mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Meromorphic.Divisor

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
V : Set 𝕜
hf : MeromorphicOn f U
hV : V ⊆ U
⊢ (divisor f U).restrict hV = divisor f V

Tactic:
ext x

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
V : Set 𝕜
hf : MeromorphicOn f U
hV : V ⊆ U
x : 𝕜
⊢ ((divisor f U).restrict hV) x = (divisor f V) x
Module: Mathlib.Analysis.Meromorphic.Divisor

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f₁ f₂ : 𝕜 → E
hf₁ : MeromorphicOn f₁ U
hf₂ : AnalyticOnNhd 𝕜 f₂ U
⊢ (divisor (f₁ + f₂) U)⁻ = (divisor f₁ U)⁻

Tactic:
ext x

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f₁ f₂ : 𝕜 → E
hf₁ : MeromorphicOn f₁ U
hf₂ : AnalyticOnNhd 𝕜 f₂ U
x : 𝕜
⊢ (divisor (f₁ + f₂) U)⁻ x = (divisor f₁ U)⁻ x
Module: Mathlib.Analysis.Meromorphic.Divisor

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
z : 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
f : 𝕜 → E
⊢ (divisor (f ∘ fun x ↦ x - c) U) (z + c) = (divisor (f ∘ fun x ↦ x + -c) U) (z - -c)

Tactic:
← sub_eq_add_neg,

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
z : 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
f : 𝕜 → E
⊢ (divisor (f ∘ fun x ↦ x - c) U) (z + c) = (divisor (f ∘ fun x ↦ x - c) U) (z - -c)
Module: Mathlib.Analysis.Meromorphic.Divisor

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
z : 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
f : 𝕜 → E
⊢ (divisor (f ∘ fun x ↦ x - c) U) (z + c) = (divisor (f ∘ fun x ↦ x - c) U) (z - -c)

Tactic:
sub_neg_eq_add

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

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : 𝕜
f : 𝕜 → E
h₁ : MeromorphicAt f x
h₂ : ¬meromorphicOrderAt f x = ⊤
g : 𝕜 → E
h₁g : AnalyticAt 𝕜 g x
h₂g : g x ≠ 0
h₃g : f =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ (meromorphicOrderAt f x).untop₀ • g z
⊢ meromorphicTrailingCoeffAt (-f) x = -meromorphicTrailingCoeffAt f x

Tactic:
rw [h₁g.meromorphicTrailingCoeffAt_of_ne_zero_of_eq_nhdsNE h₂g h₃g]

Goal state after:
case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : 𝕜
f : 𝕜 → E
h₁ : MeromorphicAt f x
h₂ : ¬meromorphicOrderAt f x = ⊤
g : 𝕜 → E
h₁g : AnalyticAt 𝕜 g x
h₂g : g x ≠ 0
h₃g : f =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ (meromorphicOrderAt f x).untop₀ • g z
⊢ meromorphicTrailingCoeffAt (-f) x = -g x
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
ho : meromorphicOrderAt f x < 0
hf : MeromorphicAt f x
m : ℤ
hm : ↑m = meromorphicOrderAt f x
m_neg : m < 0
g : 𝕜 → E
g_an : AnalyticAt 𝕜 g x
gx : g x ≠ 0
hg : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ m • g z
y : 𝕜
hy : y ∈ {x}ᶜ
⊢ y - x ∈ {0}ᶜ

Tactic:
simpa [sub_eq_zero] using hy

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

Goal state before:
pp: case mpr
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
U : Set 𝕜
h : MeromorphicNFOn f U
⊢ toMeromorphicNFOn f U = f

Tactic:
ext x

Goal state after:
case mpr
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
U : Set 𝕜
h : MeromorphicNFOn f U
x : 𝕜
⊢ toMeromorphicNFOn f U x = f x
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
h : HasFiniteSupport d
⊢ ∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ d u = fun x ↦ ∏ᶠ (u : 𝕜), (x - u) ^ d u

Tactic:
ext x

Goal state after:
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
h : HasFiniteSupport d
x : 𝕜
⊢ (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ d u) x = ∏ᶠ (u : 𝕜), (x - u) ^ d u
Module: Mathlib.Analysis.Meromorphic.TrailingCoefficient

Goal state before:
pp: case empty
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
ι : Type u_3
f : ι → 𝕜 → 𝕜
x : 𝕜
h : ∀ σ ∈ ∅, MeromorphicAt (f σ) x
⊢ meromorphicTrailingCoeffAt (∏ n ∈ ∅, f n) x = ∏ n ∈ ∅, meromorphicTrailingCoeffAt (f n) x

Tactic:
apply meromorphicTrailingCoeffAt_const

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

Goal state before:
pp: case inl
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : MeromorphicAt f x
ho : meromorphicOrderAt f x < 0
⊢ ∀ (x_1 : E), ¬Tendsto f (𝓝[≠] x) (𝓝 x_1)

Tactic:
intro c hc

Goal state after:
case inl
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : MeromorphicAt f x
ho : meromorphicOrderAt f x < 0
c : E
hc : Tendsto f (𝓝[≠] x) (𝓝 c)
⊢ False
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: case inl
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : MeromorphicAt f x
ho : 0 < meromorphicOrderAt f x
⊢ Tendsto f (𝓝[≠] x) (𝓝 0) ↔ 0 < meromorphicOrderAt f x

pp: case inr
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : MeromorphicAt f x
ho : meromorphicOrderAt f x ≤ 0
⊢ Tendsto f (𝓝[≠] x) (𝓝 0) ↔ 0 < meromorphicOrderAt f x

Tactic:
· simp [ho, tendsto_zero_of_meromorphicOrderAt_pos ho]

Goal state after:
case inr
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : MeromorphicAt f x
ho : meromorphicOrderAt f x ≤ 0
⊢ Tendsto f (𝓝[≠] x) (𝓝 0) ↔ 0 < meromorphicOrderAt f x
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
hd : ¬(Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d u).Finite
⊢ MeromorphicNFOn (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ d u) univ

Tactic:
finprod_of_infinite_mulSupport hd

Goal state after:
case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
hd : ¬(Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d u).Finite
⊢ MeromorphicNFOn 1 univ
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h : (support d).Finite
u : 𝕜
⊢ u ∉ ↑h.toFinset → u ∉ Function.mulSupport fun u ↦ (x - u) ^ update d x 0 u

Tactic:
simp only [mem_mulSupport, ne_eq, Decidable.not_not]

Goal state after:
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h : (support d).Finite
u : 𝕜
⊢ u ∉ ↑h.toFinset → (x - u) ^ update d x 0 u = 1
Module: Mathlib.Analysis.Meromorphic.TrailingCoefficient

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x c : 𝕜
⊢ meromorphicTrailingCoeffAt (f ∘ fun x ↦ x + c) x = meromorphicTrailingCoeffAt f (x + c)

Tactic:
by
  classical
  by_cases h : ¬ MeromorphicAt f (x + c)
  · simp_all [meromorphicAt_comp_add_const_iff_meromorphicAt.not.2 h]
  rw [MeromorphicAt.meromorphicTrailingCoeffAt_comp (by simp_all) (by fun_prop)
    (by simp [eventuallyConst_iff_analyticOrderAt_sub_eq_top])]
  simp [meromorphicTrailingCoeffAt_id_sub_const]

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

Goal state before:
pp: R : ℝ
w x : ℂ
hx : x ∈ {w}ᶜ
⊢ AnalyticAt ℂ (canonicalFactor R w) x

Tactic:
canonicalFactor_def

Goal state after:
R : ℝ
w x : ℂ
hx : x ∈ {w}ᶜ
⊢ AnalyticAt ℂ (fun z ↦ (↑R ^ 2 - (starRingEnd ℂ) w * z) / (↑R * (z - w))) x
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
𝕜' : Type u_4
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
x : 𝕜
f : 𝕜 → 𝕜'
hf : MeromorphicAt f x
h₂f : meromorphicOrderAt f x = ⊤
⊢ meromorphicOrderAt f⁻¹ x = -meromorphicOrderAt f x

pp: case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
𝕜' : Type u_4
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
x : 𝕜
f : 𝕜 → 𝕜'
hf : MeromorphicAt f x
h₂f : ¬meromorphicOrderAt f x = ⊤
⊢ meromorphicOrderAt f⁻¹ x = -meromorphicOrderAt f x

Tactic:
· rw [h₂f, ← LinearOrderedAddCommGroupWithTop.neg_top, neg_neg]
    rw [meromorphicOrderAt_eq_top_iff] at *
    filter_upwards [h₂f]
    simp

Goal state after:
case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
𝕜' : Type u_4
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
x : 𝕜
f : 𝕜 → 𝕜'
hf : MeromorphicAt f x
h₂f : ¬meromorphicOrderAt f x = ⊤
⊢ meromorphicOrderAt f⁻¹ x = -meromorphicOrderAt f x
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: case left.inl
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
z : ↑U
hz : z ∈ {u | meromorphicOrderAt f ↑u = ⊤}ᶜ
h : ∀ᶠ (z : 𝕜) in 𝓝[≠] ↑z, f z = 0
⊢ ∃ t ⊆ {u | meromorphicOrderAt f ↑u = ⊤}ᶜ, IsOpen[instTopologicalSpaceSubtype] t ∧ z ∈ t

Tactic:
rw [← meromorphicOrderAt_eq_top_iff] at h
      tauto

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

Goal state before:
pp: f : ℂ →ₗᵢ[ℝ] ℂ
h : f 1 = 1
z : ℂ
⊢ ∀ (z : ℂ), z + (starRingEnd ℂ) z = f z + (starRingEnd ℂ) (f z)

Tactic:
apply LinearIsometry.im_apply_eq_im h

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

Goal state before:
pp: a : Circle
⊢ LinearEquiv.det (rotation a).toLinearEquiv = 1

Tactic:
by
  rw [← Units.val_inj, LinearEquiv.coe_det, det_rotation, Units.val_one]

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

Goal state before:
pp: case h₂
f : ℂ → ℂ
z : ℂ
h : fderiv ℝ f z ≠ 0
h_diff : DifferentiableAt ℝ f ((starRingEnd ℂ) ((starRingEnd ℂ) z))
⊢ (∃ map, restrictScalars ℝ map = fderiv ℝ f z ∘SL ↑conjCLE) ↔
    DifferentiableAt ℂ (f ∘ ⇑(starRingEnd ℂ)) ((starRingEnd ℂ) z)

Tactic:
rw [differentiableAt_iff_restrictScalars ℝ (h_diff.comp _ conjCLE.differentiableAt)]

Goal state after:
case h₂
f : ℂ → ℂ
z : ℂ
h : fderiv ℝ f z ≠ 0
h_diff : DifferentiableAt ℝ f ((starRingEnd ℂ) ((starRingEnd ℂ) z))
⊢ (∃ map, restrictScalars ℝ map = fderiv ℝ f z ∘SL ↑conjCLE) ↔
    ∃ g', restrictScalars ℝ g' = fderiv ℝ (f ∘ ⇑(starRingEnd ℂ)) ((starRingEnd ℂ) z)
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
f : ℂ → E
h₁f : MeromorphicOn f (closedBall 0 R)
h₂f : ∀ (u : ↑(closedBall 0 R)), meromorphicOrderAt f ↑u ≠ ⊤
hR : 0 = R
⊢ ¬meromorphicTrailingCoeffAt f 0 = 0

Tactic:
apply MeromorphicAt.meromorphicTrailingCoeffAt_ne_zero (h₁f 0 _) _
        <;> simp_all

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

Goal state before:
pp: t : |2⁻¹| = 2⁻¹
z : |429895798851992810179897 / 620208537104218167705600 + -log 2| ≤ 1 / 17179869184
⊢ |log 2 - 287209 / 414355| ≤ 1 / 17179869184 + (1 / 10 ^ 10 - 1 / 2 ^ 34)

Tactic:
← sub_eq_add_neg,

Goal state after:
t : |2⁻¹| = 2⁻¹
z : |429895798851992810179897 / 620208537104218167705600 - log 2| ≤ 1 / 17179869184
⊢ |log 2 - 287209 / 414355| ≤ 1 / 17179869184 + (1 / 10 ^ 10 - 1 / 2 ^ 34)
Module: Mathlib.Analysis.Complex.ExponentialBounds

Goal state before:
pp: t : |2 / 3| = 2 / 3
z : |∑ i ∈ range 70, (2 / 3) ^ (i + 1) / (↑i + 1) + -log 3| ≤ (2 / 3) ^ (70 + 1) / 3⁻¹
⊢ |log 3 - 109861228867 / 100000000000| ≤ (2 / 3) ^ 71 / 3⁻¹ + (1 / 10 ^ 10 - (2 / 3) ^ 71 / 3⁻¹)

Tactic:
← sub_eq_add_neg,

Goal state after:
t : |2 / 3| = 2 / 3
z : |∑ i ∈ range 70, (2 / 3) ^ (i + 1) / (↑i + 1) - log 3| ≤ (2 / 3) ^ (70 + 1) / 3⁻¹
⊢ |log 3 - 109861228867 / 100000000000| ≤ (2 / 3) ^ 71 / 3⁻¹ + (1 / 10 ^ 10 - (2 / 3) ^ 71 / 3⁻¹)
Module: Mathlib.Analysis.Complex.ExponentialBounds

Goal state before:
pp: ⊢ |log 3 - 109861228867 / 100000000000| ≤ 1 / 10 ^ 10

Tactic:
suffices |log 3 - 109861228867 / 100000000000| ≤
      (2 / 3) ^ 71 / 3⁻¹ + (1 / 10 ^ 10 - (2 / 3) ^ 71 / 3⁻¹) by
    norm_num1 at *
    assumption
  have t : |2 / 3| = (2 : ℝ) / 3 := by norm_num
  have z := abs_log_sub_add_sum_range_le (x := 2 / 3) (by norm_num) 70
  rw [t, show (1 - (2 : ℝ) / 3) = (1 / 3 : ℝ) by norm_num, one_div (3 : ℝ), log_inv,
    ← sub_eq_add_neg, _root_.abs_sub_comm] at z
  apply le_trans (_root_.abs_sub_le _ _ _) (add_le_add z _)
  norm_num [sum_range_succ]

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

Goal state before:
pp: t : |4 / 5| = 4 / 5
z : |∑ i ∈ range 130, (4 / 5) ^ (i + 1) / (↑i + 1) + -log 5| ≤ (4 / 5) ^ (130 + 1) / 5⁻¹
⊢ |log 5 - 160943791243 / 100000000000| ≤ (4 / 5) ^ 131 / 5⁻¹ + (1 / 10 ^ 10 - (4 / 5) ^ 131 / 5⁻¹)

Tactic:
← sub_eq_add_neg,

Goal state after:
t : |4 / 5| = 4 / 5
z : |∑ i ∈ range 130, (4 / 5) ^ (i + 1) / (↑i + 1) - log 5| ≤ (4 / 5) ^ (130 + 1) / 5⁻¹
⊢ |log 5 - 160943791243 / 100000000000| ≤ (4 / 5) ^ 131 / 5⁻¹ + (1 / 10 ^ 10 - (4 / 5) ^ 131 / 5⁻¹)
Module: Mathlib.Analysis.Complex.PhragmenLindelof

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℝ
f g : ℂ → E
l : Filter ℂ
u : ℂ → ℝ
hBf : ∃ c < a, ∃ B, f =O[l] fun z ↦ expR (B * expR (c * |u z|))
hBg : ∃ c < a, ∃ B, g =O[l] fun z ↦ expR (B * expR (c * |u z|))
this :
  ∀ {c₁ c₂ B₁ B₂ : ℝ},
    c₁ ≤ c₂ → 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * |u z|))‖ ≤ ‖expR (B₂ * expR (c₂ * |u z|))‖
⊢ ∃ c < a, ∃ B, (f - g) =O[l] fun z ↦ expR (B * expR (c * |u z|))

Tactic:
rcases hBf with ⟨cf, hcf, Bf, hOf⟩

Goal state after:
E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℝ
f g : ℂ → E
l : Filter ℂ
u : ℂ → ℝ
hBg : ∃ c < a, ∃ B, g =O[l] fun z ↦ expR (B * expR (c * |u z|))
this :
  ∀ {c₁ c₂ B₁ B₂ : ℝ},
    c₁ ≤ c₂ → 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * |u z|))‖ ≤ ‖expR (B₂ * expR (c₂ * |u z|))‖
cf : ℝ
hcf : cf < a
Bf : ℝ
hOf : f =O[l] fun z ↦ expR (Bf * expR (cf * |u z|))
⊢ ∃ c < a, ∃ B, (f - g) =O[l] fun z ↦ expR (B * expR (c * |u z|))
Module: Mathlib.Analysis.Complex.PhragmenLindelof

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℝ
f g : ℂ → E
l : Filter ℂ
hBf : ∃ c < a, ∃ B, f =O[cobounded ℂ ⊓ l] fun z ↦ expR (B * ‖z‖ ^ c)
hBg : ∃ c < a, ∃ B, g =O[cobounded ℂ ⊓ l] fun z ↦ expR (B * ‖z‖ ^ c)
this :
  ∀ {c₁ c₂ B₁ B₂ : ℝ},
    c₁ ≤ c₂ → 0 ≤ B₂ → B₁ ≤ B₂ → (fun z ↦ expR (B₁ * ‖z‖ ^ c₁)) =O[cobounded ℂ ⊓ l] fun z ↦ expR (B₂ * ‖z‖ ^ c₂)
⊢ ∃ c < a, ∃ B, (f - g) =O[cobounded ℂ ⊓ l] fun z ↦ expR (B * ‖z‖ ^ c)

Tactic:
rcases hBf with ⟨cf, hcf, Bf, hOf⟩

Goal state after:
E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℝ
f g : ℂ → E
l : Filter ℂ
hBg : ∃ c < a, ∃ B, g =O[cobounded ℂ ⊓ l] fun z ↦ expR (B * ‖z‖ ^ c)
this :
  ∀ {c₁ c₂ B₁ B₂ : ℝ},
    c₁ ≤ c₂ → 0 ≤ B₂ → B₁ ≤ B₂ → (fun z ↦ expR (B₁ * ‖z‖ ^ c₁)) =O[cobounded ℂ ⊓ l] fun z ↦ expR (B₂ * ‖z‖ ^ c₂)
cf : ℝ
hcf : cf < a
Bf : ℝ
hOf : f =O[cobounded ℂ ⊓ l] fun z ↦ expR (Bf * ‖z‖ ^ cf)
⊢ ∃ c < a, ∃ B, (f - g) =O[cobounded ℂ ⊓ l] fun z ↦ expR (B * ‖z‖ ^ c)
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: E : Type u_1
X : Type u_2
A : Type u_3
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
p : E → X
cov : IsCoveringMap p
H : C(↑I × A, X)
f : C(A, E)
H_0 : ∀ (a : A), H (0, a) = p (f a)
f₀ f₁ : C(A, X)
S : Set A
F : f₀.HomotopyRel f₁ S
x✝ : FundamentalGroupoid X
⊢ ↾cov.monodromy (𝟙 x✝) = 𝟙 ↑(p ⁻¹' {x✝.as})

Tactic:
by ext x : 3; simpa using! congr_fun cov.monodromy_refl x

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

Goal state before:
pp: case refine_1
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
z : ℂ
hD : z ∈ verticalClosedStrip 0 1
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
⊢ BddAbove (norm ∘ f '' re ⁻¹' {z.re})

Tactic:
revert hB; gcongr
    exact preimage_mono (singleton_subset_iff.mpr hD)

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

Goal state before:
pp: case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C✝ : ℝ
f : ℂ → E
z : ℂ
C : ℝ
hC₀ : 0 < C
a b : ℝ
hza : a - b < z.im
hle_a : ∀ (z : ℂ), z.im = a - b → ‖f z‖ ≤ C
hzb : z.im < a + b
hle_b : ∀ (z : ℂ), z.im = a + b → ‖f z‖ ≤ C
hfd : DiffContOnCl ℂ f (im ⁻¹' Ioo (a - b) (a + b))
hab : a - b < a + b
hb : 0 < b
hπb : 0 < π / 2 / b
c : ℝ
hc : c < π / 2 / b
B : ℝ
hO : f =O[comap (abs ∘ re) atTop ⊓ 𝓟 (im ⁻¹' Ioo (a - b) (a + b))] fun z ↦ expR (B * expR (c * |z.re|))
d : ℝ
hcd : c < d
hd₀ : 0 < d
hb' : d * b < π / 2
aff : ℂ → ℂ := fun w ↦ ↑d * (w - ↑a * I)
g : ℝ → ℂ → ℂ := fun ε w ↦ cexp (↑ε * (cexp (aff w) + cexp (-aff w)))
ε : ℝ
ε₀ : ε < 0
δ : ℝ
δ₀ : δ < 0
hδ : ∀ ⦃w : ℂ⦄, w.im ∈ Icc (a - b) (a + b) → ‖g ε w‖ ≤ expR (δ * expR (d * |w.re|))
hg₁ : ∀ (w : ℂ), w.im = a - b ∨ w.im = a + b → ‖g ε w‖ ≤ 1
R : ℝ
hzR : |z.re| < R
hR : ∀ (w : ℂ), |w.re| = R → w.im ∈ Ioo (a - b) (a + b) → ‖g ε w • f w‖ ≤ C
hR₀ : 0 < R
hgd : Differentiable ℂ (g ε)
hd : DiffContOnCl ℂ (fun w ↦ g ε w • f w) (Ioo (-R) R ×ℂ Ioo (a - b) (a + b))
w : ℂ
him : ¬(w.im = a - b ∨ w.im = a + b)
hw : w ∈ {-R, R} ×ℂ Icc (a - b) (a + b)
⊢ ‖g ε w • f w‖ ≤ C

Tactic:
have hw' := eq_endpoints_or_mem_Ioo_of_mem_Icc hw.2

Goal state after:
case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C✝ : ℝ
f : ℂ → E
z : ℂ
C : ℝ
hC₀ : 0 < C
a b : ℝ
hza : a - b < z.im
hle_a : ∀ (z : ℂ), z.im = a - b → ‖f z‖ ≤ C
hzb : z.im < a + b
hle_b : ∀ (z : ℂ), z.im = a + b → ‖f z‖ ≤ C
hfd : DiffContOnCl ℂ f (im ⁻¹' Ioo (a - b) (a + b))
hab : a - b < a + b
hb : 0 < b
hπb : 0 < π / 2 / b
c : ℝ
hc : c < π / 2 / b
B : ℝ
hO : f =O[comap (abs ∘ re) atTop ⊓ 𝓟 (im ⁻¹' Ioo (a - b) (a + b))] fun z ↦ expR (B * expR (c * |z.re|))
d : ℝ
hcd : c < d
hd₀ : 0 < d
hb' : d * b < π / 2
aff : ℂ → ℂ := fun w ↦ ↑d * (w - ↑a * I)
g : ℝ → ℂ → ℂ := fun ε w ↦ cexp (↑ε * (cexp (aff w) + cexp (-aff w)))
ε : ℝ
ε₀ : ε < 0
δ : ℝ
δ₀ : δ < 0
hδ : ∀ ⦃w : ℂ⦄, w.im ∈ Icc (a - b) (a + b) → ‖g ε w‖ ≤ expR (δ * expR (d * |w.re|))
hg₁ : ∀ (w : ℂ), w.im = a - b ∨ w.im = a + b → ‖g ε w‖ ≤ 1
R : ℝ
hzR : |z.re| < R
hR : ∀ (w : ℂ), |w.re| = R → w.im ∈ Ioo (a - b) (a + b) → ‖g ε w • f w‖ ≤ C
hR₀ : 0 < R
hgd : Differentiable ℂ (g ε)
hd : DiffContOnCl ℂ (fun w ↦ g ε w • f w) (Ioo (-R) R ×ℂ Ioo (a - b) (a + b))
w : ℂ
him : ¬(w.im = a - b ∨ w.im = a + b)
hw : w ∈ {-R, R} ×ℂ Icc (a - b) (a + b)
hw' : w.im = a - b ∨ w.im = a + b ∨ w.im ∈ Ioo (a - b) (a + b)
⊢ ‖g ε w • f w‖ ≤ C
Module: Mathlib.Analysis.InnerProductSpace.Laplacian

Goal state before:
pp: E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : FiniteDimensional ℝ E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
⊢ Δ[Set.univ] f = Δ f

Tactic:
ext x

Goal state after:
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : FiniteDimensional ℝ E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
x : E
⊢ (Δ[Set.univ] f) x = Δ f x
Module: Mathlib.Analysis.InnerProductSpace.Laplacian

Goal state before:
pp: E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
f : E → F
ι : Type u_5
inst✝ : Fintype ι
v : OrthonormalBasis ι ℝ E
⊢ Δ f = fun x ↦ ∑ i, (iteratedFDeriv ℝ 2 f x) ![v i, v i]

Tactic:
ext x

Goal state after:
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
f : E → F
ι : Type u_5
inst✝ : Fintype ι
v : OrthonormalBasis ι ℝ E
x : E
⊢ Δ f x = ∑ i, (iteratedFDeriv ℝ 2 f x) ![v i, v i]
Module: Mathlib.Analysis.InnerProductSpace.Laplacian

Goal state before:
pp: E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f : E → F
l : F ≃L[ℝ] G
⊢ Δ (⇑l ∘ f) = ⇑l ∘ Δ f

Tactic:
ext x

Goal state after:
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f : E → F
l : F ≃L[ℝ] G
x : E
⊢ Δ (⇑l ∘ f) x = (⇑l ∘ Δ f) x
Module: Mathlib.MeasureTheory.Integral.CircleAverage

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℂ → E
c : ℂ
R η : ℝ
⊢ circleAverage f c R = (2 * π)⁻¹ • ∫ (θ : ℝ) in 0..2 * π, f (circleMap c R (θ + η))

Tactic:
intervalIntegral.integral_comp_add_right (fun θ ↦ f (circleMap c R θ))

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℂ → E
c : ℂ
R η : ℝ
⊢ circleAverage f c R = (2 * π)⁻¹ • ∫ (x : ℝ) in 0 + η..2 * π + η, f (circleMap c R x)
Module: Mathlib.Analysis.Complex.Harmonic.Analytic

Goal state before:
pp: case right
f : ℂ → ℝ
hf : HarmonicOnNhd f univ
g : ℂ → ℂ := ⋯
hg : Differentiable ℂ g
F : ℂ → ℂ
hF : F 0 = ↑(f 0) ∧ ∀ x ∈ univ, HasDerivAt F (g x) x
h₁F : ∀ (z₁ : ℂ), HasDerivAt F (g z₁) z₁
h₂F : Differentiable ℂ F
h₃F : Differentiable ℝ F
⊢ (fun z ↦ (F z).re) = f

Tactic:
ext x

Goal state after:
case right
f : ℂ → ℝ
hf : HarmonicOnNhd f univ
g : ℂ → ℂ := (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I
hg : Differentiable ℂ g
F : ℂ → ℂ
hF : F 0 = ↑(f 0) ∧ ∀ x ∈ univ, HasDerivAt F (g x) x
h₁F : ∀ (z₁ : ℂ), HasDerivAt F (g z₁) z₁
h₂F : Differentiable ℂ F
h₃F : Differentiable ℝ F
x : ℂ
⊢ (F x).re = f x
Module: Mathlib.Analysis.Complex.Harmonic.Analytic

Goal state before:
pp: f : ℂ → ℝ
hf : HarmonicOnNhd f univ
g : ℂ → ℂ := (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I
hg : Differentiable ℂ g
F : ℂ → ℂ
hF : F 0 = ↑(f 0) ∧ ∀ x ∈ univ, HasDerivAt F (g x) x
h₁F : ∀ (z₁ : ℂ), HasDerivAt F (g z₁) z₁
h₂F : Differentiable ℂ F
h₃F : Differentiable ℝ F
x y : ℂ
hy : y ∈ univ
⊢ fderiv ℝ (⇑reCLM) (F y) ∘SL fderiv ℝ F y = fderiv ℝ f y

Tactic:
ext x

Goal state after:
f : ℂ → ℝ
hf : HarmonicOnNhd f univ
g : ℂ → ℂ := (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I
hg : Differentiable ℂ g
F : ℂ → ℂ
hF : F 0 = ↑(f 0) ∧ ∀ x ∈ univ, HasDerivAt F (g x) x
h₁F : ∀ (z₁ : ℂ), HasDerivAt F (g z₁) z₁
h₂F : Differentiable ℂ F
h₃F : Differentiable ℝ F
x✝ y : ℂ
hy : y ∈ univ
x : ℂ
⊢ (fderiv ℝ (⇑reCLM) (F y) ∘SL fderiv ℝ F y) x = (fderiv ℝ f y) x
Module: Mathlib.Analysis.Complex.Harmonic.Analytic

Goal state before:
pp: f : ℂ → ℝ
x : ℂ
hf : HarmonicAt f x
ε : ℝ
h₁ε : ε > 0
h₂ε : ball x ε ⊆ {x | HarmonicAt f x}
F : ℂ → ℂ
h₁F : AnalyticOnNhd ℂ F (ball x ε)
h₂F : EqOn (fun z ↦ (F z).re) f (ball x ε)
⊢ AnalyticAt ℝ f x

Tactic:
rw [analyticAt_congr (Filter.eventually_of_mem (ball_mem_nhds x h₁ε) (fun y hy ↦ h₂F.symm hy))]

Goal state after:
f : ℂ → ℝ
x : ℂ
hf : HarmonicAt f x
ε : ℝ
h₁ε : ε > 0
h₂ε : ball x ε ⊆ {x | HarmonicAt f x}
F : ℂ → ℂ
h₁F : AnalyticOnNhd ℂ F (ball x ε)
h₂F : EqOn (fun z ↦ (F z).re) f (ball x ε)
⊢ AnalyticAt ℝ (fun y ↦ (F y).re) x