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

Goal state before:
pp: case e'_5.e'_5
N : ℕ
χ : DirichletCharacter ℂ N
n : ℕ
hn✝ : 2 ≤ n
x : ℝ
hx : 1 < x
y : ℝ
hn' : IsUnit ↑n
hn : ↑n ^ (-x) < 1
hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1
⊢ log (1 - ↑n ^ (-↑x)) = log (1 - ↑(↑n ^ (-x)))

pp: case e'_5.e'_6
N : ℕ
χ : DirichletCharacter ℂ N
n : ℕ
hn✝ : 2 ≤ n
x : ℝ
hx : 1 < x
y : ℝ
hn' : IsUnit ↑n
hn : ↑n ^ (-x) < 1
hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1
⊢ log (1 - χ ↑n * ↑n ^ (-(↑x + I * ↑y))) = log (1 - ↑(↑n ^ (-x)) * (χ ↑n * ↑n ^ (-(I * ↑y))))

pp: case e'_6
N : ℕ
χ : DirichletCharacter ℂ N
n : ℕ
hn✝ : 2 ≤ n
x : ℝ
hx : 1 < x
y : ℝ
hn' : IsUnit ↑n
hn : ↑n ^ (-x) < 1
hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1
⊢ χ ↑n ^ 2 * ↑n ^ (-(↑x + 2 * I * ↑y)) = ↑(↑n ^ (-x)) * (χ ↑n * ↑n ^ (-(I * ↑y))) ^ 2

Tactic:
·

Goal state after:
case e'_5.e'_5
N : ℕ
χ : DirichletCharacter ℂ N
n : ℕ
hn✝ : 2 ≤ n
x : ℝ
hx : 1 < x
y : ℝ
hn' : IsUnit ↑n
hn : ↑n ^ (-x) < 1
hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1
⊢ log (1 - ↑n ^ (-↑x)) = log (1 - ↑(↑n ^ (-x)))
Module: Mathlib.NumberTheory.LSeries.Nonvanishing

Goal state before:
pp: case e'_5.e'_6
N : ℕ
χ : DirichletCharacter ℂ N
n : ℕ
hn✝ : 2 ≤ n
x : ℝ
hx : 1 < x
y : ℝ
hn' : IsUnit ↑n
hn : ↑n ^ (-x) < 1
hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1
⊢ log (1 - χ ↑n * ↑n ^ (-(↑x + I * ↑y))) = log (1 - ↑(↑n ^ (-x)) * (χ ↑n * ↑n ^ (-(I * ↑y))))

pp: case e'_6
N : ℕ
χ : DirichletCharacter ℂ N
n : ℕ
hn✝ : 2 ≤ n
x : ℝ
hx : 1 < x
y : ℝ
hn' : IsUnit ↑n
hn : ↑n ^ (-x) < 1
hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1
⊢ χ ↑n ^ 2 * ↑n ^ (-(↑x + 2 * I * ↑y)) = ↑(↑n ^ (-x)) * (χ ↑n * ↑n ^ (-(I * ↑y))) ^ 2

Tactic:
·

Goal state after:
case e'_5.e'_6
N : ℕ
χ : DirichletCharacter ℂ N
n : ℕ
hn✝ : 2 ≤ n
x : ℝ
hx : 1 < x
y : ℝ
hn' : IsUnit ↑n
hn : ↑n ^ (-x) < 1
hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1
⊢ log (1 - χ ↑n * ↑n ^ (-(↑x + I * ↑y))) = log (1 - ↑(↑n ^ (-x)) * (χ ↑n * ↑n ^ (-(I * ↑y))))
Module: Mathlib.NumberTheory.LSeries.Nonvanishing

Goal state before:
pp: N : ℕ
χ : DirichletCharacter ℂ N
n : ℕ
hn✝ : 2 ≤ n
x : ℝ
hx : 1 < x
y : ℝ
hn' : IsUnit ↑n
hn : ↑n ^ (-x) < 1
hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1
⊢ ¬n = 0

Tactic:
by lia

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

Goal state before:
pp: case e'_6
N : ℕ
χ : DirichletCharacter ℂ N
n : ℕ
hn✝ : 2 ≤ n
x : ℝ
hx : 1 < x
y : ℝ
hn' : IsUnit ↑n
hn : ↑n ^ (-x) < 1
hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1
⊢ χ ↑n ^ 2 * ↑n ^ (-(↑x + 2 * I * ↑y)) = ↑(↑n ^ (-x)) * (χ ↑n * ↑n ^ (-(I * ↑y))) ^ 2

Tactic:
·

Goal state after:
case e'_6
N : ℕ
χ : DirichletCharacter ℂ N
n : ℕ
hn✝ : 2 ≤ n
x : ℝ
hx : 1 < x
y : ℝ
hn' : IsUnit ↑n
hn : ↑n ^ (-x) < 1
hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1
⊢ χ ↑n ^ 2 * ↑n ^ (-(↑x + 2 * I * ↑y)) = ↑(↑n ^ (-x)) * (χ ↑n * ↑n ^ (-(I * ↑y))) ^ 2
Module: Mathlib.NumberTheory.LSeries.Nonvanishing

Goal state before:
pp: case neg
N : ℕ
χ : DirichletCharacter ℂ N
n : ℕ
hn : 2 ≤ n
x : ℝ
hx : 1 < x
y : ℝ
hn' : ¬IsUnit ↑n
⊢ 0 ≤
    3 * (-log (1 - 1 ↑n * ↑n ^ (-↑x))).re + 4 * (-log (1 - χ ↑n * ↑n ^ (-(↑x + I * ↑y)))).re +
      (-log (1 - χ ↑n ^ 2 * ↑n ^ (-(↑x + 2 * I * ↑y)))).re

Tactic:
·

Goal state after:
case neg
N : ℕ
χ : DirichletCharacter ℂ N
n : ℕ
hn : 2 ≤ n
x : ℝ
hx : 1 < x
y : ℝ
hn' : ¬IsUnit ↑n
⊢ 0 ≤
    3 * (-log (1 - 1 ↑n * ↑n ^ (-↑x))).re + 4 * (-log (1 - χ ↑n * ↑n ^ (-(↑x + I * ↑y)))).re +
      (-log (1 - χ ↑n ^ 2 * ↑n ^ (-(↑x + 2 * I * ↑y)))).re
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: case e'_3.e_a.e_a.e_a
k : ℕ
hk : k ≠ 0
x : ℝ
hx : x ∈ Icc 0 1
this :
  HasSum (fun n ↦ 1 / ↑n ^ (2 * k) * ((fourier ↑n) ↑x + (fourier (-↑n)) ↑x))
    ((-1) ^ (k + 1) * (2 * ↑π) ^ (2 * k) / ↑(2 * k)! * ↑(bernoulliFun (2 * k) x))
ofReal_two : ↑2 = 2
n : ℕ
⊢ 2 * ↑π * I * ↑n * ↑x = 2 * ↑π * ↑n * ↑x * I

pp: case e'_3.e_a.e_a.e_a
k : ℕ
hk : k ≠ 0
x : ℝ
hx : x ∈ Icc 0 1
this :
  HasSum (fun n ↦ 1 / ↑n ^ (2 * k) * ((fourier ↑n) ↑x + (fourier (-↑n)) ↑x))
    ((-1) ^ (k + 1) * (2 * ↑π) ^ (2 * k) / ↑(2 * k)! * ↑(bernoulliFun (2 * k) x))
ofReal_two : ↑2 = 2
n : ℕ
⊢ 2 * ↑π * I * -↑n * ↑x = -(2 * ↑π * ↑n * ↑x) * I

Tactic:
·

Goal state after:
case e'_3.e_a.e_a.e_a
k : ℕ
hk : k ≠ 0
x : ℝ
hx : x ∈ Icc 0 1
this :
  HasSum (fun n ↦ 1 / ↑n ^ (2 * k) * ((fourier ↑n) ↑x + (fourier (-↑n)) ↑x))
    ((-1) ^ (k + 1) * (2 * ↑π) ^ (2 * k) / ↑(2 * k)! * ↑(bernoulliFun (2 * k) x))
ofReal_two : ↑2 = 2
n : ℕ
⊢ 2 * ↑π * I * ↑n * ↑x = 2 * ↑π * ↑n * ↑x * I
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: case e'_3.e_a.e_a.e_a
k : ℕ
hk : k ≠ 0
x : ℝ
hx : x ∈ Icc 0 1
this :
  HasSum (fun n ↦ 1 / ↑n ^ (2 * k) * ((fourier ↑n) ↑x + (fourier (-↑n)) ↑x))
    ((-1) ^ (k + 1) * (2 * ↑π) ^ (2 * k) / ↑(2 * k)! * ↑(bernoulliFun (2 * k) x))
ofReal_two : ↑2 = 2
n : ℕ
⊢ 2 * ↑π * I * -↑n * ↑x = -(2 * ↑π * ↑n * ↑x) * I

Tactic:
·

Goal state after:
case e'_3.e_a.e_a.e_a
k : ℕ
hk : k ≠ 0
x : ℝ
hx : x ∈ Icc 0 1
this :
  HasSum (fun n ↦ 1 / ↑n ^ (2 * k) * ((fourier ↑n) ↑x + (fourier (-↑n)) ↑x))
    ((-1) ^ (k + 1) * (2 * ↑π) ^ (2 * k) / ↑(2 * k)! * ↑(bernoulliFun (2 * k) x))
ofReal_two : ↑2 = 2
n : ℕ
⊢ 2 * ↑π * I * -↑n * ↑x = -(2 * ↑π * ↑n * ↑x) * I
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: case e'_6
k : ℕ
hk : k ≠ 0
x : ℝ
hx : x ∈ Icc 0 1
this :
  HasSum (fun n ↦ 1 / ↑n ^ (2 * k) * ((fourier ↑n) ↑x + (fourier (-↑n)) ↑x))
    ((-1) ^ (k + 1) * (2 * ↑π) ^ (2 * k) / ↑(2 * k)! * ↑(bernoulliFun (2 * k) x))
ofReal_two : ↑2 = 2
⊢ (-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 * k)! * ↑(bernoulliFun (2 * k) x) / 2).re

Tactic:
·

Goal state after:
case e'_6
k : ℕ
hk : k ≠ 0
x : ℝ
hx : x ∈ Icc 0 1
this :
  HasSum (fun n ↦ 1 / ↑n ^ (2 * k) * ((fourier ↑n) ↑x + (fourier (-↑n)) ↑x))
    ((-1) ^ (k + 1) * (2 * ↑π) ^ (2 * k) / ↑(2 * k)! * ↑(bernoulliFun (2 * k) x))
ofReal_two : ↑2 = 2
⊢ (-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 * k)! * ↑(bernoulliFun (2 * k) x) / 2).re
Module: Mathlib.NumberTheory.LSeries.PrimesInAP

Goal state before:
pp: case pos
q : ℕ
a : ZMod q
y : ℝ
hy : 1 < y
this : LSeriesSummable (fun n ↦ ↑(Λ n)) ↑y
n : ℕ
hn : ↑n = a
⊢ term (fun n ↦ ↑(residueClass a n)) (↑y) n = {n | ↑n = a}.indicator (term (fun n ↦ ↑(Λ n)) ↑y) n

pp: case neg
q : ℕ
a : ZMod q
y : ℝ
hy : 1 < y
this : LSeriesSummable (fun n ↦ ↑(Λ n)) ↑y
n : ℕ
hn : ¬↑n = a
⊢ term (fun n ↦ ↑(residueClass a n)) (↑y) n = {n | ↑n = a}.indicator (term (fun n ↦ ↑(Λ n)) ↑y) n

Tactic:
·

Goal state after:
case pos
q : ℕ
a : ZMod q
y : ℝ
hy : 1 < y
this : LSeriesSummable (fun n ↦ ↑(Λ n)) ↑y
n : ℕ
hn : ↑n = a
⊢ term (fun n ↦ ↑(residueClass a n)) (↑y) n = {n | ↑n = a}.indicator (term (fun n ↦ ↑(Λ n)) ↑y) n
Module: Mathlib.NumberTheory.LSeries.PrimesInAP

Goal state before:
pp: case neg
q : ℕ
a : ZMod q
y : ℝ
hy : 1 < y
this : LSeriesSummable (fun n ↦ ↑(Λ n)) ↑y
n : ℕ
hn : ¬↑n = a
⊢ term (fun n ↦ ↑(residueClass a n)) (↑y) n = {n | ↑n = a}.indicator (term (fun n ↦ ↑(Λ n)) ↑y) n

Tactic:
·

Goal state after:
case neg
q : ℕ
a : ZMod q
y : ℝ
hy : 1 < y
this : LSeriesSummable (fun n ↦ ↑(Λ n)) ↑y
n : ℕ
hn : ¬↑n = a
⊢ term (fun n ↦ ↑(residueClass a n)) (↑y) n = {n | ↑n = a}.indicator (term (fun n ↦ ↑(Λ n)) ↑y) n
Module: Mathlib.NumberTheory.LSeries.PrimesInAP

Goal state before:
pp: case refine_1
hp₀ : ∀ (p : Nat.Primes), 0 < (↑↑p)⁻¹
hp₁ : ∀ (p : Nat.Primes), (↑↑p)⁻¹ < 1
p : Nat.Primes
⊢ Summable fun y ↦ (↑↑(p, y).1)⁻¹ ^ (p, y).2 * (↑↑(p, y).1)⁻¹ ^ (3 / 2)

pp: case refine_2
hp₀ : ∀ (p : Nat.Primes), 0 < (↑↑p)⁻¹
hp₁ : ∀ (p : Nat.Primes), (↑↑p)⁻¹ < 1
⊢ Summable fun x ↦ ∑' (y : ℕ), (↑↑(x, y).1)⁻¹ ^ (x, y).2 * (↑↑(x, y).1)⁻¹ ^ (3 / 2)

Tactic:
·

Goal state after:
case refine_1
hp₀ : ∀ (p : Nat.Primes), 0 < (↑↑p)⁻¹
hp₁ : ∀ (p : Nat.Primes), (↑↑p)⁻¹ < 1
p : Nat.Primes
⊢ Summable fun y ↦ (↑↑(p, y).1)⁻¹ ^ (p, y).2 * (↑↑(p, y).1)⁻¹ ^ (3 / 2)
Module: Mathlib.NumberTheory.LSeries.PrimesInAP

Goal state before:
pp: case refine_2
hp₀ : ∀ (p : Nat.Primes), 0 < (↑↑p)⁻¹
hp₁ : ∀ (p : Nat.Primes), (↑↑p)⁻¹ < 1
⊢ Summable fun x ↦ ∑' (y : ℕ), (↑↑(x, y).1)⁻¹ ^ (x, y).2 * (↑↑(x, y).1)⁻¹ ^ (3 / 2)

Tactic:
·

Goal state after:
case refine_2
hp₀ : ∀ (p : Nat.Primes), 0 < (↑↑p)⁻¹
hp₁ : ∀ (p : Nat.Primes), (↑↑p)⁻¹ < 1
⊢ Summable fun x ↦ ∑' (y : ℕ), (↑↑(x, y).1)⁻¹ ^ (x, y).2 * (↑↑(x, y).1)⁻¹ ^ (3 / 2)
Module: Mathlib.NumberTheory.LSeries.PrimesInAP

Goal state before:
pp: case refine_2.refine_1
hp₀ : ∀ (p : Nat.Primes), 0 < (↑↑p)⁻¹
hp₁ : ∀ (p : Nat.Primes), (↑↑p)⁻¹ < 1
p : Nat.Primes
⊢ 0 ≤ (1 - (↑↑p)⁻¹)⁻¹ * (↑↑p)⁻¹ ^ (3 / 2)

pp: case refine_2.refine_2
hp₀ : ∀ (p : Nat.Primes), 0 < (↑↑p)⁻¹
hp₁ : ∀ (p : Nat.Primes), (↑↑p)⁻¹ < 1
p : Nat.Primes
⊢ (1 - (↑↑p)⁻¹)⁻¹ * (↑↑p)⁻¹ ^ (3 / 2) ≤ 2 * ↑↑p ^ (-(3 / 2))

Tactic:
·

Goal state after:
case refine_2.refine_1
hp₀ : ∀ (p : Nat.Primes), 0 < (↑↑p)⁻¹
hp₁ : ∀ (p : Nat.Primes), (↑↑p)⁻¹ < 1
p : Nat.Primes
⊢ 0 ≤ (1 - (↑↑p)⁻¹)⁻¹ * (↑↑p)⁻¹ ^ (3 / 2)
Module: Mathlib.NumberTheory.LSeries.PrimesInAP

Goal state before:
pp: case refine_2.refine_2
hp₀ : ∀ (p : Nat.Primes), 0 < (↑↑p)⁻¹
hp₁ : ∀ (p : Nat.Primes), (↑↑p)⁻¹ < 1
p : Nat.Primes
⊢ (1 - (↑↑p)⁻¹)⁻¹ * (↑↑p)⁻¹ ^ (3 / 2) ≤ 2 * ↑↑p ^ (-(3 / 2))

Tactic:
·

Goal state after:
case refine_2.refine_2
hp₀ : ∀ (p : Nat.Primes), 0 < (↑↑p)⁻¹
hp₁ : ∀ (p : Nat.Primes), (↑↑p)⁻¹ < 1
p : Nat.Primes
⊢ (1 - (↑↑p)⁻¹)⁻¹ * (↑↑p)⁻¹ ^ (3 / 2) ≤ 2 * ↑↑p ^ (-(3 / 2))
Module: Mathlib.NumberTheory.LSeries.PrimesInAP

Goal state before:
pp: case refine_2
q : ℕ
a : ZMod q
h₀ : ∀ (n : ℕ), 0 ≤ (if Nat.Prime n then 0 else residueClass a n) / ↑n
hleF₀ : ∀ (n : ℕ), (if Nat.Prime n then 0 else residueClass a n) / ↑n ≤ F₀ n
hF₀ : ∀ (p : Nat.Primes), F₀ ↑p = 0
n : ℕ
hn : ¬IsPrimePow n
⊢ F₀ n = 0

pp: case refine_1
q : ℕ
a : ZMod q
h₀ : ∀ (n : ℕ), 0 ≤ (if Nat.Prime n then 0 else residueClass a n) / ↑n
hleF₀ : ∀ (n : ℕ), (if Nat.Prime n then 0 else residueClass a n) / ↑n ≤ F₀ n
hF₀ : ∀ (p : Nat.Primes), F₀ ↑p = 0
⊢ Summable (F₀ ∘ Subtype.val)

Tactic:
·

Goal state after:
case refine_2
q : ℕ
a : ZMod q
h₀ : ∀ (n : ℕ), 0 ≤ (if Nat.Prime n then 0 else residueClass a n) / ↑n
hleF₀ : ∀ (n : ℕ), (if Nat.Prime n then 0 else residueClass a n) / ↑n ≤ F₀ n
hF₀ : ∀ (p : Nat.Primes), F₀ ↑p = 0
n : ℕ
hn : ¬IsPrimePow n
⊢ F₀ n = 0
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: k : ℕ
hk : k ≠ 0
x : ℝ
hx : x ∈ Icc 0 1
⊢ 2 ≤ 2 * k + 1

Tactic:
by lia

Goal state after:
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: case e'_5
k : ℕ
hk : k ≠ 0
x : ℝ
hx : x ∈ Icc 0 1
⊢ (fun n ↦ 1 / ↑n ^ (2 * k + 1) * ((fourier ↑n) ↑x - (fourier (-↑n)) ↑x)) = fun n ↦
    1 / ↑n ^ (2 * k + 1) * ((fourier ↑n) ↑x + (-1) ^ (2 * k + 1) * (fourier (-↑n)) ↑x)

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

Tactic:
·

Goal state after:
case e'_5
k : ℕ
hk : k ≠ 0
x : ℝ
hx : x ∈ Icc 0 1
⊢ (fun n ↦ 1 / ↑n ^ (2 * k + 1) * ((fourier ↑n) ↑x - (fourier (-↑n)) ↑x)) = fun n ↦
    1 / ↑n ^ (2 * k + 1) * ((fourier ↑n) ↑x + (-1) ^ (2 * k + 1) * (fourier (-↑n)) ↑x)
Module: Mathlib.NumberTheory.LSeries.Nonvanishing

Goal state before:
pp: case e'_3
N : ℕ
inst✝ : NeZero N
⊢ (fun w ↦ w * LFunctionTrivChar N (1 + w)) = (fun s ↦ (s - 1) * LFunctionTrivChar N s) ∘ fun w ↦ 1 + w

pp: case convert_2
N : ℕ
inst✝ : NeZero N
⊢ Tendsto (fun w ↦ 1 + w) (𝓝[≠] 0) (𝓝[≠] 1)

Tactic:
·

Goal state after:
case e'_3
N : ℕ
inst✝ : NeZero N
⊢ (fun w ↦ w * LFunctionTrivChar N (1 + w)) = (fun s ↦ (s - 1) * LFunctionTrivChar N s) ∘ fun w ↦ 1 + w
Module: Mathlib.NumberTheory.LSeries.Nonvanishing

Goal state before:
pp: case convert_2
N : ℕ
inst✝ : NeZero N
⊢ Tendsto (fun w ↦ 1 + w) (𝓝[≠] 0) (𝓝[≠] 1)

Tactic:
·

Goal state after:
case convert_2
N : ℕ
inst✝ : NeZero N
⊢ Tendsto (fun w ↦ 1 + w) (𝓝[≠] 0) (𝓝[≠] 1)
Module: Mathlib.NumberTheory.LSeries.PrimesInAP

Goal state before:
pp: case refine_1.refine_1
q : ℕ
a : ZMod q
h₀ : ∀ (n : ℕ), 0 ≤ (if Nat.Prime n then 0 else residueClass a n) / ↑n
hleF₀ : ∀ (n : ℕ), (if Nat.Prime n then 0 else residueClass a n) / ↑n ≤ F₀ n
hF₀ : ∀ (p : Nat.Primes), F₀ ↑p = 0
hFF' : F₀ ∘ Subtype.val = F' ∘ ⇑prodNatEquiv.symm
hF'₀ : ∀ (p : Nat.Primes), F' (p, 0) = 0
hF'₁ : F'' = F' ∘ Prod.map id fun x ↦ x + 1
⊢ Function.Injective (Prod.map id fun x ↦ x + 1)

pp: case refine_1.refine_2
q : ℕ
a : ZMod q
h₀ : ∀ (n : ℕ), 0 ≤ (if Nat.Prime n then 0 else residueClass a n) / ↑n
hleF₀ : ∀ (n : ℕ), (if Nat.Prime n then 0 else residueClass a n) / ↑n ≤ F₀ n
hF₀ : ∀ (p : Nat.Primes), F₀ ↑p = 0
hFF' : F₀ ∘ Subtype.val = F' ∘ ⇑prodNatEquiv.symm
hF'₀ : ∀ (p : Nat.Primes), F' (p, 0) = 0
hF'₁ : F'' = F' ∘ Prod.map id fun x ↦ x + 1
u : Nat.Primes × ℕ
hu : u ∉ Set.range (Prod.map id fun x ↦ x + 1)
⊢ F' u = 0

Tactic:
·

Goal state after:
case refine_1.refine_1
q : ℕ
a : ZMod q
h₀ : ∀ (n : ℕ), 0 ≤ (if Nat.Prime n then 0 else residueClass a n) / ↑n
hleF₀ : ∀ (n : ℕ), (if Nat.Prime n then 0 else residueClass a n) / ↑n ≤ F₀ n
hF₀ : ∀ (p : Nat.Primes), F₀ ↑p = 0
hFF' : F₀ ∘ Subtype.val = F' ∘ ⇑prodNatEquiv.symm
hF'₀ : ∀ (p : Nat.Primes), F' (p, 0) = 0
hF'₁ : F'' = F' ∘ Prod.map id fun x ↦ x + 1
⊢ Function.Injective (Prod.map id fun x ↦ x + 1)
Module: Mathlib.NumberTheory.LSeries.PrimesInAP

Goal state before:
pp: case refine_1.refine_2
q : ℕ
a : ZMod q
h₀ : ∀ (n : ℕ), 0 ≤ (if Nat.Prime n then 0 else residueClass a n) / ↑n
hleF₀ : ∀ (n : ℕ), (if Nat.Prime n then 0 else residueClass a n) / ↑n ≤ F₀ n
hF₀ : ∀ (p : Nat.Primes), F₀ ↑p = 0
hFF' : F₀ ∘ Subtype.val = F' ∘ ⇑prodNatEquiv.symm
hF'₀ : ∀ (p : Nat.Primes), F' (p, 0) = 0
hF'₁ : F'' = F' ∘ Prod.map id fun x ↦ x + 1
u : Nat.Primes × ℕ
hu : u ∉ Set.range (Prod.map id fun x ↦ x + 1)
⊢ F' u = 0

Tactic:
·

Goal state after:
case refine_1.refine_2
q : ℕ
a : ZMod q
h₀ : ∀ (n : ℕ), 0 ≤ (if Nat.Prime n then 0 else residueClass a n) / ↑n
hleF₀ : ∀ (n : ℕ), (if Nat.Prime n then 0 else residueClass a n) / ↑n ≤ F₀ n
hF₀ : ∀ (p : Nat.Primes), F₀ ↑p = 0
hFF' : F₀ ∘ Subtype.val = F' ∘ ⇑prodNatEquiv.symm
hF'₀ : ∀ (p : Nat.Primes), F' (p, 0) = 0
hF'₁ : F'' = F' ∘ Prod.map id fun x ↦ x + 1
u : Nat.Primes × ℕ
hu : u ∉ Set.range (Prod.map id fun x ↦ x + 1)
⊢ F' u = 0
Module: Mathlib.NumberTheory.ZetaValues

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

Tactic:
·

Goal state after:
case e'_6
k : ℕ
hk : k ≠ 0
x : ℝ
hx : x ∈ Icc 0 1
⊢ (-1) ^ (k + 1) * I * (2 * ↑π) ^ (2 * k + 1) / ↑(2 * k + 1)! * ↑(bernoulliFun (2 * k + 1) x) =
    -(2 * ↑π * I) ^ (2 * k + 1) / ↑(2 * k + 1)! * ↑(bernoulliFun (2 * k + 1) x)
Module: Mathlib.NumberTheory.LSeries.Nonvanishing

Goal state before:
pp: case inl
N : ℕ
χ : DirichletCharacter ℂ N
inst✝ : NeZero N
t : ℝ
h : χ ^ 2 ≠ 1 ∨ t ≠ 0
Hz : LFunction χ (1 + I * ↑t) = 0
hz₁ : t ≠ 0 ∨ χ ≠ 1
hz₂ : 2 * t ≠ 0 ∨ χ ^ 2 ≠ 1
⊢ (1 / ↑0) ^ 3 * ↑0 ^ 4 * 1 = ↑0

pp: case inr
N : ℕ
χ : DirichletCharacter ℂ N
inst✝ : NeZero N
t : ℝ
h✝ : χ ^ 2 ≠ 1 ∨ t ≠ 0
Hz : LFunction χ (1 + I * ↑t) = 0
hz₁ : t ≠ 0 ∨ χ ≠ 1
hz₂ : 2 * t ≠ 0 ∨ χ ^ 2 ≠ 1
x : ℝ
h : x ≠ 0
⊢ (1 / ↑x) ^ 3 * ↑x ^ 4 * 1 = ↑x

Tactic:
·

Goal state after:
case inl
N : ℕ
χ : DirichletCharacter ℂ N
inst✝ : NeZero N
t : ℝ
h : χ ^ 2 ≠ 1 ∨ t ≠ 0
Hz : LFunction χ (1 + I * ↑t) = 0
hz₁ : t ≠ 0 ∨ χ ≠ 1
hz₂ : 2 * t ≠ 0 ∨ χ ^ 2 ≠ 1
⊢ (1 / ↑0) ^ 3 * ↑0 ^ 4 * 1 = ↑0
Module: Mathlib.NumberTheory.LSeries.Nonvanishing

Goal state before:
pp: N : ℕ
χ : DirichletCharacter ℂ N
inst✝ : NeZero N
t : ℝ
h : χ ^ 2 ≠ 1 ∨ t ≠ 0
Hz : LFunction χ (1 + I * ↑t) = 0
hz₁ : t ≠ 0 ∨ χ ≠ 1
hz₂ : 2 * t ≠ 0 ∨ χ ^ 2 ≠ 1
⊢ 4 ≠ 0

Tactic:
by lia

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

Goal state before:
pp: case refine_1
q : ℕ
a : ZMod q
inst✝ : NeZero q
⊢ ContinuousOn (fun s ↦ -deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s)
    {s | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0}

pp: case refine_2
q : ℕ
a : ZMod q
inst✝ : NeZero q
⊢ ContinuousOn (fun s ↦ -∑ χ ∈ {1}ᶜ, χ a⁻¹ * deriv (LFunction χ) s / LFunction χ s)
    {s | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0}

Tactic:
·

Goal state after:
case refine_1
q : ℕ
a : ZMod q
inst✝ : NeZero q
⊢ ContinuousOn (fun s ↦ -deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s)
    {s | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0}
Module: Mathlib.NumberTheory.LSeries.Nonvanishing

Goal state before:
pp: case inr
N : ℕ
χ : DirichletCharacter ℂ N
inst✝ : NeZero N
t : ℝ
h✝ : χ ^ 2 ≠ 1 ∨ t ≠ 0
Hz : LFunction χ (1 + I * ↑t) = 0
hz₁ : t ≠ 0 ∨ χ ≠ 1
hz₂ : 2 * t ≠ 0 ∨ χ ^ 2 ≠ 1
x : ℝ
h : x ≠ 0
⊢ (1 / ↑x) ^ 3 * ↑x ^ 4 * 1 = ↑x

Tactic:
·

Goal state after:
case inr
N : ℕ
χ : DirichletCharacter ℂ N
inst✝ : NeZero N
t : ℝ
h✝ : χ ^ 2 ≠ 1 ∨ t ≠ 0
Hz : LFunction χ (1 + I * ↑t) = 0
hz₁ : t ≠ 0 ∨ χ ≠ 1
hz₂ : 2 * t ≠ 0 ∨ χ ^ 2 ≠ 1
x : ℝ
h : x ≠ 0
⊢ (1 / ↑x) ^ 3 * ↑x ^ 4 * 1 = ↑x
Module: Mathlib.NumberTheory.LSeries.PrimesInAP

Goal state before:
pp: case refine_2
q : ℕ
a : ZMod q
inst✝ : NeZero q
⊢ ContinuousOn (fun s ↦ -∑ χ ∈ {1}ᶜ, χ a⁻¹ * deriv (LFunction χ) s / LFunction χ s)
    {s | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0}

Tactic:
·

Goal state after:
case refine_2
q : ℕ
a : ZMod q
inst✝ : NeZero q
⊢ ContinuousOn (fun s ↦ -∑ χ ∈ {1}ᶜ, χ a⁻¹ * deriv (LFunction χ) s / LFunction χ s)
    {s | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0}
Module: Mathlib.NumberTheory.LSeries.PrimesInAP

Goal state before:
pp: q : ℕ
a : ZMod q
inst✝ : NeZero q
χ : DirichletCharacter ℂ q
hχ : χ ∈ {1}ᶜ
⊢ χ ≠ 1

Tactic:
simpa only [ne_eq, Finset.mem_compl, Finset.mem_singleton] using hχ

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

Goal state before:
pp: case refine_2.inl
q : ℕ
a : ZMod q
inst✝ : NeZero q
χ : DirichletCharacter ℂ q
hχ : χ ≠ 1
⊢ 1 ∈ {s | LFunction χ s ≠ 0}

pp: case refine_2.inr
q : ℕ
a : ZMod q
inst✝ : NeZero q
χ : DirichletCharacter ℂ q
hχ : χ ≠ 1
s : ℂ
hs : ∀ (χ : DirichletCharacter ℂ q), ¬LFunction χ s = 0
⊢ s ∈ {s | LFunction χ s ≠ 0}

Tactic:
·

Goal state after:
case refine_2.inl
q : ℕ
a : ZMod q
inst✝ : NeZero q
χ : DirichletCharacter ℂ q
hχ : χ ≠ 1
⊢ 1 ∈ {s | LFunction χ s ≠ 0}
Module: Mathlib.NumberTheory.LSeries.PrimesInAP

Goal state before:
pp: case refine_2.inr
q : ℕ
a : ZMod q
inst✝ : NeZero q
χ : DirichletCharacter ℂ q
hχ : χ ≠ 1
s : ℂ
hs : ∀ (χ : DirichletCharacter ℂ q), ¬LFunction χ s = 0
⊢ s ∈ {s | LFunction χ s ≠ 0}

Tactic:
·

Goal state after:
case refine_2.inr
q : ℕ
a : ZMod q
inst✝ : NeZero q
χ : DirichletCharacter ℂ q
hχ : χ ≠ 1
s : ℂ
hs : ∀ (χ : DirichletCharacter ℂ q), ¬LFunction χ s = 0
⊢ s ∈ {s | LFunction χ s ≠ 0}
Module: Mathlib.NumberTheory.LSeries.PrimesInAP

Goal state before:
pp: case inl
q : ℕ
a : ZMod q
inst✝ : NeZero q
hs : 1 ∈ {s | 1 ≤ s.re}
⊢ 1 ∈ {s | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0}

pp: case inr
q : ℕ
a : ZMod q
inst✝ : NeZero q
s : ℂ
hs : s ∈ {s | 1 ≤ s.re}
hs₁ : s ≠ 1
⊢ s ∈ {s | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0}

Tactic:
·

Goal state after:
case inl
q : ℕ
a : ZMod q
inst✝ : NeZero q
hs : 1 ∈ {s | 1 ≤ s.re}
⊢ 1 ∈ {s | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0}
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: case e'_5
k : ℕ
hk : k ≠ 0
x : ℝ
hx : x ∈ Icc 0 1
this :
  HasSum (fun n ↦ 1 / ↑n ^ (2 * k + 1) * ((fourier ↑n) ↑x - (fourier (-↑n)) ↑x))
    ((-1) ^ (k + 1) * I * (2 * ↑π) ^ (2 * k + 1) / ↑(2 * k + 1)! * ↑(bernoulliFun (2 * k + 1) x))
ofReal_two : ↑2 = 2
x✝ : ℕ
⊢ 1 / ↑x✝ ^ (2 * k + 1) * Real.sin (2 * π * ↑x✝ * x) =
    (1 / ↑x✝ ^ (2 * k + 1) * ((fourier ↑x✝) ↑x - (fourier (-↑x✝)) ↑x) / (2 * I)).re

pp: case e'_6
k : ℕ
hk : k ≠ 0
x : ℝ
hx : x ∈ Icc 0 1
this :
  HasSum (fun n ↦ 1 / ↑n ^ (2 * k + 1) * ((fourier ↑n) ↑x - (fourier (-↑n)) ↑x))
    ((-1) ^ (k + 1) * I * (2 * ↑π) ^ (2 * k + 1) / ↑(2 * k + 1)! * ↑(bernoulliFun (2 * k + 1) x))
ofReal_two : ↑2 = 2
⊢ (-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) * I * (2 * ↑π) ^ (2 * k + 1) / ↑(2 * k + 1)! * ↑(bernoulliFun (2 * k + 1) x) / (2 * I)).re

Tactic:
·

Goal state after:
case e'_5
k : ℕ
hk : k ≠ 0
x : ℝ
hx : x ∈ Icc 0 1
this :
  HasSum (fun n ↦ 1 / ↑n ^ (2 * k + 1) * ((fourier ↑n) ↑x - (fourier (-↑n)) ↑x))
    ((-1) ^ (k + 1) * I * (2 * ↑π) ^ (2 * k + 1) / ↑(2 * k + 1)! * ↑(bernoulliFun (2 * k + 1) x))
ofReal_two : ↑2 = 2
x✝ : ℕ
⊢ 1 / ↑x✝ ^ (2 * k + 1) * Real.sin (2 * π * ↑x✝ * x) =
    (1 / ↑x✝ ^ (2 * k + 1) * ((fourier ↑x✝) ↑x - (fourier (-↑x✝)) ↑x) / (2 * I)).re
Module: Mathlib.NumberTheory.LSeries.PrimesInAP

Goal state before:
pp: case inr
q : ℕ
a : ZMod q
inst✝ : NeZero q
s : ℂ
hs : s ∈ {s | 1 ≤ s.re}
hs₁ : s ≠ 1
⊢ s ∈ {s | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0}

Tactic:
·

Goal state after:
case inr
q : ℕ
a : ZMod q
inst✝ : NeZero q
s : ℂ
hs : s ∈ {s | 1 ≤ s.re}
hs₁ : s ≠ 1
⊢ s ∈ {s | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0}
Module: Mathlib.NumberTheory.LSeries.Injectivity

Goal state before:
pp: case inr
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x ↦ {m | n + 1 < m}.indicator fun m ↦ f m / (↑m / (↑n + 1)) ^ ↑x
hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0
hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m
hs : ∀ {x : ℝ}, x ≥ y → Summable fun m ↦ (↑n + 1) ^ ↑x * term f (↑x) m
key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m
hys : Summable (F y)
k : ℕ
H : k ≤ n + 1
⊢ Tendsto (fun x ↦ F x k) atTop (nhds 0)

Tactic:
·

Goal state after:
case inr
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x ↦ {m | n + 1 < m}.indicator fun m ↦ f m / (↑m / (↑n + 1)) ^ ↑x
hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0
hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m
hs : ∀ {x : ℝ}, x ≥ y → Summable fun m ↦ (↑n + 1) ^ ↑x * term f (↑x) m
key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m
hys : Summable (F y)
k : ℕ
H : k ≤ n + 1
⊢ Tendsto (fun x ↦ F x k) atTop (nhds 0)
Module: Mathlib.NumberTheory.LSeries.Nonvanishing

Goal state before:
pp: case pos
N : ℕ
χ : DirichletCharacter ℂ N
inst✝ : NeZero N
s : ℂ
hs : s.re = 1
hχs : χ ≠ 1 ∨ s ≠ 1
h : χ ^ 2 = 1 ∧ s = 1
⊢ LFunction χ s ≠ 0

pp: case neg
N : ℕ
χ : DirichletCharacter ℂ N
inst✝ : NeZero N
s : ℂ
hs : s.re = 1
hχs : χ ≠ 1 ∨ s ≠ 1
h : ¬(χ ^ 2 = 1 ∧ s = 1)
⊢ LFunction χ s ≠ 0

Tactic:
·

Goal state after:
case pos
N : ℕ
χ : DirichletCharacter ℂ N
inst✝ : NeZero N
s : ℂ
hs : s.re = 1
hχs : χ ≠ 1 ∨ s ≠ 1
h : χ ^ 2 = 1 ∧ s = 1
⊢ LFunction χ s ≠ 0
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: case e'_3.e_a
k : ℕ
hk : k ≠ 0
x : ℝ
hx : x ∈ Icc 0 1
this :
  HasSum (fun n ↦ 1 / ↑n ^ (2 * k + 1) * ((fourier ↑n) ↑x - (fourier (-↑n)) ↑x))
    ((-1) ^ (k + 1) * I * (2 * ↑π) ^ (2 * k + 1) / ↑(2 * k + 1)! * ↑(bernoulliFun (2 * k + 1) x))
ofReal_two : ↑2 = 2
x✝ : ℕ
⊢ 1 / ↑x✝ ^ (2 * k + 1) = ↑(1 / ↑x✝ ^ (2 * k + 1))

pp: case e'_3.e_a
k : ℕ
hk : k ≠ 0
x : ℝ
hx : x ∈ Icc 0 1
this :
  HasSum (fun n ↦ 1 / ↑n ^ (2 * k + 1) * ((fourier ↑n) ↑x - (fourier (-↑n)) ↑x))
    ((-1) ^ (k + 1) * I * (2 * ↑π) ^ (2 * k + 1) / ↑(2 * k + 1)! * ↑(bernoulliFun (2 * k + 1) x))
ofReal_two : ↑2 = 2
x✝ : ℕ
⊢ ((fourier ↑x✝) ↑x - (fourier (-↑x✝)) ↑x) / (2 * I) = ↑(Real.sin (2 * π * ↑x✝ * x))

Tactic:
·

Goal state after:
case e'_3.e_a
k : ℕ
hk : k ≠ 0
x : ℝ
hx : x ∈ Icc 0 1
this :
  HasSum (fun n ↦ 1 / ↑n ^ (2 * k + 1) * ((fourier ↑n) ↑x - (fourier (-↑n)) ↑x))
    ((-1) ^ (k + 1) * I * (2 * ↑π) ^ (2 * k + 1) / ↑(2 * k + 1)! * ↑(bernoulliFun (2 * k + 1) x))
ofReal_two : ↑2 = 2
x✝ : ℕ
⊢ 1 / ↑x✝ ^ (2 * k + 1) = ↑(1 / ↑x✝ ^ (2 * k + 1))
Module: Mathlib.NumberTheory.LSeries.Nonvanishing

Goal state before:
pp: case neg
N : ℕ
χ : DirichletCharacter ℂ N
inst✝ : NeZero N
s : ℂ
hs : s.re = 1
hχs : χ ≠ 1 ∨ s ≠ 1
h : ¬(χ ^ 2 = 1 ∧ s = 1)
⊢ LFunction χ s ≠ 0

Tactic:
·

Goal state after:
case neg
N : ℕ
χ : DirichletCharacter ℂ N
inst✝ : NeZero N
s : ℂ
hs : s.re = 1
hχs : χ ≠ 1 ∨ s ≠ 1
h : ¬(χ ^ 2 = 1 ∧ s = 1)
⊢ LFunction χ s ≠ 0