mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.JacobiSum.Basic

Goal state before:
pp: case e_a.e_a
F : Type u_1
R : Type u_2
inst✝⁴ : CommRing F
inst✝³ : Nontrivial F
inst✝² : Fintype F
inst✝¹ : DecidableEq F
inst✝ : CommRing R
χ ψ : MulChar F R
⊢ 0 = 0 + 0

Tactic:
add_zero

Goal state after:
case e_a.e_a
F : Type u_1
R : Type u_2
inst✝⁴ : CommRing F
inst✝³ : Nontrivial F
inst✝² : Fintype F
inst✝¹ : DecidableEq F
inst✝ : CommRing R
χ ψ : MulChar F R
⊢ 0 = 0
Module: Mathlib.NumberTheory.Harmonic.ZetaAsymp

Goal state before:
pp: this : completedRiemannZeta 1 = completedRiemannZeta₀ 1 - 1 - 0
⊢ completedRiemannZeta₀ 1 = (↑γ - log (4 * ↑π)) / 2 + 1

Tactic:
sub_zero,

Goal state after:
this : completedRiemannZeta 1 = completedRiemannZeta₀ 1 - 1
⊢ completedRiemannZeta₀ 1 = (↑γ - log (4 * ↑π)) / 2 + 1
Module: Mathlib.NumberTheory.Harmonic.ZetaAsymp

Goal state before:
pp: case refine_1
⊢ γ < 1

pp: case refine_2
⊢ 1 < Real.log (4 * π)

Tactic:
· exact Real.eulerMascheroniConstant_lt_two_thirds.trans (by norm_num)

Goal state after:
case refine_2
⊢ 1 < Real.log (4 * π)
Module: Mathlib.NumberTheory.JacobiSum.Basic

Goal state before:
pp: F : Type u_1
R : Type u_2
inst✝³ : Field F
inst✝² : Fintype F
inst✝¹ : CommRing R
inst✝ : IsDomain R
χ : MulChar F R
hχ : χ ≠ 1
⊢ ∑ x ∈ univ \ {1}, χ (x / (1 - x)) + 0 = -χ (-1)

Tactic:
add_zero

Goal state after:
F : Type u_1
R : Type u_2
inst✝³ : Field F
inst✝² : Fintype F
inst✝¹ : CommRing R
inst✝ : IsDomain R
χ : MulChar F R
hχ : χ ≠ 1
⊢ ∑ x ∈ univ \ {1}, χ (x / (1 - x)) = -χ (-1)
Module: Mathlib.NumberTheory.Harmonic.ZetaAsymp

Goal state before:
pp: s : ℂ
⊢ riemannZeta ((starRingEnd ℂ) s) = (starRingEnd ℂ) (riemannZeta s)

Tactic:
rcases eq_or_ne s 1 with rfl | hs
  · have h : riemannZeta 1 = ((γ - Real.log (4 * π)) / 2 : ℝ) := by
      rw [riemannZeta_one, ofReal_div, ofReal_sub, ofReal_log (by positivity : (0 : ℝ) ≤ 4 * π)]
      norm_cast
    rw [map_one, h, conj_ofReal]
  · -- `conj ∘ ζ ∘ conj` is analytic on `{1}ᶜ` and agrees with `ζ` termwise on `1 < re z`, so
    -- the identity principle propagates the equality to the connected set `{1}ᶜ`.
    have hg_an : AnalyticOnNhd ℂ (fun z ↦ conj (riemannZeta (conj z))) {1}ᶜ :=
      DifferentiableOn.analyticOnNhd
        (fun z hz ↦ (differentiableAt_conj_conj_iff.mpr <| differentiableAt_riemannZeta <|
          (map_ne_one_iff _ (starRingEnd ℂ).injective).mpr hz).differentiableWithinAt)
        isOpen_compl_singleton
    have hgz (z : ℂ) (hz : 1 < z.re) : conj (riemannZeta (conj z)) = riemannZeta z := by
      rw [zeta_eq_tsum_one_div_nat_cpow (by rwa [conj_re]), conj_tsum,
        zeta_eq_tsum_one_div_nat_cpow hz]
      exact tsum_congr fun n ↦ by
        rw [map_div₀, map_one, ← conj_cpow _ _ (by rw [natCast_arg]; positivity), conj_natCast]
    have heq : EqOn (fun z ↦ conj (riemannZeta (conj z))) riemannZeta {1}ᶜ :=
      hg_an.eqOn_of_preconnected_of_eventuallyEq analyticOn_riemannZeta
        (isConnected_compl_singleton_of_one_lt_rank (by simp) 1).isPreconnected
        (by norm_num : (2 : ℂ) ∈ _)
        (eventuallyEq_of_mem
          ((isOpen_lt continuous_const continuous_re).mem_nhds (by norm_num)) hgz)
    simpa using congrArg (starRingEnd ℂ) (heq hs)

Goal state after:
Module: Mathlib.NumberTheory.Height.MvPolynomial

Goal state before:
pp: case inr
K : Type u_6
inst✝³ : Field K
ι : Type u_7
ι' : Type u_8
inst✝² : Fintype ι'
inst✝¹ : AdmissibleAbsValues K
inst✝ : Finite ι
M N : ℕ
q : ι × ι' → MvPolynomial ι K
hq : ∀ (a : ι × ι'), (q a).IsHomogeneous M
p : ι' → MvPolynomial ι K
x : ι → K
h : ∀ (k : ι), ∑ j, (eval x) (q (k, j)) * (eval x) (p j) = x k ^ (M + N)
h✝ : Nonempty ι'
q' : ι × ι' → K := fun a ↦ (eval x) (q a)
H : mulHeight x ^ (M + N) ≤ ↑(Nat.card ι') ^ totalWeight K * mulHeight q' * mulHeight fun j ↦ (eval x) (p j)
⊢ mulHeight x ^ N ≤ ↑(Nat.card ι') ^ totalWeight K * max (mulHeightBound q) 1 * mulHeight fun j ↦ (eval x) (p j)

pp: case hC
K : Type u_6
inst✝³ : Field K
ι : Type u_7
ι' : Type u_8
inst✝² : Fintype ι'
inst✝¹ : AdmissibleAbsValues K
inst✝ : Finite ι
M N : ℕ
q : ι × ι' → MvPolynomial ι K
hq : ∀ (a : ι × ι'), (q a).IsHomogeneous M
p : ι' → MvPolynomial ι K
x : ι → K
h : ∀ (k : ι), ∑ j, (eval x) (q (k, j)) * (eval x) (p j) = x k ^ (M + N)
h✝ : Nonempty ι'
q' : ι × ι' → K := fun a ↦ (eval x) (q a)
H : mulHeight x ^ (M + N) ≤ ↑(Nat.card ι') ^ totalWeight K * mulHeight q' * mulHeight fun j ↦ (eval x) (p j)
⊢ 0 < ↑(Nat.card ι') ^ totalWeight K * max (mulHeightBound q) 1

Tactic:
← mul_le_mul_iff_left₀ (by positivity : 0 < mulHeight x ^ M)

Goal state after:
case inr
K : Type u_6
inst✝³ : Field K
ι : Type u_7
ι' : Type u_8
inst✝² : Fintype ι'
inst✝¹ : AdmissibleAbsValues K
inst✝ : Finite ι
M N : ℕ
q : ι × ι' → MvPolynomial ι K
hq : ∀ (a : ι × ι'), (q a).IsHomogeneous M
p : ι' → MvPolynomial ι K
x : ι → K
h : ∀ (k : ι), ∑ j, (eval x) (q (k, j)) * (eval x) (p j) = x k ^ (M + N)
h✝ : Nonempty ι'
q' : ι × ι' → K := fun a ↦ (eval x) (q a)
H : mulHeight x ^ (M + N) ≤ ↑(Nat.card ι') ^ totalWeight K * mulHeight q' * mulHeight fun j ↦ (eval x) (p j)
⊢ mulHeight x ^ N * mulHeight x ^ M ≤
    (↑(Nat.card ι') ^ totalWeight K * max (mulHeightBound q) 1 * mulHeight fun j ↦ (eval x) (p j)) * mulHeight x ^ M

case hC
K : Type u_6
inst✝³ : Field K
ι : Type u_7
ι' : Type u_8
inst✝² : Fintype ι'
inst✝¹ : AdmissibleAbsValues K
inst✝ : Finite ι
M N : ℕ
q : ι × ι' → MvPolynomial ι K
hq : ∀ (a : ι × ι'), (q a).IsHomogeneous M
p : ι' → MvPolynomial ι K
x : ι → K
h : ∀ (k : ι), ∑ j, (eval x) (q (k, j)) * (eval x) (p j) = x k ^ (M + N)
h✝ : Nonempty ι'
q' : ι × ι' → K := fun a ↦ (eval x) (q a)
H : mulHeight x ^ (M + N) ≤ ↑(Nat.card ι') ^ totalWeight K * mulHeight q' * mulHeight fun j ↦ (eval x) (p j)
⊢ 0 < ↑(Nat.card ι') ^ totalWeight K * max (mulHeightBound q) 1
Module: Mathlib.NumberTheory.JacobiSum.Basic

Goal state before:
pp: F : Type u_1
R : Type u_2
inst✝³ : Field F
inst✝² : Fintype F
inst✝¹ : CommRing R
inst✝ : IsDomain R
χ φ : MulChar F R
h : χ * φ ≠ 1
ψ : AddChar F R
⊢ gaussSum (χ * φ) ψ * jacobiSum χ φ = ∑ x ∈ univ \ {0}, ∑ x_1, χ x_1 * φ (x - x_1) * ψ x + 0

Tactic:
add_zero

Goal state after:
F : Type u_1
R : Type u_2
inst✝³ : Field F
inst✝² : Fintype F
inst✝¹ : CommRing R
inst✝ : IsDomain R
χ φ : MulChar F R
h : χ * φ ≠ 1
ψ : AddChar F R
⊢ gaussSum (χ * φ) ψ * jacobiSum χ φ = ∑ x ∈ univ \ {0}, ∑ x_1, χ x_1 * φ (x - x_1) * ψ x
Module: Mathlib.NumberTheory.JacobiSum.Basic

Goal state before:
pp: F : Type u_1
R : Type u_2
inst✝³ : Field F
inst✝² : Fintype F
inst✝¹ : CommRing R
inst✝ : IsDomain R
χ φ : MulChar F R
h : χ * φ ≠ 1
ψ : AddChar F R
sum_eq : ∀ t ∈ univ \ {0}, (∑ x, χ x * φ (t - x)) * ψ t = (∑ y, χ (t * y) * φ (t - t * y)) * ψ t
⊢ (∑ x ∈ univ \ {0}, (χ * φ) x * ψ x + 0) * jacobiSum χ φ = (∑ i ∈ univ \ {0}, (χ * φ) i * ψ i) * jacobiSum χ φ

Tactic:
add_zero

Goal state after:
F : Type u_1
R : Type u_2
inst✝³ : Field F
inst✝² : Fintype F
inst✝¹ : CommRing R
inst✝ : IsDomain R
χ φ : MulChar F R
h : χ * φ ≠ 1
ψ : AddChar F R
sum_eq : ∀ t ∈ univ \ {0}, (∑ x, χ x * φ (t - x)) * ψ t = (∑ y, χ (t * y) * φ (t - t * y)) * ψ t
⊢ (∑ x ∈ univ \ {0}, (χ * φ) x * ψ x) * jacobiSum χ φ = (∑ i ∈ univ \ {0}, (χ * φ) i * ψ i) * jacobiSum χ φ
Module: Mathlib.NumberTheory.Harmonic.ZetaAsymp

Goal state before:
pp: ⊢ DifferentiableAt ℂ (fun s ↦ s - 1) 1

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.NumberTheory.Harmonic.ZetaAsymp

Goal state before:
pp: ⊢ DifferentiableAt ℂ riemannZeta₀ 1

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.NumberTheory.Harmonic.ZetaAsymp

Goal state before:
pp: s : ℂ
hs : s ≠ 1
this : s - 1 ≠ 0
⊢ DifferentiableAt ℂ (fun s ↦ (s - 1)⁻¹) s

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.NumberTheory.Harmonic.ZetaAsymp

Goal state before:
pp: s : ℂ
hs : s ≠ 1
this : s - 1 ≠ 0
⊢ DifferentiableAt ℂ riemannZeta₀ s

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.NumberTheory.Harmonic.ZetaAsymp

Goal state before:
pp: s : ℂ
hs : s ≠ 1
this : s - 1 ≠ 0
⊢ DifferentiableAt ℂ (fun s ↦ s - 1) s

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.NumberTheory.Harmonic.ZetaAsymp

Goal state before:
pp: s : ℂ
hs : s ≠ 1
this : s - 1 ≠ 0
⊢ DifferentiableAt ℂ riemannZeta₁ s

Tactic:
by fun_prop

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

Goal state before:
pp: F : Type u_1
R : Type u_2
inst✝⁴ : Field F
inst✝³ : CommRing R
inst✝² : IsDomain R
inst✝¹ : Finite F
n : ℕ
inst✝ : NeZero n
χ ψ : MulChar F R
μ : R
hχ : χ ^ n = 1
hψ : ψ ^ n = 1
hμ : IsPrimitiveRoot μ n
⊢ (χ 0 - 1) * (ψ (1 - 0) - 1) = 0 * (μ - 1) ^ 2

Tactic:
sub_zero,

Goal state after:
F : Type u_1
R : Type u_2
inst✝⁴ : Field F
inst✝³ : CommRing R
inst✝² : IsDomain R
inst✝¹ : Finite F
n : ℕ
inst✝ : NeZero n
χ ψ : MulChar F R
μ : R
hχ : χ ^ n = 1
hψ : ψ ^ n = 1
hμ : IsPrimitiveRoot μ n
⊢ (χ 0 - 1) * (ψ 1 - 1) = 0 * (μ - 1) ^ 2
Module: Mathlib.NumberTheory.Harmonic.ZetaAsymp

Goal state before:
pp: ⊢ ContinuousAt (fun x ↦ x) 1

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.RingTheory.IsAdjoinRoot

Goal state before:
pp: R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : Ring S
f : R[X]
inst✝ : Algebra R S
h h' : IsAdjoinRoot S f
eq : h.root = h'.root
x : R[X]
⊢ h.map x = h'.map x

Tactic:
rw [← h.aeval_root_eq_map, ← h'.aeval_root_eq_map, eq]

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

Goal state before:
pp: case neg
F : Type u_1
R : Type u_2
inst✝³ : Field F
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : Fintype F
n : ℕ
hn : 2 < n
χ ψ : MulChar F R
μ : R
hχ : χ ^ n = 1
hψ : ψ ^ n = 1
hμ : IsPrimitiveRoot μ n
q : ℕ
hq : Fintype.card F = n * q + 1
z₁ : R
hz₁ : z₁ ∈ ℤ[μ]
Hz₁ : ↑n = z₁ * (μ - 1) ^ 2
hχ₀ : χ = 1
hψ₀ : ¬ψ = 1
⊢ -1 = -1 + 0

Tactic:
add_zero

Goal state after:
case neg
F : Type u_1
R : Type u_2
inst✝³ : Field F
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : Fintype F
n : ℕ
hn : 2 < n
χ ψ : MulChar F R
μ : R
hχ : χ ^ n = 1
hψ : ψ ^ n = 1
hμ : IsPrimitiveRoot μ n
q : ℕ
hq : Fintype.card F = n * q + 1
z₁ : R
hz₁ : z₁ ∈ ℤ[μ]
Hz₁ : ↑n = z₁ * (μ - 1) ^ 2
hχ₀ : χ = 1
hψ₀ : ¬ψ = 1
⊢ -1 = -1
Module: Mathlib.NumberTheory.JacobiSum.Basic

Goal state before:
pp: case pos
F : Type u_1
R : Type u_2
inst✝³ : Field F
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : Fintype F
n : ℕ
hn : 2 < n
χ ψ : MulChar F R
μ : R
hχ : χ ^ n = 1
hψ : ψ ^ n = 1
hμ : IsPrimitiveRoot μ n
q : ℕ
hq : Fintype.card F = n * q + 1
z₁ : R
hz₁ : z₁ ∈ ℤ[μ]
Hz₁ : ↑n = z₁ * (μ - 1) ^ 2
hχ₀ : ¬χ = 1
hψ₀ : ψ = 1
⊢ -1 = -1 + 0

Tactic:
add_zero

Goal state after:
case pos
F : Type u_1
R : Type u_2
inst✝³ : Field F
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : Fintype F
n : ℕ
hn : 2 < n
χ ψ : MulChar F R
μ : R
hχ : χ ^ n = 1
hψ : ψ ^ n = 1
hμ : IsPrimitiveRoot μ n
q : ℕ
hq : Fintype.card F = n * q + 1
z₁ : R
hz₁ : z₁ ∈ ℤ[μ]
Hz₁ : ↑n = z₁ * (μ - 1) ^ 2
hχ₀ : ¬χ = 1
hψ₀ : ψ = 1
⊢ -1 = -1
Module: Mathlib.NumberTheory.LSeries.ZMod

Goal state before:
pp: N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
s : ℂ
hs : s ≠ 1 ∨ ∑ j, Φ j = 0
⊢ DifferentiableAt ℂ (fun s ↦ ↑N ^ (-s)) s

Tactic:
by fun_prop

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

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
⊢ LFunctionTrivChar₁ n 1 ≠ 0

Tactic:
by
  simp only [Function.update_self]
  refine Finset.prod_ne_zero_iff.mpr fun p hp ↦ ?_
  simpa [sub_ne_zero] using (Nat.prime_of_mem_primeFactors hp).ne_one

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

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
s : ℂ
hs : s ≠ 1
H : deriv (LFunctionTrivChar₁ n) s = deriv (fun w ↦ (w - 1) * LFunctionTrivChar n w) s
⊢ DifferentiableAt ℂ (fun w ↦ w - 1) s

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: k : ℕ
⊢ bernoulliFun k 0 = ↑(bernoulli k)

Tactic:
rw [bernoulliFun, Polynomial.eval_zero_map, Polynomial.bernoulli_eval_zero, eq_ratCast]

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

Goal state before:
pp: case refine_2
k : ℕ
x : ℝ
hk : k ≠ 0
hx : x ∈ Icc 0 1
⊢ (-1) ^ (k + 1) * (2 * ↑π) ^ (2 * k) / 2 / ↑(2 * k)! *
      ↑(Polynomial.eval x (Polynomial.map (algebraMap ℚ ℝ) (Polynomial.bernoulli (2 * k)))) =
    (-1) ^ (k + 1) * (2 * ↑π) ^ (2 * k) / 2 / ↑(2 * k)! *
      Polynomial.eval (↑x) (Polynomial.map (algebraMap ℚ ℂ) (Polynomial.bernoulli (2 * k)))

Tactic:
congr 1

Goal state after:
case refine_2.e_a
k : ℕ
x : ℝ
hk : k ≠ 0
hx : x ∈ Icc 0 1
⊢ ↑(Polynomial.eval x (Polynomial.map (algebraMap ℚ ℝ) (Polynomial.bernoulli (2 * k)))) =
    Polynomial.eval (↑x) (Polynomial.map (algebraMap ℚ ℂ) (Polynomial.bernoulli (2 * k)))
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: k : ℕ
x y : ℝ
⊢ Continuous fun t ↦ ↑(k + 1) * bernoulliFun k t

Tactic:
by fun_prop

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

Goal state before:
pp: case refine_2
k : ℕ
x : ℝ
hk : k ≠ 0
hx : x ∈ Icc 0 1
⊢ (-1) ^ (k + 1) * (2 * ↑π) ^ (2 * k + 1) / 2 / ↑(2 * k + 1)! *
      ↑(Polynomial.eval x (Polynomial.map (algebraMap ℚ ℝ) (Polynomial.bernoulli (2 * k + 1)))) =
    (-1) ^ (k + 1) * (2 * ↑π) ^ (2 * k + 1) / 2 / ↑(2 * k + 1)! *
      Polynomial.eval (↑x) (Polynomial.map (algebraMap ℚ ℂ) (Polynomial.bernoulli (2 * k + 1)))

Tactic:
congr 1

Goal state after:
case refine_2.e_a
k : ℕ
x : ℝ
hk : k ≠ 0
hx : x ∈ Icc 0 1
⊢ ↑(Polynomial.eval x (Polynomial.map (algebraMap ℚ ℝ) (Polynomial.bernoulli (2 * k + 1)))) =
    Polynomial.eval (↑x) (Polynomial.map (algebraMap ℚ ℂ) (Polynomial.bernoulli (2 * k + 1)))
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaValues

Goal state before:
pp: k : ℕ
x : ℝ
hk : k ≠ 0
hx : x ∈ Icc 0 1
⊢ (-1) ^ (k + 1) * (2 * ↑π) ^ (2 * k) / 2 / ↑(2 * k)! *
      Polynomial.eval (↑x) (Polynomial.map (algebraMap ℚ ℂ) (Polynomial.bernoulli (2 * k))) =
    (-1) ^ (k + 1) / (2 * ↑k) / (2 * ↑k).Gammaℂ *
      Polynomial.eval (↑x) (Polynomial.map (algebraMap ℚ ℂ) (Polynomial.bernoulli (2 * k)))

Tactic:
congr 1

Goal state after:
case e_a
k : ℕ
x : ℝ
hk : k ≠ 0
hx : x ∈ Icc 0 1
⊢ (-1) ^ (k + 1) * (2 * ↑π) ^ (2 * k) / 2 / ↑(2 * k)! = (-1) ^ (k + 1) / (2 * ↑k) / (2 * ↑k).Gammaℂ
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaValues

Goal state before:
pp: k : ℕ
x : ℝ
hk : k ≠ 0
hx : x ∈ Icc 0 1
⊢ (-1) ^ (k + 1) * (2 * ↑π) ^ (2 * k + 1) / 2 / ↑(2 * k + 1)! *
      Polynomial.eval (↑x) (Polynomial.map (algebraMap ℚ ℂ) (Polynomial.bernoulli (2 * k + 1))) =
    (-1) ^ (k + 1) / (2 * ↑k + 1) / (2 * ↑k + 1).Gammaℂ *
      Polynomial.eval (↑x) (Polynomial.map (algebraMap ℚ ℂ) (Polynomial.bernoulli (2 * k + 1)))

Tactic:
congr 1

Goal state after:
case e_a
k : ℕ
x : ℝ
hk : k ≠ 0
hx : x ∈ Icc 0 1
⊢ (-1) ^ (k + 1) * (2 * ↑π) ^ (2 * k + 1) / 2 / ↑(2 * k + 1)! = (-1) ^ (k + 1) / (2 * ↑k + 1) / (2 * ↑k + 1).Gammaℂ
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: m : ℕ
m0 : m ≠ 0
x : ℝ
m0' : ↑m ≠ 0
f : ℕ → ℝ → ℝ := fun k x ↦ bernoulliFun k (↑m * x) - ↑m ^ k / ↑m * ∑ i ∈ Finset.range m, bernoulliFun k (x + ↑i / ↑m)
k : ℕ
h : ∀ (x : ℝ), f k x = 0
d : ∀ (x : ℝ), HasDerivAt (f (k + 1)) 0 x
c : ℝ
fc : ∀ (x : ℝ), f (k + 1) x = c
k✝ : ℕ
a✝ : k✝ < m
⊢ Continuous (bernoulliFun (k + 1))

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: m : ℕ
m0 : m ≠ 0
x : ℝ
m0' : ↑m ≠ 0
f : ℕ → ℝ → ℝ := fun k x ↦ bernoulliFun k (↑m * x) - ↑m ^ k / ↑m * ∑ i ∈ Finset.range m, bernoulliFun k (x + ↑i / ↑m)
k : ℕ
h : ∀ (x : ℝ), f k x = 0
d : ∀ (x : ℝ), HasDerivAt (f (k + 1)) 0 x
c : ℝ
fc : ∀ (x : ℝ), f (k + 1) x = c
i✝ : ℕ
a✝ : i✝ ∈ Finset.range m
⊢ Continuous fun x ↦ bernoulliFun (k + 1) (x + ↑i✝ / ↑m)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: m : ℕ
m0 : m ≠ 0
x : ℝ
m0' : ↑m ≠ 0
f : ℕ → ℝ → ℝ := fun k x ↦ bernoulliFun k (↑m * x) - ↑m ^ k / ↑m * ∑ i ∈ Finset.range m, bernoulliFun k (x + ↑i / ↑m)
k : ℕ
h : ∀ (x : ℝ), f k x = 0
d : ∀ (x : ℝ), HasDerivAt (f (k + 1)) 0 x
c : ℝ
fc : ∀ (x : ℝ), f (k + 1) x = c
⊢ Continuous fun x ↦ bernoulliFun (k + 1) (↑m * x)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.NumberTheory.KummerDedekind

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : Module.IsTorsionFree R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
hf : Irreducible (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))
mem_norm_factors :
  normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)) ∈
    normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))
⊢ ∃ y,
    Multiset.map (fun f ↦ ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))
        (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach =
      {y}

Tactic:
use ((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm
        ⟨normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)), mem_norm_factors⟩ :
      Ideal S)

Goal state after:
case h
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : Module.IsTorsionFree R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
hf : Irreducible (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))
mem_norm_factors :
  normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)) ∈
    normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))
⊢ Multiset.map (fun f ↦ ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))
      (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach =
    {↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm
          ⟨normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)), mem_norm_factors⟩)}
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: m : ℕ
m0 : m ≠ 0
x : ℝ
m0' : ↑m ≠ 0
f : ℕ → ℝ → ℝ := fun k x ↦ bernoulliFun k (↑m * x) - ↑m ^ k / ↑m * ∑ i ∈ Finset.range m, bernoulliFun k (x + ↑i / ↑m)
k : ℕ
h : ∀ (x : ℝ), f k x = 0
d : ∀ (x : ℝ), HasDerivAt (f (k + 1)) 0 x
c : ℝ
fc : ∀ (x : ℝ), f (k + 1) x = c
⊢ Continuous fun x ↦ ↑m ^ (k + 1) / ↑m * ∑ i ∈ Finset.range m, bernoulliFun (k + 1) (x + ↑i / ↑m)

Tactic:
by fun_prop

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

Goal state before:
pp: N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Even Φ
s : ℂ
⊢ ↑N ^ (-s) * (∑ x, Φ x * hurwitzZetaEven (toAddCircle x) s + ∑ x, Φ x * hurwitzZetaOdd (toAddCircle x) s) =
    ↑N ^ (-s) * ∑ j, Φ j * hurwitzZetaEven (toAddCircle j) s

Tactic:
congr 1

Goal state after:
case e_a
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Even Φ
s : ℂ
⊢ ∑ x, Φ x * hurwitzZetaEven (toAddCircle x) s + ∑ x, Φ x * hurwitzZetaOdd (toAddCircle x) s =
    ∑ j, Φ j * hurwitzZetaEven (toAddCircle j) s
Module: Mathlib.NumberTheory.LSeries.ZMod

Goal state before:
pp: N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Odd Φ
s : ℂ
⊢ ↑N ^ (-s) * (∑ x, Φ x * hurwitzZetaEven (toAddCircle x) s + ∑ x, Φ x * hurwitzZetaOdd (toAddCircle x) s) =
    ↑N ^ (-s) * ∑ j, Φ j * hurwitzZetaOdd (toAddCircle j) s

Tactic:
congr 1

Goal state after:
case e_a
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Odd Φ
s : ℂ
⊢ ∑ x, Φ x * hurwitzZetaEven (toAddCircle x) s + ∑ x, Φ x * hurwitzZetaOdd (toAddCircle x) s =
    ∑ j, Φ j * hurwitzZetaOdd (toAddCircle j) s
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: k : ℕ
n : ℤ
hn : n ≠ 0
⊢ 1 / (-2 * ↑π * I * ↑n) *
      ((fourier (-n)) ↑0 * (↑(bernoulliFun k 1) - ↑(bernoulliFun k 0)) -
        (1 - 0) * fourierCoeffOn ⋯ (fun x ↦ ↑(↑k * bernoulliFun (k - 1) x)) n) =
    1 / (-2 * ↑π * I * ↑n) *
      ((if k = 1 then 1 else 0) -
        ↑k * fourierCoeffOn bernoulliFourierCoeff._proof_1 (fun x ↦ ↑(bernoulliFun (k - 1) x)) n)

Tactic:
sub_zero,

Goal state after:
k : ℕ
n : ℤ
hn : n ≠ 0
⊢ 1 / (-2 * ↑π * I * ↑n) *
      ((fourier (-n)) ↑0 * (↑(bernoulliFun k 1) - ↑(bernoulliFun k 0)) -
        1 * fourierCoeffOn ⋯ (fun x ↦ ↑(↑k * bernoulliFun (k - 1) x)) n) =
    1 / (-2 * ↑π * I * ↑n) *
      ((if k = 1 then 1 else 0) -
        ↑k * fourierCoeffOn bernoulliFourierCoeff._proof_1 (fun x ↦ ↑(bernoulliFun (k - 1) x)) n)
Module: Mathlib.NumberTheory.LSeries.ZMod

Goal state before:
pp: N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Even Φ
s : ℂ
this : ∑ j, Φ j * completedHurwitzZetaOdd (toAddCircle j) s = 0
⊢ ↑N ^ (-s) * ∑ j, Φ j * completedHurwitzZetaEven (toAddCircle j) s + 0 =
    ↑N ^ (-s) * ∑ j, Φ j * completedHurwitzZetaEven (toAddCircle j) s

Tactic:
add_zero

Goal state after:
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Even Φ
s : ℂ
this : ∑ j, Φ j * completedHurwitzZetaOdd (toAddCircle j) s = 0
⊢ ↑N ^ (-s) * ∑ j, Φ j * completedHurwitzZetaEven (toAddCircle j) s =
    ↑N ^ (-s) * ∑ j, Φ j * completedHurwitzZetaEven (toAddCircle j) s
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: k : ℕ
hk : k ≠ 0
⊢ (1 / (1 - 0)) • ∫ (x : ℝ) in 0..1, 1 • ↑(bernoulliFun k x) = 0

Tactic:
sub_zero,

Goal state after:
k : ℕ
hk : k ≠ 0
⊢ (1 / 1) • ∫ (x : ℝ) in 0..1, 1 • ↑(bernoulliFun k x) = 0
Module: Mathlib.NumberTheory.LSeries.ZMod

Goal state before:
pp: N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Odd Φ
s : ℂ
this : ∑ j, Φ j * completedHurwitzZetaEven (toAddCircle j) s = 0
⊢ completedLFunction Φ s = ↑N ^ (-s) * ∑ j, Φ j * completedHurwitzZetaOdd (toAddCircle j) s

Tactic:
rw [completedLFunction, this, mul_zero, zero_add]

Goal state after:
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: case inr.refine_1
k : ℕ
hk : k ≠ 0
n : ℤ
hn : n ≠ 0
⊢ 1 / -(2 * ↑π * I * ↑n) * (1 - 0) = -1 / (2 * ↑π * I * ↑n)

Tactic:
sub_zero,

Goal state after:
case inr.refine_1
k : ℕ
hk : k ≠ 0
n : ℤ
hn : n ≠ 0
⊢ 1 / -(2 * ↑π * I * ↑n) * 1 = -1 / (2 * ↑π * I * ↑n)
Module: Mathlib.NumberTheory.LSeries.ZMod

Goal state before:
pp: N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
⊢ Differentiable ℂ fun s ↦ ↑N ^ (-s)

Tactic:
by fun_prop

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

Goal state before:
pp: N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
s : ℂ
hs₀ : s ≠ 0 ∨ Φ 0 = 0
hs₁ : s ≠ 1 ∨ ∑ j, Φ j = 0
⊢ DifferentiableAt ℂ (fun x ↦ ↑N ^ (-x)) s

Tactic:
by fun_prop

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

Goal state before:
pp: N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
s : ℂ
hs₀ : s ≠ 0 ∨ Φ 0 = 0
hs₁ : s ≠ 1 ∨ ∑ j, Φ j = 0
h : s ≠ 1
⊢ DifferentiableAt ℂ (fun x ↦ ∑ j, Φ j) s

Tactic:
by fun_prop

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

Goal state before:
pp: N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
s : ℂ
hs₀ : s ≠ 0 ∨ Φ 0 = 0
hs₁ : s ≠ 1 ∨ ∑ j, Φ j = 0
h : s ≠ 1
⊢ DifferentiableAt ℂ (HSub.hSub 1) s

Tactic:
by fun_prop

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

Goal state before:
pp: ⊢ riemannZeta 2 = ↑π ^ 2 / 6

Tactic:
by
  convert! congr_arg ((↑) : ℝ → ℂ) hasSum_zeta_two.tsum_eq
  · rw [← Nat.cast_two, zeta_nat_eq_tsum_of_gt_one one_lt_two]
    simp [push_cast]
  · norm_cast

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

Goal state before:
pp: N : ℕ
inst✝ : NeZero N
B : BadChar N
s : ℂ
hs : 1 < s.re
hs' : s ≠ 1
⊢ riemannZeta s * LSeries (fun x ↦ B.χ ↑x) s = LSeries (⇑↑zeta) s * LSeries (⇑(toArithmeticFunction fun x ↦ B.χ ↑x)) s

Tactic:
congr 1

Goal state after:
case e_a
N : ℕ
inst✝ : NeZero N
B : BadChar N
s : ℂ
hs : 1 < s.re
hs' : s ≠ 1
⊢ riemannZeta s = LSeries (⇑↑zeta) s

case e_a
N : ℕ
inst✝ : NeZero N
B : BadChar N
s : ℂ
hs : 1 < s.re
hs' : s ≠ 1
⊢ LSeries (fun x ↦ B.χ ↑x) s = LSeries (⇑(toArithmeticFunction fun x ↦ B.χ ↑x)) s
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: case e'_5.e_a.e_a
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
n : ℕ
⊢ (-1) ^ k / ↑n ^ k = 1 / (-1) ^ k / ↑↑n ^ k

Tactic:
congr 1

Goal state after:
case e'_5.e_a.e_a.e_a
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
n : ℕ
⊢ (-1) ^ k = 1 / (-1) ^ k
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: case e'_6
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
⊢ -(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k x) = -(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k x) + 0

Tactic:
add_zero

Goal state after:
case e'_6
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
⊢ -(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k x) = -(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k x)