mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
l₀ x : ℂ
r : NNReal
hr0 : 0 < r
hr : Metric.closedBall x ↑r ⊆ (↑L.lattice \ {l₀})ᶜ
⊢ HasFPowerSeriesOnBall ℘'[L - l₀] (L.derivWeierstrassPExceptSeries l₀ x) x ↑r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Extremal

Goal state before:
pp: n k i : ℕ
hk₁ : 0 < k
hk₂ : k ≤ n
hi : i ≤ n
x : ℝ
hx : 1 ≤ x
h✝ : ¬(1 ≤ i ∧ i ≤ n - k)
⊢ Finset.Icc 1 (n - k) ⊆ (Finset.range (n + 1)).erase i

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Extremal

Goal state before:
pp: n k i : ℕ
hk₁ : 0 < k
hk₂ : k ≤ n
hi : i ≤ n
x : ℝ
hx : 1 ≤ x
h✝ : ¬(1 ≤ i ∧ i ≤ n - k)
⊢ (Finset.Icc 1 (n - k)).card = n - k

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
l : ℂ
⊢ HasFPowerSeriesAt ℘'[L - l] (FormalMultilinearSeries.ofScalars ℂ fun i ↦ (↑i + 1) * (↑i + 2) * L.sumInvPow l (i + 3))
    l

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Extremal

Goal state before:
pp: n k i : ℕ
hk₁ : 0 < k
hk₂ : k ≤ n
hi : i ≤ n
x : ℝ
hx : 1 ≤ x
h✝ : ¬(1 ≤ i ∧ i ≤ n - k)
⊢ 0 ∉ Finset.Icc 1 (n - k)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: n : ℕ
hn : n ≠ 0
x : ℝ
hx : |x| ≤ 1
k : ℤ
hk : ↑k * π = ↑n * arccos x
⊢ ↑k = ↑n * (arccos x / π)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
l : ℂ
n : ℕ
⊢ iteratedDeriv n ℘'[L - l] l / ↑n ! = (↑n + 1) * (↑n + 2) * L.sumInvPow l (n + 3)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
l : ℂ
n : ℕ
⊢ iteratedDeriv n ℘'[L - l] l = ↑(n + 2)! * L.sumInvPow l (n + 3)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
l : ℂ
⊢ deriv ℘'[L - l] l = 6 * L.sumInvPow l 4

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Extremal

Goal state before:
pp: n k i : ℕ
hk₁ : 0 < k
hk₂ : k ≤ n
hi : i ≤ n
x : ℝ
hx : 1 ≤ x
h✝ : 1 ≤ i ∧ i ≤ n - k
⊢ (Finset.Icc 1 (n - k + 1)).erase i ⊆ (Finset.range (n + 1)).erase i

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Extremal

Goal state before:
pp: n k i : ℕ
hk₁ : 0 < k
hk₂ : k ≤ n
hi : i ≤ n
x : ℝ
hx : 1 ≤ x
h✝ : 1 ≤ i ∧ i ≤ n - k
⊢ ((Finset.Icc 1 (n - k + 1)).erase i).card = n - k

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: n : ℕ
hn : n ≠ 0
x : ℝ
hx : |x| ≤ 1
k : ℤ
hk : ↑k * π = ↑n * arccos x
hk' : ↑k = ↑n * (arccos x / π)
⊢ 0 ≤ k

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
l₀ : ℂ
hl₀ : l₀ ∉ L.lattice
⊢ L.weierstrassPExceptSeries l₀ = L.weierstrassPSeries

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
l₀ : ℂ
hl₀ : l₀ ∉ L.lattice
⊢ L.weierstrassPExceptSummand l₀ = L.weierstrassPSummand

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Extremal

Goal state before:
pp: n k i : ℕ
hk₁ : 0 < k
hk₂ : k ≤ n
hi : i ≤ n
x : ℝ
hx : 1 ≤ x
h✝ : 1 ≤ i ∧ i ≤ n - k
⊢ 0 ∉ (Finset.Icc 1 (n - k + 1)).erase i

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
x : ℂ
i : ℕ
⊢ (L.weierstrassPSeries x).coeff i = ∑' (l : ↥L.lattice), L.weierstrassPSummand x i l

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
z x : ℂ
hx : ∀ (l : ↥L.lattice), ‖z - x‖ < ‖↑l - x‖
⊢ Summable (Function.uncurry fun b c ↦ L.weierstrassPSummand x b c * (z - x) ^ b)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Extremal

Goal state before:
pp: n k i : ℕ
hk₁ : 0 < k
hk₂ : k ≤ n
hi : i ≤ n
x : ℝ
hx : 1 ≤ x
⊢ ∃ s ⊆ (Finset.range (n + 1)).erase i, s.card = n - k ∧ 0 ∉ s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: n : ℕ
hn : n ≠ 0
x : ℝ
hx : |x| ≤ 1
k : ℕ
hk : ↑k * π = ↑n * arccos x
hk' : ↑k = ↑n * (arccos x / π)
⊢ 0 ≤ π

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: n : ℕ
hn : n ≠ 0
x : ℝ
hx : |x| ≤ 1
k : ℕ
hk : ↑k * π = ↑n * arccos x
hk' : ↑k = ↑n * (arccos x / π)
⊢ ↑k ≤ ↑n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
z x : ℂ
hx : ∀ (l : ↥L.lattice), ‖z - x‖ < ‖↑l - x‖
⊢ HasSum (fun i ↦ (L.weierstrassPSeries x).coeff i * (z - x) ^ i) (℘[L] z)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Extremal

Goal state before:
pp: n k i : ℕ
hk₁ : 0 < k
hk₂ : k ≤ n
hi : i ≤ n
x : ℝ
hx : 1 ≤ x
s : Finset ℕ
hs : s ⊆ (Finset.range (n + 1)).erase i
hscard : s.card = n - k
hsn : 0 ∉ s
⊢ s ∈ Finset.powersetCard (n - k) ((Finset.range (n + 1)).erase i)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: n : ℕ
hn : n ≠ 0
x : ℝ
hx : |x| ≤ 1
k : ℕ
hk : ↑k * π = ↑n * arccos x
hk' : ↑k = ↑n * (arccos x / π)
hkn : ↑k ≤ ↑n
⊢ k ≤ n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: n : ℕ
hn : n ≠ 0
x : ℝ
hx : |x| ≤ 1
k : ℕ
hk : ↑k * π = ↑n * arccos x
hk' : ↑k = ↑n * (arccos x / π)
hkn : ↑k ≤ ↑n
⊢ ↑n ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: n : ℕ
hn : n ≠ 0
x : ℝ
hx : |x| ≤ 1
k : ℕ
hk : ↑k * π = ↑n * arccos x
hk' : ↑k = ↑n * (arccos x / π)
hkn : ↑k ≤ ↑n
⊢ -1 ≤ x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
x : ℂ
r : NNReal
hr0 : 0 < r
hr : Metric.closedBall x ↑r ⊆ (↑L.lattice)ᶜ
⊢ HasFPowerSeriesOnBall ℘[L] (L.weierstrassPSeries x) x ↑r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: n : ℕ
hn : n ≠ 0
x : ℝ
hx : |x| ≤ 1
k : ℕ
hk : ↑k * π = ↑n * arccos x
hk' : ↑k = ↑n * (arccos x / π)
hkn : ↑k ≤ ↑n
⊢ x ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable

Goal state before:
pp: z : ℍ
x : Fin 2 → ℤ
hx : x ≠ 0
⊢ r z * ‖x‖ ≤ ‖↑(x 0) * ↑z + ↑(x 1)‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
l₀ : ℂ
hl₀ : l₀ ∈ L.lattice
z : ℂ
⊢ (if z = l₀ then 1 else (z - l₀) ^ 2 * ℘[L] z) = (z - l₀) ^ 2 * ℘[L - l₀] z + 1 - (z - l₀) ^ 2 / l₀ ^ 2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Extremal

Goal state before:
pp: n k i : ℕ
hk₁ : 0 < k
hk₂ : k ≤ n
hi : i ≤ n
x : ℝ
hx : 1 ≤ x
s : Finset ℕ
hs : s ⊆ (Finset.range (n + 1)).erase i
hscard : s.card = n - k
hsn : 0 ∉ s
a : ℕ
ha : a ∈ s
⊢ a ≤ n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Extremal

Goal state before:
pp: n k i : ℕ
hk₁ : 0 < k
hk₂ : k ≤ n
hi : i ≤ n
x : ℝ
hx : 1 ≤ x
s : Finset ℕ
hs : s ⊆ (Finset.range (n + 1)).erase i
hscard : s.card = n - k
hsn : 0 ∉ s
a : ℕ
ha : a ∈ s
⊢ 0 < a

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: n : ℕ
hn : n ≠ 0
k : ℕ
hk : k ≤ n
⊢ ↑↑k * π = ↑↑n * (↑k * π / ↑n)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: n : ℕ
hn : n ≠ 0
x : ℝ
⊢ |eval x (T ℝ ↑n)| = 1 ↔ ∃ k ≤ n, x = cos (↑k * π / ↑n)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
x : ℂ
hx : x ∈ L.lattice
⊢ x ∈ (↑L.lattice \ {x})ᶜ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: k n : ℕ
hn : n ≠ 0
⊢ eval (cos (↑k * π / ↑n)) (T ℝ ↑n) = ↑↑(↑k).negOnePow

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Extremal

Goal state before:
pp: n k i : ℕ
hk₁ : 0 < k
hk₂ : k ≤ n
hi : i ≤ n
x : ℝ
hx : 1 ≤ x
⊢ 0 < (-1) ^ i * iterateDerivativeC n k x i

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
⊢ Meromorphic ℘[L]

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
⊢ Meromorphic ℘'[L]

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: n : ℕ
hn : n ≠ 0
x : ℝ
k : ℕ
hx : (-1) ^ k = 1
hk₁ : k ≤ n
hk₂ : x = cos (↑k * π / ↑n)
⊢ -1 ≠ 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Extremal

Goal state before:
pp: n : ℕ
P : ℝ[X]
k : ℕ
x : ℝ
hx : 1 ≤ x
hPdeg : P.degree ≤ ↑n
hPbnd : ∀ x ∈ Set.Icc (-1) 1, |eval x P| ≤ 1
hk : n < k
⊢ P.degree < ↑k

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: n : ℕ
hn : n ≠ 0
x : ℝ
⊢ eval x (T ℝ ↑n) = 1 ↔ ∃ k ≤ n, Even k ∧ x = cos (↑k * π / ↑n)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
l₀ : ℂ
h : l₀ ∈ L.lattice
⊢ l₀ ∈ (↑L.lattice \ {l₀})ᶜ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Extremal

Goal state before:
pp: n : ℕ
P : ℝ[X]
k : ℕ
x : ℝ
hx : 1 ≤ x
hPdeg : P.degree ≤ ↑n
hPbnd : ∀ x ∈ Set.Icc (-1) 1, |eval x P| ≤ 1
hk : n < k
⊢ (T ℝ ↑n).degree < ↑k

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
l₀ : ℂ
h : l₀ ∈ L.lattice
this✝ : AnalyticAt ℂ ℘[L - l₀] l₀
this : AnalyticAt ℂ (fun z ↦ (z - l₀) ^ 2 / l₀ ^ 2) l₀
⊢ AnalyticAt ℂ (fun z ↦ (z - l₀) ^ 2 * ℘[L - l₀] z + 1 - (z - l₀) ^ 2 / l₀ ^ 2) l₀

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable

Goal state before:
pp: z : ℍ
k : ℝ
hk : 0 ≤ k
x : Fin 2 → ℤ
⊢ ‖↑(x 0) * ↑z + ↑(x 1)‖ ^ (-k) ≤ r z ^ (-k) * ‖x‖ ^ (-k)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: n : ℕ
hn : n ≠ 0
x : ℝ
k : ℕ
hx : (-1) ^ k = -1
hk₁ : k ≤ n
hk₂ : x = cos (↑k * π / ↑n)
⊢ -1 ≠ 1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable

Goal state before:
pp: z : ℍ
k : ℝ
hk : 0 ≤ k
x : Fin 2 → ℤ
A B : ℝ
hB : 0 < B
hz : z ∈ verticalStrip A B
⊢ 0 ≤ ‖x‖ ^ (-k)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable

Goal state before:
pp: z : ℍ
k : ℝ
hk : 0 ≤ k
x : Fin 2 → ℤ
A B : ℝ
hB : 0 < B
hz : z ∈ verticalStrip A B
⊢ ‖↑(x 0) * ↑z + ↑(x 1)‖ ^ (-k) ≤ r { coe := { re := A, im := B }, coe_im_pos := hB } ^ (-k) * ‖x‖ ^ (-k)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable

Goal state before:
pp: c e : ℤ
z : ℂ
⊢ (fun d ↦ ↑c * z + ↑d + ↑e) =Θ[cofinite] fun n ↦ ↑n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: n : ℕ
hn : n ≠ 0
x : ℝ
⊢ eval x (T ℝ ↑n) = -1 ↔ ∃ k ≤ n, Odd k ∧ x = cos (↑k * π / ↑n)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable

Goal state before:
pp: d : ℤ
z : ℂ
hz : z ≠ 0
⊢ (fun c ↦ ↑c * z + ↑d) =Θ[cofinite] fun n ↦ ↑n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
l₀ : ℂ
h : l₀ ∈ L.lattice
z : ℂ
hz : z ≠ l₀
⊢ (z - l₀) ^ 2 ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable

Goal state before:
pp: c : ℤ
z : ℂ
⊢ (fun d ↦ (↑c * z + ↑d)⁻¹) =O[cofinite] fun n ↦ (↑n)⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: n : ℕ
hn : n ≠ 0
⊢ Set.InjOn (fun k ↦ (2 * ↑k + 1) * π / (2 * ↑n)) ↑(Finset.range n)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: n : ℕ
hn : n ≠ 0
k : ℕ
hk : k ∈ ↑(Finset.range n)
⊢ 0 ≤ (2 * ↑k + 1) * π / (2 * ↑n)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable

Goal state before:
pp: z : ℂ
b : ℤ
⊢ Tendsto (fun d ↦ 1 / (↑b * z + ↑d)) atTop (𝓝 0)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Extremal

Goal state before:
pp: n : ℕ
P : ℝ[X]
k : ℕ
x : ℝ
hx : 1 ≤ x
hPdeg : P.degree ≤ ↑n
hPbnd : ∀ x ∈ Set.Icc (-1) 1, |eval x P| ≤ 1
⊢ eval x ((⇑derivative)^[k] P) ≤ eval x ((⇑derivative)^[k] (T ℝ ↑n))

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable

Goal state before:
pp: z : ℂ
b : ℤ
⊢ Tendsto (fun d ↦ 1 / (↑b * z - ↑d)) atTop (𝓝 0)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: n : ℕ
⊢ (Multiset.map (fun k ↦ cos ((2 * ↑k + 1) * π / (2 * ↑n))) (Multiset.range n)).Nodup

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
l₀ : ℂ
h : l₀ ∈ L.lattice
⊢ meromorphicOrderAt ℘[L] l₀ = -2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
⊢ L.sumInvPow 0 = L.G

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
n : ℕ
hn : Odd n
⊢ L.G n = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
⊢ Meromorphic fun z ↦ ℘'[L] z ^ 2 - 4 * ℘[L] z ^ 3 + L.g₂ * ℘[L] z + L.g₃

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: n : ℕ
⊢ (T ℝ ↑n).roots = (Finset.image (fun k ↦ cos ((2 * ↑k + 1) * π / (2 * ↑n))) (Finset.range n)).val

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: n k : ℕ
hk : k < n
⊢ (Finset.image (fun k ↦ cos ((2 * ↑k + 1) * π / (2 * ↑n))) (Finset.range n)).val.Nodup

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: n k : ℕ
hk : k < n
⊢ rootMultiplicity (cos ((2 * ↑k + 1) * π / (2 * ↑n))) (T ℝ ↑n) = 1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable

Goal state before:
pp: k : ℝ
hk : 2 < k
this : Summable fun n ↦ ∑' (x : ↥(box n)), ↑n ^ (-k)
⊢ Summable fun j ↦ ∑' (i : ↑↑(box j)), ‖![(↑i).1, (↑i).2]‖ ^ (-k)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Extremal

Goal state before:
pp: n : ℕ
P : ℝ[X]
k : ℕ
hk₁ : 0 < k
hk₂ : k ≤ n
x : ℝ
hx : 1 ≤ x
hPdeg : P.degree ≤ ↑n
hPbnd : ∀ x ∈ Set.Icc (-1) 1, |eval x P| ≤ 1
⊢ eval x ((⇑derivative)^[k] P) = eval x ((⇑derivative)^[k] (T ℝ ↑n)) ↔ P = T ℝ ↑n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
this : Meromorphic fun z ↦ ℘'[L] z ^ 2 - 4 * ℘[L] z ^ 3 + L.g₂ * ℘[L] z + L.g₃
z w : ℂ
hw : w ∈ {z}ᶜ
hw' : w ∈ (↑L.lattice \ {z})ᶜ
⊢ w ∉ L.lattice

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable

Goal state before:
pp: k : ℝ
hk : 2 < k
⊢ 1 - k < -1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: n k : ℕ
hk : k ∈ ↑(Finset.range n)
⊢ 0 ≤ (fun k ↦ (↑k + 1) * π / (↑n + 1)) k

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
⊢ Meromorphic L.relation

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: n : ℕ
⊢ (Multiset.map (fun k ↦ cos ((↑k + 1) * π / (↑n + 1))) (Multiset.range n)).Nodup

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: n : ℕ
hn : n ≠ 0
x : ℝ
hx✝ : x ∈ Finset.image (fun k ↦ cos ((↑k + 1) * π / (↑n + 1))) (Finset.range n)
k : ℕ
hk : k ∈ Finset.range n
hx : cos ((↑k + 1) * π / (↑n + 1)) = x
this : eval x (U ℝ ↑n) * sin ((↑k + 1) * π / (↑n + 1)) = 0
⊢ 0 < (↑k + 1) * π / (↑n + 1)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable

Goal state before:
pp: k : ℝ
hk : 2 < k
n : ℕ
hn : 0 < n
⊢ 0 ≤ ↑(8 * n) * ↑n ^ (-k)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.TsumUniformlyOn

Goal state before:
pp: α : Type u_1
β : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : CompleteSpace F
u : α → ℝ
f : α → β → F
hu : Summable u
s : Set β
hfu : ∀ (n : α), ∀ x ∈ s, ‖f n x‖ ≤ u n
⊢ HasSumUniformlyOn f (fun x ↦ ∑' (n : α), f n x) s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.TsumUniformlyOn

Goal state before:
pp: β : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : CompleteSpace F
ι : Type u_4
f : ι → β → F
u : ι → ℝ
hu : Summable u
s : Set β
hfu : ∀ᶠ (n : ι) in cofinite, ∀ x ∈ s, ‖f n x‖ ≤ u n
⊢ HasSumUniformlyOn f (fun x ↦ ∑' (n : ι), f n x) s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.TsumUniformlyOn

Goal state before:
pp: α : Type u_1
β : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : CompleteSpace F
inst✝¹ : TopologicalSpace β
inst✝ : LocallyCompactSpace β
f : α → β → F
s : Set β
hs : IsOpen[inst✝¹] s
hu : ∀ K ⊆ s, IsCompact K → ∃ u, Summable u ∧ ∀ᶠ (n : α) in cofinite, ∀ k ∈ K, ‖f n k‖ ≤ u n
⊢ SummableLocallyUniformlyOn f s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: n : ℕ
hn : n ≠ 0
x : ℝ
hx✝ : x ∈ Finset.image (fun k ↦ cos ((↑k + 1) * π / (↑n + 1))) (Finset.range n)
k : ℕ
hk : k ∈ Finset.range n
hx : cos ((↑k + 1) * π / (↑n + 1)) = x
this : eval x (U ℝ ↑n) * sin ((↑k + 1) * π / (↑n + 1)) = 0
⊢ eval x (U ℝ ↑n) = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
z : ℂ
hz : z ∈ (↑L.lattice \ {0})ᶜ
hz0 : ¬z = 0
⊢ z ∉ L.lattice

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.TsumUniformlyOn

Goal state before:
pp: α : Type u_1
β : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : CompleteSpace F
inst✝¹ : TopologicalSpace β
inst✝ : LocallyCompactSpace β
f : α → β → F
s : Set β
hs : IsOpen[inst✝¹] s
hu : ∀ K ⊆ s, IsCompact K → ∃ u, Summable u ∧ ∀ (n : α), ∀ k ∈ K, ‖f n k‖ ≤ u n
K : Set β
hK : K ⊆ s
hKc : IsCompact K
u : α → ℝ
hu1 : Summable u
hu2 : ∀ (n : α), ∀ k ∈ K, ‖f n k‖ ≤ u n
⊢ ∀ᶠ (n : α) in cofinite, ∀ k ∈ K, ‖f n k‖ ≤ u n

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.TsumUniformlyOn

Goal state before:
pp: α : Type u_1
β : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : CompleteSpace F
inst✝¹ : TopologicalSpace β
inst✝ : LocallyCompactSpace β
f : α → β → F
s : Set β
hs : IsOpen[inst✝¹] s
hu : ∀ K ⊆ s, IsCompact K → ∃ u, Summable u ∧ ∀ (n : α), ∀ k ∈ K, ‖f n k‖ ≤ u n
⊢ SummableLocallyUniformlyOn f s

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable

Goal state before:
pp: k : ℝ
hk : 2 < k
⊢ Summable fun x ↦ ‖x‖ ^ (-k)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable

Goal state before:
pp: z : ℂ
c k : ℤ
hk : 2 ≤ k
⊢ 1 < ↑k

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable

Goal state before:
pp: z : ℂ
c k : ℤ
hk : 2 ≤ k
⊢ 0 ≤ k

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable

Goal state before:
pp: z : ℂ
c k : ℤ
hk : 2 ≤ k
⊢ Summable fun d ↦ ((↑c * z + ↑d) ^ k)⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable

Goal state before:
pp: z : ℂ
hz : z ≠ 0
d k : ℤ
hk : 2 ≤ k
⊢ 1 < ↑k

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable

Goal state before:
pp: z : ℂ
hz : z ≠ 0
d k : ℤ
hk : 2 ≤ k
⊢ 0 ≤ k

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable

Goal state before:
pp: z : ℂ
hz : z ≠ 0
d k : ℤ
hk : 2 ≤ k
⊢ Summable fun c ↦ ((↑c * z + ↑d) ^ k)⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable

Goal state before:
pp: z : ℂ
c₁ c₂ : ℤ
⊢ 1 < 2

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable

Goal state before:
pp: z : ℂ
c₁ c₂ : ℤ
⊢ Summable fun n ↦ ((↑c₁ * z - ↑n) * (↑c₂ * z + ↑n))⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable

Goal state before:
pp: z : ℂ
c₁ c₂ : ℤ
⊢ Summable fun n ↦ ((↑c₁ * z + ↑n + 1) * (↑c₂ * z + ↑n))⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable

Goal state before:
pp: z : ℂ
hz : z ≠ 0
c₁ c₂ : ℤ
⊢ 1 < 2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
⊢ L.relation * id ^ 6 =ᶠ[𝓝 0] fun z ↦
    (℘'[L - 0] z * z ^ 3 - 2) ^ 2 - 4 * (℘[L - 0] z * z ^ 2 + 1) ^ 3 + L.g₂ * (℘[L - 0] z * z ^ 6 + z ^ 4) +
      L.g₃ * z ^ 6

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: n : ℕ
⊢ (U ℝ ↑n).roots = (Finset.image (fun k ↦ cos ((↑k + 1) * π / (↑n + 1))) (Finset.range n)).val

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable

Goal state before:
pp: z : ℂ
hz : z ≠ 0
c₁ c₂ : ℤ
⊢ Summable fun n ↦ ((↑n * z + ↑c₁) * (↑n * z + ↑c₂))⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: n k : ℕ
hk : k < n
⊢ (Finset.image (fun k ↦ cos ((↑k + 1) * π / (↑n + 1))) (Finset.range n)).val.Nodup

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
⊢ AnalyticAt ℂ (L.relation * id ^ 6) 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.TsumUniformlyOn

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
F : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : IsRCLikeNormedField 𝕜
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set 𝕜
f : ι → 𝕜 → F
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
x : 𝕜
hx : x ∈ s
hf : ∀ y ∈ s, Summable fun n ↦ f n y
h : SummableLocallyUniformlyOn (fun n ↦ derivWithin (fun z ↦ f n z) s) s
hf2 : ∀ (n : ι), ∀ r ∈ s, DifferentiableAt 𝕜 (f n) r
⊢ derivWithin (fun z ↦ ∑' (n : ι), f n z) s x = ∑' (n : ι), derivWithin (f n) s x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: n k : ℕ
hk : k < n
⊢ rootMultiplicity (cos ((↑k + 1) * π / (↑n + 1))) (U ℝ ↑n) = 1

Tactic:
by

Goal state after: