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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
inst✝¹ : CompleteSpace 𝕜
f : 𝕜 → 𝕜
inst✝ : CharZero 𝕜
hf : MeromorphicAt f x
n : ℤ
hn : ↑n = meromorphicOrderAt f x
meromorphicOrderAt_logDeriv_eq_neg_one :
  ∀ [CharZero 𝕜], MeromorphicAt f x → ↑n ≠ 0 → ↑n ≠ ⊤ → meromorphicOrderAt (logDeriv f) x = -1
h₁ : ↑n ≠ 0
⊢ ↑(n - 1) - meromorphicOrderAt f x = -1

Tactic:
← hn

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
inst✝¹ : CompleteSpace 𝕜
f : 𝕜 → 𝕜
inst✝ : CharZero 𝕜
hf : MeromorphicAt f x
n : ℤ
hn : ↑n = meromorphicOrderAt f x
meromorphicOrderAt_logDeriv_eq_neg_one :
  ∀ [CharZero 𝕜], MeromorphicAt f x → ↑n ≠ 0 → ↑n ≠ ⊤ → meromorphicOrderAt (logDeriv f) x = -1
h₁ : ↑n ≠ 0
⊢ ↑(n - 1) - ↑n = -1
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
ε : ℝ
hε : ε > 0
⊢ Differentiable ℂ fun z ↦ invInterpStrip f z ε

Tactic:
apply Differentiable.mul

Goal state after:
case ha
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
ε : ℝ
hε : ε > 0
⊢ Differentiable ℂ fun z ↦ (↑ε + ↑(sSupNormIm f 0)) ^ (z - 1)

case hb
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
ε : ℝ
hε : ε > 0
⊢ Differentiable ℂ fun z ↦ (↑ε + ↑(sSupNormIm f 1)) ^ (-z)
Module: Mathlib.Analysis.Complex.HasPrimitives

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
s : Set ℂ
h : IsExactOn f s
x₀ : ℂ
y : E
⊢ ∃ g, g x₀ = y ∧ ∀ x ∈ s, HasDerivAt g (f x) x

Tactic:
obtain ⟨η, hη⟩ := h

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
s : Set ℂ
x₀ : ℂ
y : E
η : ℂ → E
hη : ∀ z ∈ s, HasDerivAt η (f z) z
⊢ ∃ g, g x₀ = y ∧ ∀ x ∈ s, HasDerivAt g (f x) x
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: case h
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
l u : ℝ
hul : l < u
e : E
z : ℂ
hz₁ : z.re = 0
hz₂ : f (↑l + z * (↑u - ↑l)) = e
⊢ (↑l + z * (↑u - ↑l)).re = l ∧ f (↑l + z * (↑u - ↑l)) = e

Tactic:
simp [hz₁, hz₂]

Goal state after:
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
hx : x ∈ Ioi 0
⊢ z.re ≠ 0

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
hx : x ∈ Ioi 0
⊢ (1 - z).re ≠ 0

Tactic:
· simpa using (ne_comm.mpr (ne_of_lt hz.1))

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
⊢ (1 - z).re ≠ 0
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
this :
  ∀ 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‖
⊢ z.re ≠ 0

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
this :
  ∀ 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‖
⊢ (1 - z).re ≠ 0

Tactic:
· simpa using (ne_comm.mpr (ne_of_lt hz.1))

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
this :
  ∀ 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‖
⊢ (1 - z).re ≠ 0
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: 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
this : ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re
⊢ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re ≤ a ^ (1 - z.re) * b ^ z.re

Tactic:
apply mul_le_mul_of_nonneg _ _ (Real.rpow_nonneg (sSupNormIm_nonneg f _) _)

Goal state after:
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
this : ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re
⊢ 0 ≤ b ^ z.re

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
this : ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re
⊢ sSupNormIm f 0 ^ (1 - z.re) ≤ a ^ (1 - z.re)

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
this : ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re
⊢ sSupNormIm f 1 ^ z.re ≤ b ^ z.re
Module: Mathlib.Analysis.InnerProductSpace.Harmonic.Basic

Goal state before:
pp: case right
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
F : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
G : Type u_3
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f : E → F
x : E
h : HarmonicAt f x
l : F →L[ℝ] G
⊢ Δ (⇑l ∘ f) =ᶠ[𝓝 x] 0

Tactic:
· filter_upwards [h.1.laplacian_CLM_comp_left_nhds (l := l), h.2]
    simp_all

Goal state after:
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
hx : x ∈ ball z R
⊢ (⇑reCLM ∘ F) x = f x

Tactic:
apply (convex_ball z R).eqOn_of_fderivWithin_eq (𝕜 := ℝ) (x := z)

Goal state after:
case right.hf
f : ℂ → ℝ
z : ℂ
R : ℝ
hf : HarmonicOnNhd f (ball z R)
hR : ¬R ≤ 0
g : ℂ → ℂ := ⋯
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
⊢ DifferentiableOn ℝ (⇑reCLM ∘ F) (ball z R)

case right.hg
f : ℂ → ℝ
z : ℂ
R : ℝ
hf : HarmonicOnNhd f (ball z R)
hR : ¬R ≤ 0
g : ℂ → ℂ := ⋯
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
⊢ DifferentiableOn ℝ f (ball z R)

case right.hs'
f : ℂ → ℝ
z : ℂ
R : ℝ
hf : HarmonicOnNhd f (ball z R)
hR : ¬R ≤ 0
g : ℂ → ℂ := ⋯
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
⊢ UniqueDiffOn ℝ (ball z R)

case right.hf'
f : ℂ → ℝ
z : ℂ
R : ℝ
hf : HarmonicOnNhd f (ball z R)
hR : ¬R ≤ 0
g : ℂ → ℂ := ⋯
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
⊢ EqOn (fderivWithin ℝ (⇑reCLM ∘ F) (ball z R)) (fderivWithin ℝ f (ball z R)) (ball z R)

case right.hx
f : ℂ → ℝ
z : ℂ
R : ℝ
hf : HarmonicOnNhd f (ball z R)
hR : ¬R ≤ 0
g : ℂ → ℂ := ⋯
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
⊢ z ∈ ball z R

case right.hfgx
f : ℂ → ℝ
z : ℂ
R : ℝ
hf : HarmonicOnNhd f (ball z R)
hR : ¬R ≤ 0
g : ℂ → ℂ := ⋯
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) z = f z

case right.a
f : ℂ → ℝ
z : ℂ
R : ℝ
hf : HarmonicOnNhd f (ball z R)
hR : ¬R ≤ 0
g : ℂ → ℂ := ⋯
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
⊢ x ∈ ball z R
Module: Mathlib.Analysis.Complex.MeanValue

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
R : ℝ
c w : ℂ
s : Set ℂ
hs : s.Countable
h₁f : ContinuousOn f (closedBall c |R|)
h₂f : ∀ z ∈ ball c |R| \ s, DifferentiableAt ℂ f z
hw : w ∈ ball c |R|
hR : 0 < |R|
⊢ circleAverage (fun z ↦ ((z - c) * (z - w)⁻¹) • f z) c |R| = (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, |R|), (z - w)⁻¹ • f z

Tactic:
simp only [circleAverage_eq_circleIntegral hR.ne', mul_inv_rev, inv_I, neg_mul, neg_smul,
      neg_inj, ne_eq, mul_eq_zero, I_ne_zero, inv_eq_zero, ofReal_eq_zero, pi_ne_zero,
      OfNat.ofNat_ne_zero, or_self, not_false_eq_true, smul_right_inj]

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
R : ℝ
c w : ℂ
s : Set ℂ
hs : s.Countable
h₁f : ContinuousOn f (closedBall c |R|)
h₂f : ∀ z ∈ ball c |R| \ s, DifferentiableAt ℂ f z
hw : w ∈ ball c |R|
hR : 0 < |R|
⊢ ∮ (z : ℂ) in C(c, |R|), (z - c)⁻¹ • ((z - c) * (z - w)⁻¹) • f z = ∮ (z : ℂ) in C(c, |R|), (z - w)⁻¹ • f 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 (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
⊢ ‖(f ∘ Neg.neg) z‖ ≤ C

Tactic:
have H : MapsTo Neg.neg (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Iio 0) := by
    intro w hw
    simpa only [mem_reProdIm, neg_re, neg_im, neg_lt_zero, mem_Iio] using! hw

Goal state after:
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)
⊢ ‖(f ∘ Neg.neg) z‖ ≤ C
Module: Mathlib.Analysis.Complex.Poisson

Goal state before:
pp: a b : ℂ
⊢ ((a.re + b.re) * (a.re - b.re) + (a.im + b.im) * (a.im - b.im)) / ‖a - b‖ ^ 2 = (‖a‖ ^ 2 - ‖b‖ ^ 2) / ‖a - b‖ ^ 2

Tactic:
calc ((a.re + b.re) * (a.re - b.re) + (a.im + b.im) * (a.im - b.im)) / ‖a - b‖ ^ 2
    _ = ((a.re * a.re + a.im * a.im) - (b.re * b.re + b.im * b.im)) / ‖a - b‖ ^ 2 := by
      congr! 1; ring
    _ = (‖a‖ ^ 2 - ‖b‖ ^ 2) / ‖a - b‖ ^ 2 := by
      simp [← normSq_apply, normSq_eq_norm_sq]

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

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

Tactic:
nlinarith [mul_pos h₁ (sub_pos.mpr h₂), Real.cos_le_one (θ - φ)]

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

Goal state before:
pp: x : ℝ
⊢ deriv^[2] negMulLog x = -x⁻¹

Tactic:
rw [negMulLog_eq_neg]
  have h := deriv2_mul_log
  simp only [Function.iterate_succ, Function.iterate_zero, Function.id_comp, deriv.fun_neg',
    Function.comp_apply] at h ⊢
  rw [h]

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

Goal state before:
pp: case refine_2
s t : ℝ
ht : 0 < t
h : -1 < s
⊢ IntegrableOn (fun x ↦ x ^ s) (Ioo 0 t) volume

Tactic:
← intervalIntegrable_iff_integrableOn_Ioo_of_le ht.le

Goal state after:
case refine_2
s t : ℝ
ht : 0 < t
h : -1 < s
⊢ IntervalIntegrable (fun x ↦ x ^ s) volume 0 t
Module: Mathlib.Analysis.SpecialFunctions.Log.InvLog

Goal state before:
pp: case inr.inl
this✝¹ : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) (-1)
this✝ : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) 1
this : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) 0
⊢ deriv (fun x ↦ (log x)⁻¹) (-1) = -(-1)⁻¹ / log (-1) ^ 2

pp: case inr.inr.inl
this✝¹ : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) (-1)
this✝ : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) 1
this : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) 0
⊢ deriv (fun x ↦ (log x)⁻¹) 0 = -0⁻¹ / log 0 ^ 2

pp: case inr.inr.inr
this✝¹ : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) (-1)
this✝ : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) 1
this : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) 0
⊢ deriv (fun x ↦ (log x)⁻¹) 1 = -1⁻¹ / log 1 ^ 2

Tactic:
all_goals rw [deriv_zero_of_not_differentiableAt ‹_›]; simp

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

Goal state before:
pp: case refine_2
s : ℂ
t : ℝ
ht : 0 < t
h : -1 < s.re
⊢ IntegrableOn (fun x ↦ ↑x ^ s) (Ioo 0 t) volume

Tactic:
← intervalIntegrable_iff_integrableOn_Ioo_of_le ht.le

Goal state after:
case refine_2
s : ℂ
t : ℝ
ht : 0 < t
h : -1 < s.re
⊢ IntervalIntegrable (fun x ↦ ↑x ^ s) volume 0 t
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: a b : ℝ
f : ℝ → ℝ
μ : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
h : ∀ x ∈ [[a, b]], f x ≠ 0
hf : ContinuousOn f [[a, b]]
⊢ IntervalIntegrable (fun x ↦ (f x)⁻¹) μ a b

Tactic:
simpa only [one_div] using intervalIntegrable_one_div h hf

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

Goal state before:
pp: α : Type u_1
s : Finset α
f : α → ℝ
hs : s.Nonempty
t_max : α
ht_max : t_max ∈ s ∧ ∀ x' ∈ s, |f x'| ≤ |f t_max|
⊢ ∑ t ∈ s, |f t| ≤ ∑ t ∈ s, |f t_max|

Tactic:
apply Finset.sum_le_sum (fun i ih ↦ ht_max.2 i ih)

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

Goal state before:
pp: x y : ℝ
⊢ log⁺ (x + y) ≤ log 2 + log⁺ x + log⁺ y

Tactic:
convert! posLog_sum Finset.univ ![x, y] using 1 <;> simp [add_assoc]

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

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

Tactic:
apply_fun Complex.re at this

Goal state after:
a b r : ℝ
h : -1 < r ∨ r ≠ -1 ∧ 0 ∉ [[a, b]]
h' : -1 < (↑r).re ∨ ↑r ≠ -1 ∧ 0 ∉ [[a, b]]
this : (∫ (x : ℝ) in a..b, ↑x ^ ↑r).re = ((↑b ^ (↑r + 1) - ↑a ^ (↑r + 1)) / (↑r + 1)).re
⊢ ∫ (x : ℝ) in a..b, x ^ r = (b ^ (r + 1) - a ^ (r + 1)) / (r + 1)
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 ≠ ⊤
⊢ IntervalIntegrable (fun x ↦ log ‖f x‖) volume a b

pp: case neg
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 ≠ ⊤
⊢ IntervalIntegrable (fun x ↦ log ‖f x‖) volume a b

Tactic:
· obtain ⟨g, h₁g, h₂g, h₃g⟩ := hf.extract_zeros_poles t₀
      ((MeromorphicOn.divisor f [[a, b]]).finiteSupport isCompact_uIcc)
    have h₄g := MeromorphicOn.extract_zeros_poles_log h₂g h₃g
    rw [intervalIntegrable_congr_codiscreteWithin
      (h₄g.filter_mono (Filter.codiscreteWithin_mono Set.uIoc_subset_uIcc))]
    apply IntervalIntegrable.add
    · apply IntervalIntegrable.finsum
      intro i
      apply IntervalIntegrable.const_mul
      rw [(by ring : a = ((a - i) + i)), (by ring : b = ((b - i) + i))]
      apply IntervalIntegrable.comp_sub_right (f := (log ‖·‖)) _ i
      simp [norm_eq_abs, log_abs]
    · apply ContinuousOn.intervalIntegrable
      apply h₁g.continuousOn.norm.log
      simp_all

Goal state after:
case neg
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 ≠ ⊤
⊢ IntervalIntegrable (fun x ↦ log ‖f x‖) volume a b
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
ha : 0 < a
hb : 0 < b
⊢ ∫ (x : ℝ) in a..b, 1 / x = log (b / a)

Tactic:
simp only [one_div, integral_inv_of_pos ha hb]

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
⊢ ∫ (x : ℝ) in a..b, sin x ^ (n + 2) =
    (sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b + (↑n + 1) * ∫ (x : ℝ) in a..b, sin x ^ n) -
      (↑n + 1) * ∫ (x : ℝ) in a..b, sin x ^ (n + 2)

Tactic:
have H := integral_mul_deriv_eq_deriv_mul hu hv ?_ ?_

Goal state after:
case refine_3
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) =
    (sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b + (↑n + 1) * ∫ (x : ℝ) in a..b, sin x ^ n) -
      (↑n + 1) * ∫ (x : ℝ) in a..b, sin x ^ (n + 2)

case refine_1
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
⊢ IntervalIntegrable (fun x ↦ ↑(n + 1) * cos x * sin x ^ n) MeasureTheory.volume a b

case refine_2
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
⊢ IntervalIntegrable sin MeasureTheory.volume a b
Module: Mathlib.Analysis.Complex.Periodic

Goal state before:
pp: h : ℝ
f : ℂ → ℂ
⊢ (𝓝[≠] 0).limUnder (f ∘ invQParam h) = (𝓝[≠] 0).limUnder (cuspFunction h f)

Tactic:
refine congr_arg lim (Filter.map_congr <| eventuallyEq_nhdsWithin_of_eqOn fun r hr ↦ ?_)

Goal state after:
h : ℝ
f : ℂ → ℂ
r : ℂ
hr : r ∈ {0}ᶜ
⊢ (f ∘ invQParam h) r = cuspFunction h f r
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
⊢ ∫ (x : ℝ) in a..b, cos x ^ (n + 2) =
    (cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a + (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ n) -
      (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ (n + 2)

Tactic:
have H := integral_mul_deriv_eq_deriv_mul hu hv ?_ ?_

Goal state after:
case refine_3
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) =
    (cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a + (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ n) -
      (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ (n + 2)

case refine_1
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
⊢ IntervalIntegrable (fun x ↦ -↑(n + 1) * sin x * cos x ^ n) MeasureTheory.volume a b

case refine_2
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
⊢ IntervalIntegrable cos MeasureTheory.volume a b
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
m n : ℕ
hc : Continuous fun u ↦ u ^ n * (1 - u ^ 2) ^ m
⊢ ∫ (x : ℝ) in a..b, sin x ^ (2 * m + 1) * cos x ^ n = -∫ (x : ℝ) in b..a, sin x ^ (2 * m + 1) * cos x ^ n

Tactic:
by rw [integral_symm]

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

Goal state before:
pp: case e'_3
a b : ℝ
h1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4
⊢ (b - a) / 8 - (sin (4 * b) - sin (4 * a)) / 32 =
    ∫ (x : ℝ) in a..b, ((1 - cos (2 * x)) / 2) ^ 1 * ((1 + cos (2 * x)) / 2) ^ 1

Tactic:
have h2 : Continuous fun x => cos (2 * x) ^ 2 := by fun_prop

Goal state after:
case e'_3
a b : ℝ
h1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4
h2 : Continuous fun x ↦ cos (2 * x) ^ 2
⊢ (b - a) / 8 - (sin (4 * b) - sin (4 * a)) / 32 =
    ∫ (x : ℝ) in a..b, ((1 - cos (2 * x)) / 2) ^ 1 * ((1 + cos (2 * x)) / 2) ^ 1
Module: Mathlib.Analysis.Complex.RiemannMapping

Goal state before:
pp: U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
hU₀ : 0 ∉ U
f : ℂ → ℂ
hfc : ContinuousOn f U
hf_inv : LeftInverse (fun x ↦ x ^ 2) f
hf₀ : ∀ z ∈ U, f z ≠ 0
hdf : ∀ z ∈ U, HasStrictDerivAt f (2 * f z)⁻¹ z
x : ℂ
hx : x ∈ U
this : f '' U ∈ 𝓝 (f x)
⊢ ∀ a ∈ f '' U, -a ∉ f '' U

Tactic:
rintro _ ⟨a, ha, rfl⟩ ⟨b, hb, hab⟩

Goal state after:
U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
hU₀ : 0 ∉ U
f : ℂ → ℂ
hfc : ContinuousOn f U
hf_inv : LeftInverse (fun x ↦ x ^ 2) f
hf₀ : ∀ z ∈ U, f z ≠ 0
hdf : ∀ z ∈ U, HasStrictDerivAt f (2 * f z)⁻¹ z
x : ℂ
hx : x ∈ U
this : f '' U ∈ 𝓝 (f x)
a : ℂ
ha : a ∈ U
b : ℂ
hb : b ∈ U
hab : f b = -f a
⊢ False
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‖)
x : ↑s
⊢ f x ∈ Metric.closedBall 0 ‖f‖

Tactic:
simpa using! f.norm_coe_le_norm x

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo

Goal state before:
pp: K : Type u_2
inst✝¹ : Field K
g : GL (Fin 2) K
inst✝ : NeZero 2
hg : g.IsParabolic
this : (↑g).trace ^ 2 = 4 * (↑g).det
⊢ (↑g).parabolicEigenvalue ≠ 0

Tactic:
parabolicEigenvalue,

Goal state after:
K : Type u_2
inst✝¹ : Field K
g : GL (Fin 2) K
inst✝ : NeZero 2
hg : g.IsParabolic
this : (↑g).trace ^ 2 = 4 * (↑g).det
⊢ (↑g).trace / 2 ≠ 0