mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: c : ℂ
hc : c ≠ 0
x : ℝ
hx : x ≠ 0
⊢ deriv (fun x ↦ ↑x ^ c) x = c * ↑x ^ (c - 1)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: c : ℂ
hc : c ≠ 0
⊢ (deriv fun x ↦ ↑x ^ c) =ᶠ[atTop] fun x ↦ c * ↑x ^ (c - 1)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: c : ℂ
hc : c ≠ 0
⊢ (fun x ↦ ‖↑x ^ (c - 1)‖) =ᶠ[atTop] fun x ↦ x ^ (c.re - 1)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: c : ℂ
hc : c ≠ 0
⊢ (deriv fun x ↦ ↑x ^ c) =Θ[atTop] fun x ↦ x ^ (c.re - 1)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: c : ℂ
⊢ (deriv fun x ↦ ↑x ^ c) =O[atTop] fun x ↦ x ^ (c.re - 1)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: R : Type u_1
inst✝ : Ring R
r s : R
h : Commute r s
k : ℕ
ih :
  (descPochhammer ℤ k).smeval (r + s) =
    ∑ ij ∈ antidiagonal k, ↑(k.choose ij.1) * ((descPochhammer ℤ ij.1).smeval r * (descPochhammer ℤ ij.2).smeval s)
ij : ℕ × ℕ
hij : ij ∈ antidiagonal k
hdx : (descPochhammer ℤ ij.1).smeval r * (X - ↑ij.2).smeval s = (X - ↑ij.2).smeval s * (descPochhammer ℤ ij.1).smeval r
⊢ (r + s - k • 1) * ↑(k.choose ij.1) = ↑(k.choose ij.1) * (X - ↑ij.2).smeval s + ↑(k.choose ij.2) * (X - ↑ij.1).smeval r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: a : ℂ
⊢ HasFPowerSeriesOnBall (fun x ↦ (1 + x) ^ a) (binomialSeries ℂ a) 0 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: p : ℝ × ℝ
hp : 0 < p.1
⊢ HasStrictFDerivAt (fun x ↦ x.1 ^ x.2)
    ((p.2 * p.1 ^ (p.2 - 1)) • ContinuousLinearMap.fst ℝ ℝ ℝ + (p.1 ^ p.2 * log p.1) • ContinuousLinearMap.snd ℝ ℝ ℝ) p

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: R : Type u_1
inst✝ : Ring R
r s : R
k : ℕ
h : Commute r s
⊢ (descPochhammer ℤ k).smeval (r + s) =
    ∑ ij ∈ antidiagonal k, ↑(k.choose ij.1) * ((descPochhammer ℤ ij.1).smeval r * (descPochhammer ℤ ij.2).smeval s)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: a : ℂ
⊢ (binomialSeries ℂ (-a)).compContinuousLinearMap (-1) =
    FormalMultilinearSeries.ofScalars ℂ fun n ↦ Ring.choose (a + ↑n - 1) n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: a : ℂ
H :
  (binomialSeries ℂ (-a)).compContinuousLinearMap (-1) =
    FormalMultilinearSeries.ofScalars ℂ fun n ↦ Ring.choose (a + ↑n - 1) n
⊢ HasFPowerSeriesOnBall (fun x ↦ (1 + x) ^ (-a)) (binomialSeries ℂ (-a)) (-0) 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: a : ℂ
⊢ HasFPowerSeriesOnBall (fun x ↦ 1 / (1 - x) ^ a)
    (FormalMultilinearSeries.ofScalars ℂ fun n ↦ Ring.choose (a + ↑n - 1) n) 0 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: a : ℕ
⊢ HasFPowerSeriesOnBall (fun x ↦ 1 / (1 - x) ^ (a + 1))
    (FormalMultilinearSeries.ofScalars ℂ fun n ↦ ↑((a + n).choose a)) 0 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.IteratedFDeriv

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : AnalyticOn 𝕜 f s
k : ℕ
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
⊢ HasFPowerSeriesWithinOnBall (iteratedFDerivWithin 𝕜 k f s) (p.iteratedFDerivSeries k) s x r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: a : ℕ
z : ℂ
hz : z ≠ 0
this✝ :
  HasFPowerSeriesOnBall (fun x ↦ 1 / (1 - x) ^ (a + 1))
    (FormalMultilinearSeries.ofScalars ℂ fun n ↦ ↑((a + n).choose a)) ((z⁻¹ • 1) 0) 1
this :
  HasFPowerSeriesOnBall ((fun x ↦ 1 / (1 - x) ^ (a + 1)) ∘ ⇑(z⁻¹ • 1))
    ((FormalMultilinearSeries.ofScalars ℂ fun n ↦ ↑((a + n).choose a)).compContinuousLinearMap (z⁻¹ • 1)) 0
    (1 / ‖z⁻¹ • 1‖ₑ)
⊢ 1 / ‖z⁻¹ • 1‖ₑ = ‖z‖ₑ

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: R : Type u_1
inst✝¹ : Ring R
inst✝ : BinomialRing R
r s : R
k : ℕ
h : Commute r s
⊢ choose (r + s) k = ∑ ij ∈ antidiagonal k, choose r ij.1 * choose s ij.2

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: R : Type u_2
S : Type u_3
F : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : Ring S
inst✝³ : BinomialRing R
inst✝² : BinomialRing S
inst✝¹ : FunLike F R S
inst✝ : RingHomClass F R S
f : F
a : R
n : ℕ
⊢ f (choose a n) = choose (f a) n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.IteratedFDeriv

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
k n : ℕ
h : p (n + k) = 0
⊢ p.iteratedFDerivSeries k n = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.IteratedFDeriv

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : AnalyticOn 𝕜 f s
hu : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
hn : p n = 0
⊢ iteratedFDerivWithin 𝕜 n f s x = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Polynomial

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
A : Type u_3
B : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : CommSemiring A
z : E
s : Set E
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : Algebra A B
f : E → B
hf : AnalyticWithinAt 𝕜 f s z
p : A[X]
⊢ AnalyticWithinAt 𝕜 (fun x ↦ (aeval (f x)) p) s z

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: p : ℝ × ℝ
hp : p.1 < 0
⊢ HasStrictFDerivAt (fun x ↦ x.1 ^ x.2)
    ((p.2 * p.1 ^ (p.2 - 1)) • ContinuousLinearMap.fst ℝ ℝ ℝ +
      (p.1 ^ p.2 * log p.1 - rexp (log p.1 * p.2) * sin (p.2 * π) * π) • ContinuousLinearMap.snd ℝ ℝ ℝ)
    p

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Polynomial

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
A : Type u_3
B : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : CommSemiring A
z : E
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : Algebra A B
f : E → B
hf : AnalyticAt 𝕜 f z
p : A[X]
⊢ AnalyticAt 𝕜 (fun x ↦ (aeval (f x)) p) z

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: p : ℝ × ℝ
hp : p.1 ≠ 0
n : WithTop ℕ∞
⊢ ContDiffAt ℝ n (fun p ↦ p.1 ^ p.2) p

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.IteratedFDeriv

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : E [×n]→L[𝕜] F
x : E
v : Fin n → E
⊢ (iteratedFDeriv 𝕜 n (fun x ↦ f fun x_1 ↦ x) x) v = ∑ σ, f fun i ↦ v (σ i)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.IteratedFDeriv

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
v : Fin n → E
h's : s ⊆ Metric.eball x r
⊢ insert x s ∩ Metric.eball x r = s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.IteratedFDeriv

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
v : Fin n → E
h's : s ⊆ Metric.eball x r
I : insert x s ∩ Metric.eball x r = s
⊢ ContDiffOn 𝕜 (↑n) f s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: x : ℝ
f g : ℝ → ℝ
f' g' : ℝ
hf : HasStrictDerivAt f f' x
hg : HasStrictDerivAt g g' x
h : 0 < f x
⊢ HasStrictDerivAt (fun x ↦ f x ^ g x) (f' * g x * f x ^ (g x - 1) + g' * f x ^ g x * log (f x)) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.IteratedFDeriv

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
v : Fin n → E
h's : s ⊆ Metric.eball x r
I : insert x s ∩ Metric.eball x r = s
fcont : ContDiffOn 𝕜 (↑n) f s
g : E → F := fun z ↦ (p n) fun x_1 ↦ z - x
⊢ ContDiff 𝕜 ω g

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Polynomial

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
A : Type u_3
B : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : CommSemiring A
z : E
inst✝² : NormedCommRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : Algebra A B
σ : Type u_5
f : E → σ → B
hf : ∀ (i : σ), AnalyticAt 𝕜 (fun x ↦ f x i) z
p : MvPolynomial σ A
⊢ AnalyticAt 𝕜 (fun x ↦ (aeval (f x)) p) z

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: a : ℕ
z : ℂ
hz : z ≠ 0
⊢ HasFPowerSeriesOnBall (fun x ↦ 1 / (z - x) ^ (a + 1))
    (FormalMultilinearSeries.ofScalars ℂ fun n ↦ (z ^ (n + a + 1))⁻¹ * ↑((a + n).choose a)) 0 ‖z‖ₑ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: z : ℂ
hz : z ≠ 0
⊢ HasFPowerSeriesOnBall (fun x ↦ 1 / (z - x)) (FormalMultilinearSeries.ofScalars ℂ fun n ↦ (z ^ (n + 1))⁻¹) 0 ‖z‖ₑ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: z : ℂ
hz : z ≠ 0
⊢ HasFPowerSeriesOnBall (fun x ↦ 1 / (z - x) ^ 2)
    (FormalMultilinearSeries.ofScalars ℂ fun n ↦ (z ^ (n + 2))⁻¹ * (↑n + 1)) 0 ‖z‖ₑ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: ⊢ HasFPowerSeriesOnBall (fun x ↦ 1 / (1 - x)) (FormalMultilinearSeries.ofScalars ℂ 1) 0 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: ⊢ HasFPowerSeriesOnBall (fun x ↦ 1 / (1 - x) ^ 2) (FormalMultilinearSeries.ofScalars ℂ fun n ↦ ↑n + 1) 0 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: x : ℝ
hx : x ≠ 0
p : ℝ
⊢ HasStrictDerivAt (fun x ↦ x ^ p) (p * x ^ (p - 1)) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: a : ℝ
ha : 0 < a
x : ℝ
⊢ HasStrictDerivAt (fun x ↦ a ^ x) (a ^ x * log a) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: r : ℝ
hr : r < 1
hr' : r ≠ 0
h : DifferentiableAt ℝ (fun x ↦ x ^ r) 0
y : ℝ := deriv (fun x ↦ x ^ r) 0
⊢ Tendsto (fun t ↦ t ^ (r - 1)) (𝓝[>] 0) (𝓝 y)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: a b : ℂ
⊢ HasFPowerSeriesOnBall (fun x ↦ (b - a) / (1 - x) + a / (1 - x) ^ 2)
    (FormalMultilinearSeries.ofScalars ℂ fun n ↦ a * ↑n + b) 0 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: r : ℝ
hr : r < 1
hr' : r ≠ 0
h✝ : DifferentiableAt ℝ (fun x ↦ x ^ r) 0
y : ℝ := deriv (fun x ↦ x ^ r) 0
h : Tendsto (fun t ↦ t ^ (r - 1)) (𝓝[>] 0) (𝓝 y)
⊢ r - 1 < 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: r : ℝ
hr : r < 1
hr' : r ≠ 0
⊢ ¬DifferentiableAt ℝ (fun x ↦ x ^ r) 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: a x : ℝ
ha : a < 0
⊢ HasStrictDerivAt (fun x ↦ a ^ x) (a ^ x * log a - rexp (log a * x) * sin (x * π) * π) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: x p : ℝ
h : x ≠ 0 ∨ 1 ≤ p
⊢ HasDerivAt (fun x ↦ x ^ p) (p * x ^ (p - 1)) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: w x : ℂ
hw : w ≠ x
⊢ w - x ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: x p : ℝ
⊢ deriv (fun x ↦ x ^ p) x = p * x ^ (p - 1)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.IteratedFDeriv

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
v : Fin n → E
h's : s ⊆ Metric.eball x r
I : insert x s ∩ Metric.eball x r = s
fcont : ContDiffOn 𝕜 (↑n) f s
g : E → F := fun z ↦ (p n) fun x_1 ↦ z - x
gcont : ContDiff 𝕜 ω g
q : FormalMultilinearSeries 𝕜 E F := fun k ↦ if h : n = k then h ▸ p n else 0
m : ℕ
hm : n + 1 ≤ m
⊢ n ≠ m

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: p : ℝ
h : ↑0 ≤ p
x : ℝ
⊢ 0 ≤ p

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: n : ℕ
ihn : ∀ {p : ℝ}, ↑n ≤ p → ContDiff ℝ ↑n fun x ↦ x ^ p
p : ℝ
h : ↑(n + 1) ≤ p
⊢ 1 ≤ ↑(n + 1)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.IteratedFDeriv

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
v : Fin n → E
h's : s ⊆ Metric.eball x r
I : insert x s ∩ Metric.eball x r = s
fcont : ContDiffOn 𝕜 (↑n) f s
g : E → F := fun z ↦ (p n) fun x_1 ↦ z - x
gcont : ContDiff 𝕜 ω g
q : FormalMultilinearSeries 𝕜 E F := fun k ↦ if h : n = k then h ▸ p n else 0
⊢ HasFiniteFPowerSeriesOnBall g q x (n + 1) r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.IteratedFDeriv

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
v : Fin n → E
h's : s ⊆ Metric.eball x r
I : insert x s ∩ Metric.eball x r = s
fcont : ContDiffOn 𝕜 (↑n) f s
g : E → F := fun z ↦ (p n) fun x_1 ↦ z - x
gcont : ContDiff 𝕜 ω g
q : FormalMultilinearSeries 𝕜 E F := fun k ↦ if h : n = k then h ▸ p n else 0
A : HasFiniteFPowerSeriesOnBall g q x (n + 1) r
B : HasFPowerSeriesWithinOnBall g q s x r
⊢ f = g + (f - g)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: w x : ℂ
hw : w ≠ x
⊢ HasFPowerSeriesOnBall (fun z ↦ 1 / (z - w) ^ 2 - 1 / w ^ 2)
    (FormalMultilinearSeries.ofScalars ℂ fun i ↦ (↑i + 1) * (w - x) ^ (-↑(i + 2)) - Nat.casesOn i (w ^ (-2)) 0) x
    ‖w - x‖ₑ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.RadiusLiminf

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
⊢ ∀ (r : ℝ≥0) {n : ℕ}, 0 < n → (↑r ≤ 1 / ↑(‖p n‖₊ ^ (1 / ↑n)) ↔ ‖p n‖₊ * r ^ n ≤ 1)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: a : ℝ
⊢ binomialSeries ℂ a = FormalMultilinearSeries.restrictScalars ℝ (binomialSeries ℂ ↑a)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: p : ℝ
n : ℕ
h : ↑n ≤ p
⊢ ContDiff ℝ ↑n fun x ↦ x ^ p

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.IteratedFDeriv

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
v : Fin n → E
h's : s ⊆ Metric.eball x r
I : insert x s ∩ Metric.eball x r = s
fcont : ContDiffOn 𝕜 (↑n) f s
g : E → F := fun z ↦ (p n) fun x_1 ↦ z - x
gcont : ContDiff 𝕜 ω g
q : FormalMultilinearSeries 𝕜 E F := fun k ↦ if h : n = k then h ▸ p n else 0
A : HasFiniteFPowerSeriesOnBall g q x (n + 1) r
B : HasFPowerSeriesWithinOnBall g q s x r
this : f = g + (f - g)
⊢ ContDiffWithinAt 𝕜 (↑n) (f - g) s x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: a : ℝ
⊢ HasFPowerSeriesOnBall (fun x ↦ (1 + x) ^ a) (binomialSeries ℝ a) 0 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.IteratedFDeriv

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
v : Fin n → E
h's : s ⊆ Metric.eball x r
I : insert x s ∩ Metric.eball x r = s
fcont : ContDiffOn 𝕜 (↑n) f s
g : E → F := fun z ↦ (p n) fun x_1 ↦ z - x
gcont : ContDiff 𝕜 ω g
q : FormalMultilinearSeries 𝕜 E F := fun k ↦ if h : n = k then h ▸ p n else 0
A : HasFiniteFPowerSeriesOnBall g q x (n + 1) r
B : HasFPowerSeriesWithinOnBall g q s x r
⊢ iteratedFDerivWithin 𝕜 n f s x = iteratedFDerivWithin 𝕜 n g s x + iteratedFDerivWithin 𝕜 n (f - g) s x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: r x : ℝ
k : ℕ
⊢ deriv^[k] (fun x ↦ x ^ r) x = Polynomial.eval r (descPochhammer ℝ k) * x ^ (r - ↑k)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: x p : ℝ
hx : x ≠ 0 ∨ 1 ≤ p
⊢ x ≠ 0 ∨ ↑1 ≤ p

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.IteratedFDeriv

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
v : Fin n → E
h's : s ⊆ Metric.eball x r
I : insert x s ∩ Metric.eball x r = s
fcont : ContDiffOn 𝕜 (↑n) f s
g : E → F := fun z ↦ (p n) fun x_1 ↦ z - x
gcont : ContDiff 𝕜 ω g
q : FormalMultilinearSeries 𝕜 E F := fun k ↦ if h : n = k then h ▸ p n else 0
A : HasFiniteFPowerSeriesOnBall g q x (n + 1) r
B : HasFPowerSeriesWithinOnBall g q s x r
⊢ (iteratedFDerivWithin 𝕜 n f s x) v = ∑ σ, (p n) fun i ↦ v (σ i)

Tactic:
have J1 : iteratedFDerivWithin 𝕜 n f s x =
      iteratedFDerivWithin 𝕜 n g s x + iteratedFDerivWithin 𝕜 n (f - g) s x := by
    have : f = g + (f - g) := by abel
    nth_rewrite 1 [this]
    rw [iteratedFDerivWithin_add_apply (gcont.of_le le_top).contDiffWithinAt
      (by exact (fcont _ hx).sub (gcont.of_le le_top).contDiffWithinAt) hs hx]

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
v : Fin n → E
h's : s ⊆ Metric.eball x r
I : insert x s ∩ Metric.eball x r = s
fcont : ContDiffOn 𝕜 (↑n) f s
g : E → F := fun z ↦ (p n) fun x_1 ↦ z - x
gcont : ContDiff 𝕜 ω g
q : FormalMultilinearSeries 𝕜 E F := fun k ↦ if h : n = k then h ▸ p n else 0
A : HasFiniteFPowerSeriesOnBall g q x (n + 1) r
B : HasFPowerSeriesWithinOnBall g q s x r
J1 : iteratedFDerivWithin 𝕜 n f s x = iteratedFDerivWithin 𝕜 n g s x + iteratedFDerivWithin 𝕜 n (f - g) s x
⊢ (iteratedFDerivWithin 𝕜 n f s x) v = ∑ σ, (p n) fun i ↦ v (σ i)
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : E → ℝ
f' g' : StrongDual ℝ E
x : E
s : Set E
hf : HasFDerivWithinAt f f' s x
hg : HasFDerivWithinAt g g' s x
h : 0 < f x
⊢ HasFDerivWithinAt (fun x ↦ f x ^ g x) ((g x * f x ^ (g x - 1)) • f' + (f x ^ g x * Real.log (f x)) • g') s x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : E → ℝ
f' g' : StrongDual ℝ E
x : E
hf : HasFDerivAt f f' x
hg : HasFDerivAt g g' x
h : 0 < f x
⊢ HasFDerivAt (fun x ↦ f x ^ g x) ((g x * f x ^ (g x - 1)) • f' + (f x ^ g x * Real.log (f x)) • g') x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : E → ℝ
x : E
s : Set E
hf : DifferentiableWithinAt ℝ f s x
hg : DifferentiableWithinAt ℝ g s x
h : f x ≠ 0
⊢ DifferentiableWithinAt ℝ (fun x ↦ f x ^ g x) s x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : E → ℝ
x : E
hf : DifferentiableAt ℝ f x
hg : DifferentiableAt ℝ g x
h : f x ≠ 0
⊢ DifferentiableAt ℝ (fun x ↦ f x ^ g x) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: a : ℝ
this :
  HasFPowerSeriesOnBall (fun x ↦ 1 / (1 - x) ^ ↑a)
    (FormalMultilinearSeries.restrictScalars ℝ
      (FormalMultilinearSeries.ofScalars ℂ fun n ↦ Ring.choose (↑a + ↑n - 1) n))
    (Complex.ofRealCLM 0) 1
x : ℝ
hx : x ∈ Metric.eball 0 (1 / ‖Complex.ofRealCLM‖ₑ)
⊢ |x| < 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: a : ℝ
this✝ :
  HasFPowerSeriesOnBall (fun x ↦ 1 / (1 - x) ^ ↑a)
    (FormalMultilinearSeries.restrictScalars ℝ
      (FormalMultilinearSeries.ofScalars ℂ fun n ↦ Ring.choose (↑a + ↑n - 1) n))
    (Complex.ofRealCLM 0) 1
x : ℝ
hx : x ∈ Metric.eball 0 (1 / ‖Complex.ofRealCLM‖ₑ)
this : |x| < 1
⊢ 0 ≤ 1 - x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : E → ℝ
x : E
s : Set E
n : WithTop ℕ∞
hf : ContDiffWithinAt ℝ n f s x
hg : ContDiffWithinAt ℝ n g s x
h : f x ≠ 0
⊢ ContDiffWithinAt ℝ n (fun x ↦ f x ^ g x) s x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : E → ℝ
x : E
n : WithTop ℕ∞
hf : ContDiffAt ℝ n f x
hg : ContDiffAt ℝ n g x
h : f x ≠ 0
⊢ ContDiffAt ℝ n (fun x ↦ f x ^ g x) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: a : ℝ
⊢ HasFPowerSeriesOnBall (fun x ↦ 1 / (1 - x) ^ a)
    (FormalMultilinearSeries.ofScalars ℝ fun n ↦ Ring.choose (a + ↑n - 1) n) 0 1

Tactic:
by

Goal state after: