mathlib-initiative/mathlib-tactics huggingface dataset:
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
φ : ℂ → E := (∏ᶠ (c : ℂ), canonicalFactor R c ^ (divisor f (ball 0 R)) c) • f
hφ : MeromorphicOn φ (closedBall 0 R)
g : ℂ → E := toMeromorphicNFOn φ (closedBall 0 R)
h₃g : divisor g (ball 0 R) = 0
h₂g : MeromorphicNFOn g (closedBall 0 R)
h₄g : ∀ {z : ℂ}, z ∈ closedBall 0 R → meromorphicOrderAt g z ≠ ⊤
η₀ : (-divisor f (ball 0 R)).support.Finite
a : ℂ
ha : a ∈ (-divisor f (ball 0 R)).supportᶜ
h₂a : a ∈ closedBall 0 R
x : ℂ
⊢ x ∈ η₀.toFinset →
    canonicalFactor R x a ^ (-(divisor f (ball 0 R)) x) • canonicalFactor R x a ^ (divisor f (ball 0 R)) x = 1

Tactic:
hx

Goal state after:
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
φ : ℂ → E := (∏ᶠ (c : ℂ), canonicalFactor R c ^ (divisor f (ball 0 R)) c) • f
hφ : MeromorphicOn φ (closedBall 0 R)
g : ℂ → E := toMeromorphicNFOn φ (closedBall 0 R)
h₃g : divisor g (ball 0 R) = 0
h₂g : MeromorphicNFOn g (closedBall 0 R)
h₄g : ∀ {z : ℂ}, z ∈ closedBall 0 R → meromorphicOrderAt g z ≠ ⊤
η₀ : (-divisor f (ball 0 R)).support.Finite
a : ℂ
ha : a ∈ (-divisor f (ball 0 R)).supportᶜ
h₂a : a ∈ closedBall 0 R
x : ℂ
hx : x ∈ η₀.toFinset
⊢ canonicalFactor R x a ^ (-(divisor f (ball 0 R)) x) • canonicalFactor R x a ^ (divisor f (ball 0 R)) x = 1
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace X
p : E → X
cov : IsCoveringMap p
γ : C(↑I, X)
e : E
γ_0 : γ 0 = p e
⊢ ∃ Γ, p ∘ ⇑Γ = ⇑γ ∧ Γ 0 = e

Tactic:
let U x := (cov x).2.choose

Goal state after:
E : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace X
p : E → X
cov : IsCoveringMap p
γ : C(↑I, X)
e : E
γ_0 : γ 0 = p e
U : X → Set X := fun x ↦ ⋯.choose
⊢ ∃ Γ, p ∘ ⇑Γ = ⇑γ ∧ Γ 0 = e
Module: Mathlib.Analysis.Complex.Isometry

Goal state before:
pp: f : ℂ ≃ₗᵢ[ℝ] ℂ
h : f 1 = 1
h0 : f I = I ∨ f I = -I
⊢ f = LinearIsometryEquiv.refl ℝ ℂ ∨ f = conjLIE

Tactic:
refine h0.imp (fun h' : f I = I => ?_) fun h' : f I = -I => ?_ <;>
    · apply LinearIsometryEquiv.toLinearEquiv_injective
      apply Complex.basisOneI.ext'
      intro i
      fin_cases i <;> simp [h, h']

Goal state after:
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
U : Set 𝕜
hf : MeromorphicOn f U
x : 𝕜
⊢ x ∈ U → Disjoint (𝓝[≠] x) (𝓟 (U \ {x | AnalyticAt 𝕜 f x}))

Tactic:
hx

Goal state after:
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
x : 𝕜
hx : x ∈ U
⊢ Disjoint (𝓝[≠] x) (𝓟 (U \ {x | AnalyticAt 𝕜 f 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
U : Set 𝕜
hf : MeromorphicOn f U
x : 𝕜
⊢ x ∈ U → Disjoint (𝓝[≠] x) (𝓟 (U \ Subtype.val '' {u | meromorphicOrderAt f ↑u = 0 ∨ meromorphicOrderAt f ↑u = ⊤}))

Tactic:
hx

Goal state after:
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
x : 𝕜
hx : x ∈ U
⊢ Disjoint (𝓝[≠] x) (𝓟 (U \ Subtype.val '' {u | meromorphicOrderAt f ↑u = 0 ∨ meromorphicOrderAt f ↑u = ⊤}))
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
w : ℂ
f h : ℂ → E
D : ECanonicalDecomp f h R
hw : w ∈ closedBall 0 R
hR : 0 < R
B₀R : Set ℂ := ball 0 R
S₀R : Set ℂ := sphere 0 R
t₁ : Finset ℂ
ht₁ : ↑t₁ = (divisor f S₀R).support
t₂ : Finset ℂ
ht₂ : ↑t₂ = (divisor f B₀R).support
this : MeromorphicAt h w
x : ℂ
⊢ x ∈ t₂ →
    meromorphicTrailingCoeffAt (canonicalFactor R x) w ^ (divisor f B₀R) x *
        meromorphicTrailingCoeffAt (canonicalFactor R x ^ (-(divisor f B₀R) x)) w =
      1

Tactic:
hx

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
w : ℂ
f h : ℂ → E
D : ECanonicalDecomp f h R
hw : w ∈ closedBall 0 R
hR : 0 < R
B₀R : Set ℂ := ball 0 R
S₀R : Set ℂ := sphere 0 R
t₁ : Finset ℂ
ht₁ : ↑t₁ = (divisor f S₀R).support
t₂ : Finset ℂ
ht₂ : ↑t₂ = (divisor f B₀R).support
this : MeromorphicAt h w
x : ℂ
hx : x ∈ t₂
⊢ meromorphicTrailingCoeffAt (canonicalFactor R x) w ^ (divisor f B₀R) x *
      meromorphicTrailingCoeffAt (canonicalFactor R x ^ (-(divisor f B₀R) x)) w =
    1
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
w : ℂ
f h : ℂ → E
D : ECanonicalDecomp f h R
hw : w ∈ closedBall 0 R
hR : 0 < R
B₀R : Set ℂ := ball 0 R
S₀R : Set ℂ := sphere 0 R
t₁ : Finset ℂ
ht₁ : ↑t₁ = (divisor f S₀R).support
t₂ : Finset ℂ
ht₂ : ↑t₂ = (divisor f B₀R).support
this : MeromorphicAt h w
x : ℂ
⊢ x ∈ t₁ →
    meromorphicTrailingCoeffAt (fun x_1 ↦ x_1 - x) w ^ (-divisor f S₀R) x *
        meromorphicTrailingCoeffAt ((fun x_1 ↦ x_1 - x) ^ (divisor f S₀R) x) w =
      1

Tactic:
hx

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
w : ℂ
f h : ℂ → E
D : ECanonicalDecomp f h R
hw : w ∈ closedBall 0 R
hR : 0 < R
B₀R : Set ℂ := ball 0 R
S₀R : Set ℂ := sphere 0 R
t₁ : Finset ℂ
ht₁ : ↑t₁ = (divisor f S₀R).support
t₂ : Finset ℂ
ht₂ : ↑t₂ = (divisor f B₀R).support
this : MeromorphicAt h w
x : ℂ
hx : x ∈ t₁
⊢ meromorphicTrailingCoeffAt (fun x_1 ↦ x_1 - x) w ^ (-divisor f S₀R) x *
      meromorphicTrailingCoeffAt ((fun x_1 ↦ x_1 - x) ^ (divisor f S₀R) x) w =
    1
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace X
p : E → X
G : Type u_4
inst✝¹ : Group G
inst✝ : MulAction G E
hp : IsQuotientCoveringMap p G
x y : X
γ γ' : Path.Homotopic.Quotient x y
e : ↑(p ⁻¹' {x})
eq : ⋯.monodromy γ e = ⋯.monodromy γ' e
⊢ ⋯.monodromy γ = ⋯.monodromy γ'

Tactic:
ext e'
    obtain ⟨g, rfl⟩ := hp.exists_toPermFiber_eq e e'
    simp_rw [monodromy_toPermFiber, eq]

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

Goal state before:
pp: case h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
ε : ℝ
hε : 0 < ε
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalClosedStrip 0 1
BF : ℝ
hBF : ∀ a ∈ verticalClosedStrip 0 1, ‖F f ε a‖ ≤ BF
x : ℂ
⊢ x ∈ re ⁻¹' Ioo 0 1 → ‖invInterpStrip f x ε • f x‖ ≤ 1 * ‖Real.exp (BF * Real.exp (0 * |x.im|))‖

Tactic:
hx

Goal state after:
case h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
ε : ℝ
hε : 0 < ε
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalClosedStrip 0 1
BF : ℝ
hBF : ∀ a ∈ verticalClosedStrip 0 1, ‖F f ε a‖ ≤ BF
x : ℂ
hx : x ∈ re ⁻¹' Ioo 0 1
⊢ ‖invInterpStrip f x ε • f x‖ ≤ 1 * ‖Real.exp (BF * Real.exp (0 * |x.im|))‖
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
f : ℂ → E
inst✝ : NormedSpace ℂ E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
x : ℝ
⊢ x ∈ Ioi 0 →
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      ‖↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z‖

Tactic:
hx

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
f : ℂ → E
inst✝ : NormedSpace ℂ E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
x : ℝ
hx : x ∈ Ioi 0
⊢ (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
    ‖↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z‖
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
h0 : 0 ≠ sSupNormIm f 0
h1 : 0 ≠ sSupNormIm f 1
⊢ ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re

Tactic:
interpStrip_eq_of_pos f _ (lt_of_le_of_ne (sSupNormIm_nonneg f 0) h0)
        (lt_of_le_of_ne (sSupNormIm_nonneg f 1) h1)

Goal state after:
case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
h0 : 0 ≠ sSupNormIm f 0
h1 : 0 ≠ sSupNormIm f 1
⊢ ‖↑(sSupNormIm f 0) ^ (1 - z) * ↑(sSupNormIm f 1) ^ z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re
Module: Mathlib.Analysis.InnerProductSpace.Harmonic.Basic

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : FiniteDimensional ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
x✝ : E
h : HarmonicAt f x✝
x : E
⊢ Δ f x = 0 x → Δ (-f) x = 0 x

Tactic:
hx

Goal state after:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : FiniteDimensional ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
x✝ : E
h : HarmonicAt f x✝
x : E
hx : Δ f x = 0 x
⊢ Δ (-f) x = 0 x
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
f₁ f₂ : E → F
x✝ : E
h : f₁ =ᶠ[𝓝 x✝] f₂
x : E
⊢ iteratedFDeriv ℝ 2 f₁ x = iteratedFDeriv ℝ 2 f₂ x → Δ f₁ x = Δ f₂ x

Tactic:
hx

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
f₁ f₂ : E → F
x✝ : E
h : f₁ =ᶠ[𝓝 x✝] f₂
x : E
hx : iteratedFDeriv ℝ 2 f₁ x = iteratedFDeriv ℝ 2 f₂ x
⊢ Δ f₁ x = Δ f₂ x
Module: Mathlib.Analysis.Complex.HasPrimitives

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
c : ℂ
r : ℝ
f : ℂ → E
inst✝ : CompleteSpace E
f_cont : ContinuousOn f (ball c r)
z : ℂ
hz : z ∈ ball c r
this : (fun w ↦ f w - f z) =o[𝓝 z] fun x ↦ 1
⊢ (fun w ↦ ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I) - f z) =o[𝓝 z] fun w ↦ w - z

Tactic:
Asymptotics.IsLittleO

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
c : ℂ
r : ℝ
f : ℂ → E
inst✝ : CompleteSpace E
f_cont : ContinuousOn f (ball c r)
z : ℂ
hz : z ∈ ball c r
this : ∀ ⦃c : ℝ⦄, 0 < c → Asymptotics.IsBigOWith c (𝓝 z) (fun w ↦ f w - f z) fun x ↦ 1
⊢ ∀ ⦃c : ℝ⦄,
    0 < c → Asymptotics.IsBigOWith c (𝓝 z) (fun w ↦ ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I) - f z) fun w ↦ w - z
Module: Mathlib.MeasureTheory.Integral.IntervalAverage

Goal state before:
pp: f : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
inst✝ : NullSingletonClass μ
hf : ContinuousOn f [[a, b]]
hμfin : μ (Ι a b) ≠ ⊤
hμ0 : μ (Ι a b) ≠ 0
hint : IntegrableOn f (Ι a b) μ
h : a ≠ b
s : Set ℝ := uIoo a b
x : ℝ
⊢ x ∈ s → x ∈ Ι a b

Tactic:
hx

Goal state after:
f : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
inst✝ : NullSingletonClass μ
hf : ContinuousOn f [[a, b]]
hμfin : μ (Ι a b) ≠ ⊤
hμ0 : μ (Ι a b) ≠ 0
hint : IntegrableOn f (Ι a b) μ
h : a ≠ b
s : Set ℝ := uIoo a b
x : ℝ
hx : x ∈ s
⊢ x ∈ Ι a b
Module: Mathlib.Analysis.Complex.Harmonic.Analytic

Goal state before:
pp: case right
f : ℂ → ℝ
z : ℂ
R : ℝ
hf : HarmonicOnNhd f (ball z R)
hR : ¬R ≤ 0
g : ℂ → ℂ := (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I
hg : DifferentiableOn ℂ g (ball z R)
F : ℂ → ℂ
hF : F z = ↑(f z) ∧ ∀ x ∈ ball z R, HasDerivAt F (g x) x
h₁F : DifferentiableOn ℂ F (ball z R)
h₂F : DifferentiableOn ℝ F (ball z R)
x : ℂ
⊢ x ∈ ball z R → (⇑reCLM ∘ F) x = f x

Tactic:
hx

Goal state after:
case right
f : ℂ → ℝ
z : ℂ
R : ℝ
hf : HarmonicOnNhd f (ball z R)
hR : ¬R ≤ 0
g : ℂ → ℂ := (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I
hg : DifferentiableOn ℂ g (ball z R)
F : ℂ → ℂ
hF : F z = ↑(f z) ∧ ∀ x ∈ ball z R, HasDerivAt F (g x) x
h₁F : DifferentiableOn ℂ F (ball z R)
h₂F : DifferentiableOn ℝ F (ball z R)
x : ℂ
hx : x ∈ ball z R
⊢ (⇑reCLM ∘ F) x = f x
Module: Mathlib.Analysis.Complex.PhragmenLindelof

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f (Ioi 0 ×ℂ Ioi 0)
hre : ∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C
him : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C
ζ : ℂ
hζ : ζ.im ∈ Icc 0 (π / 2)
hz_re : 0 ≤ (cexp ζ).re
hz_im : 0 ≤ (cexp ζ).im
hzne : cexp ζ ≠ 0
H : MapsTo cexp (im ⁻¹' Ioo 0 (π / 2)) (Ioi 0 ×ℂ Ioi 0)
c : ℝ
hc : c < 2
B : ℝ
hO : f =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Ioi 0)] fun z ↦ expR (B * ‖z‖ ^ c)
x : ℝ
⊢ 0 ≤ x → ∀ (a : ℂ), a.re = x → a ∈ im ⁻¹' Ioo 0 (π / 2) → B * expR (a.re * c) ≤ max B 0 * expR (c * |a.re|)

Tactic:
hx

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f (Ioi 0 ×ℂ Ioi 0)
hre : ∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C
him : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C
ζ : ℂ
hζ : ζ.im ∈ Icc 0 (π / 2)
hz_re : 0 ≤ (cexp ζ).re
hz_im : 0 ≤ (cexp ζ).im
hzne : cexp ζ ≠ 0
H : MapsTo cexp (im ⁻¹' Ioo 0 (π / 2)) (Ioi 0 ×ℂ Ioi 0)
c : ℝ
hc : c < 2
B : ℝ
hO : f =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Ioi 0)] fun z ↦ expR (B * ‖z‖ ^ c)
x : ℝ
hx : 0 ≤ x
⊢ ∀ (a : ℂ), a.re = x → a ∈ im ⁻¹' Ioo 0 (π / 2) → B * expR (a.re * c) ≤ max B 0 * expR (c * |a.re|)
Module: Mathlib.Analysis.Complex.Harmonic.MeanValue

Goal state before:
pp: F✝ : Type u_1
inst✝² : NormedAddCommGroup F✝
inst✝¹ : NormedSpace ℝ F✝
inst✝ : CompleteSpace F✝
f : ℂ → F✝
c : ℂ
R : ℝ
hf : HarmonicOnNhd f (closedBall c |R|)
h : CircleIntegrable f c R
g : StrongDual ℝ F✝
e : ℝ
h₁e : 0 < e
h₂e : ball c (e + |R|) ⊆ {x | HarmonicAt (⇑g ∘ f) x}
F : ℂ → ℂ
h₁F : AnalyticOnNhd ℂ F (ball c (e + |R|))
h₂F : Set.EqOn (fun z ↦ (F z).re) (⇑g ∘ f) (ball c (e + |R|))
x : ℂ
⊢ x ∈ closure (ball c |R|) → DifferentiableWithinAt ℂ F (closure (ball c |R|)) x

Tactic:
hx

Goal state after:
F✝ : Type u_1
inst✝² : NormedAddCommGroup F✝
inst✝¹ : NormedSpace ℝ F✝
inst✝ : CompleteSpace F✝
f : ℂ → F✝
c : ℂ
R : ℝ
hf : HarmonicOnNhd f (closedBall c |R|)
h : CircleIntegrable f c R
g : StrongDual ℝ F✝
e : ℝ
h₁e : 0 < e
h₂e : ball c (e + |R|) ⊆ {x | HarmonicAt (⇑g ∘ f) x}
F : ℂ → ℂ
h₁F : AnalyticOnNhd ℂ F (ball c (e + |R|))
h₂F : Set.EqOn (fun z ↦ (F z).re) (⇑g ∘ f) (ball c (e + |R|))
x : ℂ
hx : x ∈ closure (ball c |R|)
⊢ DifferentiableWithinAt ℂ F (closure (ball c |R|)) x
Module: Mathlib.Analysis.Complex.Harmonic.MeanValue

Goal state before:
pp: F : Type u_1
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℂ → F
c : ℂ
R : ℝ
hf : HarmonicContOnCl f (ball c |R|)
h : CircleIntegrable f c R
g : StrongDual ℝ F
hR : ¬R = 0
x : ℂ
⊢ x ∈ {z | ‖z - c‖ ∈ Set.Ioc 0 |R|} → x ∈ closure (ball c |R|)

Tactic:
hx

Goal state after:
F : Type u_1
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℂ → F
c : ℂ
R : ℝ
hf : HarmonicContOnCl f (ball c |R|)
h : CircleIntegrable f c R
g : StrongDual ℝ F
hR : ¬R = 0
x : ℂ
hx : x ∈ {z | ‖z - c‖ ∈ Set.Ioc 0 |R|}
⊢ x ∈ closure (ball c |R|)
Module: Mathlib.Analysis.Complex.PhragmenLindelof

Goal state before:
pp: case inr.refine_3
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f (Ioi 0 ×ℂ Ioi 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Ioi 0)] fun z ↦ expR (B * ‖z‖ ^ c)
hre : ∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C
him : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C
ζ : ℂ
hζ : ζ.im ∈ Icc 0 (π / 2)
hz_re : 0 ≤ (cexp ζ).re
hz_im : 0 ≤ (cexp ζ).im
hzne : cexp ζ ≠ 0
H : MapsTo cexp (im ⁻¹' Ioo 0 (π / 2)) (Ioi 0 ×ℂ Ioi 0)
⊢ ∀ (z : ℂ), z.im = π / 2 → ‖(f ∘ cexp) z‖ ≤ C

Tactic:
· -- If `ζ.im = π / 2`, then `Complex.exp ζ` is a purely imaginary number with positive `im`
    intro ζ hζ
    rw [← re_add_im ζ, hζ, comp_apply, exp_add_mul_I, ← ofReal_cos, ← ofReal_sin,
      Real.cos_pi_div_two, Real.sin_pi_div_two, ofReal_zero, ofReal_one, one_mul, zero_add, ←
      ofReal_exp]
    exact him _ (Real.exp_pos _).le

Goal state after:
Module: Mathlib.Analysis.Complex.Harmonic.Poisson

Goal state before:
pp: f : ℂ → ℝ
c w : ℂ
R : ℝ
hf : HarmonicOnNhd f (closedBall c R)
hw : w ∈ ball c R
e : ℝ
h₁e : 0 < e
h₂e : ball c (e + R) ⊆ {x | HarmonicAt f x}
F : ℂ → ℂ
h₁F : AnalyticOnNhd ℂ F (ball c (e + R))
h₂F : Set.EqOn (fun z ↦ (F z).re) f (ball c (e + R))
x : ℂ
⊢ x ∈ closure (ball c R) → DifferentiableWithinAt ℂ F (closure (ball c R)) x

Tactic:
hx

Goal state after:
f : ℂ → ℝ
c w : ℂ
R : ℝ
hf : HarmonicOnNhd f (closedBall c R)
hw : w ∈ ball c R
e : ℝ
h₁e : 0 < e
h₂e : ball c (e + R) ⊆ {x | HarmonicAt f x}
F : ℂ → ℂ
h₁F : AnalyticOnNhd ℂ F (ball c (e + R))
h₂F : Set.EqOn (fun z ↦ (F z).re) f (ball c (e + R))
x : ℂ
hx : x ∈ closure (ball c R)
⊢ DifferentiableWithinAt ℂ F (closure (ball c R)) x
Module: Mathlib.Analysis.Complex.Harmonic.Poisson

Goal state before:
pp: f : ℂ → ℝ
c w : ℂ
R : ℝ
hf : HarmonicOnNhd f (closedBall c R)
hw : w ∈ ball c R
e : ℝ
h₁e : 0 < e
h₂e : ball c (e + R) ⊆ {x | HarmonicAt f x}
F : ℂ → ℂ
h₁F : AnalyticOnNhd ℂ F (ball c (e + R))
h₂F : Set.EqOn (fun z ↦ (F z).re) f (ball c (e + R))
h₃F : DifferentiableOn ℂ F (closure (ball c R))
x : ℂ
⊢ x ∈ sphere c R →
    (re ∘ herglotzRieszKernel c w • f) x = (⇑reCLM ∘ fun z ↦ ((z - c + (w - c)) / (z - c - (w - c))).re • F z) x

Tactic:
hx

Goal state after:
f : ℂ → ℝ
c w : ℂ
R : ℝ
hf : HarmonicOnNhd f (closedBall c R)
hw : w ∈ ball c R
e : ℝ
h₁e : 0 < e
h₂e : ball c (e + R) ⊆ {x | HarmonicAt f x}
F : ℂ → ℂ
h₁F : AnalyticOnNhd ℂ F (ball c (e + R))
h₂F : Set.EqOn (fun z ↦ (F z).re) f (ball c (e + R))
h₃F : DifferentiableOn ℂ F (closure (ball c R))
x : ℂ
hx : x ∈ sphere c R
⊢ (re ∘ herglotzRieszKernel c w • f) x = (⇑reCLM ∘ fun z ↦ ((z - c + (w - c)) / (z - c - (w - c))).re • F z) x
Module: Mathlib.Analysis.InnerProductSpace.Harmonic.Constructions

Goal state before:
pp: z : ℂ
g : ℂ → ℂ
h₁g : AnalyticAt ℂ g z
h₂g : g z ≠ 0
h₃g : g z ∈ slitPlane
t₀ : g ⁻¹' (slitPlane ∩ {y | y ≠ 0}) ∈ 𝓝 z
x : ℂ
⊢ x ∈ g ⁻¹' (slitPlane ∩ {y | y ≠ 0}) →
    (⇑reCLM ∘ log ∘ (⇑conjCLE ∘ g * g)) x = (⇑reCLM ∘ (log ∘ ⇑conjCLE ∘ g + log ∘ g)) x

Tactic:
hx

Goal state after:
z : ℂ
g : ℂ → ℂ
h₁g : AnalyticAt ℂ g z
h₂g : g z ≠ 0
h₃g : g z ∈ slitPlane
t₀ : g ⁻¹' (slitPlane ∩ {y | y ≠ 0}) ∈ 𝓝 z
x : ℂ
hx : x ∈ g ⁻¹' (slitPlane ∩ {y | y ≠ 0})
⊢ (⇑reCLM ∘ log ∘ (⇑conjCLE ∘ g * g)) x = (⇑reCLM ∘ (log ∘ ⇑conjCLE ∘ g + log ∘ g)) x
Module: Mathlib.Analysis.SpecialFunctions.Log.NegMulLog

Goal state before:
pp: x : ℝ
⊢ x ∈ Set.Ioi 0 → deriv (fun x ↦ x * log x) x = (fun x ↦ log x + 1) x

Tactic:
hx

Goal state after:
x : ℝ
hx : x ∈ Set.Ioi 0
⊢ deriv (fun x ↦ x * log x) x = (fun x ↦ log x + 1) x
Module: Mathlib.Analysis.InnerProductSpace.Harmonic.Constructions

Goal state before:
pp: case right
z : ℂ
g : ℂ → ℂ
h₁g : AnalyticAt ℂ g z
h₂g : g z ≠ 0
h₃g : g z ∈ slitPlane
t₀ : g ⁻¹' (slitPlane ∩ {y | y ≠ 0}) ∈ 𝓝 z
x : ℂ
⊢ x ∈ g ⁻¹' (slitPlane ∩ {0}ᶜ) →
    (⇑reCLM ∘ (log ∘ ⇑conjCLE ∘ g + log ∘ g)) x = (⇑reCLM ∘ (⇑conjCLE ∘ log ∘ g + log ∘ g)) x

Tactic:
hx

Goal state after:
case right
z : ℂ
g : ℂ → ℂ
h₁g : AnalyticAt ℂ g z
h₂g : g z ≠ 0
h₃g : g z ∈ slitPlane
t₀ : g ⁻¹' (slitPlane ∩ {y | y ≠ 0}) ∈ 𝓝 z
x : ℂ
hx : x ∈ g ⁻¹' (slitPlane ∩ {0}ᶜ)
⊢ (⇑reCLM ∘ (log ∘ ⇑conjCLE ∘ g + log ∘ g)) x = (⇑reCLM ∘ (⇑conjCLE ∘ log ∘ g + log ∘ g)) x
Module: Mathlib.Analysis.SpecialFunctions.Log.NegMulLog

Goal state before:
pp: x : ℝ
⊢ x ∈ interior (Set.Ici 0) → 0 < deriv^[2] (fun x ↦ x * log x) x

Tactic:
hx

Goal state after:
x : ℝ
hx : x ∈ interior (Set.Ici 0)
⊢ 0 < deriv^[2] (fun x ↦ x * log x) x
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: a b r : ℝ
h : -1 < r
c : ℝ
hc : 0 ≤ c
x : ℝ
⊢ x ∈ Ioo 0 c → HasDerivAt (fun x ↦ x ^ (r + 1) / (r + 1)) (x ^ r) x

Tactic:
hx

Goal state after:
a b r : ℝ
h : -1 < r
c : ℝ
hc : 0 ≤ c
x : ℝ
hx : x ∈ Ioo 0 c
⊢ HasDerivAt (fun x ↦ x ^ (r + 1) / (r + 1)) (x ^ r) x
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: a b r : ℝ
h : -1 < r
c : ℝ
hc : 0 ≤ c
hderiv : ∀ x ∈ Ioo 0 c, HasDerivAt (fun x ↦ x ^ (r + 1) / (r + 1)) (x ^ r) x
x : ℝ
⊢ x ∈ Ioo 0 c → 0 ≤ x ^ r

Tactic:
hx

Goal state after:
a b r : ℝ
h : -1 < r
c : ℝ
hc : 0 ≤ c
hderiv : ∀ x ∈ Ioo 0 c, HasDerivAt (fun x ↦ x ^ (r + 1) / (r + 1)) (x ^ r) x
x : ℝ
hx : x ∈ Ioo 0 c
⊢ 0 ≤ x ^ r
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: case inr
a b r : ℝ
h : -1 < r
this : ∀ (c : ℝ), 0 ≤ c → IntervalIntegrable (fun x ↦ x ^ r) volume 0 c
c : ℝ
hc : c ≤ 0
m : IntegrableOn (cos (r * π) • fun x ↦ x ^ r) (Ι 0 (-c)) volume
x : ℝ
⊢ x ∈ Ι 0 (-c) → (cos (r * π) • fun x ↦ x ^ r) x = (fun x ↦ (-x) ^ r) x

Tactic:
hx

Goal state after:
case inr
a b r : ℝ
h : -1 < r
this : ∀ (c : ℝ), 0 ≤ c → IntervalIntegrable (fun x ↦ x ^ r) volume 0 c
c : ℝ
hc : c ≤ 0
m : IntegrableOn (cos (r * π) • fun x ↦ x ^ r) (Ι 0 (-c)) volume
x : ℝ
hx : x ∈ Ι 0 (-c)
⊢ (cos (r * π) • fun x ↦ x ^ r) x = (fun x ↦ (-x) ^ r) x
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: s t : ℝ
ht : 0 < t
h : s ≤ -1
H : IntegrableOn (fun x ↦ x ^ s) (Ioo 0 t) volume
I : 0 < min 1 t
H' : IntegrableOn (fun x ↦ x ^ s) (Ioo 0 (min 1 t)) volume
x : ℝ
⊢ x ∈ Ioo 0 (min 1 t) → ‖x⁻¹‖ ≤ x ^ s

Tactic:
hx

Goal state after:
s t : ℝ
ht : 0 < t
h : s ≤ -1
H : IntegrableOn (fun x ↦ x ^ s) (Ioo 0 t) volume
I : 0 < min 1 t
H' : IntegrableOn (fun x ↦ x ^ s) (Ioo 0 (min 1 t)) volume
x : ℝ
hx : x ∈ Ioo 0 (min 1 t)
⊢ ‖x⁻¹‖ ≤ x ^ s
Module: Mathlib.Analysis.Complex.Poisson

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
inst✝ : CompleteSpace E
hf : DiffContOnCl ℂ f (ball 0 R)
hw : w ∈ ball 0 R
hR : 0 < R
h₁w : w ≠ 0
W : ℂ := ↑R * cexp (↑w.arg * I)
q : ℝ := ‖w‖ / R
h₁q : 0 < q
h₂q : q < 1
η₀ : ∀ {x : ℂ}, ‖x‖ ≤ R → ↑q * x - W ≠ 0
h0 : ∮ (z : ℂ) in C(0, R), z⁻¹ • (↑q * z / (↑q * z - W)) • f z = 0
z : ℂ
hz : z ∈ sphere 0 |R|
hzR : ‖z‖ = R
⊢ ↑((z + ↑‖w‖ * cexp (↑w.arg * I)) / (z - ↑‖w‖ * cexp (↑w.arg * I))).re =
    z / (z - ↑‖w‖ * cexp (↑w.arg * I)) -
      ↑‖↑‖w‖ * cexp (↑w.arg * I)‖ / ↑R * z /
        (↑‖↑‖w‖ * cexp (↑w.arg * I)‖ / ↑R * z - ↑R * cexp (↑(↑‖w‖ * cexp (↑w.arg * I)).arg * I))

Tactic:
← norm_mul_exp_arg_mul_I z,

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
inst✝ : CompleteSpace E
hf : DiffContOnCl ℂ f (ball 0 R)
hw : w ∈ ball 0 R
hR : 0 < R
h₁w : w ≠ 0
W : ℂ := ↑R * cexp (↑w.arg * I)
q : ℝ := ‖w‖ / R
h₁q : 0 < q
h₂q : q < 1
η₀ : ∀ {x : ℂ}, ‖x‖ ≤ R → ↑q * x - W ≠ 0
h0 : ∮ (z : ℂ) in C(0, R), z⁻¹ • (↑q * z / (↑q * z - W)) • f z = 0
z : ℂ
hz : z ∈ sphere 0 |R|
hzR : ‖z‖ = R
⊢ ↑((↑‖z‖ * cexp (↑z.arg * I) + ↑‖w‖ * cexp (↑w.arg * I)) / (↑‖z‖ * cexp (↑z.arg * I) - ↑‖w‖ * cexp (↑w.arg * I))).re =
    ↑‖z‖ * cexp (↑z.arg * I) / (↑‖z‖ * cexp (↑z.arg * I) - ↑‖w‖ * cexp (↑w.arg * I)) -
      ↑‖↑‖w‖ * cexp (↑w.arg * I)‖ / ↑R * (↑‖z‖ * cexp (↑z.arg * I)) /
        (↑‖↑‖w‖ * cexp (↑w.arg * I)‖ / ↑R * (↑‖z‖ * cexp (↑z.arg * I)) -
          ↑R * cexp (↑(↑‖w‖ * cexp (↑w.arg * I)).arg * I))
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: a b : ℝ
μ : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
r : ℂ
h : 0 ≤ r.re
h2 : ¬0 ∉ [[a, b]]
h' : 0 = r.re
c : ℝ
hc : c < 0
x : ℝ
⊢ x ∈ Ioo c 0 → ‖Complex.exp (↑π * Complex.I * r)‖ = ‖↑x ^ r‖

Tactic:
hx

Goal state after:
a b : ℝ
μ : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
r : ℂ
h : 0 ≤ r.re
h2 : ¬0 ∉ [[a, b]]
h' : 0 = r.re
c : ℝ
hc : c < 0
x : ℝ
hx : x ∈ Ioo c 0
⊢ ‖Complex.exp (↑π * Complex.I * r)‖ = ‖↑x ^ r‖