mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.InnerProductSpace.OfNorm

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
⊢ ‖x‖ ^ 2 = re (inner_ 𝕜 x x)

Tactic:
simp only [inner_, normSq_apply, ofNat_re, ofNat_im, map_sub, map_add,
    ofReal_re, ofReal_im, mul_re, inv_re, mul_im, I_re, inv_im]

Goal state after:
𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
⊢ ‖x‖ ^ 2 =
    4 / (4 * 4 + 0 * 0) *
        (‖x + x‖ * ‖x + x‖ - 0 * 0 - (‖x - x‖ * ‖x - x‖ - 0 * 0) +
            ((0 * ‖I • x + x‖ - im I * 0) * ‖I • x + x‖ - (0 * 0 + im I * ‖I • x + x‖) * 0) -
          ((0 * ‖I • x - x‖ - im I * 0) * ‖I • x - x‖ - (0 * 0 + im I * ‖I • x - x‖) * 0)) -
      -0 / (4 * 4 + 0 * 0) *
        (‖x + x‖ * 0 + 0 * ‖x + x‖ - (‖x - x‖ * 0 + 0 * ‖x - x‖) +
            ((0 * ‖I • x + x‖ - im I * 0) * 0 + (0 * 0 + im I * ‖I • x + x‖) * ‖I • x + x‖) -
          ((0 * ‖I • x - x‖ - im I * 0) * 0 + (0 * 0 + im I * ‖I • x - x‖) * ‖I • x - x‖))
Module: Mathlib.Analysis.InnerProductSpace.NormDet

Goal state before:
pp: U : Type u_5
inst✝² : NormedAddCommGroup U
inst✝¹ : InnerProductSpace ℝ U
inst✝ : FiniteDimensional ℝ U
f : U →ₗ[ℝ] U
⊢ f.normDet = |LinearMap.det f|

Tactic:
simpa using f.normDet_eq_norm_det

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: case refine_1
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
d : ℝ
hd : 0 < d
x : X
ε : ℝ≥0∞
x✝ : 0 < ε
⊢ {x} ⊆ ⋃ n, (fun x_1 ↦ {x}) n

pp: case refine_2
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
d : ℝ
hd : 0 < d
x : X
ε : ℝ≥0∞
x✝¹ : 0 < ε
x✝ : ℕ
⊢ ediam ((fun x_1 ↦ {x}) x✝) ≤ ε

pp: case refine_3
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
d : ℝ
hd : 0 < d
x : X
ε : ℝ≥0∞
x✝ : 0 < ε
⊢ ∑' (n : ℕ), ⨆ (_ : ((fun x_1 ↦ {x}) n).Nonempty), ediam ((fun x_2 ↦ {x}) n) ^ d ≤ 0

Tactic:
· exact subset_iUnion (fun _ => {x} : ℕ → Set X) 0

Goal state after:
case refine_2
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
d : ℝ
hd : 0 < d
x : X
ε : ℝ≥0∞
x✝¹ : 0 < ε
x✝ : ℕ
⊢ ediam ((fun x_1 ↦ {x}) x✝) ≤ ε

case refine_3
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
d : ℝ
hd : 0 < d
x : X
ε : ℝ≥0∞
x✝ : 0 < ε
⊢ ∑' (n : ℕ), ⨆ (_ : ((fun x_1 ↦ {x}) n).Nonempty), ediam ((fun x_2 ↦ {x}) n) ^ d ≤ 0
Module: Mathlib.Analysis.InnerProductSpace.OfNorm

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
hI : ¬I = 0
x y : E
hI' : I * I = -1
h₁ : ‖-x - y‖ = ‖x + y‖
⊢ ‖-x + y‖ = ‖-(y - x)‖

Tactic:
norm_neg,

Goal state after:
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
hI : ¬I = 0
x y : E
hI' : I * I = -1
h₁ : ‖-x - y‖ = ‖x + y‖
⊢ ‖-x + y‖ = ‖y - x‖
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: case inr
X : Type u_2
Y : Type u_3
inst✝⁵ : EMetricSpace X
inst✝⁴ : EMetricSpace Y
inst✝³ : MeasurableSpace X
inst✝² : BorelSpace X
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
f : X → Y
K : ℝ≥0
d : ℝ
hf : AntilipschitzWith K f
hd : 0 ≤ d
s : Set Y
h0 : K ≠ 0
hKd0 : ↑K ^ d ≠ 0
hKd : ↑K ^ d ≠ ∞
⊢ ⨆ r,
      ⨆ (_ : 0 < r),
        ⨅ t,
          ⨅ (_ : f ⁻¹' s ⊆ ⋃ n, t n),
            ⨅ (_ : ∀ (n : ℕ), ediam (t n) ≤ r), ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), ediam (t n) ^ d ≤
    ⨆ i,
      ⨆ (_ : 0 < i),
        ⨅ i_1,
          ⨅ (_ : s ⊆ ⋃ n, i_1 n),
            ⨅ (_ : ∀ (n : ℕ), ediam (i_1 n) ≤ i), ∑' (i : ℕ), ⨆ (_ : (i_1 i).Nonempty), ↑K ^ d * ediam (i_1 i) ^ d

Tactic:
refine iSup₂_le fun ε ε0 => ?_

Goal state after:
case inr
X : Type u_2
Y : Type u_3
inst✝⁵ : EMetricSpace X
inst✝⁴ : EMetricSpace Y
inst✝³ : MeasurableSpace X
inst✝² : BorelSpace X
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
f : X → Y
K : ℝ≥0
d : ℝ
hf : AntilipschitzWith K f
hd : 0 ≤ d
s : Set Y
h0 : K ≠ 0
hKd0 : ↑K ^ d ≠ 0
hKd : ↑K ^ d ≠ ∞
ε : ℝ≥0∞
ε0 : 0 < ε
⊢ ⨅ t,
      ⨅ (_ : f ⁻¹' s ⊆ ⋃ n, t n),
        ⨅ (_ : ∀ (n : ℕ), ediam (t n) ≤ ε), ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), ediam (t n) ^ d ≤
    ⨆ i,
      ⨆ (_ : 0 < i),
        ⨅ i_1,
          ⨅ (_ : s ⊆ ⋃ n, i_1 n),
            ⨅ (_ : ∀ (n : ℕ), ediam (i_1 n) ≤ i), ∑' (i : ℕ), ⨆ (_ : (i_1 i).Nonempty), ↑K ^ d * ediam (i_1 i) ^ d
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: case refine_2
ι : Type u_4
inst✝ : Fintype ι
a b : ι → ℚ
H : ∀ (i : ι), a i < b i
I : ∀ (i : ι), 0 ≤ ↑(b i) - ↑(a i)
γ : ℕ → Type u_4 := fun n ↦ (i : ι) → Fin ⌈(↑(b i) - ↑(a i)) * ↑n⌉₊
t : (n : ℕ) → γ n → Set (ι → ℝ) := fun n f ↦ univ.pi fun i ↦ Icc (↑(a i) + ↑↑(f i) / ↑n) (↑(a i) + (↑↑(f i) + 1) / ↑n)
A : Tendsto (fun n ↦ 1 / ↑n) atTop (𝓝 0)
B : ∀ᶠ (n : ℕ) in atTop, ∀ (i : γ n), ediam (t n i) ≤ 1 / ↑n
n : ℕ
hn : 1 ≤ n
npos : 0 < ↑n
x : ι → ℝ
hx : ∀ (i : ι), ↑(a i) < x i ∧ x i < ↑(b i)
f : γ n := fun i ↦ ⟨⌊(x i - ↑(a i)) * ↑n⌋₊, ⋯⟩
i : ι
⊢ x i ≤ ↑(a i) + (↑↑(f i) + 1) / ↑n

Tactic:
calc
        x i = (a i : ℝ) + (x i - a i) * n / n := by field
        _ ≤ (a i : ℝ) + (⌊(x i - a i) * n⌋₊ + 1) / n := by
          gcongr
          exact (Nat.lt_floor_add_one _).le

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: ι : Type u_4
inst✝ : Fintype ι
a b : ι → ℚ
H : ∀ (i : ι), a i < b i
I : ∀ (i : ι), 0 ≤ ↑(b i) - ↑(a i)
γ : ℕ → Type u_4 := fun n ↦ (i : ι) → Fin ⌈(↑(b i) - ↑(a i)) * ↑n⌉₊
t : (n : ℕ) → γ n → Set (ι → ℝ) := fun n f ↦ univ.pi fun i ↦ Icc (↑(a i) + ↑↑(f i) / ↑n) (↑(a i) + (↑↑(f i) + 1) / ↑n)
A : Tendsto (fun n ↦ 1 / ↑n) atTop (𝓝 0)
B : ∀ᶠ (n : ℕ) in atTop, ∀ (i : γ n), ediam (t n i) ≤ 1 / ↑n
C : ∀ᶠ (n : ℕ) in atTop, (univ.pi fun i ↦ Ioo ↑(a i) ↑(b i)) ⊆ ⋃ i, t n i
⊢ liminf (fun n ↦ ∑ i, (1 / ↑n) ^ Fintype.card ι) atTop = liminf (fun n ↦ ∏ i, ↑⌈(↑(b i) - ↑(a i)) * ↑n⌉₊ / ↑n) atTop

Tactic:
by
      simp only [γ, Finset.card_univ, Nat.cast_prod, one_mul, Fintype.card_fin, Finset.sum_const,
        nsmul_eq_mul, Fintype.card_pi, div_eq_mul_inv, Finset.prod_mul_distrib, Finset.prod_const]

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.ContinuousOfBounded

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
F : Type u_4
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : IsTopologicalAddGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : TopologicalSpace F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : Module 𝕜 E
inst✝⁴ : ContinuousSMul 𝕜 E
inst✝³ : NormedField 𝕜'
inst✝² : Module 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝¹ : RingHomIsometric σ
inst✝ : FirstCountableTopology E
f : E →ₛₗ[σ] F
hf : ∀ (s : Set E), IsVonNBounded 𝕜 s → IsVonNBounded 𝕜' (⇑f '' s)
c : 𝕜
hc0 : 0 < ‖c‖
hc1 : ‖c‖ < 1
c_ne : c ≠ 0
b : ℕ → Set E
bE1 : ∀ (i : ℕ), b i ∈ 𝓝 0 ∧ Balanced 𝕜 (b i)
bE : (𝓝 0).HasAntitoneBasis fun i ↦ b i
n : ℕ
a✝ : True
⊢ ‖c ^ n‖ ≤ 1

Tactic:
norm_pow,

Goal state after:
𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
F : Type u_4
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : IsTopologicalAddGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : TopologicalSpace F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : Module 𝕜 E
inst✝⁴ : ContinuousSMul 𝕜 E
inst✝³ : NormedField 𝕜'
inst✝² : Module 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝¹ : RingHomIsometric σ
inst✝ : FirstCountableTopology E
f : E →ₛₗ[σ] F
hf : ∀ (s : Set E), IsVonNBounded 𝕜 s → IsVonNBounded 𝕜' (⇑f '' s)
c : 𝕜
hc0 : 0 < ‖c‖
hc1 : ‖c‖ < 1
c_ne : c ≠ 0
b : ℕ → Set E
bE1 : ∀ (i : ℕ), b i ∈ 𝓝 0 ∧ Balanced 𝕜 (b i)
bE : (𝓝 0).HasAntitoneBasis fun i ↦ b i
n : ℕ
a✝ : True
⊢ ‖c‖ ^ n ≤ 1
Module: Mathlib.Analysis.InnerProductSpace.TwoDim

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
hx : ¬x = 0
a : ℝ := ((o.basisRightAngleRotation x hx).repr y) 0
b : ℝ := ((o.basisRightAngleRotation x hx).repr y) 1
this : 0 ≤ a * ‖x‖ ^ 2 ∧ b * ‖x‖ ^ 2 = 0 → SameRay ℝ x (a • x + b • o.rightAngleRotation x)
⊢ 0 ≤ ((o.basisRightAngleRotation x hx).repr y) 0 * ‖x‖ ^ 2 ∧
      ((o.basisRightAngleRotation x hx).repr y) 1 * ‖x‖ ^ 2 = 0 →
    SameRay ℝ x
      (((o.basisRightAngleRotation x hx).repr y) 0 • x +
        ((o.basisRightAngleRotation x hx).repr y) 1 • o.rightAngleRotation x)

Tactic:
exact this

Goal state after:
Module: Mathlib.Analysis.MellinInversion

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
σ : ℝ
f : ℝ → E
x : ℝ
hx : 0 < x
hFf : VerticalIntegrable (mellin f) σ volume
hfx : ContinuousAt f x
g : ℝ → E := fun u ↦ rexp (-σ * u) • f (rexp (-u))
hf : Integrable g volume
h2π : 2 * π ≠ 0
this : Integrable (𝓕 fun u ↦ rexp (-σ * u) • f (rexp (-u))) volume
⊢ Integrable (𝓕 g) volume

Tactic:
exact this

Goal state after:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: case h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : 0 < a
t : ℝ
ht : t ∈ Ioi 0
⊢ ↑a ^ (s - 1) ≠ 0

Tactic:
rw [Ne, cpow_eq_zero_iff, ofReal_eq_zero, not_and_or]

Goal state after:
case h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : 0 < a
t : ℝ
ht : t ∈ Ioi 0
⊢ ¬a = 0 ∨ ¬s - 1 ≠ 0
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: n : ℕ
z : ℂ
hz : ‖z‖ < 1
help : IntervalIntegrable (fun t ↦ t ^ n * (1 - ‖z‖)⁻¹) MeasureTheory.volume 0 1
f : ℂ → ℂ := fun z ↦ log (1 + z) - logTaylor (n + 1) z
f' : ℂ → ℂ := fun z ↦ (-z) ^ n * (1 + z)⁻¹
hderiv : ∀ t ∈ Set.Icc 0 1, HasDerivAt f (f' (0 + ↑t * z)) (0 + ↑t * z)
hcont : ContinuousOn (fun t ↦ f' (0 + ↑t * z)) (Set.Icc 0 1)
H : log (1 + z) - logTaylor (n + 1) z = z * ∫ (t : ℝ) in 0..1, (-(↑t * z)) ^ n * (1 + ↑t * z)⁻¹
⊢ ‖z‖ * (‖(-1) ^ n‖ * (‖z ^ n‖ * ‖∫ (x : ℝ) in 0..1, ↑x ^ n * (1 + ↑x * z)⁻¹‖)) ≤ ‖z‖ ^ (n + 1) * (1 - ‖z‖)⁻¹ / (↑n + 1)

Tactic:
norm_pow,

Goal state after:
n : ℕ
z : ℂ
hz : ‖z‖ < 1
help : IntervalIntegrable (fun t ↦ t ^ n * (1 - ‖z‖)⁻¹) MeasureTheory.volume 0 1
f : ℂ → ℂ := fun z ↦ log (1 + z) - logTaylor (n + 1) z
f' : ℂ → ℂ := fun z ↦ (-z) ^ n * (1 + z)⁻¹
hderiv : ∀ t ∈ Set.Icc 0 1, HasDerivAt f (f' (0 + ↑t * z)) (0 + ↑t * z)
hcont : ContinuousOn (fun t ↦ f' (0 + ↑t * z)) (Set.Icc 0 1)
H : log (1 + z) - logTaylor (n + 1) z = z * ∫ (t : ℝ) in 0..1, (-(↑t * z)) ^ n * (1 + ↑t * z)⁻¹
⊢ ‖z‖ * (‖-1‖ ^ n * (‖z‖ ^ n * ‖∫ (x : ℝ) in 0..1, ↑x ^ n * (1 + ↑x * z)⁻¹‖)) ≤ ‖z‖ ^ (n + 1) * (1 - ‖z‖)⁻¹ / (↑n + 1)
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: n : ℕ
z : ℂ
hz : ‖z‖ < 1
help : IntervalIntegrable (fun t ↦ t ^ n * (1 - ‖z‖)⁻¹) MeasureTheory.volume 0 1
f : ℂ → ℂ := fun z ↦ log (1 + z) - logTaylor (n + 1) z
f' : ℂ → ℂ := fun z ↦ (-z) ^ n * (1 + z)⁻¹
hderiv : ∀ t ∈ Set.Icc 0 1, HasDerivAt f (f' (0 + ↑t * z)) (0 + ↑t * z)
hcont : ContinuousOn (fun t ↦ f' (0 + ↑t * z)) (Set.Icc 0 1)
H : log (1 + z) - logTaylor (n + 1) z = z * ∫ (t : ℝ) in 0..1, (-(↑t * z)) ^ n * (1 + ↑t * z)⁻¹
⊢ ‖z‖ * (‖-1‖ ^ n * (‖z‖ ^ n * ‖∫ (x : ℝ) in 0..1, ↑x ^ n * (1 + ↑x * z)⁻¹‖)) ≤ ‖z‖ ^ (n + 1) * (1 - ‖z‖)⁻¹ / (↑n + 1)

Tactic:
norm_neg,

Goal state after:
n : ℕ
z : ℂ
hz : ‖z‖ < 1
help : IntervalIntegrable (fun t ↦ t ^ n * (1 - ‖z‖)⁻¹) MeasureTheory.volume 0 1
f : ℂ → ℂ := fun z ↦ log (1 + z) - logTaylor (n + 1) z
f' : ℂ → ℂ := fun z ↦ (-z) ^ n * (1 + z)⁻¹
hderiv : ∀ t ∈ Set.Icc 0 1, HasDerivAt f (f' (0 + ↑t * z)) (0 + ↑t * z)
hcont : ContinuousOn (fun t ↦ f' (0 + ↑t * z)) (Set.Icc 0 1)
H : log (1 + z) - logTaylor (n + 1) z = z * ∫ (t : ℝ) in 0..1, (-(↑t * z)) ^ n * (1 + ↑t * z)⁻¹
⊢ ‖z‖ * (‖1‖ ^ n * (‖z‖ ^ n * ‖∫ (x : ℝ) in 0..1, ↑x ^ n * (1 + ↑x * z)⁻¹‖)) ≤ ‖z‖ ^ (n + 1) * (1 - ‖z‖)⁻¹ / (↑n + 1)
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: n : ℕ
z : ℂ
hz : ‖z‖ < 1
help : IntervalIntegrable (fun t ↦ t ^ n * (1 - ‖z‖)⁻¹) MeasureTheory.volume 0 1
f : ℂ → ℂ := fun z ↦ log (1 + z) - logTaylor (n + 1) z
f' : ℂ → ℂ := fun z ↦ (-z) ^ n * (1 + z)⁻¹
hderiv : ∀ t ∈ Set.Icc 0 1, HasDerivAt f (f' (0 + ↑t * z)) (0 + ↑t * z)
hcont : ContinuousOn (fun t ↦ f' (0 + ↑t * z)) (Set.Icc 0 1)
H : log (1 + z) - logTaylor (n + 1) z = z * ∫ (t : ℝ) in 0..1, (-(↑t * z)) ^ n * (1 + ↑t * z)⁻¹
t : ℝ
ht₀ : 0 < t
ht₁ : t ≤ 1
⊢ ‖↑t ^ n‖ * ‖(1 + ↑t * z)⁻¹‖ ≤ t ^ n * (1 - ‖z‖)⁻¹

Tactic:
norm_pow,

Goal state after:
n : ℕ
z : ℂ
hz : ‖z‖ < 1
help : IntervalIntegrable (fun t ↦ t ^ n * (1 - ‖z‖)⁻¹) MeasureTheory.volume 0 1
f : ℂ → ℂ := fun z ↦ log (1 + z) - logTaylor (n + 1) z
f' : ℂ → ℂ := fun z ↦ (-z) ^ n * (1 + z)⁻¹
hderiv : ∀ t ∈ Set.Icc 0 1, HasDerivAt f (f' (0 + ↑t * z)) (0 + ↑t * z)
hcont : ContinuousOn (fun t ↦ f' (0 + ↑t * z)) (Set.Icc 0 1)
H : log (1 + z) - logTaylor (n + 1) z = z * ∫ (t : ℝ) in 0..1, (-(↑t * z)) ^ n * (1 + ↑t * z)⁻¹
t : ℝ
ht₀ : 0 < t
ht₁ : t ≤ 1
⊢ ‖↑t‖ ^ n * ‖(1 + ↑t * z)⁻¹‖ ≤ t ^ n * (1 - ‖z‖)⁻¹
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: a b : ℝ
f : ℝ → ℝ
s : ℝ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x ↦ x ^ (-a)
hs_top : s < a
hf_bot : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)
hs_bot : b < s
c1 : ℝ
hc1 : 0 < c1
hc1' : IntegrableOn (fun t ↦ t ^ (s - 1) * f t) (Ioi c1) volume
c2 : ℝ
hc2 : 0 < c2
hc2' : IntegrableOn (fun t ↦ t ^ (s - 1) * f t) (Ioc 0 c2) volume
⊢ Ioi 0 = Ioc 0 c2 ∪ Ioi (min c2 c1)

Tactic:
Ioc_union_Ioi ((min_le_left _ _).trans (le_max_right _ _)),

Goal state after:
a b : ℝ
f : ℝ → ℝ
s : ℝ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x ↦ x ^ (-a)
hs_top : s < a
hf_bot : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)
hs_bot : b < s
c1 : ℝ
hc1 : 0 < c1
hc1' : IntegrableOn (fun t ↦ t ^ (s - 1) * f t) (Ioi c1) volume
c2 : ℝ
hc2 : 0 < c2
hc2' : IntegrableOn (fun t ↦ t ^ (s - 1) * f t) (Ioc 0 c2) volume
⊢ Ioi 0 = Ioi (min 0 (min c2 c1))
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: z : ℂ
hz : ‖z‖ < 1
n : ℕ
⊢ ‖log (1 + z) - logTaylor n z‖ ≤ max ‖log (1 + z)‖ (1 - ‖z‖)⁻¹ * ‖‖z‖ ^ n‖

Tactic:
norm_pow,

Goal state after:
z : ℂ
hz : ‖z‖ < 1
n : ℕ
⊢ ‖log (1 + z) - logTaylor n z‖ ≤ max ‖log (1 + z)‖ (1 - ‖z‖)⁻¹ * ‖‖z‖‖ ^ n
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: t : ℂ
⊢ Tendsto (fun x ↦ ↑x * (t / ↑x)) atTop (𝓝 t)

Tactic:
apply tendsto_nhds_of_eventually_eq

Goal state after:
t : ℂ
⊢ ∀ᶠ (x' : ℝ) in atTop, ↑x' * (t / ↑x') = t
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: t : ℝ
⊢ Tendsto (fun x ↦ x * (t / x)) atTop (𝓝 t)

Tactic:
apply tendsto_nhds_of_eventually_eq

Goal state after:
t : ℝ
⊢ ∀ᶠ (x' : ℝ) in atTop, x' * (t / x') = t
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: case refine_2
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x ↦ ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
hc : ContinuousOn F (Icc 0 1)
x : ℝ
hx : x ∈ Ioo 0 1
this : HasDerivAt (fun x ↦ x ^ u) (u * ↑x ^ (u - 1)) ↑x
⊢ HasDerivAt (fun y ↦ y ^ u) (u * ↑x ^ (u - 1)) ↑x

Tactic:
exact this

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x ↦ ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
hc : ContinuousOn F (Icc 0 1)
x : ℝ
hx : x ∈ Ioo 0 1
U : HasDerivAt (fun y ↦ y ^ u) (u * ↑x ^ (u - 1)) ↑x
⊢ HasDerivAt (fun y ↦ (1 - y) ^ v) (-v * (1 - ↑x) ^ (v - 1)) ↑x

Tactic:
have A := @HasDerivAt.cpow_const _ _ _ v (hasDerivAt_id (1 - (x : ℂ))) (Or.inl ?_)

Goal state after:
case refine_2
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x ↦ ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
hc : ContinuousOn F (Icc 0 1)
x : ℝ
hx : x ∈ Ioo 0 1
U : HasDerivAt (fun y ↦ y ^ u) (u * ↑x ^ (u - 1)) ↑x
A : HasDerivAt (fun x ↦ id x ^ v) (v * id (1 - ↑x) ^ (v - 1) * 1) (1 - ↑x)
⊢ HasDerivAt (fun y ↦ (1 - y) ^ v) (-v * (1 - ↑x) ^ (v - 1)) ↑x

case refine_1
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x ↦ ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
hc : ContinuousOn F (Icc 0 1)
x : ℝ
hx : x ∈ Ioo 0 1
U : HasDerivAt (fun y ↦ y ^ u) (u * ↑x ^ (u - 1)) ↑x
⊢ 0 < (id (1 - ↑x)).re
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: s : ℂ
hs : 0 < s.re
aux1 : -1 < (s - 1).re
aux2 : s ≠ 0
aux3 : MeasurableSet (Ioc 0 1)
⊢ Integrable ((Ioc 0 1).indicator fun a ↦ ↑a ^ (s - 1) • 1) (volume.restrict (Ioi 0)) ∧
    integral (volume.restrict (Ioi 0)) ((Ioc 0 1).indicator fun a ↦ ↑a ^ (s - 1) • 1) = 1 / s

Tactic:
integrable_indicator_iff aux3,

Goal state after:
s : ℂ
hs : 0 < s.re
aux1 : -1 < (s - 1).re
aux2 : s ≠ 0
aux3 : MeasurableSet (Ioc 0 1)
⊢ IntegrableOn (fun a ↦ ↑a ^ (s - 1) • 1) (Ioc 0 1) (volume.restrict (Ioi 0)) ∧
    integral (volume.restrict (Ioi 0)) ((Ioc 0 1).indicator fun a ↦ ↑a ^ (s - 1) • 1) = 1 / s
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: a s : ℂ
hs : 0 < s.re + a.re
this :
  MellinConvergent ((Ioc 0 1).indicator fun a_1 ↦ ↑a_1 ^ a) s ∧
    mellin ((Ioc 0 1).indicator fun a_1 ↦ ↑a_1 ^ a) s = 1 / (s + a)
⊢ HasMellin ((Ioc 0 1).indicator fun t ↦ ↑t ^ a) s (1 / (s + a))

Tactic:
exact this

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: s : ℂ
n : ℕ
hn : n ≠ 0
⊢ (s + 1).GammaSeq n / s = ↑n / (↑n + 1 + s) * s.GammaSeq n

Tactic:
by
  conv_lhs => rw [GammaSeq, Finset.prod_range_succ, div_div]
  conv_rhs =>
    rw [GammaSeq, Finset.prod_range_succ', Nat.cast_zero, add_zero, div_mul_div_comm, ← mul_assoc,
      ← mul_assoc, mul_comm _ (Finset.prod _ _)]
  congr 3
  · rw [cpow_add _ _ (Nat.cast_ne_zero.mpr hn), cpow_one, mul_comm]
  · refine Finset.prod_congr rfl fun x _ => ?_
    push_cast; ring
  · abel

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: case hl
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n ↦ (Ioc 0 ↑n).indicator fun x ↦ ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n) (volume.restrict (Ioi 0))
x : ℝ
hx : x ∈ Ioi 0
⊢ ∀ᶠ (n : ℕ) in atTop, ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1) = f n x

Tactic:
filter_upwards [eventually_ge_atTop ⌈x⌉₊] with n hn

Goal state after:
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n ↦ (Ioc 0 ↑n).indicator fun x ↦ ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n) (volume.restrict (Ioi 0))
x : ℝ
hx : x ∈ Ioi 0
n : ℕ
hn : ⌈x⌉₊ ≤ n
⊢ ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1) = f n x
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: n : ℕ
ihn : ∀ (s : ℂ), -s.re < ↑n → DifferentiableAt ℂ (fun s ↦ (Gamma s)⁻¹) s
s : ℂ
hs : -s.re < ↑(n + 1)
⊢ -s.re < ↑n + 1

Tactic:
← Nat.cast_succ

Goal state after:
n : ℕ
ihn : ∀ (s : ℂ), -s.re < ↑n → DifferentiableAt ℂ (fun s ↦ (Gamma s)⁻¹) s
s : ℂ
hs : -s.re < ↑(n + 1)
⊢ -s.re < ↑n.succ
Module: Mathlib.Analysis.Normed.Field.Approximation

Goal state before:
pp: K : Type u_1
inst✝ : NormedField K
f g : K[X]
ε : ℝ
hε : 0 < ε
a : K
ha : eval a f = 0
hfm : f.Monic
hgm : g.Monic
hdeg : g.natDegree = f.natDegree
hcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε
hg : g.Splits
⊢ (Multiset.map (fun x ↦ (NormedField.toMulRingNorm K) (a - x)) g.roots).prod =
    ‖(Multiset.map (fun x ↦ a - x) g.roots).prod‖

Tactic:
rw [g.roots.prod_hom' (NormedField.toMulRingNorm K) (fun x : K ↦ a - x)]

Goal state after:
K : Type u_1
inst✝ : NormedField K
f g : K[X]
ε : ℝ
hε : 0 < ε
a : K
ha : eval a f = 0
hfm : f.Monic
hgm : g.Monic
hdeg : g.natDegree = f.natDegree
hcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε
hg : g.Splits
⊢ (NormedField.toMulRingNorm K) (Multiset.map (fun x ↦ a - x) g.roots).prod =
    ‖(Multiset.map (fun x ↦ a - x) g.roots).prod‖
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x : R
⊢ seminormFromBounded' f x = 0 ↔ f x = 0

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

Goal state after:
case refine_1
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x : R
h : seminormFromBounded' f x = 0
⊢ f x = 0

case refine_2
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x : R
h : f x = 0
⊢ seminormFromBounded' f x = 0
Module: Mathlib.Topology.Algebra.Order.LiminfLimsup

Goal state before:
pp: case inl
ι : Type u_1
R : Type u_4
inst✝⁷ : ConditionallyCompleteLinearOrder R
inst✝⁶ : TopologicalSpace R
inst✝⁵ : OrderTopology R
inst✝⁴ : AddCommSemigroup R
inst✝³ : Sub R
inst✝² : ContinuousSub R
inst✝¹ : OrderedSub R
inst✝ : AddLeftMono R
f : ι → R
c : R
cobdd : IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) ⊥ f
bdd_below : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) ⊥ f
⊢ limsup (fun i ↦ c - f i) ⊥ = c - liminf f ⊥

pp: case inr
ι : Type u_1
R : Type u_4
inst✝⁷ : ConditionallyCompleteLinearOrder R
inst✝⁶ : TopologicalSpace R
inst✝⁵ : OrderTopology R
F : Filter ι
inst✝⁴ : AddCommSemigroup R
inst✝³ : Sub R
inst✝² : ContinuousSub R
inst✝¹ : OrderedSub R
inst✝ : AddLeftMono R
f : ι → R
c : R
cobdd : IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) F f
bdd_below : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) F f
h✝ : F.NeBot
⊢ limsup (fun i ↦ c - f i) F = c - liminf f F

Tactic:
· simp only [liminf, limsInf, limsup, limsSup, map_bot, eventually_bot, Set.ofPred_true]
    simp only [IsCoboundedUnder, IsCobounded, map_bot, eventually_bot, true_implies] at cobdd
    rcases cobdd with ⟨x, hx⟩
    refine (csInf_le ?_ (Set.mem_univ _)).antisymm
      (tsub_le_iff_tsub_le.1 (le_csSup ?_ (Set.mem_univ _)))
    · refine ⟨x - x, mem_lowerBounds.2 fun y ↦ ?_⟩
      simp only [Set.mem_univ, true_implies]
      exact tsub_le_iff_tsub_le.1 (hx (x - y))
    · refine ⟨x, mem_upperBounds.2 fun y ↦ ?_⟩
      simp only [Set.mem_univ, hx y, implies_true]

Goal state after:
case inr
ι : Type u_1
R : Type u_4
inst✝⁷ : ConditionallyCompleteLinearOrder R
inst✝⁶ : TopologicalSpace R
inst✝⁵ : OrderTopology R
F : Filter ι
inst✝⁴ : AddCommSemigroup R
inst✝³ : Sub R
inst✝² : ContinuousSub R
inst✝¹ : OrderedSub R
inst✝ : AddLeftMono R
f : ι → R
c : R
cobdd : IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) F f
bdd_below : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) F f
h✝ : F.NeBot
⊢ limsup (fun i ↦ c - f i) F = c - liminf f F
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromConst

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
c : R
f : RingSeminorm R
hf1 : f 1 ≤ 1
hc : f c ≠ 0
hpm : IsPowMul ⇑f
x : R
hx : ∀ (y : R), f (x * y) = f x * f y
⊢ Tendsto (seminormFromConst_seq c f x) atTop (𝓝 (f x))

Tactic:
have hseq : seminormFromConst_seq c f x = fun _n ↦ f x := by
      ext n
      by_cases hn : n = 0
      · simp only [seminormFromConst_seq, hn, pow_zero, mul_one, div_one]
      · simp only [seminormFromConst_seq, hx (c ^ n), hpm _ (Nat.one_le_iff_ne_zero.mpr hn),
          mul_div_assoc, div_self (pow_ne_zero n hc), mul_one]

Goal state after:
R : Type u_1
inst✝ : CommRing R
c : R
f : RingSeminorm R
hf1 : f 1 ≤ 1
hc : f c ≠ 0
hpm : IsPowMul ⇑f
x : R
hx : ∀ (y : R), f (x * y) = f x * f y
hseq : seminormFromConst_seq c f x = fun _n ↦ f x
⊢ Tendsto (seminormFromConst_seq c f x) atTop (𝓝 (f x))