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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), Measurable (f n)
h_mono : Monotone f
c : ℝ≥0 → ℝ≥0∞ := ofNNReal
F : α → ℝ≥0∞ := fun a ↦ ⨆ n, f n a
s : α →ₛ ℝ≥0
hsf : ∀ (x : α), ↑(s x) ≤ ⨆ n, f n x
r : ℝ≥0
right✝ ha✝ : ↑r < 1
ha : r < 1
rs : α →ₛ ℝ≥0 := SimpleFunc.map (fun a ↦ r * a) s
eq_rs : SimpleFunc.map c rs = const α ↑r * SimpleFunc.map c s
eq : ∀ (p : ℝ≥0∞), ⇑(SimpleFunc.map c rs) ⁻¹' {p} = ⋃ n, ⇑(SimpleFunc.map c rs) ⁻¹' {p} ∩ {a | p ≤ f n a}
⊢ ∀ (r : ℝ≥0∞), Monotone fun n ↦ ⇑(SimpleFunc.map c rs) ⁻¹' {r} ∩ {a | r ≤ f n a}

Tactic:
intro r

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), Measurable (f n)
h_mono : Monotone f
c : ℝ≥0 → ℝ≥0∞ := ofNNReal
F : α → ℝ≥0∞ := fun a ↦ ⨆ n, f n a
s : α →ₛ ℝ≥0
hsf : ∀ (x : α), ↑(s x) ≤ ⨆ n, f n x
r✝ : ℝ≥0
right✝ ha✝ : ↑r✝ < 1
ha : r✝ < 1
rs : α →ₛ ℝ≥0 := SimpleFunc.map (fun a ↦ r✝ * a) s
eq_rs : SimpleFunc.map c rs = const α ↑r✝ * SimpleFunc.map c s
eq : ∀ (p : ℝ≥0∞), ⇑(SimpleFunc.map c rs) ⁻¹' {p} = ⋃ n, ⇑(SimpleFunc.map c rs) ⁻¹' {p} ∩ {a | p ≤ f n a}
r : ℝ≥0∞
⊢ Monotone fun n ↦ ⇑(SimpleFunc.map c rs) ⁻¹' {r} ∩ {a | r ≤ f n a}
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
m₀ : MeasurableSpace α
μ : Measure α
inst✝¹ : PseudoMetrizableSpace β
inst✝ : PseudoMetrizableSpace γ
g : β → γ
f : α → β
hg : IsEmbedding g
this : PseudoMetricSpace γ := pseudoMetrizableSpacePseudoMetric γ
this✝³ : MeasurableSpace β := borel β
this✝² : BorelSpace β
this✝¹ : MeasurableSpace γ := borel γ
this✝ : BorelSpace γ
H : AEStronglyMeasurable (fun x ↦ g (f x)) μ
G : β → ↑(range g) := rangeFactorization g
⊢ IsClosed[instTopologicalSpaceSubtype] univ

Tactic:
;

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
m₀ : MeasurableSpace α
μ : Measure α
inst✝¹ : PseudoMetrizableSpace β
inst✝ : PseudoMetrizableSpace γ
g : β → γ
f : α → β
hg : IsEmbedding g
this : PseudoMetricSpace γ := pseudoMetrizableSpacePseudoMetric γ
this✝³ : MeasurableSpace β := borel β
this✝² : BorelSpace β
this✝¹ : MeasurableSpace γ := borel γ
this✝ : BorelSpace γ
H : AEStronglyMeasurable (fun x ↦ g (f x)) μ
G : β → ↑(range g) := rangeFactorization g
⊢ IsClosed[instTopologicalSpaceSubtype] univ
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : Nonempty β
inst✝ : T2Space β
hα : Nonempty α
hf : StronglyMeasurable f
fs : ℕ → α →ₛ β := hf.approx
h_fs_tendsto : ∀ (x : α), Tendsto (fun n ↦ (fs n) x) atTop (𝓝 (f x))
this : ∀ (n : ℕ), ∃ c, ∀ (x : α), (fs n) x = c
cs : ℕ → β := fun n ↦ ⋯.choose
h_cs_eq : ∀ (n : ℕ), ⇑(fs n) = fun x ↦ cs n
x : α
n : ℕ
| (fs n) x

Tactic:
;

Goal state after:
α : Type u_1
β : Type u_2
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : Nonempty β
inst✝ : T2Space β
hα : Nonempty α
hf : StronglyMeasurable f
fs : ℕ → α →ₛ β := hf.approx
h_fs_tendsto : ∀ (x : α), Tendsto (fun n ↦ (fs n) x) atTop (𝓝 (f x))
this : ∀ (n : ℕ), ∃ c, ∀ (x : α), (fs n) x = c
cs : ℕ → β := fun n ↦ ⋯.choose
h_cs_eq : ∀ (n : ℕ), ⇑(fs n) = fun x ↦ cs n
x : α
n : ℕ
| (fs n) x
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: case mp
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
h : ∫⁻ (a : α), f a ∂μ = 0
meas_levels_0 : ∀ ε > 0, μ {x | ε ≤ f x} = 0
u : ℕ → ℝ≥0∞
bu : ∀ (n : ℕ), u n ∈ Ioo 0 1
tu : ∀ ε > 0, ∃ N, ∀ n ≥ N, u n ≤ ε
x : α
h' : 0 < f x
n : ℕ
hn : ∀ n_1 ≥ n, u n_1 ≤ f x
⊢ ∃ i, x ∈ {x | u i ≤ f x}

Tactic:
;

Goal state after:
case mp
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
h : ∫⁻ (a : α), f a ∂μ = 0
meas_levels_0 : ∀ ε > 0, μ {x | ε ≤ f x} = 0
u : ℕ → ℝ≥0∞
bu : ∀ (n : ℕ), u n ∈ Ioo 0 1
tu : ∀ ε > 0, ∃ N, ∀ n ≥ N, u n ≤ ε
x : α
h' : 0 < f x
n : ℕ
hn : ∀ n_1 ≥ n, u n_1 ≤ f x
⊢ ∃ i, x ∈ {x | u i ≤ f x}
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: case mpr
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
h : ∫⁻ (a : α), f a ∂μ = 0
meas_levels_0 : ∀ ε > 0, μ {x | ε ≤ f x} = 0
u : ℕ → ℝ≥0∞
bu : ∀ (n : ℕ), u n ∈ Ioo 0 1
tu : ∀ ε > 0, ∃ N, ∀ n ≥ N, u n ≤ ε
x : α
n : ℕ
hn : x ∈ {x | u n ≤ f x}
⊢ 0 < f x

Tactic:
;

Goal state after:
case mpr
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
h : ∫⁻ (a : α), f a ∂μ = 0
meas_levels_0 : ∀ ε > 0, μ {x | ε ≤ f x} = 0
u : ℕ → ℝ≥0∞
bu : ∀ (n : ℕ), u n ∈ Ioo 0 1
tu : ∀ ε > 0, ∃ N, ∀ n ≥ N, u n ≤ ε
x : α
n : ℕ
hn : x ∈ {x | u n ≤ f x}
⊢ 0 < f x
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), AEMeasurable (f n) μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n ↦ f n x
⊢ ∫⁻ (a : α), ⨆ n, f n a ∂μ = ⨆ n, ∫⁻ (a : α), f n a ∂μ

Tactic:
by
  simp_rw [← iSup_apply]
  let p : α → (ℕ → ℝ≥0∞) → Prop := fun _ f' => Monotone f'
  have hp : ∀ᵐ x ∂μ, p x fun i => f i x := h_mono
  have h_ae_seq_mono : Monotone (aeSeq hf p) := by
    intro n m hnm x
    by_cases hx : x ∈ aeSeqSet hf p
    · exact aeSeq.prop_of_mem_aeSeqSet hf hx hnm
    · simp only [aeSeq, hx, if_false, le_rfl]
  rw [lintegral_congr_ae (aeSeq.iSup hf hp).symm]
  simp_rw [iSup_apply]
  rw [lintegral_iSup (aeSeq.measurable hf p) h_ae_seq_mono]
  congr with n
  exact lintegral_congr_ae (aeSeq.aeSeq_n_eq_fun_n_ae hf hp n)

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Markov

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hle : f ≤ᵐ[μ] g
hg : AEMeasurable g μ
ε : ℝ≥0∞
φ : α → ℝ≥0∞
hφm : Measurable φ
hφ_le : φ ≤ f
hφ_eq : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), φ a ∂μ
x : α
hx₁ : f x ≤ g x
⊢ (φ x + if x ∈ {x | φ x + ε ≤ g x} then ε else 0) ≤ g x

Tactic:
;

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hle : f ≤ᵐ[μ] g
hg : AEMeasurable g μ
ε : ℝ≥0∞
φ : α → ℝ≥0∞
hφm : Measurable φ
hφ_le : φ ≤ f
hφ_eq : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), φ a ∂μ
x : α
hx₁ : f x ≤ g x
⊢ (φ x + if x ∈ {x | φ x + ε ≤ g x} then ε else 0) ≤ g x
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: case h₁
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
h : ∫⁻ (x : α), f x ∂μ ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
ε₂ : ℝ≥0∞
hε₂0 : 0 < ε₂
hε₂ε : ε₂ < ε
ε₁ : ℝ≥0∞
hε₁0 : 0 < ε₁
hε₁₂ : ε₁ < ε₂
φ : α →ₛ ℝ≥0
left✝ : ∀ (x : α), ↑(φ x) ≤ f x
hφ : ∀ (ψ : α →ₛ ℝ≥0), (∀ (x : α), ↑(ψ x) ≤ f x) → (SimpleFunc.map ofNNReal (ψ - φ)).lintegral μ < ε₁
C : ℝ≥0
hC : ∀ (x : α), φ x ≤ C
s : Set α
hs : μ s < (ε₂ - ε₁) / ↑C
ψ : α →ₛ ℝ≥0
hψ : ∀ (x : α), ↑(ψ x) ≤ f x
⊢ ↑C * ((ε₂ - ε₁) / ↑C) ≤ ε₂ - ε₁

Tactic:
;

Goal state after:
case h₁
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
h : ∫⁻ (x : α), f x ∂μ ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
ε₂ : ℝ≥0∞
hε₂0 : 0 < ε₂
hε₂ε : ε₂ < ε
ε₁ : ℝ≥0∞
hε₁0 : 0 < ε₁
hε₁₂ : ε₁ < ε₂
φ : α →ₛ ℝ≥0
left✝ : ∀ (x : α), ↑(φ x) ≤ f x
hφ : ∀ (ψ : α →ₛ ℝ≥0), (∀ (x : α), ↑(ψ x) ≤ f x) → (SimpleFunc.map ofNNReal (ψ - φ)).lintegral μ < ε₁
C : ℝ≥0
hC : ∀ (x : α), φ x ≤ C
s : Set α
hs : μ s < (ε₂ - ε₁) / ↑C
ψ : α →ₛ ℝ≥0
hψ : ∀ (x : α), ↑(ψ x) ≤ f x
⊢ ↑C * ((ε₂ - ε₁) / ↑C) ≤ ε₂ - ε₁
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Markov

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hf : AEMeasurable f (μ.restrict s)
hμf : μ {x | x ∈ s ∧ f x = ∞} ≠ 0
⊢ μ ({a | f a = ∞} ∩ s) ≤ (μ.restrict s) {x | f x = ∞}

Tactic:
;

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hf : AEMeasurable f (μ.restrict s)
hμf : μ {x | x ∈ s ∧ f x = ∞} ≠ 0
⊢ μ ({a | f a = ∞} ∩ s) ≤ (μ.restrict s) {x | f x = ∞}
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Markov

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hf : AEMeasurable f (μ.restrict s)
hμf : μ {x | x ∈ s ∧ f x = ∞} ≠ 0
⊢ μ {x | x ∈ s ∧ f x = ∞} ≤ (μ.restrict s) {x | f x = ∞}

Tactic:
rw [← ofPred_inter_eq_sep]; exact Measure.le_restrict_apply _ _

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
f f' : α → β
h✝ : f =ᵐ[μ] f'
g : β → ℝ≥0∞
a : α
h : f a = f' a
⊢ g (f a) = g (f' a)

Tactic:
;

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
f f' : α → β
h✝ : f =ᵐ[μ] f'
g : β → ℝ≥0∞
a : α
h : f a = f' a
⊢ g (f a) = g (f' a)
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
m : MeasurableSpace α
μ : Measure α
f₁ f₁' : α → β
f₂ f₂' : α → γ
h₁✝ : f₁ =ᵐ[μ] f₁'
h₂✝ : f₂ =ᵐ[μ] f₂'
g : β → γ → ℝ≥0∞
x✝ : α
h₂ : f₂ x✝ = f₂' x✝
h₁ : f₁ x✝ = f₁' x✝
⊢ g (f₁ x✝) (f₂ x✝) = g (f₁' x✝) (f₂' x✝)

Tactic:
;

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
m : MeasurableSpace α
μ : Measure α
f₁ f₁' : α → β
f₂ f₂' : α → γ
h₁✝ : f₁ =ᵐ[μ] f₁'
h₂✝ : f₂ =ᵐ[μ] f₂'
g : β → γ → ℝ≥0∞
x✝ : α
h₂ : f₂ x✝ = f₂' x✝
h₁ : f₁ x✝ = f₁' x✝
⊢ g (f₁ x✝) (f₂ x✝) = g (f₁' x✝) (f₂' x✝)
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: case e_f
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
⊢ (fun a ↦ f a) = fun a ↦ ⨆ n, (eapprox f n) a

Tactic:
;

Goal state after:
case e_f
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
⊢ (fun a ↦ f a) = fun a ↦ ⨆ n, (eapprox f n) a
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: case e_f
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
a : α
⊢ f a = ⨆ n, (eapprox f n) a

Tactic:
;

Goal state after:
case e_f
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
a : α
⊢ f a = ⨆ n, (eapprox f n) a
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
⊢ (fun n ↦ ∫⁻ (a : α), (eapprox f n) a ∂μ) = fun n ↦ (eapprox f n).lintegral μ

Tactic:
;

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
⊢ (fun n ↦ ∫⁻ (a : α), (eapprox f n) a ∂μ) = fun n ↦ (eapprox f n).lintegral μ
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
n : ℕ
⊢ ∫⁻ (a : α), (eapprox f n) a ∂μ = (eapprox f n).lintegral μ

Tactic:
;

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
n : ℕ
⊢ ∫⁻ (a : α), (eapprox f n) a ∂μ = (eapprox f n).lintegral μ
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: case e_f
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
⊢ (fun a ↦ (⨆ n, (eapprox f n) a) + ⨆ n, (eapprox g n) a) = fun a ↦ ⨆ n, (⇑(eapprox f n) + ⇑(eapprox g n)) a

Tactic:
;

Goal state after:
case e_f
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
⊢ (fun a ↦ (⨆ n, (eapprox f n) a) + ⨆ n, (eapprox g n) a) = fun a ↦ ⨆ n, (⇑(eapprox f n) + ⇑(eapprox g n)) a
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
M : Type u_5
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : ContinuousMul M
m : MeasurableSpace α
n : MeasurableSpace β
ι : Type u_6
f : ι → α → β → M
g : α → β
s : Finset ι
hf : ∀ i ∈ s, StronglyMeasurable ↿(f i)
hg : Measurable g
⊢ StronglyMeasurable fun a ↦ ∏ c ∈ s, f c a (g a)

Tactic:
;

Goal state after:
α : Type u_1
β : Type u_2
M : Type u_5
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : ContinuousMul M
m : MeasurableSpace α
n : MeasurableSpace β
ι : Type u_6
f : ι → α → β → M
g : α → β
s : Finset ι
hf : ∀ i ∈ s, StronglyMeasurable ↿(f i)
hg : Measurable g
⊢ StronglyMeasurable fun a ↦ ∏ c ∈ s, f c a (g a)
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
f : α → β
mα : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : TopologicalSpace β
inst✝² : PseudoMetrizableSpace β
inst✝¹ : SecondCountableTopology β
inst✝ : OpensMeasurableSpace β
hf : Measurable f
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
a✝ : Nontrivial β
⊢ StronglyMeasurable f

Tactic:
;

Goal state after:
α : Type u_1
β : Type u_2
f : α → β
mα : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : TopologicalSpace β
inst✝² : PseudoMetrizableSpace β
inst✝¹ : SecondCountableTopology β
inst✝ : OpensMeasurableSpace β
hf : Measurable f
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
a✝ : Nontrivial β
⊢ StronglyMeasurable f
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Sub

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∫⁻ (a : α), f a ∂μ ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
hf₀ : ¬∫⁻ (a : α), f a ∂μ = 0
g : α → ℝ≥0∞
hgf : g ≤ f
hg_meas : Measurable g
hgsupp : μ (support g) < ∞
hgε : ∫⁻ (a : α), f a ∂μ - ε < ∫⁻ (a : α), g a ∂μ
x✝ : α
⊢ x✝ ∈ (support g)ᶜ → f x✝ = (fun a ↦ f a - g a) x✝

Tactic:
;

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∫⁻ (a : α), f a ∂μ ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
hf₀ : ¬∫⁻ (a : α), f a ∂μ = 0
g : α → ℝ≥0∞
hgf : g ≤ f
hg_meas : Measurable g
hgsupp : μ (support g) < ∞
hgε : ∫⁻ (a : α), f a ∂μ - ε < ∫⁻ (a : α), g a ∂μ
x✝ : α
⊢ x✝ ∈ (support g)ᶜ → f x✝ = (fun a ↦ f a - g a) x✝
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
s : Set α
⊢ ∫⁻ (x : α) in s, max (f x) (g x) ∂μ =
    ∫⁻ (x : α) in s ∩ {x | f x ≤ g x}, g x ∂μ + ∫⁻ (x : α) in s ∩ {x | g x < f x}, f x ∂μ

Tactic:
by
  rw [lintegral_max hf hg, restrict_restrict, restrict_restrict, inter_comm s, inter_comm s]
  exacts [measurableSet_lt hg hf, measurableSet_le hf hg]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: case hfg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
g φ : α → ℝ≥0∞
hφm : Measurable φ
hφ_le : φ ≤ fun a ↦ f a + g a
hφ_eq : ∫⁻ (a : α), f a + g a ∂μ = ∫⁻ (a : α), φ a ∂μ
a : α
⊢ φ a - f a ≤ g a

Tactic:
;

Goal state after:
case hfg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
g φ : α → ℝ≥0∞
hφm : Measurable φ
hφ_le : φ ≤ fun a ↦ f a + g a
hφ_eq : ∫⁻ (a : α), f a + g a ∂μ = ∫⁻ (a : α), φ a ∂μ
a : α
⊢ φ a - f a ≤ g a
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Sub

Goal state before:
pp: case hfg
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∫⁻ (a : α), f a ∂μ ≠ ∞
g : α → ℝ≥0∞
hmg : Measurable g
hgf : g ≤ f
hifg : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), g a ∂μ
s : Set α
hms : MeasurableSet s
x : α
⊢ g x ≤ f x

Tactic:
;

Goal state after:
case hfg
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∫⁻ (a : α), f a ∂μ ≠ ∞
g : α → ℝ≥0∞
hmg : Measurable g
hgf : g ≤ f
hifg : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), g a ∂μ
s : Set α
hms : MeasurableSet s
x : α
⊢ g x ≤ f x
Module: Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure ν
n : ℕ
hC_lt : ⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s ≤ 1 / ↑n
⊢ SigmaFinite 0

Tactic:
;

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure ν
n : ℕ
hC_lt : ⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s ≤ 1 / ↑n
⊢ SigmaFinite 0
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : TopologicalSpace β
f : α → β
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
h : univ ⊆ s ∪ t
hc : StronglyMeasurable fun a ↦ f ↑a
hd : StronglyMeasurable fun a ↦ f ↑a
a✝ : Nontrivial β
⊢ StronglyMeasurable f

Tactic:
;

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : TopologicalSpace β
f : α → β
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
h : univ ⊆ s ∪ t
hc : StronglyMeasurable fun a ↦ f ↑a
hd : StronglyMeasurable fun a ↦ f ↑a
a✝ : Nontrivial β
⊢ StronglyMeasurable f
Module: Mathlib.Topology.Perfect

Goal state before:
pp: case closed
α : Type u_1
inst✝ : TopologicalSpace α
C : Set α
hC : Preperfect C
⊢ IsClosed[inst✝] (closure[inst✝] C)

pp: case acc
α : Type u_1
inst✝ : TopologicalSpace α
C : Set α
hC : Preperfect C
⊢ Preperfect (closure[inst✝] C)

Tactic:
;

Goal state after:
case closed
α : Type u_1
inst✝ : TopologicalSpace α
C : Set α
hC : Preperfect C
⊢ IsClosed[inst✝] (closure[inst✝] C)

case acc
α : Type u_1
inst✝ : TopologicalSpace α
C : Set α
hC : Preperfect C
⊢ Preperfect (closure[inst✝] C)
Module: Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion

Goal state before:
pp: case refine_1
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
hμν : μ ≪ ν
inst✝¹ : SigmaFinite μ
inst✝ : SFinite ν
h : ν (μ.sigmaFiniteSetWRT ν)ᶜ ≠ 0 → μ (μ.sigmaFiniteSetWRT ν)ᶜ = ∞
h0 : ¬ν (μ.sigmaFiniteSetWRT ν)ᶜ = 0
i : ℕ
h_ne_zero : ¬μ ((μ.sigmaFiniteSetWRT ν)ᶜ ∩ spanningSets μ i) = 0
⊢ ν ((μ.sigmaFiniteSetWRT ν)ᶜ ∩ spanningSets μ i) ≠ 0

pp: case refine_2
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
hμν : μ ≪ ν
inst✝¹ : SigmaFinite μ
inst✝ : SFinite ν
h : ν (μ.sigmaFiniteSetWRT ν)ᶜ ≠ 0 → μ (μ.sigmaFiniteSetWRT ν)ᶜ = ∞
h0 : ¬ν (μ.sigmaFiniteSetWRT ν)ᶜ = 0
i : ℕ
h_ne_zero : ¬μ ((μ.sigmaFiniteSetWRT ν)ᶜ ∩ spanningSets μ i) = 0
h_zero_top : μ ((μ.sigmaFiniteSetWRT ν)ᶜ ∩ spanningSets μ i) = ∞
⊢ False

Tactic:
;

Goal state after:
case refine_1
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
hμν : μ ≪ ν
inst✝¹ : SigmaFinite μ
inst✝ : SFinite ν
h : ν (μ.sigmaFiniteSetWRT ν)ᶜ ≠ 0 → μ (μ.sigmaFiniteSetWRT ν)ᶜ = ∞
h0 : ¬ν (μ.sigmaFiniteSetWRT ν)ᶜ = 0
i : ℕ
h_ne_zero : ¬μ ((μ.sigmaFiniteSetWRT ν)ᶜ ∩ spanningSets μ i) = 0
⊢ ν ((μ.sigmaFiniteSetWRT ν)ᶜ ∩ spanningSets μ i) ≠ 0

case refine_2
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
hμν : μ ≪ ν
inst✝¹ : SigmaFinite μ
inst✝ : SFinite ν
h : ν (μ.sigmaFiniteSetWRT ν)ᶜ ≠ 0 → μ (μ.sigmaFiniteSetWRT ν)ᶜ = ∞
h0 : ¬ν (μ.sigmaFiniteSetWRT ν)ᶜ = 0
i : ℕ
h_ne_zero : ¬μ ((μ.sigmaFiniteSetWRT ν)ᶜ ∩ spanningSets μ i) = 0
h_zero_top : μ ((μ.sigmaFiniteSetWRT ν)ᶜ ∩ spanningSets μ i) = ∞
⊢ False
Module: Mathlib.Topology.Perfect

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : T1Space α
U : Set α
hu : U.Nontrivial
x : α
h : (U ∩ ({x} ∩ closure[inst✝¹] (U \ {x}))).Nonempty
hx : x ∈ U
⊢ AccPt x (𝓟 U)

Tactic:
· apply Set.Nonempty.right at h
    rw [Set.singleton_inter_nonempty, mem_closure_iff_clusterPt,
      ← accPt_principal_iff_clusterPt] at h
    exact h

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

Goal state before:
pp: ι : Type u_1
E : Type u_3
inst✝ : SeminormedAddCommGroup E
f : ι → E
⊢ ∀ ⦃s t : Set E⦄,
    s ⊆ t →
      (∃ s_1, ∀ (t : Finset ι), Disjoint t s_1 → ∑ b ∈ t, f b ∈ s) →
        ∃ s, ∀ (t_1 : Finset ι), Disjoint t_1 s → ∑ b ∈ t_1, f b ∈ t

Tactic:
· rintro s t hst ⟨s', hs'⟩
    exact ⟨s', fun t' ht' => hst <| hs' _ ht'⟩

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

Goal state before:
pp: α : Type u_1
β : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : CompleteSpace F
u : α → ℝ
f : α → β → F
hu : Summable u
hfu : ∀ (n : α) (x : β), ‖f n x‖ ≤ u n
⊢ TendstoUniformlyOn (fun t x ↦ ∑ n ∈ t, f n x) (fun x ↦ ∑' (n : α), f n x) atTop univ

Tactic:
;

Goal state after:
α : Type u_1
β : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : CompleteSpace F
u : α → ℝ
f : α → β → F
hu : Summable u
hfu : ∀ (n : α) (x : β), ‖f n x‖ ≤ u n
⊢ TendstoUniformlyOn (fun t x ↦ ∑ n ∈ t, f n x) (fun x ↦ ∑' (n : α), f n x) atTop univ
Module: Mathlib.Topology.Order.ProjIcc

Goal state before:
pp: α : Type u_1
inst✝² : LinearOrder α
a b : α
h : a ≤ b
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
s : Set ↑(Icc a b)
hs : IsOpen[inst✝¹] (projIcc a b h ⁻¹' s)
x✝ : { x // x ∈ Icc a b }
⊢ x✝ ∈ Subtype.val ⁻¹' projIcc a b h ⁻¹' s ↔ x✝ ∈ s

Tactic:
;

Goal state after:
α : Type u_1
inst✝² : LinearOrder α
a b : α
h : a ≤ b
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
s : Set ↑(Icc a b)
hs : IsOpen[inst✝¹] (projIcc a b h ⁻¹' s)
x✝ : { x // x ∈ Icc a b }
⊢ x✝ ∈ Subtype.val ⁻¹' projIcc a b h ⁻¹' s ↔ x✝ ∈ s
Module: Mathlib.Topology.Algebra.MetricSpace.Lipschitz

Goal state before:
pp: case h₂
α : Type u_1
β : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
s : Set α
hs : IsCompact s
hf' : ContinuousOn f s
ε : (x : α) → x ∈ s → ℝ
hε : ∀ (x : α) (a : x ∈ s), ε x a > 0
K : (x : α) → x ∈ s → ℝ≥0
hf : ∀ (x : α) (a : x ∈ s), LipschitzOnWith (K x a) f (ball x (ε x a) ∩ s)
K' : (x : α) → x ∈ s → ℝ≥0
hK' : ∀ (x : α) (hx : x ∈ s), ∀ y ∈ s.diff (ball x (ε x hx)), edist (f x) (f y) ≤ ↑(K' x hx) * edist x y
t : Finset ↑s
ht : s ⊆ ⋃ x ∈ t, s ∩ ball (↑x) (ε ↑x ⋯ / 2)
x : α
hx : x ∈ s
y : α
hy : y ∈ s
z : ↑s
hz : z ∈ t
hx' : x ∈ s ∩ ball (↑z) (ε ↑z ⋯ / 2)
hy' : y ∉ ball (↑z) (ε ↑z ⋯)
⊢ edist (↑z) y ≤ edist x ↑z + edist x y

Tactic:
;

Goal state after:
case h₂
α : Type u_1
β : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
s : Set α
hs : IsCompact s
hf' : ContinuousOn f s
ε : (x : α) → x ∈ s → ℝ
hε : ∀ (x : α) (a : x ∈ s), ε x a > 0
K : (x : α) → x ∈ s → ℝ≥0
hf : ∀ (x : α) (a : x ∈ s), LipschitzOnWith (K x a) f (ball x (ε x a) ∩ s)
K' : (x : α) → x ∈ s → ℝ≥0
hK' : ∀ (x : α) (hx : x ∈ s), ∀ y ∈ s.diff (ball x (ε x hx)), edist (f x) (f y) ≤ ↑(K' x hx) * edist x y
t : Finset ↑s
ht : s ⊆ ⋃ x ∈ t, s ∩ ball (↑x) (ε ↑x ⋯ / 2)
x : α
hx : x ∈ s
y : α
hy : y ∈ s
z : ↑s
hz : z ∈ t
hx' : x ∈ s ∩ ball (↑z) (ε ↑z ⋯ / 2)
hy' : y ∉ ball (↑z) (ε ↑z ⋯)
⊢ edist (↑z) y ≤ edist x ↑z + edist x y
Module: Mathlib.Topology.Algebra.MetricSpace.Lipschitz

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
s : Set α
hs : IsCompact s
hf' : ContinuousOn f s
ε : (x : α) → x ∈ s → ℝ
hε : ∀ (x : α) (a : x ∈ s), ε x a > 0
K : (x : α) → x ∈ s → ℝ≥0
hf : ∀ (x : α) (a : x ∈ s), LipschitzOnWith (K x a) f (ball x (ε x a) ∩ s)
K' : (x : α) → x ∈ s → ℝ≥0
hK' : ∀ (x : α) (hx : x ∈ s), ∀ y ∈ s.diff (ball x (ε x hx)), edist (f x) (f y) ≤ ↑(K' x hx) * edist x y
t : Finset ↑s
ht : s ⊆ ⋃ x ∈ t, s ∩ ball (↑x) (ε ↑x ⋯ / 2)
x : α
hx : x ∈ s
y : α
hy : y ∈ s
z : ↑s
hz : z ∈ t
hx' : x ∈ s ∩ ball (↑z) (ε ↑z ⋯ / 2)
hy' : y ∉ ball (↑z) (ε ↑z ⋯)
⊢ ↑(K ↑z ⋯) * edist x y + ↑(K' ↑z ⋯) * (edist x y + edist x y) = (↑(K ↑z ⋯) + 2 * ↑(K' ↑z ⋯)) * edist x y

Tactic:
;

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
s : Set α
hs : IsCompact s
hf' : ContinuousOn f s
ε : (x : α) → x ∈ s → ℝ
hε : ∀ (x : α) (a : x ∈ s), ε x a > 0
K : (x : α) → x ∈ s → ℝ≥0
hf : ∀ (x : α) (a : x ∈ s), LipschitzOnWith (K x a) f (ball x (ε x a) ∩ s)
K' : (x : α) → x ∈ s → ℝ≥0
hK' : ∀ (x : α) (hx : x ∈ s), ∀ y ∈ s.diff (ball x (ε x hx)), edist (f x) (f y) ≤ ↑(K' x hx) * edist x y
t : Finset ↑s
ht : s ⊆ ⋃ x ∈ t, s ∩ ball (↑x) (ε ↑x ⋯ / 2)
x : α
hx : x ∈ s
y : α
hy : y ∈ s
z : ↑s
hz : z ∈ t
hx' : x ∈ s ∩ ball (↑z) (ε ↑z ⋯ / 2)
hy' : y ∉ ball (↑z) (ε ↑z ⋯)
⊢ ↑(K ↑z ⋯) * edist x y + ↑(K' ↑z ⋯) * (edist x y + edist x y) = (↑(K ↑z ⋯) + 2 * ↑(K' ↑z ⋯)) * edist x y