mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.LSeries.Dirichlet

Goal state before:
pp: s : ℂ
hs : 1 < s.re
hf : Summable fun x ↦ ‖term (logMul 1) s x‖
⊢ Summable fun x ↦ ‖term (fun n ↦ ↑(Λ n)) s x‖

Tactic:
]

Goal state after:
s : ℂ
hs : 1 < s.re
hf : Summable fun x ↦ ‖term (logMul 1) s x‖
⊢ Summable fun x ↦ ‖term (fun n ↦ ↑(Λ n)) s x‖
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaEven

Goal state before:
pp: a : ℝ
s : ℂ
hs : 1 < s.re
hF :
  ∀ (t : ℝ),
    0 < t →
      HasSum (fun n ↦ if ↑n + a = 0 then 0 else 1 / 2 * ↑(rexp (-π * (↑n + a) ^ 2 * t)))
        ((↑(evenKernel (↑a) t) - ↑(if ↑a = 0 then 1 else 0)) / 2)
⊢ 1 < s.re

Tactic:
]

Goal state after:
a : ℝ
s : ℂ
hs : 1 < s.re
hF :
  ∀ (t : ℝ),
    0 < t →
      HasSum (fun n ↦ if ↑n + a = 0 then 0 else 1 / 2 * ↑(rexp (-π * (↑n + a) ^ 2 * t)))
        ((↑(evenKernel (↑a) t) - ↑(if ↑a = 0 then 1 else 0)) / 2)
⊢ 1 < s.re
Module: Mathlib.NumberTheory.LSeries.Dirichlet

Goal state before:
pp: N : ℕ
χ : DirichletCharacter ℂ N
⊢ (((fun n ↦ χ ↑n) * fun n ↦ ↑(Λ n)) ⍟ fun n ↦ χ ↑n) = ((fun x ↦ χ ↑x) * fun n ↦ ↑(Λ n)) ⍟ fun x ↦ χ ↑x

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaEven

Goal state before:
pp: a : ℝ
s : ℂ
hs : 1 < s.re
hF :
  ∀ (t : ℝ),
    0 < t →
      HasSum (fun n ↦ if ↑n + a = 0 then 0 else 1 / 2 * ↑(rexp (-π * (↑n + a) ^ 2 * t)))
        ((↑(evenKernel (↑a) t) - ↑(if ↑a = 0 then 1 else 0)) / 2)
⊢ HasSum (fun n ↦ s.Gammaℝ / ↑|↑n + a| ^ s / 2)
    (mellin (fun t ↦ (↑(evenKernel (↑a) t) - ↑(if ↑a = 0 then 1 else 0)) / 2) (s / 2))

Tactic:
]

Goal state after:
a : ℝ
s : ℂ
hs : 1 < s.re
hF :
  ∀ (t : ℝ),
    0 < t →
      HasSum (fun n ↦ if ↑n + a = 0 then 0 else 1 / 2 * ↑(rexp (-π * (↑n + a) ^ 2 * t)))
        ((↑(evenKernel (↑a) t) - ↑(if ↑a = 0 then 1 else 0)) / 2)
⊢ HasSum (fun n ↦ s.Gammaℝ / ↑|↑n + a| ^ s / 2)
    (mellin (fun t ↦ (↑(evenKernel (↑a) t) - ↑(if ↑a = 0 then 1 else 0)) / 2) (s / 2))
Module: Mathlib.NumberTheory.LSeries.Dirichlet

Goal state before:
pp: N : ℕ
χ : DirichletCharacter ℂ N
s : ℂ
hs : 1 < s.re
hN : N ≠ 0
hχ : LSeriesSummable (fun n ↦ χ ↑n) s
⊢ 1 < s.re

Tactic:
]

Goal state after:
N : ℕ
χ : DirichletCharacter ℂ N
s : ℂ
hs : 1 < s.re
hN : N ≠ 0
hχ : LSeriesSummable (fun n ↦ χ ↑n) s
⊢ 1 < s.re
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaOdd

Goal state before:
pp: a p : ℝ
hp : 0 < p
hp' : HurwitzKernelBounds.F_nat 1 0 =O[atTop] fun t ↦ rexp (-p * t)
t : ℝ
ht : 0 < t
⊢ ‖∑' (b : ℕ), 2 * ↑b * Real.sin (2 * π * a * ↑b) * rexp (-π * ↑b ^ 2 * t)‖ ≤
    2 * ∑' (n : ℕ), HurwitzKernelBounds.f_nat 1 0 t n

Tactic:
]

Goal state after:
a p : ℝ
hp : 0 < p
hp' : HurwitzKernelBounds.F_nat 1 0 =O[atTop] fun t ↦ rexp (-p * t)
t : ℝ
ht : 0 < t
⊢ ‖∑' (b : ℕ), 2 * ↑b * Real.sin (2 * π * a * ↑b) * rexp (-π * ↑b ^ 2 * t)‖ ≤
    2 * ∑' (n : ℕ), HurwitzKernelBounds.f_nat 1 0 t n
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaEven

Goal state before:
pp: case refine_1
a : ℝ
s : ℂ
hs : 1 < s.re
hF :
  ∀ (t : ℝ),
    0 < t →
      HasSum (fun n ↦ if ↑n + a = 0 then 0 else 1 / 2 * ↑(rexp (-π * (↑n + a) ^ 2 * t)))
        ((↑(evenKernel (↑a) t) - ↑(if ↑a = 0 then 1 else 0)) / 2)
⊢ 1 < s.re

Tactic:
]

Goal state after:
case refine_1
a : ℝ
s : ℂ
hs : 1 < s.re
hF :
  ∀ (t : ℝ),
    0 < t →
      HasSum (fun n ↦ if ↑n + a = 0 then 0 else 1 / 2 * ↑(rexp (-π * (↑n + a) ^ 2 * t)))
        ((↑(evenKernel (↑a) t) - ↑(if ↑a = 0 then 1 else 0)) / 2)
⊢ 1 < s.re
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaEven

Goal state before:
pp: case refine_1
a : ℝ
s : ℂ
hs : 1 < s.re
hF :
  ∀ (t : ℝ),
    0 < t →
      HasSum (fun n ↦ if ↑n + a = 0 then 0 else 1 / 2 * ↑(rexp (-π * (↑n + a) ^ 2 * t)))
        ((↑(evenKernel (↑a) t) - ↑(if ↑a = 0 then 1 else 0)) / 2)
⊢ Summable fun i ↦ ‖1 / 2‖ / |↑i + a| ^ s.re

Tactic:
simp_rw [← mul_one_div ‖_‖]
    apply Summable.mul_left
    rwa [summable_one_div_int_add_rpow]

Goal state after:
Module: Mathlib.NumberTheory.LSeries.Dirichlet

Goal state before:
pp: case inr
N : ℕ
χ : DirichletCharacter ℂ N
s : ℂ
hs : 1 < s.re
hN : N ≠ 0
hχ : LSeriesSummable (fun n ↦ χ ↑n) s
hs' : (abscissaOfAbsConv fun n ↦ χ ↑n) < ↑s.re
hΛ : LSeriesSummable ((fun n ↦ χ ↑n) * fun n ↦ ↑(Λ n)) s
⊢ L ((fun n ↦ χ ↑n) * fun n ↦ Complex.log ↑n) s = L (logMul fun n ↦ χ ↑n) s

Tactic:
]

Goal state after:
case inr
N : ℕ
χ : DirichletCharacter ℂ N
s : ℂ
hs : 1 < s.re
hN : N ≠ 0
hχ : LSeriesSummable (fun n ↦ χ ↑n) s
hs' : (abscissaOfAbsConv fun n ↦ χ ↑n) < ↑s.re
hΛ : LSeriesSummable ((fun n ↦ χ ↑n) * fun n ↦ ↑(Λ n)) s
⊢ L ((fun n ↦ χ ↑n) * fun n ↦ Complex.log ↑n) s = L (logMul fun n ↦ χ ↑n) s
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaEven

Goal state before:
pp: case refine_2
a : ℝ
s : ℂ
hs : 1 < s.re
hF :
  ∀ (t : ℝ),
    0 < t →
      HasSum (fun n ↦ if ↑n + a = 0 then 0 else 1 / 2 * ↑(rexp (-π * (↑n + a) ^ 2 * t)))
        ((↑(evenKernel (↑a) t) - ↑(if ↑a = 0 then 1 else 0)) / 2)
n : ℤ
⊢ s.Gammaℝ / 2 / ↑|↑n + a| ^ s = s.Gammaℝ / 2 / ↑|↑n + a| ^ s

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaEven

Goal state before:
pp: Λ : ℂ → ℂ
hf : ∀ (s : ℂ), s ≠ 0 → s ≠ 1 → DifferentiableAt ℂ Λ s
L : ℂ
h_lim : Tendsto (fun s ↦ s * Λ s) (𝓝[≠] 0) (𝓝 L)
s✝ : ℂ
hs' : s✝ ≠ 1
claim : ∀ (t : ℂ), t ≠ 0 → t ≠ 1 → DifferentiableAt ℂ (fun u ↦ Λ u / u.Gammaℝ) t
s : ℂ
hs : s ≠ 0
⊢ Λ s / s.Gammaℝ = Λ s / s.Gammaℝ

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.LSeries.Dirichlet

Goal state before:
pp: s : ℂ
hs : 1 < s.re
this : deriv (L 1) s = deriv riemannZeta s
⊢ -deriv riemannZeta s / L 1 s = -deriv riemannZeta s / L 1 s

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaEven

Goal state before:
pp: case inr
Λ : ℂ → ℂ
hf : ∀ (s : ℂ), s ≠ 0 → s ≠ 1 → DifferentiableAt ℂ Λ s
L : ℂ
h_lim : Tendsto (fun s ↦ s * Λ s) (𝓝[≠] 0) (𝓝 L)
claim : ∀ (t : ℂ), t ≠ 0 → t ≠ 1 → DifferentiableAt ℂ (fun u ↦ Λ u / u.Gammaℝ) t
claim2 : Tendsto (fun s ↦ Λ s / s.Gammaℝ) (𝓝[≠] 0) (𝓝 (L / 2))
hs' : 0 ≠ 1
S_nhds : {1}ᶜ ∈ 𝓝 0
⊢ Tendsto (fun x ↦ 1 / x⁻¹) (𝓝[≠] 0) (𝓝 0)

pp: case inr
Λ : ℂ → ℂ
hf : ∀ (s : ℂ), s ≠ 0 → s ≠ 1 → DifferentiableAt ℂ Λ s
L : ℂ
h_lim : Tendsto (fun s ↦ s * Λ s) (𝓝[≠] 0) (𝓝 L)
claim : ∀ (t : ℂ), t ≠ 0 → t ≠ 1 → DifferentiableAt ℂ (fun u ↦ Λ u / u.Gammaℝ) t
claim2 : Tendsto (fun s ↦ Λ s / s.Gammaℝ) (𝓝[≠] 0) (𝓝 (L / 2))
hs' : 0 ≠ 1
S_nhds : {1}ᶜ ∈ 𝓝 0
⊢ ∀ᶠ (x : ℂ) in 𝓝[≠] 0, x⁻¹ = 0 → 1 = 0

Tactic:
]

Goal state after:
case inr
Λ : ℂ → ℂ
hf : ∀ (s : ℂ), s ≠ 0 → s ≠ 1 → DifferentiableAt ℂ Λ s
L : ℂ
h_lim : Tendsto (fun s ↦ s * Λ s) (𝓝[≠] 0) (𝓝 L)
claim : ∀ (t : ℂ), t ≠ 0 → t ≠ 1 → DifferentiableAt ℂ (fun u ↦ Λ u / u.Gammaℝ) t
claim2 : Tendsto (fun s ↦ Λ s / s.Gammaℝ) (𝓝[≠] 0) (𝓝 (L / 2))
hs' : 0 ≠ 1
S_nhds : {1}ᶜ ∈ 𝓝 0
⊢ Tendsto (fun x ↦ 1 / x⁻¹) (𝓝[≠] 0) (𝓝 0)

case inr
Λ : ℂ → ℂ
hf : ∀ (s : ℂ), s ≠ 0 → s ≠ 1 → DifferentiableAt ℂ Λ s
L : ℂ
h_lim : Tendsto (fun s ↦ s * Λ s) (𝓝[≠] 0) (𝓝 L)
claim : ∀ (t : ℂ), t ≠ 0 → t ≠ 1 → DifferentiableAt ℂ (fun u ↦ Λ u / u.Gammaℝ) t
claim2 : Tendsto (fun s ↦ Λ s / s.Gammaℝ) (𝓝[≠] 0) (𝓝 (L / 2))
hs' : 0 ≠ 1
S_nhds : {1}ᶜ ∈ 𝓝 0
⊢ ∀ᶠ (x : ℂ) in 𝓝[≠] 0, x⁻¹ = 0 → 1 = 0
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaEven

Goal state before:
pp: Λ : ℂ → ℂ
hf : ∀ (s : ℂ), s ≠ 0 → s ≠ 1 → DifferentiableAt ℂ Λ s
L : ℂ
h_lim : Tendsto (fun s ↦ s * Λ s) (𝓝[≠] 0) (𝓝 L)
claim : ∀ (t : ℂ), t ≠ 0 → t ≠ 1 → DifferentiableAt ℂ (fun u ↦ Λ u / u.Gammaℝ) t
claim2 : Tendsto (fun s ↦ Λ s / s.Gammaℝ) (𝓝[≠] 0) (𝓝 (L / 2))
hs' : 0 ≠ 1
S_nhds : {1}ᶜ ∈ 𝓝 0
x : ℂ
⊢ x = x

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaOdd

Goal state before:
pp: case h
a p : ℝ
hp : 0 < p
hp' : HurwitzKernelBounds.F_nat 1 0 =O[atTop] fun t ↦ rexp (-p * t)
t : ℝ
ht : 0 < t
n : ℕ
⊢ ↑n * ‖Real.sin (2 * π * a * ↑n)‖ ≤ ↑n

Tactic:
]

Goal state after:
case h
a p : ℝ
hp : 0 < p
hp' : HurwitzKernelBounds.F_nat 1 0 =O[atTop] fun t ↦ rexp (-p * t)
t : ℝ
ht : 0 < t
n : ℕ
⊢ ↑n * ‖Real.sin (2 * π * a * ↑n)‖ ≤ ↑n
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaEven

Goal state before:
pp: a : UnitAddCircle
s : ℂ
h : a ≠ 0 ∨ s ≠ 0
⊢ Function.update (fun s ↦ completedHurwitzZetaEven a s / s.Gammaℝ) 0 (if a = 0 then -1 / 2 else 0) s =
    completedHurwitzZetaEven a s / s.Gammaℝ

Tactic:
]

Goal state after:
a : UnitAddCircle
s : ℂ
h : a ≠ 0 ∨ s ≠ 0
⊢ Function.update (fun s ↦ completedHurwitzZetaEven a s / s.Gammaℝ) 0 (if a = 0 then -1 / 2 else 0) s =
    completedHurwitzZetaEven a s / s.Gammaℝ
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaEven

Goal state before:
pp: case inl
a : UnitAddCircle
s : ℂ
h : a ≠ 0 ∨ s ≠ 0
h' : s ≠ 0
⊢ completedHurwitzZetaEven a s / s.Gammaℝ = completedHurwitzZetaEven a s / s.Gammaℝ

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaOdd

Goal state before:
pp: a : UnitAddCircle
r v : ℝ
hv : 0 < v
hv' : (fun x ↦ ‖oddKernel a x‖) =O[atTop] fun x ↦ rexp (-v * x)
⊢ (fun x ↦ ‖(ofReal ∘ oddKernel a) x - 0‖) =O[atTop] fun x ↦ x ^ r

Tactic:
]

Goal state after:
a : UnitAddCircle
r v : ℝ
hv : 0 < v
hv' : (fun x ↦ ‖oddKernel a x‖) =O[atTop] fun x ↦ rexp (-v * x)
⊢ (fun x ↦ ‖(ofReal ∘ oddKernel a) x - 0‖) =O[atTop] fun x ↦ x ^ r
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaEven

Goal state before:
pp: a : UnitAddCircle
n : ℕ
this : -2 * (↑n + 1) ≠ 0
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaOdd

Goal state before:
pp: a : UnitAddCircle
r v : ℝ
hv : 0 < v
hv' : (fun x ↦ ‖sinKernel a x‖) =O[atTop] fun x ↦ rexp (-v * x)
⊢ (fun x ↦ ‖(ofReal ∘ sinKernel a) x - 0‖) =O[atTop] fun x ↦ x ^ r

Tactic:
]

Goal state after:
a : UnitAddCircle
r v : ℝ
hv : 0 < v
hv' : (fun x ↦ ‖sinKernel a x‖) =O[atTop] fun x ↦ rexp (-v * x)
⊢ (fun x ↦ ‖(ofReal ∘ sinKernel a) x - 0‖) =O[atTop] fun x ↦ x ^ r
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaEven

Goal state before:
pp: a : UnitAddCircle
this : DifferentiableAt ℂ (fun s ↦ completedHurwitzZetaEven a s / s.Gammaℝ - 1 / (s - 1) / s.Gammaℝ) 1
x : ℂ
hx : x ≠ 0
⊢ completedHurwitzZetaEven a x / x.Gammaℝ - 1 / (x - 1) / x.Gammaℝ =
    completedHurwitzZetaEven a x / x.Gammaℝ - 1 / (x - 1) / x.Gammaℝ

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.FLT.Basic

Goal state before:
pp: n : ℕ
tfae_1_to_2 : FermatLastTheoremWith ℕ n → FermatLastTheoremWith ℤ n
h : FermatLastTheoremWith ℤ n
a b c : ℚ
ha : a.num ≠ 0
hb : b.num ≠ 0
hc : c.num ≠ 0
habc : a ^ n + b ^ n = c ^ n
⊢ False

Tactic:
]

Goal state after:
n : ℕ
tfae_1_to_2 : FermatLastTheoremWith ℕ n → FermatLastTheoremWith ℤ n
h : FermatLastTheoremWith ℤ n
a b c : ℚ
ha : a.num ≠ 0
hb : b.num ≠ 0
hc : c.num ≠ 0
habc : a ^ n + b ^ n = c ^ n
⊢ False
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaOdd

Goal state before:
pp: a : UnitAddCircle
s : ℂ
⊢ (hurwitzOddFEPair a).symm.Λ ((s + 1) / 2) / 2 = (hurwitzOddFEPair a).symm.Λ ((s + 1) / 2) / 2

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaEven

Goal state before:
pp: a : ℝ
s : ℂ
hs : 1 < s.re
⊢ HasSum (fun n ↦ 1 / ↑|↑n + a| ^ s / 2) (completedHurwitzZetaEven (↑a) s / s.Gammaℝ)

Tactic:
]

Goal state after:
a : ℝ
s : ℂ
hs : 1 < s.re
⊢ HasSum (fun n ↦ 1 / ↑|↑n + a| ^ s / 2) (completedHurwitzZetaEven (↑a) s / s.Gammaℝ)
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaEven

Goal state before:
pp: case e_a.e_a
a : ℝ
ha : a ∈ Icc 0 1
s : ℂ
hs : 1 < s.re
n : ℕ
⊢ 1 / (↑n + ↑a) ^ s = 1 / ↑(↑n + a) ^ s

Tactic:
]

Goal state after:
case e_a.e_a
a : ℝ
ha : a ∈ Icc 0 1
s : ℂ
hs : 1 < s.re
n : ℕ
⊢ 1 / (↑n + ↑a) ^ s = 1 / ↑(↑n + a) ^ s
Module: Mathlib.NumberTheory.FLT.Basic

Goal state before:
pp: R : Type u_2
inst✝¹ : CommSemiring R
inst✝ : IsDomain R
n : ℕ
h : FermatLastTheoremWith' R n
hn : ∀ (a b c : R), IsUnit a → IsUnit b → IsUnit c → a ^ n + b ^ n ≠ c ^ n
d a' b' c' : R
ha : a' * d ≠ 0
hb : b' * d ≠ 0
hc : c' * d ≠ 0
heq : (a' ^ n + b' ^ n) * d ^ n = c' ^ n * d ^ n
ua : IsUnit a'
ub : IsUnit b'
uc : IsUnit c'
⊢ False

Tactic:
]

Goal state after:
R : Type u_2
inst✝¹ : CommSemiring R
inst✝ : IsDomain R
n : ℕ
h : FermatLastTheoremWith' R n
hn : ∀ (a b c : R), IsUnit a → IsUnit b → IsUnit c → a ^ n + b ^ n ≠ c ^ n
d a' b' c' : R
ha : a' * d ≠ 0
hb : b' * d ≠ 0
hc : c' * d ≠ 0
heq : (a' ^ n + b' ^ n) * d ^ n = c' ^ n * d ^ n
ua : IsUnit a'
ub : IsUnit b'
uc : IsUnit c'
⊢ False
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaEven

Goal state before:
pp: case e_a.e_a
a : ℝ
ha : a ∈ Icc 0 1
s : ℂ
hs : 1 < s.re
n : ℕ
⊢ 1 / (↑n + 1 - ↑a) ^ s = 1 / ↑(↑n + 1 - a) ^ s

Tactic:
]

Goal state after:
case e_a.e_a
a : ℝ
ha : a ∈ Icc 0 1
s : ℂ
hs : 1 < s.re
n : ℕ
⊢ 1 / (↑n + 1 - ↑a) ^ s = 1 / ↑(↑n + 1 - a) ^ s
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaOdd

Goal state before:
pp: a : ℝ
s : ℂ
hs : 1 < s.re
c : ℤ → ℂ := fun n ↦ -I * cexp (2 * ↑π * I * ↑a * ↑n) / 2
hc : ∀ (n : ℤ), ‖c n‖ = 1 / 2
t : ℝ
ht : 0 < t
n : ℤ
⊢ -I * ↑n * cexp (2 * ↑π * I * ↑a * ↑n) * ↑(rexp (-π * ↑n ^ 2 * t)) / 2 =
    -I * ↑n * cexp (2 * ↑π * I * ↑a * ↑n) * ↑(rexp (-π * ↑n ^ 2 * t)) / 2

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.FLT.Basic

Goal state before:
pp: n a✝ b✝ c✝ : ℕ
ha : a✝ = 1
hb : b✝ = 1
hc : c✝ = 1
⊢ 1 ^ n + 1 ^ n ≠ 1 ^ n

Tactic:
]

Goal state after:
n a✝ b✝ c✝ : ℕ
ha : a✝ = 1
hb : b✝ = 1
hc : c✝ = 1
⊢ 1 ^ n + 1 ^ n ≠ 1 ^ n
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaEven

Goal state before:
pp: a : UnitAddCircle
n : ℕ
this : -2 * (↑n + 1) ≠ 0
⊢ -(2 * (↑n + 1)) = -(2 * (↑n + 1))

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.FLT.Basic

Goal state before:
pp: n : ℕ
tfae_2_iff_1 : FermatLastTheoremWith' ℕ n ↔ FermatLastTheoremFor n
⊢ FermatLastTheoremWith' ℤ n ↔ FermatLastTheoremWith ℤ n

Tactic:
]

Goal state after:
n : ℕ
tfae_2_iff_1 : FermatLastTheoremWith' ℕ n ↔ FermatLastTheoremFor n
⊢ FermatLastTheoremWith' ℤ n ↔ FermatLastTheoremWith ℤ n
Module: Mathlib.NumberTheory.FLT.Basic

Goal state before:
pp: case neg
n : ℕ
tfae_2_iff_1 : FermatLastTheoremWith' ℕ n ↔ FermatLastTheoremFor n
a b c : ℤ
ha : a ^ n = 1 ∨ a ^ n = -1
hb : b ^ n = 1 ∨ b ^ n = -1
hc : c ^ n = 1 ∨ c ^ n = -1
hn : ¬n = 0
⊢ a ^ n + b ^ n ≠ c ^ n

Tactic:
]

Goal state after:
case neg
n : ℕ
tfae_2_iff_1 : FermatLastTheoremWith' ℕ n ↔ FermatLastTheoremFor n
a b c : ℤ
ha : a ^ n = 1 ∨ a ^ n = -1
hb : b ^ n = 1 ∨ b ^ n = -1
hc : c ^ n = 1 ∨ c ^ n = -1
hn : ¬n = 0
⊢ a ^ n + b ^ n ≠ c ^ n
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaEven

Goal state before:
pp: a : UnitAddCircle
hs' : 1 ≠ 1 ∨ a ≠ 0
x : ℂ
hx : x ∈ {0}ᶜ
⊢ completedCosZeta a x * x.Gammaℝ⁻¹ = completedCosZeta a x * x.Gammaℝ⁻¹

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaEven

Goal state before:
pp: a : ℝ
s : ℂ
hs : 1 < s.re
⊢ HasSum (fun n ↦ cexp (2 * ↑π * I * ↑a * ↑n) / ↑|n| ^ s / 2) (completedCosZeta (↑a) s / s.Gammaℝ)

Tactic:
]

Goal state after:
a : ℝ
s : ℂ
hs : 1 < s.re
⊢ HasSum (fun n ↦ cexp (2 * ↑π * I * ↑a * ↑n) / ↑|n| ^ s / 2) (completedCosZeta (↑a) s / s.Gammaℝ)
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaEven

Goal state before:
pp: a : ℝ
s : ℂ
hs : 1 < s.re
n : ℤ
⊢ cexp (2 * ↑π * I * ↑a * ↑n) / ↑|n| ^ s / 2 = cexp (2 * ↑π * I * ↑a * ↑n) / ↑|n| ^ s / 2

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaOdd

Goal state before:
pp: a : ℝ
s : ℂ
hs : 1 < s.re
c : ℤ → ℂ := fun n ↦ -I * cexp (2 * ↑π * I * ↑a * ↑n) / 2
hc : ∀ (n : ℤ), ‖c n‖ = 1 / 2
hF : ∀ (t : ℝ), 0 < t → HasSum (fun n ↦ c n * ↑n * ↑(rexp (-π * ↑n ^ 2 * t))) (↑(sinKernel (↑a) t) / 2)
h_sum : Summable fun i ↦ ‖c i‖ / |↑i| ^ s.re
⊢ HasSum (fun n ↦ (s + 1).Gammaℝ * -I * ↑n.sign * cexp (2 * ↑π * I * ↑a * ↑n) / ↑|n| ^ s / 2)
    (mellin (hurwitzOddFEPair ↑a).g ((s + 1) / 2) / 2)

Tactic:
]

Goal state after:
a : ℝ
s : ℂ
hs : 1 < s.re
c : ℤ → ℂ := fun n ↦ -I * cexp (2 * ↑π * I * ↑a * ↑n) / 2
hc : ∀ (n : ℤ), ‖c n‖ = 1 / 2
hF : ∀ (t : ℝ), 0 < t → HasSum (fun n ↦ c n * ↑n * ↑(rexp (-π * ↑n ^ 2 * t))) (↑(sinKernel (↑a) t) / 2)
h_sum : Summable fun i ↦ ‖c i‖ / |↑i| ^ s.re
⊢ HasSum (fun n ↦ (s + 1).Gammaℝ * -I * ↑n.sign * cexp (2 * ↑π * I * ↑a * ↑n) / ↑|n| ^ s / 2)
    (mellin (hurwitzOddFEPair ↑a).g ((s + 1) / 2) / 2)
Module: Mathlib.NumberTheory.FLT.Basic

Goal state before:
pp: case neg.inl.inl.inl
n : ℕ
tfae_2_iff_1 : FermatLastTheoremWith' ℕ n ↔ FermatLastTheoremFor n
a b c : ℤ
hn : ¬n = 0
ha : a ^ n = 1
hb : b ^ n = 1
hc : c ^ n = 1
⊢ 1 + 1 ≠ 1

Tactic:
]

Goal state after:
case neg.inl.inl.inl
n : ℕ
tfae_2_iff_1 : FermatLastTheoremWith' ℕ n ↔ FermatLastTheoremFor n
a b c : ℤ
hn : ¬n = 0
ha : a ^ n = 1
hb : b ^ n = 1
hc : c ^ n = 1
⊢ 1 + 1 ≠ 1
Module: Mathlib.NumberTheory.EulerProduct.DirichletLSeries

Goal state before:
pp: s : ℂ
hs : 1 < s.re
n : ℕ
⊢ (↑n ^ s.re)⁻¹ = (↑n ^ s.re)⁻¹

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.FLT.Basic

Goal state before:
pp: case neg.inl.inl.inr
n : ℕ
tfae_2_iff_1 : FermatLastTheoremWith' ℕ n ↔ FermatLastTheoremFor n
a b c : ℤ
hn : ¬n = 0
ha : a ^ n = 1
hb : b ^ n = 1
hc : c ^ n = -1
⊢ 1 + 1 ≠ -1

Tactic:
]

Goal state after:
case neg.inl.inl.inr
n : ℕ
tfae_2_iff_1 : FermatLastTheoremWith' ℕ n ↔ FermatLastTheoremFor n
a b c : ℤ
hn : ¬n = 0
ha : a ^ n = 1
hb : b ^ n = 1
hc : c ^ n = -1
⊢ 1 + 1 ≠ -1
Module: Mathlib.NumberTheory.FLT.Basic

Goal state before:
pp: case neg.inl.inr.inl
n : ℕ
tfae_2_iff_1 : FermatLastTheoremWith' ℕ n ↔ FermatLastTheoremFor n
a b c : ℤ
hn : ¬n = 0
ha : a ^ n = 1
hb : b ^ n = -1
hc : c ^ n = 1
⊢ 1 + -1 ≠ 1

Tactic:
]

Goal state after:
case neg.inl.inr.inl
n : ℕ
tfae_2_iff_1 : FermatLastTheoremWith' ℕ n ↔ FermatLastTheoremFor n
a b c : ℤ
hn : ¬n = 0
ha : a ^ n = 1
hb : b ^ n = -1
hc : c ^ n = 1
⊢ 1 + -1 ≠ 1
Module: Mathlib.NumberTheory.EulerProduct.DirichletLSeries

Goal state before:
pp: s : ℂ
hs : 1 < s.re
hsum : Summable fun n ↦ ‖(riemannZetaSummandHom ⋯) n‖
⊢ ∑' (b : ℕ), (riemannZetaSummandHom ⋯) (b + 1) = ∑' (n : ℕ), 1 / (↑n + 1) ^ s

Tactic:
]

Goal state after:
s : ℂ
hs : 1 < s.re
hsum : Summable fun n ↦ ‖(riemannZetaSummandHom ⋯) n‖
⊢ ∑' (b : ℕ), (riemannZetaSummandHom ⋯) (b + 1) = ∑' (n : ℕ), 1 / (↑n + 1) ^ s
Module: Mathlib.NumberTheory.FLT.Basic

Goal state before:
pp: case neg.inl.inr.inr
n : ℕ
tfae_2_iff_1 : FermatLastTheoremWith' ℕ n ↔ FermatLastTheoremFor n
a b c : ℤ
hn : ¬n = 0
ha : a ^ n = 1
hb : b ^ n = -1
hc : c ^ n = -1
⊢ 1 + -1 ≠ -1

Tactic:
]

Goal state after:
case neg.inl.inr.inr
n : ℕ
tfae_2_iff_1 : FermatLastTheoremWith' ℕ n ↔ FermatLastTheoremFor n
a b c : ℤ
hn : ¬n = 0
ha : a ^ n = 1
hb : b ^ n = -1
hc : c ^ n = -1
⊢ 1 + -1 ≠ -1
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaOdd

Goal state before:
pp: a : ℝ
s : ℂ
hs : 1 < s.re
this :
  HasSum
    (fun n ↦
      ((s + 1).Gammaℝ * -I * ↑(↑n).sign * cexp (2 * ↑π * I * ↑a * ↑n) +
            (s + 1).Gammaℝ * -I * -↑(↑n).sign * cexp (2 * ↑π * I * ↑a * -↑n)) /
          ↑n ^ s /
        2)
    (completedSinZeta (↑a) s)
n : ℕ
⊢ (s + 1).Gammaℝ * ↑(Real.sin (2 * π * a * ↑n)) / ↑n ^ s =
    ((s + 1).Gammaℝ * -I * ↑(↑n).sign * cexp (2 * ↑π * I * ↑a * ↑n) +
          (s + 1).Gammaℝ * -I * -↑(↑n).sign * cexp (2 * ↑π * I * ↑a * -↑n)) /
        2 /
      ↑n ^ s

Tactic:
]

Goal state after:
a : ℝ
s : ℂ
hs : 1 < s.re
this :
  HasSum
    (fun n ↦
      ((s + 1).Gammaℝ * -I * ↑(↑n).sign * cexp (2 * ↑π * I * ↑a * ↑n) +
            (s + 1).Gammaℝ * -I * -↑(↑n).sign * cexp (2 * ↑π * I * ↑a * -↑n)) /
          ↑n ^ s /
        2)
    (completedSinZeta (↑a) s)
n : ℕ
⊢ (s + 1).Gammaℝ * ↑(Real.sin (2 * π * a * ↑n)) / ↑n ^ s =
    ((s + 1).Gammaℝ * -I * ↑(↑n).sign * cexp (2 * ↑π * I * ↑a * ↑n) +
          (s + 1).Gammaℝ * -I * -↑(↑n).sign * cexp (2 * ↑π * I * ↑a * -↑n)) /
        2 /
      ↑n ^ s
Module: Mathlib.NumberTheory.FLT.Basic

Goal state before:
pp: case neg.inr.inl.inl
n : ℕ
tfae_2_iff_1 : FermatLastTheoremWith' ℕ n ↔ FermatLastTheoremFor n
a b c : ℤ
hn : ¬n = 0
ha : a ^ n = -1
hb : b ^ n = 1
hc : c ^ n = 1
⊢ -1 + 1 ≠ 1

Tactic:
]

Goal state after:
case neg.inr.inl.inl
n : ℕ
tfae_2_iff_1 : FermatLastTheoremWith' ℕ n ↔ FermatLastTheoremFor n
a b c : ℤ
hn : ¬n = 0
ha : a ^ n = -1
hb : b ^ n = 1
hc : c ^ n = 1
⊢ -1 + 1 ≠ 1
Module: Mathlib.NumberTheory.FLT.Basic

Goal state before:
pp: case neg.inr.inl.inr
n : ℕ
tfae_2_iff_1 : FermatLastTheoremWith' ℕ n ↔ FermatLastTheoremFor n
a b c : ℤ
hn : ¬n = 0
ha : a ^ n = -1
hb : b ^ n = 1
hc : c ^ n = -1
⊢ -1 + 1 ≠ -1

Tactic:
]

Goal state after:
case neg.inr.inl.inr
n : ℕ
tfae_2_iff_1 : FermatLastTheoremWith' ℕ n ↔ FermatLastTheoremFor n
a b c : ℤ
hn : ¬n = 0
ha : a ^ n = -1
hb : b ^ n = 1
hc : c ^ n = -1
⊢ -1 + 1 ≠ -1
Module: Mathlib.NumberTheory.FLT.Basic

Goal state before:
pp: case neg.inr.inr.inl
n : ℕ
tfae_2_iff_1 : FermatLastTheoremWith' ℕ n ↔ FermatLastTheoremFor n
a b c : ℤ
hn : ¬n = 0
ha : a ^ n = -1
hb : b ^ n = -1
hc : c ^ n = 1
⊢ -1 + -1 ≠ 1

Tactic:
]

Goal state after:
case neg.inr.inr.inl
n : ℕ
tfae_2_iff_1 : FermatLastTheoremWith' ℕ n ↔ FermatLastTheoremFor n
a b c : ℤ
hn : ¬n = 0
ha : a ^ n = -1
hb : b ^ n = -1
hc : c ^ n = 1
⊢ -1 + -1 ≠ 1
Module: Mathlib.NumberTheory.EulerProduct.DirichletLSeries

Goal state before:
pp: s : ℂ
hs : 1 < s.re
⊢ HasProd (fun p ↦ (1 - ↑↑p ^ (-s))⁻¹) (∑' (n : ℕ), (riemannZetaSummandHom ⋯) n)

Tactic:
]

Goal state after:
s : ℂ
hs : 1 < s.re
⊢ HasProd (fun p ↦ (1 - ↑↑p ^ (-s))⁻¹) (∑' (n : ℕ), (riemannZetaSummandHom ⋯) n)
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaEven

Goal state before:
pp: a : ℝ
s : ℂ
hs : 1 < s.re
this : HasSum (fun n ↦ (cexp (2 * ↑π * I * ↑a * ↑n) + cexp (-(2 * ↑π * I * ↑a * ↑n))) / 2 / ↑n ^ s) (cosZeta (↑a) s)
n : ℕ
⊢ (cexp (2 * ↑π * I * ↑a * ↑n) + cexp (-(2 * ↑π * I * ↑a * ↑n))) / 2 / ↑n ^ s =
    (cexp (2 * ↑π * I * ↑a * ↑n) + cexp (-(2 * ↑π * I * ↑a * ↑n))) / 2 / ↑n ^ s

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.FLT.Basic

Goal state before:
pp: case neg.inr.inr.inr
n : ℕ
tfae_2_iff_1 : FermatLastTheoremWith' ℕ n ↔ FermatLastTheoremFor n
a b c : ℤ
hn : ¬n = 0
ha : a ^ n = -1
hb : b ^ n = -1
hc : c ^ n = -1
⊢ -1 + -1 ≠ -1

Tactic:
]

Goal state after:
case neg.inr.inr.inr
n : ℕ
tfae_2_iff_1 : FermatLastTheoremWith' ℕ n ↔ FermatLastTheoremFor n
a b c : ℤ
hn : ¬n = 0
ha : a ^ n = -1
hb : b ^ n = -1
hc : c ^ n = -1
⊢ -1 + -1 ≠ -1
Module: Mathlib.NumberTheory.EulerProduct.DirichletLSeries

Goal state before:
pp: s : ℂ
hs : 1 < s.re
⊢ Tendsto (fun n ↦ ∏ p ∈ n.primesBelow, (1 - ↑p ^ (-s))⁻¹) atTop (𝓝 (∑' (n : ℕ), (riemannZetaSummandHom ⋯) n))

Tactic:
]

Goal state after:
s : ℂ
hs : 1 < s.re
⊢ Tendsto (fun n ↦ ∏ p ∈ n.primesBelow, (1 - ↑p ^ (-s))⁻¹) atTop (𝓝 (∑' (n : ℕ), (riemannZetaSummandHom ⋯) n))
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaOdd

Goal state before:
pp: case inr
a : ℝ
s : ℂ
hs : 1 < s.re
this :
  HasSum
    (fun n ↦
      ((s + 1).Gammaℝ * -I * ↑(↑n).sign * cexp (2 * ↑π * I * ↑a * ↑n) +
            (s + 1).Gammaℝ * -I * -↑(↑n).sign * cexp (2 * ↑π * I * ↑a * -↑n)) /
          ↑n ^ s /
        2)
    (completedSinZeta (↑a) s)
n : ℕ
h : n ≠ 0
⊢ (s + 1).Gammaℝ * ↑(Real.sin (2 * π * a * ↑n)) / ↑n ^ s =
    ((s + 1).Gammaℝ * -I * ↑(↑n).sign * cexp (2 * ↑π * I * ↑a * ↑n) +
          (s + 1).Gammaℝ * -I * -↑(↑n).sign * cexp (2 * ↑π * I * ↑a * -↑n)) /
        2 /
      ↑n ^ s

Tactic:
Int.sign_natCast_of_ne_zero h,

Goal state after:
case inr
a : ℝ
s : ℂ
hs : 1 < s.re
this :
  HasSum
    (fun n ↦
      ((s + 1).Gammaℝ * -I * ↑(↑n).sign * cexp (2 * ↑π * I * ↑a * ↑n) +
            (s + 1).Gammaℝ * -I * -↑(↑n).sign * cexp (2 * ↑π * I * ↑a * -↑n)) /
          ↑n ^ s /
        2)
    (completedSinZeta (↑a) s)
n : ℕ
h : n ≠ 0
⊢ (s + 1).Gammaℝ * ↑(Real.sin (2 * π * a * ↑n)) / ↑n ^ s =
    ((s + 1).Gammaℝ * -I * ↑1 * cexp (2 * ↑π * I * ↑a * ↑n) +
          (s + 1).Gammaℝ * -I * -↑1 * cexp (2 * ↑π * I * ↑a * -↑n)) /
        2 /
      ↑n ^ s
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaEven

Goal state before:
pp: a : UnitAddCircle
s : ℂ
hs : ∀ (n : ℕ), s ≠ -↑n
hs' : a ≠ 0 ∨ s ≠ 1
⊢ completedHurwitzZetaEven a (1 - s) * (1 - s).Gammaℝ⁻¹ = completedHurwitzZetaEven a (1 - s) * (1 - s).Gammaℝ⁻¹

pp: a : UnitAddCircle
s : ℂ
hs : ∀ (n : ℕ), s ≠ -↑n
hs' : a ≠ 0 ∨ s ≠ 1
⊢ a ≠ 0 ∨ 1 - s ≠ 0

Tactic:
]

Goal state after:
a : UnitAddCircle
s : ℂ
hs : ∀ (n : ℕ), s ≠ -↑n
hs' : a ≠ 0 ∨ s ≠ 1
⊢ a ≠ 0 ∨ 1 - s ≠ 0
Module: Mathlib.NumberTheory.EulerProduct.DirichletLSeries

Goal state before:
pp: s : ℂ
N : ℕ
χ : DirichletCharacter ℂ N
hs : 1 < s.re
⊢ HasProd (fun p ↦ (1 - χ ↑↑p * ↑↑p ^ (-s))⁻¹) (∑' (n : ℕ), (dirichletSummandHom χ ⋯) n)

Tactic:
]

Goal state after:
s : ℂ
N : ℕ
χ : DirichletCharacter ℂ N
hs : 1 < s.re
⊢ HasProd (fun p ↦ (1 - χ ↑↑p * ↑↑p ^ (-s))⁻¹) (∑' (n : ℕ), (dirichletSummandHom χ ⋯) n)