l3lab/ntp-mathlib huggingface dataset:
/-- An auxiliary result en route to `ModularGroup.c_eq_zero`. -/
theorem abs_c_le_one (hz : z ∈ 𝒟ᵒ) (hg : g • z ∈ 𝒟ᵒ) : |(↑ₘg) 1 0| ≤ 1 := by
  let c' : ℤ := (↑ₘg) 1 0
  let c : ℝ := (c' : ℝ)
  suffices 3 * c ^ 2 < 4 by
    rw [← Int.cast_pow, ← Int.cast_three, ← Int.cast_four, ← Int.cast_mul, Int.cast_lt] at this
    replace this : c' ^ 2 ≤ 1 ^ 2; · linarith
    rwa [sq_le_sq, abs_one] at this
  suffices c ≠ 0 → 9 * c ^ 4 < 16 by
    rcases eq_or_ne c 0 with (hc | hc)
    · rw [hc]; norm_num
    · refine' (abs_lt_of_sq_lt_sq' _ (by norm_num)).2
      specialize this hc
      linarith
  intro hc
  replace hc : 0 < c ^ 4;
  · change 0 < c ^ (2 * 2); rw [pow_mul]; apply sq_pos_of_pos (sq_pos_of_ne_zero _ hc)
  have h₁ :=
    mul_lt_mul_of_pos_right
      (mul_lt_mul'' (three_lt_four_mul_im_sq_of_mem_fdo hg) (three_lt_four_mul_im_sq_of_mem_fdo hz)
        (by linarith) (by linarith))
      hc
  

Proof state:
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
c' : ℤ := ↑g 1 0
c : ℝ := ↑c'
hc : 0 < c ^ 4
h₁ : 3 * 3 * c ^ 4 < 4 * UpperHalfPlane.im (g • z) ^ 2 * (4 * UpperHalfPlane.im z ^ 2) * c ^ 4
⊢ 9 * c ^ 4 < 16

Next tactic:
have h₂ : (c * z.im) ^ 4 / normSq (denom (↑g) z) ^ 2 ≤ 1 :=
    div_le_one_of_le
      (pow_four_le_pow_two_of_pow_two_le (UpperHalfPlane.c_mul_im_sq_le_normSq_denom z g))
      (sq_nonneg _)
/-- An auxiliary result en route to `ModularGroup.c_eq_zero`. -/
theorem abs_c_le_one (hz : z ∈ 𝒟ᵒ) (hg : g • z ∈ 𝒟ᵒ) : |(↑ₘg) 1 0| ≤ 1 := by
  let c' : ℤ := (↑ₘg) 1 0
  let c : ℝ := (c' : ℝ)
  suffices 3 * c ^ 2 < 4 by
    rw [← Int.cast_pow, ← Int.cast_three, ← Int.cast_four, ← Int.cast_mul, Int.cast_lt] at this
    replace this : c' ^ 2 ≤ 1 ^ 2; · linarith
    rwa [sq_le_sq, abs_one] at this
  suffices c ≠ 0 → 9 * c ^ 4 < 16 by
    rcases eq_or_ne c 0 with (hc | hc)
    · rw [hc]; norm_num
    · refine' (abs_lt_of_sq_lt_sq' _ (by norm_num)).2
      specialize this hc
      linarith
  intro hc
  replace hc : 0 < c ^ 4;
  · change 0 < c ^ (2 * 2); rw [pow_mul]; apply sq_pos_of_pos (sq_pos_of_ne_zero _ hc)
  have h₁ :=
    mul_lt_mul_of_pos_right
      (mul_lt_mul'' (three_lt_four_mul_im_sq_of_mem_fdo hg) (three_lt_four_mul_im_sq_of_mem_fdo hz)
        (by linarith) (by linarith))
      hc
  have h₂ : (c * z.im) ^ 4 / normSq (denom (↑g) z) ^ 2 ≤ 1 :=
    div_le_one_of_le
      (pow_four_le_pow_two_of_pow_two_le (UpperHalfPlane.c_mul_im_sq_le_normSq_denom z g))
      (sq_nonneg _)
  

Proof state:
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
c' : ℤ := ↑g 1 0
c : ℝ := ↑c'
hc : 0 < c ^ 4
h₁ : 3 * 3 * c ^ 4 < 4 * UpperHalfPlane.im (g • z) ^ 2 * (4 * UpperHalfPlane.im z ^ 2) * c ^ 4
h₂ : (c * UpperHalfPlane.im z) ^ 4 / normSq (denom (↑g) z) ^ 2 ≤ 1
⊢ 9 * c ^ 4 < 16

Next tactic:
let nsq := normSq (denom g z)
/-- An auxiliary result en route to `ModularGroup.c_eq_zero`. -/
theorem abs_c_le_one (hz : z ∈ 𝒟ᵒ) (hg : g • z ∈ 𝒟ᵒ) : |(↑ₘg) 1 0| ≤ 1 := by
  let c' : ℤ := (↑ₘg) 1 0
  let c : ℝ := (c' : ℝ)
  suffices 3 * c ^ 2 < 4 by
    rw [← Int.cast_pow, ← Int.cast_three, ← Int.cast_four, ← Int.cast_mul, Int.cast_lt] at this
    replace this : c' ^ 2 ≤ 1 ^ 2; · linarith
    rwa [sq_le_sq, abs_one] at this
  suffices c ≠ 0 → 9 * c ^ 4 < 16 by
    rcases eq_or_ne c 0 with (hc | hc)
    · rw [hc]; norm_num
    · refine' (abs_lt_of_sq_lt_sq' _ (by norm_num)).2
      specialize this hc
      linarith
  intro hc
  replace hc : 0 < c ^ 4;
  · change 0 < c ^ (2 * 2); rw [pow_mul]; apply sq_pos_of_pos (sq_pos_of_ne_zero _ hc)
  have h₁ :=
    mul_lt_mul_of_pos_right
      (mul_lt_mul'' (three_lt_four_mul_im_sq_of_mem_fdo hg) (three_lt_four_mul_im_sq_of_mem_fdo hz)
        (by linarith) (by linarith))
      hc
  have h₂ : (c * z.im) ^ 4 / normSq (denom (↑g) z) ^ 2 ≤ 1 :=
    div_le_one_of_le
      (pow_four_le_pow_two_of_pow_two_le (UpperHalfPlane.c_mul_im_sq_le_normSq_denom z g))
      (sq_nonneg _)
  let nsq := normSq (denom g z)
  

Proof state:
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
c' : ℤ := ↑g 1 0
c : ℝ := ↑c'
hc : 0 < c ^ 4
h₁ : 3 * 3 * c ^ 4 < 4 * UpperHalfPlane.im (g • z) ^ 2 * (4 * UpperHalfPlane.im z ^ 2) * c ^ 4
h₂ : (c * UpperHalfPlane.im z) ^ 4 / normSq (denom (↑g) z) ^ 2 ≤ 1
nsq : ℝ := normSq (denom (↑g) z)
⊢ 9 * c ^ 4 < 16

Next tactic:
calc
    9 * c ^ 4 < c ^ 4 * z.im ^ 2 * (g • z).im ^ 2 * 16 := by linarith
    _ = c ^ 4 * z.im ^ 4 / nsq ^ 2 * 16 := by
      rw [SpecialLinearGroup.im_smul_eq_div_normSq, div_pow]
      ring
    _ ≤ 16 := by rw [← mul_pow]; linarith
/-- An auxiliary result en route to `ModularGroup.c_eq_zero`. -/
theorem abs_c_le_one (hz : z ∈ 𝒟ᵒ) (hg : g • z ∈ 𝒟ᵒ) : |(↑ₘg) 1 0| ≤ 1 := by
  let c' : ℤ := (↑ₘg) 1 0
  let c : ℝ := (c' : ℝ)
  suffices 3 * c ^ 2 < 4 by
    rw [← Int.cast_pow, ← Int.cast_three, ← Int.cast_four, ← Int.cast_mul, Int.cast_lt] at this
    replace this : c' ^ 2 ≤ 1 ^ 2; · linarith
    rwa [sq_le_sq, abs_one] at this
  suffices c ≠ 0 → 9 * c ^ 4 < 16 by
    rcases eq_or_ne c 0 with (hc | hc)
    · rw [hc]; norm_num
    · refine' (abs_lt_of_sq_lt_sq' _ (by norm_num)).2
      specialize this hc
      linarith
  intro hc
  replace hc : 0 < c ^ 4;
  · change 0 < c ^ (2 * 2); rw [pow_mul]; apply sq_pos_of_pos (sq_pos_of_ne_zero _ hc)
  have h₁ :=
    mul_lt_mul_of_pos_right
      (mul_lt_mul'' (three_lt_four_mul_im_sq_of_mem_fdo hg) (three_lt_four_mul_im_sq_of_mem_fdo hz)
        (by linarith) (by linarith))
      hc
  have h₂ : (c * z.im) ^ 4 / normSq (denom (↑g) z) ^ 2 ≤ 1 :=
    div_le_one_of_le
      (pow_four_le_pow_two_of_pow_two_le (UpperHalfPlane.c_mul_im_sq_le_normSq_denom z g))
      (sq_nonneg _)
  let nsq := normSq (denom g z)
  calc
    9 * c ^ 4 < c ^ 4 * z.im ^ 2 * (g • z).im ^ 2 * 16 := by 

Proof state:
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
c' : ℤ := ↑g 1 0
c : ℝ := ↑c'
hc : 0 < c ^ 4
h₁ : 3 * 3 * c ^ 4 < 4 * UpperHalfPlane.im (g • z) ^ 2 * (4 * UpperHalfPlane.im z ^ 2) * c ^ 4
h₂ : (c * UpperHalfPlane.im z) ^ 4 / normSq (denom (↑g) z) ^ 2 ≤ 1
nsq : ℝ := normSq (denom (↑g) z)
⊢ 9 * c ^ 4 < c ^ 4 * UpperHalfPlane.im z ^ 2 * UpperHalfPlane.im (g • z) ^ 2 * 16

Next tactic:
linarith
/-- An auxiliary result en route to `ModularGroup.c_eq_zero`. -/
theorem abs_c_le_one (hz : z ∈ 𝒟ᵒ) (hg : g • z ∈ 𝒟ᵒ) : |(↑ₘg) 1 0| ≤ 1 := by
  let c' : ℤ := (↑ₘg) 1 0
  let c : ℝ := (c' : ℝ)
  suffices 3 * c ^ 2 < 4 by
    rw [← Int.cast_pow, ← Int.cast_three, ← Int.cast_four, ← Int.cast_mul, Int.cast_lt] at this
    replace this : c' ^ 2 ≤ 1 ^ 2; · linarith
    rwa [sq_le_sq, abs_one] at this
  suffices c ≠ 0 → 9 * c ^ 4 < 16 by
    rcases eq_or_ne c 0 with (hc | hc)
    · rw [hc]; norm_num
    · refine' (abs_lt_of_sq_lt_sq' _ (by norm_num)).2
      specialize this hc
      linarith
  intro hc
  replace hc : 0 < c ^ 4;
  · change 0 < c ^ (2 * 2); rw [pow_mul]; apply sq_pos_of_pos (sq_pos_of_ne_zero _ hc)
  have h₁ :=
    mul_lt_mul_of_pos_right
      (mul_lt_mul'' (three_lt_four_mul_im_sq_of_mem_fdo hg) (three_lt_four_mul_im_sq_of_mem_fdo hz)
        (by linarith) (by linarith))
      hc
  have h₂ : (c * z.im) ^ 4 / normSq (denom (↑g) z) ^ 2 ≤ 1 :=
    div_le_one_of_le
      (pow_four_le_pow_two_of_pow_two_le (UpperHalfPlane.c_mul_im_sq_le_normSq_denom z g))
      (sq_nonneg _)
  let nsq := normSq (denom g z)
  calc
    9 * c ^ 4 < c ^ 4 * z.im ^ 2 * (g • z).im ^ 2 * 16 := by linarith
    _ = c ^ 4 * z.im ^ 4 / nsq ^ 2 * 16 := by
      

Proof state:
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
c' : ℤ := ↑g 1 0
c : ℝ := ↑c'
hc : 0 < c ^ 4
h₁ : 3 * 3 * c ^ 4 < 4 * UpperHalfPlane.im (g • z) ^ 2 * (4 * UpperHalfPlane.im z ^ 2) * c ^ 4
h₂ : (c * UpperHalfPlane.im z) ^ 4 / normSq (denom (↑g) z) ^ 2 ≤ 1
nsq : ℝ := normSq (denom (↑g) z)
⊢ c ^ 4 * UpperHalfPlane.im z ^ 2 * UpperHalfPlane.im (g • z) ^ 2 * 16 = c ^ 4 * UpperHalfPlane.im z ^ 4 / nsq ^ 2 * 16

Next tactic:
rw [SpecialLinearGroup.im_smul_eq_div_normSq, div_pow]
/-- An auxiliary result en route to `ModularGroup.c_eq_zero`. -/
theorem abs_c_le_one (hz : z ∈ 𝒟ᵒ) (hg : g • z ∈ 𝒟ᵒ) : |(↑ₘg) 1 0| ≤ 1 := by
  let c' : ℤ := (↑ₘg) 1 0
  let c : ℝ := (c' : ℝ)
  suffices 3 * c ^ 2 < 4 by
    rw [← Int.cast_pow, ← Int.cast_three, ← Int.cast_four, ← Int.cast_mul, Int.cast_lt] at this
    replace this : c' ^ 2 ≤ 1 ^ 2; · linarith
    rwa [sq_le_sq, abs_one] at this
  suffices c ≠ 0 → 9 * c ^ 4 < 16 by
    rcases eq_or_ne c 0 with (hc | hc)
    · rw [hc]; norm_num
    · refine' (abs_lt_of_sq_lt_sq' _ (by norm_num)).2
      specialize this hc
      linarith
  intro hc
  replace hc : 0 < c ^ 4;
  · change 0 < c ^ (2 * 2); rw [pow_mul]; apply sq_pos_of_pos (sq_pos_of_ne_zero _ hc)
  have h₁ :=
    mul_lt_mul_of_pos_right
      (mul_lt_mul'' (three_lt_four_mul_im_sq_of_mem_fdo hg) (three_lt_four_mul_im_sq_of_mem_fdo hz)
        (by linarith) (by linarith))
      hc
  have h₂ : (c * z.im) ^ 4 / normSq (denom (↑g) z) ^ 2 ≤ 1 :=
    div_le_one_of_le
      (pow_four_le_pow_two_of_pow_two_le (UpperHalfPlane.c_mul_im_sq_le_normSq_denom z g))
      (sq_nonneg _)
  let nsq := normSq (denom g z)
  calc
    9 * c ^ 4 < c ^ 4 * z.im ^ 2 * (g • z).im ^ 2 * 16 := by linarith
    _ = c ^ 4 * z.im ^ 4 / nsq ^ 2 * 16 := by
      rw [SpecialLinearGroup.im_smul_eq_div_normSq, div_pow]
      

Proof state:
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
c' : ℤ := ↑g 1 0
c : ℝ := ↑c'
hc : 0 < c ^ 4
h₁ : 3 * 3 * c ^ 4 < 4 * UpperHalfPlane.im (g • z) ^ 2 * (4 * UpperHalfPlane.im z ^ 2) * c ^ 4
h₂ : (c * UpperHalfPlane.im z) ^ 4 / normSq (denom (↑g) z) ^ 2 ≤ 1
nsq : ℝ := normSq (denom (↑g) z)
⊢ c ^ 4 * UpperHalfPlane.im z ^ 2 * (UpperHalfPlane.im z ^ 2 / normSq (denom (↑g) z) ^ 2) * 16 =
    c ^ 4 * UpperHalfPlane.im z ^ 4 / nsq ^ 2 * 16

Next tactic:
ring
/-- An auxiliary result en route to `ModularGroup.c_eq_zero`. -/
theorem abs_c_le_one (hz : z ∈ 𝒟ᵒ) (hg : g • z ∈ 𝒟ᵒ) : |(↑ₘg) 1 0| ≤ 1 := by
  let c' : ℤ := (↑ₘg) 1 0
  let c : ℝ := (c' : ℝ)
  suffices 3 * c ^ 2 < 4 by
    rw [← Int.cast_pow, ← Int.cast_three, ← Int.cast_four, ← Int.cast_mul, Int.cast_lt] at this
    replace this : c' ^ 2 ≤ 1 ^ 2; · linarith
    rwa [sq_le_sq, abs_one] at this
  suffices c ≠ 0 → 9 * c ^ 4 < 16 by
    rcases eq_or_ne c 0 with (hc | hc)
    · rw [hc]; norm_num
    · refine' (abs_lt_of_sq_lt_sq' _ (by norm_num)).2
      specialize this hc
      linarith
  intro hc
  replace hc : 0 < c ^ 4;
  · change 0 < c ^ (2 * 2); rw [pow_mul]; apply sq_pos_of_pos (sq_pos_of_ne_zero _ hc)
  have h₁ :=
    mul_lt_mul_of_pos_right
      (mul_lt_mul'' (three_lt_four_mul_im_sq_of_mem_fdo hg) (three_lt_four_mul_im_sq_of_mem_fdo hz)
        (by linarith) (by linarith))
      hc
  have h₂ : (c * z.im) ^ 4 / normSq (denom (↑g) z) ^ 2 ≤ 1 :=
    div_le_one_of_le
      (pow_four_le_pow_two_of_pow_two_le (UpperHalfPlane.c_mul_im_sq_le_normSq_denom z g))
      (sq_nonneg _)
  let nsq := normSq (denom g z)
  calc
    9 * c ^ 4 < c ^ 4 * z.im ^ 2 * (g • z).im ^ 2 * 16 := by linarith
    _ = c ^ 4 * z.im ^ 4 / nsq ^ 2 * 16 := by
      rw [SpecialLinearGroup.im_smul_eq_div_normSq, div_pow]
      ring
    _ ≤ 16 := by 

Proof state:
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
c' : ℤ := ↑g 1 0
c : ℝ := ↑c'
hc : 0 < c ^ 4
h₁ : 3 * 3 * c ^ 4 < 4 * UpperHalfPlane.im (g • z) ^ 2 * (4 * UpperHalfPlane.im z ^ 2) * c ^ 4
h₂ : (c * UpperHalfPlane.im z) ^ 4 / normSq (denom (↑g) z) ^ 2 ≤ 1
nsq : ℝ := normSq (denom (↑g) z)
⊢ c ^ 4 * UpperHalfPlane.im z ^ 4 / nsq ^ 2 * 16 ≤ 16

Next tactic:
rw [← mul_pow]
/-- An auxiliary result en route to `ModularGroup.c_eq_zero`. -/
theorem abs_c_le_one (hz : z ∈ 𝒟ᵒ) (hg : g • z ∈ 𝒟ᵒ) : |(↑ₘg) 1 0| ≤ 1 := by
  let c' : ℤ := (↑ₘg) 1 0
  let c : ℝ := (c' : ℝ)
  suffices 3 * c ^ 2 < 4 by
    rw [← Int.cast_pow, ← Int.cast_three, ← Int.cast_four, ← Int.cast_mul, Int.cast_lt] at this
    replace this : c' ^ 2 ≤ 1 ^ 2; · linarith
    rwa [sq_le_sq, abs_one] at this
  suffices c ≠ 0 → 9 * c ^ 4 < 16 by
    rcases eq_or_ne c 0 with (hc | hc)
    · rw [hc]; norm_num
    · refine' (abs_lt_of_sq_lt_sq' _ (by norm_num)).2
      specialize this hc
      linarith
  intro hc
  replace hc : 0 < c ^ 4;
  · change 0 < c ^ (2 * 2); rw [pow_mul]; apply sq_pos_of_pos (sq_pos_of_ne_zero _ hc)
  have h₁ :=
    mul_lt_mul_of_pos_right
      (mul_lt_mul'' (three_lt_four_mul_im_sq_of_mem_fdo hg) (three_lt_four_mul_im_sq_of_mem_fdo hz)
        (by linarith) (by linarith))
      hc
  have h₂ : (c * z.im) ^ 4 / normSq (denom (↑g) z) ^ 2 ≤ 1 :=
    div_le_one_of_le
      (pow_four_le_pow_two_of_pow_two_le (UpperHalfPlane.c_mul_im_sq_le_normSq_denom z g))
      (sq_nonneg _)
  let nsq := normSq (denom g z)
  calc
    9 * c ^ 4 < c ^ 4 * z.im ^ 2 * (g • z).im ^ 2 * 16 := by linarith
    _ = c ^ 4 * z.im ^ 4 / nsq ^ 2 * 16 := by
      rw [SpecialLinearGroup.im_smul_eq_div_normSq, div_pow]
      ring
    _ ≤ 16 := by rw [← mul_pow]; 

Proof state:
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
c' : ℤ := ↑g 1 0
c : ℝ := ↑c'
hc : 0 < c ^ 4
h₁ : 3 * 3 * c ^ 4 < 4 * UpperHalfPlane.im (g • z) ^ 2 * (4 * UpperHalfPlane.im z ^ 2) * c ^ 4
h₂ : (c * UpperHalfPlane.im z) ^ 4 / normSq (denom (↑g) z) ^ 2 ≤ 1
nsq : ℝ := normSq (denom (↑g) z)
⊢ (c * UpperHalfPlane.im z) ^ 4 / nsq ^ 2 * 16 ≤ 16

Next tactic:
linarith
/-- An auxiliary result en route to `ModularGroup.eq_smul_self_of_mem_fdo_mem_fdo`. -/
theorem c_eq_zero (hz : z ∈ 𝒟ᵒ) (hg : g • z ∈ 𝒟ᵒ) : (↑ₘg) 1 0 = 0 := by
  

Proof state:
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
⊢ ↑g 1 0 = 0

Next tactic:
have hp : ∀ {g' : SL(2, ℤ)}, g' • z ∈ 𝒟ᵒ → (↑ₘg') 1 0 ≠ 1 := by
    intro g' hg'
    by_contra hc
    let a := (↑ₘg') 0 0
    let d := (↑ₘg') 1 1
    have had : T ^ (-a) * g' = S * T ^ d := by rw [g_eq_of_c_eq_one hc]; group
    let w := T ^ (-a) • g' • z
    have h₁ : w = S • T ^ d • z := by simp only [← mul_smul, had]
    replace h₁ : normSq w < 1 := h₁.symm ▸ normSq_S_smul_lt_one (one_lt_normSq_T_zpow_smul hz d)
    have h₂ : 1 < normSq w := one_lt_normSq_T_zpow_smul hg' (-a)
    linarith
/-- An auxiliary result en route to `ModularGroup.eq_smul_self_of_mem_fdo_mem_fdo`. -/
theorem c_eq_zero (hz : z ∈ 𝒟ᵒ) (hg : g • z ∈ 𝒟ᵒ) : (↑ₘg) 1 0 = 0 := by
  have hp : ∀ {g' : SL(2, ℤ)}, g' • z ∈ 𝒟ᵒ → (↑ₘg') 1 0 ≠ 1 := by
    

Proof state:
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
⊢ ∀ {g' : SL(2, ℤ)}, g' • z ∈ 𝒟ᵒ → ↑g' 1 0 ≠ 1

Next tactic:
intro g' hg'
/-- An auxiliary result en route to `ModularGroup.eq_smul_self_of_mem_fdo_mem_fdo`. -/
theorem c_eq_zero (hz : z ∈ 𝒟ᵒ) (hg : g • z ∈ 𝒟ᵒ) : (↑ₘg) 1 0 = 0 := by
  have hp : ∀ {g' : SL(2, ℤ)}, g' • z ∈ 𝒟ᵒ → (↑ₘg') 1 0 ≠ 1 := by
    intro g' hg'
    

Proof state:
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
g' : SL(2, ℤ)
hg' : g' • z ∈ 𝒟ᵒ
⊢ ↑g' 1 0 ≠ 1

Next tactic:
by_contra hc
/-- An auxiliary result en route to `ModularGroup.eq_smul_self_of_mem_fdo_mem_fdo`. -/
theorem c_eq_zero (hz : z ∈ 𝒟ᵒ) (hg : g • z ∈ 𝒟ᵒ) : (↑ₘg) 1 0 = 0 := by
  have hp : ∀ {g' : SL(2, ℤ)}, g' • z ∈ 𝒟ᵒ → (↑ₘg') 1 0 ≠ 1 := by
    intro g' hg'
    by_contra hc
    

Proof state:
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
g' : SL(2, ℤ)
hg' : g' • z ∈ 𝒟ᵒ
hc : ↑g' 1 0 = 1
⊢ False

Next tactic:
let a := (↑ₘg') 0 0
/-- An auxiliary result en route to `ModularGroup.eq_smul_self_of_mem_fdo_mem_fdo`. -/
theorem c_eq_zero (hz : z ∈ 𝒟ᵒ) (hg : g • z ∈ 𝒟ᵒ) : (↑ₘg) 1 0 = 0 := by
  have hp : ∀ {g' : SL(2, ℤ)}, g' • z ∈ 𝒟ᵒ → (↑ₘg') 1 0 ≠ 1 := by
    intro g' hg'
    by_contra hc
    let a := (↑ₘg') 0 0
    

Proof state:
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
g' : SL(2, ℤ)
hg' : g' • z ∈ 𝒟ᵒ
hc : ↑g' 1 0 = 1
a : ℤ := ↑g' 0 0
⊢ False

Next tactic:
let d := (↑ₘg') 1 1
/-- An auxiliary result en route to `ModularGroup.eq_smul_self_of_mem_fdo_mem_fdo`. -/
theorem c_eq_zero (hz : z ∈ 𝒟ᵒ) (hg : g • z ∈ 𝒟ᵒ) : (↑ₘg) 1 0 = 0 := by
  have hp : ∀ {g' : SL(2, ℤ)}, g' • z ∈ 𝒟ᵒ → (↑ₘg') 1 0 ≠ 1 := by
    intro g' hg'
    by_contra hc
    let a := (↑ₘg') 0 0
    let d := (↑ₘg') 1 1
    

Proof state:
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
g' : SL(2, ℤ)
hg' : g' • z ∈ 𝒟ᵒ
hc : ↑g' 1 0 = 1
a : ℤ := ↑g' 0 0
d : ℤ := ↑g' 1 1
⊢ False

Next tactic:
have had : T ^ (-a) * g' = S * T ^ d := by rw [g_eq_of_c_eq_one hc]; group
/-- An auxiliary result en route to `ModularGroup.eq_smul_self_of_mem_fdo_mem_fdo`. -/
theorem c_eq_zero (hz : z ∈ 𝒟ᵒ) (hg : g • z ∈ 𝒟ᵒ) : (↑ₘg) 1 0 = 0 := by
  have hp : ∀ {g' : SL(2, ℤ)}, g' • z ∈ 𝒟ᵒ → (↑ₘg') 1 0 ≠ 1 := by
    intro g' hg'
    by_contra hc
    let a := (↑ₘg') 0 0
    let d := (↑ₘg') 1 1
    have had : T ^ (-a) * g' = S * T ^ d := by 

Proof state:
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
g' : SL(2, ℤ)
hg' : g' • z ∈ 𝒟ᵒ
hc : ↑g' 1 0 = 1
a : ℤ := ↑g' 0 0
d : ℤ := ↑g' 1 1
⊢ T ^ (-a) * g' = S * T ^ d

Next tactic:
rw [g_eq_of_c_eq_one hc]
/-- An auxiliary result en route to `ModularGroup.eq_smul_self_of_mem_fdo_mem_fdo`. -/
theorem c_eq_zero (hz : z ∈ 𝒟ᵒ) (hg : g • z ∈ 𝒟ᵒ) : (↑ₘg) 1 0 = 0 := by
  have hp : ∀ {g' : SL(2, ℤ)}, g' • z ∈ 𝒟ᵒ → (↑ₘg') 1 0 ≠ 1 := by
    intro g' hg'
    by_contra hc
    let a := (↑ₘg') 0 0
    let d := (↑ₘg') 1 1
    have had : T ^ (-a) * g' = S * T ^ d := by rw [g_eq_of_c_eq_one hc]; 

Proof state:
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
g' : SL(2, ℤ)
hg' : g' • z ∈ 𝒟ᵒ
hc : ↑g' 1 0 = 1
a : ℤ := ↑g' 0 0
d : ℤ := ↑g' 1 1
⊢ T ^ (-a) * (T ^ ↑g' 0 0 * S * T ^ ↑g' 1 1) = S * T ^ d

Next tactic:
group
/-- An auxiliary result en route to `ModularGroup.eq_smul_self_of_mem_fdo_mem_fdo`. -/
theorem c_eq_zero (hz : z ∈ 𝒟ᵒ) (hg : g • z ∈ 𝒟ᵒ) : (↑ₘg) 1 0 = 0 := by
  have hp : ∀ {g' : SL(2, ℤ)}, g' • z ∈ 𝒟ᵒ → (↑ₘg') 1 0 ≠ 1 := by
    intro g' hg'
    by_contra hc
    let a := (↑ₘg') 0 0
    let d := (↑ₘg') 1 1
    have had : T ^ (-a) * g' = S * T ^ d := by rw [g_eq_of_c_eq_one hc]; group
    

Proof state:
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
g' : SL(2, ℤ)
hg' : g' • z ∈ 𝒟ᵒ
hc : ↑g' 1 0 = 1
a : ℤ := ↑g' 0 0
d : ℤ := ↑g' 1 1
had : T ^ (-a) * g' = S * T ^ d
⊢ False

Next tactic:
let w := T ^ (-a) • g' • z
/-- An auxiliary result en route to `ModularGroup.eq_smul_self_of_mem_fdo_mem_fdo`. -/
theorem c_eq_zero (hz : z ∈ 𝒟ᵒ) (hg : g • z ∈ 𝒟ᵒ) : (↑ₘg) 1 0 = 0 := by
  have hp : ∀ {g' : SL(2, ℤ)}, g' • z ∈ 𝒟ᵒ → (↑ₘg') 1 0 ≠ 1 := by
    intro g' hg'
    by_contra hc
    let a := (↑ₘg') 0 0
    let d := (↑ₘg') 1 1
    have had : T ^ (-a) * g' = S * T ^ d := by rw [g_eq_of_c_eq_one hc]; group
    let w := T ^ (-a) • g' • z
    

Proof state:
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
g' : SL(2, ℤ)
hg' : g' • z ∈ 𝒟ᵒ
hc : ↑g' 1 0 = 1
a : ℤ := ↑g' 0 0
d : ℤ := ↑g' 1 1
had : T ^ (-a) * g' = S * T ^ d
w : ℍ := T ^ (-a) • g' • z
⊢ False

Next tactic:
have h₁ : w = S • T ^ d • z := by simp only [← mul_smul, had]
/-- An auxiliary result en route to `ModularGroup.eq_smul_self_of_mem_fdo_mem_fdo`. -/
theorem c_eq_zero (hz : z ∈ 𝒟ᵒ) (hg : g • z ∈ 𝒟ᵒ) : (↑ₘg) 1 0 = 0 := by
  have hp : ∀ {g' : SL(2, ℤ)}, g' • z ∈ 𝒟ᵒ → (↑ₘg') 1 0 ≠ 1 := by
    intro g' hg'
    by_contra hc
    let a := (↑ₘg') 0 0
    let d := (↑ₘg') 1 1
    have had : T ^ (-a) * g' = S * T ^ d := by rw [g_eq_of_c_eq_one hc]; group
    let w := T ^ (-a) • g' • z
    have h₁ : w = S • T ^ d • z := by 

Proof state:
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
g' : SL(2, ℤ)
hg' : g' • z ∈ 𝒟ᵒ
hc : ↑g' 1 0 = 1
a : ℤ := ↑g' 0 0
d : ℤ := ↑g' 1 1
had : T ^ (-a) * g' = S * T ^ d
w : ℍ := T ^ (-a) • g' • z
⊢ w = S • T ^ d • z

Next tactic:
simp only [← mul_smul, had]
/-- An auxiliary result en route to `ModularGroup.eq_smul_self_of_mem_fdo_mem_fdo`. -/
theorem c_eq_zero (hz : z ∈ 𝒟ᵒ) (hg : g • z ∈ 𝒟ᵒ) : (↑ₘg) 1 0 = 0 := by
  have hp : ∀ {g' : SL(2, ℤ)}, g' • z ∈ 𝒟ᵒ → (↑ₘg') 1 0 ≠ 1 := by
    intro g' hg'
    by_contra hc
    let a := (↑ₘg') 0 0
    let d := (↑ₘg') 1 1
    have had : T ^ (-a) * g' = S * T ^ d := by rw [g_eq_of_c_eq_one hc]; group
    let w := T ^ (-a) • g' • z
    have h₁ : w = S • T ^ d • z := by simp only [← mul_smul, had]
    

Proof state:
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
g' : SL(2, ℤ)
hg' : g' • z ∈ 𝒟ᵒ
hc : ↑g' 1 0 = 1
a : ℤ := ↑g' 0 0
d : ℤ := ↑g' 1 1
had : T ^ (-a) * g' = S * T ^ d
w : ℍ := T ^ (-a) • g' • z
h₁ : w = S • T ^ d • z
⊢ False

Next tactic:
replace h₁ : normSq w < 1 := h₁.symm ▸ normSq_S_smul_lt_one (one_lt_normSq_T_zpow_smul hz d)
/-- An auxiliary result en route to `ModularGroup.eq_smul_self_of_mem_fdo_mem_fdo`. -/
theorem c_eq_zero (hz : z ∈ 𝒟ᵒ) (hg : g • z ∈ 𝒟ᵒ) : (↑ₘg) 1 0 = 0 := by
  have hp : ∀ {g' : SL(2, ℤ)}, g' • z ∈ 𝒟ᵒ → (↑ₘg') 1 0 ≠ 1 := by
    intro g' hg'
    by_contra hc
    let a := (↑ₘg') 0 0
    let d := (↑ₘg') 1 1
    have had : T ^ (-a) * g' = S * T ^ d := by rw [g_eq_of_c_eq_one hc]; group
    let w := T ^ (-a) • g' • z
    have h₁ : w = S • T ^ d • z := by simp only [← mul_smul, had]
    replace h₁ : normSq w < 1 := h₁.symm ▸ normSq_S_smul_lt_one (one_lt_normSq_T_zpow_smul hz d)
    

Proof state:
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
g' : SL(2, ℤ)
hg' : g' • z ∈ 𝒟ᵒ
hc : ↑g' 1 0 = 1
a : ℤ := ↑g' 0 0
d : ℤ := ↑g' 1 1
had : T ^ (-a) * g' = S * T ^ d
w : ℍ := T ^ (-a) • g' • z
h₁ : normSq ↑w < 1
⊢ False

Next tactic:
have h₂ : 1 < normSq w := one_lt_normSq_T_zpow_smul hg' (-a)
/-- An auxiliary result en route to `ModularGroup.eq_smul_self_of_mem_fdo_mem_fdo`. -/
theorem c_eq_zero (hz : z ∈ 𝒟ᵒ) (hg : g • z ∈ 𝒟ᵒ) : (↑ₘg) 1 0 = 0 := by
  have hp : ∀ {g' : SL(2, ℤ)}, g' • z ∈ 𝒟ᵒ → (↑ₘg') 1 0 ≠ 1 := by
    intro g' hg'
    by_contra hc
    let a := (↑ₘg') 0 0
    let d := (↑ₘg') 1 1
    have had : T ^ (-a) * g' = S * T ^ d := by rw [g_eq_of_c_eq_one hc]; group
    let w := T ^ (-a) • g' • z
    have h₁ : w = S • T ^ d • z := by simp only [← mul_smul, had]
    replace h₁ : normSq w < 1 := h₁.symm ▸ normSq_S_smul_lt_one (one_lt_normSq_T_zpow_smul hz d)
    have h₂ : 1 < normSq w := one_lt_normSq_T_zpow_smul hg' (-a)
    

Proof state:
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
g' : SL(2, ℤ)
hg' : g' • z ∈ 𝒟ᵒ
hc : ↑g' 1 0 = 1
a : ℤ := ↑g' 0 0
d : ℤ := ↑g' 1 1
had : T ^ (-a) * g' = S * T ^ d
w : ℍ := T ^ (-a) • g' • z
h₁ : normSq ↑w < 1
h₂ : 1 < normSq ↑w
⊢ False

Next tactic:
linarith
/-- An auxiliary result en route to `ModularGroup.eq_smul_self_of_mem_fdo_mem_fdo`. -/
theorem c_eq_zero (hz : z ∈ 𝒟ᵒ) (hg : g • z ∈ 𝒟ᵒ) : (↑ₘg) 1 0 = 0 := by
  have hp : ∀ {g' : SL(2, ℤ)}, g' • z ∈ 𝒟ᵒ → (↑ₘg') 1 0 ≠ 1 := by
    intro g' hg'
    by_contra hc
    let a := (↑ₘg') 0 0
    let d := (↑ₘg') 1 1
    have had : T ^ (-a) * g' = S * T ^ d := by rw [g_eq_of_c_eq_one hc]; group
    let w := T ^ (-a) • g' • z
    have h₁ : w = S • T ^ d • z := by simp only [← mul_smul, had]
    replace h₁ : normSq w < 1 := h₁.symm ▸ normSq_S_smul_lt_one (one_lt_normSq_T_zpow_smul hz d)
    have h₂ : 1 < normSq w := one_lt_normSq_T_zpow_smul hg' (-a)
    linarith
  

Proof state:
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
hp : ∀ {g' : SL(2, ℤ)}, g' • z ∈ 𝒟ᵒ → ↑g' 1 0 ≠ 1
⊢ ↑g 1 0 = 0

Next tactic:
have hn : (↑ₘg) 1 0 ≠ -1 := by
    intro hc
    replace hc : (↑ₘ(-g)) 1 0 = 1; · simp [← neg_eq_iff_eq_neg.mpr hc]
    replace hg : -g • z ∈ 𝒟ᵒ := (SL_neg_smul g z).symm ▸ hg
    exact hp hg hc
/-- An auxiliary result en route to `ModularGroup.eq_smul_self_of_mem_fdo_mem_fdo`. -/
theorem c_eq_zero (hz : z ∈ 𝒟ᵒ) (hg : g • z ∈ 𝒟ᵒ) : (↑ₘg) 1 0 = 0 := by
  have hp : ∀ {g' : SL(2, ℤ)}, g' • z ∈ 𝒟ᵒ → (↑ₘg') 1 0 ≠ 1 := by
    intro g' hg'
    by_contra hc
    let a := (↑ₘg') 0 0
    let d := (↑ₘg') 1 1
    have had : T ^ (-a) * g' = S * T ^ d := by rw [g_eq_of_c_eq_one hc]; group
    let w := T ^ (-a) • g' • z
    have h₁ : w = S • T ^ d • z := by simp only [← mul_smul, had]
    replace h₁ : normSq w < 1 := h₁.symm ▸ normSq_S_smul_lt_one (one_lt_normSq_T_zpow_smul hz d)
    have h₂ : 1 < normSq w := one_lt_normSq_T_zpow_smul hg' (-a)
    linarith
  have hn : (↑ₘg) 1 0 ≠ -1 := by
    

Proof state:
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
hp : ∀ {g' : SL(2, ℤ)}, g' • z ∈ 𝒟ᵒ → ↑g' 1 0 ≠ 1
⊢ ↑g 1 0 ≠ -1

Next tactic:
intro hc