mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
inst✝ : CompleteSpace E
s : Set E
c : ℝ≥0
f' : E →L[𝕜] F
hf : ApproximatesLinearOn f f' s c
f'symm : f'.NonlinearRightInverse
ε : ℝ
b : E
ε0 : 0 ≤ ε
hε : closedBall b ε ⊆ s
y : F
hy : y ∈ closedBall (f b) (((↑f'symm.nnnorm)⁻¹ - ↑c) * ε)
hc : ↑c < (↑f'symm.nnnorm)⁻¹
If' : 0 < ↑f'symm.nnnorm
Icf' : ↑c * ↑f'symm.nnnorm < 1
Jcf' : 1 - ↑c * ↑f'symm.nnnorm ≠ 0
g : E → E := fun x ↦ x + f'symm.toFun (y - f x)
hg : g = fun x ↦ x + f'symm.toFun (y - f x)
u : ℕ → E := fun n ↦ g^[n] b
hu : u = fun n ↦ g^[n] b
usucc : ∀ (n : ℕ), u (n + 1) = g (u n)
A : ∀ (z : E), dist (g z) z ≤ ↑f'symm.nnnorm * dist (f z) y
⊢ ∀ z ∈ closedBall b ε, g z ∈ closedBall b ε → dist (f (g z)) y ≤ ↑c * ↑f'symm.nnnorm * dist (f z) y

Tactic:
intro z

Goal state after:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
inst✝ : CompleteSpace E
s : Set E
c : ℝ≥0
f' : E →L[𝕜] F
hf : ApproximatesLinearOn f f' s c
f'symm : f'.NonlinearRightInverse
ε : ℝ
b : E
ε0 : 0 ≤ ε
hε : closedBall b ε ⊆ s
y : F
hy : y ∈ closedBall (f b) (((↑f'symm.nnnorm)⁻¹ - ↑c) * ε)
hc : ↑c < (↑f'symm.nnnorm)⁻¹
If' : 0 < ↑f'symm.nnnorm
Icf' : ↑c * ↑f'symm.nnnorm < 1
Jcf' : 1 - ↑c * ↑f'symm.nnnorm ≠ 0
g : E → E := fun x ↦ x + f'symm.toFun (y - f x)
hg : g = fun x ↦ x + f'symm.toFun (y - f x)
u : ℕ → E := fun n ↦ g^[n] b
hu : u = fun n ↦ g^[n] b
usucc : ∀ (n : ℕ), u (n + 1) = g (u n)
A : ∀ (z : E), dist (g z) z ≤ ↑f'symm.nnnorm * dist (f z) y
z : E
⊢ z ∈ closedBall b ε → g z ∈ closedBall b ε → dist (f (g z)) y ≤ ↑c * ↑f'symm.nnnorm * dist (f z) y
Module: Mathlib.Analysis.AbsoluteValue.Equivalence

Goal state before:
pp: case neg
F : Type u_1
inst✝¹ : Field F
ι : Type u_2
inst✝ : Finite ι
v : ι → AbsoluteValue F ℝ
h : ∀ (i : ι), (v i).IsNontrivial
hv : Pairwise fun i j ↦ ¬(v i).IsEquiv (v j)
this✝ : Fintype ι
z : (i : ι) → WithAbs (v i)
r : ℝ
hr : r > 0
a : ι → F
hx : ∀ (i : ι), 1 < (v i) (a i) ∧ ∀ (j : ι), j ≠ i → (v j) (a i) < 1
y : ℕ → F := fun n ↦ ∑ i, 1 / (1 + (a i)⁻¹ ^ n) * (WithAbs.equiv (v i)) (z i)
u w : ι
x✝ : w ∈ Finset.univ
hw : ¬u = w
hgt : 1 < (v u) (a w)⁻¹
this : Tendsto (fun n ↦ (v u) (1 / (1 + (a w)⁻¹ ^ n))) atTop (𝓝 0)
⊢ Tendsto
    (fun i ↦
      (WithAbs.equiv (v u)).symm (1 / (1 + (a w)⁻¹ ^ i)) * (WithAbs.equiv (v u)).symm ((WithAbs.equiv (v w)) (z w)))
    atTop (𝓝 0)

Tactic:
simp_rw [← WithAbs.norm_toAbs_eq] at this

Goal state after:
case neg
F : Type u_1
inst✝¹ : Field F
ι : Type u_2
inst✝ : Finite ι
v : ι → AbsoluteValue F ℝ
h : ∀ (i : ι), (v i).IsNontrivial
hv : Pairwise fun i j ↦ ¬(v i).IsEquiv (v j)
this✝ : Fintype ι
z : (i : ι) → WithAbs (v i)
r : ℝ
hr : r > 0
a : ι → F
hx : ∀ (i : ι), 1 < (v i) (a i) ∧ ∀ (j : ι), j ≠ i → (v j) (a i) < 1
y : ℕ → F := fun n ↦ ∑ i, 1 / (1 + (a i)⁻¹ ^ n) * (WithAbs.equiv (v i)) (z i)
u w : ι
x✝ : w ∈ Finset.univ
hw : ¬u = w
hgt : 1 < (v u) (a w)⁻¹
this : Tendsto (fun n ↦ ‖WithAbs.toAbs (v u) (1 / (1 + (a w)⁻¹ ^ n))‖) atTop (𝓝 0)
⊢ Tendsto
    (fun i ↦
      (WithAbs.equiv (v u)).symm (1 / (1 + (a w)⁻¹ ^ i)) * (WithAbs.equiv (v u)).symm ((WithAbs.equiv (v w)) (z w)))
    atTop (𝓝 0)
Module: Mathlib.Analysis.Analytic.Order

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : AnalyticAt 𝕜 f x
hfx : f x = 0
hf' : deriv f x ≠ 0
⊢ analyticOrderAt f x = 1

Tactic:
by
  simpa [hfx] using hf.analyticOrderAt_sub_eq_one_of_deriv_ne_zero hf'

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CharZero 𝕜
inst✝ : CompleteSpace E
f : 𝕜 → E
hf : AnalyticAt 𝕜 f 0
n : ℕ
⊢ ∃ F,
    AnalyticAt 𝕜 F 0 ∧
      ∀ᶠ (z : 𝕜) in 𝓝 0, f z - ∑ i ∈ Finset.range n, (z ^ i / ↑i.factorial) • iteratedDeriv i f 0 = z ^ n • F z

Tactic:
have : AnalyticAt 𝕜
      (fun z : 𝕜 ↦ ∑ i ∈ .range n, (z ^ i / i.factorial) • iteratedDeriv i f 0) 0 := by
    refine Finset.analyticAt_fun_sum _ fun i hi ↦ ?_
    fun_prop

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CharZero 𝕜
inst✝ : CompleteSpace E
f : 𝕜 → E
hf : AnalyticAt 𝕜 f 0
n : ℕ
this : AnalyticAt 𝕜 (fun z ↦ ∑ i ∈ Finset.range n, (z ^ i / ↑i.factorial) • iteratedDeriv i f 0) 0
⊢ ∃ F,
    AnalyticAt 𝕜 F 0 ∧
      ∀ᶠ (z : 𝕜) in 𝓝 0, f z - ∑ i ∈ Finset.range n, (z ^ i / ↑i.factorial) • iteratedDeriv i f 0 = z ^ n • F z
Module: Mathlib.Analysis.SpecialFunctions.Complex.Analytic

Goal state before:
pp: z : ℂ
m : z ∈ slitPlane
⊢ ∀ a ∈ slitPlane, DifferentiableAt ℂ log a

Tactic:
intro z

Goal state after:
z✝ : ℂ
m : z✝ ∈ slitPlane
z : ℂ
⊢ z ∈ slitPlane → DifferentiableAt ℂ log z
Module: Mathlib.Analysis.SpecialFunctions.Complex.Analytic

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f g : E → ℂ
x : E
s : Set E
fa : AnalyticWithinAt ℂ f s x
ga : AnalyticWithinAt ℂ g s x
m : f x ∈ slitPlane
⊢ ∀ (a : E), f a ≠ 0 → f a ^ g a = cexp (log (f a) * g a)

Tactic:
intro z

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f g : E → ℂ
x : E
s : Set E
fa : AnalyticWithinAt ℂ f s x
ga : AnalyticWithinAt ℂ g s x
m : f x ∈ slitPlane
z : E
⊢ f z ≠ 0 → f z ^ g z = cexp (log (f z) * g z)
Module: Mathlib.Analysis.Calculus.FDeriv.Extend

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
s : Set E
x : E
f' : E →L[ℝ] F
f_diff : DifferentiableOn ℝ f s
s_conv : Convex ℝ s
s_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
f_cont : ∀ y ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s, ContinuousWithinAt f s y
h : Tendsto (fun y ↦ fderiv ℝ f y) (𝓝[s] x) (𝓝 f')
hx : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
ε : ℝ
ε_pos : 0 < ε
δ : ℝ
δ_pos : δ > 0
hδ : ∀ y ∈ s, dist y x < δ → ‖fderiv ℝ f y - f'‖ < ε
B : Set E := ball x δ
u v : E
u_in : (u, v).1 ∈ B ∩ s
v_in : (u, v).2 ∈ B ∩ s
conv : Convex ℝ (B ∩ s)
diff : DifferentiableOn ℝ f (B ∩ s)
⊢ ∀ z ∈ B ∩ s, ‖fderivWithin ℝ f (B ∩ s) z - f'‖ ≤ ε

Tactic:
intro z

Goal state after:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
s : Set E
x : E
f' : E →L[ℝ] F
f_diff : DifferentiableOn ℝ f s
s_conv : Convex ℝ s
s_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
f_cont : ∀ y ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s, ContinuousWithinAt f s y
h : Tendsto (fun y ↦ fderiv ℝ f y) (𝓝[s] x) (𝓝 f')
hx : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
ε : ℝ
ε_pos : 0 < ε
δ : ℝ
δ_pos : δ > 0
hδ : ∀ y ∈ s, dist y x < δ → ‖fderiv ℝ f y - f'‖ < ε
B : Set E := ball x δ
u v : E
u_in : (u, v).1 ∈ B ∩ s
v_in : (u, v).2 ∈ B ∩ s
conv : Convex ℝ (B ∩ s)
diff : DifferentiableOn ℝ f (B ∩ s)
z : E
⊢ z ∈ B ∩ s → ‖fderivWithin ℝ f (B ∩ s) z - f'‖ ≤ ε
Module: Mathlib.Analysis.Analytic.Order

Goal state before:
pp: case left
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f : 𝕜 → E
hf : AnalyticOnNhd 𝕜 f U
⊢ ∀ x ∈ {u | analyticOrderAt f ↑u = ⊤}ᶜ,
    ∃ t ⊆ {u | analyticOrderAt f ↑u = ⊤}ᶜ, IsOpen[instTopologicalSpaceSubtype] t ∧ x ∈ t

Tactic:
intro z

Goal state after:
case left
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f : 𝕜 → E
hf : AnalyticOnNhd 𝕜 f U
z : ↑U
⊢ z ∈ {u | analyticOrderAt f ↑u = ⊤}ᶜ →
    ∃ t ⊆ {u | analyticOrderAt f ↑u = ⊤}ᶜ, IsOpen[instTopologicalSpaceSubtype] t ∧ z ∈ t
Module: Mathlib.Analysis.Analytic.Order

Goal state before:
pp: case right
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f : 𝕜 → E
hf : AnalyticOnNhd 𝕜 f U
⊢ ∀ x ∈ {u | analyticOrderAt f ↑u = ⊤},
    ∃ t ⊆ {u | analyticOrderAt f ↑u = ⊤}, IsOpen[instTopologicalSpaceSubtype] t ∧ x ∈ t

Tactic:
intro z

Goal state after:
case right
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f : 𝕜 → E
hf : AnalyticOnNhd 𝕜 f U
z : ↑U
⊢ z ∈ {u | analyticOrderAt f ↑u = ⊤} →
    ∃ t ⊆ {u | analyticOrderAt f ↑u = ⊤}, IsOpen[instTopologicalSpaceSubtype] t ∧ z ∈ t
Module: Mathlib.Analysis.Analytic.Order

Goal state before:
pp: case right
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f : 𝕜 → E
hf : AnalyticOnNhd 𝕜 f U
⊢ IsOpen[instTopologicalSpaceSubtype] {u | analyticOrderAt f ↑u = ⊤}

Tactic:
apply isOpen_iff_forall_mem_open.mpr
    intro z hz
    conv =>
      arg 1; intro; left; right; arg 1; intro
      rw [analyticOrderAt_eq_top, eventually_nhds_iff]
    simp only [mem_ofPred_eq] at hz
    rw [analyticOrderAt_eq_top, eventually_nhds_iff] at hz
    obtain ⟨t', h₁t', h₂t', h₃t'⟩ := hz
    use Subtype.val ⁻¹' t'
    simp only [isOpen_induced h₂t', mem_preimage, h₃t', and_self, and_true]
    grind

Goal state after:
Module: Mathlib.Analysis.Meromorphic.Basic

Goal state before:
pp: case refine_2
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
x : 𝕜
inst✝ : CompleteSpace E
f : 𝕜 → E
n : ℕ
h : ∀ᶠ (y : 𝕜) in 𝓝 x, AnalyticAt 𝕜 (fun z ↦ (z - x) ^ n • f z) y
y : 𝕜
hy : ¬y = x
hf : AnalyticAt 𝕜 (fun z ↦ (z - x) ^ n • f z) y
this : AnalyticAt 𝕜 ((_root_.id - fun x_1 ↦ x) ^ n)⁻¹ y
⊢ ∀ (x_1 : 𝕜), x_1 ≠ x → (((_root_.id - fun x_2 ↦ x) ^ n)⁻¹ • fun z ↦ (z - x) ^ n • f z) x_1 = f x_1

Tactic:
intro z

Goal state after:
case refine_2
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
x : 𝕜
inst✝ : CompleteSpace E
f : 𝕜 → E
n : ℕ
h : ∀ᶠ (y : 𝕜) in 𝓝 x, AnalyticAt 𝕜 (fun z ↦ (z - x) ^ n • f z) y
y : 𝕜
hy : ¬y = x
hf : AnalyticAt 𝕜 (fun z ↦ (z - x) ^ n • f z) y
this : AnalyticAt 𝕜 ((_root_.id - fun x_1 ↦ x) ^ n)⁻¹ y
z : 𝕜
⊢ z ≠ x → (((_root_.id - fun x_1 ↦ x) ^ n)⁻¹ • fun z ↦ (z - x) ^ n • f z) z = f z
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: n : ℕ
⊢ -(↑n + 1) * (ascPochhammer ℕ n).smeval (-(↑n + 1) + 1) = (-1) ^ (n + 1) * ↑(n + 1).factorial

Tactic:
neg_add_rev,

Goal state after:
n : ℕ
⊢ (-1 + -↑n) * (ascPochhammer ℕ n).smeval (-1 + -↑n + 1) = (-1) ^ (n + 1) * ↑(n + 1).factorial
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: case inl
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 < x
⊢ HasDerivAt (fun y ↦ ↑y ^ (r + 1) / (r + 1)) (↑x ^ r) x

Tactic:
apply HasDerivAt.comp_ofReal (e := fun y => (y : ℂ) ^ (r + 1) / (r + 1))

Goal state after:
case inl
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 < x
⊢ HasDerivAt (fun y ↦ y ^ (r + 1) / (r + 1)) (↑x ^ r) ↑x
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: case convert_7
a : ℂ
this : binomialSeries ℂ a = FormalMultilinearSeries.ofScalars ℂ fun n ↦ iteratedDeriv n (fun x ↦ (1 + x) ^ a) 0 / ↑n !
⊢ ∀ z ∈ Metric.eball 0 1, ((fun x ↦ 1) + fun x ↦ x) z ∈ slitPlane

Tactic:
intro z

Goal state after:
case convert_7
a : ℂ
this : binomialSeries ℂ a = FormalMultilinearSeries.ofScalars ℂ fun n ↦ iteratedDeriv n (fun x ↦ (1 + x) ^ a) 0 / ↑n !
z : ℂ
⊢ z ∈ Metric.eball 0 1 → ((fun x ↦ 1) + fun x ↦ x) z ∈ slitPlane
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: 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
⊢ 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

Tactic:
intro z

Goal state after:
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
z : ℂ
⊢ z ∈ B →
    derivWithin (iteratedDerivWithin n (fun x ↦ (1 + x) ^ a) B) B z =
      derivWithin (fun x ↦ (descPochhammer ℤ n).smeval a * (1 + x) ^ (a - ↑n)) B z
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: case succ
a : ℂ
B : Set ℂ := ⋯
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
⊢ Set.EqOn (derivWithin (fun x ↦ (descPochhammer ℤ n).smeval a * (1 + x) ^ (a - ↑n)) B)
    (fun x ↦ (descPochhammer ℤ (n + 1)).smeval a * (1 + x) ^ (a - ↑(n + 1))) B

Tactic:
intro z

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
⊢ z ∈ B →
    derivWithin (fun x ↦ (descPochhammer ℤ n).smeval a * (1 + x) ^ (a - ↑n)) B z =
      (fun x ↦ (descPochhammer ℤ (n + 1)).smeval a * (1 + x) ^ (a - ↑(n + 1))) z
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: case refine_1
w x : ℂ
hw : w ≠ x
⊢ HasFPowerSeriesOnBall (fun z ↦ 1 / (z - w) ^ 2)
    (FormalMultilinearSeries.ofScalars ℂ fun i ↦ (↑i + 1) * (w - x) ^ (-↑(i + 2))) x ‖w - x‖ₑ

pp: case refine_2
w x : ℂ
hw : w ≠ x
⊢ HasFPowerSeriesOnBall (fun z ↦ 1 / w ^ 2) (FormalMultilinearSeries.ofScalars ℂ fun i ↦ Nat.casesOn i (w ^ (-2)) 0) x
    ‖w - x‖ₑ

Tactic:
· simpa only [sub_sub_sub_cancel_right, zero_add, sub_sq_comm w, zpow_neg, zpow_natCast, mul_comm]
      using (one_div_sub_sq_hasFPowerSeriesOnBall_zero
        (z := w - x) (by simp [sub_eq_zero, hw])).comp_sub x

Goal state after:
case refine_2
w x : ℂ
hw : w ≠ x
⊢ HasFPowerSeriesOnBall (fun z ↦ 1 / w ^ 2) (FormalMultilinearSeries.ofScalars ℂ fun i ↦ Nat.casesOn i (w ^ (-2)) 0) x
    ‖w - x‖ₑ
Module: Mathlib.Analysis.Analytic.RadiusLiminf

Goal state before:
pp: case a.hf
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
⊢ autoParam (IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) atTop fun n ↦ 1 / ↑(‖p n‖₊ ^ (1 / ↑n))) le_liminf_of_le._auto_1

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 (n : ℕ) in atTop, ↑r ≤ 1 / ↑(‖p n‖₊ ^ (1 / ↑n))

Tactic:
· infer_param

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 (n : ℕ) in atTop, ↑r ≤ 1 / ↑(‖p n‖₊ ^ (1 / ↑n))
Module: Mathlib.Analysis.Asymptotics.ExpGrowth

Goal state before:
pp: b : ℝ≥0∞
⊢ limsup (fun n ↦ (b ^ n).log / ↑n) atTop = limsup (fun x ↦ b.log) atTop

Tactic:
refine limsup_congr (eventually_atTop.2 ⟨1, fun n n_1 ↦ ?_⟩)

Goal state after:
b : ℝ≥0∞
n : ℕ
n_1 : 1 ≤ n
⊢ (b ^ n).log / ↑n = b.log
Module: Mathlib.MeasureTheory.Measure.Sub

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
⊢ μ - 0 = μ

Tactic:
sub_def

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
⊢ sInf {d | μ ≤ d + 0} = μ
Module: Mathlib.MeasureTheory.Measure.Sub

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
⊢ (μ - ν).restrict s = μ.restrict s - ν.restrict s

Tactic:
sub_def

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
⊢ (sInf {d | μ ≤ d + ν}).restrict s = μ.restrict s - ν.restrict s
Module: Mathlib.MeasureTheory.Measure.Sub

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
⊢ (sInf {d | μ ≤ d + ν}).restrict s = μ.restrict s - ν.restrict s

Tactic:
sub_def

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
⊢ (sInf {d | μ ≤ d + ν}).restrict s = sInf {d | μ.restrict s ≤ d + ν.restrict s}
Module: Mathlib.MeasureTheory.Measure.Sub

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
⊢ (sInf {d | μ ≤ d + ν}).restrict s = sInf {d | μ.restrict s ≤ d + ν.restrict s}

Tactic:
sub_def

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
⊢ (sInf {d | μ ≤ d + ν}).restrict s = sInf {d | μ.restrict s ≤ d + ν.restrict s}
Module: Mathlib.MeasureTheory.Covering.Vitali

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)
ht : ∀ a ∈ t, (interior (B a)).Nonempty
h't : ∀ a ∈ t, IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (B a)
hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ∞
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint B
hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : u.Countable
x : α
x✝ : x ∈ s \ ⋃ a ∈ u, B a
v : Set ι := {a | a ∈ u ∧ (B a ∩ ball x (R x)).Nonempty}
vu : v ⊆ u
K : ℝ
μK : μ (closedBall x K) < ∞
hK : ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
⊢ μ ((s \ ⋃ a ∈ u, B a) ∩ ball x (R x)) ≤ ε

Tactic:
have I : (∑' a : v, μ (B a)) < ∞ := by
    calc
      (∑' a : v, μ (B a)) = μ (⋃ a ∈ v, B a) := by
        rw [measure_biUnion (u_count.mono vu) _ fun a ha => (h't _ (vu.trans ut ha)).measurableSet]
        exact u_disj.subset vu
      _ ≤ μ (closedBall x K) := (measure_mono (iUnion₂_subset fun a ha => hK a (vu ha) ha.2))
      _ < ∞ := μK

Goal state after:
α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)
ht : ∀ a ∈ t, (interior (B a)).Nonempty
h't : ∀ a ∈ t, IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (B a)
hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ∞
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint B
hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : u.Countable
x : α
x✝ : x ∈ s \ ⋃ a ∈ u, B a
v : Set ι := {a | a ∈ u ∧ (B a ∩ ball x (R x)).Nonempty}
vu : v ⊆ u
K : ℝ
μK : μ (closedBall x K) < ∞
hK : ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : ∑' (a : ↑v), μ (B ↑a) < ∞
⊢ μ ((s \ ⋃ a ∈ u, B a) ∩ ball x (R x)) ≤ ε
Module: Mathlib.MeasureTheory.Covering.Vitali

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)
ht : ∀ a ∈ t, (interior (B a)).Nonempty
h't : ∀ a ∈ t, IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (B a)
hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ∞
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint B
hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : u.Countable
x : α
x✝ : x ∈ s \ ⋃ a ∈ u, B a
v : Set ι := {a | a ∈ u ∧ (B a ∩ ball x (R x)).Nonempty}
vu : v ⊆ u
K : ℝ
μK : μ (closedBall x K) < ∞
hK : ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : ∑' (a : ↑v), μ (B ↑a) < ∞
w : Finset ↑v
hw : ∑' (a : { a // a ∉ w }), μ (B ↑↑a) < ε / ↑C
⊢ (s \ ⋃ a ∈ u, B a) ∩ ball x (R x) ⊆ ⋃ a, closedBall (c ↑↑a) (3 * r ↑↑a)

Tactic:
intro z

Goal state after:
α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)
ht : ∀ a ∈ t, (interior (B a)).Nonempty
h't : ∀ a ∈ t, IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (B a)
hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ∞
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint B
hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : u.Countable
x : α
x✝ : x ∈ s \ ⋃ a ∈ u, B a
v : Set ι := {a | a ∈ u ∧ (B a ∩ ball x (R x)).Nonempty}
vu : v ⊆ u
K : ℝ
μK : μ (closedBall x K) < ∞
hK : ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : ∑' (a : ↑v), μ (B ↑a) < ∞
w : Finset ↑v
hw : ∑' (a : { a // a ∉ w }), μ (B ↑↑a) < ε / ↑C
z : α
⊢ z ∈ (s \ ⋃ a ∈ u, B a) ∩ ball x (R x) → z ∈ ⋃ a, closedBall (c ↑↑a) (3 * r ↑↑a)