awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
z_1 w : ℍ
r : ℝ
this : MetricSpace ℍ := metricSpaceAux
z : ℍ
R : ℝ
hR : 0 < R
h₁ : 1 < R / z.im + 1
h₀ : 0 < R / z.im + 1
⊢ ∃ (i : ℝ), 0 < i ∧ ball z i ⊆ UpperHalfPlane.coe ⁻¹' ball (↑z : ℂ) R
have refine_2 : ball z (log (R / z.im + 1)) ⊆ UpperHalfPlane.coe ⁻¹' ball (↑z : ℂ) R
Tactic: refine ⟨log (R / im z + 1), Real.log_pos h₁, ?_⟩
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
z_1 w : ℍ
r : ℝ
this : MetricSpace ℍ := metricSpaceAux
z : ℍ
R : ℝ
hR : 0 < R
h₁ : 1 < R / z.im + 1
h₀ : 0 < R / z.im + 1
⊢ ball z (log (R / z.im + 1)) ⊆ UpperHalfPlane.coe ⁻¹' ball (↑z : ℂ) R
have refine_2 : ∀ (w : ℍ) (hw : w ∈ ball z (log (R / z.im + 1))), z.im * (rexp (dist w z) - 1) < R
Tactic: refine fun w hw => (dist_coe_le w z).trans_lt ?_
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
z_1 w_1 : ℍ
r : ℝ
this : MetricSpace ℍ := metricSpaceAux
z : ℍ
R : ℝ
hR : 0 < R
h₁ : 1 < R / z.im + 1
h₀ : 0 < R / z.im + 1
w : ℍ
hw : w ∈ ball z (log (R / z.im + 1))
⊢ z.im * (rexp (dist w z) - 1) < R
have refine_2 : rexp (dist w z) - 1 < R / z.im
Tactic: rwa [← lt_div_iff₀' z.im_pos, sub_lt_iff_lt_add, ← Real.lt_log_iff_exp_lt h₀]
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
z_1 w_1 : ℍ
r : ℝ
this : MetricSpace ℍ := metricSpaceAux
z : ℍ
R : ℝ
hR : 0 < R
h₁ : 1 < R / z.im + 1
h₀ : 0 < R / z.im + 1
w : ℍ
hw : w ∈ ball z (log (R / z.im + 1))
refine_2 : rexp (dist w z) - 1 < R / z.im
⊢ z.im * (rexp (dist w z) - 1) < R
have refine_2₁ : rexp (dist w z) < R / z.im + 1
Tactic: rwa [← lt_div_iff₀' z.im_pos, sub_lt_iff_lt_add, ← Real.lt_log_iff_exp_lt h₀]
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
z_1 w_1 : ℍ
r : ℝ
this : MetricSpace ℍ := metricSpaceAux
z : ℍ
R : ℝ
hR : 0 < R
h₁ : 1 < R / z.im + 1
h₀ : 0 < R / z.im + 1
w : ℍ
hw : w ∈ ball z (log (R / z.im + 1))
refine_2 : rexp (dist w z) - 1 < R / z.im
refine_2₁ : rexp (dist w z) < R / z.im + 1
⊢ z.im * (rexp (dist w z) - 1) < R
have refine_2₂ : dist w z < log (R / z.im + 1)
Tactic: rwa [← lt_div_iff₀' z.im_pos, sub_lt_iff_lt_add, ← Real.lt_log_iff_exp_lt h₀]
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
z : ℍ
r : ℝ
⊢ UpperHalfPlane.coe '' closedBall z r = closedBall (↑(z.center r) : ℂ) (z.im * Real.sinh r)
have h : ∀ (w : ℂ), w ∈ UpperHalfPlane.coe '' closedBall z r ↔ w ∈ closedBall (↑(z.center r) : ℂ) (z.im * Real.sinh r)
Tactic: ext w
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
z : ℍ
r : ℝ
w : ℂ
⊢ w ∈ UpperHalfPlane.coe '' closedBall z r ↔ w ∈ closedBall (↑(z.center r) : ℂ) (z.im * Real.sinh r)
have h : w ∈ UpperHalfPlane.coe '' closedBall z r → w ∈ closedBall (↑(z.center r) : ℂ) (z.im * Real.sinh r)
Tactic: constructor
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
z : ℍ
r : ℝ
w : ℂ
h : w ∈ UpperHalfPlane.coe '' closedBall z r → w ∈ closedBall (↑(z.center r) : ℂ) (z.im * Real.sinh r)
⊢ w ∈ UpperHalfPlane.coe '' closedBall z r ↔ w ∈ closedBall (↑(z.center r) : ℂ) (z.im * Real.sinh r)
have h₁ : w ∈ closedBall (↑(z.center r) : ℂ) (z.im * Real.sinh r) → w ∈ UpperHalfPlane.coe '' closedBall z r
Tactic: constructor
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
z : ℍ
r : ℝ
w : ℂ
hw : w ∈ closedBall (↑(z.center r) : ℂ) (z.im * Real.sinh r)
⊢ w ∈ UpperHalfPlane.coe '' closedBall z r
have h : ∀ (w : ℍ) (hw : (↑w : ℂ) ∈ closedBall (↑(z.center r) : ℂ) (z.im * Real.sinh r)),
  (↑w : ℂ) ∈ UpperHalfPlane.coe '' closedBall z r
Tactic: lift w to ℍ using im_pos_of_dist_center_le hw
Cleared hypotheses: w, hw
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
z : ℍ
r : ℝ
⊢ UpperHalfPlane.coe '' ball z r = ball (↑(z.center r) : ℂ) (z.im * Real.sinh r)
have h : ∀ (w : ℂ), w ∈ UpperHalfPlane.coe '' ball z r ↔ w ∈ ball (↑(z.center r) : ℂ) (z.im * Real.sinh r)
Tactic: ext w
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
z : ℍ
r : ℝ
w : ℂ
⊢ w ∈ UpperHalfPlane.coe '' ball z r ↔ w ∈ ball (↑(z.center r) : ℂ) (z.im * Real.sinh r)
have h : w ∈ UpperHalfPlane.coe '' ball z r → w ∈ ball (↑(z.center r) : ℂ) (z.im * Real.sinh r)
Tactic: constructor
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
z : ℍ
r : ℝ
w : ℂ
h : w ∈ UpperHalfPlane.coe '' ball z r → w ∈ ball (↑(z.center r) : ℂ) (z.im * Real.sinh r)
⊢ w ∈ UpperHalfPlane.coe '' ball z r ↔ w ∈ ball (↑(z.center r) : ℂ) (z.im * Real.sinh r)
have h₁ : w ∈ ball (↑(z.center r) : ℂ) (z.im * Real.sinh r) → w ∈ UpperHalfPlane.coe '' ball z r
Tactic: constructor
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
z : ℍ
r : ℝ
w : ℂ
hw : w ∈ ball (↑(z.center r) : ℂ) (z.im * Real.sinh r)
⊢ w ∈ UpperHalfPlane.coe '' ball z r
have h : ∀ (w : ℍ) (hw : (↑w : ℂ) ∈ ball (↑(z.center r) : ℂ) (z.im * Real.sinh r)), (↑w : ℂ) ∈ UpperHalfPlane.coe '' ball z r
Tactic: lift w to ℍ using im_pos_of_dist_center_le (ball_subset_closedBall hw)
Cleared hypotheses: w, hw
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
z : ℍ
r : ℝ
⊢ UpperHalfPlane.coe '' sphere z r = sphere (↑(z.center r) : ℂ) (z.im * Real.sinh r)
have h : ∀ (w : ℂ), w ∈ UpperHalfPlane.coe '' sphere z r ↔ w ∈ sphere (↑(z.center r) : ℂ) (z.im * Real.sinh r)
Tactic: ext w
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
z : ℍ
r : ℝ
w : ℂ
⊢ w ∈ UpperHalfPlane.coe '' sphere z r ↔ w ∈ sphere (↑(z.center r) : ℂ) (z.im * Real.sinh r)
have h : w ∈ UpperHalfPlane.coe '' sphere z r → w ∈ sphere (↑(z.center r) : ℂ) (z.im * Real.sinh r)
Tactic: constructor
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
z : ℍ
r : ℝ
w : ℂ
h : w ∈ UpperHalfPlane.coe '' sphere z r → w ∈ sphere (↑(z.center r) : ℂ) (z.im * Real.sinh r)
⊢ w ∈ UpperHalfPlane.coe '' sphere z r ↔ w ∈ sphere (↑(z.center r) : ℂ) (z.im * Real.sinh r)
have h₁ : w ∈ sphere (↑(z.center r) : ℂ) (z.im * Real.sinh r) → w ∈ UpperHalfPlane.coe '' sphere z r
Tactic: constructor
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
z : ℍ
r : ℝ
w : ℂ
hw : w ∈ sphere (↑(z.center r) : ℂ) (z.im * Real.sinh r)
⊢ w ∈ UpperHalfPlane.coe '' sphere z r
have h : ∀ (w : ℍ) (hw : (↑w : ℂ) ∈ sphere (↑(z.center r) : ℂ) (z.im * Real.sinh r)), (↑w : ℂ) ∈ UpperHalfPlane.coe '' sphere z r
Tactic: lift w to ℍ using im_pos_of_dist_center_le (sphere_subset_closedBall hw)
Cleared hypotheses: w, hw
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
z w : ℍ
r : ℝ
⊢ ProperSpace ℍ
have refine : ∀ (z : ℍ) (r : ℝ), IsCompact (closedBall z r)
Tactic: refine ⟨fun z r => ?_⟩
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
z_1 w : ℍ
r_1 : ℝ
z : ℍ
r : ℝ
⊢ IsCompact (closedBall z r)
have rw : IsCompact (closedBall (↑(z.center r) : ℂ) (z.im * Real.sinh r))
Tactic: rw [isEmbedding_coe.isCompact_iff (f := ((↑) : ℍ → ℂ)), image_coe_closedBall]
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
a : ℝ
⊢ Isometry (β := ℍ) fun (y : ℝ) ↦ { coe := { re := a, im := rexp y }, coe_im_pos := ⋯ }
have refine : ∀ (y₁ y₂ : ℝ),
  dist { coe := { re := a, im := rexp y₁ : ℂ }, coe_im_pos := sorry : ℍ }
      { coe := { re := a, im := rexp y₂ : ℂ }, coe_im_pos := sorry : ℍ } =
    dist y₁ y₂
Tactic: refine Isometry.of_dist_eq fun y₁ y₂ => ?_
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
a y₁ y₂ : ℝ
⊢ dist { coe := { re := a, im := rexp y₁ : ℂ }, coe_im_pos := ⋯ : ℍ }
      { coe := { re := a, im := rexp y₂ : ℂ }, coe_im_pos := ⋯ : ℍ } =
    dist y₁ y₂
have rw : dist (log { coe := { re := a, im := rexp y₁ : ℂ }, coe_im_pos := sorry : ℍ }.im)
    (log { coe := { re := a, im := rexp y₂ : ℂ }, coe_im_pos := sorry : ℍ }.im) =
  dist y₁ y₂
Tactic: rw [dist_of_re_eq]
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
a y₁ y₂ : ℝ
rw :
  dist (log { coe := { re := a, im := rexp y₁ : ℂ }, coe_im_pos := ⋯ : ℍ }.im)
      (log { coe := { re := a, im := rexp y₂ : ℂ }, coe_im_pos := ⋯ : ℍ }.im) =
    dist y₁ y₂
⊢ dist { coe := { re := a, im := rexp y₁ : ℂ }, coe_im_pos := ⋯ : ℍ }
      { coe := { re := a, im := rexp y₂ : ℂ }, coe_im_pos := ⋯ : ℍ } =
    dist y₁ y₂
have rw₁ : { coe := { re := a, im := rexp y₁ : ℂ }, coe_im_pos := sorry : ℍ }.re =
  { coe := { re := a, im := rexp y₂ }, coe_im_pos := sorry }.re
Tactic: rw [dist_of_re_eq]
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
a : { x : ℝ // 0 < x }
⊢ Isometry (β := ℍ) fun (x : ℍ) ↦ a • x
have refine : ∀ (y₁ y₂ : ℍ), dist (a • y₁) (a • y₂) = dist y₁ y₂
Tactic: refine Isometry.of_dist_eq fun y₁ y₂ => ?_
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
a : { x : ℝ // 0 < x }
y₁ y₂ : ℍ
⊢ dist (a • y₁) (a • y₂) = dist y₁ y₂
have simp : 2 * arsinh (dist ((↑a : ℝ) • (↑y₁ : ℂ)) ((↑a : ℝ) • (↑y₂ : ℂ)) / (2 * √((↑a : ℝ) * y₁.im * ((↑a : ℝ) * y₂.im)))) =
  2 * arsinh (dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (2 * √(y₁.im * y₂.im)))
Tactic: simp only [dist_eq, coe_pos_real_smul, pos_real_im]
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
a : { x : ℝ // 0 < x }
y₁ y₂ : ℍ
⊢ 2 * arsinh (dist ((↑a : ℝ) • (↑y₁ : ℂ)) ((↑a : ℝ) • (↑y₂ : ℂ)) / (2 * √((↑a : ℝ) * y₁.im * ((↑a : ℝ) * y₂.im)))) =
    2 * arsinh (dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (2 * √(y₁.im * y₂.im)))
have e_a : dist ((↑a : ℝ) • (↑y₁ : ℂ)) ((↑a : ℝ) • (↑y₂ : ℂ)) / (2 * √((↑a : ℝ) * y₁.im * ((↑a : ℝ) * y₂.im))) =
  dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (2 * √(y₁.im * y₂.im))
Tactic: congr 2
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
a : { x : ℝ // 0 < x }
y₁ y₂ : ℍ
⊢ dist ((↑a : ℝ) • (↑y₁ : ℂ)) ((↑a : ℝ) • (↑y₂ : ℂ)) / (2 * √((↑a : ℝ) * y₁.im * ((↑a : ℝ) * y₂.im))) =
    dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (2 * √(y₁.im * y₂.im))
have e_a : |(↑a : ℝ)| * dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (|(↑a : ℝ)| * (2 * √(y₁.im * y₂.im))) =
  dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (2 * √(y₁.im * y₂.im))
Tactic: rw [dist_smul₀, mul_mul_mul_comm, Real.sqrt_mul (mul_self_nonneg _), Real.sqrt_mul_self_eq_abs,
    Real.norm_eq_abs, mul_left_comm]
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (hypothesis)
Goal:
z w : ℍ
r : ℝ
g : SL(2, ℝ)
y₁ y₂ : ℍ
⊢ dist (ModularGroup.S • y₁) (ModularGroup.S • y₂) = dist y₁ y₂
have h₁ : 0 ≤ y₁.im * y₂.im
Tactic: have h₁ : 0 ≤ im y₁ * im y₂ := by positivity
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (hypothesis)
Goal:
z w : ℍ
r : ℝ
g : SL(2, ℝ)
y₁ y₂ : ℍ
h₁ : 0 ≤ y₁.im * y₂.im
⊢ dist (ModularGroup.S • y₁) (ModularGroup.S • y₂) = dist y₁ y₂
have h₂ : ‖(↑y₁ : ℂ) * (↑y₂ : ℂ)‖ ≠ 0
Tactic: have h₂ : ‖(y₁ * y₂ : ℂ)‖ ≠ 0 := by simp [y₁.ne_zero, y₂.ne_zero]
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
z w : ℍ
r : ℝ
g : SL(2, ℝ)
y₁ y₂ : ℍ
h₁ : 0 ≤ y₁.im * y₂.im
h₂ : ‖(↑y₁ : ℂ) * (↑y₂ : ℂ)‖ ≠ 0
⊢ dist (ModularGroup.S • y₁) (ModularGroup.S • y₂) = dist y₁ y₂
have simp_rw : 2 * arsinh (dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (‖(↑y₁ : ℂ)‖ * ‖(↑y₂ : ℂ)‖) / (2 * √(y₁.im * y₂.im) / ‖(↑y₁ : ℂ) * (↑y₂ : ℂ)‖)) =
  2 * arsinh (dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (2 * √(y₁.im * y₂.im)))
Tactic: simp_rw [modular_S_smul, inv_neg, dist_eq, dist_neg_neg,
          dist_inv_inv₀ y₁.ne_zero y₂.ne_zero, mk_im, neg_im, inv_im, coe_im, neg_div, neg_neg,
          div_mul_div_comm, ← normSq_mul, Real.sqrt_div h₁, ← norm_def, mul_div (2 : ℝ)]
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
z w : ℍ
r : ℝ
g : SL(2, ℝ)
y₁ y₂ : ℍ
h₁ : 0 ≤ y₁.im * y₂.im
h₂ : ‖(↑y₁ : ℂ) * (↑y₂ : ℂ)‖ ≠ 0
⊢ 2 *
      arsinh
        (dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (‖(↑y₁ : ℂ)‖ * ‖(↑y₂ : ℂ)‖) / (2 * √(y₁.im * y₂.im) / ‖(↑y₁ : ℂ) * (↑y₂ : ℂ)‖)) =
    2 * arsinh (dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (2 * √(y₁.im * y₂.im)))
have rw : 2 * arsinh (dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (2 * √(y₁.im * y₂.im)) / (‖(↑y₁ : ℂ)‖ * ‖(↑y₂ : ℂ)‖ / ‖(↑y₁ : ℂ) * (↑y₂ : ℂ)‖)) =
  2 * arsinh (dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (2 * √(y₁.im * y₂.im)))
Tactic: rw [div_div_div_comm, ← norm_mul, div_self h₂, div_one]
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
z w : ℍ
r : ℝ
g : SL(2, ℝ)
y₁ y₂ : ℍ
h₁ : 0 ≤ y₁.im * y₂.im
h₂ : ‖(↑y₁ : ℂ) * (↑y₂ : ℂ)‖ ≠ 0
rw :
  2 *
      arsinh
        (dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (2 * √(y₁.im * y₂.im)) / (‖(↑y₁ : ℂ)‖ * ‖(↑y₂ : ℂ)‖ / ‖(↑y₁ : ℂ) * (↑y₂ : ℂ)‖)) =
    2 * arsinh (dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (2 * √(y₁.im * y₂.im)))
⊢ 2 *
      arsinh
        (dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (‖(↑y₁ : ℂ)‖ * ‖(↑y₂ : ℂ)‖) / (2 * √(y₁.im * y₂.im) / ‖(↑y₁ : ℂ) * (↑y₂ : ℂ)‖)) =
    2 * arsinh (dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (2 * √(y₁.im * y₂.im)))
have rw₁ : 2 * arsinh (dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (2 * √(y₁.im * y₂.im)) / (‖(↑y₁ : ℂ) * (↑y₂ : ℂ)‖ / ‖(↑y₁ : ℂ) * (↑y₂ : ℂ)‖)) =
  2 * arsinh (dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (2 * √(y₁.im * y₂.im)))
Tactic: rw [div_div_div_comm, ← norm_mul, div_self h₂, div_one]
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
z w : ℍ
r : ℝ
g : SL(2, ℝ)
y₁ y₂ : ℍ
h₁ : 0 ≤ y₁.im * y₂.im
h₂ : ‖(↑y₁ : ℂ) * (↑y₂ : ℂ)‖ ≠ 0
rw :
  2 *
      arsinh
        (dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (2 * √(y₁.im * y₂.im)) / (‖(↑y₁ : ℂ)‖ * ‖(↑y₂ : ℂ)‖ / ‖(↑y₁ : ℂ) * (↑y₂ : ℂ)‖)) =
    2 * arsinh (dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (2 * √(y₁.im * y₂.im)))
rw₁ :
  2 * arsinh (dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (2 * √(y₁.im * y₂.im)) / (‖(↑y₁ : ℂ) * (↑y₂ : ℂ)‖ / ‖(↑y₁ : ℂ) * (↑y₂ : ℂ)‖)) =
    2 * arsinh (dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (2 * √(y₁.im * y₂.im)))
⊢ 2 *
      arsinh
        (dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (‖(↑y₁ : ℂ)‖ * ‖(↑y₂ : ℂ)‖) / (2 * √(y₁.im * y₂.im) / ‖(↑y₁ : ℂ) * (↑y₂ : ℂ)‖)) =
    2 * arsinh (dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (2 * √(y₁.im * y₂.im)))
have rw₂ : 2 * arsinh (dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (2 * √(y₁.im * y₂.im)) / 1) =
  2 * arsinh (dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (2 * √(y₁.im * y₂.im)))
Tactic: rw [div_div_div_comm, ← norm_mul, div_self h₂, div_one]
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
z w : ℍ
r : ℝ
g : SL(2, ℝ)
y₁ y₂ : ℍ
h₁ : 0 ≤ y₁.im * y₂.im
h₂ : ‖(↑y₁ : ℂ) * (↑y₂ : ℂ)‖ ≠ 0
rw :
  2 *
      arsinh
        (dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (2 * √(y₁.im * y₂.im)) / (‖(↑y₁ : ℂ)‖ * ‖(↑y₂ : ℂ)‖ / ‖(↑y₁ : ℂ) * (↑y₂ : ℂ)‖)) =
    2 * arsinh (dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (2 * √(y₁.im * y₂.im)))
rw₁ :
  2 * arsinh (dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (2 * √(y₁.im * y₂.im)) / (‖(↑y₁ : ℂ) * (↑y₂ : ℂ)‖ / ‖(↑y₁ : ℂ) * (↑y₂ : ℂ)‖)) =
    2 * arsinh (dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (2 * √(y₁.im * y₂.im)))
rw₂ :
  2 * arsinh (dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (2 * √(y₁.im * y₂.im)) / 1) =
    2 * arsinh (dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (2 * √(y₁.im * y₂.im)))
⊢ 2 *
      arsinh
        (dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (‖(↑y₁ : ℂ)‖ * ‖(↑y₂ : ℂ)‖) / (2 * √(y₁.im * y₂.im) / ‖(↑y₁ : ℂ) * (↑y₂ : ℂ)‖)) =
    2 * arsinh (dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (2 * √(y₁.im * y₂.im)))
have rw₃ : 2 * arsinh (dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (2 * √(y₁.im * y₂.im))) =
  2 * arsinh (dist (↑y₁ : ℂ) (↑y₂ : ℂ) / (2 * √(y₁.im * y₂.im)))
Tactic: rw [div_div_div_comm, ← norm_mul, div_self h₂, div_one]
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (hypothesis)
Goal:
z w : ℍ
r : ℝ
g : SL(2, ℝ)
⊢ Isometry (β := ℍ) fun (x : ℍ) ↦ g • x
have h₀ : Isometry (β := ℍ) fun (z : ℍ) ↦ ModularGroup.S • z
Tactic: have h₀ : Isometry (fun z => ModularGroup.S • z : ℍ → ℍ) :=
      Isometry.of_dist_eq fun y₁ y₂ => by
        have h₁ : 0 ≤ im y₁ * im y₂ := by positivity
        have h₂ : ‖(y₁ * y₂ : ℂ)‖ ≠ 0 := by simp [y₁.ne_zero, y₂.ne_zero]
        simp_rw [modular_S_smul, inv_neg, dist_eq, dist_neg_neg,
          dist_inv_inv₀ y₁.ne_zero y₂.ne_zero, mk_im, neg_im, inv_im, coe_im, neg_div, neg_neg,
          div_mul_div_comm, ← normSq_mul, Real.sqrt_div h₁, ← norm_def, mul_div (2 : ℝ)]
        rw [div_div_div_comm, ← norm_mul, div_self h₂, div_one]
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
z w : ℍ
r : ℝ
g : SL(2, ℝ)
h₀ : Isometry (β := ℍ) fun (z : ℍ) ↦ ModularGroup.S • z
⊢ Isometry (β := ℍ) fun (x : ℍ) ↦ g • x
have pos : ∀ (hc : (↑g : Matrix (Fin 2) (Fin 2) ℝ) 1 0 = 0), Isometry (β := ℍ) fun (x : ℍ) ↦ g • x
Tactic: by_cases hc : g 1 0 = 0
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
z w : ℍ
r : ℝ
g : SL(2, ℝ)
h₀ : Isometry (β := ℍ) fun (z : ℍ) ↦ ModularGroup.S • z
pos : ∀ (hc : (↑g : Matrix (Fin 2) (Fin 2) ℝ) 1 0 = 0), Isometry (β := ℍ) fun (x : ℍ) ↦ g • x
⊢ Isometry (β := ℍ) fun (x : ℍ) ↦ g • x
have neg : ∀ (hc : ¬(↑g : Matrix (Fin 2) (Fin 2) ℝ) 1 0 = 0), Isometry (β := ℍ) fun (x : ℍ) ↦ g • x
Tactic: by_cases hc : g 1 0 = 0
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
z w : ℍ
r : ℝ
g : SL(2, ℝ)
h₀ : Isometry (β := ℍ) fun (z : ℍ) ↦ ModularGroup.S • z
hc : (↑g : Matrix (Fin 2) (Fin 2) ℝ) 1 0 = 0
⊢ Isometry (β := ℍ) fun (x : ℍ) ↦ g • x
have pos : ∀ (u : { x : ℝ // 0 < x }) (v : ℝ)
  (h : Eq (α := ℍ → ℍ) (fun (x : ℍ) ↦ g • x) ((fun (x : ℍ) ↦ v +ᵥ x) ∘ fun (x : ℍ) ↦ u • x)),
  Isometry (β := ℍ) fun (x : ℍ) ↦ g • x
Tactic: obtain ⟨u, v, h⟩ := exists_SL2_smul_eq_of_apply_zero_one_eq_zero g hc
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
z w : ℍ
r : ℝ
g : SL(2, ℝ)
h₀ : Isometry (β := ℍ) fun (z : ℍ) ↦ ModularGroup.S • z
hc : (↑g : Matrix (Fin 2) (Fin 2) ℝ) 1 0 = 0
u : { x : ℝ // 0 < x }
v : ℝ
h : Eq (α := ℍ → ℍ) (fun (x : ℍ) ↦ g • x) ((fun (x : ℍ) ↦ v +ᵥ x) ∘ fun (x : ℍ) ↦ u • x)
⊢ Isometry (β := ℍ) fun (x : ℍ) ↦ g • x
have pos : Isometry (α := ℍ) (β := ℍ) ((fun (x : ℍ) ↦ v +ᵥ x) ∘ fun (x : ℍ) ↦ u • x)
Tactic: rw [h]
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
z w : ℍ
r : ℝ
g : SL(2, ℝ)
h₀ : Isometry (β := ℍ) fun (z : ℍ) ↦ ModularGroup.S • z
hc : ¬(↑g : Matrix (Fin 2) (Fin 2) ℝ) 1 0 = 0
⊢ Isometry (β := ℍ) fun (x : ℍ) ↦ g • x
have neg : ∀ (u : { x : ℝ // 0 < x }) (v w : ℝ)
  (h :
    Eq (α := ℍ → ℍ) (fun (x : ℍ) ↦ g • x)
      ((fun (x : ℍ) ↦ w +ᵥ x) ∘ (fun (x : ℍ) ↦ ModularGroup.S • x) ∘ (fun (x : ℍ) ↦ v +ᵥ x) ∘ fun (x : ℍ) ↦ u • x)),
  Isometry (β := ℍ) fun (x : ℍ) ↦ g • x
Tactic: obtain ⟨u, v, w, h⟩ := exists_SL2_smul_eq_of_apply_zero_one_ne_zero g hc
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.Metric (goal)
Goal:
z w_1 : ℍ
r : ℝ
g : SL(2, ℝ)
h₀ : Isometry (β := ℍ) fun (z : ℍ) ↦ ModularGroup.S • z
hc : ¬(↑g : Matrix (Fin 2) (Fin 2) ℝ) 1 0 = 0
u : { x : ℝ // 0 < x }
v w : ℝ
h :
  Eq (α := ℍ → ℍ) (fun (x : ℍ) ↦ g • x)
    ((fun (x : ℍ) ↦ w +ᵥ x) ∘ (fun (x : ℍ) ↦ ModularGroup.S • x) ∘ (fun (x : ℍ) ↦ v +ᵥ x) ∘ fun (x : ℍ) ↦ u • x)
⊢ Isometry (β := ℍ) fun (x : ℍ) ↦ g • x
have neg : Isometry (α := ℍ) (β := ℍ)
  ((fun (x : ℍ) ↦ w +ᵥ x) ∘ (fun (x : ℍ) ↦ ModularGroup.S • x) ∘ (fun (x : ℍ) ↦ v +ᵥ x) ∘ fun (x : ℍ) ↦ u • x)
Tactic: rw [h]
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction (goal)
Goal:
g : GL (Fin 2) ℝ
z : ℂ
⊢ num (-g) z = -num g z
have simp : -((↑((↑g : Matrix (Fin 2) (Fin 2) ℝ) 0 0) : ℂ) * z) + -(↑((↑g : Matrix (Fin 2) (Fin 2) ℝ) 0 1) : ℂ) =
  -(↑((↑g : Matrix (Fin 2) (Fin 2) ℝ) 0 1) : ℂ) + -((↑((↑g : Matrix (Fin 2) (Fin 2) ℝ) 0 0) : ℂ) * z)
Tactic: simp [num]
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction (goal)
Goal:
g : GL (Fin 2) ℝ
z : ℂ
⊢ denom (-g) z = -denom g z
have simp : -((↑((↑g : Matrix (Fin 2) (Fin 2) ℝ) 1 0) : ℂ) * z) + -(↑((↑g : Matrix (Fin 2) (Fin 2) ℝ) 1 1) : ℂ) =
  -(↑((↑g : Matrix (Fin 2) (Fin 2) ℝ) 1 1) : ℂ) + -((↑((↑g : Matrix (Fin 2) (Fin 2) ℝ) 1 0) : ℂ) * z)
Tactic: simp [denom]
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction (goal)
Goal:
cd : Fin 2 → ℝ
z : ℂ
hz : z.im ≠ 0
h : cd ≠ 0
⊢ (↑(cd 0) : ℂ) * z + (↑(cd 1) : ℂ) ≠ 0
have contrapose : ∀ (h : (↑(cd 0) : ℂ) * z + (↑(cd 1) : ℂ) = 0), cd = 0
Tactic: contrapose h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction (hypothesis)
Goal:
cd : Fin 2 → ℝ
z : ℂ
hz : z.im ≠ 0
h : (↑(cd 0) : ℂ) * z + (↑(cd 1) : ℂ) = 0
⊢ cd 0 = 0
have h₁ : ((↑(cd 0) : ℂ) * z + (↑(cd 1) : ℂ)).im = Complex.im 0
Tactic: apply_fun Complex.im at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction (hypothesis)
Goal:
cd : Fin 2 → ℝ
z : ℂ
hz : z.im ≠ 0
h : (↑(cd 0) : ℂ) * z + (↑(cd 1) : ℂ) = 0
⊢ cd = 0
have this : cd 0 = 0
Tactic: have : cd 0 = 0 := by
    -- we will need this twice
    apply_fun Complex.im at h
    simpa only [Complex.add_im, Complex.mul_im, Complex.ofReal_im, zero_mul, add_zero,
      Complex.zero_im, mul_eq_zero, hz, or_false] using h
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction (hypothesis)
Goal:
cd : Fin 2 → ℝ
z : ℂ
hz : z.im ≠ 0
h : (↑(cd 0) : ℂ) * z + (↑(cd 1) : ℂ) = 0
this : cd 0 = 0
⊢ cd = 0
have h₁ : cd 1 = 0
Tactic: simp only [this, zero_mul, Complex.ofReal_zero, zero_add, Complex.ofReal_eq_zero] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction (goal)
Goal:
cd : Fin 2 → ℝ
z : ℂ
hz : z.im ≠ 0
this : cd 0 = 0
h : cd 1 = 0
⊢ cd = 0
have h₁ : ∀ (i : Fin 2), cd i = 0 i
Tactic: ext i
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction (goal)
Goal:
cd : Fin 2 → ℝ
z : ℂ
hz : z.im ≠ 0
this : cd 0 = 0
h : cd 1 = 0
i : Fin 2
⊢ cd i = 0 i
have h₁ : cd ((fun (i : Fin 2) ↦ i) (⟨0, sorry⟩ : Fin 2)) = 0 ((fun (i : Fin 2) ↦ i) ⟨0, sorry⟩)
Tactic: fin_cases i
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction (goal)
Goal:
cd : Fin 2 → ℝ
z : ℂ
hz : z.im ≠ 0
this : cd 0 = 0
h : cd 1 = 0
i : Fin 2
h₁ : cd ((fun (i : Fin 2) ↦ i) (⟨0, ⋯⟩ : Fin 2)) = 0 ((fun (i : Fin 2) ↦ i) ⟨0, ⋯⟩)
⊢ cd i = 0 i
have h₂ : cd ((fun (i : Fin 2) ↦ i) (⟨1, sorry⟩ : Fin 2)) = 0 ((fun (i : Fin 2) ↦ i) ⟨1, sorry⟩)
Tactic: fin_cases i
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction (goal)
Goal:
g : GL (Fin 2) ℝ
z : ℂ
hz : z.im ≠ 0
⊢ denom g z ≠ 0
have refine : ∀ (H : (↑g : Matrix (Fin 2) (Fin 2) ℝ) 1 = 0),
  Eq (α := ℝ) (↑((GeneralLinearGroup.det : (a : GL (Fin 2) ℝ) → ℝˣ) g) : ℝ) 0
Tactic: refine linear_ne_zero_of_im hz fun H ↦ g.det.ne_zero ?_
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction (goal)
Goal:
g h : GL (Fin 2) ℝ
z : ℂ
hz : z.im ≠ 0
⊢ denom (g * h) z =
    ((↑((↑g : Matrix (Fin 2) (Fin 2) ℝ) 1 0) : ℂ) * (num h z / denom h z) +
        (↑((↑g : Matrix (Fin 2) (Fin 2) ℝ) 1 1) : ℂ)) *
      denom h z
have field_simp : denom (g * h) z =
  (↑((↑g : Matrix (Fin 2) (Fin 2) ℝ) 1 0) : ℂ) * num h z + denom h z * (↑((↑g : Matrix (Fin 2) (Fin 2) ℝ) 1 1) : ℂ)
Tactic: field_simp [denom_ne_zero_of_im h hz]
-- Lean 4 proof step from Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction (goal)
Goal:
g h : GL (Fin 2) ℝ
z : ℂ
hz : z.im ≠ 0
⊢ denom (g * h) z =
    (↑((↑g : Matrix (Fin 2) (Fin 2) ℝ) 1 0) : ℂ) * num h z + denom h z * (↑((↑g : Matrix (Fin 2) (Fin 2) ℝ) 1 1) : ℂ)
have simp : ((↑((↑g : Matrix (Fin 2) (Fin 2) ℝ) 1 0) : ℂ) * (↑((↑h : Matrix (Fin 2) (Fin 2) ℝ) 0 0) : ℂ) +
        (↑((↑g : Matrix (Fin 2) (Fin 2) ℝ) 1 1) : ℂ) * (↑((↑h : Matrix (Fin 2) (Fin 2) ℝ) 1 0) : ℂ)) *
      z +
    ((↑((↑g : Matrix (Fin 2) (Fin 2) ℝ) 1 0) : ℂ) * (↑((↑h : Matrix (Fin 2) (Fin 2) ℝ) 0 1) : ℂ) +
      (↑((↑g : Matrix (Fin 2) (Fin 2) ℝ) 1 1) : ℂ) * (↑((↑h : Matrix (Fin 2) (Fin 2) ℝ) 1 1) : ℂ)) =
  (↑((↑g : Matrix (Fin 2) (Fin 2) ℝ) 1 0) : ℂ) *
      ((↑((↑h : Matrix (Fin 2) (Fin 2) ℝ) 0 0) : ℂ) * z + (↑((↑h : Matrix (Fin 2) (Fin 2) ℝ) 0 1) : ℂ)) +
    ((↑((↑h : Matrix (Fin 2) (Fin 2) ℝ) 1 0) : ℂ) * z + (↑((↑h : Matrix (Fin 2) (Fin 2) ℝ) 1 1) : ℂ)) *
      (↑((↑g : Matrix (Fin 2) (Fin 2) ℝ) 1 1) : ℂ)
Tactic: simp only [denom, Units.val_mul, mul_apply, Fin.sum_univ_succ, Finset.univ_unique,
    Fin.default_eq_zero, Finset.sum_singleton, Fin.succ_zero_eq_one, Complex.ofReal_add,
    Complex.ofReal_mul, num]