mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: case mpr
n : ℕ
hn : n ≠ 0
k : ℕ
hk : k ≤ n
⊢ ∃ k_1, ↑k_1 * π = ↑↑n * (↑k * π / ↑n)

Tactic:
]

Goal state after:
case mpr
n : ℕ
hn : n ≠ 0
k : ℕ
hk : k ≤ n
⊢ ∃ k_1, ↑k_1 * π = ↑↑n * (↑k * π / ↑n)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: k n : ℕ
hn : n ≠ 0
⊢ cos (↑↑n * (↑k * π / ↑n)) = (-1) ^ ↑k

Tactic:
]

Goal state after:
k n : ℕ
hn : n ≠ 0
⊢ cos (↑↑n * (↑k * π / ↑n)) = (-1) ^ ↑k
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

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

Tactic:
]

Goal state after:
L : PeriodPair
⊢ Meromorphic (deriv ℘[L])
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable

Goal state before:
pp: case pos
z : ℍ
k : ℝ
hk : 0 ≤ k
x : Fin 2 → ℤ
hx : x = 0
h : -k = 0
⊢ 1 ≤ 1

Tactic:
]

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

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

Tactic:
]

Goal state after:
case h
n : ℕ
hn : n ≠ 0
x : ℝ
k : ℕ
hx : (-1) ^ k = 1
hk₁ : k ≤ n
hk₂ : x = cos (↑k * π / ↑n)
⊢ Even k
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
l₀ : ℂ
h : l₀ ∈ L.lattice
⊢ ∃ g, AnalyticAt ℂ g l₀ ∧ g l₀ ≠ 0 ∧ ∀ᶠ (z : ℂ) in 𝓝[≠] l₀, ℘[L] z = (z - l₀) ^ (-2) • g z

Tactic:
]

Goal state after:
L : PeriodPair
l₀ : ℂ
h : l₀ ∈ L.lattice
⊢ ∃ g, AnalyticAt ℂ g l₀ ∧ g l₀ ≠ 0 ∧ ∀ᶠ (z : ℂ) in 𝓝[≠] l₀, ℘[L] z = (z - l₀) ^ (-2) • g z
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
⊢ ↑n < ↑k

Tactic:
]

Goal state after:
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
⊢ ↑n < ↑k
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable

Goal state before:
pp: case neg
z : ℍ
k : ℝ
hk : 0 ≤ k
x : Fin 2 → ℤ
hx : x = 0
h : ¬-k = 0
⊢ 0 ≤ 0

Tactic:
]

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

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

Tactic:
]

Goal state after:
case mpr
n : ℕ
hn : n ≠ 0
x : ℝ
k : ℕ
hk₁ : k ≤ n
hk₂ : Even k
hx : x = cos (↑k * π / ↑n)
⊢ ↑↑1 = 1
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable

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

Tactic:
]

Goal state after:
case neg
z : ℍ
k : ℝ
hk : 0 ≤ k
x : Fin 2 → ℤ
hx : ¬x = 0
⊢ ‖↑(x 0) * ↑z + ↑(x 1)‖ ^ (-k) ≤ (r z * ‖x‖) ^ (-k)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Extremal

Goal state before:
pp: case pos
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
⊢ eval x 0 ≤ eval x 0

Tactic:
]

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

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

Tactic:
]

Goal state after:
case h
n : ℕ
hn : n ≠ 0
x : ℝ
k : ℕ
hx : (-1) ^ k = -1
hk₁ : k ≤ n
hk₂ : x = cos (↑k * π / ↑n)
⊢ Odd k
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

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

Tactic:
]

Goal state after:
case mpr
n : ℕ
hn : n ≠ 0
x : ℝ
k : ℕ
hk₁ : k ≤ n
hk₂ : Odd k
hx : x = cos (↑k * π / ↑n)
⊢ ↑↑(-1) = -1
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Extremal

Goal state before:
pp: case e'_3
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 : k ≤ n
⊢ eval x ((⇑derivative)^[k] P) = eval x ((⇑derivative)^[k] P)

Tactic:
]

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

Goal state before:
pp: case e'_4
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 : k ≤ n
⊢ eval x ((⇑derivative)^[k] (T ℝ ↑n)) = eval x ((⇑derivative)^[k] (T ℝ ↑n))

Tactic:
]

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

Goal state before:
pp: L : PeriodPair
n : ℕ
hn : Odd n
⊢ ∑' (c : ↥L.lattice), (↑((Equiv.neg ↥L.lattice) c) ^ n)⁻¹ = ∑' (b : ↥L.lattice), -(↑b ^ n)⁻¹

Tactic:
]

Goal state after:
L : PeriodPair
n : ℕ
hn : Odd n
⊢ ∑' (c : ↥L.lattice), (↑((Equiv.neg ↥L.lattice) c) ^ n)⁻¹ = ∑' (b : ↥L.lattice), -(↑b ^ n)⁻¹
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable

Goal state before:
pp: k : ℝ
hk : 2 < k
⊢ (∀ (j : ℕ), Summable fun i ↦ ((fun x ↦ ‖x‖ ^ (-k)) ∘ ⇑(finTwoArrowEquiv ℤ).symm) ↑i) ∧
    Summable fun j ↦ ∑' (i : ↑↑(box j)), ((fun x ↦ ‖x‖ ^ (-k)) ∘ ⇑(finTwoArrowEquiv ℤ).symm) ↑i

pp: k : ℝ
hk : 2 < k
⊢ 0 ≤ (fun x ↦ ‖x‖ ^ (-k)) ∘ ⇑(finTwoArrowEquiv ℤ).symm

Tactic:
]

Goal state after:
k : ℝ
hk : 2 < k
⊢ (∀ (j : ℕ), Summable fun i ↦ ((fun x ↦ ‖x‖ ^ (-k)) ∘ ⇑(finTwoArrowEquiv ℤ).symm) ↑i) ∧
    Summable fun j ↦ ∑' (i : ↑↑(box j)), ((fun x ↦ ‖x‖ ^ (-k)) ∘ ⇑(finTwoArrowEquiv ℤ).symm) ↑i

k : ℝ
hk : 2 < k
⊢ 0 ≤ (fun x ↦ ‖x‖ ^ (-k)) ∘ ⇑(finTwoArrowEquiv ℤ).symm
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: case refine_1
n : ℕ
hn : n ≠ 0
x : ℝ
hx✝ : x ∈ Finset.image (fun k ↦ cos ((2 * ↑k + 1) * π / (2 * ↑n))) (Finset.range n)
k : ℕ
hk : k ∈ Finset.range n
hx : cos ((2 * ↑k + 1) * π / (2 * ↑n)) = x
⊢ ∃ k_1, ↑↑n * ((2 * ↑k + 1) * π / (2 * ↑n)) = (2 * ↑k_1 + 1) * π / 2

Tactic:
]

Goal state after:
case refine_1
n : ℕ
hn : n ≠ 0
x : ℝ
hx✝ : x ∈ Finset.image (fun k ↦ cos ((2 * ↑k + 1) * π / (2 * ↑n))) (Finset.range n)
k : ℕ
hk : k ∈ Finset.range n
hx : cos ((2 * ↑k + 1) * π / (2 * ↑n)) = x
⊢ ∃ k_1, ↑↑n * ((2 * ↑k + 1) * π / (2 * ↑n)) = (2 * ↑k_1 + 1) * π / 2
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Extremal

Goal state before:
pp: case e'_2
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] P)

Tactic:
]

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

Goal state before:
pp: case refine_2
n : ℕ
hn : n ≠ 0
⊢ ↑n = ↑n

pp: case refine_2
n : ℕ
hn : n ≠ 0
⊢ Set.InjOn (fun k ↦ cos ((2 * ↑k + 1) * π / (2 * ↑n))) ↑(Finset.range n)

Tactic:
]

Goal state after:
case refine_2
n : ℕ
hn : n ≠ 0
⊢ Set.InjOn (fun k ↦ cos ((2 * ↑k + 1) * π / (2 * ↑n))) ↑(Finset.range n)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Extremal

Goal state before:
pp: case e'_3
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] (T ℝ ↑n)) = eval x ((⇑derivative)^[k] (T ℝ ↑n))

Tactic:
]

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

Goal state before:
pp: n k : ℕ
hk : k < n
⊢ 1 = 1

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

Tactic:
]

Goal state after:
n k : ℕ
hk : k < n
⊢ cos ((2 * ↑k + 1) * π / (2 * ↑n)) ∈ (Finset.image (fun k ↦ cos ((2 * ↑k + 1) * π / (2 * ↑n))) (Finset.range n)).val
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
      refine this.congr fun n ↦ tsum_congr fun p ↦ ?_
      simp only [← Int.mem_box.mp p.2, Nat.cast_max, norm_eq_max_natAbs, Matrix.cons_val_zero,
        Matrix.cons_val_one]

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
⊢ ℘'[L] w ^ 2 - 4 * ℘[L] w ^ 3 + L.g₂ * ℘[L] w + L.g₃ = ℘'[L] w ^ 2 - 4 * ℘[L] w ^ 3 + L.g₂ * ℘[L] w + L.g₃

Tactic:
]

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
⊢ ∀ K ⊆ s, IsCompact K → TendstoUniformlyOn (fun x1 x2 ↦ ∑ i ∈ x1, f i x2) (fun x ↦ ∑' (n : α), f n x) atTop K

Tactic:
]

Goal state after:
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
⊢ ∀ K ⊆ s, IsCompact K → TendstoUniformlyOn (fun x1 x2 ↦ ∑ i ∈ x1, f i x2) (fun x ↦ ∑' (n : α), f n x) atTop K
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable

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

Tactic:
]

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

Goal state before:
pp: case refine_1
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
⊢ ∃ n_1, ↑n_1 * π = (↑↑n + 1) * ((↑k + 1) * π / (↑n + 1))

Tactic:
]

Goal state after:
case refine_1
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
⊢ ∃ n_1, ↑n_1 * π = (↑↑n + 1) * ((↑k + 1) * π / (↑n + 1))
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: case refine_2
n : ℕ
hn : n ≠ 0
⊢ ↑n = ↑n

pp: case refine_2
n : ℕ
hn : n ≠ 0
⊢ Set.InjOn (fun k ↦ cos ((↑k + 1) * π / (↑n + 1))) ↑(Finset.range n)

Tactic:
]

Goal state after:
case refine_2
n : ℕ
hn : n ≠ 0
⊢ Set.InjOn (fun k ↦ cos ((↑k + 1) * π / (↑n + 1))) ↑(Finset.range n)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: n k : ℕ
hk : k < n
⊢ 1 = 1

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

Tactic:
]

Goal state after:
n k : ℕ
hk : k < n
⊢ cos ((↑k + 1) * π / (↑n + 1)) ∈ (Finset.image (fun k ↦ cos ((↑k + 1) * π / (↑n + 1))) (Finset.range n)).val
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable

Goal state before:
pp: z : ℍ
a b : ℤ
⊢ ∃ c, ∀ᶠ (x : Fin 2 → ℤ) in cofinite, ‖((↑(x 0) + ↑a) * ↑z + ↑(x 1) + ↑b)⁻¹‖ ≤ c * ‖‖![x 0 + a, x 1 + b]‖⁻¹‖

Tactic:
]

Goal state after:
z : ℍ
a b : ℤ
⊢ ∃ c, ∀ᶠ (x : Fin 2 → ℤ) in cofinite, ‖((↑(x 0) + ↑a) * ↑z + ↑(x 1) + ↑b)⁻¹‖ ≤ c * ‖‖![x 0 + a, x 1 + b]‖⁻¹‖
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
⊢ iteratedDeriv 6
      (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)
      0 =
    0

Tactic:
]

Goal state after:
L : PeriodPair
⊢ iteratedDeriv 6
      (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)
      0 =
    0
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: case refine_1
n k : ℕ
hn : n ≠ 0
hk₀ : 0 < k
hk₁ : k < n
hk₂ : Even k
zero_lt : 0 < ↑k * π / ↑n
lt_pi : ↑k * π / ↑n < π
x : ℝ
hx : x ∈ Set.Ioo (-1) 1
⊢ eval x (T ℝ ↑n) ≤ 1

Tactic:
]

Goal state after:
case refine_1
n k : ℕ
hn : n ≠ 0
hk₀ : 0 < k
hk₁ : k < n
hk₂ : Even k
zero_lt : 0 < ↑k * π / ↑n
lt_pi : ↑k * π / ↑n < π
x : ℝ
hx : x ∈ Set.Ioo (-1) 1
⊢ eval x (T ℝ ↑n) ≤ 1
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: case refine_2
n k : ℕ
hn : n ≠ 0
hk₀ : 0 < k
hk₁ : k < n
hk₂ : Even k
zero_lt : 0 < ↑k * π / ↑n
lt_pi : ↑k * π / ↑n < π
⊢ cos π < cos (↑k * π / ↑n)

Tactic:
]

Goal state after:
case refine_2
n k : ℕ
hn : n ≠ 0
hk₀ : 0 < k
hk₁ : k < n
hk₂ : Even k
zero_lt : 0 < ↑k * π / ↑n
lt_pi : ↑k * π / ↑n < π
⊢ cos π < cos (↑k * π / ↑n)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: case refine_3
n k : ℕ
hn : n ≠ 0
hk₀ : 0 < k
hk₁ : k < n
hk₂ : Even k
zero_lt : 0 < ↑k * π / ↑n
lt_pi : ↑k * π / ↑n < π
⊢ cos (↑k * π / ↑n) < cos 0

Tactic:
]

Goal state after:
case refine_3
n k : ℕ
hn : n ≠ 0
hk₀ : 0 < k
hk₁ : k < n
hk₂ : Even k
zero_lt : 0 < ↑k * π / ↑n
lt_pi : ↑k * π / ↑n < π
⊢ cos (↑k * π / ↑n) < cos 0
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable

Goal state before:
pp: a b : ℤ
this : ∀ (x : Fin 2 → ℤ), ![x 0 + a, x 1 + b] = x + ![a, b]
⊢ (fun x ↦ ‖x‖) =Θ[cofinite] fun x ↦ ‖x‖

Tactic:
]

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

Goal state before:
pp: case refine_1
n k : ℕ
hn : n ≠ 0
hk₁ : k < n
hk₂ : Odd k
k_pos : 0 < k
zero_lt : 0 < ↑k * π / ↑n
lt_pi : ↑k * π / ↑n < π
x : ℝ
hx : x ∈ Set.Ioo (-1) 1
⊢ -1 ≤ eval x (T ℝ ↑n)

Tactic:
]

Goal state after:
case refine_1
n k : ℕ
hn : n ≠ 0
hk₁ : k < n
hk₂ : Odd k
k_pos : 0 < k
zero_lt : 0 < ↑k * π / ↑n
lt_pi : ↑k * π / ↑n < π
x : ℝ
hx : x ∈ Set.Ioo (-1) 1
⊢ -1 ≤ eval x (T ℝ ↑n)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: case refine_2
n k : ℕ
hn : n ≠ 0
hk₁ : k < n
hk₂ : Odd k
k_pos : 0 < k
zero_lt : 0 < ↑k * π / ↑n
lt_pi : ↑k * π / ↑n < π
⊢ cos π < cos (↑k * π / ↑n)

Tactic:
]

Goal state after:
case refine_2
n k : ℕ
hn : n ≠ 0
hk₁ : k < n
hk₂ : Odd k
k_pos : 0 < k
zero_lt : 0 < ↑k * π / ↑n
lt_pi : ↑k * π / ↑n < π
⊢ cos π < cos (↑k * π / ↑n)
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
this : 7 ≤ meromorphicOrderAt L.relation 0 + ↑6 * meromorphicOrderAt id 0
⊢ 1 ≤ meromorphicOrderAt L.relation 0

Tactic:
]

Goal state after:
L : PeriodPair
this : 7 ≤ meromorphicOrderAt L.relation 0 + ↑6 * meromorphicOrderAt id 0
⊢ 1 ≤ meromorphicOrderAt L.relation 0
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: case refine_3
n k : ℕ
hn : n ≠ 0
hk₁ : k < n
hk₂ : Odd k
k_pos : 0 < k
zero_lt : 0 < ↑k * π / ↑n
lt_pi : ↑k * π / ↑n < π
⊢ cos (↑k * π / ↑n) < cos 0

Tactic:
]

Goal state after:
case refine_3
n k : ℕ
hn : n ≠ 0
hk₁ : k < n
hk₂ : Odd k
k_pos : 0 < k
zero_lt : 0 < ↑k * π / ↑n
lt_pi : ↑k * π / ↑n < π
⊢ cos (↑k * π / ↑n) < cos 0
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
this : 7 ≤ meromorphicOrderAt L.relation 0 + ↑6 * meromorphicOrderAt id 0
⊢ 1 + 6 ≤ meromorphicOrderAt L.relation 0 + 6

Tactic:
]

Goal state after:
L : PeriodPair
this : 7 ≤ meromorphicOrderAt L.relation 0 + ↑6 * meromorphicOrderAt id 0
⊢ 1 + 6 ≤ meromorphicOrderAt L.relation 0 + 6
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
⊢ 7 ≤ ENat.map Nat.cast (analyticOrderAt (L.relation * id ^ 6) 0)

Tactic:
]

Goal state after:
L : PeriodPair
⊢ 7 ≤ ENat.map Nat.cast (analyticOrderAt (L.relation * id ^ 6) 0)
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: case neg
L : PeriodPair
i : ℕ
hi₁ : i < 7
hi₂ : ¬Odd i
hi₃ : ¬i = 0
hi₄ : ¬i = 6
⊢ iteratedDeriv i
      (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)
      0 =
    0

Tactic:
]

Goal state after:
case neg
L : PeriodPair
i : ℕ
hi₁ : i < 7
hi₂ : ¬Odd i
hi₃ : ¬i = 0
hi₄ : ¬i = 6
⊢ iteratedDeriv i
      (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)
      0 =
    0
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: case mp
n : ℕ
hn : 2 ≤ n
x : ℝ
hx : eval x (U ℝ (↑n - 1)) = 0
⊢ ∃ k ∈ Finset.Ioo 0 n, x = cos (↑k * π / ↑n)

Tactic:
]

Goal state after:
case mp
n : ℕ
hn : 2 ≤ n
x : ℝ
hx : eval x (U ℝ (↑n - 1)) = 0
⊢ ∃ k ∈ Finset.Ioo 0 n, x = cos (↑k * π / ↑n)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: case mp
n : ℕ
hn : 2 ≤ n
x : ℝ
hx : x ∈ Finset.image (fun k ↦ cos ((↑k + 1) * π / (↑(n - 1) + 1))) (Finset.range (n - 1))
⊢ ∃ k ∈ Finset.Ioo 0 n, x = cos (↑k * π / ↑n)

Tactic:
]

Goal state after:
case mp
n : ℕ
hn : 2 ≤ n
x : ℝ
hx : x ∈ Finset.image (fun k ↦ cos ((↑k + 1) * π / (↑(n - 1) + 1))) (Finset.range (n - 1))
⊢ ∃ k ∈ Finset.Ioo 0 n, x = cos (↑k * π / ↑n)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: case mp
n : ℕ
hn : 2 ≤ n
x : ℝ
hx✝ : x ∈ Finset.image (fun k ↦ cos ((↑k + 1) * π / (↑(n - 1) + 1))) (Finset.range (n - 1))
k : ℕ
hk₁ : k ∈ Finset.range (n - 1)
hx : cos ((↑k + 1) * π / (↑(n - 1) + 1)) = x
⊢ cos ((↑k + 1) * π / (↑(n - 1) + 1)) = cos (↑(k + 1) * π / ↑n)

Tactic:
]

Goal state after:
case mp
n : ℕ
hn : 2 ≤ n
x : ℝ
hx✝ : x ∈ Finset.image (fun k ↦ cos ((↑k + 1) * π / (↑(n - 1) + 1))) (Finset.range (n - 1))
k : ℕ
hk₁ : k ∈ Finset.range (n - 1)
hx : cos ((↑k + 1) * π / (↑(n - 1) + 1)) = x
⊢ cos ((↑k + 1) * π / (↑(n - 1) + 1)) = cos (↑(k + 1) * π / ↑n)
Module: Mathlib.Topology.Algebra.InfiniteSum.TsumUniformlyOn

Goal state before:
pp: case pos
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
hsum : ∀ t ∈ s, Summable fun n ↦ f n t
m : ℕ
hm :
  ∀ {x : 𝕜},
    x ∈ s →
      (∀ (k : ℕ), 1 ≤ k → k ≤ m → SummableLocallyUniformlyOn (fun n ↦ iteratedDerivWithin k (fun z ↦ f n z) s) s) →
        (∀ (n : ι) (k : ℕ) (r : 𝕜), k ≤ m → r ∈ s → DifferentiableAt 𝕜 (iteratedDerivWithin k (fun z ↦ f n z) s) r) →
          iteratedDerivWithin m (fun z ↦ ∑' (n : ι), f n z) s x = ∑' (n : ι), iteratedDerivWithin m (f n) s x
x : 𝕜
hx : x ∈ s
h : ∀ (k : ℕ), 1 ≤ k → k ≤ m + 1 → SummableLocallyUniformlyOn (fun n ↦ iteratedDerivWithin k (fun z ↦ f n z) s) s
hf2 : ∀ (n : ι) (k : ℕ) (r : 𝕜), k ≤ m + 1 → r ∈ s → DifferentiableAt 𝕜 (iteratedDerivWithin k (fun z ↦ f n z) s) r
r : 𝕜
hr : r ∈ s
hm2 : m = 0
⊢ Summable fun n ↦ iteratedDerivWithin m (fun z ↦ f n z) s r

Tactic:
simp [hm2, hsum r hr]

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

Goal state before:
pp: case mpr
n : ℕ
hn : 2 ≤ n
x : ℝ
k : ℕ
hk : k ∈ Finset.Ioo 0 n
hx : x = cos (↑k * π / ↑n)
⊢ IsLocalExtr (fun x ↦ eval x (T ℝ ↑n)) (cos (↑k * π / ↑n))

Tactic:
]

Goal state after:
case mpr
n : ℕ
hn : 2 ≤ n
x : ℝ
k : ℕ
hk : k ∈ Finset.Ioo 0 n
hx : x = cos (↑k * π / ↑n)
⊢ IsLocalExtr (fun x ↦ eval x (T ℝ ↑n)) (cos (↑k * π / ↑n))
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
hsum : ∀ t ∈ s, Summable fun n ↦ f n t
m : ℕ
hm :
  ∀ {x : 𝕜},
    x ∈ s →
      (∀ (k : ℕ), 1 ≤ k → k ≤ m → SummableLocallyUniformlyOn (fun n ↦ iteratedDerivWithin k (fun z ↦ f n z) s) s) →
        (∀ (n : ι) (k : ℕ) (r : 𝕜), k ≤ m → r ∈ s → DifferentiableAt 𝕜 (iteratedDerivWithin k (fun z ↦ f n z) s) r) →
          iteratedDerivWithin m (fun z ↦ ∑' (n : ι), f n z) s x = ∑' (n : ι), iteratedDerivWithin m (f n) s x
x : 𝕜
hx : x ∈ s
h : ∀ (k : ℕ), 1 ≤ k → k ≤ m + 1 → SummableLocallyUniformlyOn (fun n ↦ iteratedDerivWithin k (fun z ↦ f n z) s) s
hf2 : ∀ (n : ι) (k : ℕ) (r : 𝕜), k ≤ m + 1 → r ∈ s → DifferentiableAt 𝕜 (iteratedDerivWithin k (fun z ↦ f n z) s) r
x✝¹ : ι
x✝ : 𝕜
ht : x✝ ∈ s
⊢ derivWithin (iteratedDerivWithin m (fun z ↦ f x✝¹ z) s) s x✝ =
    derivWithin (fun z ↦ iteratedDerivWithin m (fun z ↦ f x✝¹ z) s z) s x✝

Tactic:
]

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

Goal state before:
pp: case mp.refine_1
n : ℕ
hn : n ≠ 0
x : ℝ
hx : x ∈ Set.Icc (-1) 1
h✝ : IsExtrOn (fun x ↦ eval x (T ℝ ↑n)) (Set.Icc (-1) 1) x
h : IsMinOn (fun x ↦ eval x (T ℝ ↑n)) (Set.Icc (-1) 1) x
this : eval x (T ℝ ↑n) ≤ -1
⊢ eval x (T ℝ ↑n) ≤ -1

Tactic:
]

Goal state after:
case mp.refine_1
n : ℕ
hn : n ≠ 0
x : ℝ
hx : x ∈ Set.Icc (-1) 1
h✝ : IsExtrOn (fun x ↦ eval x (T ℝ ↑n)) (Set.Icc (-1) 1) x
h : IsMinOn (fun x ↦ eval x (T ℝ ↑n)) (Set.Icc (-1) 1) x
this : eval x (T ℝ ↑n) ≤ -1
⊢ eval x (T ℝ ↑n) ≤ -1
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
x : ℂ
l : ↥L.lattice
⊢ L.relation x = L.relation x

Tactic:
]

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

Goal state before:
pp: case pos
L : PeriodPair
x : ↥L.lattice
this : AnalyticAt ℂ L.relation (↑x - ↑x)
⊢ AnalyticAt ℂ L.relation ↑x

Tactic:
]

Goal state after:
case pos
L : PeriodPair
x : ↥L.lattice
this : AnalyticAt ℂ L.relation (↑x - ↑x)
⊢ AnalyticAt ℂ L.relation ↑x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: case mp.refine_2
n : ℕ
hn : n ≠ 0
x : ℝ
hx : x ∈ Set.Icc (-1) 1
h✝ : IsExtrOn (fun x ↦ eval x (T ℝ ↑n)) (Set.Icc (-1) 1) x
h : IsMaxOn (fun x ↦ eval x (T ℝ ↑n)) (Set.Icc (-1) 1) x
this : 1 ≤ eval x (T ℝ ↑n)
⊢ 1 ≤ eval x (T ℝ ↑n)

Tactic:
]

Goal state after:
case mp.refine_2
n : ℕ
hn : n ≠ 0
x : ℝ
hx : x ∈ Set.Icc (-1) 1
h✝ : IsExtrOn (fun x ↦ eval x (T ℝ ↑n)) (Set.Icc (-1) 1) x
h : IsMaxOn (fun x ↦ eval x (T ℝ ↑n)) (Set.Icc (-1) 1) x
this : 1 ≤ eval x (T ℝ ↑n)
⊢ 1 ≤ eval x (T ℝ ↑n)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent

Goal state before:
pp: z : ℂ
⊢ (cexp (z * I) + cexp (-z * I)) / 2 / ((cexp (-z * I) - cexp (z * I)) * I / 2) =
    (cexp (2 * I * z) + 1) / (I * (1 - cexp (2 * I * z)))

Tactic:
]

Goal state after:
z : ℂ
⊢ (cexp (z * I) + cexp (-z * I)) / 2 / ((cexp (-z * I) - cexp (z * I)) * I / 2) =
    (cexp (2 * I * z) + 1) / (I * (1 - cexp (2 * I * z)))
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent

Goal state before:
pp: z : ℂ
⊢ cexp (z * I) + cexp (-z * I) = cexp (-(z * I) + 2 * I * z) + cexp (-(z * I)) * 1

Tactic:
]

Goal state after:
z : ℂ
⊢ cexp (z * I) + cexp (-z * I) = cexp (-(z * I) + 2 * I * z) + cexp (-(z * I)) * 1
Module: Mathlib.Analysis.SpecificLimits.ArithmeticGeometric

Goal state before:
pp: case succ
R : Type u_1
a b u₀ : R
inst✝ : CommSemiring R
n : ℕ
hn : arithGeom a b u₀ n = a ^ n * u₀ + b * ∑ k ∈ Finset.range n, a ^ k
⊢ a * a ^ n * u₀ + (a * (b * ∑ k ∈ Finset.range n, a ^ k) + b) = a * a ^ n * u₀ + b * ∑ k ∈ Finset.range (1 + n), a ^ k

Tactic:
]

Goal state after:
case succ
R : Type u_1
a b u₀ : R
inst✝ : CommSemiring R
n : ℕ
hn : arithGeom a b u₀ n = a ^ n * u₀ + b * ∑ k ∈ Finset.range n, a ^ k
⊢ a * a ^ n * u₀ + (a * (b * ∑ k ∈ Finset.range n, a ^ k) + b) = a * a ^ n * u₀ + b * ∑ k ∈ Finset.range (1 + n), a ^ k
Module: Mathlib.Analysis.SpecificLimits.ArithmeticGeometric

Goal state before:
pp: case succ.e_a
R : Type u_1
a b u₀ : R
inst✝ : CommSemiring R
n : ℕ
hn : arithGeom a b u₀ n = a ^ n * u₀ + b * ∑ k ∈ Finset.range n, a ^ k
⊢ ∑ i ∈ Finset.range n, a * (b * a ^ i) + b = ∑ i ∈ Finset.range n, b * a ^ (i + 1) + b

Tactic:
]

Goal state after:
case succ.e_a
R : Type u_1
a b u₀ : R
inst✝ : CommSemiring R
n : ℕ
hn : arithGeom a b u₀ n = a ^ n * u₀ + b * ∑ k ∈ Finset.range n, a ^ k
⊢ ∑ i ∈ Finset.range n, a * (b * a ^ i) + b = ∑ i ∈ Finset.range n, b * a ^ (i + 1) + b