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

Goal state before:
pp: case pos.refine_1
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 : Measurable f
g : X → E
hE : CompleteSpace E
hg : Integrable g (μ.withDensity fun x ↦ ↑(f x))
c : E
s : Set X
s_meas : MeasurableSet s
hs : (μ.withDensity fun x ↦ ↑(f x)) s < ∞
⊢ (∫ (x : X) in s, c ∂μ.withDensity fun x ↦ ↑(f x)) = ∫ (x : X), f x • s.indicator (fun x ↦ c) x ∂μ

Tactic:
simp_rw [← Set.indicator_smul_apply, integral_indicator s_meas]

Goal state after:
case pos.refine_1
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 : Measurable f
g : X → E
hE : CompleteSpace E
hg : Integrable g (μ.withDensity fun x ↦ ↑(f x))
c : E
s : Set X
s_meas : MeasurableSet s
hs : (μ.withDensity fun x ↦ ↑(f x)) s < ∞
⊢ (∫ (x : X) in s, c ∂μ.withDensity fun x ↦ ↑(f x)) = ∫ (a : X) in s, f a • c ∂μ
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)
this : CauchySeq L0
f' : F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
e : ℕ
he : (1 / 2) ^ e < ε / 16
y : ℝ
hy : y ∈ Icc x (x + (1 / 2) ^ (n e + 1))
xy : x < y
yzero : 0 < y - x
y_le : y - x ≤ (1 / 2) ^ (n e + 1)
yone : y - x ≤ 1
k : ℕ
k_gt : n e < k
m : ℕ := k - 1
h'k : y - x ≤ (1 / 2) ^ (m + 1)
hk : (1 / 2) ^ (m + 1 + 1) < y - x
m_ge : n e ≤ m
km : k = m + 1
⊢ ‖f y - f x - (y - x) • L e (n e) m‖ ≤ (1 / 2) ^ e * (1 / 2) ^ m

Tactic:
apply le_of_mem_A (hn e (n e) m le_rfl m_ge).2.2

Goal state after:
case hy
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 := ⋯
this : CauchySeq L0
f' : F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
e : ℕ
he : (1 / 2) ^ e < ε / 16
y : ℝ
hy : y ∈ Icc x (x + (1 / 2) ^ (n e + 1))
xy : x < y
yzero : 0 < y - x
y_le : y - x ≤ (1 / 2) ^ (n e + 1)
yone : y - x ≤ 1
k : ℕ
k_gt : n e < k
m : ℕ := ⋯
h'k : y - x ≤ (1 / 2) ^ (m + 1)
hk : (1 / 2) ^ (m + 1 + 1) < y - x
m_ge : n e ≤ m
km : k = m + 1
⊢ x ∈ Icc x (x + (1 / 2) ^ m / 2)

case hz
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 := ⋯
this : CauchySeq L0
f' : F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
e : ℕ
he : (1 / 2) ^ e < ε / 16
y : ℝ
hy : y ∈ Icc x (x + (1 / 2) ^ (n e + 1))
xy : x < y
yzero : 0 < y - x
y_le : y - x ≤ (1 / 2) ^ (n e + 1)
yone : y - x ≤ 1
k : ℕ
k_gt : n e < k
m : ℕ := ⋯
h'k : y - x ≤ (1 / 2) ^ (m + 1)
hk : (1 / 2) ^ (m + 1 + 1) < y - x
m_ge : n e ≤ m
km : k = m + 1
⊢ y ∈ Icc x (x + (1 / 2) ^ m / 2)
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: case refine_3
α : Type u_1
E : Type u_2
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : ℕ → Set α
f : α → E
hsm : ∀ (i : ℕ), MeasurableSet (s i)
h_anti : Antitone s
hfi : IntegrableOn f (s 0) μ
bound : α → ℝ := (s 0).indicator fun a ↦ ‖f a‖
h_int_eq : (fun i ↦ ∫ (a : α) in s i, f a ∂μ) = fun i ↦ ∫ (a : α), (s i).indicator f a ∂μ
⊢ ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖(s n).indicator f a‖ ≤ bound a

pp: case refine_4
α : Type u_1
E : Type u_2
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : ℕ → Set α
f : α → E
hsm : ∀ (i : ℕ), MeasurableSet (s i)
h_anti : Antitone s
hfi : IntegrableOn f (s 0) μ
bound : α → ℝ := (s 0).indicator fun a ↦ ‖f a‖
h_int_eq : (fun i ↦ ∫ (a : α) in s i, f a ∂μ) = fun i ↦ ∫ (a : α), (s i).indicator f a ∂μ
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ (s n).indicator f a) atTop (𝓝 ((⋂ b, s b).indicator f a))

Tactic:
· simp_rw [norm_indicator_eq_indicator_norm]
    refine fun n => Eventually.of_forall fun x => ?_
    grw [h_anti zero_le]

Goal state after:
case refine_4
α : Type u_1
E : Type u_2
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : ℕ → Set α
f : α → E
hsm : ∀ (i : ℕ), MeasurableSet (s i)
h_anti : Antitone s
hfi : IntegrableOn f (s 0) μ
bound : α → ℝ := (s 0).indicator fun a ↦ ‖f a‖
h_int_eq : (fun i ↦ ∫ (a : α) in s i, f a ∂μ) = fun i ↦ ∫ (a : α), (s i).indicator f a ∂μ
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ (s n).indicator f a) atTop (𝓝 ((⋂ b, s b).indicator f a))
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
ht : y ∈ closure[inst✝²] t
z : E
hz : z ∈ tangentConeAt 𝕜 s x
ι : Type (max u_1 u_2)
l : Filter ι
hl : l.NeBot
c : ι → 𝕜
d : ι → E
hd₀ : Tendsto d l (𝓝 0)
hds : ∀ᶠ (n : ι) in l, x + d n ∈ s
hcd : Tendsto (fun n ↦ c n • d n) l (𝓝 z)
n : ι
hn : x + d n ∈ s
⊢ (x, y) + (d n, 0) ∈ closure[inst✝⁷] s ×ˢ closure[inst✝²] t

Tactic:
simp [ht, subset_closure hn]

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

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a : ℝ
⊢ FTCFilter a (Pure.pure a) ⊥

Tactic:
infer_instance

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

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a : ℝ
⊢ FTCFilter a (𝓝 a) (𝓝 a)

Tactic:
infer_instance

Goal state after:
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
⊢ ∀ (b : ℕ),
    MeasurableSet (⋃ n, ⋂ p, ⋂ (_ : p ≥ n), ⋂ q, ⋂ (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ b))

Tactic:
intro

Goal state after:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
b✝ : ℕ
⊢ MeasurableSet (⋃ n, ⋂ p, ⋂ (_ : p ≥ n), ⋂ q, ⋂ (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ b✝))
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
b✝ : ℕ
⊢ ∀ (b : ℕ), MeasurableSet (⋂ p, ⋂ (_ : p ≥ b), ⋂ q, ⋂ (_ : q ≥ b), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ b✝))

Tactic:
intro

Goal state after:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
b✝¹ b✝ : ℕ
⊢ MeasurableSet (⋂ p, ⋂ (_ : p ≥ b✝), ⋂ q, ⋂ (_ : q ≥ b✝), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ b✝¹))
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 (c * x)) volume (a / c) (b / c)

Tactic:
rw [hc]; simp

Goal state after:
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
b✝¹ b✝ : ℕ
⊢ ∀ (b : ℕ), MeasurableSet (⋂ (_ : b ≥ b✝), ⋂ q, ⋂ (_ : q ≥ b✝), B f K ((1 / 2) ^ b) ((1 / 2) ^ q) ((1 / 2) ^ b✝¹))

Tactic:
intro

Goal state after:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
b✝² b✝¹ b✝ : ℕ
⊢ MeasurableSet (⋂ (_ : b✝ ≥ b✝¹), ⋂ q, ⋂ (_ : q ≥ b✝¹), B f K ((1 / 2) ^ b✝) ((1 / 2) ^ q) ((1 / 2) ^ b✝²))
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
b✝² b✝¹ b✝ : ℕ
⊢ b✝ ≥ b✝¹ → MeasurableSet (⋂ q, ⋂ (_ : q ≥ b✝¹), B f K ((1 / 2) ^ b✝) ((1 / 2) ^ q) ((1 / 2) ^ b✝²))

Tactic:
intro

Goal state after:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
b✝³ b✝² b✝¹ : ℕ
b✝ : b✝¹ ≥ b✝²
⊢ MeasurableSet (⋂ q, ⋂ (_ : q ≥ b✝²), B f K ((1 / 2) ^ b✝¹) ((1 / 2) ^ q) ((1 / 2) ^ b✝³))
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
b✝³ b✝² b✝¹ : ℕ
b✝ : b✝¹ ≥ b✝²
⊢ ∀ (b : ℕ), MeasurableSet (⋂ (_ : b ≥ b✝²), B f K ((1 / 2) ^ b✝¹) ((1 / 2) ^ b) ((1 / 2) ^ b✝³))

Tactic:
intro

Goal state after:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
b✝⁴ b✝³ b✝² : ℕ
b✝¹ : b✝² ≥ b✝³
b✝ : ℕ
⊢ MeasurableSet (⋂ (_ : b✝ ≥ b✝³), B f K ((1 / 2) ^ b✝²) ((1 / 2) ^ b✝) ((1 / 2) ^ b✝⁴))
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
b✝⁴ b✝³ b✝² : ℕ
b✝¹ : b✝² ≥ b✝³
b✝ : ℕ
⊢ b✝ ≥ b✝³ → MeasurableSet (B f K ((1 / 2) ^ b✝²) ((1 / 2) ^ b✝) ((1 / 2) ^ b✝⁴))

Tactic:
intro

Goal state after:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
b✝⁵ b✝⁴ b✝³ : ℕ
b✝² : b✝³ ≥ b✝⁴
b✝¹ : ℕ
b✝ : b✝¹ ≥ b✝⁴
⊢ MeasurableSet (B f K ((1 / 2) ^ b✝³) ((1 / 2) ^ b✝¹) ((1 / 2) ^ b✝⁵))
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
b✝⁵ b✝⁴ b✝³ : ℕ
b✝² : b✝³ ≥ b✝⁴
b✝¹ : ℕ
b✝ : b✝¹ ≥ b✝⁴
⊢ ∀ (b : F),
    MeasurableSet (⋃ (_ : b ∈ K), A f b ((1 / 2) ^ b✝³) ((1 / 2) ^ b✝⁵) ∩ A f b ((1 / 2) ^ b✝¹) ((1 / 2) ^ b✝⁵))

Tactic:
intro

Goal state after:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
b✝⁶ b✝⁵ b✝⁴ : ℕ
b✝³ : b✝⁴ ≥ b✝⁵
b✝² : ℕ
b✝¹ : b✝² ≥ b✝⁵
b✝ : F
⊢ MeasurableSet (⋃ (_ : b✝ ∈ K), A f b✝ ((1 / 2) ^ b✝⁴) ((1 / 2) ^ b✝⁶) ∩ A f b✝ ((1 / 2) ^ b✝²) ((1 / 2) ^ b✝⁶))
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
b✝⁵ b✝⁴ b✝³ : ℕ
b✝² : b✝³ ≥ b✝⁴
b✝¹ : ℕ
b✝ : b✝¹ ≥ b✝⁴
⊢ Countable F

Tactic:
intro

Goal state after:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
b✝⁵ b✝⁴ b✝³ : ℕ
b✝² : b✝³ ≥ b✝⁴
b✝¹ : ℕ
b✝ : b✝¹ ≥ b✝⁴
⊢ Countable F
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: case refine_3
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
⊢ ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι b₁ b₂ → ContinuousWithinAt (fun x ↦ {x_1 | x_1 ≤ x}.indicator f t) (Icc b₁ b₂) b₀

Tactic:
have : ∀ᵐ t ∂μ, t < b₀ ∨ b₀ < t := by
        filter_upwards [compl_mem_ae_iff.mpr hb₀] with x hx using Ne.lt_or_gt hx

Goal state after:
case refine_3
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
this : ∀ᵐ (t : ℝ) ∂μ, t < b₀ ∨ b₀ < t
⊢ ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι b₁ b₂ → ContinuousWithinAt (fun x ↦ {x_1 | x_1 ≤ x}.indicator f t) (Icc b₁ b₂) b₀
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: case refine_3.inl
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✝¹ :
  (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
this✝ : ∀ᵐ (t : ℝ) ∂μ, t < b₀ ∨ b₀ < t
x₀ : ℝ
hx₀ : x₀ < b₀
this : ∀ᶠ (x : ℝ) in 𝓝[Icc b₁ b₂] b₀, {t | t ≤ x}.indicator f x₀ = f x₀
⊢ {t | t ≤ b₀}.indicator f x₀ = f x₀

Tactic:
simp [hx₀.le]

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
μ : Measure ℝ
f : ℝ → E
⊢ ‖∫ (x : ℝ) in a..b, f x ∂μ‖ = ‖∫ (x : ℝ) in Ι a b, f x ∂μ‖

Tactic:
← norm_integral_min_max,

Goal state after:
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℝ
μ : Measure ℝ
f : ℝ → E
⊢ ‖∫ (x : ℝ) in min a b..max a b, f x ∂μ‖ = ‖∫ (x : ℝ) in Ι a b, f x ∂μ‖
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)
A : ∀ᶠ (δ : ℝ) in 𝓝[>] 0, δ ∈ Ioo 0 1
B : ∀ᶠ (δ : ℝ) in 𝓝 0, Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
C : ∀ᶠ (δ : ℝ) in 𝓝 0, (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
⊢ ∃ δ,
    0 < δ ∧
      δ < 1 ∧ Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b ∧ (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε

Tactic:
rcases (A.and ((B.and C).filter_mono nhdsWithin_le_nhds)).exists with ⟨δ, hδ, h'δ, h''δ⟩

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)
A : ∀ᶠ (δ : ℝ) in 𝓝[>] 0, δ ∈ Ioo 0 1
B : ∀ᶠ (δ : ℝ) in 𝓝 0, Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
C : ∀ᶠ (δ : ℝ) in 𝓝 0, (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
δ : ℝ
hδ : δ ∈ Ioo 0 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
⊢ ∃ δ,
    0 < δ ∧
      δ < 1 ∧ Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b ∧ (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
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
⊢ ∫ (x : ℝ) in Iio b, f x ∂μ - ∫ (x : ℝ) in Iio a, f x ∂μ = ∫ (x : ℝ) in Ico a b, f x ∂μ

Tactic:
rw [sub_eq_iff_eq_add', ← setIntegral_union (by grind) measurableSet_Ico ha h,
      Iio_union_Ico_eq_Iio hab]

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

Goal state before:
pp: case refine_2
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
inst✝ : CompleteSpace E
f f' : ℝ → E
hab : a < b
fa fb : E
hderiv : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hint : IntervalIntegrable f' volume a b
ha : Tendsto f (𝓝[>] a) (𝓝 fa)
hb : Tendsto f (𝓝[<] b) (𝓝 fb)
F : ℝ → E := update (update f a fa) b fb
Fderiv : ∀ x ∈ Ioo a b, HasDerivAt F (f' x) x
⊢ b ∈ Icc a b → Tendsto (update f a fa) (𝓝[Ico a b] b) (𝓝 fb)

Tactic:
· rintro -
      refine (hb.congr' ?_).mono_left (nhdsWithin_mono _ Ico_subset_Iio_self)
      filter_upwards [Ioo_mem_nhdsLT hab] with _ hz using (update_of_ne hz.1.ne' _ _).symm

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
a c : K
d : ℕ
ε : K
hε : ε < 0
n : ℕ
h : ↑(n - d)! * ε < a * c ^ n
⊢ ε < a * c ^ n / ↑(n - d)!

Tactic:
rwa [lt_div_iff₀' (Nat.cast_pos.mpr (Nat.factorial_pos _))]

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

Goal state before:
pp: α : Type u_1
inst✝³ : Ring α
inst✝² : LinearOrder α
inst✝¹ : FloorRing α
inst✝ : IsStrictOrderedRing α
b : ℤ
⊢ b ≤ b + 1

Tactic:
exact (lt_add_one _).le

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 ≠ ∞
⊢ QuotientMeasureEqMeasurePreimage ν (ν (QuotientGroup.mk ⁻¹' ↑K ∩ 𝓕) • haarMeasure K)

Tactic:
set c := ν ((π ⁻¹' (K : Set (G ⧸ Γ))) ∩ 𝓕)
  have c_ne_top : c ≠ ∞ := measure_inter_ne_top_of_right_ne_top h𝓕_finite
  set μ := c • haarMeasure K
  have hμK : μ K = c := by simp [μ, haarMeasure_self]
  have : SigmaFinite μ := by
    clear_value c
    lift c to NNReal using c_ne_top
    exact SMul.sigmaFinite c
  apply IsFundamentalDomain.QuotientMeasureEqMeasurePreimage_HaarMeasure (h𝓕 := h𝓕)
    (meas_V := K.isCompact.measurableSet) (μ := μ)
  · exact K.interior_nonempty
  · exact hμK
  · rw [hμK]
    exact c_ne_top

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

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
t : ℝ
f : ℝ → ℂ
p : ℝ≥0∞
hLp : MemLp f p (volume.restrict (Ioc t (t + T)))
⊢ MeasurePreserving (⇑(AddCircle.equivIoc T t)) volume (Measure.comap Subtype.val volume)

Tactic:
exact AddCircle.measurePreserving_equivIoc T

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

Goal state before:
pp: c : ℂ
R : ℝ
⊢ AnalyticOnNhd ℝ (circleMap c R) univ

Tactic:
intro z hz
  apply analyticAt_const.add
  apply analyticAt_const.mul
  rw [← Function.comp_def]
  apply analyticAt_cexp.restrictScalars.comp ((ofRealCLM.analyticAt z).mul (by fun_prop))

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

Goal state before:
pp: case left
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure ℝ
f : ℝ → E
inst✝ : NullSingletonClass μ
a₀ : ℝ
hf : IntegrableOn f (Ioi a₀) μ
a : ℝ
ha : a₀ ≤ a
⊢ ContinuousWithinAt (fun b ↦ ∫ (x : ℝ) in Ioi b, f x ∂μ) (Ici a₀ ∩ Iic a) a

Tactic:
rw [Ici_inter_Iic]

Goal state after:
case left
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure ℝ
f : ℝ → E
inst✝ : NullSingletonClass μ
a₀ : ℝ
hf : IntegrableOn f (Ioi a₀) μ
a : ℝ
ha : a₀ ≤ a
⊢ ContinuousWithinAt (fun b ↦ ∫ (x : ℝ) in Ioi b, f x ∂μ) (Icc a₀ a) a
Module: Mathlib.MeasureTheory.Integral.Prod

Goal state before:
pp: case hbc
α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝² : NormedAddCommGroup E
F : Type u_4
G : Type u_5
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
op : E → F → G
hop : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (uncurry op)
f : α → E
g : β → F
hf : Integrable f μ
hg : Integrable g ν
C : ℝ
hC : ∀ (x : E) (y : F), ‖op x y‖ ≤ C * ‖x‖ * ‖y‖
z : α × β
⊢ C ≤ ↑C.toNNReal

Tactic:
apply le_coe_toNNReal

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 : ℝ≥0
n : ℕ
⊢ ((2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, ‖f (circleMap c (↑R) θ)‖) * |↑R|⁻¹ ^ n * ↑R ^ n ≤
    (2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, ‖f (circleMap c (↑R) θ)‖

Tactic:
abs_of_nonneg R.coe_nonneg

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ≥0
n : ℕ
⊢ ((2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, ‖f (circleMap c (↑R) θ)‖) * (↑R)⁻¹ ^ n * ↑R ^ n ≤
    (2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, ‖f (circleMap c (↑R) θ)‖
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Complex

Goal state before:
pp: ⊢ MeasurePreserving (⇑measurableEquivPi) volume volume

Tactic:
convert! (measurableEquivPi.symm.measurable.measurePreserving volume).symm
  rw [← addHaarMeasure_eq_volume_pi, ← Basis.parallelepiped_basisFun, ← Basis.addHaar,
    measurableEquivPi, Homeomorph.toMeasurableEquiv_symm_coe,
    ContinuousLinearEquiv.coe_symm_toHomeomorph, Basis.map_addHaar, eq_comm]
  exact (Basis.addHaar_eq_iff _ _).mpr Complex.orthonormalBasisOneI.volume_parallelepiped

Goal state after: