l3lab/ntp-mathlib huggingface dataset:
/-- First Fundamental Domain Lemma: Any `z : ℍ` can be moved to `𝒟` by an element of
`SL(2,ℤ)` -/
theorem exists_smul_mem_fd (z : ℍ) : ∃ g : SL(2, ℤ), g • z ∈ 𝒟 := by
  -- obtain a g₀ which maximizes im (g • z),
  obtain ⟨g₀, hg₀⟩ := exists_max_im z
  -- then among those, minimize re
  obtain ⟨g, hg, hg'⟩ := exists_row_one_eq_and_min_re z (bottom_row_coprime g₀)
  refine' ⟨g, _⟩
  -- `g` has same max im property as `g₀`
  have hg₀' : ∀ g' : SL(2, ℤ), (g' • z).im ≤ (g • z).im := by
    have hg'' : (g • z).im = (g₀ • z).im := by
      rw [SpecialLinearGroup.im_smul_eq_div_normSq, SpecialLinearGroup.im_smul_eq_div_normSq,
        denom_apply, denom_apply, hg]
    simpa only [hg''] using hg₀
  constructor
  · -- Claim: `1 ≤ ⇑norm_sq ↑(g • z)`. If not, then `S•g•z` has larger imaginary part
    contrapose! hg₀'
    refine' ⟨S * g, _⟩
    rw [mul_smul]
    exact im_lt_im_S_smul hg₀'
  · show |(g • z).re| ≤ 1 / 2
    -- if not, then either `T` or `T'` decrease |Re|.
    rw [abs_le]
    constructor
    · contrapose! hg'
      refine' ⟨T * g, (T_mul_apply_one _).symm, _⟩
      rw [mul_smul, re_T_smul]
      cases abs_cases ((g • z).re + 1) <;> cases abs_cases (g • z).re <;> linarith
    · contrapose! hg'
      refine' ⟨T⁻¹ * g, (T_inv_mul_apply_one _).symm, _⟩
      rw [mul_smul, re_T_inv_smul]
      cases abs_cases ((g • z).re - 1) <;> cases abs_cases (g • z).re <;> 

Proof state:
case intro.intro.intro.right.right.inl.inl
g✝ : SL(2, ℤ)
z✝ z : ℍ
g₀ : SL(2, ℤ)
hg₀ : ∀ (g' : SL(2, ℤ)), UpperHalfPlane.im (g' • z) ≤ UpperHalfPlane.im (g₀ • z)
g : SL(2, ℤ)
hg : ↑g 1 = ↑g₀ 1
hg₀' : ∀ (g' : SL(2, ℤ)), UpperHalfPlane.im (g' • z) ≤ UpperHalfPlane.im (g • z)
hg' : 1 / 2 < UpperHalfPlane.re (g • z)
h✝¹ : |UpperHalfPlane.re (g • z) - 1| = UpperHalfPlane.re (g • z) - 1 ∧ 0 ≤ UpperHalfPlane.re (g • z) - 1
h✝ : |UpperHalfPlane.re (g • z)| = UpperHalfPlane.re (g • z) ∧ 0 ≤ UpperHalfPlane.re (g • z)
⊢ |UpperHalfPlane.re (g • z) - 1| < |UpperHalfPlane.re (g • z)|

Next tactic:
linarith
/-- First Fundamental Domain Lemma: Any `z : ℍ` can be moved to `𝒟` by an element of
`SL(2,ℤ)` -/
theorem exists_smul_mem_fd (z : ℍ) : ∃ g : SL(2, ℤ), g • z ∈ 𝒟 := by
  -- obtain a g₀ which maximizes im (g • z),
  obtain ⟨g₀, hg₀⟩ := exists_max_im z
  -- then among those, minimize re
  obtain ⟨g, hg, hg'⟩ := exists_row_one_eq_and_min_re z (bottom_row_coprime g₀)
  refine' ⟨g, _⟩
  -- `g` has same max im property as `g₀`
  have hg₀' : ∀ g' : SL(2, ℤ), (g' • z).im ≤ (g • z).im := by
    have hg'' : (g • z).im = (g₀ • z).im := by
      rw [SpecialLinearGroup.im_smul_eq_div_normSq, SpecialLinearGroup.im_smul_eq_div_normSq,
        denom_apply, denom_apply, hg]
    simpa only [hg''] using hg₀
  constructor
  · -- Claim: `1 ≤ ⇑norm_sq ↑(g • z)`. If not, then `S•g•z` has larger imaginary part
    contrapose! hg₀'
    refine' ⟨S * g, _⟩
    rw [mul_smul]
    exact im_lt_im_S_smul hg₀'
  · show |(g • z).re| ≤ 1 / 2
    -- if not, then either `T` or `T'` decrease |Re|.
    rw [abs_le]
    constructor
    · contrapose! hg'
      refine' ⟨T * g, (T_mul_apply_one _).symm, _⟩
      rw [mul_smul, re_T_smul]
      cases abs_cases ((g • z).re + 1) <;> cases abs_cases (g • z).re <;> linarith
    · contrapose! hg'
      refine' ⟨T⁻¹ * g, (T_inv_mul_apply_one _).symm, _⟩
      rw [mul_smul, re_T_inv_smul]
      cases abs_cases ((g • z).re - 1) <;> cases abs_cases (g • z).re <;> 

Proof state:
case intro.intro.intro.right.right.inl.inr
g✝ : SL(2, ℤ)
z✝ z : ℍ
g₀ : SL(2, ℤ)
hg₀ : ∀ (g' : SL(2, ℤ)), UpperHalfPlane.im (g' • z) ≤ UpperHalfPlane.im (g₀ • z)
g : SL(2, ℤ)
hg : ↑g 1 = ↑g₀ 1
hg₀' : ∀ (g' : SL(2, ℤ)), UpperHalfPlane.im (g' • z) ≤ UpperHalfPlane.im (g • z)
hg' : 1 / 2 < UpperHalfPlane.re (g • z)
h✝¹ : |UpperHalfPlane.re (g • z) - 1| = UpperHalfPlane.re (g • z) - 1 ∧ 0 ≤ UpperHalfPlane.re (g • z) - 1
h✝ : |UpperHalfPlane.re (g • z)| = -UpperHalfPlane.re (g • z) ∧ UpperHalfPlane.re (g • z) < 0
⊢ |UpperHalfPlane.re (g • z) - 1| < |UpperHalfPlane.re (g • z)|

Next tactic:
linarith
/-- First Fundamental Domain Lemma: Any `z : ℍ` can be moved to `𝒟` by an element of
`SL(2,ℤ)` -/
theorem exists_smul_mem_fd (z : ℍ) : ∃ g : SL(2, ℤ), g • z ∈ 𝒟 := by
  -- obtain a g₀ which maximizes im (g • z),
  obtain ⟨g₀, hg₀⟩ := exists_max_im z
  -- then among those, minimize re
  obtain ⟨g, hg, hg'⟩ := exists_row_one_eq_and_min_re z (bottom_row_coprime g₀)
  refine' ⟨g, _⟩
  -- `g` has same max im property as `g₀`
  have hg₀' : ∀ g' : SL(2, ℤ), (g' • z).im ≤ (g • z).im := by
    have hg'' : (g • z).im = (g₀ • z).im := by
      rw [SpecialLinearGroup.im_smul_eq_div_normSq, SpecialLinearGroup.im_smul_eq_div_normSq,
        denom_apply, denom_apply, hg]
    simpa only [hg''] using hg₀
  constructor
  · -- Claim: `1 ≤ ⇑norm_sq ↑(g • z)`. If not, then `S•g•z` has larger imaginary part
    contrapose! hg₀'
    refine' ⟨S * g, _⟩
    rw [mul_smul]
    exact im_lt_im_S_smul hg₀'
  · show |(g • z).re| ≤ 1 / 2
    -- if not, then either `T` or `T'` decrease |Re|.
    rw [abs_le]
    constructor
    · contrapose! hg'
      refine' ⟨T * g, (T_mul_apply_one _).symm, _⟩
      rw [mul_smul, re_T_smul]
      cases abs_cases ((g • z).re + 1) <;> cases abs_cases (g • z).re <;> linarith
    · contrapose! hg'
      refine' ⟨T⁻¹ * g, (T_inv_mul_apply_one _).symm, _⟩
      rw [mul_smul, re_T_inv_smul]
      cases abs_cases ((g • z).re - 1) <;> cases abs_cases (g • z).re <;> 

Proof state:
case intro.intro.intro.right.right.inr.inl
g✝ : SL(2, ℤ)
z✝ z : ℍ
g₀ : SL(2, ℤ)
hg₀ : ∀ (g' : SL(2, ℤ)), UpperHalfPlane.im (g' • z) ≤ UpperHalfPlane.im (g₀ • z)
g : SL(2, ℤ)
hg : ↑g 1 = ↑g₀ 1
hg₀' : ∀ (g' : SL(2, ℤ)), UpperHalfPlane.im (g' • z) ≤ UpperHalfPlane.im (g • z)
hg' : 1 / 2 < UpperHalfPlane.re (g • z)
h✝¹ : |UpperHalfPlane.re (g • z) - 1| = -(UpperHalfPlane.re (g • z) - 1) ∧ UpperHalfPlane.re (g • z) - 1 < 0
h✝ : |UpperHalfPlane.re (g • z)| = UpperHalfPlane.re (g • z) ∧ 0 ≤ UpperHalfPlane.re (g • z)
⊢ |UpperHalfPlane.re (g • z) - 1| < |UpperHalfPlane.re (g • z)|

Next tactic:
linarith
/-- First Fundamental Domain Lemma: Any `z : ℍ` can be moved to `𝒟` by an element of
`SL(2,ℤ)` -/
theorem exists_smul_mem_fd (z : ℍ) : ∃ g : SL(2, ℤ), g • z ∈ 𝒟 := by
  -- obtain a g₀ which maximizes im (g • z),
  obtain ⟨g₀, hg₀⟩ := exists_max_im z
  -- then among those, minimize re
  obtain ⟨g, hg, hg'⟩ := exists_row_one_eq_and_min_re z (bottom_row_coprime g₀)
  refine' ⟨g, _⟩
  -- `g` has same max im property as `g₀`
  have hg₀' : ∀ g' : SL(2, ℤ), (g' • z).im ≤ (g • z).im := by
    have hg'' : (g • z).im = (g₀ • z).im := by
      rw [SpecialLinearGroup.im_smul_eq_div_normSq, SpecialLinearGroup.im_smul_eq_div_normSq,
        denom_apply, denom_apply, hg]
    simpa only [hg''] using hg₀
  constructor
  · -- Claim: `1 ≤ ⇑norm_sq ↑(g • z)`. If not, then `S•g•z` has larger imaginary part
    contrapose! hg₀'
    refine' ⟨S * g, _⟩
    rw [mul_smul]
    exact im_lt_im_S_smul hg₀'
  · show |(g • z).re| ≤ 1 / 2
    -- if not, then either `T` or `T'` decrease |Re|.
    rw [abs_le]
    constructor
    · contrapose! hg'
      refine' ⟨T * g, (T_mul_apply_one _).symm, _⟩
      rw [mul_smul, re_T_smul]
      cases abs_cases ((g • z).re + 1) <;> cases abs_cases (g • z).re <;> linarith
    · contrapose! hg'
      refine' ⟨T⁻¹ * g, (T_inv_mul_apply_one _).symm, _⟩
      rw [mul_smul, re_T_inv_smul]
      cases abs_cases ((g • z).re - 1) <;> cases abs_cases (g • z).re <;> 

Proof state:
case intro.intro.intro.right.right.inr.inr
g✝ : SL(2, ℤ)
z✝ z : ℍ
g₀ : SL(2, ℤ)
hg₀ : ∀ (g' : SL(2, ℤ)), UpperHalfPlane.im (g' • z) ≤ UpperHalfPlane.im (g₀ • z)
g : SL(2, ℤ)
hg : ↑g 1 = ↑g₀ 1
hg₀' : ∀ (g' : SL(2, ℤ)), UpperHalfPlane.im (g' • z) ≤ UpperHalfPlane.im (g • z)
hg' : 1 / 2 < UpperHalfPlane.re (g • z)
h✝¹ : |UpperHalfPlane.re (g • z) - 1| = -(UpperHalfPlane.re (g • z) - 1) ∧ UpperHalfPlane.re (g • z) - 1 < 0
h✝ : |UpperHalfPlane.re (g • z)| = -UpperHalfPlane.re (g • z) ∧ UpperHalfPlane.re (g • z) < 0
⊢ |UpperHalfPlane.re (g • z) - 1| < |UpperHalfPlane.re (g • z)|

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
  

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

Next tactic:
let c' : ℤ := (↑ₘg) 1 0
/-- 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
  

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

Next tactic:
let c : ℝ := (c' : ℝ)
/-- 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)
  

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

Next tactic:
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
/-- 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
    

Proof state:
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
c' : ℤ := ↑g 1 0
c : ℝ := ↑c'
this : 3 * c ^ 2 < 4
⊢ |↑g 1 0| ≤ 1

Next tactic:
rw [← Int.cast_pow, ← Int.cast_three, ← Int.cast_four, ← Int.cast_mul, Int.cast_lt] at this
/-- 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
    

Proof state:
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
c' : ℤ := ↑g 1 0
c : ℝ := ↑c'
this : 3 * c' ^ 2 < 4
⊢ |↑g 1 0| ≤ 1

Next tactic:
replace this : c' ^ 2 ≤ 1 ^ 2
/-- 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; · 

Proof state:
case this
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
c' : ℤ := ↑g 1 0
c : ℝ := ↑c'
this : 3 * c' ^ 2 < 4
⊢ c' ^ 2 ≤ 1 ^ 2

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
    

Proof state:
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
c' : ℤ := ↑g 1 0
c : ℝ := ↑c'
this : c' ^ 2 ≤ 1 ^ 2
⊢ |↑g 1 0| ≤ 1

Next tactic:
rwa [sq_le_sq, abs_one] at this
/-- 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
  

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

Next tactic:
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
/-- 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
    

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

Next tactic:
rcases eq_or_ne c 0 with (hc | hc)
/-- 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)
    · 

Proof state:
case inl
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
c' : ℤ := ↑g 1 0
c : ℝ := ↑c'
this : c ≠ 0 → 9 * c ^ 4 < 16
hc : c = 0
⊢ 3 * c ^ 2 < 4

Next tactic:
rw [hc]
/-- 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]; 

Proof state:
case inl
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
c' : ℤ := ↑g 1 0
c : ℝ := ↑c'
this : c ≠ 0 → 9 * c ^ 4 < 16
hc : c = 0
⊢ 3 * 0 ^ 2 < 4

Next tactic:
norm_num
/-- 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
    · 

Proof state:
case inr
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
c' : ℤ := ↑g 1 0
c : ℝ := ↑c'
this : c ≠ 0 → 9 * c ^ 4 < 16
hc : c ≠ 0
⊢ 3 * c ^ 2 < 4

Next tactic:
refine' (abs_lt_of_sq_lt_sq' _ (by norm_num)).2
/-- 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 

Proof state:
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
c' : ℤ := ↑g 1 0
c : ℝ := ↑c'
this : c ≠ 0 → 9 * c ^ 4 < 16
hc : c ≠ 0
⊢ 0 ≤ 4

Next tactic:
norm_num
/-- 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
      

Proof state:
case inr
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
c' : ℤ := ↑g 1 0
c : ℝ := ↑c'
this : c ≠ 0 → 9 * c ^ 4 < 16
hc : c ≠ 0
⊢ (3 * c ^ 2) ^ 2 < 4 ^ 2

Next tactic:
specialize this hc
/-- 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
      

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

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
  

Proof state:
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
c' : ℤ := ↑g 1 0
c : ℝ := ↑c'
⊢ c ≠ 0 → 9 * c ^ 4 < 16

Next tactic:
intro hc
/-- 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
  

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

Next tactic:
replace hc : 0 < c ^ 4
/-- 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;
  · 

Proof state:
case hc
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
c' : ℤ := ↑g 1 0
c : ℝ := ↑c'
hc : c ≠ 0
⊢ 0 < c ^ 4

Next tactic:
change 0 < c ^ (2 * 2)
/-- 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); 

Proof state:
case hc
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
c' : ℤ := ↑g 1 0
c : ℝ := ↑c'
hc : c ≠ 0
⊢ 0 < c ^ (2 * 2)

Next tactic:
rw [pow_mul]
/-- 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]; 

Proof state:
case hc
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
c' : ℤ := ↑g 1 0
c : ℝ := ↑c'
hc : c ≠ 0
⊢ 0 < (c ^ 2) ^ 2

Next tactic:
apply sq_pos_of_pos (sq_pos_of_ne_zero _ hc)
/-- 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)
  

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
⊢ 9 * c ^ 4 < 16

Next tactic:
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
/-- 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 

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
⊢ 0 ≤ 3

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 

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
⊢ 0 ≤ 3

Next tactic:
linarith