero {z w} h := by
    

Proof state:
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
h : dist z w = 0
⊢ z = w

Next tactic:
simpa [dist_eq, Real.sqrt_eq_zero', (mul_pos z.im_pos w.im_pos).not_le, ext_iff] using h
theorem cosh_dist' (z w : ℍ) :
    Real.cosh (dist z w) = ((z.re - w.re) ^ 2 + z.im ^ 2 + w.im ^ 2) / (2 * z.im * w.im) := by
  

Proof state:
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
⊢ Real.cosh (dist z w) = ((re z - re w) ^ 2 + im z ^ 2 + im w ^ 2) / (2 * im z * im w)

Next tactic:
have H : 0 < 2 * z.im * w.im := mul_pos (mul_pos two_pos z.im_pos) w.im_pos
theorem cosh_dist' (z w : ℍ) :
    Real.cosh (dist z w) = ((z.re - w.re) ^ 2 + z.im ^ 2 + w.im ^ 2) / (2 * z.im * w.im) := by
  have H : 0 < 2 * z.im * w.im := mul_pos (mul_pos two_pos z.im_pos) w.im_pos
  

Proof state:
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
H : 0 < 2 * im z * im w
⊢ Real.cosh (dist z w) = ((re z - re w) ^ 2 + im z ^ 2 + im w ^ 2) / (2 * im z * im w)

Next tactic:
field_simp [cosh_dist, Complex.dist_eq, Complex.sq_abs, normSq_apply, H, H.ne']
theorem cosh_dist' (z w : ℍ) :
    Real.cosh (dist z w) = ((z.re - w.re) ^ 2 + z.im ^ 2 + w.im ^ 2) / (2 * z.im * w.im) := by
  have H : 0 < 2 * z.im * w.im := mul_pos (mul_pos two_pos z.im_pos) w.im_pos
  field_simp [cosh_dist, Complex.dist_eq, Complex.sq_abs, normSq_apply, H, H.ne']
  

Proof state:
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
H : 0 < 2 * im z * im w
⊢ 2 * im z * im w + ((re z - re w) * (re z - re w) + (im z - im w) * (im z - im w)) =
    (re z - re w) ^ 2 + im z ^ 2 + im w ^ 2

Next tactic:
ring
@[simp]
theorem center_zero (z : ℍ) : center z 0 = z :=
  ext' rfl <| by 

Proof state:
z✝ w : ℍ
r R : ℝ
z : ℍ
⊢ im (center z 0) = im z

Next tactic:
rw [center_im, Real.cosh_zero, mul_one]
theorem dist_coe_center_sq (z w : ℍ) (r : ℝ) : dist (z : ℂ) (w.center r) ^ 2 =
    2 * z.im * w.im * (Real.cosh (dist z w) - Real.cosh r) + (w.im * Real.sinh r) ^ 2 := by
  

Proof state:
z✝ w✝ : ℍ
r✝ R : ℝ
z w : ℍ
r : ℝ
⊢ dist ↑z ↑(center w r) ^ 2 = 2 * im z * im w * (Real.cosh (dist z w) - Real.cosh r) + (im w * Real.sinh r) ^ 2

Next tactic:
have H : 2 * z.im * w.im ≠ 0 := by apply_rules [mul_ne_zero, two_ne_zero, im_ne_zero]
theorem dist_coe_center_sq (z w : ℍ) (r : ℝ) : dist (z : ℂ) (w.center r) ^ 2 =
    2 * z.im * w.im * (Real.cosh (dist z w) - Real.cosh r) + (w.im * Real.sinh r) ^ 2 := by
  have H : 2 * z.im * w.im ≠ 0 := by 

Proof state:
z✝ w✝ : ℍ
r✝ R : ℝ
z w : ℍ
r : ℝ
⊢ 2 * im z * im w ≠ 0

Next tactic:
apply_rules [mul_ne_zero, two_ne_zero, im_ne_zero]
theorem dist_coe_center_sq (z w : ℍ) (r : ℝ) : dist (z : ℂ) (w.center r) ^ 2 =
    2 * z.im * w.im * (Real.cosh (dist z w) - Real.cosh r) + (w.im * Real.sinh r) ^ 2 := by
  have H : 2 * z.im * w.im ≠ 0 := by apply_rules [mul_ne_zero, two_ne_zero, im_ne_zero]
  

Proof state:
z✝ w✝ : ℍ
r✝ R : ℝ
z w : ℍ
r : ℝ
H : 2 * im z * im w ≠ 0
⊢ dist ↑z ↑(center w r) ^ 2 = 2 * im z * im w * (Real.cosh (dist z w) - Real.cosh r) + (im w * Real.sinh r) ^ 2

Next tactic:
simp only [Complex.dist_eq, Complex.sq_abs, normSq_apply, coe_re, coe_im, center_re, center_im,
    cosh_dist', mul_div_cancel' _ H, sub_sq z.im, mul_pow, Real.cosh_sq, sub_re, sub_im, mul_sub, ←
    sq]
theorem dist_coe_center_sq (z w : ℍ) (r : ℝ) : dist (z : ℂ) (w.center r) ^ 2 =
    2 * z.im * w.im * (Real.cosh (dist z w) - Real.cosh r) + (w.im * Real.sinh r) ^ 2 := by
  have H : 2 * z.im * w.im ≠ 0 := by apply_rules [mul_ne_zero, two_ne_zero, im_ne_zero]
  simp only [Complex.dist_eq, Complex.sq_abs, normSq_apply, coe_re, coe_im, center_re, center_im,
    cosh_dist', mul_div_cancel' _ H, sub_sq z.im, mul_pow, Real.cosh_sq, sub_re, sub_im, mul_sub, ←
    sq]
  

Proof state:
z✝ w✝ : ℍ
r✝ R : ℝ
z w : ℍ
r : ℝ
H : 2 * im z * im w ≠ 0
⊢ (re z - re w) ^ 2 + (im z ^ 2 - 2 * im z * (im w * Real.cosh r) + im w ^ 2 * (Real.sinh r ^ 2 + 1)) =
    (re z - re w) ^ 2 + im z ^ 2 + im w ^ 2 - 2 * im z * im w * Real.cosh r + im w ^ 2 * Real.sinh r ^ 2

Next tactic:
ring
theorem dist_coe_center (z w : ℍ) (r : ℝ) : dist (z : ℂ) (w.center r) =
    sqrt (2 * z.im * w.im * (Real.cosh (dist z w) - Real.cosh r) + (w.im * Real.sinh r) ^ 2) := by
  

Proof state:
z✝ w✝ : ℍ
r✝ R : ℝ
z w : ℍ
r : ℝ
⊢ dist ↑z ↑(center w r) = sqrt (2 * im z * im w * (Real.cosh (dist z w) - Real.cosh r) + (im w * Real.sinh r) ^ 2)

Next tactic:
rw [← sqrt_sq dist_nonneg, dist_coe_center_sq]
theorem cmp_dist_eq_cmp_dist_coe_center (z w : ℍ) (r : ℝ) :
    cmp (dist z w) r = cmp (dist (z : ℂ) (w.center r)) (w.im * Real.sinh r) := by
  

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

Next tactic:
letI := metricSpaceAux
theorem cmp_dist_eq_cmp_dist_coe_center (z w : ℍ) (r : ℝ) :
    cmp (dist z w) r = cmp (dist (z : ℂ) (w.center r)) (w.im * Real.sinh r) := by
  letI := metricSpaceAux
  

Proof state:
z✝ w✝ : ℍ
r✝ R : ℝ
z w : ℍ
r : ℝ
this : MetricSpace ℍ := metricSpaceAux
⊢ cmp (dist z w) r = cmp (dist ↑z ↑(center w r)) (im w * Real.sinh r)

Next tactic:
cases' lt_or_le r 0 with hr₀ hr₀
theorem cmp_dist_eq_cmp_dist_coe_center (z w : ℍ) (r : ℝ) :
    cmp (dist z w) r = cmp (dist (z : ℂ) (w.center r)) (w.im * Real.sinh r) := by
  letI := metricSpaceAux
  cases' lt_or_le r 0 with hr₀ hr₀
  · 

Proof state:
case inl
z✝ w✝ : ℍ
r✝ R : ℝ
z w : ℍ
r : ℝ
this : MetricSpace ℍ := metricSpaceAux
hr₀ : r < 0
⊢ cmp (dist z w) r = cmp (dist ↑z ↑(center w r)) (im w * Real.sinh r)

Next tactic:
trans Ordering.gt
theorem cmp_dist_eq_cmp_dist_coe_center (z w : ℍ) (r : ℝ) :
    cmp (dist z w) r = cmp (dist (z : ℂ) (w.center r)) (w.im * Real.sinh r) := by
  letI := metricSpaceAux
  cases' lt_or_le r 0 with hr₀ hr₀
  · trans Ordering.gt
    

Proof state:
z✝ w✝ : ℍ
r✝ R : ℝ
z w : ℍ
r : ℝ
this : MetricSpace ℍ := metricSpaceAux
hr₀ : r < 0
⊢ cmp (dist z w) r = Ordering.gt
z✝ w✝ : ℍ
r✝ R : ℝ
z w : ℍ
r : ℝ
this : MetricSpace ℍ := metricSpaceAux
hr₀ : r < 0
⊢ Ordering.gt = cmp (dist ↑z ↑(center w r)) (im w * Real.sinh r)

Next tactic:
exacts [(hr₀.trans_le dist_nonneg).cmp_eq_gt,
      ((mul_neg_of_pos_of_neg w.im_pos (sinh_neg_iff.2 hr₀)).trans_le dist_nonneg).cmp_eq_gt.symm]
theorem cmp_dist_eq_cmp_dist_coe_center (z w : ℍ) (r : ℝ) :
    cmp (dist z w) r = cmp (dist (z : ℂ) (w.center r)) (w.im * Real.sinh r) := by
  letI := metricSpaceAux
  cases' lt_or_le r 0 with hr₀ hr₀
  · trans Ordering.gt
    exacts [(hr₀.trans_le dist_nonneg).cmp_eq_gt,
      ((mul_neg_of_pos_of_neg w.im_pos (sinh_neg_iff.2 hr₀)).trans_le dist_nonneg).cmp_eq_gt.symm]
  

Proof state:
case inr
z✝ w✝ : ℍ
r✝ R : ℝ
z w : ℍ
r : ℝ
this : MetricSpace ℍ := metricSpaceAux
hr₀ : 0 ≤ r
⊢ cmp (dist z w) r = cmp (dist ↑z ↑(center w r)) (im w * Real.sinh r)

Next tactic:
have hr₀' : 0 ≤ w.im * Real.sinh r := mul_nonneg w.im_pos.le (sinh_nonneg_iff.2 hr₀)
theorem cmp_dist_eq_cmp_dist_coe_center (z w : ℍ) (r : ℝ) :
    cmp (dist z w) r = cmp (dist (z : ℂ) (w.center r)) (w.im * Real.sinh r) := by
  letI := metricSpaceAux
  cases' lt_or_le r 0 with hr₀ hr₀
  · trans Ordering.gt
    exacts [(hr₀.trans_le dist_nonneg).cmp_eq_gt,
      ((mul_neg_of_pos_of_neg w.im_pos (sinh_neg_iff.2 hr₀)).trans_le dist_nonneg).cmp_eq_gt.symm]
  have hr₀' : 0 ≤ w.im * Real.sinh r := mul_nonneg w.im_pos.le (sinh_nonneg_iff.2 hr₀)
  

Proof state:
case inr
z✝ w✝ : ℍ
r✝ R : ℝ
z w : ℍ
r : ℝ
this : MetricSpace ℍ := metricSpaceAux
hr₀ : 0 ≤ r
hr₀' : 0 ≤ im w * Real.sinh r
⊢ cmp (dist z w) r = cmp (dist ↑z ↑(center w r)) (im w * Real.sinh r)

Next tactic:
have hzw₀ : 0 < 2 * z.im * w.im := mul_pos (mul_pos two_pos z.im_pos) w.im_pos
theorem cmp_dist_eq_cmp_dist_coe_center (z w : ℍ) (r : ℝ) :
    cmp (dist z w) r = cmp (dist (z : ℂ) (w.center r)) (w.im * Real.sinh r) := by
  letI := metricSpaceAux
  cases' lt_or_le r 0 with hr₀ hr₀
  · trans Ordering.gt
    exacts [(hr₀.trans_le dist_nonneg).cmp_eq_gt,
      ((mul_neg_of_pos_of_neg w.im_pos (sinh_neg_iff.2 hr₀)).trans_le dist_nonneg).cmp_eq_gt.symm]
  have hr₀' : 0 ≤ w.im * Real.sinh r := mul_nonneg w.im_pos.le (sinh_nonneg_iff.2 hr₀)
  have hzw₀ : 0 < 2 * z.im * w.im := mul_pos (mul_pos two_pos z.im_pos) w.im_pos
  

Proof state:
case inr
z✝ w✝ : ℍ
r✝ R : ℝ
z w : ℍ
r : ℝ
this : MetricSpace ℍ := metricSpaceAux
hr₀ : 0 ≤ r
hr₀' : 0 ≤ im w * Real.sinh r
hzw₀ : 0 < 2 * im z * im w
⊢ cmp (dist z w) r = cmp (dist ↑z ↑(center w r)) (im w * Real.sinh r)

Next tactic:
simp only [← cosh_strictMonoOn.cmp_map_eq dist_nonneg hr₀, ←
    (pow_left_strictMonoOn two_ne_zero).cmp_map_eq dist_nonneg hr₀', dist_coe_center_sq]
theorem cmp_dist_eq_cmp_dist_coe_center (z w : ℍ) (r : ℝ) :
    cmp (dist z w) r = cmp (dist (z : ℂ) (w.center r)) (w.im * Real.sinh r) := by
  letI := metricSpaceAux
  cases' lt_or_le r 0 with hr₀ hr₀
  · trans Ordering.gt
    exacts [(hr₀.trans_le dist_nonneg).cmp_eq_gt,
      ((mul_neg_of_pos_of_neg w.im_pos (sinh_neg_iff.2 hr₀)).trans_le dist_nonneg).cmp_eq_gt.symm]
  have hr₀' : 0 ≤ w.im * Real.sinh r := mul_nonneg w.im_pos.le (sinh_nonneg_iff.2 hr₀)
  have hzw₀ : 0 < 2 * z.im * w.im := mul_pos (mul_pos two_pos z.im_pos) w.im_pos
  simp only [← cosh_strictMonoOn.cmp_map_eq dist_nonneg hr₀, ←
    (pow_left_strictMonoOn two_ne_zero).cmp_map_eq dist_nonneg hr₀', dist_coe_center_sq]
  

Proof state:
case inr
z✝ w✝ : ℍ
r✝ R : ℝ
z w : ℍ
r : ℝ
this : MetricSpace ℍ := metricSpaceAux
hr₀ : 0 ≤ r
hr₀' : 0 ≤ im w * Real.sinh r
hzw₀ : 0 < 2 * im z * im w
⊢ cmp (Real.cosh (dist z w)) (Real.cosh r) =
    cmp (2 * im z * im w * (Real.cosh (dist z w) - Real.cosh r) + (im w * Real.sinh r) ^ 2) ((im w * Real.sinh r) ^ 2)

Next tactic:
rw [← cmp_mul_pos_left hzw₀, ← cmp_sub_zero, ← mul_sub, ← cmp_add_right, zero_add]
@[simp]
theorem dist_self_center (z : ℍ) (r : ℝ) :
    dist (z : ℂ) (z.center r) = z.im * (Real.cosh r - 1) := by
  

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

Next tactic:
rw [dist_of_re_eq (z.center_re r).symm, dist_comm, Real.dist_eq, mul_sub, mul_one]
@[simp]
theorem dist_self_center (z : ℍ) (r : ℝ) :
    dist (z : ℂ) (z.center r) = z.im * (Real.cosh r - 1) := by
  rw [dist_of_re_eq (z.center_re r).symm, dist_comm, Real.dist_eq, mul_sub, mul_one]
  

Proof state:
z✝ w : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
⊢ |(↑(center z r)).im - (↑z).im| = im z * Real.cosh r - im z

Next tactic:
exact abs_of_nonneg (sub_nonneg.2 <| le_mul_of_one_le_right z.im_pos.le (one_le_cosh _))
/-- 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
  

Proof state:
z w : ℍ
r R : ℝ
h : re z = re w
⊢ dist z w = dist (log (im z)) (log (im w))

Next tactic:
have h₀ : 0 < z.im / w.im := div_pos z.im_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
  

Proof state:
z w : ℍ
r R : ℝ
h : re z = re w
h₀ : 0 < im z / im w
⊢ dist z w = dist (log (im z)) (log (im w))

Next tactic:
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]
/-- 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
  

Proof state:
z w : ℍ
r R : ℝ
h : re z = re w
h₀ : 0 < im z / im w
⊢ dist z w = dist (log (im z)) (log (im w))

Next tactic:
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]
/-- 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] <;>
  [

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:
skip
/-- 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; 

Proof state:
z w : ℍ
r R : ℝ
h : re z = re w
h₀ : 0 < im z / im w
⊢ (↑z).re = (↑(center w |log (im z / im w)|)).re

Next tactic:
exact h