mathlib-initiative/mathlib-tactics huggingface dataset:
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
hg : ∀ x ∈ s, ∀ y ∈ s, ∀ z ∈ [[x, y]], ‖deriv f z‖ ≤ C * ‖g z‖
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‖
⊢ False

Tactic:
replace hg : ∀ x ∈ Ι c d, ‖deriv f x‖ ≤ C * ‖g x‖ :=
    fun z hz => hg c hc d hd z ⟨hz.1.le, hz.2⟩

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‖
⊢ False
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Goal state before:
pp: case e'_4
g' g φ : ℝ → ℝ
a b : ℝ
hab : a ≤ b
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x
φint : IntegrableOn φ (Icc a b) volume
hφg : ∀ x ∈ Ioo a b, φ x ≤ g' x
⊢ ∫ (x : ℝ) in a..b, -φ x = ∫ (y : ℝ) in a..b, (-φ) y

Tactic:
rfl

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

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
hf : IntegrableOn f (Iio b) μ
hab : a ≤ b
ha : IntegrableOn f (Iio a) μ
h : IntegrableOn f (Ico a b) μ
⊢ Disjoint (Iio a) (Ico a b)

Tactic:
grind

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

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
hf : IntegrableOn f (Ici a) μ
hab : a ≤ b
ha : IntegrableOn f (Ici b) μ
h : IntegrableOn f (Ico a b) μ
⊢ Disjoint (Ici b) (Ico a b)

Tactic:
grind

Goal state after:
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
x : ℝ
hx : DifferentiableAt ℝ f x
this : fderiv ℝ (⇑a) (f x) = a.toContinuousLinearMap
⊢ a.toContinuousLinearMap (deriv f x) = a (deriv f x)

Tactic:
rfl

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.Quotient

Goal state before:
pp: G : Type u_1
inst✝⁹ : Group G
inst✝⁸ : MeasurableSpace G
ν : Measure G
Γ : Subgroup G
μ : Measure (G ⧸ Γ)
inst✝⁷ : QuotientMeasureEqMeasurePreimage ν μ
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : BorelSpace G
inst✝³ : PolishSpace G
inst✝² : T2Space (G ⧸ Γ)
inst✝¹ : SecondCountableTopology (G ⧸ Γ)
inst✝ : ν.IsMulLeftInvariant
hasFun : HasFundamentalDomain (↥Γ.op) G ν
g : G
A : Set (G ⧸ Γ)
hA : MeasurableSet A
meas_π : Measurable QuotientGroup.mk
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 ν
h𝓕_translate_fundom : IsFundamentalDomain (↥Γ.op) (g • 𝓕) ν
π_preA : Set G := QuotientGroup.mk ⁻¹' A
this✝ : QuotientGroup.mk ⁻¹' (fun x ↦ g • x) ⁻¹' A = (fun x ↦ g * x) ⁻¹' π_preA
this : ν ((fun x ↦ g * x) ⁻¹' π_preA ∩ 𝓕) = ν (π_preA ∩ (fun x ↦ g⁻¹ * x) ⁻¹' 𝓕)
⊢ ν (π_preA ∩ (fun x ↦ g⁻¹ * x) ⁻¹' 𝓕) = ν (π_preA ∩ (fun x ↦ g⁻¹ • x) ⁻¹' 𝓕)

Tactic:
rfl

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.Quotient

Goal state before:
pp: G : Type u_1
inst✝¹⁴ : Group G
inst✝¹³ : MeasurableSpace G
inst✝¹² : TopologicalSpace G
inst✝¹¹ : IsTopologicalGroup G
inst✝¹⁰ : BorelSpace G
inst✝⁹ : PolishSpace G
Γ : Subgroup G
inst✝⁸ : Γ.Normal
inst✝⁷ : T2Space (G ⧸ Γ)
inst✝⁶ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝⁵ : ν.IsMulLeftInvariant
inst✝⁴ : Countable ↥Γ
inst✝³ : ν.IsMulRightInvariant
inst✝² : SigmaFinite ν
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : SigmaFinite μ
s : Set G
fund_dom_s : IsFundamentalDomain (↥Γ.op) s ν
V : Set (G ⧸ Γ)
meas_V : MeasurableSet V
neZeroV : μ V ≠ 0
hV : μ V = ν (QuotientGroup.mk ⁻¹' V ∩ s)
neTopV : μ V ≠ ∞
U : Set (G ⧸ Γ)
a✝ : MeasurableSet U
meas_π : Measurable QuotientGroup.mk
μ' : Measure (G ⧸ Γ) := map QuotientGroup.mk (ν.restrict s)
has_fund : HasFundamentalDomain (↥Γ.op) G ν
i : QuotientMeasureEqMeasurePreimage ν μ'
this✝ : μ'.IsMulLeftInvariant
this : μ = μ'
⊢ μ' U = (map (Quotient.mk (MulAction.orbitRel (↥Γ.op) G)) (ν.restrict s)) U

Tactic:
rfl

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

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
t : ℝ
f : AddCircle T → ℝ≥0∞
m : MeasurableSet (Ioc t (t + T))
this : ∫⁻ (a : AddCircle T), f a ∂Measure.map (fun x ↦ ↑↑x) volume = ∫⁻ (a : ↑(Ioc t (t + T))), f ↑↑a
x : ↑(Ioc t (t + T))
⊢ ↑↑x = (QuotientAddGroup.mk ∘ Subtype.val) x

Tactic:
rfl

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

Goal state before:
pp: case e'_3.e_μ
T : ℝ
hT : Fact (0 < T)
t : ℝ
f : AddCircle T → ℝ≥0∞
m : MeasurableSet (Ioc t (t + T))
this✝ : ∫⁻ (a : AddCircle T), f a ∂Measure.map (fun x ↦ ↑↑x) volume = ∫⁻ (a : ↑(Ioc t (t + T))), f ↑↑a
this : (fun x ↦ ↑↑x) = QuotientAddGroup.mk ∘ Subtype.val
⊢ Measure.map QuotientAddGroup.mk (Measure.map Subtype.val (Measure.comap Subtype.val volume)) =
    Measure.map QuotientAddGroup.mk (Measure.map Subtype.val volume)

Tactic:
rfl

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

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
t : ℝ
f : AddCircle T → E
m : MeasurableSet (Ioc t (t + T))
this : ∫ (y : AddCircle T), f y ∂Measure.map (fun x ↦ ↑↑x) volume = ∫ (x : ↑(Ioc t (t + T))), f ↑↑x
x : ↑(Ioc t (t + T))
⊢ ↑↑x = (QuotientAddGroup.mk ∘ Subtype.val) x

Tactic:
rfl

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

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
t : ℝ
f : AddCircle T → E
m : MeasurableSet (Ioc t (t + T))
this✝ : ∫ (y : AddCircle T), f y ∂Measure.map (fun x ↦ ↑↑x) volume = ∫ (x : ↑(Ioc t (t + T))), f ↑↑x
this : (fun x ↦ ↑↑x) = QuotientAddGroup.mk ∘ Subtype.val
⊢ ∫ (y : AddCircle T), f y ∂Measure.map QuotientAddGroup.mk (Measure.map Subtype.val volume) =
    ∫ (y : AddCircle T),
      f y ∂Measure.map QuotientAddGroup.mk (Measure.map Subtype.val (Measure.comap Subtype.val volume))

Tactic:
rfl

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
μ : Measure ℝ
f : ℝ → E
a₀ : ℝ
hf : IntegrableOn f (Ioi a₀) μ
a : ℝ
ha : a ∈ Ici a₀
x : ℝ
⊢ a₀ ≤ a

Tactic:
grind

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
μ : Measure ℝ
f : ℝ → E
a₀ : ℝ
hf : IntegrableOn f (Ioi a₀) μ
x : ℝ
hx : a₀ < x
a : ℝ
ha : a ∈ Iio x
⊢ (if a₀ < x then f x else 0) = if a < x then f x else 0

Tactic:
grind

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
μ : Measure ℝ
f : ℝ → E
a₀ : ℝ
hf : IntegrableOn f (Ioi a₀) μ
x : ℝ
hx : ¬a₀ < x
a : ℝ
ha : a ∈ Ici a₀
⊢ (if a₀ < x then f x else 0) = if a < x then f x else 0

Tactic:
grind

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

Goal state before:
pp: 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
a : ℝ
ha : a ∈ Iic a₀
x : ℝ
⊢ a ≤ a₀

Tactic:
grind

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

Goal state before:
pp: case mp
c w : ℂ
R : ℝ
n : ℤ
h : R ≠ 0 ∧ n < 0 ∧ w ∈ sphere c |R|
⊢ ¬CircleIntegrable (fun z ↦ (z - w) ^ n) c R

Tactic:
rcases h with ⟨hR, hn, hw⟩

Goal state after:
case mp
c w : ℂ
R : ℝ
n : ℤ
hR : R ≠ 0
hn : n < 0
hw : w ∈ sphere c |R|
⊢ ¬CircleIntegrable (fun z ↦ (z - w) ^ n) c R
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: 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
x : ℝ
hx : x < a₀
a : ℝ
ha : a ∈ Ioi x
⊢ (if x < a₀ then f x else 0) = if x < a then f x else 0

Tactic:
grind

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

Goal state before:
pp: 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
x : ℝ
hx : ¬x < a₀
a : ℝ
ha : a ∈ Iic a₀
⊢ (if x < a₀ then f x else 0) = if x < a then f x else 0

Tactic:
grind

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

Goal state before:
pp: case mp
c : ℂ
R : ℝ
n : ℤ
hR : R ≠ 0
hn : n < 0
θ : ℝ
hθ : θ ∈ [[0, 2 * π]]
f : ℝ → ℂ := fun θ' ↦ circleMap c R θ' - circleMap c R θ
this✝ : ∀ᶠ (θ' : ℝ) in 𝓝[≠] θ, f θ' ∈ ball 0 1 \ {0}
θ' : ℝ
hθ' : f θ' ∈ ball 0 1 \ {0}
x : ℝ := ‖f θ'‖
this : x ∈ Ioo 0 1
⊢ x⁻¹ ≤ x ^ n

Tactic:
rw [← zpow_neg_one]

Goal state after:
case mp
c : ℂ
R : ℝ
n : ℤ
hR : R ≠ 0
hn : n < 0
θ : ℝ
hθ : θ ∈ [[0, 2 * π]]
f : ℝ → ℂ := fun θ' ↦ circleMap c R θ' - circleMap c R θ
this✝ : ∀ᶠ (θ' : ℝ) in 𝓝[≠] θ, f θ' ∈ ball 0 1 \ {0}
θ' : ℝ
hθ' : f θ' ∈ ball 0 1 \ {0}
x : ℝ := ‖f θ'‖
this : x ∈ Ioo 0 1
⊢ x ^ (-1) ≤ x ^ n
Module: Mathlib.Analysis.BoxIntegral.DivergenceTheorem

Goal state before:
pp: E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
i : Fin (n + 1)
f : (Fin (n + 1) → ℝ) → E
f' : (Fin (n + 1) → ℝ) →L[ℝ] E
hfc : ContinuousOn f (Box.Icc I)
x : Fin (n + 1) → ℝ
hxI : x ∈ Box.Icc I
a : E
ε : ℝ
h0 : 0 < ε
hε : ∀ y ∈ Box.Icc I, ‖f y - a - f' (y - x)‖ ≤ ε * ‖y - x‖
c : ℝ≥0
hc : I.distortion ≤ c
e : ℝ → (Fin n → ℝ) → Fin (n + 1) → ℝ := i.insertNth
Hl : I.lower i ∈ Set.Icc (I.lower i) (I.upper i)
Hu : I.upper i ∈ Set.Icc (I.lower i) (I.upper i)
Hi : ∀ x ∈ Set.Icc (I.lower i) (I.upper i), Integrable (I.face i) ⊥ (f ∘ e x) BoxAdditiveMap.volume
this :
  ∀ y ∈ Box.Icc (I.face i),
    ‖f' (Pi.single i (I.upper i - I.lower i)) - (f (e (I.upper i) y) - f (e (I.lower i) y))‖ ≤ 2 * ε * diam (Box.Icc I)
⊢ ‖volume.toBoxAdditive (I.face i) • (I.upper i - I.lower i) • f' (Pi.single i 1) -
        integral (I.face i) ⊥ (f ∘ e (I.upper i) - f ∘ e (I.lower i)) BoxAdditiveMap.volume‖ =
    ‖integral (I.face i) ⊥
        (fun x ↦ f' (Pi.single i (I.upper i - I.lower i)) - (f (e (I.upper i) x) - f (e (I.lower i) x)))
        BoxAdditiveMap.volume‖

Tactic:
← BoxAdditiveMap.toSMul_apply,

Goal state after:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
i : Fin (n + 1)
f : (Fin (n + 1) → ℝ) → E
f' : (Fin (n + 1) → ℝ) →L[ℝ] E
hfc : ContinuousOn f (Box.Icc I)
x : Fin (n + 1) → ℝ
hxI : x ∈ Box.Icc I
a : E
ε : ℝ
h0 : 0 < ε
hε : ∀ y ∈ Box.Icc I, ‖f y - a - f' (y - x)‖ ≤ ε * ‖y - x‖
c : ℝ≥0
hc : I.distortion ≤ c
e : ℝ → (Fin n → ℝ) → Fin (n + 1) → ℝ := i.insertNth
Hl : I.lower i ∈ Set.Icc (I.lower i) (I.upper i)
Hu : I.upper i ∈ Set.Icc (I.lower i) (I.upper i)
Hi : ∀ x ∈ Set.Icc (I.lower i) (I.upper i), Integrable (I.face i) ⊥ (f ∘ e x) BoxAdditiveMap.volume
this :
  ∀ y ∈ Box.Icc (I.face i),
    ‖f' (Pi.single i (I.upper i - I.lower i)) - (f (e (I.upper i) y) - f (e (I.lower i) y))‖ ≤ 2 * ε * diam (Box.Icc I)
⊢ ‖(volume.toBoxAdditive.toSMul (I.face i)) ((I.upper i - I.lower i) • f' (Pi.single i 1)) -
        integral (I.face i) ⊥ (f ∘ e (I.upper i) - f ∘ e (I.lower i)) BoxAdditiveMap.volume‖ =
    ‖integral (I.face i) ⊥
        (fun x ↦ f' (Pi.single i (I.upper i - I.lower i)) - (f (e (I.upper i) x) - f (e (I.lower i) x)))
        BoxAdditiveMap.volume‖
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic

Goal state before:
pp: T : ℝ
g : ℝ → ℝ
hg : Periodic g T
h₀ : 0 < ∫ (x : ℝ) in 0..T, g x
hT : 0 < T
h_int : IntervalIntegrable g volume 0 T
⊢ Tendsto (fun x ↦ ⌊x / T⌋ • ∫ (x : ℝ) in 0..T, g x) atTop atTop

Tactic:
apply Tendsto.atTop_zsmul_const h₀

Goal state after:
T : ℝ
g : ℝ → ℝ
hg : Periodic g T
h₀ : 0 < ∫ (x : ℝ) in 0..T, g x
hT : 0 < T
h_int : IntervalIntegrable g volume 0 T
⊢ Tendsto (fun x ↦ ⌊x / T⌋) atTop atTop
Module: Mathlib.MeasureTheory.Integral.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : StronglyMeasurable f
this : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖
⊢ ∫⁻ (x : α), ∫⁻ (y : β), ‖f (x, y)‖ₑ ∂ν ∂μ < ∞ ↔
    (∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ∞) ∧ ∫⁻ (a : α), ‖∫ (y : β), ‖f (a, y)‖ ∂ν‖ₑ ∂μ < ∞

Tactic:
integral_eq_lintegral_of_nonneg_ae (this _)
      (h1f.norm.comp_measurable measurable_prodMk_left).aestronglyMeasurable,

Goal state after:
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : StronglyMeasurable f
this : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖
⊢ ∫⁻ (x : α), ∫⁻ (y : β), ‖f (x, y)‖ₑ ∂ν ∂μ < ∞ ↔
    (∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ∞) ∧
      ∫⁻ (a : α), ‖(∫⁻ (a_1 : β), ENNReal.ofReal ‖f (a, a_1)‖ ∂ν).toReal‖ₑ ∂μ < ∞
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 : ℝ
n : ℕ
w : ℂ
⊢ ((cauchyPowerSeries f c R n) fun x ↦ w) = (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (w / (z - c)) ^ n • (z - c)⁻¹ • f z

Tactic:
simp only [cauchyPowerSeries, ContinuousMultilinearMap.mkPiRing_apply, Fin.prod_const,
    div_eq_mul_inv, mul_pow, mul_smul, circleIntegral.integral_smul]
  rw [← smul_comm (w ^ n)]

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

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
a b : Fin (n + 1) → ℝ
hle : a ≤ b
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 (Set.Icc a b)
Hd : ∀ x ∈ (Set.univ.pi fun i ↦ Set.Ioo (a i) (b i)) \ s, HasFDerivAt f (f' x) x
Hi : IntegrableOn (fun x ↦ ∑ i, (f' x) (e i) i) (Set.Icc a b) volume
this : (Set.univ.pi fun j ↦ Set.Ioc (a j) (b j)) = ∅
j : Fin (n + 1)
i : Fin n
hi : a (j.succAbove i) = b (j.succAbove i)
hi' : Set.Ioc (a (j.succAbove i)) (b (j.succAbove i)) = ∅
hne : j.succAbove i ≠ j
⊢ Set.Icc (a ∘ j.succAbove) (b ∘ j.succAbove) =ᵐ[volume] ∅

Tactic:
rw [ae_eq_empty, Real.volume_Icc_pi, prod_eq_zero (Finset.mem_univ i)]

Goal state after:
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
a b : Fin (n + 1) → ℝ
hle : a ≤ b
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 (Set.Icc a b)
Hd : ∀ x ∈ (Set.univ.pi fun i ↦ Set.Ioo (a i) (b i)) \ s, HasFDerivAt f (f' x) x
Hi : IntegrableOn (fun x ↦ ∑ i, (f' x) (e i) i) (Set.Icc a b) volume
this : (Set.univ.pi fun j ↦ Set.Ioc (a j) (b j)) = ∅
j : Fin (n + 1)
i : Fin n
hi : a (j.succAbove i) = b (j.succAbove i)
hi' : Set.Ioc (a (j.succAbove i)) (b (j.succAbove i)) = ∅
hne : j.succAbove i ≠ j
⊢ ENNReal.ofReal ((b ∘ j.succAbove) i - (a ∘ j.succAbove) i) = 0
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: F : Type u_1
inst✝² : Field F
p q : F[X]
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
x✝² x✝¹ : p.Gal
x✝ : ↑(p.rootSet p.SplittingField)
⊢ ↑((x✝² * x✝¹) • x✝) = ↑(x✝² • x✝¹ • x✝)

Tactic:
rfl

Goal state after:
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: F : Type u_1
inst✝² : Field F
p q : F[X]
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
x✝ : ↑(p.rootSet p.SplittingField)
⊢ ↑(1 • x✝) = ↑x✝

Tactic:
rfl

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

Goal state before:
pp: F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
c : ℂ
R C : ℝ
f : ℂ → F
n : ℕ
hR : 0 < R
hf : DiffContOnCl ℂ f (ball c R)
hC : ∀ z ∈ sphere c R, ‖f z‖ ≤ C
hp : ∀ z ∈ sphere c R, ‖(z - c)⁻¹ ^ (n + 1) • f z‖ ≤ C / (R ^ n * R)
hq : iteratedDeriv n f c = n.factorial • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ (n + 1) • f z
⊢ ↑n.factorial * (R * (C / R ^ (n + 1))) = ↑n.factorial * C / R ^ n

Tactic:
grind

Goal state after:
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.Defs

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : Monoid S
inst✝² : DistribMulAction S R
inst✝¹ : IsScalarTower S R R
inst✝ : SMulCommClass S R R
s : S
n : ℕ
m : Multiset R
⊢ (map (fun x ↦ (map (fun x ↦ s • x) x).prod) (powersetCard n m)).sum =
    (map ((fun x ↦ x.prod) ∘ map fun x ↦ s • x) (powersetCard n m)).sum

Tactic:
Function.comp_def

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Vieta

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
s : Multiset R
x✝ : ℕ
a✝ : x✝ ∈ range (s.card + 1)
⊢ (map ((fun p ↦ (map (fun r ↦ X) p.1).prod * (map (⇑C) p.2).prod) ∘ fun t ↦ (s - t, t)) (powersetCard x✝ s)).sum =
    (map (fun i ↦ C i.prod) (powersetCard x✝ s)).sum * X ^ (s.card - x✝)

Tactic:
← sum_map_mul_right,

Goal state after:
R : Type u_1
inst✝ : CommSemiring R
s : Multiset R
x✝ : ℕ
a✝ : x✝ ∈ range (s.card + 1)
⊢ (map ((fun p ↦ (map (fun r ↦ X) p.1).prod * (map (⇑C) p.2).prod) ∘ fun t ↦ (s - t, t)) (powersetCard x✝ s)).sum =
    (map (fun i ↦ C i.prod * X ^ (s.card - x✝)) (powersetCard x✝ s)).sum
Module: Mathlib.RingTheory.Polynomial.Vieta

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
s : Multiset R
k : ℕ
h : k ≤ s.card
⊢ s.esymm (s.card - k) = if k = s.card - (s.card - k) then s.esymm (s.card - k) else 0

Tactic:
grind

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Vieta

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
s : Multiset R
k : ℕ
h : k ≤ s.card
⊢ ∀ b ∈ Finset.range (s.card + 1), b ≠ s.card - k → (if k = s.card - b then s.esymm b else 0) = 0

Tactic:
grind

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Vieta

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
s : Multiset R
k : ℕ
h : k ≤ s.card
⊢ s.card - k ∈ Finset.range (s.card + 1)

Tactic:
grind

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Vieta

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
σ : Type u_2
s : Finset σ
r : σ → R
k : ℕ
h : k ≤ #s
⊢ ∑ t ∈ Finset.powersetCard (s.val.card - k) s, t.prod r = ∑ t ∈ Finset.powersetCard (#s - k) s, ∏ i ∈ t, r i

Tactic:
rfl

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Vieta

Goal state before:
pp: case e'_2
R : Type u_1
inst✝ : CommRing R
s : Multiset R
⊢ (map (fun x ↦ X + C (-x)) s).prod = (map ((fun r ↦ X + C r) ∘ fun t ↦ -t) s).prod

Tactic:
rfl

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

Goal state before:
pp: case inl
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
⊢ ‖(map f p).coeff i‖ ≤ B ^ (p.natDegree - i) * ↑(p.natDegree.choose i)

pp: case inr
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 : 0 ≤ B
⊢ ‖(map f p).coeff i‖ ≤ B ^ (p.natDegree - i) * ↑(p.natDegree.choose i)

Tactic:
· rw [eq_one_of_roots_le hB h1 h2 h3, Polynomial.map_one, natDegree_one, zero_tsub, pow_zero,
      one_mul, coeff_one]
    split_ifs with h <;> simp [h]

Goal state after:
case inr
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 : 0 ≤ B
⊢ ‖(map f p).coeff i‖ ≤ B ^ (p.natDegree - i) * ↑(p.natDegree.choose i)
Module: Mathlib.RingTheory.Polynomial.Vieta

Goal state before:
pp: case e'_2
R : Type u_1
inst✝ : CommRing R
s : Multiset R
k : ℕ
h : k ≤ s.card
⊢ (map (fun x ↦ X + C (-x)) s).prod.coeff k = (map ((fun r ↦ X + C r) ∘ fun t ↦ -t) s).prod.coeff ?convert_1

Tactic:
rfl

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

Goal state before:
pp: case refine_5
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : ℝ × ℝ → E
f' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E
a b : ℝ × ℝ
hle : a ≤ b
s : Set (ℝ × ℝ)
hs : s.Countable
Hcf : ContinuousOn f (Set.Icc a b)
Hcg : ContinuousOn g (Set.Icc a b)
Hdf : ∀ x ∈ Set.Ioo a.1 b.1 ×ˢ Set.Ioo a.2 b.2 \ s, HasFDerivAt f (f' x) x
Hdg : ∀ x ∈ Set.Ioo a.1 b.1 ×ˢ Set.Ioo a.2 b.2 \ s, HasFDerivAt g (g' x) x
Hi : IntegrableOn (fun x ↦ (f' x) (1, 0) + (g' x) (0, 1)) (Set.Icc a b) volume
e : (ℝ × ℝ) ≃L[ℝ] Fin 2 → ℝ := (ContinuousLinearEquiv.finTwoArrow ℝ ℝ).symm
x : ℝ × ℝ
⊢ (f' x) (1, 0) + (g' x) (0, 1) = (![f', g'] 0 x) (e.symm (Pi.single 0 1)) + (![f', g'] 1 x) (e.symm (Pi.single 1 1))

Tactic:
rfl

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

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : ℝ × ℝ → E
f' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E
a b : ℝ × ℝ
hle : a ≤ b
s : Set (ℝ × ℝ)
hs : s.Countable
Hcf : ContinuousOn f (Set.Icc a b)
Hcg : ContinuousOn g (Set.Icc a b)
Hdf : ∀ x ∈ Set.Ioo a.1 b.1 ×ˢ Set.Ioo a.2 b.2 \ s, HasFDerivAt f (f' x) x
Hdg : ∀ x ∈ Set.Ioo a.1 b.1 ×ˢ Set.Ioo a.2 b.2 \ s, HasFDerivAt g (g' x) x
Hi : IntegrableOn (fun x ↦ (f' x) (1, 0) + (g' x) (0, 1)) (Set.Icc a b) volume
e : (ℝ × ℝ) ≃L[ℝ] Fin 2 → ℝ := (ContinuousLinearEquiv.finTwoArrow ℝ ℝ).symm
this :
  ∀ (a b : Fin 1 → ℝ) (f : (Fin 1 → ℝ) → E),
    ∫ (x : Fin 1 → ℝ) in Set.Icc a b, f x = ∫ (x : ℝ) in Set.Icc (a 0) (b 0), f fun x_1 ↦ x
⊢ ((∫ (x : ℝ) in Set.Icc ((e a ∘ Fin.succAbove 0) 0) ((e b ∘ Fin.succAbove 0) 0),
          ![f, g] 0 (e.symm (Fin.insertNth 0 (e b 0) fun x_1 ↦ x))) -
        ∫ (x : ℝ) in Set.Icc ((e a ∘ Fin.succAbove 0) 0) ((e b ∘ Fin.succAbove 0) 0),
          ![f, g] 0 (e.symm (Fin.insertNth 0 (e a 0) fun x_1 ↦ x))) +
      ((∫ (x : ℝ) in Set.Icc ((e a ∘ Fin.succAbove 1) 0) ((e b ∘ Fin.succAbove 1) 0),
          ![f, g] 1 (e.symm (Fin.insertNth 1 (e b 1) fun x_1 ↦ x))) -
        ∫ (x : ℝ) in Set.Icc ((e a ∘ Fin.succAbove 1) 0) ((e b ∘ Fin.succAbove 1) 0),
          ![f, g] 1 (e.symm (Fin.insertNth 1 (e a 1) fun x_1 ↦ x))) =
    ((∫ (y : ℝ) in Set.Icc a.2 b.2, f (b.1, y)) - ∫ (y : ℝ) in Set.Icc a.2 b.2, f (a.1, y)) +
      ((∫ (x : ℝ) in Set.Icc a.1 b.1, g (x, b.2)) - ∫ (x : ℝ) in Set.Icc a.1 b.1, g (x, a.2))

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Complex.Polynomial.Basic

Goal state before:
pp: case a
p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
a : Finset ℂ := ⋯
b : Finset ℂ := ⋯
c : Finset ℂ := ⋯
ha : ∀ (z : ℂ), z ∈ a ↔ (aeval z) p = 0
z : ℂ
hz1 : (aeval z) p = 0
hz2 : z.im = 0
⊢ ((IsScalarTower.toAlgHom ℚ ℝ ℂ) z.re).re = z.re

Tactic:
rfl

Goal state after: