mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ DifferentiableAt ℝ F y

Tactic:
by fun_prop

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

Goal state before:
pp: case refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)] fun z ↦ expR (B * ‖z‖ ^ c)
hre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C
him : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C
z : ℂ
hz_re : 0 ≤ z.re
hz_im : 0 ≤ z.im
H : MapsTo Neg.neg (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Iio 0)
x : ℝ
hx : 0 ≤ x
⊢ ‖(f ∘ Neg.neg) ↑x‖ ≤ C

Tactic:
rw [comp_apply, ← ofReal_neg]

Goal state after:
case refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)] fun z ↦ expR (B * ‖z‖ ^ c)
hre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C
him : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C
z : ℂ
hz_re : 0 ≤ z.re
hz_im : 0 ≤ z.im
H : MapsTo Neg.neg (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Iio 0)
x : ℝ
hx : 0 ≤ x
⊢ ‖f ↑(-x)‖ ≤ C
Module: Mathlib.Analysis.Complex.Poisson

Goal state before:
pp: φ θ r R : ℝ
h₁ : 0 < r
h₂ : r < R
h_cos : R ^ 2 + r ^ 2 - 2 * R * r * Real.cos (θ - φ) ≥ (R - r) ^ 2
⊢ (R ^ 2 - r ^ 2) / (R ^ 2 + r ^ 2 - 2 * R * r * Real.cos (θ - φ)) ≤ (R + r) / (R - r)

Tactic:
rw [div_le_div_iff₀] <;> nlinarith [mul_pos h₁ (sub_pos.mpr h₂)]

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

Goal state before:
pp: case refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f (Ioi 0 ×ℂ Iio 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Iio 0)] fun z ↦ expR (B * ‖z‖ ^ c)
hre : ∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C
him : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C
z : ℂ
hz_re : z.re ≤ 0
hz_im : 0 ≤ z.im
H : MapsTo Neg.neg (Iio 0 ×ℂ Ioi 0) (Ioi 0 ×ℂ Iio 0)
x : ℝ
hx : x ≤ 0
⊢ ‖(f ∘ Neg.neg) ↑x‖ ≤ C

Tactic:
rw [comp_apply, ← ofReal_neg]

Goal state after:
case refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f (Ioi 0 ×ℂ Iio 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Iio 0)] fun z ↦ expR (B * ‖z‖ ^ c)
hre : ∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C
him : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C
z : ℂ
hz_re : z.re ≤ 0
hz_im : 0 ≤ z.im
H : MapsTo Neg.neg (Iio 0 ×ℂ Ioi 0) (Ioi 0 ×ℂ Iio 0)
x : ℝ
hx : x ≤ 0
⊢ ‖f ↑(-x)‖ ≤ C
Module: Mathlib.Analysis.SpecialFunctions.Log.NegMulLog

Goal state before:
pp: x : ℝ
hx : x ≠ 0
⊢ deriv (fun x ↦ x * log x) x = log x + 1

Tactic:
by
  simp [hx]

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

Goal state before:
pp: R : ℝ
w c : ℂ
hw : w ∈ ball c R
⊢ ContinuousOn (fun z ↦ z - c + (w - c)) (sphere c |R|)

Tactic:
by fun_prop

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

Goal state before:
pp: R : ℝ
w c : ℂ
hw : w ∈ ball c R
⊢ ContinuousOn (fun z ↦ z - c - (w - c)) (sphere c |R|)

Tactic:
by fun_prop

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

Goal state before:
pp: ⊢ (deriv fun x ↦ x * log x) =ᶠ[𝓝[>] 0] fun x ↦ log x + 1

Tactic:
apply eventuallyEq_nhdsWithin_of_eqOn

Goal state after:
⊢ Set.EqOn (deriv fun x ↦ x * log x) (fun x ↦ log x + 1) (Set.Ioi 0)
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
hx : x ∈ g ⁻¹' (slitPlane ∩ {0}ᶜ)
⊢ reCLM (log ((starRingEnd ℂ) (g x)) + log (g x)) = reCLM ((starRingEnd ℂ) (log (g x)) + log (g x))

Tactic:
congr 1

Goal state after:
case right.e_6
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)
⊢ log ((starRingEnd ℂ) (g x)) + log (g x) = (starRingEnd ℂ) (log (g x)) + log (g x)
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
c : ℂ
hf : DiffContOnCl ℂ f (ball c R)
hw : w ∈ ball c R
hR : 0 < R
⊢ DifferentiableOn ℂ (fun z ↦ z + c) (closure (ball 0 R))

Tactic:
by fun_prop

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)))
x : ℂ
⊢ Measurable fun θ ↦ (circleMap 0 R θ + x) / (circleMap 0 R θ - x)

Tactic:
by fun_prop

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)))
⊢ Measurable fun θ ↦ 2 * circleMap 0 R θ / (circleMap 0 R θ - w) ^ 2

Tactic:
by fun_prop

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

Goal state before:
pp: x : ℝ
hx : x = 0
⊢ ContinuousAt (fun x ↦ 0) x

Tactic:
by fun_prop

Goal state after:
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
z : ℂ
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 : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) atTop fun x ↦ ‖f ↑x‖
him : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
hz : 0 ≤ z.re
this : ∀ᶠ (ε : ℝ) in 𝓝[<] 0, ‖cexp (↑ε * z) • f z‖ ≤ C
⊢ Continuous fun c ↦ ‖cexp (↑c * z) • f z‖

Tactic:
by fun_prop

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

Goal state before:
pp: a b : ℝ
r : ℂ
h : -1 < r.re ∨ r ≠ -1 ∧ 0 ∉ [[a, b]]
⊢ ∫ (x : ℝ) in a..b, ↑x ^ r = (↑b ^ (r + 1) - ↑a ^ (r + 1)) / (r + 1)

Tactic:
rw [sub_div]
  have hr : r + 1 ≠ 0 := by
    rcases h with h | h
    · apply_fun Complex.re
      rw [Complex.add_re, Complex.one_re, Complex.zero_re, Ne, add_eq_zero_iff_eq_neg]
      exact h.ne'
    · rw [Ne, ← add_eq_zero_iff_eq_neg] at h; exact h.1
  by_cases hab : (0 : ℝ) ∉ [[a, b]]
  · apply integral_eq_sub_of_hasDerivAt (fun x hx => ?_)
      (intervalIntegrable_cpow (r := r) <| Or.inr hab)
    refine hasDerivAt_ofReal_cpow_const' (ne_of_mem_of_not_mem hx hab) ?_
    contrapose hr; rwa [add_eq_zero_iff_eq_neg]
  replace h : -1 < r.re := by tauto
  suffices ∀ c : ℝ, (∫ x : ℝ in 0..c, (x : ℂ) ^ r) =
      (c : ℂ) ^ (r + 1) / (r + 1) - (0 : ℂ) ^ (r + 1) / (r + 1) by
    rw [← integral_add_adjacent_intervals (@intervalIntegrable_cpow' a 0 r h)
      (@intervalIntegrable_cpow' 0 b r h), integral_symm, this a, this b, Complex.zero_cpow hr]
    ring
  intro c
  apply integral_eq_sub_of_hasDeriv_right
  · refine ((Complex.continuous_ofReal_cpow_const ?_).div_const _).continuousOn
    rwa [Complex.add_re, Complex.one_re, ← neg_lt_iff_pos_add]
  · refine fun x hx => (hasDerivAt_ofReal_cpow_const' ?_ ?_).hasDerivWithinAt
    · rcases le_total c 0 with (hc | hc)
      · rw [max_eq_left hc] at hx; exact hx.2.ne
      · rw [min_eq_left hc] at hx; exact hx.1.ne'
    · contrapose hr; rw [hr]; ring
  · exact intervalIntegrable_cpow' h

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

Goal state before:
pp: a c : ℂ
r : ℝ
z : ℂ
hz : z ∈ sphere c |r|
⊢ MeromorphicAt (fun x ↦ x - a) z

Tactic:
by fun_prop

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

Goal state before:
pp: a : ℂ
h : ‖a‖ < 1
z : ℂ
hz : ‖z‖ = 1
⊢ log ‖z - a‖ = log ‖1 - z⁻¹ * a‖

Tactic:
congr 1

Goal state after:
a : ℂ
h : ‖a‖ < 1
z : ℂ
hz : ‖z‖ = 1
⊢ ‖z - a‖ = ‖1 - z⁻¹ * a‖
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: a : ℂ
h : ‖a‖ < 1
⊢ log ‖1 - 0‖ = 0

pp: a : ℂ
h : ‖a‖ < 1
⊢ InnerProductSpace.HarmonicOnNhd (fun x ↦ log ‖1 - x * a‖) (closedBall 0 |1|)

Tactic:
sub_zero,

Goal state after:
a : ℂ
h : ‖a‖ < 1
⊢ log ‖1‖ = 0

a : ℂ
h : ‖a‖ < 1
⊢ InnerProductSpace.HarmonicOnNhd (fun x ↦ log ‖1 - x * a‖) (closedBall 0 |1|)
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: a : ℂ
h : ‖a‖ < 1
x : ℂ
hx : x ∈ closedBall 0 |1|
this : ‖x * a‖ < 1
⊢ AnalyticAt ℂ (fun x ↦ 1 - x * a) x

Tactic:
by fun_prop

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

Goal state before:
pp: x✝¹ : ℝ
x✝ : x✝¹ ∈ Set.univ
⊢ AnalyticAt ℝ (fun x ↦ 4 * sin x ^ 2) x✝¹

Tactic:
by fun_prop

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

Goal state before:
pp: case hab
b : ℝ
ht : 0 < b
⊢ 0 < b

Tactic:
exact ht

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

Goal state before:
pp: b : ℝ
ht : 0 < b
⊢ ContinuousAt (fun x ↦ x * log x - x) b

Tactic:
by fun_prop

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

Goal state before:
pp: z : ℂ
hz : z ≠ 0
a b : ℝ
⊢ Continuous fun y ↦ Complex.cos (z * ↑y)

Tactic:
by fun_prop

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
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 : SuperpolynomialDecay atTop expR fun x ↦ ‖f ↑x‖
C : ℝ
hC : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
g : ℕ → ℂ → E := fun n z ↦ cexp z ^ n • f z
hg : ∀ (n : ℕ) (z : ℂ), ‖g n z‖ = expR z.re ^ n * ‖f z‖
z : ℂ
hz : 0 < z.re
⊢ f z = 0

Tactic:
suffices H : ∀ n : ℕ, ‖g n z‖ ≤ C by
    contrapose! H
    simp only [hg]
    exact (((tendsto_pow_atTop_atTop_of_one_lt (Real.one_lt_exp_iff.2 hz)).atTop_mul_const
      (norm_pos_iff.2 H)).eventually (eventually_gt_atTop C)).exists

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
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 : SuperpolynomialDecay atTop expR fun x ↦ ‖f ↑x‖
C : ℝ
hC : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
g : ℕ → ℂ → E := fun n z ↦ cexp z ^ n • f z
hg : ∀ (n : ℕ) (z : ℂ), ‖g n z‖ = expR z.re ^ n * ‖f z‖
z : ℂ
hz : 0 < z.re
⊢ ∀ (n : ℕ), ‖g n z‖ ≤ C
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: a : ℂ
h : 1 < ‖a‖
x : ℂ
hx : x ∈ closedBall 0 |1|
⊢ AnalyticAt ℂ (fun x ↦ x - a) x

Tactic:
by fun_prop

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

Goal state before:
pp: a b : ℝ
t : ℂ
ht : t ≠ -1
this : t + 1 ≠ 0
⊢ Continuous fun x ↦ 1 + ↑x ^ 2

Tactic:
by fun_prop

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

Goal state before:
pp: a c : ℂ
R : ℝ
hR : R ≠ 0
x✝¹ : ℂ
x✝ : x✝¹ ∈ sphere 0 |1|
⊢ MeromorphicAt (fun z ↦ z + ↑R⁻¹ * (c - a)) x✝¹

Tactic:
by fun_prop

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

Goal state before:
pp: a c : ℂ
R : ℝ
hR : R ≠ 0
⊢ log R + log⁺ (|R|⁻¹ * ‖c - a‖) = log R + log⁺ (R⁻¹ * ‖c - a‖)

Tactic:
congr 1

Goal state after:
case e_a
a c : ℂ
R : ℝ
hR : R ≠ 0
⊢ log⁺ (|R|⁻¹ * ‖c - a‖) = log⁺ (R⁻¹ * ‖c - a‖)
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: case convert_3
a b t : ℝ
ht : t ≠ -1
this : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (1 + ↑x ^ 2) ^ ↑s
⊢ ¬t = -1

Tactic:
exact ht

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

Goal state before:
pp: a b : ℝ
n : ℕ
C : ℝ := sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b
h : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)
hu : ∀ x ∈ [[a, b]], HasDerivAt (fun y ↦ sin y ^ (n + 1)) (↑(n + 1) * cos x * sin x ^ n) x
hv : ∀ x ∈ [[a, b]], HasDerivAt (-cos) (sin x) x
H :
  ∫ (x : ℝ) in a..b, sin x ^ (n + 1) * sin x =
    sin b ^ (n + 1) * (-cos) b - sin a ^ (n + 1) * (-cos) a - ∫ (x : ℝ) in a..b, ↑(n + 1) * cos x * sin x ^ n * (-cos) x
⊢ ∫ (x : ℝ) in a..b, sin x ^ (n + 2) = ∫ (x : ℝ) in a..b, sin x ^ (n + 1) * sin x

Tactic:
simp only [_root_.pow_succ]

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

Goal state before:
pp: f : ℂ → ℂ
z₀ : ℂ
ε r : ℝ
h : DiffContOnCl ℂ f (ball z₀ r)
hr : 0 < r
hf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖
hz₀ : ∃ᶠ (z : ℂ) in 𝓝 z₀, f z ≠ f z₀
v : ℂ
hv : v ∈ ball (f z₀) (ε / 2)
h1 : DiffContOnCl ℂ (fun z ↦ f z - v) (ball z₀ r)
h2 : ContinuousOn (fun z ↦ ‖f z - v‖) (closedBall z₀ r)
h3 : AnalyticOnNhd ℂ f (ball z₀ r)
h4 : ∀ z ∈ sphere z₀ r, ε / 2 ≤ ‖f z - v‖
h5 : ‖f z₀ - v‖ < ε / 2
z : ℂ
hz1 : z ∈ ball z₀ r
hz2 : IsLocalMin (fun z ↦ ‖f z - v‖) z
h6 : ∀ᶠ (y : ℂ) in 𝓝 z, DifferentiableAt ℂ (fun z ↦ f z - v) y
key : ∀ᶠ (z_1 : ℂ) in 𝓝 z, f z_1 - v = f z - v
h7 : ∀ᶠ (w : ℂ) in 𝓝 z, f w = f z
⊢ False

Tactic:
replace h7 : ∃ᶠ w in 𝓝[≠] z, f w = f z := (h7.filter_mono nhdsWithin_le_nhds).frequently

Goal state after:
f : ℂ → ℂ
z₀ : ℂ
ε r : ℝ
h : DiffContOnCl ℂ f (ball z₀ r)
hr : 0 < r
hf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖
hz₀ : ∃ᶠ (z : ℂ) in 𝓝 z₀, f z ≠ f z₀
v : ℂ
hv : v ∈ ball (f z₀) (ε / 2)
h1 : DiffContOnCl ℂ (fun z ↦ f z - v) (ball z₀ r)
h2 : ContinuousOn (fun z ↦ ‖f z - v‖) (closedBall z₀ r)
h3 : AnalyticOnNhd ℂ f (ball z₀ r)
h4 : ∀ z ∈ sphere z₀ r, ε / 2 ≤ ‖f z - v‖
h5 : ‖f z₀ - v‖ < ε / 2
z : ℂ
hz1 : z ∈ ball z₀ r
hz2 : IsLocalMin (fun z ↦ ‖f z - v‖) z
h6 : ∀ᶠ (y : ℂ) in 𝓝 z, DifferentiableAt ℂ (fun z ↦ f z - v) y
key : ∀ᶠ (z_1 : ℂ) in 𝓝 z, f z_1 - v = f z - v
h7 : ∃ᶠ (w : ℂ) in 𝓝[≠] z, f w = f z
⊢ False
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: w ρ : ℂ
R r : ℝ
hρ : ‖ρ‖ = R
hr_lt : r < R
hwr : ‖w‖ < r
θ : ℝ
⊢ ContinuousAt (circleMap 0 r) θ

Tactic:
by fun_prop

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

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
⊢ Continuous fun z ↦ ⟨↑z ^ n, ⋯⟩

Tactic:
by fun_prop

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

Goal state before:
pp: a b : ℝ
n : ℕ
C : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a
h : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)
hu : ∀ x ∈ [[a, b]], HasDerivAt (fun y ↦ cos y ^ (n + 1)) (-↑(n + 1) * sin x * cos x ^ n) x
hv : ∀ x ∈ [[a, b]], HasDerivAt sin (cos x) x
H :
  ∫ (x : ℝ) in a..b, cos x ^ (n + 1) * cos x =
    cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a - ∫ (x : ℝ) in a..b, -↑(n + 1) * sin x * cos x ^ n * sin x
⊢ ∫ (x : ℝ) in a..b, cos x ^ (n + 2) = ∫ (x : ℝ) in a..b, cos x ^ (n + 1) * cos x

Tactic:
simp only [_root_.pow_succ]

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

Goal state before:
pp: h : ℝ
f : ℂ → ℂ
hh : 0 < h
hf : Periodic f ↑h
h_hol : ∀ᶠ (z : ℂ) in I∞, DifferentiableAt ℂ f z
h_bd : I∞.BoundedAtFilter f
c : ℝ
t : ∀ᶠ (x : ℂ) in 𝓝[≠] 0, DifferentiableAt ℂ (cuspFunction h f) x ∧ ‖cuspFunction h f x‖ ≤ c
S : Set ℂ
hS1 : ∀ y ∈ S, y ∈ {0}ᶜ → DifferentiableAt ℂ (cuspFunction h f) y ∧ ‖cuspFunction h f y‖ ≤ c
hS2 : IsOpen S
hS3 : 0 ∈ S
h_diff : DifferentiableOn ℂ (cuspFunction h f) (S \ {0})
hF_bd : BddAbove (norm ∘ cuspFunction h f '' (S \ {0}))
this : DifferentiableOn ℂ (update (cuspFunction h f) 0 ((𝓝[≠] 0).limUnder (cuspFunction h f))) S
⊢ DifferentiableAt ℂ (cuspFunction h f) 0

Tactic:
rw [← cuspFunction_zero_eq_limUnder_nhds_ne, update_eq_self] at this

Goal state after:
h : ℝ
f : ℂ → ℂ
hh : 0 < h
hf : Periodic f ↑h
h_hol : ∀ᶠ (z : ℂ) in I∞, DifferentiableAt ℂ f z
h_bd : I∞.BoundedAtFilter f
c : ℝ
t : ∀ᶠ (x : ℂ) in 𝓝[≠] 0, DifferentiableAt ℂ (cuspFunction h f) x ∧ ‖cuspFunction h f x‖ ≤ c
S : Set ℂ
hS1 : ∀ y ∈ S, y ∈ {0}ᶜ → DifferentiableAt ℂ (cuspFunction h f) y ∧ ‖cuspFunction h f y‖ ≤ c
hS2 : IsOpen S
hS3 : 0 ∈ S
h_diff : DifferentiableOn ℂ (cuspFunction h f) (S \ {0})
hF_bd : BddAbove (norm ∘ cuspFunction h f '' (S \ {0}))
this : DifferentiableOn ℂ (cuspFunction h f) S
⊢ DifferentiableAt ℂ (cuspFunction h f) 0
Module: Mathlib.Analysis.Complex.TaylorSeries

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
hf : DifferentiableOn ℂ f (Metric.ball c r)
z : ℂ
hz : z ∈ Metric.ball c r
r' : NNReal
hr' : ↑r' < r
hr'₀ : 0 < ↑r'
hzr' : z ∈ Metric.ball c ↑r'
⊢ z - c ∈ Metric.ball 0 ↑r'

Tactic:
simpa only [mem_ball_iff_norm, sub_zero] using hzr'

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

Goal state before:
pp: a b : ℝ
m n : ℕ
⊢ Continuous fun u ↦ u ^ m * (1 - u ^ 2) ^ n

Tactic:
by fun_prop

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

Goal state before:
pp: a b : ℝ
m n : ℕ
⊢ Continuous fun u ↦ u ^ n * (1 - u ^ 2) ^ m

Tactic:
by fun_prop

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

Goal state before:
pp: ⊢ ∫ (x : ℝ) in -1..1, √(1 - x ^ 2) = ∫ (x : ℝ) in sin (-(π / 2))..sin (π / 2), √(1 - x ^ 2)

Tactic:
sin_neg,

Goal state after:
⊢ ∫ (x : ℝ) in -1..1, √(1 - x ^ 2) = ∫ (x : ℝ) in -sin (π / 2)..sin (π / 2), √(1 - x ^ 2)
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: ⊢ Continuous fun x ↦ √(1 - x ^ 2)

Tactic:
by fun_prop

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

Goal state before:
pp: w ρ : ℂ
R : ℝ
hρ : ‖ρ‖ = R
hw : ‖w‖ < R
hR : 0 < R
r : ℕ → ℝ := fun n ↦ R - (R - ‖w‖) / (↑n + 2)
hr_lt : ∀ (n : ℕ), r n < R
hr_pos : ∀ (n : ℕ), 0 < r n
hr_tendsto : Tendsto r atTop (𝓝 R)
DCT :
  Tendsto (fun n ↦ circleAverage (herglotzLogIntegrand w ρ) 0 (r n)) atTop
    (𝓝 (circleAverage (herglotzLogIntegrand w ρ) 0 R))
n : ℕ
z : ℂ
hz : z ∈ ball 0 (r n)
⊢ AnalyticAt ℂ (fun x ↦ x - ρ) z

Tactic:
by fun_prop

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

Goal state before:
pp: w ρ : ℂ
R : ℝ
hρ : ‖ρ‖ = R
hw : ‖w‖ < R
hR : 0 < R
r : ℕ → ℝ := fun n ↦ R - (R - ‖w‖) / (↑n + 2)
hr_lt : ∀ (n : ℕ), r n < R
hr_pos : ∀ (n : ℕ), 0 < r n
hr_tendsto : Tendsto r atTop (𝓝 R)
DCT :
  Tendsto (fun n ↦ circleAverage (herglotzLogIntegrand w ρ) 0 (r n)) atTop
    (𝓝 (circleAverage (herglotzLogIntegrand w ρ) 0 R))
n : ℕ
x : ℂ
hx : x ∈ closure (ball 0 (r n))
this : ‖x - ρ‖ ≠ 0
⊢ ContinuousWithinAt (fun x ↦ log ‖x - ρ‖) (closure (ball 0 (r n))) x

Tactic:
by fun_prop

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

Goal state before:
pp: case inr.refine_1
U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
hU₀ : ¬0 ∉ U
a : ℂ
ha : a ∉ U
f : ℂ → ℂ
hf_inj : Injective f
hf_dense : ¬Dense (f '' (-a +ᵥ U))
hdf : ∀ z ∈ -a +ᵥ U, deriv f z ≠ 0
⊢ ¬Dense ((f ∘ fun x ↦ -a + x) '' U)

Tactic:
simpa only [← image_vadd, Set.image_image] using! hf_dense

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

Goal state before:
pp: c : ℂ
R : ℝ
f : ℂ → ℂ
hR : R ≠ 0
h₁f : MeromorphicOn f (closedBall c |R|)
CB : Set ℂ := closedBall c |R|
h₂f : ∀ u ∈ CB, meromorphicOrderAt f u ≠ ⊤
h₃f : (divisor f CB).support.Finite
g : ℂ → ℂ
h₁g : AnalyticOnNhd ℂ g (closedBall c |R|)
h₂g : ∀ (u : ↑(closedBall c |R|)), g ↑u ≠ 0
h₃g : f =ᶠ[codiscreteWithin (closedBall c |R|)] (∏ᶠ (u : ℂ), (fun x ↦ x - u) ^ (divisor f (closedBall c |R|)) u) • g
h₄g :
  (fun x ↦ log ‖f x‖) =ᶠ[codiscreteWithin (closedBall c |R|)]
    (∑ᶠ (u : ℂ), fun x ↦ ↑((divisor f (closedBall c |R|)) u) * log ‖x - u‖) + fun x ↦ log ‖g x‖
⊢ circleAverage (fun x ↦ log ‖f x‖) c R =
    circleAverage ((∑ᶠ (u : ℂ), fun x ↦ ↑((divisor f CB) u) * log ‖x - u‖) + fun x ↦ log ‖g x‖) c R

Tactic:
circleAverage_congr_codiscreteWithin (codiscreteWithin_mono sphere_subset_closedBall h₄g)
        hR

Goal state after:
c : ℂ
R : ℝ
f : ℂ → ℂ
hR : R ≠ 0
h₁f : MeromorphicOn f (closedBall c |R|)
CB : Set ℂ := closedBall c |R|
h₂f : ∀ u ∈ CB, meromorphicOrderAt f u ≠ ⊤
h₃f : (divisor f CB).support.Finite
g : ℂ → ℂ
h₁g : AnalyticOnNhd ℂ g (closedBall c |R|)
h₂g : ∀ (u : ↑(closedBall c |R|)), g ↑u ≠ 0
h₃g : f =ᶠ[codiscreteWithin (closedBall c |R|)] (∏ᶠ (u : ℂ), (fun x ↦ x - u) ^ (divisor f (closedBall c |R|)) u) • g
h₄g :
  (fun x ↦ log ‖f x‖) =ᶠ[codiscreteWithin (closedBall c |R|)]
    (∑ᶠ (u : ℂ), fun x ↦ ↑((divisor f (closedBall c |R|)) u) * log ‖x - u‖) + fun x ↦ log ‖g x‖
⊢ circleAverage ((∑ᶠ (u : ℂ), fun x ↦ ↑((divisor f (closedBall c |R|)) u) * log ‖x - u‖) + fun x ↦ log ‖g x‖) c R =
    circleAverage ((∑ᶠ (u : ℂ), fun x ↦ ↑((divisor f CB) u) * log ‖x - u‖) + fun x ↦ log ‖g x‖) c R
Module: Mathlib.Analysis.Complex.Tietze

Goal state before:
pp: 𝕜 : Type v
inst✝³ : RCLike 𝕜
E : Type w
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
this✝¹ : NormedSpace ℝ E
this✝ : IsScalarTower ℝ 𝕜 E
g : E → E := fun x ↦ ‖x‖⁻¹ • x
this : Continuous ((Metric.closedBall 0 1).piecewise id g)
⊢ Continuous Subtype.val

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Topology.TietzeExtension

Goal state before:
pp: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : NormalSpace X
s : Set X
hs : IsClosed[inst✝²] s
Y : Type v
inst✝ : TopologicalSpace Y
f : C(↑s, Y)
t : Set Y
hf : ∀ (x : ↑s), f x ∈ t
ht : TietzeExtension ↑t
g : C(X, ↑t)
hg : restrict s g = { toFun := Set.codRestrict (⇑f) t hf, continuous_toFun := ⋯ }
⊢ Continuous[instTopologicalSpaceSubtype, inst✝] Subtype.val

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Topology.TietzeExtension

Goal state before:
pp: X₁ : Type u₁
inst✝³ : TopologicalSpace X₁
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : NormalSpace X
e : X₁ → X
he : IsClosedEmbedding e
Y : Type v
inst✝ : TopologicalSpace Y
f : C(X₁, Y)
t : Set Y
hf : ∀ (x : X₁), f x ∈ t
ht : TietzeExtension ↑t
g : C(X, ↑t)
hg : g.comp { toFun := e, continuous_toFun := ⋯ } = { toFun := Set.codRestrict (⇑f) t hf, continuous_toFun := ⋯ }
⊢ Continuous[instTopologicalSpaceSubtype, inst✝] Subtype.val

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction

Goal state before:
pp: g h : GL (Fin 2) ℝ
z : ℍ
u : ℂ
hu✝ : (σ g) ((σ h) ↑z) = u
hu : u.im ≠ 0
hu' : (num h u / denom h u).im ≠ 0
⊢ num (g * h) u * (↑(↑g 1 0) * num h u + ↑(↑g 1 1) * denom h u) / denom h u =
    (↑(↑g 0 0) * num h u + ↑(↑g 0 1) * denom h u) * denom (g * h) u / denom h u

Tactic:
congr 1

Goal state after:
case e_a
g h : GL (Fin 2) ℝ
z : ℍ
u : ℂ
hu✝ : (σ g) ((σ h) ↑z) = u
hu : u.im ≠ 0
hu' : (num h u / denom h u).im ≠ 0
⊢ num (g * h) u * (↑(↑g 1 0) * num h u + ↑(↑g 1 1) * denom h u) =
    (↑(↑g 0 0) * num h u + ↑(↑g 0 1) * denom h u) * denom (g * h) u
Module: Mathlib.Topology.TietzeExtension

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : NormalSpace Y
inst✝ : Nonempty X
f : X →ᵇ ℝ
e : X → Y
he : IsClosedEmbedding e
inhabited_h : Inhabited X
a : ℝ
ha : IsGLB (range ⇑f) a
b : ℝ
hb : IsLUB (range ⇑f) b
hmem : ∀ (x : X), f x ∈ Icc a b
hle : a ≤ b
⊢ ∃ g, (∀ (y : Y), ∃ x₁ x₂, g y ∈ Icc (f x₁) (f x₂)) ∧ ⇑g ∘ e = ⇑f

Tactic:
rcases hle.eq_or_lt with (rfl | hlt)

Goal state after:
case inl
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : NormalSpace Y
inst✝ : Nonempty X
f : X →ᵇ ℝ
e : X → Y
he : IsClosedEmbedding e
inhabited_h : Inhabited X
a : ℝ
ha : IsGLB (range ⇑f) a
hb : IsLUB (range ⇑f) a
hmem : ∀ (x : X), f x ∈ Icc a a
hle : a ≤ a
⊢ ∃ g, (∀ (y : Y), ∃ x₁ x₂, g y ∈ Icc (f x₁) (f x₂)) ∧ ⇑g ∘ e = ⇑f

case inr
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : NormalSpace Y
inst✝ : Nonempty X
f : X →ᵇ ℝ
e : X → Y
he : IsClosedEmbedding e
inhabited_h : Inhabited X
a : ℝ
ha : IsGLB (range ⇑f) a
b : ℝ
hb : IsLUB (range ⇑f) b
hmem : ∀ (x : X), f x ∈ Icc a b
hle : a ≤ b
hlt : a < b
⊢ ∃ g, (∀ (y : Y), ∃ x₁ x₂, g y ∈ Icc (f x₁) (f x₂)) ∧ ⇑g ∘ e = ⇑f