mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.Height.NumberField

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
m n : ℕ
hm : m ≠ 0
hn : n ≠ 0
a b : 𝓞 K
h : ↑n * a = ↑m * b
⊢ (Submodule.toAddSubgroup (span {↑m * ↑n})).relIndex (Submodule.toAddSubgroup (span {↑(m * n), ↑m * b})) =
    (Submodule.toAddSubgroup (span {↑n})).relIndex (Submodule.toAddSubgroup (span {↑n, b}))

Tactic:
exact (relIndex_span_span_nat_mul n hm b).symm

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
m n : ℕ
hm : m ≠ 0
hn : n ≠ 0
a b : 𝓞 K
h : ↑n * a = ↑m * b
⊢ (Submodule.toAddSubgroup (span {↑m})).relIndex (Submodule.toAddSubgroup (span {↑m, a})) =
    (Submodule.toAddSubgroup (span {↑n})).relIndex (Submodule.toAddSubgroup (span {↑n, b}))

Tactic:
by

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

Goal state before:
pp: 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
h✝ : IsEmpty ι'
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)
⊢ 1 * mulHeight x✝ ^ N ≤ mulHeight fun j ↦ (eval x✝) (p j)

Tactic:
by

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

Goal state before:
pp: h₁ :
  HasDerivAt (id * completedRiemannZeta₀ - 1 - id / (1 - id))
    (1 * completedRiemannZeta₀ 0 + id 0 * deriv completedRiemannZeta₀ 0 - 0 -
      (1 * (1 - id) 0 - id 0 * (0 - 1)) / (1 - id) 0 ^ 2)
    0
h₂ :
  HasDerivAt ((fun x ↦ 2) * fun x ↦ ↑π ^ (-x / 2))
    (0 * ↑π ^ (-0 / 2) + 2 * (-0 / 2 * ↑π ^ (-0 / 2 - 1) * 0 + ↑π ^ (-0 / 2) * log ↑π * (-1 / 2))) 0
⊢ HasDerivAt (Gamma ∘ fun x ↦ x / 2 + 1) (deriv Gamma (0 / 2 + 1) * (1 / 2 + 0)) 0

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
hx : IsAlgebraic ℤ x
m : ℕ
r : 𝓞 K
hm : m ≠ 0
hmr : m • x = ↑r
n : ℕ := (Submodule.toAddSubgroup (span {↑m})).relIndex (Submodule.toAddSubgroup (span {↑m, r}))
hndef : n = (Submodule.toAddSubgroup (span {↑m})).relIndex (Submodule.toAddSubgroup (span {↑m, r}))
⊢ span {↑m} ≠ ⊥

Tactic:
by

Goal state after:
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
⊢ jacobiSum χ χ⁻¹ = -χ (-1)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
hx : IsAlgebraic ℤ x
m : ℕ
r : 𝓞 K
hm : m ≠ 0
hmr : m • x = ↑r
n : ℕ := (Submodule.toAddSubgroup (span {↑m})).relIndex (Submodule.toAddSubgroup (span {↑m, r}))
hndef : n = (Submodule.toAddSubgroup (span {↑m})).relIndex (Submodule.toAddSubgroup (span {↑m, r}))
hn : n ≠ 0
⊢ r ∈ {↑m, r}

Tactic:
by

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

Goal state before:
pp: 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
h✝ : Nonempty ι'
this : 0 < Nat.card ι'
⊢ (↑(Nat.card ι') ^ totalWeight K * max (mulHeightBound q) 1)⁻¹ > 0

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
hx : IsAlgebraic ℤ x
m : ℕ
r : 𝓞 K
hm : m ≠ 0
hmr : m • x = ↑r
n : ℕ := (Submodule.toAddSubgroup (span {↑m})).relIndex (Submodule.toAddSubgroup (span {↑m, r}))
hndef : n = (Submodule.toAddSubgroup (span {↑m})).relIndex (Submodule.toAddSubgroup (span {↑m, r}))
hn : n ≠ 0
⊢ ∃ a, ↑m * a = ↑n * r

Tactic:
by

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

Goal state before:
pp: h₁ :
  HasDerivAt (id * completedRiemannZeta₀ - 1 - id / (1 - id))
    (1 * completedRiemannZeta₀ 0 + id 0 * deriv completedRiemannZeta₀ 0 - 0 -
      (1 * (1 - id) 0 - id 0 * (0 - 1)) / (1 - id) 0 ^ 2)
    0
h₂ :
  HasDerivAt ((fun x ↦ 2) * fun x ↦ ↑π ^ (-x / 2))
    (0 * ↑π ^ (-0 / 2) + 2 * (-0 / 2 * ↑π ^ (-0 / 2 - 1) * 0 + ↑π ^ (-0 / 2) * log ↑π * (-1 / 2))) 0
h₃ : HasDerivAt (Gamma ∘ fun x ↦ x / 2 + 1) (deriv Gamma (0 / 2 + 1) * (1 / 2 + 0)) 0
h : -(log (2 * ↑π) * 2) = ↑γ - log (4 * ↑π) + (-log ↑π + -↑γ)
⊢ (((fun x ↦ 2) * fun x ↦ ↑π ^ (-x / 2)) * Gamma ∘ fun x ↦ x / 2 + 1) 0 ≠ 0

Tactic:
by

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

Goal state before:
pp: 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
⊢ ∃ C > 0,
    ∀ (p : ι' → MvPolynomial ι K) {x : ι → K},
      (∀ (k : ι), ∑ j, (eval x) (q (k, j)) * (eval x) (p j) = x k ^ (M + N)) →
        C * mulHeight x ^ N ≤ mulHeight fun j ↦ (eval x) (p j)

Tactic:
by

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

Goal state before:
pp: 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'
⊢ ↑(Nat.card ι') ^ totalWeight K ≠ 0

Tactic:
by

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

Goal state before:
pp: h₁ :
  HasDerivAt (id * completedRiemannZeta₀ - 1 - id / (1 - id))
    (1 * completedRiemannZeta₀ 0 + id 0 * deriv completedRiemannZeta₀ 0 - 0 -
      (1 * (1 - id) 0 - id 0 * (0 - 1)) / (1 - id) 0 ^ 2)
    0
h₂ :
  HasDerivAt ((fun x ↦ 2) * fun x ↦ ↑π ^ (-x / 2))
    (0 * ↑π ^ (-0 / 2) + 2 * (-0 / 2 * ↑π ^ (-0 / 2 - 1) * 0 + ↑π ^ (-0 / 2) * log ↑π * (-1 / 2))) 0
h₃ : HasDerivAt (Gamma ∘ fun x ↦ x / 2 + 1) (deriv Gamma (0 / 2 + 1) * (1 / 2 + 0)) 0
h : -(log (2 * ↑π) * 2) = ↑γ - log (2 * 2 * ↑π) + (-log ↑π + -↑γ)
⊢ HasDerivAt (fun x ↦ (x * completedRiemannZeta₀ x - 1 - x / (1 - x)) / (2 * ↑π ^ (-x / 2) * Gamma (x / 2 + 1)))
    (-log (2 * ↑π) / 2) 0

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
hx : IsAlgebraic ℤ x
m : ℕ
r : 𝓞 K
hm : m ≠ 0
hmr : m • x = ↑r
n : ℕ := (Submodule.toAddSubgroup (span {↑m})).relIndex (Submodule.toAddSubgroup (span {↑m, r}))
hndef : n = (Submodule.toAddSubgroup (span {↑m})).relIndex (Submodule.toAddSubgroup (span {↑m, r}))
hn : n ≠ 0
a : 𝓞 K
ha' : ↑m * a = ↑n * r
⊢ ↑n * x = ↑a

Tactic:
by

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

Goal state before:
pp: 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'
this : ↑(Nat.card ι') ^ totalWeight K ≠ 0
⊢ 0 < (↑(Nat.card ι') ^ totalWeight K * max (mulHeightBound q) 1)⁻¹ * mulHeight x ^ N

Tactic:
by

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

Goal state before:
pp: 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)
⊢ -log (↑(Nat.card ι') ^ totalWeight K * max (mulHeightBound q) 1) + ↑N * logHeight x ≤ logHeight fun j ↦ (eval x) (p j)

Tactic:
by

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

Goal state before:
pp: h₁ :
  HasDerivAt (id * completedRiemannZeta₀ - 1 - id / (1 - id))
    (1 * completedRiemannZeta₀ 0 + id 0 * deriv completedRiemannZeta₀ 0 - 0 -
      (1 * (1 - id) 0 - id 0 * (0 - 1)) / (1 - id) 0 ^ 2)
    0
h₂ :
  HasDerivAt ((fun x ↦ 2) * fun x ↦ ↑π ^ (-x / 2))
    (0 * ↑π ^ (-0 / 2) + 2 * (-0 / 2 * ↑π ^ (-0 / 2 - 1) * 0 + ↑π ^ (-0 / 2) * log ↑π * (-1 / 2))) 0
h₃ : HasDerivAt (Gamma ∘ fun x ↦ x / 2 + 1) (deriv Gamma (0 / 2 + 1) * (1 / 2 + 0)) 0
⊢ 2 ≠ 0

Tactic:
by

Goal state after:
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
⊢ ∀ t ∈ univ \ {0}, (∑ x, χ x * φ (t - x)) * ψ t = (∑ y, χ (t * y) * φ (t - t * y)) * ψ t

Tactic:
by

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

Goal state before:
pp: h₁ :
  HasDerivAt (id * completedRiemannZeta₀ - 1 - id / (1 - id))
    (1 * completedRiemannZeta₀ 0 + id 0 * deriv completedRiemannZeta₀ 0 - 0 -
      (1 * (1 - id) 0 - id 0 * (0 - 1)) / (1 - id) 0 ^ 2)
    0
h₂ :
  HasDerivAt ((fun x ↦ 2) * fun x ↦ ↑π ^ (-x / 2))
    (0 * ↑π ^ (-0 / 2) + 2 * (-0 / 2 * ↑π ^ (-0 / 2 - 1) * 0 + ↑π ^ (-0 / 2) * log ↑π * (-1 / 2))) 0
h₃ : HasDerivAt (Gamma ∘ fun x ↦ x / 2 + 1) (deriv Gamma (0 / 2 + 1) * (1 / 2 + 0)) 0
⊢ ↑π ≠ 0

Tactic:
by

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

Goal state before:
pp: h₁ :
  HasDerivAt (id * completedRiemannZeta₀ - 1 - id / (1 - id))
    (1 * completedRiemannZeta₀ 0 + id 0 * deriv completedRiemannZeta₀ 0 - 0 -
      (1 * (1 - id) 0 - id 0 * (0 - 1)) / (1 - id) 0 ^ 2)
    0
h₂ :
  HasDerivAt ((fun x ↦ 2) * fun x ↦ ↑π ^ (-x / 2))
    (0 * ↑π ^ (-0 / 2) + 2 * (-0 / 2 * ↑π ^ (-0 / 2 - 1) * 0 + ↑π ^ (-0 / 2) * log ↑π * (-1 / 2))) 0
h₃ : HasDerivAt (Gamma ∘ fun x ↦ x / 2 + 1) (deriv Gamma (0 / 2 + 1) * (1 / 2 + 0)) 0
⊢ arg 2 + (↑π).arg ∈ Ioc (-π) π

Tactic:
by

Goal state after:
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
| gaussSum (χ * φ) ψ * jacobiSum χ φ = ∑ x ∈ univ \ {0}, (∑ x_1, χ x * (χ x_1 * (φ x * φ (1 - x_1)))) * ψ x

Tactic:
enter [2, 2, t, 1, 2, x, 2]

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
t x : F
| χ x * (φ t * φ (1 - x))
Module: Mathlib.NumberTheory.Height.MvPolynomial

Goal state before:
pp: K : Type u_4
inst✝¹ : Field K
inst✝ : AdmissibleAbsValues K
a b c d : K
hab : ![a, b] ≠ 0
hcd : ![c, d] ≠ 0
⊢ mulHeight ![a, b] * mulHeight ![c, d] = mulHeight ![a * c, a * d, b * c, b * d]

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
hx : IsAlgebraic ℤ x
m : ℕ
r : 𝓞 K
hm : m ≠ 0
hmr : m • x = ↑r
n : ℕ := (Submodule.toAddSubgroup (span {↑m})).relIndex (Submodule.toAddSubgroup (span {↑m, r}))
hndef : n = (Submodule.toAddSubgroup (span {↑m})).relIndex (Submodule.toAddSubgroup (span {↑m, r}))
hn : n ≠ 0
a : 𝓞 K
ha' : ↑m * a = ↑n * r
ha : ↑n * x = ↑a
⊢ {↑n} ⊆ {↑n, a}

Tactic:
by

Goal state after:
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
t x : F
| χ x * φ t * φ (1 - x)

Tactic:
mul_comm (χ x) (φ t)

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
t x : F
| φ t * χ x * φ (1 - x)
Module: Mathlib.NumberTheory.Height.NumberField

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
⊢ ∃ n, n ≠ 0 ∧ ∃ a, ↑n * x = ↑a ∧ absNorm (span {↑n, a}) = n ^ (finrank ℚ K - 1)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
n : ℕ
hn : n ≠ 0
a : 𝓞 K
⊢ 0 < ↑n ^ totalWeight K

Tactic:
by

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

Goal state before:
pp: K : Type u_4
inst✝¹ : Field K
inst✝ : AdmissibleAbsValues K
p : Fin 3 → MvPolynomial (Fin 4) K := ![X 0, X 1 + X 2, X 3]
i : Fin 3
⊢ (p i).IsHomogeneous 1

Tactic:
by

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

Goal state before:
pp: h₁ :
  HasDerivAt (id * completedRiemannZeta₀ - 1 - id / (1 - id))
    (1 * completedRiemannZeta₀ 0 + id 0 * deriv completedRiemannZeta₀ 0 - 0 -
      (1 * (1 - id) 0 - id 0 * (0 - 1)) / (1 - id) 0 ^ 2)
    0
h₂ :
  HasDerivAt ((fun x ↦ 2) * fun x ↦ ↑π ^ (-x / 2))
    (0 * ↑π ^ (-0 / 2) + 2 * (-0 / 2 * ↑π ^ (-0 / 2 - 1) * 0 + ↑π ^ (-0 / 2) * log ↑π * (-1 / 2))) 0
h₃ : HasDerivAt (Gamma ∘ fun x ↦ x / 2 + 1) (deriv Gamma (0 / 2 + 1) * (1 / 2 + 0)) 0
⊢ 2 * 2 ≠ 0

Tactic:
by

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

Goal state before:
pp: K : Type u_4
inst✝¹ : Field K
inst✝ : AdmissibleAbsValues K
p : Fin 3 → MvPolynomial (Fin 4) K := ![X 0, X 1 + X 2, X 3]
hom : ∀ (i : Fin 3), (p i).IsHomogeneous 1
C : ℝ
hC₀ : C > 0
hC : ∀ (x : Fin 4 → K), (mulHeight fun j ↦ (eval x) (p j)) ≤ C * mulHeight x
⊢ max C 1 > 0

Tactic:
by

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

Goal state before:
pp: h₁ :
  HasDerivAt (id * completedRiemannZeta₀ - 1 - id / (1 - id))
    (1 * completedRiemannZeta₀ 0 + id 0 * deriv completedRiemannZeta₀ 0 - 0 -
      (1 * (1 - id) 0 - id 0 * (0 - 1)) / (1 - id) 0 ^ 2)
    0
h₂ :
  HasDerivAt ((fun x ↦ 2) * fun x ↦ ↑π ^ (-x / 2))
    (0 * ↑π ^ (-0 / 2) + 2 * (-0 / 2 * ↑π ^ (-0 / 2 - 1) * 0 + ↑π ^ (-0 / 2) * log ↑π * (-1 / 2))) 0
h₃ : HasDerivAt (Gamma ∘ fun x ↦ x / 2 + 1) (deriv Gamma (0 / 2 + 1) * (1 / 2 + 0)) 0
⊢ (2 * 2).arg + (↑π).arg ∈ Ioc (-π) π

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
n : ℕ
hn : ¬n = 0
a : 𝓞 K
Hw : 0 < absNorm (span {↑n})
⊢ {↑n} ⊆ Set.range ![a, ↑n]

Tactic:
by

Goal state after:
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 = gaussSum χ ψ * gaussSum φ ψ

Tactic:
by

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

Goal state before:
pp: F : Type u_1
F' : Type u_2
inst✝² : Fintype F
inst✝¹ : Field F
inst✝ : Field F'
h : ↑(Fintype.card F) ≠ 0
χ φ : MulChar F F'
hχφ : χ * φ ≠ 1
ψ : AddChar F F'
hψ : ψ.IsPrimitive
⊢ jacobiSum χ φ = gaussSum χ ψ * gaussSum φ ψ / gaussSum (χ * φ) ψ

Tactic:
by

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

Goal state before:
pp: F : Type u_1
F' : Type u_2
inst✝² : Fintype F
inst✝¹ : Field F
inst✝ : Field F'
h : ↑(Fintype.card F) ≠ 0
χ φ : MulChar F F'
hχφ : χ * φ ≠ 1
ψ : AddChar F F'
hψ : ψ.IsPrimitive
⊢ jacobiSum χ φ = gaussSum χ ψ * gaussSum φ ψ / gaussSum (χ * φ) ψ

Tactic:
by
  rw [eq_div_iff <| gaussSum_ne_zero_of_nontrivial h hχφ hψ, mul_comm]
  exact jacobiSum_mul_nontrivial hχφ ψ

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
n : ℕ
hn : n ≠ 0
a : 𝓞 K
⊢ 1 ≤ ↑n ^ totalWeight K * ∏ᶠ (v : FinitePlace K), ⨆ i, v (![↑a, ↑n] i)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
n : ℕ
hn : n ≠ 0
a : 𝓞 K
ha₁ : ↑n * x = ↑a
ha₂ : Ideal.absNorm (Ideal.span {↑n, a}) = n ^ (totalWeight K - 1)
i : Fin 2
⊢ ↑(![a, ↑n] i) = ![↑a, ↑n] i

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
n : ℕ
hn : n ≠ 0
a : 𝓞 K
ha₁ : ↑n * x = ↑a
ha₂ : Ideal.absNorm (Ideal.span {↑n, a}) = n ^ (totalWeight K - 1)
hv : ∀ (i : Fin 2), ↑(![a, ↑n] i) = ![↑a, ↑n] i
⊢ ![↑a, ↑n] ≠ 0

Tactic:
by

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

Goal state before:
pp: h₁ :
  HasDerivAt (id * completedRiemannZeta₀ - 1 - id / (1 - id))
    (1 * completedRiemannZeta₀ 0 + id 0 * deriv completedRiemannZeta₀ 0 - 0 -
      (1 * (1 - id) 0 - id 0 * (0 - 1)) / (1 - id) 0 ^ 2)
    0
h₂ :
  HasDerivAt ((fun x ↦ 2) * fun x ↦ ↑π ^ (-x / 2))
    (0 * ↑π ^ (-0 / 2) + 2 * (-0 / 2 * ↑π ^ (-0 / 2 - 1) * 0 + ↑π ^ (-0 / 2) * log ↑π * (-1 / 2))) 0
h₃ : HasDerivAt (Gamma ∘ fun x ↦ x / 2 + 1) (deriv Gamma (0 / 2 + 1) * (1 / 2 + 0)) 0
⊢ arg 2 + arg 2 ∈ Ioc (-π) π

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
n : ℕ
hn : n ≠ 0
a : 𝓞 K
ha₁ : ↑n * x = ↑a
ha₂ : Ideal.absNorm (Ideal.span {↑n, a}) = n ^ (totalWeight K - 1)
hv : ∀ (i : Fin 2), ↑(![a, ↑n] i) = ![↑a, ↑n] i
⊢ ↑n ^ (totalWeight K - 1) * ∏ᶠ (v : FinitePlace K), ⨆ i, v (![↑a, ↑n] i) = 1

Tactic:
by

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

Goal state before:
pp: F : Type u_1
F' : Type u_2
inst✝² : Fintype F
inst✝¹ : Field F
inst✝ : Field F'
h : ringChar F' ≠ ringChar F
χ φ : MulChar F F'
hχ : χ ≠ 1
hφ : φ ≠ 1
hχφ : χ * φ ≠ 1
n : ℕ+
hp : Nat.Prime (ringChar F)
hc : Fintype.card F = ringChar F ^ ↑n
ψ : PrimitiveAddChar F F' := FiniteField.primitiveChar F F' h
FF' : Type u_2 := CyclotomicField (↑ψ.n) F'
χ' : MulChar F FF' := χ.ringHomComp (algebraMap F' FF')
φ' : MulChar F FF' := φ.ringHomComp (algebraMap F' FF')
hinj : Function.Injective ⇑(algebraMap F' FF')
⊢ χ' * φ' ≠ 1

Tactic:
by

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

Goal state before:
pp: F : Type u_1
F' : Type u_2
inst✝² : Fintype F
inst✝¹ : Field F
inst✝ : Field F'
h : ringChar F' ≠ ringChar F
χ φ : MulChar F F'
hχ : χ ≠ 1
hφ : φ ≠ 1
hχφ : χ * φ ≠ 1
n : ℕ+
hp : Nat.Prime (ringChar F)
hc : Fintype.card F = ringChar F ^ ↑n
ψ : PrimitiveAddChar F F' := FiniteField.primitiveChar F F' h
FF' : Type u_2 := CyclotomicField (↑ψ.n) F'
χ' : MulChar F FF' := χ.ringHomComp (algebraMap F' FF')
φ' : MulChar F FF' := φ.ringHomComp (algebraMap F' FF')
hinj : Function.Injective ⇑(algebraMap F' FF')
Hχφ : χ' * φ' ≠ 1
⊢ χ'⁻¹ * φ'⁻¹ ≠ 1

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
n : ℕ
hn : n ≠ 0
a : 𝓞 K
ha₁ : ↑n * x = ↑a
ha₂ : Ideal.absNorm (Ideal.span {↑n, a}) = n ^ (totalWeight K - 1)
hv : ∀ (i : Fin 2), ↑(![a, ↑n] i) = ![↑a, ↑n] i
this : ↑n ^ (totalWeight K - 1) * ∏ᶠ (v : FinitePlace K), ⨆ i, v (![↑a, ↑n] i) = 1
i : InfinitePlace K
a✝ : i ∈ univ
⊢ ↑n ≤ i (![↑a, ↑n] 1)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
⊢ ∃ n, n ≠ 0 ∧ ↑n ≤ mulHeight₁ x ∧ IsIntegral ℤ (↑n * x)

Tactic:
by

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

Goal state before:
pp: ⊢ deriv riemannZeta 0 = -log (2 * ↑π) / 2

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : DecidableEq (InfinitePlace K)
n : ℕ
hn : n ≠ 0
v : InfinitePlace K
⊢ ↑n ^ (totalWeight K - 1) = (∏ w ∈ univ.erase v, ↑n ^ w.mult) * ↑n ^ (v.mult - 1)

Tactic:
by

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

Goal state before:
pp: ⊢ riemannZeta₀ 1 = ↑γ

Tactic:
by

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

Goal state before:
pp: ⊢ riemannZeta₁ 1 = 1

Tactic:
by

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

Goal state before:
pp: s : ℂ
hs : s ≠ 1
⊢ riemannZeta s = (s - 1)⁻¹ + riemannZeta₀ s

Tactic:
by

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

Goal state before:
pp: s : ℂ
hs : s ≠ 1
⊢ riemannZeta s = (s - 1)⁻¹ * riemannZeta₁ s

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
n : ℕ
hn : n ≠ 0
B : ℝ
x : 𝓞 K
h : ∏ v, (⨆ i, v (![↑x, ↑n] i)) ^ v.mult ≤ B
v : InfinitePlace K
⊢ 0 < ↑n ^ (totalWeight K - 1)

Tactic:
by

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

Goal state before:
pp: F : Type u_1
F' : Type u_2
inst✝² : Fintype F
inst✝¹ : Field F
inst✝ : Field F'
h : ringChar F' ≠ ringChar F
χ φ : MulChar F F'
hχ : χ ≠ 1
hφ : φ ≠ 1
hχφ : χ * φ ≠ 1
n : ℕ+
hp : Nat.Prime (ringChar F)
hc : Fintype.card F = ringChar F ^ ↑n
ψ : PrimitiveAddChar F F' := FiniteField.primitiveChar F F' h
FF' : Type u_2 := CyclotomicField (↑ψ.n) F'
χ' : MulChar F FF' := χ.ringHomComp (algebraMap F' FF')
φ' : MulChar F FF' := φ.ringHomComp (algebraMap F' FF')
hinj : Function.Injective ⇑(algebraMap F' FF')
Hχφ : χ' * φ' ≠ 1
Hχφ' : χ'⁻¹ * φ'⁻¹ ≠ 1
Hχ : χ' ≠ 1
Hφ : φ' ≠ 1
⊢ ↑(Fintype.card F) ≠ 0

Tactic:
by

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

Goal state before:
pp: ⊢ ∀ x ∈ {1}ᶜ, riemannZeta₀ x = riemannZeta x - (x - 1)⁻¹

Tactic:
by

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

Goal state before:
pp: ⊢ DifferentiableOn ℂ (fun s ↦ (s - 1)⁻¹) {1}ᶜ

Tactic:
by

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

Goal state before:
pp: ⊢ Differentiable ℂ riemannZeta₀

Tactic:
by

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

Goal state before:
pp: ⊢ Differentiable ℂ riemannZeta₁

Tactic:
by

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

Goal state before:
pp: ⊢ ∀ᶠ (s : ℂ) in 𝓝 1, riemannZeta₁ s ≠ 0

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
n : ℕ
hn : n ≠ 0
B : ℝ
x : 𝓞 K
h : ∏ v, (⨆ i, v (![↑x, ↑n] i)) ^ v.mult ≤ B
v : InfinitePlace K
⊢ ↑n ^ (totalWeight K - 1) * v ↑x ≤ ↑n ^ (totalWeight K - 1) * ⨆ i, v (![↑x, ↑n] i)

Tactic:
by

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

Goal state before:
pp: ⊢ DifferentiableAt ℂ riemannZeta₀ 1

Tactic:
by

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

Goal state before:
pp: ⊢ deriv riemannZeta₁ 1 = ↑γ

Tactic:
by

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

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

Goal state after: