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

Goal state before:
pp: ι : Type u_1
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
l : Filter ι
inst✝¹ : l.IsCountablyGenerated
F : ι → ℝ → E
inst✝ : IsLocallyFiniteMeasure μ
hF : ∀ᶠ (i : ι) in l, ContinuousOn (F i) [[a, b]]
h_lim : TendstoUniformlyOn F f l [[a, b]]
hl : l.NeBot
C : ℝ
hC : C ∈ upperBounds ((fun x ↦ ‖f x‖) '' [[a, b]])
⊢ ∀ x ∈ [[a, b]], (norm ∘ f) x ≤ C

Tactic:
simpa [upperBounds] using hC

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

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
a b : ℝ
μ : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
c : E
⊢ ‖c‖ₑ ≠ ∞

Tactic:
simp

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

Goal state before:
pp: case left
ε : Type u_3
inst✝¹ : TopologicalSpace ε
inst✝ : ENormedAddMonoid ε
f : ℝ → ε
a : ℝ
μ : Measure ℝ
⊢ IntegrableOn f (Ioc a a) μ

Tactic:
simp

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

Goal state before:
pp: case right
ε : Type u_3
inst✝¹ : TopologicalSpace ε
inst✝ : ENormedAddMonoid ε
f : ℝ → ε
a : ℝ
μ : Measure ℝ
⊢ IntegrableOn f (Ioc a a) μ

Tactic:
simp

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

Goal state before:
pp: ε : Type u_3
inst✝¹ : TopologicalSpace ε
inst✝ : ENormedAddMonoid ε
f : ℝ → ε
a : ℝ
μ : Measure ℝ
⊢ IntervalIntegrable f μ a a

Tactic:
constructor <;> simp

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

Goal state before:
pp: case refine_1
ε : Type u_3
inst✝² : TopologicalSpace ε
inst✝¹ : ENormedAddMonoid ε
f : ℝ → ε
μ : Measure ℝ
inst✝ : PseudoMetrizableSpace ε
a : ℕ → ℝ
m n : ℕ
hmn : m ≤ n
⊢ (∀ k ∈ Ico m m, IntervalIntegrable f μ (a k) (a (k + 1))) → IntervalIntegrable f μ (a m) (a m)

Tactic:
simp

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

Goal state before:
pp: ι : Type u_1
a b : ℝ
μ : Measure ℝ
ε : Type u_8
inst✝³ : TopologicalSpace ε
inst✝² : ENormedAddCommMonoid ε
inst✝¹ : ContinuousAdd ε
inst✝ : PseudoMetrizableSpace ε
f : ι → ℝ → ε
h : ∀ (i : ι), IntervalIntegrable (f i) μ a b
h₁ : ¬(support f).Finite
⊢ ‖0‖ₑ ≠ ∞

Tactic:
simp

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

Goal state before:
pp: case inl
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 ≠ ∞
hc : c = 0
⊢ IntervalIntegrable (fun x ↦ f (0 * x)) volume (a / 0) (b / 0)

Tactic:
simp

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

Goal state before:
pp: F : Type u_1
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
f : ℝ → F
inst✝ : CompleteSpace F
this : IsComplete univ
x✝ : ℝ
⊢ DifferentiableWithinAt ℝ f (Ici x✝) x✝ ↔ DifferentiableWithinAt ℝ f (Ici x✝) x✝ ∧ derivWithin f (Ici x✝) x✝ ∈ univ

Tactic:
simp

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

Goal state before:
pp: E : Type u_5
inst✝ : NormedAddCommGroup E
a b : ℝ
f : ℝ → E
c : ℝ
hc : c ≠ 0
h✝ : ‖f (min a b)‖ₑ ≠ ∞
h' : ‖f (c * min (a / c) (b / c))‖ₑ ≠ ∞
h : IntervalIntegrable (fun x ↦ f (c * x)) volume (a / c) (b / c)
⊢ ‖f (c * (c⁻¹ * min (a / c / c⁻¹) (b / c / c⁻¹)))‖ₑ ≠ ∞

Tactic:
simp

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)‖
⊢ (closedBall x t ×ˢ closedBall x t).Nonempty

Tactic:
simp

Goal state after:
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)‖
⊢ 0 ≤ t
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: case pos
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b₀ b₁ b₂ : ℝ
μ : Measure ℝ
f : ℝ → E
hb₀ : μ {b₀} = 0
h_int : IntervalIntegrable f μ (min a b₁) (max a b₂)
h₀ : b₀ ∈ Icc b₁ b₂
h₁₂ : b₁ ≤ b₂
min₁₂ : min b₁ b₂ = b₁
h_int' : ∀ {x : ℝ}, x ∈ Icc b₁ b₂ → IntervalIntegrable f μ b₁ x
this :
  (fun b ↦ ∫ (x : ℝ) in b₁..b, f x ∂μ) =ᶠ[𝓝[Icc b₁ b₂] b₀] fun b ↦ ∫ (x : ℝ) in b₁..b₂, {x | x ≤ b}.indicator f x ∂μ
x t : ℝ
h✝ : t ≤ x
⊢ t ∈ Ι b₁ b₂ → ‖f t‖ ≤ ‖f t‖

Tactic:
simp

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

Goal state before:
pp: case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b₀ b₁ b₂ : ℝ
μ : Measure ℝ
f : ℝ → E
hb₀ : μ {b₀} = 0
h_int : IntervalIntegrable f μ (min a b₁) (max a b₂)
h₀ : b₀ ∈ Icc b₁ b₂
h₁₂ : b₁ ≤ b₂
min₁₂ : min b₁ b₂ = b₁
h_int' : ∀ {x : ℝ}, x ∈ Icc b₁ b₂ → IntervalIntegrable f μ b₁ x
this :
  (fun b ↦ ∫ (x : ℝ) in b₁..b, f x ∂μ) =ᶠ[𝓝[Icc b₁ b₂] b₀] fun b ↦ ∫ (x : ℝ) in b₁..b₂, {x | x ≤ b}.indicator f x ∂μ
x t : ℝ
h✝ : ¬t ≤ x
⊢ t ∈ Ι b₁ b₂ → ‖0‖ ≤ ‖f t‖

Tactic:
simp

Goal state after:
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
c : ℝ
h : ‖f (min a b)‖ₑ ≠ ∞
⊢ IntervalIntegrable f volume (a - c + c) (b - c + c) ↔ IntervalIntegrable f volume a b

Tactic:
simp

Goal state after:
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
c : ℝ
h : ‖f (min a b)‖ₑ ≠ ∞
⊢ IntervalIntegrable f volume (a + c + -c) (b + c + -c) ↔ IntervalIntegrable f volume a b

Tactic:
simp

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

Goal state before:
pp: E : Type u_5
inst✝ : NormedAddCommGroup E
a b : ℝ
f : ℝ → E
h : ‖f (min a b)‖ₑ ≠ ∞
⊢ ‖f (-1 * min (a / -1) (b / -1))‖ₑ ≠ ∞

Tactic:
simp

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

Goal state before:
pp: E : Type u_5
inst✝ : NormedAddCommGroup E
a b : ℝ
f : ℝ → E
hf : IntervalIntegrable f volume a b
c : ℝ
h : ‖f (min a b)‖ₑ ≠ ∞
⊢ ‖f (c + min (a - c) (b - c))‖ₑ ≠ ∞

Tactic:
simp

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

Goal state before:
pp: case pos
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
μ : Measure ℝ
f : ℝ → E
a b : ℝ
h✝ : a ≤ b
⊢ 1 • ∫ (x : ℝ) in Ι a b, f x ∂μ ∈ {∫ (x : ℝ) in Ι a b, f x ∂μ, -∫ (x : ℝ) in Ι a b, f x ∂μ}

Tactic:
simp

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

Goal state before:
pp: case neg
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
μ : Measure ℝ
f : ℝ → E
a b : ℝ
h✝ : ¬a ≤ b
⊢ -1 • ∫ (x : ℝ) in Ι a b, f x ∂μ ∈ {∫ (x : ℝ) in Ι a b, f x ∂μ, -∫ (x : ℝ) in Ι a b, f x ∂μ}

Tactic:
simp

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

Goal state before:
pp: case hf
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
⊢ IntegrableOn (fun x ↦ y) (Icc t u) volume

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : LocallyCompactSpace E
F : Type u_3
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁵ : TopologicalSpace α
f : α → E → F
inst✝⁴ : MeasurableSpace α
inst✝³ : OpensMeasurableSpace α
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
inst✝ : CompleteSpace F
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
this : IsComplete univ
x✝ : α × E
⊢ DifferentiableAt 𝕜 (f x✝.1) x✝.2 ↔ DifferentiableAt 𝕜 (f x✝.1) x✝.2 ∧ fderiv 𝕜 (f x✝.1) x✝.2 ∈ univ

Tactic:
simp

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

Goal state before:
pp: case refine_1
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
μ : Measure ℝ
a : ℕ → ℝ
m n : ℕ
hmn : m ≤ n
⊢ (∀ k ∈ Ico m m, IntervalIntegrable f μ (a k) (a (k + 1))) →
    ∑ k ∈ Finset.Ico m m, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ = ∫ (x : ℝ) in a m..a m, f x ∂μ

Tactic:
simp

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

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

Tactic:
simp

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

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : TopologicalSpace X
μ : Measure ℝ
inst✝¹ : NullSingletonClass μ
inst✝ : IsLocallyFiniteMeasure μ
f : X → ℝ → E
a₀ : ℝ
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
q : X
b₀ ε : ℝ
εpos : ε > 0
a : ℝ
a_lt : a < a₀ ∧ a < b₀
b : ℝ
lt_b : a₀ < b ∧ b₀ < b
this✝ : IsCompact ({q} ×ˢ Icc a b)
M : ℝ
hM : M ∈ upperBounds ((fun x ↦ ‖Function.uncurry f x‖) '' {q} ×ˢ Icc a b)
δ : ℝ
δpos : 0 < δ
hδ : δ < 1
h'δ : Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
h''δ : (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
v : Set X
v_mem : v ∈ 𝓝[univ] q
this : v ×ˢ Ioo (b₀ - δ) (b₀ + δ) ∈ 𝓝 (q, b₀)
p : X
s : ℝ
hp : p ∈ v
hs : s ∈ Ioo (b₀ - δ) (b₀ + δ)
hv : ∀ p ∈ v, ∀ x ∈ Icc a b, ‖f p x - f q x‖ < δ
⊢ ‖∫ (t : ℝ) in a₀..s, f p t ∂μ - ∫ (t : ℝ) in a₀..b₀, f q t ∂μ‖ < ε

Tactic:
have J r u v : IntervalIntegrable (f r) μ u v := (hf.uncurry_left _).intervalIntegrable _ _

Goal state after:
E : Type u_1
X : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : TopologicalSpace X
μ : Measure ℝ
inst✝¹ : NullSingletonClass μ
inst✝ : IsLocallyFiniteMeasure μ
f : X → ℝ → E
a₀ : ℝ
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
q : X
b₀ ε : ℝ
εpos : ε > 0
a : ℝ
a_lt : a < a₀ ∧ a < b₀
b : ℝ
lt_b : a₀ < b ∧ b₀ < b
this✝ : IsCompact ({q} ×ˢ Icc a b)
M : ℝ
hM : M ∈ upperBounds ((fun x ↦ ‖Function.uncurry f x‖) '' {q} ×ˢ Icc a b)
δ : ℝ
δpos : 0 < δ
hδ : δ < 1
h'δ : Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
h''δ : (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
v : Set X
v_mem : v ∈ 𝓝[univ] q
this : v ×ˢ Ioo (b₀ - δ) (b₀ + δ) ∈ 𝓝 (q, b₀)
p : X
s : ℝ
hp : p ∈ v
hs : s ∈ Ioo (b₀ - δ) (b₀ + δ)
hv : ∀ p ∈ v, ∀ x ∈ Icc a b, ‖f p x - f q x‖ < δ
J : ∀ (r : X) (u v : ℝ), IntervalIntegrable (f r) μ u v
⊢ ‖∫ (t : ℝ) in a₀..s, f p t ∂μ - ∫ (t : ℝ) in a₀..b₀, f q t ∂μ‖ < ε
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
b : ℝ
f : ℝ → E
μ : Measure ℝ
h_left : IntegrableOn f (Iic b) μ
h_right : IntegrableOn f (Ioi b) μ
⊢ μ = μ.restrict (Iic b ∪ Ioi b)

Tactic:
rw [Iic_union_Ioi, Measure.restrict_univ]

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

Goal state before:
pp: case refine_1
E : Type u_5
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
a b : ℝ
μ : Measure ℝ
inst✝¹ : CompleteSpace E
inst✝ : IsFiniteMeasure μ
c : E
⊢ IntegrableOn (fun x ↦ c) (Iic a) μ

Tactic:
simp

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

Goal state before:
pp: case refine_2
E : Type u_5
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
a b : ℝ
μ : Measure ℝ
inst✝¹ : CompleteSpace E
inst✝ : IsFiniteMeasure μ
c : E
⊢ IntegrableOn (fun x ↦ c) (Iic b) μ

Tactic:
simp

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 : deriv f' x = a (deriv f x)
⊢ ‖a (deriv f x)‖ = ‖deriv f x‖

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : RCLike 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : IsScalarTower ℝ 𝕜 E
f f' : 𝕜 → E
z₀ z₁ : 𝕜
hcont : ContinuousOn (fun t ↦ f' (z₀ + t • z₁)) (Icc 0 1)
hderiv : ∀ t ∈ Icc 0 1, HasDerivAt f (f' (z₀ + t • z₁)) (z₀ + t • z₁)
γ : ℝ → 𝕜 := fun t ↦ z₀ + t • z₁
hint : IntervalIntegrable (z₁ • f' ∘ γ) volume 0 1
⊢ z₁ • ∫ (t : ℝ) in 0..1, f' (z₀ + t • z₁) = f (z₀ + z₁) - f z₀

Tactic:
have hderiv' (t) (ht : t ∈ Set.uIcc (0 : ℝ) 1) : HasDerivAt (f ∘ γ) (z₁ • (f' ∘ γ) t) t := by
    refine (hderiv t <| (Set.uIcc_of_le (α := ℝ) zero_le_one).symm ▸ ht).scomp t <| .const_add _ ?_
    simp [hasDerivAt_iff_isLittleO, sub_smul]

Goal state after:
𝕜 : Type u_2
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : RCLike 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : IsScalarTower ℝ 𝕜 E
f f' : 𝕜 → E
z₀ z₁ : 𝕜
hcont : ContinuousOn (fun t ↦ f' (z₀ + t • z₁)) (Icc 0 1)
hderiv : ∀ t ∈ Icc 0 1, HasDerivAt f (f' (z₀ + t • z₁)) (z₀ + t • z₁)
γ : ℝ → 𝕜 := fun t ↦ z₀ + t • z₁
hint : IntervalIntegrable (z₁ • f' ∘ γ) volume 0 1
hderiv' : ∀ t ∈ [[0, 1]], HasDerivAt (f ∘ γ) (z₁ • (f' ∘ γ) t) t
⊢ z₁ • ∫ (t : ℝ) in 0..1, f' (z₀ + t • z₁) = f (z₀ + z₁) - f z₀
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: f : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hle : ∀ x ∈ Ioc a b, 0 ≤ f x
hlt : ∃ c ∈ Icc a b, 0 < f c
⊢ ∫ (x : ℝ) in a..b, 0 = 0

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.Algebra.Order.Floor

Goal state before:
pp: K : Type u_1
inst✝⁵ : Field K
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : FloorSemiring K
inst✝¹ : TopologicalSpace K
inst✝ : OrderTopology K
c : K
⊢ Tendsto (fun n ↦ c ^ n / ↑n !) atTop (𝓝 0)

Tactic:
by
  convert! tendsto_mul_pow_div_factorial_sub_atTop 1 c 0
  rw [one_mul]

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

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
⊢ volume univ < ∞

Tactic:
simp

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

Goal state before:
pp: case neg.hV
G : Type u_1
inst✝¹⁵ : Group G
inst✝¹⁴ : MeasurableSpace G
inst✝¹³ : TopologicalSpace G
inst✝¹² : IsTopologicalGroup G
inst✝¹¹ : BorelSpace G
inst✝¹⁰ : PolishSpace G
Γ : Subgroup G
inst✝⁹ : Γ.Normal
inst✝⁸ : T2Space (G ⧸ Γ)
inst✝⁷ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝⁶ : ν.IsMulLeftInvariant
inst✝⁵ : Countable ↥Γ
inst✝⁴ : ν.IsMulRightInvariant
inst✝³ : SigmaFinite ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : SigmaFinite μ
inst✝ : IsFiniteMeasure μ
hasFun : HasFundamentalDomain (↥Γ.op) G ν
s : Set G
h : ν s = μ univ
fund_dom_s : IsFundamentalDomain (↥Γ.op) s ν
finiteCovol : μ univ < ∞
meas_s_ne_zero : ¬ν s = 0
⊢ ν s = ν (QuotientGroup.mk ⁻¹' univ ∩ s)

Tactic:
simp

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)
inst✝⁵ : Countable ↥Γ
ν : Measure G
inst✝⁴ : ν.IsHaarMeasure
inst✝³ : ν.IsMulRightInvariant
inst✝² : SigmaFinite ν
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 ν
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : SigmaFinite μ
V : Set (G ⧸ Γ)
hV : (interior V).Nonempty
meas_V : MeasurableSet V
hμK : μ V = ν (QuotientGroup.mk ⁻¹' V ∩ 𝓕)
neTopV : μ V ≠ ∞
⊢ QuotientMeasureEqMeasurePreimage ν μ

Tactic:
apply IsMulLeftInvariant.quotientMeasureEqMeasurePreimage_of_set (fund_dom_s := h𝓕)
    (meas_V := meas_V)

Goal state after:
case neZeroV
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)
inst✝⁵ : Countable ↥Γ
ν : Measure G
inst✝⁴ : ν.IsHaarMeasure
inst✝³ : ν.IsMulRightInvariant
inst✝² : SigmaFinite ν
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 ν
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : SigmaFinite μ
V : Set (G ⧸ Γ)
hV : (interior V).Nonempty
meas_V : MeasurableSet V
hμK : μ V = ν (QuotientGroup.mk ⁻¹' V ∩ 𝓕)
neTopV : μ V ≠ ∞
⊢ μ V ≠ 0

case hV
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)
inst✝⁵ : Countable ↥Γ
ν : Measure G
inst✝⁴ : ν.IsHaarMeasure
inst✝³ : ν.IsMulRightInvariant
inst✝² : SigmaFinite ν
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 ν
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : SigmaFinite μ
V : Set (G ⧸ Γ)
hV : (interior V).Nonempty
meas_V : MeasurableSet V
hμK : μ V = ν (QuotientGroup.mk ⁻¹' V ∩ 𝓕)
neTopV : μ V ≠ ∞
⊢ μ V = ν (QuotientGroup.mk ⁻¹' V ∩ 𝓕)

case neTopV
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)
inst✝⁵ : Countable ↥Γ
ν : Measure G
inst✝⁴ : ν.IsHaarMeasure
inst✝³ : ν.IsMulRightInvariant
inst✝² : SigmaFinite ν
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 ν
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : SigmaFinite μ
V : Set (G ⧸ Γ)
hV : (interior V).Nonempty
meas_V : MeasurableSet V
hμK : μ V = ν (QuotientGroup.mk ⁻¹' V ∩ 𝓕)
neTopV : μ V ≠ ∞
⊢ μ V ≠ ∞
Module: Mathlib.MeasureTheory.Measure.Haar.Quotient

Goal state before:
pp: case neZeroV.a
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)
inst✝⁵ : Countable ↥Γ
ν : Measure G
inst✝⁴ : ν.IsHaarMeasure
inst✝³ : ν.IsMulRightInvariant
inst✝² : SigmaFinite ν
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 ν
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : SigmaFinite μ
V : Set (G ⧸ Γ)
hV : (interior V).Nonempty
meas_V : MeasurableSet V
hμK : μ V = ν (QuotientGroup.mk ⁻¹' V ∩ 𝓕)
neTopV : μ V ≠ ∞
c_eq_zero : ν (QuotientGroup.mk ⁻¹' V ∩ 𝓕) = 0
⊢ IsOpen (interior (QuotientGroup.mk ⁻¹' V))

Tactic:
simp

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

Goal state before:
pp: case neZeroV.a
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)
inst✝⁵ : Countable ↥Γ
ν : Measure G
inst✝⁴ : ν.IsHaarMeasure
inst✝³ : ν.IsMulRightInvariant
inst✝² : SigmaFinite ν
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 ν
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : SigmaFinite μ
V : Set (G ⧸ Γ)
hV : (interior V).Nonempty
meas_V : MeasurableSet V
hμK : μ V = ν (QuotientGroup.mk ⁻¹' V ∩ 𝓕)
neTopV : μ V ≠ ∞
c_eq_zero : ν (QuotientGroup.mk ⁻¹' V ∩ 𝓕) = 0
⊢ interior V ⊆ range QuotientGroup.mk

Tactic:
simp

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

Goal state before:
pp: G : Type u_1
inst✝¹² : Group G
inst✝¹¹ : MeasurableSpace G
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : IsTopologicalGroup G
inst✝⁸ : BorelSpace G
inst✝⁷ : PolishSpace G
Γ : Subgroup G
inst✝⁶ : Γ.Normal
inst✝⁵ : T2Space (G ⧸ Γ)
inst✝⁴ : SecondCountableTopology (G ⧸ Γ)
inst✝³ : Countable ↥Γ
ν : Measure G
inst✝² : ν.IsHaarMeasure
inst✝¹ : ν.IsMulRightInvariant
inst✝ : SigmaFinite ν
K : PositiveCompacts (G ⧸ Γ)
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 ν
h𝓕_finite : ν 𝓕 ≠ ∞
c : ℝ≥0∞ := ν (QuotientGroup.mk ⁻¹' ↑K ∩ 𝓕)
c_ne_top : c ≠ ∞
μ : Measure (G ⧸ Γ) := c • haarMeasure K
hμK : μ ↑K = c
⊢ SigmaFinite μ

Tactic:
clear_value c

Goal state after:
G : Type u_1
inst✝¹² : Group G
inst✝¹¹ : MeasurableSpace G
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : IsTopologicalGroup G
inst✝⁸ : BorelSpace G
inst✝⁷ : PolishSpace G
Γ : Subgroup G
inst✝⁶ : Γ.Normal
inst✝⁵ : T2Space (G ⧸ Γ)
inst✝⁴ : SecondCountableTopology (G ⧸ Γ)
inst✝³ : Countable ↥Γ
ν : Measure G
inst✝² : ν.IsHaarMeasure
inst✝¹ : ν.IsMulRightInvariant
inst✝ : SigmaFinite ν
K : PositiveCompacts (G ⧸ Γ)
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 ν
h𝓕_finite : ν 𝓕 ≠ ∞
c : ℝ≥0∞
c_ne_top : c ≠ ∞
μ : Measure (G ⧸ Γ) := c • haarMeasure K
hμK : μ ↑K = c
⊢ SigmaFinite μ
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic

Goal state before:
pp: ⊢ volume univ = 1

Tactic:
simp

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

Goal state before:
pp: c : ℂ
R : ℝ
⊢ c +ᵥ sphere (R • 0) (‖R‖ * 1) = sphere c |R|

Tactic:
simp

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

Goal state before:
pp: c : ℂ
R θ : ℝ
⊢ ↑‖deriv (circleMap c R) θ‖₊ ≤ ↑(Real.nnabs R)

Tactic:
simp

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

Goal state before:
pp: case zero
E : Type u_1
inst✝¹ : NormedAddCommGroup E
f : ℝ → E
T : ℝ
inst✝ : NormedSpace ℝ E
hf : Periodic f T
n : ℤ
t : ℝ
h_int : ∀ (t₁ t₂ : ℝ), IntervalIntegrable f volume t₁ t₂
⊢ ∫ (x : ℝ) in 0..0 • T, f x = 0 • ∫ (x : ℝ) in 0..T, f x

Tactic:
simp

Goal state after: