mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Complex.Periodic

Goal state before:
pp: h : ℝ
f : ℂ → ℂ
hh : 0 < h
h_bd : I∞.BoundedAtFilter f
⊢ 1 ∘ invQParam h =ᶠ[𝓝[≠] 0] 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.OpenMapping

Goal state before:
pp: f : ℂ → ℂ
U : Set ℂ
c₀ : ℝ
h₁f : AnalyticOnNhd ℂ f U
h₂f : ∀ x ∈ U, (f x).re = c₀
h₁U : IsOpen U
h₂U : IsConnected U
z₀ : ℂ
h✝ : z₀ ∈ U
h₅ : ¬∃ c, ∀ x ∈ U, f x = c
⊢ re '' f '' U = {c₀}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.OpenMapping

Goal state before:
pp: f : ℂ → ℂ
U : Set ℂ
c₀ : ℝ
h₁f : AnalyticOnNhd ℂ f U
h₂f : ∀ x ∈ U, (f x).re = c₀
h₁U : IsOpen U
h₂U : IsConnected U
z₀ : ℂ
h✝ : z₀ ∈ U
h₅ : ¬∃ c, ∀ x ∈ U, f x = c
⊢ U ⊆ U

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: n : ℕ
⊢ 0 < ∫ (x : ℝ) in 0..π, sin x ^ n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.OpenMapping

Goal state before:
pp: f : ℂ → ℂ
U : Set ℂ
c₀ : ℝ
h₁f : AnalyticOnNhd ℂ f U
h₂f : ∀ x ∈ U, (f x).re = c₀
h₁U : IsOpen U
h₂U : IsConnected U
⊢ ∃ c, ∀ x ∈ U, f x = c

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.Periodic

Goal state before:
pp: h : ℝ
f : ℂ → ℂ
hh : 0 < h
h_bd : I∞.BoundedAtFilter f
⊢ (𝓝[≠] 0).BoundedAtFilter (cuspFunction h f)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.OpenMapping

Goal state before:
pp: f : ℂ → ℂ
U : Set ℂ
c₀ : ℝ
h₁f : AnalyticOnNhd ℂ f U
h₂f : ∀ x ∈ U, (f x).re = c₀
h₁U : IsOpen U
h₂U : IsConnected U
⊢ ∃ c, ∀ x ∈ U, f x = ↑c₀ + ↑c * I

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.OpenMapping

Goal state before:
pp: f : ℂ → ℂ
U : Set ℂ
c₀ : ℝ
h₁f : AnalyticOnNhd ℂ f U
h₂f : ∀ x ∈ U, (f x).im = c₀
h₁U : IsOpen U
h₂U : IsConnected U
z₀ : ℂ
h✝ : z₀ ∈ U
h₅ : ¬∃ c, ∀ x ∈ U, f x = c
⊢ im '' f '' U = {c₀}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.Periodic

Goal state before:
pp: h : ℝ
f : ℂ → ℂ
hh : 0 < h
h_zer : I∞.ZeroAtFilter f
⊢ cuspFunction h f 0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.OpenMapping

Goal state before:
pp: f : ℂ → ℂ
U : Set ℂ
c₀ : ℝ
h₁f : AnalyticOnNhd ℂ f U
h₂f : ∀ x ∈ U, (f x).im = c₀
h₁U : IsOpen U
h₂U : IsConnected U
z₀ : ℂ
h✝ : z₀ ∈ U
h₅ : ¬∃ c, ∀ x ∈ U, f x = c
⊢ U ⊆ U

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.OpenMapping

Goal state before:
pp: f : ℂ → ℂ
U : Set ℂ
c₀ : ℝ
h₁f : AnalyticOnNhd ℂ f U
h₂f : ∀ x ∈ U, (f x).im = c₀
h₁U : IsOpen U
h₂U : IsConnected U
⊢ ∃ c, ∀ x ∈ U, f x = c

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: n : ℕ
⊢ ∫ (x : ℝ) in 0..π, sin x ^ (n + 1) ≤ ∫ (x : ℝ) in 0..π, sin x ^ n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.OpenMapping

Goal state before:
pp: f : ℂ → ℂ
U : Set ℂ
c₀ : ℝ
h₁f : AnalyticOnNhd ℂ f U
h₂f : ∀ x ∈ U, (f x).im = c₀
h₁U : IsOpen U
h₂U : IsConnected U
⊢ ∃ c, ∀ x ∈ U, f x = ↑c + ↑c₀ * I

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.OpenMapping

Goal state before:
pp: p : ℂ[X]
h : ¬∃ x, C x = p
x : ℂ
eq : ∀ (z : ℂ), eval z p = x
⊢ ∀ (r : ℂ), eval r (C x) = eval r p

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.OpenMapping

Goal state before:
pp: p : ℂ[X]
⊢ (∃ x, C x = p) ∨ IsOpenQuotientMap fun x ↦ eval x p

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
n : ℕ
C : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a
α β γ : ℝ
⊢ β * α * γ * α = β * (α * α * γ)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.OpenMapping

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
⊢ IsOpenQuotientMap fun x ↦ x ^ n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.OpenMapping

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
z : { z // z ≠ 0 }
h : (fun x ↦ x ^ n) 0 = ↑z
⊢ ↑z = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.OpenMapping

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
z : { z // z ≠ 0 }
w : ℂ
h : (fun x ↦ x ^ n) w = ↑z
⊢ w ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.OpenMapping

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
⊢ Continuous fun z ↦ ⟨↑z ^ n, ⋯⟩

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
n : ℕ
C : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a
h : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)
x : ℝ
x✝ : x ∈ [[a, b]]
⊢ HasDerivAt (fun y ↦ cos y ^ (n + 1)) (-↑(n + 1) * sin x * cos x ^ n) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: w ρ : ℂ
R r₀ r : ℝ
hR : 0 < R
hρ : ‖ρ‖ = R
hr₀ : 0 < r₀
hw : ‖w‖ < r₀
hr₀r : r₀ ≤ r
hrR : r ≤ R
θ : ℝ
hdR : 0 < ‖circleMap 0 R θ - ρ‖
hrw : 0 < r₀ - ‖w‖
hr : 0 < r
h_norm_sub₁ : √(r₀ / R) * ‖circleMap 0 R θ - ρ‖ ≤ ‖circleMap 0 r θ - ρ‖
h_norm_sub₂ : 0 < ‖circleMap 0 r θ - ρ‖
⊢ log ‖circleMap 0 r θ - ρ‖ ≤ |log (2 * R)| + 0 + 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.OpenMapping

Goal state before:
pp: n : ℤ
inst✝ : NeZero n
⊢ IsOpenQuotientMap fun z ↦ ⟨↑z ^ n, ⋯⟩

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.Polynomial.GaussLucas

Goal state before:
pp: P : ℂ[X]
z w : ℂ
⊢ 0 ≤ P.derivRootWeight z w

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.Polynomial.GaussLucas

Goal state before:
pp: P : ℂ[X]
hP : 0 < P.degree
z : ℂ
⊢ P ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.Polynomial.GaussLucas

Goal state before:
pp: case pos
P : ℂ[X]
hP : 0 < P.degree
z : ℂ
hP₀ : P ≠ 0
hPz : eval z P = 0
⊢ 0 < ∑ w ∈ P.roots.toFinset, P.derivRootWeight z w

Tactic:
simp [derivRootWeight, hPz, hP₀]

Goal state after:
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: w ρ : ℂ
R r₀ r : ℝ
hR : 0 < R
hρ : ‖ρ‖ = R
hr₀ : 0 < r₀
hw : ‖w‖ < r₀
hr₀r : r₀ ≤ r
hrR : r ≤ R
θ : ℝ
hdR : 0 < ‖circleMap 0 R θ - ρ‖
hrw : 0 < r₀ - ‖w‖
hr : 0 < r
h_norm_sub₁ : √(r₀ / R) * ‖circleMap 0 R θ - ρ‖ ≤ ‖circleMap 0 r θ - ρ‖
h_norm_sub₂ : 0 < ‖circleMap 0 r θ - ρ‖
⊢ |log (2 * R)| + 0 + 0 ≤ |log (2 * R)| + |log √(r₀ / R)| + |log ‖circleMap 0 R θ - ρ‖|

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.Polynomial.GaussLucas

Goal state before:
pp: P : ℂ[X]
hP : 0 < P.degree
z : ℂ
hP₀ : P ≠ 0
hPz : ¬eval z P = 0
w : ℂ
hw : w ∈ P.roots.toFinset
⊢ 0 < ↑(rootMultiplicity w P)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.Polynomial.GaussLucas

Goal state before:
pp: P : ℂ[X]
hP : 0 < P.degree
z : ℂ
hP₀ : P ≠ 0
hPz : ¬eval z P = 0
w : ℂ
hw : w ∈ P.roots.toFinset
this : z ≠ w
⊢ 0 < ‖z - w‖ ^ 2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
n : ℕ
C : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a
h : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)
hu : ∀ x ∈ [[a, b]], HasDerivAt (fun y ↦ cos y ^ (n + 1)) (-↑(n + 1) * sin x * cos x ^ n) x
hv : ∀ x ∈ [[a, b]], HasDerivAt sin (cos x) x
H :
  ∫ (x : ℝ) in a..b, cos x ^ (n + 1) * cos x =
    cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a - ∫ (x : ℝ) in a..b, -↑(n + 1) * sin x * cos x ^ n * sin x
⊢ ∫ (x : ℝ) in a..b, cos x ^ (n + 2) = ∫ (x : ℝ) in a..b, cos x ^ (n + 1) * cos x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: w ρ : ℂ
R r₀ r : ℝ
hR : 0 < R
hρ : ‖ρ‖ = R
hr₀ : 0 < r₀
hw : ‖w‖ < r₀
hr₀r : r₀ ≤ r
hrR : r ≤ R
θ : ℝ
hdR : 0 < ‖circleMap 0 R θ - ρ‖
⊢ ‖herglotzLogIntegrand w ρ (circleMap 0 r θ)‖ ≤
    (R + ‖w‖) / (r₀ - ‖w‖) * (|log (2 * R)| + |log √(r₀ / R)| + |log ‖circleMap 0 R θ - ρ‖|)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.Polynomial.GaussLucas

Goal state before:
pp: case neg.hs
P : ℂ[X]
hP : 0 < P.degree
z : ℂ
hP₀ : P ≠ 0
hPz : ¬eval z P = 0
⊢ P.roots ≠ 0

Tactic:
apply Splits.roots_ne_zero (IsAlgClosed.splits _)

Goal state after:
case neg.hs
P : ℂ[X]
hP : 0 < P.degree
z : ℂ
hP₀ : P ≠ 0
hPz : ¬eval z P = 0
⊢ P.natDegree ≠ 0
Module: Mathlib.Analysis.Complex.Polynomial.GaussLucas

Goal state before:
pp: P : ℂ[X]
hP : 0 < P.degree
z : ℂ
⊢ 0 < ∑ w ∈ P.roots.toFinset, P.derivRootWeight z w

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
n : ℕ
C : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a
h : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)
hu : ∀ x ∈ [[a, b]], HasDerivAt (fun y ↦ cos y ^ (n + 1)) (-↑(n + 1) * sin x * cos x ^ n) x
hv : ∀ x ∈ [[a, b]], HasDerivAt sin (cos x) x
H :
  ∫ (x : ℝ) in a..b, cos x ^ (n + 1) * cos x =
    cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a - ∫ (x : ℝ) in a..b, -↑(n + 1) * sin x * cos x ^ n * sin x
⊢ ∫ (x : ℝ) in a..b, cos x ^ (n + 1) * cos x = C + (↑n + 1) * ∫ (x : ℝ) in a..b, sin x ^ 2 * cos x ^ n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.Periodic

Goal state before:
pp: h : ℝ
f : ℂ → ℂ
hh : 0 < h
hf : Periodic f ↑h
h_hol : ∀ᶠ (z : ℂ) in I∞, DifferentiableAt ℂ f z
h_bd : I∞.BoundedAtFilter f
c : ℝ
t : ∀ᶠ (x : ℂ) in 𝓝[≠] 0, DifferentiableAt ℂ (cuspFunction h f) x ∧ ‖cuspFunction h f x‖ ≤ c
S : Set ℂ
hS1 : ∀ y ∈ S, y ∈ {0}ᶜ → DifferentiableAt ℂ (cuspFunction h f) y ∧ ‖cuspFunction h f y‖ ≤ c
hS2 : IsOpen S
hS3 : 0 ∈ S
h_diff : DifferentiableOn ℂ (cuspFunction h f) (S \ {0})
⊢ BddAbove (norm ∘ cuspFunction h f '' (S \ {0}))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
n : ℕ
C : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a
h : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)
hu : ∀ x ∈ [[a, b]], HasDerivAt (fun y ↦ cos y ^ (n + 1)) (-↑(n + 1) * sin x * cos x ^ n) x
hv : ∀ x ∈ [[a, b]], HasDerivAt sin (cos x) x
H :
  ∫ (x : ℝ) in a..b, cos x ^ (n + 1) * cos x =
    cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a - ∫ (x : ℝ) in a..b, -↑(n + 1) * sin x * cos x ^ n * sin x
⊢ C + (↑n + 1) * ∫ (x : ℝ) in a..b, sin x ^ 2 * cos x ^ n =
    C + (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ n - cos x ^ (n + 2)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.Polynomial.GaussLucas

Goal state before:
pp: P : ℂ[X]
z : ℂ
hP : 0 < P.degree
hz : eval z (derivative P) = 0
weight : ℂ → ℝ := P.derivRootWeight z
s : Finset ℂ := P.roots.toFinset
this : ∑ x ∈ s, weight x • (z - x) = 0
⊢ z = s.centerMass weight fun x ↦ z

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.Polynomial.GaussLucas

Goal state before:
pp: P : ℂ[X]
z : ℂ
hP : 0 < P.degree
hz : eval z (derivative P) = 0
weight : ℂ → ℝ := P.derivRootWeight z
s : Finset ℂ := P.roots.toFinset
this : ∑ x ∈ s, weight x • (z - x) = 0
⊢ (s.centerMass weight fun x ↦ z) = (s.centerMass weight fun x ↦ z - x) + s.centerMass weight id

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.Polynomial.GaussLucas

Goal state before:
pp: P : ℂ[X]
z : ℂ
hP : 0 < P.degree
hz : eval z (derivative P) = 0
weight : ℂ → ℝ := P.derivRootWeight z
s : Finset ℂ := P.roots.toFinset
this : ∑ x ∈ s, weight x • (z - x) = 0
⊢ (s.centerMass weight fun x ↦ z - x) + s.centerMass weight id = s.centerMass weight id

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.Polynomial.GaussLucas

Goal state before:
pp: P : ℂ[X]
z : ℂ
hP : 0 < P.degree
hz : eval z (derivative P) = 0
weight : ℂ → ℝ := P.derivRootWeight z
s : Finset ℂ := P.roots.toFinset
this : ∑ x ∈ s, weight x • (z - x) = 0
⊢ z = s.centerMass weight id

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.Periodic

Goal state before:
pp: h : ℝ
f : ℂ → ℂ
hh : 0 < h
hf : Periodic f ↑h
h_hol : ∀ᶠ (z : ℂ) in I∞, DifferentiableAt ℂ f z
h_bd : I∞.BoundedAtFilter f
⊢ DifferentiableAt ℂ (cuspFunction h f) 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.Periodic

Goal state before:
pp: h : ℝ
f : ℂ → ℂ
hh : 0 < h
hf : Periodic f ↑h
h_hol : ∀ᶠ (z : ℂ) in I∞, DifferentiableAt ℂ f z
h_bd : I∞.BoundedAtFilter f
this : Tendsto (cuspFunction h f) (𝓝[≠] 0) (𝓝 (cuspFunction h f 0))
⊢ Tendsto f I∞ (𝓝 (cuspFunction h f 0))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.Periodic

Goal state before:
pp: h : ℝ
f : ℂ → ℂ
hh : 0 < h
hf : Periodic f ↑h
h_hol : ∀ᶠ (z : ℂ) in I∞, DifferentiableAt ℂ f z
h_bd : I∞.BoundedAtFilter f
⊢ Tendsto f I∞ (𝓝 (cuspFunction h f 0))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.Periodic

Goal state before:
pp: h : ℝ
f : ℂ → ℂ
hh : 0 < h
hf : Periodic f ↑h
h_hol : ∀ᶠ (z : ℂ) in I∞, DifferentiableAt ℂ f z
h_bd : I∞.BoundedAtFilter f
⊢ (fun z ↦ f z - cuspFunction h f 0) =O[I∞] fun z ↦ rexp (-2 * π * z.im / h)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.Periodic

Goal state before:
pp: h : ℝ
f : ℂ → ℂ
hh : 0 < h
hf : Periodic f ↑h
h_hol : ∀ᶠ (z : ℂ) in I∞, DifferentiableAt ℂ f z
h_zer : I∞.ZeroAtFilter f
⊢ f =O[I∞] fun z ↦ rexp (-2 * π * z.im / h)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.Periodic

Goal state before:
pp: h : ℝ
f : ℂ → ℂ
hfcts : ContinuousAt (update (f ∘ invQParam h) 0 ((𝓝[≠] 0).limUnder (f ∘ invQParam h))) 0
a : ℂ
⊢ Tendsto (fun x ↦ f (invQParam h x)) (𝓝[≠] 0) (𝓝 ((𝓝[≠] 0).limUnder (f ∘ invQParam h)))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.Periodic

Goal state before:
pp: h : ℝ
f : ℂ → ℂ
hfcts : ContinuousAt (cuspFunction h f) 0
a : ℂ
⊢ cuspFunction h (a • f) = a • cuspFunction h f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: ρ w : ℂ
R : ℝ
hR : 0 < R
hρ : ‖ρ‖ = R
hw : ‖w‖ < R
r : ℕ → ℝ
hr_lt : ∀ (n : ℕ), r n < R
hr_tendsto : Tendsto r atTop (𝓝 R)
bound : ℝ → ℝ :=
  fun θ ↦ (R + ‖w‖) / ((R + ‖w‖) / 2 - ‖w‖) * (|log (2 * R)| + |log √((R + ‖w‖) / 2 / R)| + |log ‖circleMap 0 R θ - ρ‖|)
⊢ (R + ‖w‖) / 2 < R

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.Periodic

Goal state before:
pp: h : ℝ
f : ℂ → ℂ
hfcts : ContinuousAt (cuspFunction h f) 0
⊢ cuspFunction h (-f) = -cuspFunction h f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.Polynomial.GaussLucas

Goal state before:
pp: P : ℂ[X]
z : ℂ
hP : 0 < P.degree
hz : eval z (derivative P) = 0
weight : ℂ → ℝ := P.derivRootWeight z
s : Finset ℂ := P.roots.toFinset
hzP : ¬eval z P = 0
x : ℂ
hx : x ∈ s
⊢ z - x ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.Polynomial.GaussLucas

Goal state before:
pp: P : ℂ[X]
z : ℂ
hP : 0 < P.degree
hz : eval z (derivative P) = 0
weight : ℂ → ℝ := P.derivRootWeight z
s : Finset ℂ := P.roots.toFinset
hzP : ¬eval z P = 0
⊢ ∑ x ∈ s, weight x • (z - x) = (starRingEnd ℂ) (∑ x ∈ s, rootMultiplicity x P • (1 / (z - x)))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.Polynomial.GaussLucas

Goal state before:
pp: P : ℂ[X]
z : ℂ
hP : 0 < P.degree
hz : eval z (derivative P) = 0
weight : ℂ → ℝ := P.derivRootWeight z
s : Finset ℂ := P.roots.toFinset
hzP : ¬eval z P = 0
⊢ (starRingEnd ℂ) (∑ x ∈ s, rootMultiplicity x P • (1 / (z - x))) =
    (starRingEnd ℂ) (Multiset.map (fun x ↦ 1 / (z - x)) P.roots).sum

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.Polynomial.GaussLucas

Goal state before:
pp: P : ℂ[X]
z : ℂ
hP : 0 < P.degree
hz : eval z (derivative P) = 0
weight : ℂ → ℝ := P.derivRootWeight z
s : Finset ℂ := P.roots.toFinset
hzP : ¬eval z P = 0
⊢ (starRingEnd ℂ) (Multiset.map (fun x ↦ 1 / (z - x)) P.roots).sum = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.Polynomial.GaussLucas

Goal state before:
pp: P : ℂ[X]
z : ℂ
hP : 0 < P.degree
hz : eval z (derivative P) = 0
⊢ z = P.roots.toFinset.centerMass (P.derivRootWeight z) id

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
n : ℕ
C : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a
h : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)
hu : ∀ x ∈ [[a, b]], HasDerivAt (fun y ↦ cos y ^ (n + 1)) (-↑(n + 1) * sin x * cos x ^ n) x
hv : ∀ x ∈ [[a, b]], HasDerivAt sin (cos x) x
H :
  ∫ (x : ℝ) in a..b, cos x ^ (n + 1) * cos x =
    cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a - ∫ (x : ℝ) in a..b, -↑(n + 1) * sin x * cos x ^ n * sin x
⊢ C + (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ n - cos x ^ (n + 2) =
    (C + (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ n) - (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ (n + 2)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: ρ w : ℂ
R : ℝ
hR : 0 < R
hρ : ‖ρ‖ = R
hw : ‖w‖ < R
r : ℕ → ℝ
hr_lt : ∀ (n : ℕ), r n < R
hr_tendsto : Tendsto r atTop (𝓝 R)
bound : ℝ → ℝ :=
  fun θ ↦ (R + ‖w‖) / ((R + ‖w‖) / 2 - ‖w‖) * (|log (2 * R)| + |log √((R + ‖w‖) / 2 / R)| + |log ‖circleMap 0 R θ - ρ‖|)
n : ℕ
hn : (R + ‖w‖) / 2 ≤ r n
θ : ℝ
h : ¬‖circleMap 0 R θ - ρ‖ = 0
⊢ 0 < (R + ‖w‖) / 2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.Polynomial.GaussLucas

Goal state before:
pp: P : ℂ[X]
h₀ : 0 < P.degree
⊢ (derivative P).rootSet ℂ ⊆ (convexHull ℝ) (P.rootSet ℂ)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.Periodic

Goal state before:
pp: h : ℝ
f g : ℂ → ℂ
hfcts : ContinuousAt (cuspFunction h f) 0
hgcts : ContinuousAt (cuspFunction h g) 0
⊢ cuspFunction h (f + g) = cuspFunction h f + cuspFunction h g

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: ρ w : ℂ
R : ℝ
hR : 0 < R
hρ : ‖ρ‖ = R
hw : ‖w‖ < R
r : ℕ → ℝ
hr_lt : ∀ (n : ℕ), r n < R
hr_tendsto : Tendsto r atTop (𝓝 R)
bound : ℝ → ℝ :=
  fun θ ↦ (R + ‖w‖) / ((R + ‖w‖) / 2 - ‖w‖) * (|log (2 * R)| + |log √((R + ‖w‖) / 2 / R)| + |log ‖circleMap 0 R θ - ρ‖|)
n : ℕ
hn : (R + ‖w‖) / 2 ≤ r n
θ : ℝ
h : ¬‖circleMap 0 R θ - ρ‖ = 0
⊢ ‖w‖ < (R + ‖w‖) / 2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.Periodic

Goal state before:
pp: h : ℝ
f g : ℂ → ℂ
hfcts : ContinuousAt (cuspFunction h f) 0
hgcts : ContinuousAt (cuspFunction h g) 0
⊢ ContinuousAt (cuspFunction h (-g)) 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: ρ w : ℂ
R : ℝ
hR : 0 < R
hρ : ‖ρ‖ = R
hw : ‖w‖ < R
r : ℕ → ℝ
hr_lt : ∀ (n : ℕ), r n < R
hr_tendsto : Tendsto r atTop (𝓝 R)
bound : ℝ → ℝ :=
  fun θ ↦ (R + ‖w‖) / ((R + ‖w‖) / 2 - ‖w‖) * (|log (2 * R)| + |log √((R + ‖w‖) / 2 / R)| + |log ‖circleMap 0 R θ - ρ‖|)
n : ℕ
hn : (R + ‖w‖) / 2 ≤ r n
θ : ℝ
⊢ ‖herglotzLogIntegrand w ρ (circleMap 0 (r n) θ)‖ ≤ bound θ ∨ ‖circleMap 0 R θ - ρ‖ = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.Periodic

Goal state before:
pp: h : ℝ
f g : ℂ → ℂ
hfcts : ContinuousAt (cuspFunction h f) 0
hgcts : ContinuousAt (cuspFunction h g) 0
⊢ cuspFunction h (f - g) = cuspFunction h f - cuspFunction h g

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.Polynomial.UnitTrinomial

Goal state before:
pp: p : ℤ[X]
hp : p.IsUnitTrinomial
h : ∀ (z : ℂ), ¬((aeval z) p = 0 ∧ (aeval z) p.mirror = 0)
q : ℤ[X]
hq : q ∣ p
hq' : q ∣ p.mirror
this : ¬0 < q.natDegree
⊢ IsUnit q

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
n : ℕ
⊢ ∫ (x : ℝ) in a..b, cos x ^ (n + 2) =
    (cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a + (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ n) -
      (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ (n + 2)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.TaylorSeries

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
hf : DifferentiableOn ℂ f (Metric.ball c r)
z : ℂ
hz : z ∈ Metric.ball c r
⊢ ∃ r' < r, 0 < r' ∧ z ∈ Metric.ball c r'

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.TaylorSeries

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
hf : DifferentiableOn ℂ f (Metric.ball c r)
z : ℂ
hz : z ∈ Metric.ball c r
r' : NNReal
hr' : ↑r' < r
hr'₀ : 0 < ↑r'
hzr' : z ∈ Metric.ball c ↑r'
⊢ z - c ∈ Metric.eball 0 ↑r'

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: ρ w : ℂ
R : ℝ
hR : 0 < R
hρ : ‖ρ‖ = R
hw : ‖w‖ < R
r : ℕ → ℝ
hr_lt : ∀ (n : ℕ), r n < R
hr_tendsto : Tendsto r atTop (𝓝 R)
bound : ℝ → ℝ :=
  fun θ ↦ (R + ‖w‖) / ((R + ‖w‖) / 2 - ‖w‖) * (|log (2 * R)| + |log √((R + ‖w‖) / 2 / R)| + |log ‖circleMap 0 R θ - ρ‖|)
n : ℕ
hn : (R + ‖w‖) / 2 ≤ r n
h_bound : ∀ {θ : ℝ}, ‖herglotzLogIntegrand w ρ (circleMap 0 (r n) θ)‖ ≤ bound θ ∨ ‖circleMap 0 R θ - ρ‖ = 0
⊢ {x | (fun x ↦ x ∈ uIoc 0 (2 * π) → ‖herglotzLogIntegrand w ρ (circleMap 0 (r n) x)‖ ≤ bound x) x}ᶜ ⊆
    {θ | ‖circleMap 0 R θ - ρ‖ = 0}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
n : ℕ
⊢ ∫ (x : ℝ) in a..b, cos x ^ (n + 2) =
    (cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a) / (↑n + 2) + (↑n + 1) / (↑n + 2) * ∫ (x : ℝ) in a..b, cos x ^ n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
⊢ ∫ (x : ℝ) in a..b, cos x ^ 2 = (cos b * sin b - cos a * sin a + b - a) / 2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
m n : ℕ
⊢ Continuous fun u ↦ u ^ m * (1 - u ^ 2) ^ n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.TaylorSeries

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
hf : DifferentiableOn ℂ f (Metric.ball c r)
z : ℂ
hz : z ∈ Metric.ball c r
⊢ HasSum (fun n ↦ (↑n !)⁻¹ • (z - c) ^ n • iteratedDeriv n f c) (f z)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
m n : ℕ
hc : Continuous fun u ↦ u ^ m * (1 - u ^ 2) ^ n
⊢ ∫ (x : ℝ) in a..b, sin x ^ m * cos x ^ (2 * n + 1) = ∫ (x : ℝ) in a..b, sin x ^ m * (1 - sin x ^ 2) ^ n * cos x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.Polynomial.UnitTrinomial

Goal state before:
pp: p : ℤ[X]
hp : p.IsUnitTrinomial
h : ∀ (z : ℂ), ¬((aeval z) p = 0 ∧ (aeval z) p.mirror = 0)
⊢ Irreducible p

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
⊢ ∫ (x : ℝ) in a..b, sin x * cos x = (sin b ^ 2 - sin a ^ 2) / 2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: ρ w : ℂ
R : ℝ
hR : 0 < R
hρ : ‖ρ‖ = R
hw : ‖w‖ < R
r : ℕ → ℝ
hr_lt : ∀ (n : ℕ), r n < R
hr_tendsto : Tendsto r atTop (𝓝 R)
bound : ℝ → ℝ :=
  fun θ ↦ (R + ‖w‖) / ((R + ‖w‖) / 2 - ‖w‖) * (|log (2 * R)| + |log √((R + ‖w‖) / 2 / R)| + |log ‖circleMap 0 R θ - ρ‖|)
⊢ IntervalIntegrable (fun x ↦ |log (2 * R)|) volume 0 (2 * π)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.TaylorSeries

Goal state before:
pp: c : ℂ
r : ℝ
z : ℂ
hz : z ∈ Metric.ball c r
f : ℂ → ℂ
hf : DifferentiableOn ℂ f (Metric.ball c r)
⊢ ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f c * (z - c) ^ n = f z

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
⊢ ∫ (x : ℝ) in a..b, sin x ^ 2 * cos x = (sin b ^ 3 - sin a ^ 3) / 3

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.Positivity

Goal state before:
pp: f : ℂ → ℂ
c : ℂ
r : ℝ
hf : DifferentiableOn ℂ f (Metric.ball c r)
h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f c
z : ℂ
hz₁ : c ≤ z
hz₂ : z ∈ Metric.ball c r
⊢ 0 ≤ f z

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: ρ w : ℂ
R : ℝ
hR : 0 < R
hρ : ‖ρ‖ = R
hw : ‖w‖ < R
r : ℕ → ℝ
hr_lt : ∀ (n : ℕ), r n < R
hr_tendsto : Tendsto r atTop (𝓝 R)
bound : ℝ → ℝ :=
  fun θ ↦ (R + ‖w‖) / ((R + ‖w‖) / 2 - ‖w‖) * (|log (2 * R)| + |log √((R + ‖w‖) / 2 / R)| + |log ‖circleMap 0 R θ - ρ‖|)
⊢ IntervalIntegrable (fun x ↦ |log √((R + ‖w‖) / 2 / R)|) volume 0 (2 * π)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
⊢ ∫ (x : ℝ) in a..b, cos x ^ 3 = sin b - sin a - (sin b ^ 3 - sin a ^ 3) / 3

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
m n : ℕ
⊢ Continuous fun u ↦ u ^ n * (1 - u ^ 2) ^ m

Tactic:
by

Goal state after: