mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.Bochner.FundThmCalculus

Goal state before:
pp: X : Type u_1
E : Type u_2
ι : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure X
l : Filter X
inst✝ : l.IsMeasurablyGenerated
f : X → E
b : E
h : Tendsto f (l ⊓ ae μ) (𝓝 b)
hfm : StronglyMeasurableAtFilter f l μ
hμ : μ.FiniteAtFilter l
s✝ : ι → Set X
li : Filter ι
hs : Tendsto s✝ li l.smallSets
m : ι → ℝ
hsμ : (fun i ↦ μ.real (s✝ i)) =ᶠ[li] m
ε : ℝ
ε₀ : 0 < ε
this : ∀ᶠ (s : Set X) in l.smallSets, ∀ᵐ (x : X) ∂μ, x ∈ s → f x ∈ closedBall b ε
s : Set X
⊢ μ s < ⊤ →
    IntegrableOn f s μ →
      AEStronglyMeasurable f (μ.restrict s) →
        (∀ᵐ (x : X) ∂μ, x ∈ s → ‖f x - b‖ ≤ ε) → ‖∫ (x : X) in s, f x ∂μ - μ.real s • b‖ ≤ ε * ‖μ.real s‖

Tactic:
hμs

Goal state after:
X : Type u_1
E : Type u_2
ι : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure X
l : Filter X
inst✝ : l.IsMeasurablyGenerated
f : X → E
b : E
h : Tendsto f (l ⊓ ae μ) (𝓝 b)
hfm : StronglyMeasurableAtFilter f l μ
hμ : μ.FiniteAtFilter l
s✝ : ι → Set X
li : Filter ι
hs : Tendsto s✝ li l.smallSets
m : ι → ℝ
hsμ : (fun i ↦ μ.real (s✝ i)) =ᶠ[li] m
ε : ℝ
ε₀ : 0 < ε
this : ∀ᶠ (s : Set X) in l.smallSets, ∀ᵐ (x : X) ∂μ, x ∈ s → f x ∈ closedBall b ε
s : Set X
hμs : μ s < ⊤
⊢ IntegrableOn f s μ →
    AEStronglyMeasurable f (μ.restrict s) →
      (∀ᵐ (x : X) ∂μ, x ∈ s → ‖f x - b‖ ≤ ε) → ‖∫ (x : X) in s, f x ∂μ - μ.real s • b‖ ≤ ε * ‖μ.real s‖
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: case inl
x : ℝ
hx✝ : x ≠ 0
hx : x < 0
⊢ HasStrictDerivAt log x⁻¹ x

Tactic:
convert! (hasStrictDerivAt_log_of_pos (neg_pos.mpr hx)).comp x (hasStrictDerivAt_neg x) using 1

Goal state after:
case e'_8
x : ℝ
hx✝ : x ≠ 0
hx : x < 0
⊢ log = log ∘ Neg.neg

case e'_9
x : ℝ
hx✝ : x ≠ 0
hx : x < 0
⊢ x⁻¹ = (-x)⁻¹ * -1
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ),
    n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun e ↦ L e (n e) (n e)
⊢ CauchySeq L0

Tactic:
Metric.cauchySeq_iff'

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ),
    n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun e ↦ L e (n e) (n e)
⊢ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (L0 n) (L0 N) < ε
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
r : ℝ
hr : 0 ≤ r
x : E
hr' : 0 < r
this : ∀ᶠ (R : ℝ) in 𝓝[>] r, ‖x‖ / R ≤ gauge (closedBall 0 r) x
⊢ Filter.Tendsto (HDiv.hDiv ‖x‖) (𝓝[>] r) (𝓝 (‖x‖ / r))

Tactic:
exact tendsto_const_nhds.div inf_le_left hr'.ne'

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: case refine_5
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ
hf : Integrable f μ
ε : ℝ
εpos : 0 < ε
g : α → EReal
g_lt_f : ∀ (x : α), ↑(-f x) < g x
gcont : LowerSemicontinuous g
g_integrable : Integrable (fun x ↦ (g x).toReal) μ
g_lt_top : ∀ᵐ (x : α) ∂μ, g x < ⊤
gint : ∫ (x : α), (g x).toReal ∂μ < ∫ (x : α), -f x ∂μ + ε
⊢ ∫ (x : α), f x ∂μ < ∫ (x : α), ((fun x ↦ -g x) x).toReal ∂μ + ε

Tactic:
simp_rw [integral_neg, lt_neg_add_iff_add_lt] at gint
    rw [add_comm] at gint
    simpa [integral_neg] using gint

Goal state after:
Module: Mathlib.MeasureTheory.Topology

Goal state before:
pp: α : Type u_1
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : NullSingletonClass μ
U : Set α
hU : MeasurableSet U
s : Set α
hs : s ∈ codiscreteWithin U
this : DiscreteTopology ↑(sᶜ ∩ U)
⊢ (μ.restrict U) sᶜ = 0

Tactic:
Measure.restrict_apply' hU

Goal state after:
α : Type u_1
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : NullSingletonClass μ
U : Set α
hU : MeasurableSet U
s : Set α
hs : s ∈ codiscreteWithin U
this : DiscreteTopology ↑(sᶜ ∩ U)
⊢ μ (sᶜ ∩ U) = 0
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
x : ℝ
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ), n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * (1 / 2) ^ e
L0 : ℕ → F := fun e ↦ L e (n e) (n e)
⊢ CauchySeq L0

Tactic:
Metric.cauchySeq_iff'

Goal state after:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
x : ℝ
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ), n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * (1 / 2) ^ e
L0 : ℕ → F := fun e ↦ L e (n e) (n e)
⊢ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (L0 n) (L0 N) < ε
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: α : Type u_1
G : Type u_3
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
ι : Type u_4
l : Filter ι
inst✝¹ : l.IsCountablyGenerated
F : ι → α → G
inst✝ : IsFiniteMeasure μ
f : α → G
h_meas : ∀ᶠ (n : ι) in l, AEStronglyMeasurable (F n) μ
h_bound : ∃ C, ∀ᶠ (n : ι) in l, ∀ᵐ (ω : α) ∂μ, ‖F n ω‖ ≤ C
h_lim : ∀ᵐ (ω : α) ∂μ, Tendsto (fun n ↦ F n ω) l (𝓝 (f ω))
⊢ Tendsto (fun n ↦ ∫ (ω : α), F n ω ∂μ) l (𝓝 (∫ (ω : α), f ω ∂μ))

Tactic:
simp only [integral_eq_setToFun]
  exact tendsto_setToFun_filter_of_norm_le_const _ h_meas h_bound h_lim

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

Goal state before:
pp: X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0∞
f_meas : AEMeasurable f μ
hf_lt_top : ∀ᵐ (x : X) ∂μ, f x < ∞
g : X → E
⊢ ∫ (x : X), g x ∂μ.withDensity f = ∫ (x : X), (f x).toReal • g x ∂μ

Tactic:
dsimp only [ENNReal.toReal, ← NNReal.smul_def]

Goal state after:
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0∞
f_meas : AEMeasurable f μ
hf_lt_top : ∀ᵐ (x : X) ∂μ, f x < ∞
g : X → E
⊢ ∫ (x : X), g x ∂μ.withDensity f = ∫ (x : X), (f x).toNNReal • g x ∂μ
Module: Mathlib.Analysis.Calculus.TangentCone.Prod

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁰ : Semiring 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : TopologicalSpace E
inst✝⁶ : ContinuousAdd E
inst✝⁵ : ContinuousConstSMul 𝕜 E
inst✝⁴ : AddCommGroup F
inst✝³ : Module 𝕜 F
inst✝² : TopologicalSpace F
inst✝¹ : ContinuousAdd F
inst✝ : ContinuousConstSMul 𝕜 F
x : E
s : Set E
y : F
t : Set F
hs : Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 s x)) ∧ x ∈ closure[inst✝⁷] s
ht : Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 t y)) ∧ y ∈ closure[inst✝²] t
this :
  Submodule.span 𝕜 (⇑(LinearMap.inl 𝕜 E F) '' tangentConeAt 𝕜 s x ∪ ⇑(LinearMap.inr 𝕜 E F) '' tangentConeAt 𝕜 t y) ≤
    Submodule.span 𝕜 (tangentConeAt 𝕜 (s ×ˢ t) (x, y))
⊢ Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 (s ×ˢ t) (x, y)))

Tactic:
rw [LinearMap.span_inl_union_inr, SetLike.le_def] at this

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁰ : Semiring 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : TopologicalSpace E
inst✝⁶ : ContinuousAdd E
inst✝⁵ : ContinuousConstSMul 𝕜 E
inst✝⁴ : AddCommGroup F
inst✝³ : Module 𝕜 F
inst✝² : TopologicalSpace F
inst✝¹ : ContinuousAdd F
inst✝ : ContinuousConstSMul 𝕜 F
x : E
s : Set E
y : F
t : Set F
hs : Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 s x)) ∧ x ∈ closure[inst✝⁷] s
ht : Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 t y)) ∧ y ∈ closure[inst✝²] t
this :
  ∀ ⦃x_1 : E × F⦄,
    x_1 ∈ (Submodule.span 𝕜 (tangentConeAt 𝕜 s x)).prod (Submodule.span 𝕜 (tangentConeAt 𝕜 t y)) →
      x_1 ∈ Submodule.span 𝕜 (tangentConeAt 𝕜 (s ×ˢ t) (x, y))
⊢ Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 (s ×ˢ t) (x, y)))
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: ε : Type u_3
inst✝² : TopologicalSpace ε
inst✝¹ : ENormedAddMonoid ε
f : ℝ → ε
a b : ℝ
inst✝ : PseudoMetrizableSpace ε
hf : IntervalIntegrable f volume a b
c : ℝ
h : ‖f (min a b)‖ₑ ≠ ∞
h' : ‖f (c * min (a / c) (b / c))‖ₑ ≠ ∞
⊢ IntervalIntegrable (fun x ↦ f (x * c)) volume (a / c) (b / c)

Tactic:
simpa only [mul_comm] using comp_mul_left hf h h'

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : LocallyCompactSpace E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
α : Type u_4
inst✝ : TopologicalSpace α
f : α → E → F
r s : ℝ
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
L : E →L[𝕜] F
this : ProperSpace E
a : α
x : E
r' : ℝ
hr :
  ∀ (y : E),
    dist y (a, x).2 < r' → ∀ (z : E), dist z (a, x).2 < r' → ‖f (a, x).1 z - f (a, x).1 y - (L z - L y)‖ < s * r
Irr' : r / 2 < r'
Ir'r : r' ≤ r
t : ℝ
hrt : r / 2 < t
htr' : t < r'
t' : ℝ
hrt' : r / 2 < t'
ht't : t' < t
B : Continuous[instTopologicalSpaceProd, _] fun p ↦ ‖f a p.2 - f a p.1 - (L p.2 - L p.1)‖
C : (closedBall x t ×ˢ closedBall x t).Nonempty
p : E × E
hp : IsMaxOn (fun p ↦ ‖f a p.2 - f a p.1 - (L p.2 - L p.1)‖) (closedBall x t ×ˢ closedBall x t) p
pt : dist p.1 x ≤ t ∧ dist p.2 x ≤ t
y : E
hy : y ∈ closedBall x t
z : E
hz : z ∈ closedBall x t
D : (y, z) ∈ closedBall x t ×ˢ closedBall x t
⊢ ‖f a z - f a y - (L z - L y)‖ ≤ ‖f a p.2 - f a p.1 - (L p.2 - L p.1)‖

Tactic:
exact hp D

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure ℝ
f : ℝ → E
inst✝ : NullSingletonClass μ
h_int : ∀ (a b : ℝ), IntervalIntegrable f μ a b
a b₀ b₁ : ℝ
hb₁ : b₁ < b₀
b₂ : ℝ
hb₂ : b₀ < b₂
⊢ ContinuousWithinAt (fun b ↦ ∫ (x : ℝ) in a..b, f x ∂μ) (Icc b₁ b₂) b₀

Tactic:
exact continuousWithinAt_primitive (measure_singleton b₀) (h_int _ _)

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 ((fun x ↦ g * x) ⁻¹' π_preA ∩ 𝓕) = ν (π_preA ∩ g • 𝓕)

Tactic:
this,

Goal state after:
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 ∩ g • 𝓕)
Module: Mathlib.Topology.Algebra.Order.Floor

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁷ : Ring α
inst✝⁶ : LinearOrder α
inst✝⁵ : FloorRing α
inst✝⁴ : TopologicalSpace α
inst✝³ : IsStrictOrderedRing α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : β → α → γ
h : ContinuousOn (uncurry f) (univ ×ˢ Icc 0 1)
hf : ∀ (s : β), f s 0 = f s 1
s : β
n : ℤ
⊢ Tendsto (uncurry f ∘ Prod.map id fract) (𝓝 s ×ˢ (𝓝[<] ↑n ⊔ 𝓝[≥] ↑n)) (𝓝 ((uncurry f ∘ Prod.map id fract) (s, ↑n)))

Tactic:
prod_sup,

Goal state after:
case inl
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁷ : Ring α
inst✝⁶ : LinearOrder α
inst✝⁵ : FloorRing α
inst✝⁴ : TopologicalSpace α
inst✝³ : IsStrictOrderedRing α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : β → α → γ
h : ContinuousOn (uncurry f) (univ ×ˢ Icc 0 1)
hf : ∀ (s : β), f s 0 = f s 1
s : β
n : ℤ
⊢ Tendsto (uncurry f ∘ Prod.map id fract) (𝓝 s ×ˢ 𝓝[<] ↑n ⊔ 𝓝 s ×ˢ 𝓝[≥] ↑n)
    (𝓝 ((uncurry f ∘ Prod.map id fract) (s, ↑n)))
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✝¹ : μ'.IsMulLeftInvariant
this✝ : SigmaFinite μ'
this : μ' V / ν (QuotientGroup.mk ⁻¹' V ∩ s) = 1
⊢ (μ' V / ν (QuotientGroup.mk ⁻¹' V ∩ s)) • μ = μ

Tactic:
this,

Goal state after:
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✝¹ : μ'.IsMulLeftInvariant
this✝ : SigmaFinite μ'
this : μ' V / ν (QuotientGroup.mk ⁻¹' V ∩ s) = 1
⊢ 1 • μ = μ
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
a : ℝ
⊢ MeasurePreserving (⇑(equivIoc T a)) volume (Measure.comap Subtype.val volume)

Tactic:
have h := (measurableEquivIoc T a).measurable
  refine ⟨h, ?_⟩
  ext s hs
  rw [comap_apply _ Subtype.val_injective (fun _ ↦ measurableSet_Ioc.subtype_image) _ hs,
    map_apply (by measurability) hs, add_projection_respects_measure T a (by exact h hs)]
  congr!
  ext x
  simp only [mem_inter_iff, mem_preimage, mem_image, Subtype.exists, exists_and_right,
    exists_eq_right]
  rw [and_comm, ← exists_prop]
  congr! with hx
  rw [equivIoc_coe_eq hx]

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

Goal state before:
pp: case h_lim
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
⊢ Tendsto (fun n ↦ if n < x then f x else 0) (𝓝[≥] a₀) (𝓝 (if a₀ < x then f x else 0))

Tactic:
by_cases hx : a₀ < x

Goal state after:
case pos
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
⊢ Tendsto (fun n ↦ if n < x then f x else 0) (𝓝[≥] a₀) (𝓝 (if a₀ < x then f x else 0))

case neg
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
⊢ Tendsto (fun n ↦ if n < x then f x else 0) (𝓝[≥] a₀) (𝓝 (if a₀ < x then f x else 0))
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|
⊢ ¬CircleIntegrable (fun z ↦ (z - w) ^ n) c R

Tactic:
simp only [circleIntegrable_iff R, deriv_circleMap]

Goal state after:
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)
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
hf : ∀ z ∈ sphere c R, ‖f z‖ ≤ C
this : ‖(2 * ↑π * I)⁻¹‖ = (2 * π)⁻¹
⊢ ‖(2 * ↑π * I)⁻¹‖ * ‖∮ (z : ℂ) in C(c, R), f z‖ ≤ R * C

Tactic:
this,

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
hf : ∀ z ∈ sphere c R, ‖f z‖ ≤ C
this : ‖(2 * ↑π * I)⁻¹‖ = (2 * π)⁻¹
⊢ (2 * π)⁻¹ * ‖∮ (z : ℂ) in C(c, R), f z‖ ≤ R * C
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))
f : (Fin (n + 1) → ℝ) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x
Hd : ∀ x ∈ Box.Icc I \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x
i : Fin (n + 1)
x : Fin (n + 1) → ℝ
hx : x ∈ Box.Icc I
⊢ ContinuousWithinAt f (Box.Icc I) x

Tactic:
by_cases hxs : x ∈ s
    exacts [Hs x hxs, (Hd x ⟨hx, hxs⟩).continuousWithinAt]

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 ∈ circleMap c R '' Ioc 0 (2 * π), ‖f z‖ < C
⊢ ‖∮ (z : ℂ) in C(c, R), f z‖ < 2 * π * R * C

Tactic:
rcases hlt with ⟨_, ⟨θ₀, hmem, rfl⟩, hlt⟩

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
θ₀ : ℝ
hmem : θ₀ ∈ Ioc 0 (2 * π)
hlt : ‖f (circleMap c R θ₀)‖ < C
⊢ ‖∮ (z : ℂ) in C(c, R), f z‖ < 2 * π * R * C
Module: Mathlib.MeasureTheory.Integral.Prod

Goal state before:
pp: case inl
α : Type u_1
E : Type u_3
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : SFinite μ
a b : ℝ
f : ℝ → α → E
h_int : Integrable (uncurry f) ((volume.restrict (uIoc a b)).prod μ)
hab : a ≤ b
⊢ ∫ (x : ℝ) in a..b, ∫ (y : α), f x y ∂μ = ∫ (y : α), ∫ (x : ℝ) in a..b, f x y ∂μ

Tactic:
simp_rw [intervalIntegral.integral_of_le hab]
    simp only [hab, Set.uIoc_of_le] at h_int
    exact integral_integral_swap h_int

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

Goal state before:
pp: case inl
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
i : Fin (n + 1)
hi : a i = b i
hi' : Set.Ioc (a i) (b i) = ∅
this : (Set.univ.pi fun j ↦ Set.Ioc (a j) (b j)) = ∅
⊢ (∫ (x : Fin (n + 1) → ℝ) in Set.univ.pi fun i ↦ Set.Ioc (a i) (b i), ∑ i, (f' x) (e i) i ∂Measure.pi fun x ↦ volume) =
    ∑ i,
      ((∫ (x : Fin n → ℝ) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), f (i.insertNth (b i) x) i) -
        ∫ (x : Fin n → ℝ) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), f (i.insertNth (a i) x) i)

Tactic:
this,

Goal state after:
case inl
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
i : Fin (n + 1)
hi : a i = b i
hi' : Set.Ioc (a i) (b i) = ∅
this : (Set.univ.pi fun j ↦ Set.Ioc (a j) (b j)) = ∅
⊢ (∫ (x : Fin (n + 1) → ℝ) in ∅, ∑ i, (f' x) (e i) i ∂Measure.pi fun x ↦ volume) =
    ∑ i,
      ((∫ (x : Fin n → ℝ) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), f (i.insertNth (b i) x) i) -
        ∫ (x : Fin n → ℝ) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), f (i.insertNth (a i) x) i)
Module: Mathlib.Analysis.Complex.CauchyIntegral

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
c : ℂ
f : ℂ → E
s : Set ℂ
a : ℝ
h0 : 0 < rexp a
b : ℝ
hle : a ≤ b
hd : ∀ z ∈ (ball c (rexp b) \ closedBall c (rexp a)) \ s, DifferentiableAt ℂ f z
A : Set ℂ := closedBall c (rexp b) \ ball c (rexp a)
hc : ContinuousOn f A
R : Set ℂ := [[a, b]] ×ℂ [[0, 2 * π]]
g : ℂ → ℂ := fun x ↦ c + cexp x
hdg : Differentiable ℂ g
hs : (g ⁻¹' s).Countable
⊢ ∫ (θ : ℝ) in 0..2 * π, I • f (circleMap c (rexp b) θ) = ∫ (θ : ℝ) in 0..2 * π, I • f (circleMap c (rexp a) θ)

Tactic:
have h_maps : MapsTo g R A := by
    rintro z ⟨h, -⟩; simpa [g, A, dist_eq, norm_exp, hle] using h.symm

Goal state after:
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
c : ℂ
f : ℂ → E
s : Set ℂ
a : ℝ
h0 : 0 < rexp a
b : ℝ
hle : a ≤ b
hd : ∀ z ∈ (ball c (rexp b) \ closedBall c (rexp a)) \ s, DifferentiableAt ℂ f z
A : Set ℂ := closedBall c (rexp b) \ ball c (rexp a)
hc : ContinuousOn f A
R : Set ℂ := [[a, b]] ×ℂ [[0, 2 * π]]
g : ℂ → ℂ := fun x ↦ c + cexp x
hdg : Differentiable ℂ g
hs : (g ⁻¹' s).Countable
h_maps : MapsTo g R A
⊢ ∫ (θ : ℝ) in 0..2 * π, I • f (circleMap c (rexp b) θ) = ∫ (θ : ℝ) in 0..2 * π, I • f (circleMap c (rexp a) θ)