mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: case inr.inr
E : Type u_5
inst✝ : NormedAddCommGroup E
f : ℝ → E
h₁f : ∀ (x : ℝ), f x = f (-x)
h₂f : ∀ (x : ℝ), 0 < x → IntervalIntegrable f volume 0 x
t : ℝ
ht : ‖f (min 0 t)‖ₑ ≠ ∞
h : 0 < t
⊢ IntervalIntegrable f volume 0 t

Tactic:
· exact h₂f t h

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝ : NormedAddCommGroup E
f : ℝ → E
h₁f : ∀ (x : ℝ), -f x = f (-x)
h₂f : ∀ (x : ℝ), 0 < x → IntervalIntegrable f volume 0 x
t : ℝ
ht : ‖f (min 0 t)‖ₑ ≠ ∞
h : t < 0
⊢ 0 < -t

Tactic:
simp [h]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: case inr.inr
E : Type u_5
inst✝ : NormedAddCommGroup E
f : ℝ → E
h₁f : ∀ (x : ℝ), -f x = f (-x)
h₂f : ∀ (x : ℝ), 0 < x → IntervalIntegrable f volume 0 x
t : ℝ
ht : ‖f (min 0 t)‖ₑ ≠ ∞
h : 0 < t
⊢ IntervalIntegrable f volume 0 t

Tactic:
· exact h₂f t h

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Goal state before:
pp: case h
g' g φ : ℝ → ℝ
a b : ℝ
hab : a ≤ b
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt g (g' x) (Ioi x) x
φint : IntegrableOn φ (Icc a b) volume
hφg : ∀ x ∈ Ico a b, g' x ≤ φ x
ε : ℝ
εpos : 0 < ε
G' : ℝ → EReal
f_lt_G' : ∀ (x : ℝ), ↑(φ x) < G' x
G'cont : LowerSemicontinuous G'
G'int : Integrable (fun x ↦ (G' x).toReal) (volume.restrict (Icc a b))
G'lt_top : ∀ᵐ (x : ℝ) ∂volume.restrict (Icc a b), G' x < ⊤
hG' : ∫ (x : ℝ) in Icc a b, (G' x).toReal < (∫ (x : ℝ) in Icc a b, φ x) + ε
s : Set ℝ := ⋯
s_closed : IsClosed s
t : ℝ
ht : t ∈ {t | g t - g a ≤ ∫ (u : ℝ) in a..t, (G' u).toReal} ∩ Ico a b
v : ℝ
t_lt_v : v ∈ Ioi t
y : ℝ
g'_lt_y' : ↑(g' t) < ↑y
y_lt_G' : ↑y < G' t
B : ∀ᶠ (u : ℝ) in 𝓝 t, ↑y < G' u
m M : ℝ
H : Ioo m M ⊆ {x | (fun u ↦ ↑y < G' u) x}
hm : m < t
hM : t < M
this : Ioo t (min M b) ∈ 𝓝[>] t
u : ℝ
hu : u ∈ Ioo t (min M b)
I : Icc t u ⊆ Icc a b
⊢ (fun x ↦ y) ≤ᵐ[volume.restrict (Icc t u)] fun x ↦ (G' x).toReal

Tactic:
have C1 : ∀ᵐ x : ℝ ∂volume.restrict (Icc t u), G' x < ∞ :=
              ae_mono (Measure.restrict_mono I le_rfl) G'lt_top

Goal state after:
case h
g' g φ : ℝ → ℝ
a b : ℝ
hab : a ≤ b
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt g (g' x) (Ioi x) x
φint : IntegrableOn φ (Icc a b) volume
hφg : ∀ x ∈ Ico a b, g' x ≤ φ x
ε : ℝ
εpos : 0 < ε
G' : ℝ → EReal
f_lt_G' : ∀ (x : ℝ), ↑(φ x) < G' x
G'cont : LowerSemicontinuous G'
G'int : Integrable (fun x ↦ (G' x).toReal) (volume.restrict (Icc a b))
G'lt_top : ∀ᵐ (x : ℝ) ∂volume.restrict (Icc a b), G' x < ⊤
hG' : ∫ (x : ℝ) in Icc a b, (G' x).toReal < (∫ (x : ℝ) in Icc a b, φ x) + ε
s : Set ℝ := {t | g t - g a ≤ ∫ (u : ℝ) in a..t, (G' u).toReal} ∩ Icc a b
s_closed : IsClosed s
t : ℝ
ht : t ∈ {t | g t - g a ≤ ∫ (u : ℝ) in a..t, (G' u).toReal} ∩ Ico a b
v : ℝ
t_lt_v : v ∈ Ioi t
y : ℝ
g'_lt_y' : ↑(g' t) < ↑y
y_lt_G' : ↑y < G' t
B : ∀ᶠ (u : ℝ) in 𝓝 t, ↑y < G' u
m M : ℝ
H : Ioo m M ⊆ {x | (fun u ↦ ↑y < G' u) x}
hm : m < t
hM : t < M
this : Ioo t (min M b) ∈ 𝓝[>] t
u : ℝ
hu : u ∈ Ioo t (min M b)
I : Icc t u ⊆ Icc a b
C1 : ∀ᵐ (x : ℝ) ∂volume.restrict (Icc t u), G' x < ↑∞
⊢ (fun x ↦ y) ≤ᵐ[volume.restrict (Icc t u)] fun x ↦ (G' x).toReal
Module: Mathlib.Analysis.SpecialFunctions.NonIntegrable

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : CompleteSpace E
f : ℝ → E
g : ℝ → F
k : Set ℝ
l : Filter ℝ
inst✝¹ : l.NeBot
inst✝ : TendstoIxxClass Icc l l
hl : k ∈ l
hd✝ : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x
hf : Tendsto (fun x ↦ ‖f x‖) l atTop
hfg : deriv f =O[l] g
C : ℝ
hC₀ : 0 ≤ C
s : Set ℝ
hsl : s ∈ l
hgi : IntegrableOn (fun x ↦ C * ‖g x‖) k volume
c : ℝ
hc : c ∈ s
d : ℝ
hfd : ∀ z ∈ [[c, d]], DifferentiableAt ℝ f z
hsub : [[c, d]] ⊆ k
hd : d ∈ s
hlt : ‖f c‖ + ∫ (y : ℝ) in k, C * ‖g y‖ < ‖f d‖
hg : ∀ x ∈ Ι c d, ‖deriv f x‖ ≤ C * ‖g x‖
hg_ae : ∀ᵐ (x : ℝ) ∂volume.restrict (Ι c d), ‖deriv f x‖ ≤ C * ‖g x‖
hsub' : Ι c d ⊆ k
hfi : IntervalIntegrable (deriv f) volume c d
⊢ False

Tactic:
refine hlt.not_ge (sub_le_iff_le_add'.1 ?_)

Goal state after:
E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : CompleteSpace E
f : ℝ → E
g : ℝ → F
k : Set ℝ
l : Filter ℝ
inst✝¹ : l.NeBot
inst✝ : TendstoIxxClass Icc l l
hl : k ∈ l
hd✝ : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x
hf : Tendsto (fun x ↦ ‖f x‖) l atTop
hfg : deriv f =O[l] g
C : ℝ
hC₀ : 0 ≤ C
s : Set ℝ
hsl : s ∈ l
hgi : IntegrableOn (fun x ↦ C * ‖g x‖) k volume
c : ℝ
hc : c ∈ s
d : ℝ
hfd : ∀ z ∈ [[c, d]], DifferentiableAt ℝ f z
hsub : [[c, d]] ⊆ k
hd : d ∈ s
hlt : ‖f c‖ + ∫ (y : ℝ) in k, C * ‖g y‖ < ‖f d‖
hg : ∀ x ∈ Ι c d, ‖deriv f x‖ ≤ C * ‖g x‖
hg_ae : ∀ᵐ (x : ℝ) ∂volume.restrict (Ι c d), ‖deriv f x‖ ≤ C * ‖g x‖
hsub' : Ι c d ⊆ k
hfi : IntervalIntegrable (deriv f) volume c d
⊢ ‖f d‖ - ‖f c‖ ≤ ∫ (y : ℝ) in k, C * ‖g y‖
Module: Mathlib.Analysis.SpecialFunctions.NonIntegrable

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
f : ℝ → E
g : ℝ → F
k : Set ℝ
l : Filter ℝ
inst✝¹ : l.NeBot
inst✝ : TendstoIxxClass Icc l l
hl : k ∈ l
hd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x
hf : Tendsto (fun x ↦ ‖f x‖) l atTop
hfg : deriv f =O[l] g
a : E →ₗᵢ[ℝ] UniformSpace.Completion E := UniformSpace.Completion.toComplₗᵢ
f' : ℝ → UniformSpace.Completion E := ⇑a ∘ f
h'd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f' x
h'f : Tendsto (fun x ↦ ‖f' x‖) l atTop
this : (fun x ↦ ‖deriv f' x‖) =ᶠ[l] fun x ↦ ‖deriv f x‖
⊢ (fun x ↦ ‖deriv f' x‖) =O[l] fun x ↦ ‖deriv f x‖

Tactic:
by exact this.isBigO

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: case inr
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
μ : Measure ℝ
a b : ℝ
h : f = 0
hab : b ≤ a
⊢ ∫ (x : ℝ) in a..b, f x ∂μ = ∫ (x : ℝ), f x ∂μ

Tactic:
simp [h]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Goal state before:
pp: case pos
g' g : ℝ → ℝ
a b : ℝ
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x
g'pos : ∀ x ∈ Ioo a b, 0 ≤ g' x
hab : a < b
⊢ IntegrableOn g' (Ioc a b) volume

pp: case neg
g' g : ℝ → ℝ
a b : ℝ
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x
g'pos : ∀ x ∈ Ioo a b, 0 ≤ g' x
hab : ¬a < b
⊢ IntegrableOn g' (Ioc a b) volume

Tactic:
swap

Goal state after:
case neg
g' g : ℝ → ℝ
a b : ℝ
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x
g'pos : ∀ x ∈ Ioo a b, 0 ≤ g' x
hab : ¬a < b
⊢ IntegrableOn g' (Ioc a b) volume

case pos
g' g : ℝ → ℝ
a b : ℝ
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x
g'pos : ∀ x ∈ Ioo a b, 0 ≤ g' x
hab : a < b
⊢ IntegrableOn g' (Ioc a b) volume
Module: Mathlib.Analysis.SpecialFunctions.NonIntegrable

Goal state before:
pp: case mp
a b c : ℝ
⊢ IntervalIntegrable (fun x ↦ (x - c)⁻¹) volume a b → a = b ∨ c ∉ [[a, b]]

pp: case mpr
a b c : ℝ
⊢ a = b ∨ c ∉ [[a, b]] → IntervalIntegrable (fun x ↦ (x - c)⁻¹) volume a b

Tactic:
· refine fun h => or_iff_not_imp_left.2 fun hne hc => ?_
    exact not_intervalIntegrable_of_sub_inv_isBigO_punctured (isBigO_refl _ _) hne hc h

Goal state after:
case mpr
a b c : ℝ
⊢ a = b ∨ c ∉ [[a, b]] → IntervalIntegrable (fun x ↦ (x - c)⁻¹) volume a b
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: case h_bound
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
μ : Measure ℝ
f : ℝ → E
a₀ : ℝ
hf : IntegrableOn f (Iio a₀) μ
⊢ ∀ᶠ (n : ℝ) in 𝓝[≤] a₀, ∀ᵐ (a : ℝ) ∂μ, ‖(Iio n).indicator f a‖ ≤ (Iio a₀).indicator (norm ∘ f) a

Tactic:
filter_upwards [self_mem_nhdsWithin] with a ha
    refine ae_of_all _ fun x ↦ ?_
    rw [norm_indicator_eq_indicator_norm]
    apply indicator_le_indicator_of_subset (Iio_subset_Iio (by grind)) (fun a ↦ norm_nonneg (f a))

Goal state after:
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: case mp
c w : ℂ
R : ℝ
n : ℤ
hR : R ≠ 0
hn : n < 0
hw : w ∈ sphere c |R|
⊢ ¬IntervalIntegrable (fun θ ↦ (circleMap 0 R θ * I) • (circleMap c R θ - w) ^ n) volume 0 (2 * π)

Tactic:
← image_circleMap_Ioc

Goal state after:
case mp
c w : ℂ
R : ℝ
n : ℤ
hR : R ≠ 0
hn : n < 0
hw : w ∈ circleMap c R '' Ioc 0 (2 * π)
⊢ ¬IntervalIntegrable (fun θ ↦ (circleMap 0 R θ * I) • (circleMap c R θ - w) ^ n) volume 0 (2 * π)
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f g : ℂ → E
c : ℂ
R : ℝ
hR : 0 ≤ R
h : EqOn f g (sphere c R)
θ : ℝ
x✝ : θ ∈ [[0, 2 * π]]
⊢ deriv (circleMap c R) θ • (fun z ↦ f z) (circleMap c R θ) = deriv (circleMap c R) θ • (fun z ↦ g z) (circleMap c R θ)

Tactic:
by simp only [h (circleMap_mem_sphere _ hR _)]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R C : ℝ
hR : 0 < R
hc : ContinuousOn f (sphere c R)
hf : ∀ z ∈ sphere c R, ‖f z‖ ≤ C
hlt : ∃ z ∈ sphere c |R|, ‖f z‖ < C
⊢ ‖∮ (z : ℂ) in C(c, R), f z‖ < 2 * π * R * C

Tactic:
← image_circleMap_Ioc

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R C : ℝ
hR : 0 < R
hc : ContinuousOn f (sphere c R)
hf : ∀ z ∈ sphere c R, ‖f z‖ ≤ C
hlt : ∃ z ∈ circleMap c R '' Ioc 0 (2 * π), ‖f z‖ < C
⊢ ‖∮ (z : ℂ) in C(c, R), f z‖ < 2 * π * R * C
Module: Mathlib.MeasureTheory.Integral.DivergenceTheorem

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
I : Box (Fin (n + 1))
f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hc : ContinuousOn f (Box.Icc I)
Hd : ∀ x ∈ Box.Ioo I \ s, HasFDerivAt f (f' x) x
J : ℕ →o Box (Fin (n + 1))
hJ_sub : ∀ (n_1 : ℕ), Box.Icc (J n_1) ⊆ Box.Ioo I
hJl : Tendsto (Box.lower ∘ ⇑J) atTop (𝓝 I.lower)
hJu : Tendsto (Box.upper ∘ ⇑J) atTop (𝓝 I.upper)
hJ_sub' : ∀ (k : ℕ), Box.Icc (J k) ⊆ Box.Icc I
hJ_le : ∀ (k : ℕ), J k ≤ I
HcJ : ∀ (k : ℕ), ContinuousOn f (Box.Icc (J k))
HdJ : ∀ (k : ℕ), ∀ x ∈ Box.Icc (J k) \ s, HasFDerivWithinAt f (f' x) (Box.Icc (J k)) x
HiJ : ∀ (k : ℕ), IntegrableOn (fun x ↦ ∑ i, (f' x) (e i) i) (Box.Icc (J k)) volume
HJ_eq :
  ∀ (k : ℕ),
    ∫ (x : Fin (n + 1) → ℝ) in Box.Icc (J k), ∑ i, (f' x) (e i) i =
      ∑ i,
        ((∫ (x : Fin n → ℝ) in Box.Icc ((J k).face i), f (i.insertNth ((J k).upper i) x) i) -
          ∫ (x : Fin n → ℝ) in Box.Icc ((J k).face i), f (i.insertNth ((J k).lower i) x) i)
Hi : Integrable (fun x ↦ ∑ i, (f' x) (e i) i) (volume.restrict (⋃ n_1, Box.Ioo (J n_1)))
⊢ Tendsto (fun k ↦ ∫ (x : Fin (n + 1) → ℝ) in Box.Ioo (J k), ∑ i, (f' x) (e i) i) atTop
    (𝓝 (∫ (x : Fin (n + 1) → ℝ) in ⋃ n_1, Box.Ioo (J n_1), ∑ i, (f' x) (e i) i))

Tactic:
exact tendsto_setIntegral_of_monotone (fun k => (J k).measurableSet_Ioo)
      (Box.Ioo.comp J).monotone Hi

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝³ : NormedAddCommGroup E
inst✝² : SFinite ν
inst✝¹ : NormedSpace ℝ E
inst✝ : SFinite μ
g : ↥(Lp E 1 (μ.prod ν))
⊢ Tendsto (fun i ↦ ∫⁻ (x : α), ∫⁻ (y : β), ‖↑↑i (x, y) - ↑↑g (x, y)‖ₑ ∂ν ∂μ) (𝓝 g) (𝓝 0)

pp: α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝³ : NormedAddCommGroup E
inst✝² : SFinite ν
inst✝¹ : NormedSpace ℝ E
inst✝ : SFinite μ
g : ↥(Lp E 1 (μ.prod ν))
⊢ (fun i ↦ ∫⁻ (x : α), ‖∫ (y : β), ↑↑i (x, y) - ↑↑g (x, y) ∂ν‖ₑ ∂μ) ≤ fun i ↦
    ∫⁻ (x : α), ∫⁻ (y : β), ‖↑↑i (x, y) - ↑↑g (x, y)‖ₑ ∂ν ∂μ

Tactic:
swap

Goal state after:
α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝³ : NormedAddCommGroup E
inst✝² : SFinite ν
inst✝¹ : NormedSpace ℝ E
inst✝ : SFinite μ
g : ↥(Lp E 1 (μ.prod ν))
⊢ (fun i ↦ ∫⁻ (x : α), ‖∫ (y : β), ↑↑i (x, y) - ↑↑g (x, y) ∂ν‖ₑ ∂μ) ≤ fun i ↦
    ∫⁻ (x : α), ∫⁻ (y : β), ‖↑↑i (x, y) - ↑↑g (x, y)‖ₑ ∂ν ∂μ

α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝³ : NormedAddCommGroup E
inst✝² : SFinite ν
inst✝¹ : NormedSpace ℝ E
inst✝ : SFinite μ
g : ↥(Lp E 1 (μ.prod ν))
⊢ Tendsto (fun i ↦ ∫⁻ (x : α), ∫⁻ (y : β), ‖↑↑i (x, y) - ↑↑g (x, y)‖ₑ ∂ν ∂μ) (𝓝 g) (𝓝 0)
Module: Mathlib.MeasureTheory.Integral.Prod

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝³ : NormedAddCommGroup E
inst✝² : SFinite ν
inst✝¹ : NormedSpace ℝ E
inst✝ : SFinite μ
f : α × β → E
hf : Integrable f (μ.prod ν)
hE : CompleteSpace E
⊢ ∫ (z : α × β), f z ∂μ.prod ν = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ

pp: case neg
α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝³ : NormedAddCommGroup E
inst✝² : SFinite ν
inst✝¹ : NormedSpace ℝ E
inst✝ : SFinite μ
f : α × β → E
hf : Integrable f (μ.prod ν)
hE : ¬CompleteSpace E
⊢ ∫ (z : α × β), f z ∂μ.prod ν = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ

Tactic:
swap

Goal state after:
case neg
α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝³ : NormedAddCommGroup E
inst✝² : SFinite ν
inst✝¹ : NormedSpace ℝ E
inst✝ : SFinite μ
f : α × β → E
hf : Integrable f (μ.prod ν)
hE : ¬CompleteSpace E
⊢ ∫ (z : α × β), f z ∂μ.prod ν = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ

case pos
α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝³ : NormedAddCommGroup E
inst✝² : SFinite ν
inst✝¹ : NormedSpace ℝ E
inst✝ : SFinite μ
f : α × β → E
hf : Integrable f (μ.prod ν)
hE : CompleteSpace E
⊢ ∫ (z : α × β), f z ∂μ.prod ν = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ
Module: Mathlib.MeasureTheory.Integral.Prod

Goal state before:
pp: case h_ae
α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝³ : NormedAddCommGroup E
inst✝² : SFinite ν
inst✝¹ : NormedSpace ℝ E
inst✝ : SFinite μ
hE : CompleteSpace E
⊢ ∀ ⦃f g : α × β → E⦄,
    f =ᵐ[μ.prod ν] g →
      Integrable f (μ.prod ν) →
        ∫ (z : α × β), f z ∂μ.prod ν = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ →
          ∫ (z : α × β), g z ∂μ.prod ν = ∫ (x : α), ∫ (y : β), g (x, y) ∂ν ∂μ

Tactic:
· rintro f g hfg - hf; convert! hf using 1
    · exact integral_congr_ae hfg.symm
    · apply integral_congr_ae
      filter_upwards [ae_ae_of_ae_prod hfg] with x hfgx using integral_congr_ae (ae_eq_symm hfgx)

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Prod

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : SFinite ν
inst✝³ : NormedSpace ℝ E
inst✝² : SFinite μ
𝕜 : Type u_5
inst✝¹ : RCLike 𝕜
inst✝ : NormedSpace 𝕜 E
f : α → 𝕜
g : β → E
hE : CompleteSpace E
⊢ ∫ (z : α × β), f z.1 • g z.2 ∂μ.prod ν = (∫ (x : α), f x ∂μ) • ∫ (y : β), g y ∂ν

pp: case neg
α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : SFinite ν
inst✝³ : NormedSpace ℝ E
inst✝² : SFinite μ
𝕜 : Type u_5
inst✝¹ : RCLike 𝕜
inst✝ : NormedSpace 𝕜 E
f : α → 𝕜
g : β → E
hE : ¬CompleteSpace E
⊢ ∫ (z : α × β), f z.1 • g z.2 ∂μ.prod ν = (∫ (x : α), f x ∂μ) • ∫ (y : β), g y ∂ν

Tactic:
swap

Goal state after:
case neg
α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : SFinite ν
inst✝³ : NormedSpace ℝ E
inst✝² : SFinite μ
𝕜 : Type u_5
inst✝¹ : RCLike 𝕜
inst✝ : NormedSpace 𝕜 E
f : α → 𝕜
g : β → E
hE : ¬CompleteSpace E
⊢ ∫ (z : α × β), f z.1 • g z.2 ∂μ.prod ν = (∫ (x : α), f x ∂μ) • ∫ (y : β), g y ∂ν

case pos
α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : SFinite ν
inst✝³ : NormedSpace ℝ E
inst✝² : SFinite μ
𝕜 : Type u_5
inst✝¹ : RCLike 𝕜
inst✝ : NormedSpace 𝕜 E
f : α → 𝕜
g : β → E
hE : CompleteSpace E
⊢ ∫ (z : α × β), f z.1 • g z.2 ∂μ.prod ν = (∫ (x : α), f x ∂μ) • ∫ (y : β), g y ∂ν
Module: Mathlib.Analysis.Complex.Liouville

Goal state before:
pp: F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : ℂ → F
hf : Differentiable ℂ f
hb : Bornology.IsBounded (range f)
c : ℂ
⊢ deriv f c = 0

Tactic:
obtain ⟨C, C₀, hC⟩ : ∃ C > (0 : ℝ), ∀ z, ‖f z‖ ≤ C := by
    rcases isBounded_iff_forall_norm_le.1 hb with ⟨C, hC⟩
    exact
      ⟨max C 1, lt_max_iff.2 (Or.inr zero_lt_one), fun z =>
        (hC (f z) (mem_range_self _)).trans (le_max_left _ _)⟩

Goal state after:
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : ℂ → F
hf : Differentiable ℂ f
hb : Bornology.IsBounded (range f)
c : ℂ
C : ℝ
C₀ : C > 0
hC : ∀ (z : ℂ), ‖f z‖ ≤ C
⊢ deriv f c = 0
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: case neg
F : Type u_1
inst✝ : Field F
p q : F[X]
hpq : ¬p * q = 0
f g : (p * q).Gal
hfg :
  (restrict p (p * q).SplittingField) f = (restrict p (p * q).SplittingField) g ∧
    (restrict q (p * q).SplittingField) f = (restrict q (p * q).SplittingField) g
x : (p * q).SplittingField
hx : x ∈ (p * q).rootSet (p * q).SplittingField
⊢ f x = g x

Tactic:
rootSet_def,

Goal state after:
case neg
F : Type u_1
inst✝ : Field F
p q : F[X]
hpq : ¬p * q = 0
f g : (p * q).Gal
hfg :
  (restrict p (p * q).SplittingField) f = (restrict p (p * q).SplittingField) g ∧
    (restrict q (p * q).SplittingField) f = (restrict q (p * q).SplittingField) g
x : (p * q).SplittingField
hx : x ∈ ↑((p * q).aroots (p * q).SplittingField).toFinset
⊢ f x = g x
Module: Mathlib.Analysis.Complex.CauchyIntegral

Goal state before:
pp: E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ
c w : ℂ
f : ℂ → E
s : Set ℂ
hs : s.Countable
hw : w ∈ ball c R \ s
hc : ContinuousOn f (closedBall c R)
hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x
hR : 0 < R
F : ℂ → E := dslope f w
hws : (insert w s).Countable
hcF : ContinuousOn F (closedBall c R)
⊢ ∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z = (2 * ↑π * I) • f w

Tactic:
have hdF : ∀ z ∈ ball (c : ℂ) R \ insert w s, DifferentiableAt ℂ F z := fun z hz =>
    (differentiableAt_dslope_of_ne (ne_of_mem_of_not_mem (mem_insert _ _) hz.2).symm).2
      (hd _ (sdiff_subset_sdiff_right (subset_insert _ _) hz))

Goal state after:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ
c w : ℂ
f : ℂ → E
s : Set ℂ
hs : s.Countable
hw : w ∈ ball c R \ s
hc : ContinuousOn f (closedBall c R)
hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x
hR : 0 < R
F : ℂ → E := dslope f w
hws : (insert w s).Countable
hcF : ContinuousOn F (closedBall c R)
hdF : ∀ z ∈ ball c R \ insert w s, DifferentiableAt ℂ F z
⊢ ∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z = (2 * ↑π * I) • f w
Module: Mathlib.Topology.Algebra.Polynomial

Goal state before:
pp: case pos
F : Type u_3
K : Type u_4
inst✝¹ : CommRing F
inst✝ : NormedField K
p : F[X]
f : F →+* K
B : ℝ
i : ℕ
h1 : p.Monic
h2 : (map f p).Splits
h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B
hB : B < 0
h : i = 0
⊢ ‖1‖ ≤ ↑(Nat.choose 0 i)

Tactic:
simp [h]

Goal state after:
Module: Mathlib.Topology.Algebra.Polynomial

Goal state before:
pp: case neg
F : Type u_3
K : Type u_4
inst✝¹ : CommRing F
inst✝ : NormedField K
p : F[X]
f : F →+* K
B : ℝ
i : ℕ
h1 : p.Monic
h2 : (map f p).Splits
h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B
hB : B < 0
h : ¬i = 0
⊢ ‖0‖ ≤ ↑(Nat.choose 0 i)

Tactic:
simp [h]

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

Goal state before:
pp: E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ
f : ℂ → E
c : ℂ
s : Set ℂ
h0 : 0 < R
n : ℕ
hs : s.Countable
hc : ContinuousOn f (closedBall c R)
hd : ∀ z ∈ ball c R \ s, DifferentiableAt ℂ f z
this : (n.factorial • (cauchyPowerSeries f c (↑(NNReal.mk R ⋯)) n) fun x ↦ 1) = iteratedDeriv n f c
⊢ ∮ (z : ℂ) in C(c, R), (1 / (z - c) ^ (n + 1)) • f z =
    1 • ∮ (z : ℂ) in C(c, ↑(NNReal.mk R ⋯)), (1 / (z - c)) ^ n • (z - c)⁻¹ • f z

Tactic:
simp [← mul_smul, pow_succ, mul_comm]

Goal state after:
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Embeddings

Goal state before:
pp: K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
A : Type u_2
inst✝² : NormedField A
inst✝¹ : IsAlgClosed A
inst✝ : NormedAlgebra ℚ A
B : ℝ
C : ℕ := ⌈max B 1 ^ finrank ℚ K * ↑((finrank ℚ K).choose (finrank ℚ K / 2))⌉₊
⊢ {x | IsIntegral ℤ x ∧ ∀ (φ : K →+* A), ‖φ x‖ ≤ B}.Finite

Tactic:
have := bUnion_roots_finite (algebraMap ℤ K) (finrank ℚ K) (finite_Icc (-C : ℤ) C)

Goal state after:
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
A : Type u_2
inst✝² : NormedField A
inst✝¹ : IsAlgClosed A
inst✝ : NormedAlgebra ℚ A
B : ℝ
C : ℕ := ⌈max B 1 ^ finrank ℚ K * ↑((finrank ℚ K).choose (finrank ℚ K / 2))⌉₊
this :
  (⋃ f,
      ⋃ (_ : f.natDegree ≤ finrank ℚ K ∧ ∀ (i : ℕ), f.coeff i ∈ Icc (-↑C) ↑C),
        ↑(map (algebraMap ℤ K) f).roots.toFinset).Finite
⊢ {x | IsIntegral ℤ x ∧ ∀ (φ : K →+* A), ‖φ x‖ ≤ B}.Finite
Module: Mathlib.Analysis.Complex.Polynomial.Basic

Goal state before:
pp: p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
⊢ Fintype.card ↑(p.rootSet ℂ) = p.natDegree

Tactic:
rootSet_def,

Goal state after:
p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
⊢ Fintype.card ↑↑(p.aroots ℂ).toFinset = p.natDegree
Module: Mathlib.RingTheory.Valuation.RankOne

Goal state before:
pp: Γ₀ : Type u_2
inst✝² : LinearOrderedCommGroupWithZero Γ₀
K : Type u_3
inst✝¹ : DivisionRing K
v : Valuation K Γ₀
inst✝ : v.RankOne
γ : ℝ≥0
hγ : γ ≠ 0
hγ_pos : 0 < γ
x : (ofClass v.restrict).ValueGroup₀ˣ
h : (hom v.restrict) ↑x < γ
k : K
hk : (restrict₀ (ofClass v.restrict)) k = ↑x
⊢ ∃ x, x ≠ 0 ∧ (hom v) (v.restrict x) < γ

Tactic:
refine ⟨k, ?_, ?_⟩

Goal state after:
case refine_1
Γ₀ : Type u_2
inst✝² : LinearOrderedCommGroupWithZero Γ₀
K : Type u_3
inst✝¹ : DivisionRing K
v : Valuation K Γ₀
inst✝ : v.RankOne
γ : ℝ≥0
hγ : γ ≠ 0
hγ_pos : 0 < γ
x : (ofClass v.restrict).ValueGroup₀ˣ
h : (hom v.restrict) ↑x < γ
k : K
hk : (restrict₀ (ofClass v.restrict)) k = ↑x
⊢ k ≠ 0

case refine_2
Γ₀ : Type u_2
inst✝² : LinearOrderedCommGroupWithZero Γ₀
K : Type u_3
inst✝¹ : DivisionRing K
v : Valuation K Γ₀
inst✝ : v.RankOne
γ : ℝ≥0
hγ : γ ≠ 0
hγ_pos : 0 < γ
x : (ofClass v.restrict).ValueGroup₀ˣ
h : (hom v.restrict) ↑x < γ
k : K
hk : (restrict₀ (ofClass v.restrict)) k = ↑x
⊢ (hom v) (v.restrict k) < γ
Module: Mathlib.RingTheory.Valuation.RankOne

Goal state before:
pp: case refine_1
Γ₀ : Type u_2
inst✝² : LinearOrderedCommGroupWithZero Γ₀
K : Type u_3
inst✝¹ : DivisionRing K
v : Valuation K Γ₀
inst✝ : v.RankOne
γ : ℝ≥0
hγ : γ ≠ 0
hγ_pos : 0 < γ
x : (ofClass v.restrict).ValueGroup₀ˣ
h : (hom v.restrict) ↑x < γ
k : K
hk : ↑x = 0
h0 : k = 0
⊢ False

Tactic:
simp at hk

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Ultra

Goal state before:
pp: S : Type u_1
inst✝¹ : SeminormedGroup S
inst✝ : IsUltrametricDist S
x y : S
h : ‖x‖ ≠ ‖y‖
⊢ dist x⁻¹ 1 ≠ dist 1 y

Tactic:
simp [h]

Goal state after: