awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.EulerProduct.DirichletLSeries (goal)
Goal:
M N : ℕ
inst : NeZero N
hMN : M ∣ N
χ : DirichletCharacter ℂ M
s : ℂ
hs : 1 < s.re
this : ∀ (f : Primes → ℂ), Eq (α := ℂ) (∏' (p : Primes), f p) (∏' (p : (↑{p : ℕ | Nat.Prime p} : Type)), f p)
p : ℕ
h : Nat.Prime p
hp' : ¬p ∣ N
⊢ (1 -
        HMul.hMul (α := ℂ)
          ((((changeLevel hMN : (a : DirichletCharacter ℂ M) → DirichletCharacter ℂ N) χ : DirichletCharacter ℂ N) :
                (a : ZMod N) → ℂ)
              (↑p : ZMod N) :
            ℂ)
          (HPow.hPow (α := ℂ) (↑p : ℂ) (-s)))⁻¹ =
    if p ∣ N ∧ ¬N = 0 then
      (1 - HMul.hMul (α := ℂ) ((χ : (a : ZMod M) → ℂ) (↑p : ZMod M) : ℂ) (HPow.hPow (α := ℂ) (↑p : ℂ) (-s)))⁻¹ *
        (1 - HMul.hMul (α := ℂ) ((χ : (a : ZMod M) → ℂ) (↑p : ZMod M) : ℂ) (HPow.hPow (α := ℂ) (↑p : ℂ) (-s)))
    else (1 - HMul.hMul (α := ℂ) ((χ : (a : ZMod M) → ℂ) (↑p : ZMod M) : ℂ) (HPow.hPow (α := ℂ) (↑p : ℂ) (-s)))⁻¹
have neg : Eq (α := ℂ)
  ((((changeLevel hMN : (a : DirichletCharacter ℂ M) → DirichletCharacter ℂ N) χ : DirichletCharacter ℂ N) :
        (a : ZMod N) → ℂ)
      (↑p : ZMod N) :
    ℂ)
  ((χ : (a : ZMod M) → ℂ) (↑p : ZMod M) : ℂ)
Tactic: simp only [hp', false_and, ↓reduceIte, inv_inj, sub_right_inj, mul_eq_mul_right_iff,
        cpow_eq_zero_iff, Nat.cast_eq_zero, h.ne_zero, ne_eq, neg_eq_zero, or_false]
-- Lean 4 proof step from Mathlib.NumberTheory.EulerProduct.DirichletLSeries (hypothesis)
Goal:
M N : ℕ
inst : NeZero N
hMN : M ∣ N
χ : DirichletCharacter ℂ M
s : ℂ
hs : 1 < s.re
this : ∀ (f : Primes → ℂ), Eq (α := ℂ) (∏' (p : Primes), f p) (∏' (p : (↑{p : ℕ | Nat.Prime p} : Type)), f p)
p : ℕ
h : Nat.Prime p
hp' : ¬p ∣ N
⊢ Eq (α := ℂ)
    ((((changeLevel hMN : (a : DirichletCharacter ℂ M) → DirichletCharacter ℂ N) χ : DirichletCharacter ℂ N) :
          (a : ZMod N) → ℂ)
        (↑p : ZMod N) :
      ℂ)
    ((χ : (a : ZMod M) → ℂ) (↑p : ZMod M) : ℂ)
have hq : IsUnit (M := ZMod N) (↑p : ZMod N)
Tactic: have hq : IsUnit (p : ZMod N) := (ZMod.isUnit_prime_iff_not_dvd h).mpr hp'
-- Lean 4 proof step from Mathlib.NumberTheory.EulerProduct.DirichletLSeries (goal)
Goal:
M N : ℕ
inst : NeZero N
hMN : M ∣ N
χ : DirichletCharacter ℂ M
s : ℂ
hs : 1 < s.re
this : ∀ (f : Primes → ℂ), Eq (α := ℂ) (∏' (p : Primes), f p) (∏' (p : (↑{p : ℕ | Nat.Prime p} : Type)), f p)
p : ℕ
h : Nat.Prime p
hp' : p ∣ N
⊢ (1 -
        HMul.hMul (α := ℂ)
          ((((changeLevel hMN : (a : DirichletCharacter ℂ M) → DirichletCharacter ℂ N) χ : DirichletCharacter ℂ N) :
                (a : ZMod N) → ℂ)
              (↑p : ZMod N) :
            ℂ)
          (HPow.hPow (α := ℂ) (↑p : ℂ) (-s)))⁻¹ =
    if p ∣ N ∧ ¬N = 0 then
      (1 - HMul.hMul (α := ℂ) ((χ : (a : ZMod M) → ℂ) (↑p : ZMod M) : ℂ) (HPow.hPow (α := ℂ) (↑p : ℂ) (-s)))⁻¹ *
        (1 - HMul.hMul (α := ℂ) ((χ : (a : ZMod M) → ℂ) (↑p : ZMod M) : ℂ) (HPow.hPow (α := ℂ) (↑p : ℂ) (-s)))
    else (1 - HMul.hMul (α := ℂ) ((χ : (a : ZMod M) → ℂ) (↑p : ZMod M) : ℂ) (HPow.hPow (α := ℂ) (↑p : ℂ) (-s)))⁻¹
have pos : (1 -
      HMul.hMul (α := ℂ)
        ((((changeLevel hMN : (a : DirichletCharacter ℂ M) → DirichletCharacter ℂ N) χ : DirichletCharacter ℂ N) :
              (a : ZMod N) → ℂ)
            (↑p : ZMod N) :
          ℂ)
        (HPow.hPow (α := ℂ) (↑p : ℂ) (-s)))⁻¹ =
  (1 - HMul.hMul (α := ℂ) ((χ : (a : ZMod M) → ℂ) (↑p : ZMod M) : ℂ) (HPow.hPow (α := ℂ) (↑p : ℂ) (-s)))⁻¹ *
    (1 - HMul.hMul (α := ℂ) ((χ : (a : ZMod M) → ℂ) (↑p : ZMod M) : ℂ) (HPow.hPow (α := ℂ) (↑p : ℂ) (-s)))
Tactic: simp only [hp', NeZero.ne N, not_false_eq_true, and_self, ↓reduceIte]
-- Lean 4 proof step from Mathlib.NumberTheory.EulerProduct.DirichletLSeries (goal)
Goal:
M N : ℕ
inst : NeZero N
hMN : M ∣ N
χ : DirichletCharacter ℂ M
s : ℂ
hs : 1 < s.re
this : ∀ (f : Primes → ℂ), Eq (α := ℂ) (∏' (p : Primes), f p) (∏' (p : (↑{p : ℕ | Nat.Prime p} : Type)), f p)
p : ℕ
h : Nat.Prime p
hp' : p ∣ N
⊢ ¬IsUnit (M := ZMod N) (↑p : ZMod N)
have rwa : ¬¬p ∣ N
Tactic: rwa [ZMod.isUnit_prime_iff_not_dvd h, not_not]
-- Lean 4 proof step from Mathlib.NumberTheory.EulerProduct.DirichletLSeries (hypothesis)
Goal:
M N : ℕ
inst : NeZero N
hMN : M ∣ N
χ : DirichletCharacter ℂ M
s : ℂ
hs : 1 < s.re
this : ∀ (f : Primes → ℂ), Eq (α := ℂ) (∏' (p : Primes), f p) (∏' (p : (↑{p : ℕ | Nat.Prime p} : Type)), f p)
p : ℕ
h : Nat.Prime p
hp' : p ∣ N
⊢ (1 -
        HMul.hMul (α := ℂ)
          ((((changeLevel hMN : (a : DirichletCharacter ℂ M) → DirichletCharacter ℂ N) χ : DirichletCharacter ℂ N) :
                (a : ZMod N) → ℂ)
              (↑p : ZMod N) :
            ℂ)
          (HPow.hPow (α := ℂ) (↑p : ℂ) (-s)))⁻¹ =
    (1 - HMul.hMul (α := ℂ) ((χ : (a : ZMod M) → ℂ) (↑p : ZMod M) : ℂ) (HPow.hPow (α := ℂ) (↑p : ℂ) (-s)))⁻¹ *
      (1 - HMul.hMul (α := ℂ) ((χ : (a : ZMod M) → ℂ) (↑p : ZMod M) : ℂ) (HPow.hPow (α := ℂ) (↑p : ℂ) (-s)))
have this₁ : ¬IsUnit (M := ZMod N) (↑p : ZMod N)
Tactic: have : ¬IsUnit (p : ZMod N) := by rwa [ZMod.isUnit_prime_iff_not_dvd h, not_not]
-- Lean 4 proof step from Mathlib.NumberTheory.EulerProduct.DirichletLSeries (goal)
Goal:
M N : ℕ
inst : NeZero N
hMN : M ∣ N
χ : DirichletCharacter ℂ M
s : ℂ
hs : 1 < s.re
this_1 : ∀ (f : Primes → ℂ), Eq (α := ℂ) (∏' (p : Primes), f p) (∏' (p : (↑{p : ℕ | Nat.Prime p} : Type)), f p)
p : ℕ
h : Nat.Prime p
hp' : p ∣ N
this : ¬IsUnit (M := ZMod N) (↑p : ZMod N)
⊢ (1 -
        HMul.hMul (α := ℂ)
          ((((changeLevel hMN : (a : DirichletCharacter ℂ M) → DirichletCharacter ℂ N) χ : DirichletCharacter ℂ N) :
                (a : ZMod N) → ℂ)
              (↑p : ZMod N) :
            ℂ)
          (HPow.hPow (α := ℂ) (↑p : ℂ) (-s)))⁻¹ =
    (1 - HMul.hMul (α := ℂ) ((χ : (a : ZMod M) → ℂ) (↑p : ZMod M) : ℂ) (HPow.hPow (α := ℂ) (↑p : ℂ) (-s)))⁻¹ *
      (1 - HMul.hMul (α := ℂ) ((χ : (a : ZMod M) → ℂ) (↑p : ZMod M) : ℂ) (HPow.hPow (α := ℂ) (↑p : ℂ) (-s)))
have pos : 1 =
  (1 - HMul.hMul (α := ℂ) ((χ : (a : ZMod M) → ℂ) (↑p : ZMod M) : ℂ) (HPow.hPow (α := ℂ) (↑p : ℂ) (-s)))⁻¹ *
    (1 - HMul.hMul (α := ℂ) ((χ : (a : ZMod M) → ℂ) (↑p : ZMod M) : ℂ) (HPow.hPow (α := ℂ) (↑p : ℂ) (-s)))
Tactic: rw [MulChar.map_nonunit _ this, zero_mul, sub_zero, inv_one]
-- Lean 4 proof step from Mathlib.NumberTheory.EulerProduct.DirichletLSeries (goal)
Goal:
M N : ℕ
inst : NeZero N
hMN : M ∣ N
χ : DirichletCharacter ℂ M
s : ℂ
hs : 1 < s.re
this_1 : ∀ (f : Primes → ℂ), Eq (α := ℂ) (∏' (p : Primes), f p) (∏' (p : (↑{p : ℕ | Nat.Prime p} : Type)), f p)
p : ℕ
h : Nat.Prime p
hp' : p ∣ N
this : ¬IsUnit (M := ZMod N) (↑p : ZMod N)
⊢ 1 =
    (1 - HMul.hMul (α := ℂ) ((χ : (a : ZMod M) → ℂ) (↑p : ZMod M) : ℂ) (HPow.hPow (α := ℂ) (↑p : ℂ) (-s)))⁻¹ *
      (1 - HMul.hMul (α := ℂ) ((χ : (a : ZMod M) → ℂ) (↑p : ZMod M) : ℂ) (HPow.hPow (α := ℂ) (↑p : ℂ) (-s)))
have pos : 1 - HMul.hMul (α := ℂ) ((χ : (a : ZMod M) → ℂ) (↑p : ZMod M) : ℂ) (HPow.hPow (α := ℂ) (↑p : ℂ) (-s)) ≠ 0
Tactic: refine (inv_mul_cancel₀ ?_).symm
-- Lean 4 proof step from Mathlib.NumberTheory.EulerProduct.DirichletLSeries (goal)
Goal:
M N : ℕ
inst : NeZero N
hMN : M ∣ N
χ : DirichletCharacter ℂ M
s : ℂ
hs : 1 < s.re
this_1 : ∀ (f : Primes → ℂ), Eq (α := ℂ) (∏' (p : Primes), f p) (∏' (p : (↑{p : ℕ | Nat.Prime p} : Type)), f p)
p : ℕ
h : Nat.Prime p
hp' : p ∣ N
this : ¬IsUnit (M := ZMod N) (↑p : ZMod N)
⊢ 1 - HMul.hMul (α := ℂ) ((χ : (a : ZMod M) → ℂ) (↑p : ZMod M) : ℂ) (HPow.hPow (α := ℂ) (↑p : ℂ) (-s)) ≠ 0
have pos : HMul.hMul (α := ℂ) ((χ : (a : ZMod M) → ℂ) (↑p : ZMod M) : ℂ) (HPow.hPow (α := ℂ) (↑p : ℂ) (-s)) ≠ 1
Tactic: rw [sub_ne_zero, ne_comm]
-- Lean 4 proof step from Mathlib.NumberTheory.EulerProduct.DirichletLSeries (goal)
Goal:
M N : ℕ
inst : NeZero N
hMN : M ∣ N
χ : DirichletCharacter ℂ M
s : ℂ
hs : 1 < s.re
this_1 : ∀ (f : Primes → ℂ), Eq (α := ℂ) (∏' (p : Primes), f p) (∏' (p : (↑{p : ℕ | Nat.Prime p} : Type)), f p)
p : ℕ
h : Nat.Prime p
hp' : p ∣ N
this : ¬IsUnit (M := ZMod N) (↑p : ZMod N)
⊢ HMul.hMul (α := ℂ) ((χ : (a : ZMod M) → ℂ) (↑p : ZMod M) : ℂ) (HPow.hPow (α := ℂ) (↑p : ℂ) (-s)) ≠ 1
have apply_fun : Ne (α := ℝ)
  ((fun (x : ℂ) ↦ ‖x‖)
    (HMul.hMul (α := ℂ) ((χ : (a : ZMod M) → ℂ) (↑p : ZMod M) : ℂ) (HPow.hPow (α := ℂ) (↑p : ℂ) (-s))))
  ((fun (x : ℂ) ↦ ‖x‖) 1)
Tactic: apply_fun (‖·‖)
-- Lean 4 proof step from Mathlib.NumberTheory.EulerProduct.DirichletLSeries (goal)
Goal:
M N : ℕ
inst : NeZero N
hMN : M ∣ N
χ : DirichletCharacter ℂ M
s : ℂ
hs : 1 < s.re
this_1 : ∀ (f : Primes → ℂ), Eq (α := ℂ) (∏' (p : Primes), f p) (∏' (p : (↑{p : ℕ | Nat.Prime p} : Type)), f p)
p : ℕ
h : Nat.Prime p
hp' : p ∣ N
this : ¬IsUnit (M := ZMod N) (↑p : ZMod N)
⊢ Ne (α := ℝ)
    ((fun (x : ℂ) ↦ ‖x‖)
      (HMul.hMul (α := ℂ) ((χ : (a : ZMod M) → ℂ) (↑p : ZMod M) : ℂ) (HPow.hPow (α := ℂ) (↑p : ℂ) (-s))))
    ((fun (x : ℂ) ↦ ‖x‖) 1)
have simp : norm (E := ℂ) ((χ : (a : ZMod M) → ℂ) (↑p : ZMod M) : ℂ) * ‖HPow.hPow (α := ℂ) (↑p : ℂ) (-s)‖ ≠ 1
Tactic: simp only [norm_mul, norm_one]
-- Lean 4 proof step from Mathlib.NumberTheory.EulerProduct.DirichletLSeries (hypothesis)
Goal:
M N : ℕ
inst : NeZero N
hMN : M ∣ N
χ : DirichletCharacter ℂ M
s : ℂ
hs : 1 < s.re
this_1 : ∀ (f : Primes → ℂ), Eq (α := ℂ) (∏' (p : Primes), f p) (∏' (p : (↑{p : ℕ | Nat.Prime p} : Type)), f p)
p : ℕ
h : Nat.Prime p
hp' : p ∣ N
this : ¬IsUnit (M := ZMod N) (↑p : ZMod N)
⊢ norm (E := ℂ) ((χ : (a : ZMod M) → ℂ) (↑p : ZMod M) : ℂ) * ‖HPow.hPow (α := ℂ) (↑p : ℂ) (-s)‖ ≠ 1
have ha : norm (E := ℂ) ((χ : (a : ZMod M) → ℂ) (↑p : ZMod M) : ℂ) ≤ 1
Tactic: have ha : ‖χ p‖ ≤ 1 := χ.norm_le_one p
-- Lean 4 proof step from Mathlib.NumberTheory.EulerProduct.DirichletLSeries (hypothesis)
Goal:
M N : ℕ
inst : NeZero N
hMN : M ∣ N
χ : DirichletCharacter ℂ M
s : ℂ
hs : 1 < s.re
this_1 : ∀ (f : Primes → ℂ), Eq (α := ℂ) (∏' (p : Primes), f p) (∏' (p : (↑{p : ℕ | Nat.Prime p} : Type)), f p)
p : ℕ
h : Nat.Prime p
hp' : p ∣ N
this : ¬IsUnit (M := ZMod N) (↑p : ZMod N)
ha : norm (E := ℂ) ((χ : (a : ZMod M) → ℂ) (↑p : ZMod M) : ℂ) ≤ 1
⊢ norm (E := ℂ) ((χ : (a : ZMod M) → ℂ) (↑p : ZMod M) : ℂ) * ‖HPow.hPow (α := ℂ) (↑p : ℂ) (-s)‖ ≠ 1
have hb : ‖HPow.hPow (α := ℂ) (↑p : ℂ) (-s)‖ ≤ 1 / 2
Tactic: have hb : ‖(p : ℂ) ^ (-s)‖ ≤ 1 / 2 := norm_prime_cpow_le_one_half ⟨p, h⟩ hs
-- Lean 4 proof step from Mathlib.NumberTheory.EulerProduct.ExpLog (hypothesis)
Goal:
α : Type u_1
f : α → ℂ
hsum : Summable f
⊢ DifferentiableAt ℂ (F := ℂ) (fun (z : ℂ) ↦ log (1 - z)) 0
have this : HSub.hSub (α := ℂ) 1 0 ∈ slitPlane
Tactic: have : 1 - 0 ∈ slitPlane := (sub_zero (1 : ℂ)).symm ▸ one_mem_slitPlane
-- Lean 4 proof step from Mathlib.NumberTheory.EulerProduct.ExpLog (hypothesis)
Goal:
α : Type u_1
f : α → ℂ
hsum : Summable f
⊢ Summable (α := ℂ) fun (n : α) ↦ log (1 - f n)
have hg : DifferentiableAt ℂ (F := ℂ) (fun (z : ℂ) ↦ log (1 - z)) 0
Tactic: have hg : DifferentiableAt ℂ (fun z ↦ log (1 - z)) 0 := by
    have : 1 - 0 ∈ slitPlane := (sub_zero (1 : ℂ)).symm ▸ one_mem_slitPlane
    fun_prop
-- Lean 4 proof step from Mathlib.NumberTheory.EulerProduct.ExpLog (hypothesis)
Goal:
α : Type u_1
f : α → ℂ
hsum : Summable f
hg : DifferentiableAt ℂ (F := ℂ) (fun (z : ℂ) ↦ log (1 - z)) 0
⊢ Summable (α := ℂ) fun (n : α) ↦ log (1 - f n)
have this : Asymptotics.IsBigO (E := ℂ) (𝓝 0) (fun (z : ℂ) ↦ log (1 - z)) id
Tactic: have : (fun z ↦ log (1 - z)) =O[𝓝 0] id := by
    simpa only [sub_zero, log_one] using hg.isBigO_sub
-- Lean 4 proof step from Mathlib.NumberTheory.EulerProduct.ExpLog (hypothesis)
Goal:
f : ℕ →*₀ ℂ
hsum : Summable (α := ℝ) fun (x : ℕ) ↦ norm (E := ℂ) ((f : (a : ℕ) → ℂ) x : ℂ)
⊢ cexp (∑' (p : Nat.Primes), -log (HSub.hSub (α := ℂ) (β := ℂ) (1 : ℂ) ((f : (a : ℕ) → ℂ) (↑p : ℕ) : ℂ))) =
    ∑' (n : ℕ), (f : (a : ℕ) → ℂ) n
have hs : ∀ {p : ℕ} (hp : 1 < p), norm (E := ℂ) ((f : (a : ℕ) → ℂ) p : ℂ) < 1
Tactic: have hs {p : ℕ} (hp : 1 < p) : ‖f p‖ < 1 := hsum.of_norm.norm_lt_one (f := f.toMonoidHom) hp
-- Lean 4 proof step from Mathlib.NumberTheory.EulerProduct.ExpLog (hypothesis)
Goal:
f : ℕ →*₀ ℂ
hsum : Summable (α := ℝ) fun (x : ℕ) ↦ norm (E := ℂ) ((f : (a : ℕ) → ℂ) x : ℂ)
hs : ∀ {p : ℕ} (hp : 1 < p), norm (E := ℂ) ((f : (a : ℕ) → ℂ) p : ℂ) < 1
⊢ cexp (∑' (p : Nat.Primes), -log (HSub.hSub (α := ℂ) (β := ℂ) (1 : ℂ) ((f : (a : ℕ) → ℂ) (↑p : ℕ) : ℂ))) =
    ∑' (n : ℕ), (f : (a : ℕ) → ℂ) n
have hp : ∀ (p : Nat.Primes), HSub.hSub (α := ℂ) (β := ℂ) (1 : ℂ) ((f : (a : ℕ) → ℂ) (↑p : ℕ) : ℂ) ≠ 0
Tactic: have hp (p : Nat.Primes) : 1 - f p ≠ 0 :=
    fun h ↦ (norm_one (α := ℂ) ▸ (sub_eq_zero.mp h) ▸ hs p.prop.one_lt).false
-- Lean 4 proof step from Mathlib.NumberTheory.EulerProduct.ExpLog (hypothesis)
Goal:
f : ℕ →*₀ ℂ
hsum : Summable (α := ℝ) fun (x : ℕ) ↦ norm (E := ℂ) ((f : (a : ℕ) → ℂ) x : ℂ)
hs : ∀ {p : ℕ} (hp : 1 < p), norm (E := ℂ) ((f : (a : ℕ) → ℂ) p : ℂ) < 1
hp : ∀ (p : Nat.Primes), HSub.hSub (α := ℂ) (β := ℂ) (1 : ℂ) ((f : (a : ℕ) → ℂ) (↑p : ℕ) : ℂ) ≠ 0
⊢ cexp (∑' (p : Nat.Primes), -log (HSub.hSub (α := ℂ) (β := ℂ) (1 : ℂ) ((f : (a : ℕ) → ℂ) (↑p : ℕ) : ℂ))) =
    ∑' (n : ℕ), (f : (a : ℕ) → ℂ) n
have H : ∏' (b : Subtype Nat.Prime),
    (cexp ∘
        (fun (b : ℕ) ↦ -log (HSub.hSub (α := ℂ) (β := ℂ) (1 : ℂ) ((f : (a : ℕ) → ℂ) b : ℂ))) ∘
          Subtype.val (p := Nat.Prime))
      b =
  cexp
    (∑' (b : Subtype Nat.Prime),
      ((fun (b : ℕ) ↦ -log (HSub.hSub (α := ℂ) (β := ℂ) (1 : ℂ) ((f : (a : ℕ) → ℂ) b : ℂ))) ∘
          Subtype.val (p := Nat.Prime))
        b)
Tactic: have H := hsum.of_norm.clog_one_sub.neg.subtype Nat.Prime |>.hasSum.cexp.tprod_eq
-- Lean 4 proof step from Mathlib.NumberTheory.EulerProduct.ExpLog (hypothesis)
Goal:
f : ℕ →*₀ ℂ
hsum : Summable (α := ℝ) fun (x : ℕ) ↦ norm (E := ℂ) ((f : (a : ℕ) → ℂ) x : ℂ)
hs : ∀ {p : ℕ} (hp : 1 < p), norm (E := ℂ) ((f : (a : ℕ) → ℂ) p : ℂ) < 1
hp : ∀ (p : Nat.Primes), HSub.hSub (α := ℂ) (β := ℂ) (1 : ℂ) ((f : (a : ℕ) → ℂ) (↑p : ℕ) : ℂ) ≠ 0
H :
  ∏' (b : Subtype Nat.Prime),
      (cexp ∘
          (fun (b : ℕ) ↦ -log (HSub.hSub (α := ℂ) (β := ℂ) (1 : ℂ) ((f : (a : ℕ) → ℂ) b : ℂ))) ∘
            Subtype.val (p := Nat.Prime))
        b =
    cexp
      (∑' (b : Subtype Nat.Prime),
        ((fun (b : ℕ) ↦ -log (HSub.hSub (α := ℂ) (β := ℂ) (1 : ℂ) ((f : (a : ℕ) → ℂ) b : ℂ))) ∘
            Subtype.val (p := Nat.Prime))
          b)
⊢ cexp (∑' (p : Nat.Primes), -log (HSub.hSub (α := ℂ) (β := ℂ) (1 : ℂ) ((f : (a : ℕ) → ℂ) (↑p : ℕ) : ℂ))) =
    ∑' (n : ℕ), (f : (a : ℕ) → ℂ) n
have H₁ : ∏' (b : Subtype Nat.Prime), (HSub.hSub (α := ℂ) (β := ℂ) (1 : ℂ) ((f : (a : ℕ) → ℂ) (↑b : ℕ) : ℂ))⁻¹ =
  cexp (∑' (b : Subtype Nat.Prime), -log (HSub.hSub (α := ℂ) (β := ℂ) (1 : ℂ) ((f : (a : ℕ) → ℂ) (↑b : ℕ) : ℂ)))
Tactic: simp only [Function.comp_apply, exp_neg, exp_log (hp _)] at H
Cleared hypotheses: H
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
inst_1 : NoZeroDivisors R
m n : ℕ
hmn : m ∣ n
hm : FermatLastTheoremWith R m
a b c : R
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
⊢ a ^ n + b ^ n ≠ c ^ n
have obtain : ∀ (k : ℕ), a ^ (m * k) + b ^ (m * k) ≠ c ^ (m * k)
Tactic: obtain ⟨k, rfl⟩ := hmn
Cleared hypotheses: n, hmn
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
inst_1 : NoZeroDivisors R
m : ℕ
hm : FermatLastTheoremWith R m
a b c : R
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
k : ℕ
⊢ a ^ (m * k) + b ^ (m * k) ≠ c ^ (m * k)
have simp_rw : (a ^ k) ^ m + (b ^ k) ^ m ≠ (c ^ k) ^ m
Tactic: simp_rw [pow_mul']
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
inst_1 : NoZeroDivisors R
m : ℕ
hm : FermatLastTheoremWith R m
a b c : R
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
k : ℕ
⊢ (a ^ k) ^ m + (b ^ k) ^ m ≠ (c ^ k) ^ m
have refine_1 : a ^ k ≠ 0
Tactic: refine hm _ _ _ ?_ ?_ ?_
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
inst_1 : NoZeroDivisors R
m : ℕ
hm : FermatLastTheoremWith R m
a b c : R
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
k : ℕ
refine_1 : a ^ k ≠ 0
⊢ (a ^ k) ^ m + (b ^ k) ^ m ≠ (c ^ k) ^ m
have refine_2 : b ^ k ≠ 0
Tactic: refine hm _ _ _ ?_ ?_ ?_
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
inst_1 : NoZeroDivisors R
m : ℕ
hm : FermatLastTheoremWith R m
a b c : R
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
k : ℕ
refine_1 : a ^ k ≠ 0
refine_2 : b ^ k ≠ 0
⊢ (a ^ k) ^ m + (b ^ k) ^ m ≠ (c ^ k) ^ m
have refine_3 : c ^ k ≠ 0
Tactic: refine hm _ _ _ ?_ ?_ ?_
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Basic (goal)
Goal:
n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
⊢ False
have inl : ∀ (hn : Even n), False
Tactic: obtain hn | hn := n.even_or_odd
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Basic (goal)
Goal:
n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
inl : ∀ (hn : Even n), False
⊢ False
have inr : ∀ (hn : Odd n), False
Tactic: obtain hn | hn := n.even_or_odd
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Basic (goal)
Goal:
n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Even n
⊢ False
have inl : Eq (α := ℤ) (↑(a.natAbs ^ n + b.natAbs ^ n) : ℤ) (↑(c.natAbs ^ n) : ℤ)
Tactic: refine h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.natCast_inj.1 ?_)
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Basic (goal)
Goal:
n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Even n
⊢ Eq (α := ℤ) (↑(a.natAbs ^ n + b.natAbs ^ n) : ℤ) (↑(c.natAbs ^ n) : ℤ)
have inl : |a| ^ n + |b| ^ n = |c| ^ n
Tactic: push_cast
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Basic (goal)
Goal:
n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
⊢ False
have inr : ∀ (ha : a < 0), False
Tactic: obtain ha | ha := ha.lt_or_gt
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Basic (goal)
Goal:
n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
inr : ∀ (ha : a < 0), False
⊢ False
have inr₁ : ∀ (ha : 0 < a), False
Tactic: obtain ha | ha := ha.lt_or_gt
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Basic (goal)
Goal:
n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha_1 : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : a < 0
⊢ False
have inr : ∀ (hb : b < 0), False
Tactic: obtain hb | hb := hb.lt_or_gt
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Basic (goal)
Goal:
n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha_1 : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : a < 0
inr : ∀ (hb : b < 0), False
⊢ False
have inr₁ : ∀ (hb : 0 < b), False
Tactic: obtain hb | hb := hb.lt_or_gt
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Basic (goal)
Goal:
n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha_1 : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : 0 < a
⊢ False
have inr : ∀ (hb : b < 0), False
Tactic: obtain hb | hb := hb.lt_or_gt
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Basic (goal)
Goal:
n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha_1 : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : 0 < a
inr : ∀ (hb : b < 0), False
⊢ False
have inr₁ : ∀ (hb : 0 < b), False
Tactic: obtain hb | hb := hb.lt_or_gt
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Basic (goal)
Goal:
n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha_1 : a ≠ 0
hb_1 : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : a < 0
hb : b < 0
⊢ False
have inr : ∀ (hc : c < 0), False
Tactic: obtain hc | hc := hc.lt_or_gt
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Basic (goal)
Goal:
n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha_1 : a ≠ 0
hb_1 : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : a < 0
hb : b < 0
inr : ∀ (hc : c < 0), False
⊢ False
have inr₁ : ∀ (hc : 0 < c), False
Tactic: obtain hc | hc := hc.lt_or_gt
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Basic (goal)
Goal:
n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha_1 : a ≠ 0
hb_1 : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : a < 0
hb : 0 < b
⊢ False
have inr : ∀ (hc : c < 0), False
Tactic: obtain hc | hc := hc.lt_or_gt
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Basic (goal)
Goal:
n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha_1 : a ≠ 0
hb_1 : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : a < 0
hb : 0 < b
inr : ∀ (hc : c < 0), False
⊢ False
have inr₁ : ∀ (hc : 0 < c), False
Tactic: obtain hc | hc := hc.lt_or_gt
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Basic (goal)
Goal:
n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha_1 : a ≠ 0
hb_1 : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : 0 < a
hb : b < 0
⊢ False
have inr : ∀ (hc : c < 0), False
Tactic: obtain hc | hc := hc.lt_or_gt
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Basic (goal)
Goal:
n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha_1 : a ≠ 0
hb_1 : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : 0 < a
hb : b < 0
inr : ∀ (hc : c < 0), False
⊢ False
have inr₁ : ∀ (hc : 0 < c), False
Tactic: obtain hc | hc := hc.lt_or_gt
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Basic (goal)
Goal:
n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha_1 : a ≠ 0
hb_1 : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : 0 < a
hb : 0 < b
⊢ False
have inr : ∀ (hc : c < 0), False
Tactic: obtain hc | hc := hc.lt_or_gt
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Basic (goal)
Goal:
n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha_1 : a ≠ 0
hb_1 : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : 0 < a
hb : 0 < b
inr : ∀ (hc : c < 0), False
⊢ False
have inr₁ : ∀ (hc : 0 < c), False
Tactic: obtain hc | hc := hc.lt_or_gt
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Basic (goal)
Goal:
n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha_1 : a ≠ 0
hb_1 : b ≠ 0
hc_1 : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : a < 0
hb : b < 0
hc : c < 0
⊢ False
have inr : Eq (α := ℤ) (↑(a.natAbs ^ n + b.natAbs ^ n) : ℤ) (↑(c.natAbs ^ n) : ℤ)
Tactic: refine h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.natCast_inj.1 ?_)
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Basic (goal)
Goal:
n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha_1 : a ≠ 0
hb_1 : b ≠ 0
hc_1 : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : a < 0
hb : b < 0
hc : c < 0
⊢ Eq (α := ℤ) (↑(a.natAbs ^ n + b.natAbs ^ n) : ℤ) (↑(c.natAbs ^ n) : ℤ)
have inr : |a| ^ n + |b| ^ n = |c| ^ n
Tactic: push_cast
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Basic (goal)
Goal:
n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha_1 : a ≠ 0
hb_1 : b ≠ 0
hc_1 : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : a < 0
hb : 0 < b
hc : c < 0
⊢ False
have inr : Eq (α := ℤ) (↑(b.natAbs ^ n + c.natAbs ^ n) : ℤ) (↑(a.natAbs ^ n) : ℤ)
Tactic: refine h b.natAbs c.natAbs a.natAbs (by positivity) (by positivity) (by positivity)
        (Int.natCast_inj.1 ?_)
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Basic (goal)
Goal:
n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha_1 : a ≠ 0
hb_1 : b ≠ 0
hc_1 : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : a < 0
hb : 0 < b
hc : c < 0
⊢ Eq (α := ℤ) (↑(b.natAbs ^ n + c.natAbs ^ n) : ℤ) (↑(a.natAbs ^ n) : ℤ)
have inr : |b| ^ n + |c| ^ n = |a| ^ n
Tactic: push_cast
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Basic (goal)
Goal:
n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha_1 : a ≠ 0
hb_1 : b ≠ 0
hc_1 : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : a < 0
hb : 0 < b
hc : 0 < c
⊢ False
have inr : Eq (α := ℤ) (↑(a.natAbs ^ n + c.natAbs ^ n) : ℤ) (↑(b.natAbs ^ n) : ℤ)
Tactic: refine h a.natAbs c.natAbs b.natAbs (by positivity) (by positivity) (by positivity)
        (Int.natCast_inj.1 ?_)
-- Lean 4 proof step from Mathlib.NumberTheory.FLT.Basic (goal)
Goal:
n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha_1 : a ≠ 0
hb_1 : b ≠ 0
hc_1 : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : a < 0
hb : 0 < b
hc : 0 < c
⊢ Eq (α := ℤ) (↑(a.natAbs ^ n + c.natAbs ^ n) : ℤ) (↑(b.natAbs ^ n) : ℤ)
have inr : |a| ^ n + |c| ^ n = |b| ^ n
Tactic: push_cast