mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: R : ℝ
w : ℂ
hw : w ∈ ball 0 R
z : ℂ
⊢ (divisor (canonicalFactor R w) (ball 0 R)) z = (-locallyFinsuppWithin.restrict (locallyFinsuppWithin.single w 1) ⋯) z

Tactic:
by_cases hz : z ∈ ball 0 R

Goal state after:
case pos
R : ℝ
w : ℂ
hw : w ∈ ball 0 R
z : ℂ
hz : z ∈ ball 0 R
⊢ (divisor (canonicalFactor R w) (ball 0 R)) z = (-locallyFinsuppWithin.restrict (locallyFinsuppWithin.single w 1) ⋯) z

case neg
R : ℝ
w : ℂ
hw : w ∈ ball 0 R
z : ℂ
hz : z ∉ ball 0 R
⊢ (divisor (canonicalFactor R w) (ball 0 R)) z = (-locallyFinsuppWithin.restrict (locallyFinsuppWithin.single w 1) ⋯) z
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: R : ℝ
w : ℂ
hw : w ∈ ball 0 R
z : ℂ
hz : z ∈ ball 0 R
h₂z : z ≠ w
⊢ meromorphicOrderAt (canonicalFactor R w) z = 0

Tactic:
rw [(meromorphicNFOn_canonicalFactor hw (Set.mem_univ z)).meromorphicOrderAt_eq_zero_iff]

Goal state after:
R : ℝ
w : ℂ
hw : w ∈ ball 0 R
z : ℂ
hz : z ∈ ball 0 R
h₂z : z ≠ w
⊢ canonicalFactor R w z ≠ 0
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: case succ
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
𝕜' : Type u_4
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
f : 𝕜 → 𝕜'
x : 𝕜
hf : MeromorphicAt f x
n✝ : ℕ
a✝ : meromorphicOrderAt (f ^ n✝) x = ↑n✝ * meromorphicOrderAt f x
⊢ meromorphicOrderAt (f ^ (n✝ + 1)) x = ↑(n✝ + 1) * meromorphicOrderAt f x

Tactic:
case succ n hn =>
    simp only [pow_add, pow_one, meromorphicOrderAt_mul (hf.pow n) hf, hn, Nat.cast_add,
      Nat.cast_one]
    cases meromorphicOrderAt f x
    · aesop
    · norm_cast
      simp only [Nat.cast_add, Nat.cast_one]
      ring

Goal state after:
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
z : ℂ
hR : 0 < R
c : ℂ
⊢ meromorphicOrderAt (canonicalFactor R c ^ (divisor f (ball 0 R)) c) z ≠ ⊤

Tactic:
rw [meromorphicOrderAt_zpow (meromorphic_canonicalFactor R c z)]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
f : ℂ → E
z : ℂ
hR : 0 < R
c : ℂ
⊢ ↑((divisor f (ball 0 R)) c) * meromorphicOrderAt (canonicalFactor R c) z ≠ ⊤
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f g : 𝕜 → E
D : locallyFinsuppWithin U ℤ
hg : ∀ (u : ↑U), g ↑u ≠ 0
h : f =ᶠ[codiscreteWithin U] (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ D u) • g
⊢ (fun x ↦ log ‖f x‖) =ᶠ[codiscreteWithin U] (∑ᶠ (u : 𝕜), fun x ↦ ↑(D u) * log ‖x - u‖) + fun x ↦ log ‖g x‖

Tactic:
have t₁ : (fun u ↦ (D u * log ‖· - u‖)).support = D.support := by
    ext u
    rw [← not_iff_not]
    simp only [ne_eq, not_not, Function.mem_support]
    constructor <;> intro hx
    · obtain ⟨y, hy⟩ := NormedField.exists_one_lt_norm 𝕜
      have := congrFun hx (y + u)
      simp only [add_sub_cancel_right, Pi.zero_apply, mul_eq_zero, Int.cast_eq_zero, log_eq_zero,
        norm_eq_zero] at this
      rcases this with h | h | h | h
      · assumption
      · simp only [h, norm_zero] at hy
        linarith
      · simp only [h, lt_self_iff_false] at hy
      · simp only [h, lt_neg_self_iff] at hy
        linarith
    · simp_all [Pi.zero_def]

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f g : 𝕜 → E
D : locallyFinsuppWithin U ℤ
hg : ∀ (u : ↑U), g ↑u ≠ 0
h : f =ᶠ[codiscreteWithin U] (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ D u) • g
t₁ : (support fun u x ↦ ↑(D u) * log ‖x - u‖) = D.support
⊢ (fun x ↦ log ‖f x‖) =ᶠ[codiscreteWithin U] (∑ᶠ (u : 𝕜), fun x ↦ ↑(D u) * log ‖x - u‖) + fun x ↦ log ‖g x‖
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
z : ℂ
hz : z ∈ ball 0 R
this : (divisor g (ball 0 R)) z = 0
⊢ meromorphicOrderAt g z = 0

Tactic:
rw [divisor_apply (fun x hx ↦ (h₂g (ball_subset_closedBall hx)).meromorphicAt) hz] at this

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
z : ℂ
hz : z ∈ ball 0 R
this : (meromorphicOrderAt g z).untop₀ = 0
⊢ meromorphicOrderAt g z = 0
Module: Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Basic

Goal state before:
pp: case mp.inr
R : Type u_1
n : Type u_2
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommRing R
g : GL n R
hg : g ∈ Subgroup.center (GL n R)
hn : Nonempty n
a : R
ha : (Matrix.scalar n) a = ↑g
b : R
hb : (Matrix.scalar n) b = ↑g⁻¹
hab : a * b = 1
⊢ (scalar n) { val := a, inv := b, val_inv := hab, inv_val := ⋯ } = g

Tactic:
simp [Units.ext_iff, ← ha]

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

Goal state before:
pp: f : ℂ → ℂ
z : ℂ
⊢ ConformalAt f z ↔
    (DifferentiableAt ℂ f z ∨ DifferentiableAt ℂ (f ∘ ⇑(starRingEnd ℂ)) ((starRingEnd ℂ) z)) ∧ fderiv ℝ f z ≠ 0

Tactic:
conformalAt_iff_isConformalMap_fderiv

Goal state after:
f : ℂ → ℂ
z : ℂ
⊢ IsConformalMap (fderiv ℝ f z) ↔
    (DifferentiableAt ℂ f z ∨ DifferentiableAt ℂ (f ∘ ⇑(starRingEnd ℂ)) ((starRingEnd ℂ) z)) ∧ fderiv ℝ f z ≠ 0
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
⊢ ∀ u ∈ closedBall 0 R, meromorphicTrailingCoeffAt f 0 ≠ 0

Tactic:
simp only [hR.symm, closedBall_zero, mem_singleton_iff, ne_eq, forall_eq]

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
⊢ ¬meromorphicTrailingCoeffAt f 0 = 0
Module: Mathlib.Analysis.Complex.Conformal

Goal state before:
pp: case refine_1
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
x : ℂ
s : Set ℂ
hs : UniqueDiffWithinAt ℝ s x
h : DifferentiableWithinAt ℂ f s x
⊢ (fderivWithin ℝ f s x) I = I • (fderivWithin ℝ f s x) 1

Tactic:
simp only [← h.restrictScalars_fderivWithin ℝ hs, ContinuousLinearMap.coe_restrictScalars']
    rw [(by simp : I = I • 1), (fderivWithin ℂ f s x).map_smul]
    simp

Goal state after:
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
e₀ e₁ : E
⊢ Injective fun γ ↦ γ.map { toFun := p, continuous_toFun := ⋯ }

Tactic:
refine Quotient.ind₂ fun γ₀ γ₁ ↦ ?_
  dsimp only
  simp only [Path.Homotopic.Quotient.mk''_eq_mk]
  simp_rw [← Path.Homotopic.Quotient.mk_map]
  iterate 2 rw [Path.Homotopic.Quotient.eq]
  exact (cov.homotopicRel_iff_comp ⟨0, .inl rfl, γ₀.source.trans γ₁.source.symm⟩).mpr

Goal state after:
Module: Mathlib.Analysis.Distribution.DerivNotation

Goal state before:
pp: case succ
V : Type u_11
E : Type u_12
inst✝ : LineDeriv V E E
y : V
f : E
n : ℕ
IH : ∂^{fun x ↦ y} f = ∂_{y}^[n] f
⊢ ∂_{y} (∂^{tail fun x ↦ y} f) = ∂_{y}^[n + 1] f

Tactic:
Function.iterate_succ_apply'

Goal state after:
case succ
V : Type u_11
E : Type u_12
inst✝ : LineDeriv V E E
y : V
f : E
n : ℕ
IH : ∂^{fun x ↦ y} f = ∂_{y}^[n] f
⊢ ∂_{y} (∂^{tail fun x ↦ y} f) = ∂_{y} (∂_{y}^[n] f)
Module: Mathlib.Analysis.InnerProductSpace.Laplacian

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
e : E
s : Set E
f : E → F
hs : UniqueDiffOn 𝕜 s
he : e ∈ s
e₁ e₂ : E
⊢ ((bilinearIteratedFDerivWithinTwo 𝕜 f s e) e₁) e₂ = (iteratedFDerivWithin 𝕜 2 f s e) ![e₁, e₂]

Tactic:
simp [iteratedFDerivWithin_two_apply f hs he ![e₁, e₂], bilinearIteratedFDerivWithinTwo]

Goal state after:
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
s : Set E
h₁ : ContDiffWithinAt ℝ 2 f₁ s x
h₂ : ContDiffWithinAt ℝ 2 f₂ s x
hs : UniqueDiffOn ℝ s
hx : x ∈ s
⊢ (Δ[s] f₁ + f₂) x = (Δ[s] f₁) x + (Δ[s] f₂) x

Tactic:
by
  simp [laplacianWithin_eq_iteratedFDerivWithin_stdOrthonormalBasis _ hs hx,
    ← Finset.sum_add_distrib, iteratedFDerivWithin_add_apply h₁ h₂ hs hx]

Goal state after:
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
s : Set E
h₁ : ContDiffWithinAt ℝ 2 f₁ s x
h₂ : ContDiffWithinAt ℝ 2 f₂ s x
hs : UniqueDiffOn ℝ s
hx : x ∈ s
⊢ Δ[s] f₁ + f₂ =ᶠ[𝓝[s] x] (Δ[s] f₁) + Δ[s] f₂

Tactic:
← s.insert_eq_of_mem 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
s : Set E
h₁ : ContDiffWithinAt ℝ 2 f₁ s x
h₂ : ContDiffWithinAt ℝ 2 f₂ s x
hs : UniqueDiffOn ℝ s
hx : x ∈ s
⊢ Δ[s] f₁ + f₂ =ᶠ[𝓝[insert x s] x] (Δ[s] f₁) + Δ[s] f₂
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
s : Set E
h₁ : ContDiffWithinAt ℝ 2 f₁ s x
h₂ : ContDiffWithinAt ℝ 2 f₂ s x
hs : UniqueDiffOn ℝ s
hx : x ∈ s
⊢ Δ[s] f₁ - f₂ =ᶠ[𝓝[s] x] (Δ[s] f₁) - Δ[s] f₂

Tactic:
← s.insert_eq_of_mem 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
s : Set E
h₁ : ContDiffWithinAt ℝ 2 f₁ s x
h₂ : ContDiffWithinAt ℝ 2 f₂ s x
hs : UniqueDiffOn ℝ s
hx : x ∈ s
⊢ Δ[s] f₁ - f₂ =ᶠ[𝓝[insert x s] x] (Δ[s] f₁) - Δ[s] f₂
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
h : IsConservativeOn f (ball c r)
⊢ HasDerivAt (fun w ↦ c.wedgeIntegral w f) (f z) z

Tactic:
hasDerivAt_iff_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
h : IsConservativeOn f (ball c r)
⊢ (fun x' ↦ c.wedgeIntegral x' f - c.wedgeIntegral z f - (x' - z) • f z) =o[𝓝 z] fun x' ↦ x' - z
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)
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
z : ℂ
hz : z ∈ im ⁻¹' Ioo 0 (π / 2)
⊢ 0 < expR z.re * Real.cos z.im ∧ 0 < expR z.re * Real.sin z.im

Tactic:
have : 0 < Real.cos z.im := Real.cos_pos_of_mem_Ioo ⟨by linarith [hz.1, hz.2], hz.2⟩

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)
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
z : ℂ
hz : z ∈ im ⁻¹' Ioo 0 (π / 2)
this : 0 < Real.cos z.im
⊢ 0 < expR z.re * Real.cos z.im ∧ 0 < expR z.re * Real.sin z.im
Module: Mathlib.Analysis.Complex.UpperHalfPlane.Basic

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
z : ℍ
⊢ 0 < (-↑n / ↑z).im

Tactic:
by
  suffices 0 < n * z.im / Complex.normSq z by simpa [Complex.div_im, neg_div]
  positivity [NeZero.ne n, z.normSq_pos]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.NegMulLog

Goal state before:
pp: case inr
this : Set.univ = Set.Iio 0 ∪ Set.Ioi 0 ∪ {0}
⊢ Filter.Tendsto (fun x ↦ log x * x) (𝓝[<] 0 ⊔ 𝓝[>] 0 ⊔ 𝓝[{0}] 0) (𝓝 0)

Tactic:
simp only [nhdsWithin_singleton, Filter.tendsto_sup]

Goal state after:
case inr
this : Set.univ = Set.Iio 0 ∪ Set.Ioi 0 ∪ {0}
⊢ (Filter.Tendsto (fun x ↦ log x * x) (𝓝[<] 0) (𝓝 0) ∧ Filter.Tendsto (fun x ↦ log x * x) (𝓝[>] 0) (𝓝 0)) ∧
    Filter.Tendsto (fun x ↦ log x * x) (pure 0) (𝓝 0)
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: s t : ℝ
ht : 0 < t
⊢ IntegrableOn (fun x ↦ x ^ s) (Ioo 0 t) volume ↔ -1 < s

Tactic:
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩

Goal state after:
case refine_1
s t : ℝ
ht : 0 < t
h : IntegrableOn (fun x ↦ x ^ s) (Ioo 0 t) volume
⊢ -1 < s

case refine_2
s t : ℝ
ht : 0 < t
h : -1 < s
⊢ IntegrableOn (fun x ↦ x ^ s) (Ioo 0 t) volume
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
hd : DiffContOnCl ℂ f {z | 0 < z.re}
hexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)
hre : Tendsto (fun x ↦ f ↑x) atTop (𝓝 0)
him : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
hle : ∀ (C' : ℝ), (∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C') → ∀ (z : ℂ), 0 ≤ z.re → ‖f z‖ ≤ max C C'
hfc : ContinuousOn (fun x ↦ f ↑x) (Ici 0)
h₀ : ∃ x, 0 ≤ x ∧ f ↑x ≠ 0
⊢ ∃ x, 0 ≤ x ∧ ∀ (y : ℝ), 0 ≤ y → ‖f ↑y‖ ≤ ‖f ↑x‖

Tactic:
rcases h₀ with ⟨x₀, hx₀, hne⟩

Goal state after:
case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f {z | 0 < z.re}
hexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)
hre : Tendsto (fun x ↦ f ↑x) atTop (𝓝 0)
him : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
hle : ∀ (C' : ℝ), (∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C') → ∀ (z : ℂ), 0 ≤ z.re → ‖f z‖ ≤ max C C'
hfc : ContinuousOn (fun x ↦ f ↑x) (Ici 0)
x₀ : ℝ
hx₀ : 0 ≤ x₀
hne : f ↑x₀ ≠ 0
⊢ ∃ x, 0 ≤ x ∧ ∀ (y : ℝ), 0 ≤ y → ‖f ↑y‖ ≤ ‖f ↑x‖
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: s : ℂ
t : ℝ
ht : 0 < t
⊢ IntegrableOn (fun x ↦ ↑x ^ s) (Ioo 0 t) volume ↔ -1 < s.re

Tactic:
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩

Goal state after:
case refine_1
s : ℂ
t : ℝ
ht : 0 < t
h : IntegrableOn (fun x ↦ ↑x ^ s) (Ioo 0 t) volume
⊢ -1 < s.re

case refine_2
s : ℂ
t : ℝ
ht : 0 < t
h : -1 < s.re
⊢ IntegrableOn (fun x ↦ ↑x ^ s) (Ioo 0 t) volume
Module: Mathlib.Analysis.Complex.Poisson

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
hg : CircleIntegrable f 0 R
hw : w ∈ ball 0 R
hR : 0 < R
d : ℝ
hd : d > 0
hsub : ball w d ⊆ ball 0 R
hdist : ∀ x ∈ ball w d, ∀ (θ : ℝ), d ≤ ‖circleMap 0 R θ - x‖
hgm : AEStronglyMeasurable (fun θ ↦ f (circleMap 0 R θ)) (volume.restrict (uIoc 0 (2 * π)))
⊢ AEStronglyMeasurable (fun θ ↦ (2 * circleMap 0 R θ / (circleMap 0 R θ - w) ^ 2) • f (circleMap 0 R θ))
    (volume.restrict (uIoc 0 (2 * π)))

Tactic:
exact (Measurable.aestronglyMeasurable (by fun_prop)).smul hgm

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
hg : CircleIntegrable f 0 R
hw : w ∈ ball 0 R
hR : 0 < R
d : ℝ
hd : d > 0
hsub : ball w d ⊆ ball 0 R
hdist : ∀ x ∈ ball w d, ∀ (θ : ℝ), d ≤ ‖circleMap 0 R θ - x‖
hgm : AEStronglyMeasurable (fun θ ↦ f (circleMap 0 R θ)) (volume.restrict (uIoc 0 (2 * π)))
θ : ℝ
a✝ : θ ∈ uIoc 0 (2 * π)
x : ℂ
hx : x ∈ ball w d
h₁ : ‖2‖ = 2
⊢ ‖2‖ * R / ‖(circleMap 0 R θ - x) ^ 2‖ * ‖f (circleMap 0 R θ)‖ ≤ 2 * R * (d ^ 2)⁻¹ * ‖f (circleMap 0 R θ)‖

Tactic:
norm_pow,

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
hg : CircleIntegrable f 0 R
hw : w ∈ ball 0 R
hR : 0 < R
d : ℝ
hd : d > 0
hsub : ball w d ⊆ ball 0 R
hdist : ∀ x ∈ ball w d, ∀ (θ : ℝ), d ≤ ‖circleMap 0 R θ - x‖
hgm : AEStronglyMeasurable (fun θ ↦ f (circleMap 0 R θ)) (volume.restrict (uIoc 0 (2 * π)))
θ : ℝ
a✝ : θ ∈ uIoc 0 (2 * π)
x : ℂ
hx : x ∈ ball w d
h₁ : ‖2‖ = 2
⊢ ‖2‖ * R / ‖circleMap 0 R θ - x‖ ^ 2 * ‖f (circleMap 0 R θ)‖ ≤ 2 * R * (d ^ 2)⁻¹ * ‖f (circleMap 0 R θ)‖
Module: Mathlib.Analysis.SpecialFunctions.Integrability.LogMeromorphic

Goal state before:
pp: case pos
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
hf : MeromorphicOn f [[a, b]]
t₀ : ∀ (u : ↑[[a, b]]), meromorphicOrderAt f ↑u ≠ ⊤
g : ℝ → E
h₁g : AnalyticOnNhd ℝ g [[a, b]]
h₂g : ∀ (u : ↑[[a, b]]), g ↑u ≠ 0
h₃g : f =ᶠ[codiscreteWithin [[a, b]]] (∏ᶠ (u : ℝ), (fun x ↦ x - u) ^ (divisor f [[a, b]]) u) • g
h₄g :
  (fun x ↦ log ‖f x‖) =ᶠ[codiscreteWithin [[a, b]]]
    (∑ᶠ (u : ℝ), fun x ↦ ↑((divisor f [[a, b]]) u) * log ‖x - u‖) + fun x ↦ log ‖g x‖
⊢ IntervalIntegrable (fun x ↦ log ‖f x‖) volume a b

Tactic:
intervalIntegrable_congr_codiscreteWithin
      (h₄g.filter_mono (Filter.codiscreteWithin_mono Set.uIoc_subset_uIcc))

Goal state after:
case pos
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
hf : MeromorphicOn f [[a, b]]
t₀ : ∀ (u : ↑[[a, b]]), meromorphicOrderAt f ↑u ≠ ⊤
g : ℝ → E
h₁g : AnalyticOnNhd ℝ g [[a, b]]
h₂g : ∀ (u : ↑[[a, b]]), g ↑u ≠ 0
h₃g : f =ᶠ[codiscreteWithin [[a, b]]] (∏ᶠ (u : ℝ), (fun x ↦ x - u) ^ (divisor f [[a, b]]) u) • g
h₄g :
  (fun x ↦ log ‖f x‖) =ᶠ[codiscreteWithin [[a, b]]]
    (∑ᶠ (u : ℝ), fun x ↦ ↑((divisor f [[a, b]]) u) * log ‖x - u‖) + fun x ↦ log ‖g x‖
⊢ IntervalIntegrable ((∑ᶠ (u : ℝ), fun x ↦ ↑((divisor f [[a, b]]) u) * log ‖x - u‖) + fun x ↦ log ‖g x‖) volume a b
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: case inr
a b r : ℝ
h✝ : r ≠ -1 ∧ 0 ∉ [[a, b]]
⊢ -1 < (↑r).re ∨ ↑r ≠ -1 ∧ 0 ∉ [[a, b]]

Tactic:
right; rwa [← Complex.ofReal_one, ← Complex.ofReal_neg, Ne, Complex.ofReal_inj]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
this : ∫ (x : ℝ) in a..b, x = (b ^ 2 - a ^ 2) / 2
⊢ ∫ (x : ℝ) in a..b, x = (b ^ 2 - a ^ 2) / 2

Tactic:
exact this

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

Goal state before:
pp: f : ℂ → ℂ
z₀ : ℂ
hf : AnalyticAt ℂ f z₀
h : ¬∀ᶠ (z : ℂ) in 𝓝 z₀, f z = f z₀
R : ℝ
hR : 0 < R
h1 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ f z₀
h2 : ∀ᶠ (z : ℂ) in 𝓝 z₀, AnalyticAt ℂ f z
ρ : ℝ
hρ : ρ > 0
h4 : ∀ z ∈ closedBall z₀ ρ, z ≠ z₀ → f z ≠ f z₀
h3 : DiffContOnCl ℂ f (ball z₀ ρ)
r : ℝ := min ρ R
hr : 0 < r
h5 : closedBall z₀ r ⊆ closedBall z₀ ρ
⊢ ∃ i, 0 < i ∧ ball (f z₀) i ⊆ f '' closedBall z₀ R

Tactic:
have h6 : DiffContOnCl ℂ f (ball z₀ r) := h3.mono (ball_subset_ball inf_le_left)

Goal state after:
f : ℂ → ℂ
z₀ : ℂ
hf : AnalyticAt ℂ f z₀
h : ¬∀ᶠ (z : ℂ) in 𝓝 z₀, f z = f z₀
R : ℝ
hR : 0 < R
h1 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ f z₀
h2 : ∀ᶠ (z : ℂ) in 𝓝 z₀, AnalyticAt ℂ f z
ρ : ℝ
hρ : ρ > 0
h4 : ∀ z ∈ closedBall z₀ ρ, z ≠ z₀ → f z ≠ f z₀
h3 : DiffContOnCl ℂ f (ball z₀ ρ)
r : ℝ := min ρ R
hr : 0 < r
h5 : closedBall z₀ r ⊆ closedBall z₀ ρ
h6 : DiffContOnCl ℂ f (ball z₀ r)
⊢ ∃ i, 0 < i ∧ ball (f z₀) i ⊆ f '' closedBall z₀ R
Module: Mathlib.Analysis.Complex.OpenMapping

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
U : Set E
g : E → ℂ
hg : AnalyticOnNhd ℂ g U
hU : IsPreconnected U
⊢ (∃ w, ∀ z ∈ U, g z = w) ∨ ∀ s ⊆ U, IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s → IsOpen (g '' s)

Tactic:
by_cases h : ∃ z₀ ∈ U, ∀ᶠ z in 𝓝 z₀, g z = g z₀
  · obtain ⟨z₀, hz₀, h⟩ := h
    exact Or.inl ⟨g z₀, hg.eqOn_of_preconnected_of_eventuallyEq analyticOnNhd_const hU hz₀ h⟩
  · simp only [not_exists, not_and] at h
    refine Or.inr fun s hs1 hs2 => isOpen_iff_mem_nhds.mpr ?_
    rintro z ⟨w, hw1, rfl⟩
    exact (hg w (hs1 hw1)).eventually_constant_or_nhds_le_map_nhds.resolve_left (h w (hs1 hw1))
        (image_mem_map (hs2.mem_nhds hw1))

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
this : ∫ (x : ℝ) in a..b, sin x ^ 2 * cos x = (sin b ^ 3 - sin a ^ 3) / 3
⊢ ∫ (x : ℝ) in a..b, sin x ^ 2 * cos x = (sin b ^ 3 - sin a ^ 3) / 3

Tactic:
exact this

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
this : ∫ (x : ℝ) in a..b, cos x ^ 3 = sin b - sin a - (sin b ^ 3 - sin a ^ 3) / 3
⊢ ∫ (x : ℝ) in a..b, cos x ^ 3 = sin b - sin a - (sin b ^ 3 - sin a ^ 3) / 3

Tactic:
exact this

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
this : ∫ (x : ℝ) in a..b, sin x * cos x ^ 2 = (cos a ^ 3 - cos b ^ 3) / 3
⊢ ∫ (x : ℝ) in a..b, sin x * cos x ^ 2 = (cos a ^ 3 - cos b ^ 3) / 3

Tactic:
exact this

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
this : ∫ (x : ℝ) in a..b, sin x ^ 3 = cos a - cos b - (cos a ^ 3 - cos b ^ 3) / 3
⊢ ∫ (x : ℝ) in a..b, sin x ^ 3 = cos a - cos b - (cos a ^ 3 - cos b ^ 3) / 3

Tactic:
exact this

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

Goal state before:
pp: case neg
c : ℂ
R : ℝ
D : ℂ → ℤ
hR : R ≠ 0
hD : Function.HasFiniteSupport D
h : c ∉ Function.support D
⊢ ∑ᶠ (u : ℂ), ↑(D u) * (log R - log ‖c - u‖) = ∑ᶠ (u : ℂ), ↑(D u) * log (R * ‖c - u‖⁻¹) + ↑(D c) * log R

Tactic:
simp_all only [Function.mem_support, Decidable.not_not, Int.cast_zero, zero_mul, add_zero]

Goal state after:
case neg
c : ℂ
R : ℝ
D : ℂ → ℤ
hR : R ≠ 0
hD : Function.HasFiniteSupport D
h : D c = 0
⊢ ∑ᶠ (u : ℂ), ↑(D u) * (log R - log ‖c - u‖) = ∑ᶠ (u : ℂ), ↑(D u) * log (R * ‖c - u‖⁻¹)
Module: Mathlib.Analysis.RCLike.Sqrt

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : RCLike 𝕜
⊢ sqrt I = ↑√2⁻¹ * (1 - I) * I

Tactic:
by
  rw [sqrt_eq_ite]
  split_ifs with h
  · simp_rw [RingEquiv.symm_apply_eq, map_mul]
    simp [h, mul_assoc, mul_add, add_comm, Complex.sqrt_I, add_mul]
  grind [I_eq_zero_or_im_I_eq_one]

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

Goal state before:
pp: X : Type u
inst✝⁵ : TopologicalSpace X
inst✝⁴ : NormalSpace X
s : Set X
hs : IsClosed s
𝕜 : Type v
inst✝³ : RCLike 𝕜
E : Type w
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
f : ↑s →ᵇ E
hf : ¬‖f‖ = 0
this : TietzeExtension ↑(Metric.closedBall 0 ‖f‖)
hf' : ∀ (x : ↑s), f x ∈ Metric.closedBall 0 ‖f‖
g : C(X, E)
hg : ContinuousMap.restrict s g = ↑f
hg_mem : ∀ (x : X), ‖g x‖ ≤ ‖f‖
g' : X →ᵇ E := ofNormedAddCommGroup ⇑g ⋯ ‖f‖ hg_mem
⊢ g'.domRestrict s = f

Tactic:
by ext x; congrm($(hg) x)

Goal state after:
Module: Mathlib.Topology.TietzeExtension

Goal state before:
pp: X₁ : Type u₁
inst✝⁴ : TopologicalSpace X₁
X : Type u
inst✝³ : TopologicalSpace X
inst✝² : NormalSpace X
e : X₁ → X
Y : Type v
inst✝¹ : TopologicalSpace Y
inst✝ : TietzeExtension Y
he : IsClosedEmbedding e
f : C(X₁, Y)
g : C(X, Y)
hg : g.comp { toFun := e, continuous_toFun := ⋯ } = f
⊢ ⇑g ∘ e = ⇑f

Tactic:
by ext x; congrm($(hg) x)

Goal state after: