mathlib-initiative/mathlib-tactics huggingface dataset:
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, χ x * χ⁻¹ (1 - x) = -χ (-1)

Tactic:
]

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, χ x * χ⁻¹ (1 - x) = -χ (-1)
Module: Mathlib.NumberTheory.Harmonic.ZetaAsymp

Goal state before:
pp: s : ℂ
hs : s ≠ 1
hg_an : AnalyticOnNhd ℂ (fun z ↦ (starRingEnd ℂ) (riemannZeta ((starRingEnd ℂ) z))) {1}ᶜ
z : ℂ
hz : 1 < z.re
⊢ ∑' (a : ℕ), (starRingEnd ℂ) (1 / ↑a ^ (starRingEnd ℂ) z) = ∑' (n : ℕ), 1 / ↑n ^ z

Tactic:
]

Goal state after:
s : ℂ
hs : s ≠ 1
hg_an : AnalyticOnNhd ℂ (fun z ↦ (starRingEnd ℂ) (riemannZeta ((starRingEnd ℂ) z))) {1}ᶜ
z : ℂ
hz : 1 < z.re
⊢ ∑' (a : ℕ), (starRingEnd ℂ) (1 / ↑a ^ (starRingEnd ℂ) z) = ∑' (n : ℕ), 1 / ↑n ^ z
Module: Mathlib.NumberTheory.Height.MvPolynomial

Goal state before:
pp: case hbc
K : Type u_4
inst✝³ : Field K
ι : Type u_5
ι' : Type u_6
inst✝² : AdmissibleAbsValues K
inst✝¹ : Finite ι'
inst✝ : Finite ι
N : ℕ
p : ι' → MvPolynomial ι K
hp : ∀ (i : ι'), (p i).IsHomogeneous N
x : ι → K
hx : x ≠ 0
h₀ : (fun j ↦ (eval x) (p j)) ≠ 0
H₀ : ∀ (v : AbsoluteValue K ℝ), 0 ≤ ⨆ j, (p j).coeff.sum fun x c ↦ v c
H₁ : 0 ≤ (Multiset.map (fun v ↦ ⨆ j, (p j).coeff.sum fun x c ↦ v c) archAbsVal).prod
H₂ : 0 ≤ (Multiset.map (fun v ↦ ⨆ i, v (x i)) archAbsVal).prod
H₃ : 0 ≤ ∏ᶠ (v : ↑nonarchAbsVal), ⨆ i, ↑v ((eval x) (p i))
H₄ : 0 ≤ ∏ᶠ (v : ↑nonarchAbsVal), ⨆ i, ↑v (x i)
HH₁ : ∀ (v : AbsoluteValue K ℝ), 0 ≤ (⨆ i, v (x i)) ^ N
HH₂ : ∀ (f : ι' → ℝ) (j : ι'), f j ≤ ⨆ j, f j
this : Nonempty ι'
F : Function.HasFiniteMulSupport fun v ↦ ⨆ i, ↑v (x i)
v : ↑nonarchAbsVal
j : ι'
⊢ ⨆ s, ↑v (coeff (↑s) (p j)) ≤ ⨆ s, ↑v (coeff (↑s) (p ?j))

pp: case j
K : Type u_4
inst✝³ : Field K
ι : Type u_5
ι' : Type u_6
inst✝² : AdmissibleAbsValues K
inst✝¹ : Finite ι'
inst✝ : Finite ι
N : ℕ
p : ι' → MvPolynomial ι K
hp : ∀ (i : ι'), (p i).IsHomogeneous N
x : ι → K
hx : x ≠ 0
h₀ : (fun j ↦ (eval x) (p j)) ≠ 0
H₀ : ∀ (v : AbsoluteValue K ℝ), 0 ≤ ⨆ j, (p j).coeff.sum fun x c ↦ v c
H₁ : 0 ≤ (Multiset.map (fun v ↦ ⨆ j, (p j).coeff.sum fun x c ↦ v c) archAbsVal).prod
H₂ : 0 ≤ (Multiset.map (fun v ↦ ⨆ i, v (x i)) archAbsVal).prod
H₃ : 0 ≤ ∏ᶠ (v : ↑nonarchAbsVal), ⨆ i, ↑v ((eval x) (p i))
H₄ : 0 ≤ ∏ᶠ (v : ↑nonarchAbsVal), ⨆ i, ↑v (x i)
HH₁ : ∀ (v : AbsoluteValue K ℝ), 0 ≤ (⨆ i, v (x i)) ^ N
HH₂ : ∀ (f : ι' → ℝ) (j : ι'), f j ≤ ⨆ j, f j
this : Nonempty ι'
F : Function.HasFiniteMulSupport fun v ↦ ⨆ i, ↑v (x i)
v : ↑nonarchAbsVal
j : ι'
⊢ ι'

Tactic:
]

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

Goal state before:
pp: s : ℂ
hs : s ≠ 1
hg_an : AnalyticOnNhd ℂ (fun z ↦ (starRingEnd ℂ) (riemannZeta ((starRingEnd ℂ) z))) {1}ᶜ
z : ℂ
hz : 1 < z.re
n : ℕ
⊢ 0 ≠ π

Tactic:
]

Goal state after:
s : ℂ
hs : s ≠ 1
hg_an : AnalyticOnNhd ℂ (fun z ↦ (starRingEnd ℂ) (riemannZeta ((starRingEnd ℂ) z))) {1}ᶜ
z : ℂ
hz : 1 < z.re
n : ℕ
⊢ 0 ≠ π
Module: Mathlib.NumberTheory.Harmonic.ZetaAsymp

Goal state before:
pp: s : ℂ
hs : s ≠ 1
hg_an : AnalyticOnNhd ℂ (fun z ↦ (starRingEnd ℂ) (riemannZeta ((starRingEnd ℂ) z))) {1}ᶜ
z : ℂ
hz : 1 < z.re
n : ℕ
⊢ 1 / ↑n ^ z = 1 / ↑n ^ z

Tactic:
]

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
⊢ ∑ x ∈ univ \ {1}, χ (x / (1 - x)) = -χ (-1)

Tactic:
]

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.Height.NumberField

Goal state before:
pp: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
ι : Type u_2
inst✝ : Finite ι
x : ι → 𝓞 K
hx : x ≠ 0
i₀ : ι
hi₀ : x i₀ ≠ 0 i₀
i' : { j // ↑(x j) ≠ 0 } := ⟨i₀, ⋯⟩
this : Nonempty { j // ↑(x j) ≠ 0 }
hI : span (Set.range x) = span (Set.range fun i ↦ x ↑i)
hx₀ : (fun i ↦ ↑(x i)) ≠ 0
⊢ ↑(absNorm (span (Set.range x))) * ∏ᶠ (v : FinitePlace K), ⨆ i, v ↑(x ↑i) = 1

Tactic:
hI

Goal state after:
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
ι : Type u_2
inst✝ : Finite ι
x : ι → 𝓞 K
hx : x ≠ 0
i₀ : ι
hi₀ : x i₀ ≠ 0 i₀
i' : { j // ↑(x j) ≠ 0 } := ⟨i₀, ⋯⟩
this : Nonempty { j // ↑(x j) ≠ 0 }
hI : span (Set.range x) = span (Set.range fun i ↦ x ↑i)
hx₀ : (fun i ↦ ↑(x i)) ≠ 0
⊢ ↑(absNorm (span (Set.range fun i ↦ x ↑i))) * ∏ᶠ (v : FinitePlace K), ⨆ i, v ↑(x ↑i) = 1
Module: Mathlib.NumberTheory.Harmonic.ZetaAsymp

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

Tactic:
]

Goal state after:
⊢ completedRiemannZeta₀ (1 - 0) = (↑γ - log (4 * ↑π)) / 2 + 1
Module: Mathlib.NumberTheory.Height.MvPolynomial

Goal state before:
pp: K : Type u_4
inst✝³ : Field K
ι : Type u_5
ι' : Type u_6
inst✝² : AdmissibleAbsValues K
inst✝¹ : Finite ι'
inst✝ : Finite ι
N : ℕ
p : ι' → MvPolynomial ι K
hp : ∀ (i : ι'), (p i).IsHomogeneous N
x : ι → K
⊢ log (mulHeight fun j ↦ (eval x) (p j)) ≤ log (max (mulHeightBound p) 1) + ↑N * log (mulHeight x)

Tactic:
pull (disch := positivity) log

Goal state after:
K : Type u_4
inst✝³ : Field K
ι : Type u_5
ι' : Type u_6
inst✝² : AdmissibleAbsValues K
inst✝¹ : Finite ι'
inst✝ : Finite ι
N : ℕ
p : ι' → MvPolynomial ι K
hp : ∀ (i : ι'), (p i).IsHomogeneous N
x : ι → K
⊢ log (mulHeight fun j ↦ (eval x) (p j)) ≤ log (max (mulHeightBound p) 1 * mulHeight x ^ N)
Module: Mathlib.NumberTheory.JacobiSum.Basic

Goal state before:
pp: case refine_1
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 : F
hx : ¬x = 1
⊢ ¬-1 = 0

Tactic:
neg_eq_zero

Goal state after:
case refine_1
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 : F
hx : ¬x = 1
⊢ ¬1 = 0
Module: Mathlib.NumberTheory.JacobiSum.Basic

Goal state before:
pp: case refine_1
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 : F
hx : ¬x = 1
⊢ ¬1 = 0

Tactic:
]

Goal state after:
case refine_1
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 : F
hx : ¬x = 1
⊢ ¬1 = 0
Module: Mathlib.NumberTheory.Harmonic.ZetaAsymp

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

Tactic:
]

Goal state after:
⊢ deriv (fun x ↦ (x * completedRiemannZeta₀ x - 1 - x / (1 - x)) / (2 * ↑π ^ (-x / 2) * Gamma (x / 2 + 1))) 0 =
    -log (2 * ↑π) / 2
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'
q' : ι × ι' → K := fun a ↦ (eval x) (q a)
⊢ mulHeight (x ^ (M + N)) ≤ ↑(Nat.card ι') ^ totalWeight K * mulHeight q' * mulHeight fun j ↦ (eval x) (p j)

Tactic:
]

Goal state after:
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)
⊢ mulHeight (x ^ (M + N)) ≤ ↑(Nat.card ι') ^ totalWeight K * mulHeight q' * mulHeight fun j ↦ (eval x) (p j)
Module: Mathlib.NumberTheory.JacobiSum.Basic

Goal state before:
pp: case refine_2
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
y : F
hy : ¬y = -1
⊢ ¬1 = 0

Tactic:
]

Goal state after:
case refine_2
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
y : F
hy : ¬y = -1
⊢ ¬1 = 0
Module: Mathlib.NumberTheory.JacobiSum.Basic

Goal state before:
pp: case refine_3
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 : F
hx : ¬1 - x = 0
⊢ x / (1 - x) / (1 + x / (1 - x)) = x

Tactic:
]

Goal state after:
case refine_3
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 : F
hx : ¬1 - x = 0
⊢ x / (1 - x) / (1 + x / (1 - x)) = x
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 * mulHeight x ^ M ≤
    (↑(Nat.card ι') ^ totalWeight K * max (mulHeightBound q) 1 * mulHeight fun j ↦ (eval x) (p j)) * mulHeight x ^ M

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:
]

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.Height.NumberField

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
m n : ℕ
hn : n ≠ 0
a : 𝓞 K
f : 𝓞 K →ₗ[𝓞 K] 𝓞 K := LinearMap.mulLeft (𝓞 K) ↑n
hf : Function.Injective ⇑↑f
H₁ : span {↑n * ↑m} = Submodule.map f (span {↑m})
H₂ : span {↑(n * m), ↑n * a} = Submodule.map f (span {↑m, a})
⊢ (Submodule.toAddSubgroup (span {↑m})).relIndex (Submodule.toAddSubgroup (span {↑m, a})) =
    (Submodule.map f (span {↑m})).toAddSubgroup.relIndex (Submodule.map f (span {↑m, a})).toAddSubgroup

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
m n : ℕ
hn : n ≠ 0
a : 𝓞 K
f : 𝓞 K →ₗ[𝓞 K] 𝓞 K := LinearMap.mulLeft (𝓞 K) ↑n
hf : Function.Injective ⇑↑f
H₁ : span {↑n * ↑m} = Submodule.map f (span {↑m})
H₂ : span {↑(n * m), ↑n * a} = Submodule.map f (span {↑m, a})
⊢ (Submodule.toAddSubgroup (span {↑m})).relIndex (Submodule.toAddSubgroup (span {↑m, a})) =
    (Submodule.map f (span {↑m})).toAddSubgroup.relIndex (Submodule.map f (span {↑m, a})).toAddSubgroup
Module: Mathlib.NumberTheory.JacobiSum.Basic

Goal state before:
pp: case refine_4
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
y : F
hy : ¬1 + y = 0
⊢ y / (1 + y) / (1 - y / (1 + y)) = y

Tactic:
]

Goal state after:
case refine_4
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
y : F
hy : ¬1 + y = 0
⊢ y / (1 + y) / (1 - y / (1 + y)) = y
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 ^ (M + N) ≤
    (↑(Nat.card ι') ^ totalWeight K * max (mulHeightBound q) 1 * mulHeight fun j ↦ (eval x) (p j)) * mulHeight x ^ M

Tactic:
]

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 ^ (M + N) ≤
    (↑(Nat.card ι') ^ totalWeight K * max (mulHeightBound q) 1 * mulHeight fun j ↦ (eval x) (p j)) * mulHeight x ^ M
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)
⊢ (↑(Nat.card ι') ^ totalWeight K * (max (mulHeightBound q) 1 * mulHeight x ^ M) * mulHeight fun j ↦ (eval x) (p j)) ≤
    (↑(Nat.card ι') ^ totalWeight K * max (mulHeightBound q) 1 * mulHeight fun j ↦ (eval x) (p j)) * mulHeight x ^ M

Tactic:
]

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)
⊢ (↑(Nat.card ι') ^ totalWeight K * (max (mulHeightBound q) 1 * mulHeight x ^ M) * mulHeight fun j ↦ (eval x) (p j)) ≤
    (↑(Nat.card ι') ^ totalWeight K * max (mulHeightBound q) 1 * mulHeight fun j ↦ (eval x) (p j)) * mulHeight x ^ M
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'
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)
⊢ (↑(Nat.card ι') ^ totalWeight K * (max (mulHeightBound q) 1 * mulHeight x ^ M) * mulHeight fun j ↦ (eval x) (p j)) =
    (↑(Nat.card ι') ^ totalWeight K * max (mulHeightBound q) 1 * mulHeight fun j ↦ (eval x) (p j)) * mulHeight x ^ M

Tactic:
by ring

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

Goal state before:
pp: case refine_1
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
⊢ ∀ (m : ℕ), 0 / 2 + 1 ≠ -↑m

Tactic:
simp only [zero_div]
      norm_cast
      simp

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 * ↑n})).relIndex (Submodule.toAddSubgroup (span {↑(m * n), ↑m * b})) =
    (Submodule.toAddSubgroup (span {↑n})).relIndex (Submodule.toAddSubgroup (span {↑n, b}))

Tactic:
]

Goal state after:
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}))
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
this : ∑ x ∈ univ \ {1}, χ (x / (1 - x)) = ∑ x ∈ univ \ {-1}, χ x
⊢ ∑ x, χ x = 0

Tactic:
]

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
this : ∑ x ∈ univ \ {1}, χ (x / (1 - x)) = ∑ x ∈ univ \ {-1}, χ x
⊢ ∑ x, χ x = 0
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 ≠ 0 ∧ ∃ a, ↑n * x = ↑a ∧ absNorm (span {↑n, a}) = n ^ (finrank ℚ K - 1)

Tactic:
]

Goal state after:
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 ≠ 0 ∧ ∃ a, ↑n * x = ↑a ∧ absNorm (span {↑n, a}) = n ^ (finrank ℚ K - 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 + ∑ x, χ x * φ (0 - x) * ψ 0

Tactic:
]

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 + ∑ x, χ x * φ (0 - x) * ψ 0
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 * ↑r = ↑m * ↑a

Tactic:
]

Goal state after:
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 * ↑r = ↑m * ↑a
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

Tactic:
]

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.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
⊢ (Submodule.toAddSubgroup (span {↑n})).relIndex (Submodule.toAddSubgroup (span {↑n, a})) *
      (Submodule.toAddSubgroup (span {↑n, a})).index =
    (Submodule.toAddSubgroup (span {↑n})).index

Tactic:
]

Goal state after:
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
⊢ (Submodule.toAddSubgroup (span {↑n})).relIndex (Submodule.toAddSubgroup (span {↑n, a})) *
      (Submodule.toAddSubgroup (span {↑n, a})).index =
    (Submodule.toAddSubgroup (span {↑n})).index
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
⊢ -((log 2 + log ↑π) * 2) = ↑γ - log (2 * 2 * ↑π) + (-log ↑π + -↑γ)

Tactic:
]

Goal state after:
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
⊢ -((log 2 + log ↑π) * 2) = ↑γ - log (2 * 2 * ↑π) + (-log ↑π + -↑γ)
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})
⊢ ↑(absNorm (span (Set.range ![a, ↑n]))) * ∏ᶠ (v : FinitePlace K), ⨆ i, v ↑(![a, ↑n] i) ≤
    ↑(absNorm (span {↑n})) * ∏ᶠ (v : FinitePlace K), ⨆ i, v (![↑a, ↑n] i)

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
n : ℕ
hn : ¬n = 0
a : 𝓞 K
Hw : 0 < absNorm (span {↑n})
⊢ ↑(absNorm (span (Set.range ![a, ↑n]))) * ∏ᶠ (v : FinitePlace K), ⨆ i, v ↑(![a, ↑n] i) ≤
    ↑(absNorm (span {↑n})) * ∏ᶠ (v : FinitePlace K), ⨆ i, v (![↑a, ↑n] i)