awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Derivative (goal)
Goal:
k₁ k₂ : ℂ
F G : ℍ → ℂ
hF : MDiff F
hG : MDiff G
z : ℍ
⊢ serreDerivative (k₁ + k₂) (F * G) z = (serreDerivative k₁ F * G + F * serreDerivative k₂ G) z
have h : D F z * G z + F z * D G z - (k₁ + k₂) * 12⁻¹ * EisensteinSeries.E2 z * (F z * G z) =
  (D F z - k₁ * 12⁻¹ * EisensteinSeries.E2 z * F z) * G z + F z * (D G z - k₂ * 12⁻¹ * EisensteinSeries.E2 z * G z)
Tactic: simp [serreDerivative, normalizedDerivOfComplex_mul F G hF hG]
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Derivative (goal)
Goal:
F : ℍ → ℂ
k : ℂ
hF : MDiff F
⊢ MDiff (serreDerivative k F)
have refine : MDiff fun (z : ℍ) ↦ k * 12⁻¹ * EisensteinSeries.E2 z * F z
Tactic: refine (normalizedDerivOfComplex_mdifferentiable hF).sub ?_
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Derivative (goal)
Goal:
F : ℍ → ℂ
k : ℂ
hF : MDiff F
⊢ MDiff fun (z : ℍ) ↦ k * 12⁻¹ * EisensteinSeries.E2 z * F z
have h : ∀ (x : ℍ),
  k * 12⁻¹ * EisensteinSeries.E2 x * F x =
    HMul.hMul (α := ℍ → ℂ) (β := ℍ → ℂ) (fun (z : ℍ) ↦ k * 12⁻¹) (fun (z : ℍ) ↦ EisensteinSeries.E2 z * F z) x
Tactic: convert!
    (MDifferentiable.mul mdifferentiable_const (E2_mdifferentiable.mul hF) :
      MDiff (fun z ↦ (k * 12⁻¹) * (EisensteinSeries.E2 z * F z)))
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
z : ℂ
hz : z ≠ 0
⊢ HPow.hPow (β := ℕ) z.sqrt 24 = HPow.hPow (β := ℕ) z 12
have rw : cexp ((↑24 : ℂ) * (log z / 2)) = HPow.hPow (β := ℕ) z 12
Tactic: rw [sqrt_eq_exp hz, ← exp_nat_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
z : ℂ
hz : z ≠ 0
⊢ cexp ((↑24 : ℂ) * (log z / 2)) = HPow.hPow (β := ℕ) z 12
have ring_nf : cexp (log z * 12) = HPow.hPow (β := ℕ) z 12
Tactic: ring_nf
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
z : ℂ
hz : z ≠ 0
⊢ cexp (log z * 12) = HPow.hPow (β := ℕ) z 12
have rw : cexp ((↑12 : ℂ) * log z) = HPow.hPow (β := ℕ) z 12
Tactic: rw [show (log z * 12) = (12 : ℕ) * log z by ring, exp_nat_mul, exp_log hz]
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
z : ℂ
hz : z ≠ 0
rw : cexp ((↑12 : ℂ) * log z) = HPow.hPow (β := ℕ) z 12
⊢ cexp (log z * 12) = HPow.hPow (β := ℕ) z 12
have rw₁ : HPow.hPow (β := ℕ) (cexp (log z)) 12 = HPow.hPow (β := ℕ) z 12
Tactic: rw [show (log z * 12) = (12 : ℕ) * log z by ring, exp_nat_mul, exp_log hz]
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
z : ℂ
hz : z ≠ 0
rw : cexp ((↑12 : ℂ) * log z) = HPow.hPow (β := ℕ) z 12
rw₁ : HPow.hPow (β := ℕ) (cexp (log z)) 12 = HPow.hPow (β := ℕ) z 12
⊢ cexp (log z * 12) = HPow.hPow (β := ℕ) z 12
have rw₂ : HPow.hPow (β := ℕ) z 12 = HPow.hPow (β := ℕ) z 12
Tactic: rw [show (log z * 12) = (12 : ℕ) * log z by ring, exp_nat_mul, exp_log hz]
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
⊢ HPow.hPow (β := ℕ) I.sqrt 24 = 1
have rw : HPow.hPow (β := ℕ) I 12 = 1
Tactic: rw [csqrt_pow_24_eq I_ne_zero, show 12 = 4 * 3 by lia, pow_mul, I_pow_four, one_pow]
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
rw : HPow.hPow (β := ℕ) I 12 = 1
rw₁ : I ^ HMul.hMul (α := ℕ) 4 3 = 1
⊢ HPow.hPow (β := ℕ) I.sqrt 24 = 1
have rw₂ : HPow.hPow (β := ℕ) (HPow.hPow (β := ℕ) I 4) 3 = 1
Tactic: rw [csqrt_pow_24_eq I_ne_zero, show 12 = 4 * 3 by lia, pow_mul, I_pow_four, one_pow]
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
rw : HPow.hPow (β := ℕ) I 12 = 1
rw₁ : I ^ HMul.hMul (α := ℕ) 4 3 = 1
rw₂ : HPow.hPow (β := ℕ) (HPow.hPow (β := ℕ) I 4) 3 = 1
⊢ HPow.hPow (β := ℕ) I.sqrt 24 = 1
have rw₃ : HPow.hPow (α := ℂ) (β := ℕ) 1 3 = 1
Tactic: rw [csqrt_pow_24_eq I_ne_zero, show 12 = 4 * 3 by lia, pow_mul, I_pow_four, one_pow]
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
rw : HPow.hPow (β := ℕ) I 12 = 1
rw₁ : I ^ HMul.hMul (α := ℕ) 4 3 = 1
rw₂ : HPow.hPow (β := ℕ) (HPow.hPow (β := ℕ) I 4) 3 = 1
rw₃ : HPow.hPow (α := ℂ) (β := ℕ) 1 3 = 1
⊢ HPow.hPow (β := ℕ) I.sqrt 24 = 1
have rw₄ : Eq (α := ℂ) 1 1
Tactic: rw [csqrt_pow_24_eq I_ne_zero, show 12 = 4 * 3 by lia, pow_mul, I_pow_four, one_pow]
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
z : ℍ
⊢ logDeriv (η ∘ fun (x : ℂ) ↦ -1 / x) (↑z : ℂ) = (HPow.hPow (β := ℤ) (↑z : ℂ) 2)⁻¹ * logDeriv η (-(↑z : ℂ))⁻¹
have simp : logDeriv (η ∘ fun (x : ℂ) ↦ -x⁻¹) (↑z : ℂ) = (HPow.hPow (β := ℤ) (↑z : ℂ) 2)⁻¹ * logDeriv η (-(↑z : ℂ)⁻¹)
Tactic: simp only [neg_div, one_div, inv_neg]
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
z : ℍ
⊢ logDeriv (η ∘ fun (x : ℂ) ↦ -x⁻¹) (↑z : ℂ) = (HPow.hPow (β := ℤ) (↑z : ℂ) 2)⁻¹ * logDeriv η (-(↑z : ℂ)⁻¹)
have rw : deriv (fun (x : ℂ) ↦ -x⁻¹) (↑z : ℂ) * logDeriv η (-(↑z : ℂ)⁻¹) =
  (HPow.hPow (β := ℤ) (↑z : ℂ) 2)⁻¹ * logDeriv η (-(↑z : ℂ)⁻¹)
Tactic: rw [logDeriv_comp, mul_comm]
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
z : ℍ
rw :
  deriv (fun (x : ℂ) ↦ -x⁻¹) (↑z : ℂ) * logDeriv η (-(↑z : ℂ)⁻¹) =
    (HPow.hPow (β := ℤ) (↑z : ℂ) 2)⁻¹ * logDeriv η (-(↑z : ℂ)⁻¹)
⊢ logDeriv (η ∘ fun (x : ℂ) ↦ -x⁻¹) (↑z : ℂ) = (HPow.hPow (β := ℤ) (↑z : ℂ) 2)⁻¹ * logDeriv η (-(↑z : ℂ)⁻¹)
have hf : DifferentiableAt ℂ η (-(↑z : ℂ)⁻¹)
Tactic: rw [logDeriv_comp, mul_comm]
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
z : ℍ
rw :
  deriv (fun (x : ℂ) ↦ -x⁻¹) (↑z : ℂ) * logDeriv η (-(↑z : ℂ)⁻¹) =
    (HPow.hPow (β := ℤ) (↑z : ℂ) 2)⁻¹ * logDeriv η (-(↑z : ℂ)⁻¹)
hf : DifferentiableAt ℂ η (-(↑z : ℂ)⁻¹)
⊢ logDeriv (η ∘ fun (x : ℂ) ↦ -x⁻¹) (↑z : ℂ) = (HPow.hPow (β := ℤ) (↑z : ℂ) 2)⁻¹ * logDeriv η (-(↑z : ℂ)⁻¹)
have hg : DifferentiableAt ℂ (F := ℂ) (fun (x : ℂ) ↦ -x⁻¹) (↑z : ℂ)
Tactic: rw [logDeriv_comp, mul_comm]
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
z : ℍ
⊢ DifferentiableAt ℂ (F := ℂ) (fun (x : ℂ) ↦ -x⁻¹) (↑z : ℂ)
have funProp : (↑z : ℂ) ≠ 0
Tactic: fun_prop (disch := exact z.ne_zero)
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
z : ℍ
⊢ logDeriv (η ∘ fun (x : ℂ) ↦ -1 / x) (↑z : ℂ) = logDeriv (sqrt * η) (↑z : ℂ)
have rw : (HPow.hPow (β := ℤ) (↑z : ℂ) 2)⁻¹ * logDeriv η (-(↑z : ℂ))⁻¹ = deriv sqrt (↑z : ℂ) / (↑z : ℂ).sqrt + logDeriv η (↑z : ℂ)
Tactic: rw [logDeriv_eta_comp_div_eq z, Pi.mul_def,
      logDeriv_mul _ (by simp [sqrt, ne_zero z]) (eta_ne_zero z.2)
      (differentiableAt_sqrt (mem_slitPlane z))
      (differentiableAt_eta_of_mem_upperHalfPlaneSet z.2), logDeriv_apply sqrt]
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (hypothesis)
Goal:
z : ℍ
⊢ (HPow.hPow (β := ℤ) (↑z : ℂ) 2)⁻¹ * logDeriv η (-(↑z : ℂ))⁻¹ =
    deriv sqrt (↑z : ℂ) / (↑z : ℂ).sqrt + logDeriv η (↑z : ℂ)
have hE2 : @SlashAction.map ℤ _ _ Matrix.SpecialLinearGroup.monoid (Pi.addMonoid (f := fun (a : ℍ) ↦ ℂ)) _ 2 ModularGroup.S E2 z =
  (E2 - (1 / (2 * riemannZeta 2)) • D2 ModularGroup.S) z
Tactic: have hE2 := congrFun (E2_slash_action ModularGroup.S) z
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (hypothesis)
Goal:
z : ℍ
hE2 :
  @SlashAction.map ℤ _ _ Matrix.SpecialLinearGroup.monoid (Pi.addMonoid (f := fun (a : ℍ) ↦ ℂ)) _ 2 ModularGroup.S E2
      z =
    (E2 - (1 / (2 * riemannZeta 2)) • D2 ModularGroup.S) z
⊢ (HPow.hPow (β := ℤ) (↑z : ℂ) 2)⁻¹ * logDeriv η (-(↑z : ℂ))⁻¹ =
    deriv sqrt (↑z : ℂ) / (↑z : ℂ).sqrt + logDeriv η (↑z : ℂ)
have hE2₁ : E2 { coe := (-(↑z : ℂ))⁻¹, coe_im_pos := sorry : ℍ } * (HPow.hPow (β := ℤ) (↑z : ℂ) 2)⁻¹ =
  E2 z -
    6 / HPow.hPow (β := ℕ) (↑π : ℂ) 2 * 2⁻¹ *
      (2 * (↑π : ℂ) * I * (↑((↑ModularGroup.S : Matrix (Fin 2) (Fin 2) ℤ) 1 0) : ℂ) / (↑z : ℂ))
Tactic: simp only [one_div, SL_slash_def, modular_S_smul, ModularGroup.denom_S,
    Int.reduceNeg, zpow_neg, riemannZeta_two, mul_inv_rev, inv_div, Pi.sub_apply, Pi.smul_apply,
    D2, ModularGroup.denom_S, smul_eq_mul] at hE2
Cleared hypotheses: hE2
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
z : ℍ
hE2 :
  E2 { coe := (-(↑z : ℂ))⁻¹, coe_im_pos := ⋯ : ℍ } * (HPow.hPow (β := ℤ) (↑z : ℂ) 2)⁻¹ =
    E2 z -
      6 / HPow.hPow (β := ℕ) (↑π : ℂ) 2 * 2⁻¹ *
        (2 * (↑π : ℂ) * I * (↑((↑ModularGroup.S : Matrix (Fin 2) (Fin 2) ℤ) 1 0) : ℂ) / (↑z : ℂ))
⊢ (HPow.hPow (β := ℤ) (↑z : ℂ) 2)⁻¹ * logDeriv η (-(↑z : ℂ))⁻¹ =
    deriv sqrt (↑z : ℂ) / (↑z : ℂ).sqrt + logDeriv η (↑z : ℂ)
have rw : (E2 z - 6 / HPow.hPow (β := ℕ) (↑π : ℂ) 2 * 2⁻¹ * (2 * (↑π : ℂ) * I * (↑1 : ℂ) / (↑z : ℂ))) * ((↑π : ℂ) * I / 12) =
  (↑z : ℂ) ^ (-1 / 2) / 2 * (HPow.hPow (β := ℂ) (↑z : ℂ) 2⁻¹)⁻¹ + (↑π : ℂ) * I / 12 * E2 z
Tactic: rw [deriv_sqrt (mem_slitPlane z), div_eq_mul_inv, logDeriv_eta_eq_E2 z,
    logDeriv_eta_eq_E2 (.mk _ z.im_inv_neg_coe_pos), ← mul_assoc, mul_comm, ← mul_assoc, hE2, sqrt,
    show ModularGroup.S 1 0 = 1 by simp [ModularGroup.S]]
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
z : ℍ
hE2 :
  E2 { coe := (-(↑z : ℂ))⁻¹, coe_im_pos := ⋯ : ℍ } * (HPow.hPow (β := ℤ) (↑z : ℂ) 2)⁻¹ =
    E2 z -
      6 / HPow.hPow (β := ℕ) (↑π : ℂ) 2 * 2⁻¹ *
        (2 * (↑π : ℂ) * I * (↑((↑ModularGroup.S : Matrix (Fin 2) (Fin 2) ℤ) 1 0) : ℂ) / (↑z : ℂ))
⊢ (E2 z - 6 / HPow.hPow (β := ℕ) (↑π : ℂ) 2 * 2⁻¹ * (2 * (↑π : ℂ) * I * (↑1 : ℂ) / (↑z : ℂ))) * ((↑π : ℂ) * I / 12) =
    (↑z : ℂ) ^ (-1 / 2) / 2 * (HPow.hPow (β := ℂ) (↑z : ℂ) 2⁻¹)⁻¹ + (↑π : ℂ) * I / 12 * E2 z
have transitivity : (E2 z - 6 / HPow.hPow (β := ℕ) (↑π : ℂ) 2 * 2⁻¹ * (2 * (↑π : ℂ) * I * (↑1 : ℂ) / (↑z : ℂ))) * ((↑π : ℂ) * I / 12) =
  1 / (↑z : ℂ) / 2 + (↑π : ℂ) * I / 12 * E2 z
Tactic: transitivity 1 / z / 2 + π * I / 12 * E2 z
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
z : ℍ
hE2 :
  E2 { coe := (-(↑z : ℂ))⁻¹, coe_im_pos := ⋯ : ℍ } * (HPow.hPow (β := ℤ) (↑z : ℂ) 2)⁻¹ =
    E2 z -
      6 / HPow.hPow (β := ℕ) (↑π : ℂ) 2 * 2⁻¹ *
        (2 * (↑π : ℂ) * I * (↑((↑ModularGroup.S : Matrix (Fin 2) (Fin 2) ℤ) 1 0) : ℂ) / (↑z : ℂ))
transitivity :
  (E2 z - 6 / HPow.hPow (β := ℕ) (↑π : ℂ) 2 * 2⁻¹ * (2 * (↑π : ℂ) * I * (↑1 : ℂ) / (↑z : ℂ))) * ((↑π : ℂ) * I / 12) =
    1 / (↑z : ℂ) / 2 + (↑π : ℂ) * I / 12 * E2 z
⊢ (E2 z - 6 / HPow.hPow (β := ℕ) (↑π : ℂ) 2 * 2⁻¹ * (2 * (↑π : ℂ) * I * (↑1 : ℂ) / (↑z : ℂ))) * ((↑π : ℂ) * I / 12) =
    (↑z : ℂ) ^ (-1 / 2) / 2 * (HPow.hPow (β := ℂ) (↑z : ℂ) 2⁻¹)⁻¹ + (↑π : ℂ) * I / 12 * E2 z
have transitivity₁ : 1 / (↑z : ℂ) / 2 + (↑π : ℂ) * I / 12 * E2 z =
  (↑z : ℂ) ^ (-1 / 2) / 2 * (HPow.hPow (β := ℂ) (↑z : ℂ) 2⁻¹)⁻¹ + (↑π : ℂ) * I / 12 * E2 z
Tactic: transitivity 1 / z / 2 + π * I / 12 * E2 z
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
z : ℍ
hE2 :
  E2 { coe := (-(↑z : ℂ))⁻¹, coe_im_pos := ⋯ : ℍ } * (HPow.hPow (β := ℤ) (↑z : ℂ) 2)⁻¹ =
    E2 z -
      6 / HPow.hPow (β := ℕ) (↑π : ℂ) 2 * 2⁻¹ *
        (2 * (↑π : ℂ) * I * (↑((↑ModularGroup.S : Matrix (Fin 2) (Fin 2) ℤ) 1 0) : ℂ) / (↑z : ℂ))
⊢ (E2 z - 6 / HPow.hPow (β := ℕ) (↑π : ℂ) 2 * 2⁻¹ * (2 * (↑π : ℂ) * I * (↑1 : ℂ) / (↑z : ℂ))) * ((↑π : ℂ) * I / 12) =
    1 / (↑z : ℂ) / 2 + (↑π : ℂ) * I / 12 * E2 z
have field_simp : 2 * I * (E2 z * (↑π : ℂ) - 6 * I * (↑1 : ℂ) / (↑z : ℂ)) = 12 / (↑z : ℂ) + E2 z * (↑π : ℂ) * 2 * I
Tactic: field_simp
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
z : ℍ
hE2 :
  E2 { coe := (-(↑z : ℂ))⁻¹, coe_im_pos := ⋯ : ℍ } * (HPow.hPow (β := ℤ) (↑z : ℂ) 2)⁻¹ =
    E2 z -
      6 / HPow.hPow (β := ℕ) (↑π : ℂ) 2 * 2⁻¹ *
        (2 * (↑π : ℂ) * I * (↑((↑ModularGroup.S : Matrix (Fin 2) (Fin 2) ℤ) 1 0) : ℂ) / (↑z : ℂ))
⊢ 1 / (↑z : ℂ) / 2 + (↑π : ℂ) * I / 12 * E2 z =
    (↑z : ℂ) ^ (-1 / 2) / 2 * (HPow.hPow (β := ℂ) (↑z : ℂ) 2⁻¹)⁻¹ + (↑π : ℂ) * I / 12 * E2 z
have rw : 1 / (↑z : ℂ) / 2 + (↑π : ℂ) * I / 12 * E2 z = ((↑z : ℂ) ^ (1 / 2 + 2⁻¹))⁻¹ / 2 + (↑π : ℂ) * I / 12 * E2 z
Tactic: rw [div_mul_eq_mul_div₀ _ _ (2 : ℂ), neg_div, cpow_neg, ← mul_inv, ← cpow_add _ _ z.ne_zero]
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
⊢ ∃ (c : ℂ), c ≠ 0 ∧ Set.EqOn (η ∘ fun (z : ℂ) ↦ -1 / z) (c • (sqrt * η)) upperHalfPlaneSet
have rw : Set.EqOn (logDeriv (η ∘ fun (z : ℂ) ↦ -1 / z)) (logDeriv (sqrt * η)) upperHalfPlaneSet
Tactic: rw [← logDeriv_eqOn_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
rw : Set.EqOn (logDeriv (η ∘ fun (z : ℂ) ↦ -1 / z)) (logDeriv (sqrt * η)) upperHalfPlaneSet
⊢ ∃ (c : ℂ), c ≠ 0 ∧ Set.EqOn (η ∘ fun (z : ℂ) ↦ -1 / z) (c • (sqrt * η)) upperHalfPlaneSet
have hf : DifferentiableOn ℂ (η ∘ fun (z : ℂ) ↦ -1 / z) upperHalfPlaneSet
Tactic: rw [← logDeriv_eqOn_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
rw : Set.EqOn (logDeriv (η ∘ fun (z : ℂ) ↦ -1 / z)) (logDeriv (sqrt * η)) upperHalfPlaneSet
hf : DifferentiableOn ℂ (η ∘ fun (z : ℂ) ↦ -1 / z) upperHalfPlaneSet
⊢ ∃ (c : ℂ), c ≠ 0 ∧ Set.EqOn (η ∘ fun (z : ℂ) ↦ -1 / z) (c • (sqrt * η)) upperHalfPlaneSet
have hg : DifferentiableOn ℂ (sqrt * η) upperHalfPlaneSet
Tactic: rw [← logDeriv_eqOn_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
rw : Set.EqOn (logDeriv (η ∘ fun (z : ℂ) ↦ -1 / z)) (logDeriv (sqrt * η)) upperHalfPlaneSet
hf : DifferentiableOn ℂ (η ∘ fun (z : ℂ) ↦ -1 / z) upperHalfPlaneSet
hg : DifferentiableOn ℂ (sqrt * η) upperHalfPlaneSet
⊢ ∃ (c : ℂ), c ≠ 0 ∧ Set.EqOn (η ∘ fun (z : ℂ) ↦ -1 / z) (c • (sqrt * η)) upperHalfPlaneSet
have hs2 : IsOpen upperHalfPlaneSet
Tactic: rw [← logDeriv_eqOn_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
rw : Set.EqOn (logDeriv (η ∘ fun (z : ℂ) ↦ -1 / z)) (logDeriv (sqrt * η)) upperHalfPlaneSet
hf : DifferentiableOn ℂ (η ∘ fun (z : ℂ) ↦ -1 / z) upperHalfPlaneSet
hg : DifferentiableOn ℂ (sqrt * η) upperHalfPlaneSet
hs2 : IsOpen upperHalfPlaneSet
⊢ ∃ (c : ℂ), c ≠ 0 ∧ Set.EqOn (η ∘ fun (z : ℂ) ↦ -1 / z) (c • (sqrt * η)) upperHalfPlaneSet
have hsc : IsPreconnected upperHalfPlaneSet
Tactic: rw [← logDeriv_eqOn_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
rw : Set.EqOn (logDeriv (η ∘ fun (z : ℂ) ↦ -1 / z)) (logDeriv (sqrt * η)) upperHalfPlaneSet
hf : DifferentiableOn ℂ (η ∘ fun (z : ℂ) ↦ -1 / z) upperHalfPlaneSet
hg : DifferentiableOn ℂ (sqrt * η) upperHalfPlaneSet
hs2 : IsOpen upperHalfPlaneSet
hsc : IsPreconnected upperHalfPlaneSet
⊢ ∃ (c : ℂ), c ≠ 0 ∧ Set.EqOn (η ∘ fun (z : ℂ) ↦ -1 / z) (c • (sqrt * η)) upperHalfPlaneSet
have hgn : ∀ x ∈ upperHalfPlaneSet, (sqrt * η) x ≠ 0
Tactic: rw [← logDeriv_eqOn_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
rw : Set.EqOn (logDeriv (η ∘ fun (z : ℂ) ↦ -1 / z)) (logDeriv (sqrt * η)) upperHalfPlaneSet
hf : DifferentiableOn ℂ (η ∘ fun (z : ℂ) ↦ -1 / z) upperHalfPlaneSet
hg : DifferentiableOn ℂ (sqrt * η) upperHalfPlaneSet
hs2 : IsOpen upperHalfPlaneSet
hsc : IsPreconnected upperHalfPlaneSet
hgn : ∀ x ∈ upperHalfPlaneSet, (sqrt * η) x ≠ 0
⊢ ∃ (c : ℂ), c ≠ 0 ∧ Set.EqOn (η ∘ fun (z : ℂ) ↦ -1 / z) (c • (sqrt * η)) upperHalfPlaneSet
have hfn : ∀ x ∈ upperHalfPlaneSet, (η ∘ fun (z : ℂ) ↦ -1 / z) x ≠ 0
Tactic: rw [← logDeriv_eqOn_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
⊢ DifferentiableOn ℂ (η ∘ fun (z : ℂ) ↦ -1 / z) upperHalfPlaneSet
have hf : DifferentiableOn ℂ η upperHalfPlaneSet
Tactic: apply DifferentiableOn.comp (t := upperHalfPlaneSet)
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
hf : DifferentiableOn ℂ η upperHalfPlaneSet
⊢ DifferentiableOn ℂ (η ∘ fun (z : ℂ) ↦ -1 / z) upperHalfPlaneSet
have hf₁ : DifferentiableOn ℂ (F := ℂ) (fun (z : ℂ) ↦ -1 / z) upperHalfPlaneSet
Tactic: apply DifferentiableOn.comp (t := upperHalfPlaneSet)
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
hf : DifferentiableOn ℂ η upperHalfPlaneSet
hf₁ : DifferentiableOn ℂ (F := ℂ) (fun (z : ℂ) ↦ -1 / z) upperHalfPlaneSet
⊢ DifferentiableOn ℂ (η ∘ fun (z : ℂ) ↦ -1 / z) upperHalfPlaneSet
have hf₂ : Set.MapsTo (fun (z : ℂ) ↦ -1 / z) upperHalfPlaneSet upperHalfPlaneSet
Tactic: apply DifferentiableOn.comp (t := upperHalfPlaneSet)
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
z : ℍ
⊢ Δ z = 𝕢 1 (↑z : ℂ) * ∏' (n : ℕ), HPow.hPow (β := ℕ) (1 - eta_q n (↑z : ℂ)) 24
have simp : HPow.hPow (β := ℕ) (𝕢 24 (↑z : ℂ)) 24 * HPow.hPow (α := ℂ) (β := ℕ) (∏' (n : ℕ), (1 - eta_q n (↑z : ℂ))) 24 =
  𝕢 1 (↑z : ℂ) * ∏' (n : ℕ), HPow.hPow (β := ℕ) (1 - eta_q n (↑z : ℂ)) 24
Tactic: simp only [discriminant, eta, mul_pow]
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
z : ℍ
⊢ HPow.hPow (β := ℕ) (𝕢 24 (↑z : ℂ)) 24 * HPow.hPow (α := ℂ) (β := ℕ) (∏' (n : ℕ), (1 - eta_q n (↑z : ℂ))) 24 =
    𝕢 1 (↑z : ℂ) * ∏' (n : ℕ), HPow.hPow (β := ℕ) (1 - eta_q n (↑z : ℂ)) 24
have e_a : HPow.hPow (β := ℕ) (𝕢 24 (↑z : ℂ)) 24 = 𝕢 1 (↑z : ℂ)
Tactic: congr
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
z : ℍ
e_a : HPow.hPow (β := ℕ) (𝕢 24 (↑z : ℂ)) 24 = 𝕢 1 (↑z : ℂ)
⊢ HPow.hPow (β := ℕ) (𝕢 24 (↑z : ℂ)) 24 * HPow.hPow (α := ℂ) (β := ℕ) (∏' (n : ℕ), (1 - eta_q n (↑z : ℂ))) 24 =
    𝕢 1 (↑z : ℂ) * ∏' (n : ℕ), HPow.hPow (β := ℕ) (1 - eta_q n (↑z : ℂ)) 24
have e_a₁ : HPow.hPow (α := ℂ) (β := ℕ) (∏' (n : ℕ), (1 - eta_q n (↑z : ℂ))) 24 =
  ∏' (n : ℕ), HPow.hPow (β := ℕ) (1 - eta_q n (↑z : ℂ)) 24
Tactic: congr
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
z : ℍ
⊢ HPow.hPow (β := ℕ) (𝕢 24 (↑z : ℂ)) 24 = 𝕢 1 (↑z : ℂ)
have e_a : cexp (24 * (2 * (↑π : ℂ) * I * (↑z : ℂ) / 24)) = cexp (2 * (↑π : ℂ) * I * (↑z : ℂ))
Tactic: simp [Periodic.qParam, ← exp_nsmul, nsmul_eq_mul, Nat.cast_ofNat]
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
⊢ @SlashAction.map ℤ _ _ Matrix.SpecialLinearGroup.monoid (Pi.addMonoid (f := fun (a : ℍ) ↦ ℂ)) _ 12 ModularGroup.T Δ =
    Δ
have h : ∀ (z : ℍ),
  @SlashAction.map ℤ _ _ Matrix.SpecialLinearGroup.monoid (Pi.addMonoid (f := fun (a : ℍ) ↦ ℂ)) _ 12 ModularGroup.T Δ
      z =
    Δ z
Tactic: ext z
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
z : ℍ
⊢ @SlashAction.map ℤ _ _ Matrix.SpecialLinearGroup.monoid (Pi.addMonoid (f := fun (a : ℍ) ↦ ℂ)) _ 12 ModularGroup.T Δ
      z =
    Δ z
have h : Δ (HVAdd.hVAdd (α := ℝ) 1 z) *
    HPow.hPow (β := ℤ)
      ((↑((↑((Matrix.SpecialLinearGroup.toGL : (a : SL(2, ℝ)) → GL (Fin 2) ℝ)
                      ((Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ) : (a : SL(2, ℤ)) → SL(2, ℝ))
                        (Subtype.mk (p := fun (A : Matrix (Fin 2) (Fin 2) ℤ) ↦ A.det = 1) !![1, 1; 0, 1]
                          ModularGroup.T._proof_1))) :
                  Matrix (Fin 2) (Fin 2) ℝ)
                1 0) :
            ℂ) *
          (↑z : ℂ) +
        (↑((↑((Matrix.SpecialLinearGroup.toGL : (a : SL(2, ℝ)) → GL (Fin 2) ℝ)
                    ((Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ) : (a : SL(2, ℤ)) → SL(2, ℝ))
                      (Subtype.mk (p := fun (A : Matrix (Fin 2) (Fin 2) ℤ) ↦ A.det = 1) !![1, 1; 0, 1]
                        ModularGroup.T._proof_1))) :
                Matrix (Fin 2) (Fin 2) ℝ)
              1 1) :
          ℂ))
      (-12) =
  Δ z
Tactic: rw [SL_slash_apply, denom, modular_T_smul, ModularGroup.T]
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
z : ℍ
⊢ Δ (HVAdd.hVAdd (α := ℝ) 1 z) *
      HPow.hPow (β := ℤ)
        ((↑((↑((Matrix.SpecialLinearGroup.toGL : (a : SL(2, ℝ)) → GL (Fin 2) ℝ)
                        ((Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ) : (a : SL(2, ℤ)) → SL(2, ℝ))
                          (Subtype.mk (p := fun (A : Matrix (Fin 2) (Fin 2) ℤ) ↦ A.det = 1) !![1, 1; 0, 1]
                            ModularGroup.T._proof_1))) :
                    Matrix (Fin 2) (Fin 2) ℝ)
                  1 0) :
              ℂ) *
            (↑z : ℂ) +
          (↑((↑((Matrix.SpecialLinearGroup.toGL : (a : SL(2, ℝ)) → GL (Fin 2) ℝ)
                      ((Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ) : (a : SL(2, ℤ)) → SL(2, ℝ))
                        (Subtype.mk (p := fun (A : Matrix (Fin 2) (Fin 2) ℤ) ↦ A.det = 1) !![1, 1; 0, 1]
                          ModularGroup.T._proof_1))) :
                  Matrix (Fin 2) (Fin 2) ℝ)
                1 1) :
            ℂ))
        (-12) =
    Δ z
have h : cexp (2 * (↑π : ℂ) * I * (1 + (↑z : ℂ))) *
    ∏' (n : ℕ), HPow.hPow (β := ℕ) (1 - cexp (2 * (↑π : ℂ) * I * (1 + (↑z : ℂ))) ^ (n + 1)) 24 =
  cexp (2 * (↑π : ℂ) * I * (↑z : ℂ) + 2 * (↑π : ℂ) * I) *
    ∏' (n : ℕ), HPow.hPow (β := ℕ) (1 - cexp (2 * (↑π : ℂ) * I * (↑z : ℂ) + 2 * (↑π : ℂ) * I) ^ (n + 1)) 24
Tactic: simp [discriminant_eq_q_prod, eta_q, Periodic.qParam, ← exp_periodic (2 * π * I * z)]
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
⊢ Set.EqOn (η ∘ fun (x : ℂ) ↦ -1 / x) (I.sqrt⁻¹ • (sqrt * η)) upperHalfPlaneSet
have obtain : ∀ (z : ℂ) (hz : z ≠ 0) (h : Set.EqOn (η ∘ fun (z : ℂ) ↦ -1 / z) (z • (sqrt * η)) upperHalfPlaneSet),
  Set.EqOn (η ∘ fun (x : ℂ) ↦ -1 / x) (I.sqrt⁻¹ • (sqrt * η)) upperHalfPlaneSet
Tactic: obtain ⟨z, hz, h⟩ := eta_comp_eqOn_const_mul_csqrt_eta
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (hypothesis)
Goal:
z : ℂ
hz : z ≠ 0
h : Set.EqOn (η ∘ fun (z : ℂ) ↦ -1 / z) (z • (sqrt * η)) upperHalfPlaneSet
⊢ Set.EqOn (η ∘ fun (x : ℂ) ↦ -1 / x) (I.sqrt⁻¹ • (sqrt * η)) upperHalfPlaneSet
have h3 : η I = z * I.sqrt * η I
Tactic: have h3 : η I = z * sqrt I * η I := by simpa [← mul_assoc] using h (show I ∈ _ by simp)
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
⊢ @SlashAction.map ℤ _ _ Matrix.SpecialLinearGroup.monoid (Pi.addMonoid (f := fun (a : ℍ) ↦ ℂ)) _ 12 ModularGroup.S Δ =
    Δ
have h : ∀ (z : ℍ),
  @SlashAction.map ℤ _ _ Matrix.SpecialLinearGroup.monoid (Pi.addMonoid (f := fun (a : ℍ) ↦ ℂ)) _ 12 ModularGroup.S Δ
      z =
    Δ z
Tactic: ext z
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
z : ℍ
this : HPow.hPow (β := ℕ) (η (-(↑z : ℂ)⁻¹)) 24 * (HPow.hPow (β := ℕ) (↑z : ℂ) 12)⁻¹ = HPow.hPow (β := ℕ) (η (↑z : ℂ)) 24
⊢ @SlashAction.map ℤ _ _ Matrix.SpecialLinearGroup.monoid (Pi.addMonoid (f := fun (a : ℍ) ↦ ℂ)) _ 12 ModularGroup.S Δ
      z =
    Δ z
have rw : Δ { coe := (-(↑z : ℂ))⁻¹, coe_im_pos := sorry : ℍ } *
    HPow.hPow (β := ℤ)
      (denom
        ((Matrix.SpecialLinearGroup.toGL : (a : SL(2, ℝ)) → GL (Fin 2) ℝ)
          ((Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ) : (a : SL(2, ℤ)) → SL(2, ℝ)) ModularGroup.S))
        (↑z : ℂ))
      (-12) =
  Δ z
Tactic: rw [SL_slash_apply, UpperHalfPlane.modular_S_smul]
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
z : ℍ
⊢ @SlashAction.map ℤ _ _ Matrix.SpecialLinearGroup.monoid (Pi.addMonoid (f := fun (a : ℍ) ↦ ℂ)) _ 12 ModularGroup.S Δ
      z =
    Δ z
have h : HPow.hPow (β := ℕ) (η (-(↑z : ℂ)⁻¹)) 24 * (HPow.hPow (β := ℕ) (↑z : ℂ) 12)⁻¹ = HPow.hPow (β := ℕ) (η (↑z : ℂ)) 24
Tactic: suffices η (-(↑z)⁻¹) ^ 24 * ((z : ℂ) ^ 12)⁻¹ = η z ^ 24 by
    rw [SL_slash_apply, UpperHalfPlane.modular_S_smul]
    simpa [denom, ModularGroup.S]
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (hypothesis)
Goal:
z : ℍ
⊢ HPow.hPow (β := ℕ) (η (-(↑z : ℂ)⁻¹)) 24 * (HPow.hPow (β := ℕ) (↑z : ℂ) 12)⁻¹ = HPow.hPow (β := ℕ) (η (↑z : ℂ)) 24
have he : η (-(↑z : ℂ)⁻¹) = I.sqrt⁻¹ * ((↑z : ℂ).sqrt * η (↑z : ℂ))
Tactic: have he : η (-(↑z)⁻¹) = (sqrt I)⁻¹ * (sqrt z * η z) := by
    simpa [neg_div] using eta_comp_eq_csqrt_I_inv z.2
-- Lean 4 proof step from Mathlib.NumberTheory.ModularForms.Discriminant (goal)
Goal:
z : ℍ
he : η (-(↑z : ℂ)⁻¹) = I.sqrt⁻¹ * ((↑z : ℂ).sqrt * η (↑z : ℂ))
⊢ HPow.hPow (β := ℕ) (η (-(↑z : ℂ)⁻¹)) 24 * (HPow.hPow (β := ℕ) (↑z : ℂ) 12)⁻¹ = HPow.hPow (β := ℕ) (η (↑z : ℂ)) 24
have h : 1⁻¹ * (HPow.hPow (β := ℕ) (↑z : ℂ) 12 * HPow.hPow (β := ℕ) (η (↑z : ℂ)) 24) * (HPow.hPow (β := ℕ) (↑z : ℂ) 12)⁻¹ =
  HPow.hPow (β := ℕ) (η (↑z : ℂ)) 24
Tactic: simp only [he, mul_pow, mul_pow, inv_pow, csqrt_I_pow_24, csqrt_pow_24_eq (ne_zero z)]