mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: case succ
a : ℂ
B : Set ℂ := Metric.ball 0 1
n : ℕ
ih :
  Set.EqOn (iteratedDerivWithin n (fun x ↦ (1 + x) ^ a) B) (fun x ↦ (descPochhammer ℤ n).smeval a * (1 + x) ^ (a - ↑n))
    B
this✝ :
  iteratedDerivWithin (n + 1) (fun x ↦ (1 + x) ^ a) B = derivWithin (iteratedDerivWithin n (fun x ↦ (1 + x) ^ a) B) B
this :
  Set.EqOn (derivWithin (iteratedDerivWithin n (fun x ↦ (1 + x) ^ a) B) B)
    (derivWithin (fun x ↦ (descPochhammer ℤ n).smeval a * (1 + x) ^ (a - ↑n)) B) B
z : ℂ
hz : z ∈ B
⊢ (descPochhammer ℤ n).smeval a * (a - ↑n) * (1 + z) ^ (a - ↑n - 1) =
    (descPochhammer ℤ (n + 1)).smeval a * (1 + z) ^ (a - ↑n - 1)

pp: case succ
a : ℂ
B : Set ℂ := Metric.ball 0 1
n : ℕ
ih :
  Set.EqOn (iteratedDerivWithin n (fun x ↦ (1 + x) ^ a) B) (fun x ↦ (descPochhammer ℤ n).smeval a * (1 + x) ^ (a - ↑n))
    B
this✝ :
  iteratedDerivWithin (n + 1) (fun x ↦ (1 + x) ^ a) B = derivWithin (iteratedDerivWithin n (fun x ↦ (1 + x) ^ a) B) B
this :
  Set.EqOn (derivWithin (iteratedDerivWithin n (fun x ↦ (1 + x) ^ a) B) B)
    (derivWithin (fun x ↦ (descPochhammer ℤ n).smeval a * (1 + x) ^ (a - ↑n)) B) B
z : ℂ
hz : z ∈ B
⊢ 1 + z ∈ slitPlane

Tactic:
]

Goal state after:
case succ
a : ℂ
B : Set ℂ := Metric.ball 0 1
n : ℕ
ih :
  Set.EqOn (iteratedDerivWithin n (fun x ↦ (1 + x) ^ a) B) (fun x ↦ (descPochhammer ℤ n).smeval a * (1 + x) ^ (a - ↑n))
    B
this✝ :
  iteratedDerivWithin (n + 1) (fun x ↦ (1 + x) ^ a) B = derivWithin (iteratedDerivWithin n (fun x ↦ (1 + x) ^ a) B) B
this :
  Set.EqOn (derivWithin (iteratedDerivWithin n (fun x ↦ (1 + x) ^ a) B) B)
    (derivWithin (fun x ↦ (descPochhammer ℤ n).smeval a * (1 + x) ^ (a - ↑n)) B) B
z : ℂ
hz : z ∈ B
⊢ (descPochhammer ℤ n).smeval a * (a - ↑n) * (1 + z) ^ (a - ↑n - 1) =
    (descPochhammer ℤ (n + 1)).smeval a * (1 + z) ^ (a - ↑n - 1)

case succ
a : ℂ
B : Set ℂ := Metric.ball 0 1
n : ℕ
ih :
  Set.EqOn (iteratedDerivWithin n (fun x ↦ (1 + x) ^ a) B) (fun x ↦ (descPochhammer ℤ n).smeval a * (1 + x) ^ (a - ↑n))
    B
this✝ :
  iteratedDerivWithin (n + 1) (fun x ↦ (1 + x) ^ a) B = derivWithin (iteratedDerivWithin n (fun x ↦ (1 + x) ^ a) B) B
this :
  Set.EqOn (derivWithin (iteratedDerivWithin n (fun x ↦ (1 + x) ^ a) B) B)
    (derivWithin (fun x ↦ (descPochhammer ℤ n).smeval a * (1 + x) ^ (a - ↑n)) B) B
z : ℂ
hz : z ∈ B
⊢ 1 + z ∈ slitPlane
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: c : ℂ
hc : c ≠ 0
x : ℝ
hx : 0 < x
⊢ x ^ (c.re - 1) = x ^ (c.re - 1)

Tactic:
]

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 - (ij.1 • 1 + ij.2 • 1)) * ↑((ij.1 + ij.2).choose ij.1) =
    (s - ij.2 • 1 + (r - ij.1 • 1)) * ↑((ij.1 + ij.2).choose ij.1)

Tactic:
]

Goal state after:
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 - (ij.1 • 1 + ij.2 • 1)) * ↑((ij.1 + ij.2).choose ij.1) =
    (s - ij.2 • 1 + (r - ij.1 • 1)) * ↑((ij.1 + ij.2).choose ij.1)
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: case succ
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
hl :
  (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
⊢ (↑(k.choose ij.1) * (X - ↑ij.2).smeval s + ↑(k.choose ij.2) * (X - ↑ij.1).smeval r) *
      ((descPochhammer ℤ ij.1).smeval r * (descPochhammer ℤ ij.2).smeval s) =
    (↑(k.choose ij.1) * (X - ↑ij.2).smeval s + ↑(k.choose ij.2) * (X - ↑ij.1).smeval r) *
      ((descPochhammer ℤ ij.1).smeval r * (descPochhammer ℤ ij.2).smeval s)

Tactic:
]

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

Goal state before:
pp: case e'_12
p : ℝ × ℝ
hp : 0 < p.1
this : (fun x ↦ x.1 ^ x.2) =ᶠ[𝓝 p] fun x ↦ rexp (log x.1 * x.2)
e_4✝ : Prod.instAddCommGroup = Prod.normedAddCommGroup.toAddCommGroup
e_5✝ : Prod.instModule ≍ Prod.normedSpace.toModule
e_6✝ : instTopologicalSpaceProd = PseudoMetricSpace.toUniformSpace.toTopologicalSpace
⊢ (p.1 ^ p.2 * log p.1) • ContinuousLinearMap.snd ℝ ℝ ℝ + (p.1 ^ p.2 * (p.2 * p.1⁻¹)) • ContinuousLinearMap.fst ℝ ℝ ℝ =
    (p.1 ^ p.2 * log p.1) • ContinuousLinearMap.snd ℝ ℝ ℝ + (p.1 ^ p.2 * (p.2 * p.1⁻¹)) • ContinuousLinearMap.fst ℝ ℝ ℝ

Tactic:
]

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
⊢ ∑ ij ∈ antidiagonal k, ↑(k.choose ij.1) * ((descPochhammer ℤ ij.1).smeval r * (descPochhammer ℤ ij.2).smeval s) =
    ∑ x ∈ antidiagonal k, k.factorial • (choose r x.1 * choose s x.2)

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : Ring R
inst✝ : BinomialRing R
r s : R
k : ℕ
h : Commute r s
⊢ ∑ ij ∈ antidiagonal k, ↑(k.choose ij.1) * ((descPochhammer ℤ ij.1).smeval r * (descPochhammer ℤ ij.2).smeval s) =
    ∑ x ∈ antidiagonal k, k.factorial • (choose r x.1 * choose s x.2)
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: case e'_12
p : ℝ × ℝ
hp : p.1 < 0
this : (fun x ↦ x.1 ^ x.2) =ᶠ[𝓝 p] fun x ↦ rexp (log x.1 * x.2) * cos (x.2 * π)
e_4✝ : Prod.instAddCommGroup = Prod.normedAddCommGroup.toAddCommGroup
e_5✝ : Prod.instModule ≍ Prod.normedSpace.toModule
e_6✝ : instTopologicalSpaceProd = PseudoMetricSpace.toUniformSpace.toTopologicalSpace
e_8✝ : instAddCommGroup = normedCommRing.toCommRing.toAddCommGroup
e_9✝ : Semiring.toModule ≍ (NormedAlgebra.toNormedSpace ℝ).toModule
⊢ (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 =
    rexp (log p.1 * p.2) • -sin (p.2 * π) • π • ContinuousLinearMap.snd ℝ ℝ ℝ +
      cos (p.2 * π) •
        rexp (log p.1 * p.2) • (log p.1 • ContinuousLinearMap.snd ℝ ℝ ℝ + p.2 • p.1⁻¹ • ContinuousLinearMap.fst ℝ ℝ ℝ)

Tactic:
rpow_sub_one hp.ne,

Goal state after:
case e'_12
p : ℝ × ℝ
hp : p.1 < 0
this : (fun x ↦ x.1 ^ x.2) =ᶠ[𝓝 p] fun x ↦ rexp (log x.1 * x.2) * cos (x.2 * π)
e_4✝ : Prod.instAddCommGroup = Prod.normedAddCommGroup.toAddCommGroup
e_5✝ : Prod.instModule ≍ Prod.normedSpace.toModule
e_6✝ : instTopologicalSpaceProd = PseudoMetricSpace.toUniformSpace.toTopologicalSpace
e_8✝ : instAddCommGroup = normedCommRing.toCommRing.toAddCommGroup
e_9✝ : Semiring.toModule ≍ (NormedAlgebra.toNormedSpace ℝ).toModule
⊢ (p.2 * (p.1 ^ p.2 / p.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 =
    rexp (log p.1 * p.2) • -sin (p.2 * π) • π • ContinuousLinearMap.snd ℝ ℝ ℝ +
      cos (p.2 * π) •
        rexp (log p.1 * p.2) • (log p.1 • ContinuousLinearMap.snd ℝ ℝ ℝ + p.2 • p.1⁻¹ • ContinuousLinearMap.fst ℝ ℝ ℝ)
Module: Mathlib.Analysis.Analytic.IteratedFDeriv

Goal state before:
pp: case succ
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
k : ℕ
ih : HasFPowerSeriesWithinOnBall (iteratedFDerivWithin 𝕜 k f s) (p.iteratedFDerivSeries k) s x r
⊢ HasFPowerSeriesWithinOnBall
    (⇑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x ↦ E) F).symm ∘ fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 k f s) s)
    (p.iteratedFDerivSeries (k + 1)) s x r

Tactic:
]

Goal state after:
case succ
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
k : ℕ
ih : HasFPowerSeriesWithinOnBall (iteratedFDerivWithin 𝕜 k f s) (p.iteratedFDerivSeries k) s x r
⊢ HasFPowerSeriesWithinOnBall
    (⇑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x ↦ E) F).symm ∘ fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 k f s) s)
    (p.iteratedFDerivSeries (k + 1)) s x r
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: case e'_10
a n : ℕ
⊢ ↑((a + n).choose n) = ↑((a + n).choose n)

Tactic:
]

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
⊢ 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:
]

Goal state after:
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
⊢ 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
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
x : ℕ × ℕ
hx : x ∈ antidiagonal k
⊢ ↑(k.choose x.1) * ((descPochhammer ℤ x.1).smeval r * (descPochhammer ℤ x.2).smeval s) =
    ↑(k.choose x.1) * ↑x.1.factorial * (choose r x.1 * x.2.factorial • choose s x.2)

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : Ring R
inst✝ : BinomialRing R
r s : R
k : ℕ
h : Commute r s
x : ℕ × ℕ
hx : x ∈ antidiagonal k
⊢ ↑(k.choose x.1) * ((descPochhammer ℤ x.1).smeval r * (descPochhammer ℤ x.2).smeval s) =
    ↑(k.choose x.1) * ↑x.1.factorial * (choose r x.1 * x.2.factorial • choose s x.2)
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: case e'_12
p : ℝ × ℝ
hp : p.1 < 0
this : (fun x ↦ x.1 ^ x.2) =ᶠ[𝓝 p] fun x ↦ rexp (log x.1 * x.2) * cos (x.2 * π)
e_4✝ : Prod.instAddCommGroup = Prod.normedAddCommGroup.toAddCommGroup
e_5✝ : Prod.instModule ≍ Prod.normedSpace.toModule
e_6✝ : instTopologicalSpaceProd = PseudoMetricSpace.toUniformSpace.toTopologicalSpace
e_8✝ : instAddCommGroup = normedCommRing.toCommRing.toAddCommGroup
e_9✝ : Semiring.toModule ≍ (NormedAlgebra.toNormedSpace ℝ).toModule
⊢ (p.1 ^ p.2 * log p.1 - rexp (log p.1 * p.2) * sin (p.2 * π) * π) • ContinuousLinearMap.snd ℝ ℝ ℝ +
      (p.2 * (p.1 ^ p.2 * p.1⁻¹)) • ContinuousLinearMap.fst ℝ ℝ ℝ =
    (rexp (log p.1 * p.2) * -sin (p.2 * π) * π + p.1 ^ p.2 * log p.1) • ContinuousLinearMap.snd ℝ ℝ ℝ +
      (p.1 ^ p.2 * p.2 * p.1⁻¹) • ContinuousLinearMap.fst ℝ ℝ ℝ

Tactic:
]

Goal state after:
case e'_12
p : ℝ × ℝ
hp : p.1 < 0
this : (fun x ↦ x.1 ^ x.2) =ᶠ[𝓝 p] fun x ↦ rexp (log x.1 * x.2) * cos (x.2 * π)
e_4✝ : Prod.instAddCommGroup = Prod.normedAddCommGroup.toAddCommGroup
e_5✝ : Prod.instModule ≍ Prod.normedSpace.toModule
e_6✝ : instTopologicalSpaceProd = PseudoMetricSpace.toUniformSpace.toTopologicalSpace
e_8✝ : instAddCommGroup = normedCommRing.toCommRing.toAddCommGroup
e_9✝ : Semiring.toModule ≍ (NormedAlgebra.toNormedSpace ℝ).toModule
⊢ (p.1 ^ p.2 * log p.1 - rexp (log p.1 * p.2) * sin (p.2 * π) * π) • ContinuousLinearMap.snd ℝ ℝ ℝ +
      (p.2 * (p.1 ^ p.2 * p.1⁻¹)) • ContinuousLinearMap.fst ℝ ℝ ℝ =
    (rexp (log p.1 * p.2) * -sin (p.2 * π) * π + p.1 ^ p.2 * log p.1) • ContinuousLinearMap.snd ℝ ℝ ℝ +
      (p.1 ^ p.2 * p.2 * p.1⁻¹) • ContinuousLinearMap.fst ℝ ℝ ℝ
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
this : iteratedFDerivWithin 𝕜 n f s x = (p.iteratedFDerivSeries n 0) fun x ↦ 0
⊢ 0 = 0

Tactic:
]

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 = ∑ i, f fun i_1 ↦ v (((Equiv.inv (Perm (Fin n))) i) i_1)

Tactic:
]

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
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 = ∑ i, f fun i_1 ↦ v (((Equiv.inv (Perm (Fin n))) i) i_1)
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
g : E →L[𝕜] Fin n → E := ContinuousLinearMap.pi fun i ↦ ContinuousLinearMap.id 𝕜 E
⊢ ((f.iteratedFDeriv n (g x)).compContinuousLinearMap fun x ↦ g) v =
    ∑ i, f fun i_1 ↦ v (((Equiv.inv (Perm (Fin n))) i) i_1)

Tactic:
]

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
n : ℕ
f : E [×n]→L[𝕜] F
x : E
v : Fin n → E
g : E →L[𝕜] Fin n → E := ContinuousLinearMap.pi fun i ↦ ContinuousLinearMap.id 𝕜 E
⊢ ((f.iteratedFDeriv n (g x)).compContinuousLinearMap fun x ↦ g) v =
    ∑ i, f fun i_1 ↦ v (((Equiv.inv (Perm (Fin n))) i) i_1)
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 : AnalyticWithinAt 𝕜 f Set.univ z
p : A[X]
⊢ AnalyticWithinAt 𝕜 (fun x ↦ (aeval (f x)) p) Set.univ z

Tactic:
]

Goal state after:
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 : AnalyticWithinAt 𝕜 f Set.univ z
p : A[X]
⊢ AnalyticWithinAt 𝕜 (fun x ↦ (aeval (f x)) p) Set.univ z
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
g : E →L[𝕜] Fin n → E := ContinuousLinearMap.pi fun i ↦ ContinuousLinearMap.id 𝕜 E
⊢ (∑ x_1, f fun j ↦ if h : ∃ y, x_1 y = j then g (v (x_1.toEquivRange.symm ⟨j, ⋯⟩)) j else g x j) =
    ∑ i, f fun i_1 ↦ v (((Equiv.inv (Perm (Fin n))) ((embeddingEquivOfFinite (Fin n)) i)) i_1)

Tactic:
]

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
n : ℕ
f : E [×n]→L[𝕜] F
x : E
v : Fin n → E
g : E →L[𝕜] Fin n → E := ContinuousLinearMap.pi fun i ↦ ContinuousLinearMap.id 𝕜 E
⊢ (∑ x_1, f fun j ↦ if h : ∃ y, x_1 y = j then g (v (x_1.toEquivRange.symm ⟨j, ⋯⟩)) j else g x j) =
    ∑ i, f fun i_1 ↦ v (((Equiv.inv (Perm (Fin n))) ((embeddingEquivOfFinite (Fin n)) i)) i_1)
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
⊢ s ∩ Metric.eball x r = s

Tactic:
]

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
⊢ s ∩ Metric.eball x r = s
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
x : ℝ
hx : 0 < x
⊢ x ^ 1 * x ^ (r - 1) = x * x ^ (r - 1)

Tactic:
]

Goal state after:
r : ℝ
hr : r < 1
hr' : r ≠ 0
h : DifferentiableAt ℝ (fun x ↦ x ^ r) 0
y : ℝ := deriv (fun x ↦ x ^ r) 0
x : ℝ
hx : 0 < x
⊢ x ^ 1 * x ^ (r - 1) = x * x ^ (r - 1)
Module: Mathlib.Analysis.Analytic.Binomial

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

Tactic:
]

Goal state after:
w x : ℂ
hw : w ≠ x
⊢ HasFPowerSeriesOnBall ((fun z ↦ 1 / (z - w) ^ 2) - fun z ↦ 1 / w ^ 2)
    ((FormalMultilinearSeries.ofScalars ℂ fun i ↦ (↑i + 1) * (w - x) ^ (-↑(i + 2))) -
      FormalMultilinearSeries.ofScalars ℂ fun i ↦ Nat.casesOn i (w ^ (-2)) 0)
    x ‖w - x‖ₑ
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: case zero
p : ℝ
h : ↑0 ≤ p
⊢ ContDiff ℝ ↑0 fun x ↦ x ^ p

Tactic:
exact contDiff_zero.2 (continuous_id.rpow_const fun x => Or.inr <| by simpa using h)

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

Goal state before:
pp: case succ
n : ℕ
ihn : ∀ {p : ℝ}, ↑n ≤ p → ContDiff ℝ ↑n fun x ↦ x ^ p
p : ℝ
h : ↑n ≤ p - 1
h1 : 1 ≤ p
⊢ ContDiff ℝ ↑(n + 1) fun x ↦ x ^ p

Tactic:
]

Goal state after:
case succ
n : ℕ
ihn : ∀ {p : ℝ}, ↑n ≤ p → ContDiff ℝ ↑n fun x ↦ x ^ p
p : ℝ
h : ↑n ≤ p - 1
h1 : 1 ≤ p
⊢ ContDiff ℝ ↑(n + 1) fun x ↦ x ^ p
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: case succ
n : ℕ
ihn : ∀ {p : ℝ}, ↑n ≤ p → ContDiff ℝ ↑n fun x ↦ x ^ p
p : ℝ
h : ↑n ≤ p - 1
h1 : 1 ≤ p
⊢ (Differentiable ℝ fun x ↦ x ^ p) ∧
    (↑n = ⊤ → AnalyticOn ℝ (fun x ↦ x ^ p) Set.univ) ∧ ContDiff ℝ ↑n fun x ↦ p * x ^ (p - 1)

Tactic:
]

Goal state after:
case succ
n : ℕ
ihn : ∀ {p : ℝ}, ↑n ≤ p → ContDiff ℝ ↑n fun x ↦ x ^ p
p : ℝ
h : ↑n ≤ p - 1
h1 : 1 ≤ p
⊢ (Differentiable ℝ fun x ↦ x ^ p) ∧
    (↑n = ⊤ → AnalyticOn ℝ (fun x ↦ x ^ p) Set.univ) ∧ ContDiff ℝ ↑n fun x ↦ p * x ^ (p - 1)
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)
⊢ iteratedFDerivWithin 𝕜 n g s x + iteratedFDerivWithin 𝕜 n (f - g) s x =
    iteratedFDerivWithin 𝕜 n g s x + iteratedFDerivWithin 𝕜 n (f - g) s x

Tactic:
]

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

Tactic:
]

Goal state after:
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
⊢ HasFPowerSeriesOnBall (fun x ↦ 1 / (1 - x) ^ a)
    (FormalMultilinearSeries.ofScalars ℝ fun n ↦ Ring.choose (a + ↑n - 1) n) 0 1
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

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

Tactic:
]

Goal state after:
x p : ℝ
hx : x ≠ 0 ∨ ↑1 ≤ p
⊢ x ≠ 0 ∨ ↑1 ≤ p
Module: Mathlib.Analysis.Analytic.RadiusLiminf

Goal state before:
pp: case a.h
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
this : ∀ (r : ℝ≥0) {n : ℕ}, 0 < n → (↑r ≤ 1 / ↑(‖p n‖₊ ^ (1 / ↑n)) ↔ ‖p n‖₊ * r ^ n ≤ 1)
r : ℝ≥0
hr : ↑r < p.radius
a : ℝ
ha : a ∈ Set.Ioo 0 1
H : ∀ᶠ (n : ℕ) in atTop, |‖p n‖ * ↑r ^ n| ≤ a ^ n
⊢ ∀ᶠ (b : ℝ≥0∞) in map (fun n ↦ 1 / ↑(‖p n‖₊ ^ (1 / ↑n))) atTop, ↑r ≤ b

Tactic:
]

Goal state after:
case a.h
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
this : ∀ (r : ℝ≥0) {n : ℕ}, 0 < n → (↑r ≤ 1 / ↑(‖p n‖₊ ^ (1 / ↑n)) ↔ ‖p n‖₊ * r ^ n ≤ 1)
r : ℝ≥0
hr : ↑r < p.radius
a : ℝ
ha : a ∈ Set.Ioo 0 1
H : ∀ᶠ (n : ℕ) in atTop, |‖p n‖ * ↑r ^ n| ≤ a ^ n
⊢ ∀ᶠ (b : ℝ≥0∞) in map (fun n ↦ 1 / ↑(‖p n‖₊ ^ (1 / ↑n))) atTop, ↑r ≤ b
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
x : E
p : ℝ
m : ℕ
hf : ContDiffWithinAt ℝ (↑m) f Set.univ x
h : ↑m ≤ p
⊢ ContDiffWithinAt ℝ (↑m) (fun x ↦ f x ^ p) Set.univ x

Tactic:
]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
x : E
p : ℝ
m : ℕ
hf : ContDiffWithinAt ℝ (↑m) f Set.univ x
h : ↑m ≤ p
⊢ ContDiffWithinAt ℝ (↑m) (fun x ↦ f x ^ p) Set.univ x
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: a : ℕ
r : ℝ
hr : r ≠ 0
this :
  HasFPowerSeriesOnBall (fun x ↦ 1 / (↑r - x) ^ (a + 1))
    (FormalMultilinearSeries.restrictScalars ℝ
      (FormalMultilinearSeries.ofScalars ℂ fun n ↦ (↑r ^ (n + a + 1))⁻¹ * ↑((a + n).choose a)))
    (Complex.ofRealCLM 0) ‖↑r‖ₑ
⊢ HasFPowerSeriesOnBall (fun x ↦ 1 / (r - x) ^ (a + 1))
    (FormalMultilinearSeries.ofScalars ℝ fun n ↦ (r ^ (n + a + 1))⁻¹ * ↑((a + n).choose a)) 0 ‖r‖ₑ

Tactic:
]

Goal state after:
a : ℕ
r : ℝ
hr : r ≠ 0
this :
  HasFPowerSeriesOnBall (fun x ↦ 1 / (↑r - x) ^ (a + 1))
    (FormalMultilinearSeries.restrictScalars ℝ
      (FormalMultilinearSeries.ofScalars ℂ fun n ↦ (↑r ^ (n + a + 1))⁻¹ * ↑((a + n).choose a)))
    (Complex.ofRealCLM 0) ‖↑r‖ₑ
⊢ HasFPowerSeriesOnBall (fun x ↦ 1 / (r - x) ^ (a + 1))
    (FormalMultilinearSeries.ofScalars ℝ fun n ↦ (r ^ (n + a + 1))⁻¹ * ↑((a + n).choose a)) 0 ‖r‖ₑ