mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Constructions.Cylinders

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
s : Finset ι
S T : Set ((i : ↥s) → α ↑i)
f : (i : ι) → α i
⊢ f ∈ cylinder s S \ cylinder s T ↔ f ∈ cylinder s (S \ T)

Tactic:
simp only [mem_sdiff, mem_cylinder]

Goal state after:
Module: Mathlib.MeasureTheory.SetSemiring

Goal state before:
pp: case cons
α : Type u_1
C : Set (Set α)
ι : Type u_2
hC : IsSetRing C
s : ι → Set α
S✝ : Finset ι
i : ι
S : Finset ι
hiS : i ∉ S
hs✝ : S.Nonempty
h : (∀ n ∈ S, s n ∈ C) → ⋂ i ∈ S, s i ∈ C
hs : ∀ n ∈ cons i S hiS, s n ∈ C
⊢ ⋂ i_1 ∈ cons i S hiS, s i_1 ∈ C

Tactic:
simp_rw [← Finset.mem_coe, Finset.coe_cons, Set.biInter_insert]

Goal state after:
case cons
α : Type u_1
C : Set (Set α)
ι : Type u_2
hC : IsSetRing C
s : ι → Set α
S✝ : Finset ι
i : ι
S : Finset ι
hiS : i ∉ S
hs✝ : S.Nonempty
h : (∀ n ∈ S, s n ∈ C) → ⋂ i ∈ S, s i ∈ C
hs : ∀ n ∈ cons i S hiS, s n ∈ C
⊢ s i ∩ ⋂ x ∈ ↑S, s x ∈ C
Module: Mathlib.LinearAlgebra.Transvection.Generation

Goal state before:
pp: K : Type u_1
inst✝³ : DivisionRing K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
e : V ≃ₗ[K] V
f g : Dual K V
v : V
a b : K
hv : LinearIndependent K ![(↑e).fixedSubmodule.mkQ v, e.fixedReduce ((↑e).fixedSubmodule.mkQ v)]
hf : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ≤ ker f
hfv : f v = 1
hg : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ≤ ker (f + g)
hfgv : (f + g) v = 1
hg0 : g ≠ 0
hne_top : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ⊔ K ∙ v < ⊤
ha✝ : ∀ (v_1 : V), (auxTransvection hf * e) v_1 - a • v_1 ∈ (↑e).fixedSubmodule ⊔ K ∙ v
hb✝ : ∀ (v_1 : V), (auxTransvection hg * e) v_1 - b • v_1 ∈ (↑e).fixedSubmodule ⊔ K ∙ v
c : K := b - a
hc : c = b - a
x : V
ha : e x + f (e x) • (v - e v) - a • x ∈ (↑e).fixedSubmodule ⊔ K ∙ v
hb : e x + (f (e x) + g (e x)) • (v - e v) - b • x ∈ (↑e).fixedSubmodule ⊔ K ∙ v
⊢ g (e x) • (v - e v) - c • x ∈ (↑e).fixedSubmodule ⊔ K ∙ v

Tactic:
add_smul,

Goal state after:
K : Type u_1
inst✝³ : DivisionRing K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
e : V ≃ₗ[K] V
f g : Dual K V
v : V
a b : K
hv : LinearIndependent K ![(↑e).fixedSubmodule.mkQ v, e.fixedReduce ((↑e).fixedSubmodule.mkQ v)]
hf : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ≤ ker f
hfv : f v = 1
hg : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ≤ ker (f + g)
hfgv : (f + g) v = 1
hg0 : g ≠ 0
hne_top : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ⊔ K ∙ v < ⊤
ha✝ : ∀ (v_1 : V), (auxTransvection hf * e) v_1 - a • v_1 ∈ (↑e).fixedSubmodule ⊔ K ∙ v
hb✝ : ∀ (v_1 : V), (auxTransvection hg * e) v_1 - b • v_1 ∈ (↑e).fixedSubmodule ⊔ K ∙ v
c : K := b - a
hc : c = b - a
x : V
ha : e x + f (e x) • (v - e v) - a • x ∈ (↑e).fixedSubmodule ⊔ K ∙ v
hb : e x + (f (e x) • (v - e v) + g (e x) • (v - e v)) - b • x ∈ (↑e).fixedSubmodule ⊔ K ∙ v
⊢ g (e x) • (v - e v) - c • x ∈ (↑e).fixedSubmodule ⊔ K ∙ v
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: case a.inl.coe.top
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
basis : Set (Set ι) := {s | ∃ f g, f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a}
h_basis : IsTopologicalBasis basis
a : ι
hb : ⊤ ∈ Ioi ↑a
⊢ ∃ t ⊆ Ioi ↑a, IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] t ∧ ⊤ ∈ t

Tactic:
rcases eq_empty_or_nonempty (Ioi a) with ha | ha

Goal state after:
case a.inl.coe.top.inl
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
basis : Set (Set ι) := {s | ∃ f g, f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a}
h_basis : IsTopologicalBasis basis
a : ι
hb : ⊤ ∈ Ioi ↑a
ha : Ioi a = ∅
⊢ ∃ t ⊆ Ioi ↑a, IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] t ∧ ⊤ ∈ t

case a.inl.coe.top.inr
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
basis : Set (Set ι) := {s | ∃ f g, f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a}
h_basis : IsTopologicalBasis basis
a : ι
hb : ⊤ ∈ Ioi ↑a
ha : (Ioi a).Nonempty
⊢ ∃ t ⊆ Ioi ↑a, IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] t ∧ ⊤ ∈ t
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: case a.inl.coe.coe.refine_1
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
basis : Set (Set ι) := {s | ∃ f g, f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a}
h_basis : IsTopologicalBasis basis
a b : ι
hb✝ : a < b
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Ioi a
w : WithTop ι
hw : w ∈ (⋂ z ∈ f, Ioi ↑z) ∩ ⋂ z ∈ g, Iio ↑z
⊢ w ∈ Ioi ↑a

Tactic:
induction w with

Goal state after:
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: case a.inr.coe.coe.refine_1
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
basis : Set (Set ι) := {s | ∃ f g, f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a}
h_basis : IsTopologicalBasis basis
a b : ι
hb✝ : b < a
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Iio a
w : WithTop ι
hw : w ∈ ((⋂ z ∈ f, Ioi ↑z) ∩ ⋂ z ∈ g, Iio ↑z) ∩ Iio ⊤
⊢ w ∈ Iio ↑a

Tactic:
induction w with

Goal state after:
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
hf : MemLp f 1 μ
hmeas✝ : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M ↦ {x | ↑M ≤ ↑‖f x‖₊}.indicator f x) atTop (𝓝 0)
hmeas : ∀ (M : ℕ), AEStronglyMeasurable ({x | ↑M ≤ ↑‖f x‖₊}.indicator f) μ
hbound : HasFiniteIntegral (fun x ↦ ‖f x‖) μ
n : ℕ
x : α
hx : ¬↑n ≤ ‖f x‖
⊢ 0 ≤ ‖f x‖

Tactic:
exact norm_nonneg _

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : TopologicalSpace M
inst✝ : T2Space M
X : Type u_5
mX : MeasurableSpace X
μ ν : VectorMeasure X M
C : Set (Set X)
hμν : ∀ s ∈ C, μ s = ν s
hA : mX = MeasurableSpace.generateFrom C
hC : IsPiSystem C
h_univ : μ univ = ν univ
⊢ μ = ν

Tactic:
ext s hs

Goal state after:
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : TopologicalSpace M
inst✝ : T2Space M
X : Type u_5
mX : MeasurableSpace X
μ ν : VectorMeasure X M
C : Set (Set X)
hμν : ∀ s ∈ C, μ s = ν s
hA : mX = MeasurableSpace.generateFrom C
hC : IsPiSystem C
h_univ : μ univ = ν univ
s : Set X
hs : MeasurableSet s
⊢ μ s = ν s
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : MeasurableSpace β
x✝ : β
v✝ : M
s : Set β
x : β
v : M
f : ℕ → Set β
f_meas : ∀ (i : ℕ), MeasurableSet (f i)
f_disj : Pairwise (Disjoint on f)
⊢ HasSum (fun i ↦ if MeasurableSet (f i) ∧ x ∈ f i then v else 0)
    (if MeasurableSet (⋃ i, f i) ∧ x ∈ ⋃ i, f i then v else 0)

Tactic:
by_cases hx : x ∈ ⋃ i, f i

Goal state after:
case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : MeasurableSpace β
x✝ : β
v✝ : M
s : Set β
x : β
v : M
f : ℕ → Set β
f_meas : ∀ (i : ℕ), MeasurableSet (f i)
f_disj : Pairwise (Disjoint on f)
hx : x ∈ ⋃ i, f i
⊢ HasSum (fun i ↦ if MeasurableSet (f i) ∧ x ∈ f i then v else 0)
    (if MeasurableSet (⋃ i, f i) ∧ x ∈ ⋃ i, f i then v else 0)

case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : MeasurableSpace β
x✝ : β
v✝ : M
s : Set β
x : β
v : M
f : ℕ → Set β
f_meas : ∀ (i : ℕ), MeasurableSet (f i)
f_disj : Pairwise (Disjoint on f)
hx : x ∉ ⋃ i, f i
⊢ HasSum (fun i ↦ if MeasurableSet (f i) ∧ x ∈ f i then v else 0)
    (if MeasurableSet (⋃ i, f i) ∧ x ∈ ⋃ i, f i then v else 0)
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: β : Type u_2
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : MeasurableSpace β
x : β
⊢ dirac x 0 = 0

Tactic:
ext s hs

Goal state after:
β : Type u_2
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : MeasurableSpace β
x : β
s : Set β
hs : MeasurableSet s
⊢ (dirac x 0) s = 0 s
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_top : p ≠ ∞
ε : ℝ
hε : 0 < ε
M : ℝ
hf : ∀ (x : α), ‖f x‖ < M
hM : 0 < M
s : Set α
hs : MeasurableSet s
hμ : μ s ≤ ENNReal.ofReal ((ε / M) ^ p.toReal)
hp : ¬p = 0
haebdd : ∀ᵐ (x : α) ∂μ.restrict s, ‖f x‖ ≤ M
⊢ μ s ^ p.toReal⁻¹ ≤ ENNReal.ofReal ε / ENNReal.ofReal M

Tactic:
rw [ENNReal.rpow_inv_le_iff (ENNReal.toReal_pos hp hp_top)]
    refine le_trans hμ ?_
    rw [← ENNReal.ofReal_rpow_of_pos (div_pos hε hM)]
    gcongr
    rw [ENNReal.ofReal_div_of_pos hM]

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case pos
α : Type u_1
mα : MeasurableSpace α
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
v : VectorMeasure α M
a : α
h : MeasurableSet {a}
⊢ v.restrict {a} = dirac a (v {a})

Tactic:
ext s hs

Goal state after:
case pos
α : Type u_1
mα : MeasurableSpace α
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
v : VectorMeasure α M
a : α
h : MeasurableSet {a}
s : Set α
hs : MeasurableSet s
⊢ (v.restrict {a}) s = (dirac a (v {a})) s
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
n m : ℕ
h : n ≠ m
⊢ (Disjoint on s.restrictNonposSeq i) n m

Tactic:
rw [Function.onFun, Set.disjoint_iff_inter_eq_empty]

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
n m : ℕ
h : n ≠ m
⊢ s.restrictNonposSeq i n ∩ s.restrictNonposSeq i m = ∅
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
⊢ ∃ i,
    ∃ (hi₁ : MeasurableSet i) (hi₂ : 0 ≤[i] s) (hi₃ : s ≤[iᶜ] 0),
      s.toJordanDecomposition.posPart = s.toMeasureOfZeroLE i hi₁ hi₂ ∧
        s.toJordanDecomposition.negPart = s.toMeasureOfLEZero iᶜ ⋯ hi₃

Tactic:
set i := s.exists_compl_positive_negative.choose

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α := ⋯.choose
⊢ ∃ i,
    ∃ (hi₁ : MeasurableSet i) (hi₂ : 0 ≤[i] s) (hi₃ : s ≤[iᶜ] 0),
      s.toJordanDecomposition.posPart = s.toMeasureOfZeroLE i hi₁ hi₂ ∧
        s.toJordanDecomposition.negPart = s.toMeasureOfLEZero iᶜ ⋯ hi₃
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case refine_3
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid M
inst✝² : PartialOrder M
inst✝¹ : IsOrderedAddMonoid M
inst✝ : OrderClosedTopology M
v w : VectorMeasure α M
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ), v ≤[f n] w
a : Set α
ha₁ : MeasurableSet a
ha₂ : a ⊆ ⋃ n, f n
ha₃ : ⋃ n, a ∩ disjointed f n = a
ha₄ : Pairwise (Disjoint on fun n ↦ a ∩ disjointed f n)
⊢ Summable fun i ↦ v (a ∩ disjointed f i)

Tactic:
refine (v.m_iUnion (fun n => ?_) ?_).summable

Goal state after:
case refine_3.refine_1
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid M
inst✝² : PartialOrder M
inst✝¹ : IsOrderedAddMonoid M
inst✝ : OrderClosedTopology M
v w : VectorMeasure α M
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ), v ≤[f n] w
a : Set α
ha₁ : MeasurableSet a
ha₂ : a ⊆ ⋃ n, f n
ha₃ : ⋃ n, a ∩ disjointed f n = a
ha₄ : Pairwise (Disjoint on fun n ↦ a ∩ disjointed f n)
n : ℕ
⊢ MeasurableSet (a ∩ disjointed f n)

case refine_3.refine_2
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid M
inst✝² : PartialOrder M
inst✝¹ : IsOrderedAddMonoid M
inst✝ : OrderClosedTopology M
v w : VectorMeasure α M
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ), v ≤[f n] w
a : Set α
ha₁ : MeasurableSet a
ha₂ : a ⊆ ⋃ n, f n
ha₃ : ⋃ n, a ∩ disjointed f n = a
ha₄ : Pairwise (Disjoint on fun n ↦ a ∩ disjointed f n)
⊢ Pairwise (Disjoint on fun i ↦ a ∩ disjointed f i)
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
j₁ j₂ : JordanDecomposition α
hj : j₁.posPart = j₂.posPart
hj' : j₁.toSignedMeasure = j₂.toSignedMeasure
⊢ j₁ = j₂

Tactic:
ext1

Goal state after:
case posPart
α : Type u_1
inst✝ : MeasurableSpace α
j₁ j₂ : JordanDecomposition α
hj : j₁.posPart = j₂.posPart
hj' : j₁.toSignedMeasure = j₂.toSignedMeasure
⊢ j₁.posPart = j₂.posPart

case negPart
α : Type u_1
inst✝ : MeasurableSpace α
j₁ j₂ : JordanDecomposition α
hj : j₁.posPart = j₂.posPart
hj' : j₁.toSignedMeasure = j₂.toSignedMeasure
⊢ j₁.negPart = j₂.negPart
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ∞
ε : ℝ
hε : 0 < ε
hℒp : MemLp f p μ
right✝ : eLpNorm f p μ < ∞
f' : α → β
hf' : StronglyMeasurable f'
heq : f =ᵐ[μ] f'
δ : ℝ
hδpos : 0 < δ
hδ : ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator f') p μ ≤ ENNReal.ofReal ε
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal δ
⊢ eLpNorm (s.indicator f) p μ = eLpNorm (s.indicator f') p μ

Tactic:
rw [eLpNorm_indicator_eq_eLpNorm_restrict hs, eLpNorm_indicator_eq_eLpNorm_restrict hs]

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ∞
ε : ℝ
hε : 0 < ε
hℒp : MemLp f p μ
right✝ : eLpNorm f p μ < ∞
f' : α → β
hf' : StronglyMeasurable f'
heq : f =ᵐ[μ] f'
δ : ℝ
hδpos : 0 < δ
hδ : ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator f') p μ ≤ ENNReal.ofReal ε
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal δ
⊢ eLpNorm f p (μ.restrict s) = eLpNorm f' p (μ.restrict s)
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan

Goal state before:
pp: case neg
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
r : ℝ
hr : r < 0
⊢ (r • s).toJordanDecomposition = r • s.toJordanDecomposition

Tactic:
ext1

Goal state after:
case neg.posPart
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
r : ℝ
hr : r < 0
⊢ (r • s).toJordanDecomposition.posPart = (r • s.toJordanDecomposition).posPart

case neg.negPart
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
r : ℝ
hr : r < 0
⊢ (r • s).toJordanDecomposition.negPart = (r • s.toJordanDecomposition).negPart
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan

Goal state before:
pp: case neg.negPart
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
r : ℝ
hr : r < 0
⊢ (r • s).toJordanDecomposition.negPart = (r • s.toJordanDecomposition).negPart

Tactic:
rw [real_smul_negPart_neg _ _ hr,
        show r • s = -(-r • s) by rw [neg_smul, neg_neg], toJordanDecomposition_neg, neg_negPart,
        toJordanDecomposition_smul_real_nonneg, ← smul_posPart, real_smul_nonneg]

Goal state after:
case neg.negPart.hr
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
r : ℝ
hr : r < 0
⊢ 0 ≤ -r

case neg.negPart.hr
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
r : ℝ
hr : r < 0
⊢ 0 ≤ -r
Module: Mathlib.MeasureTheory.VectorMeasure.WithDensity

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
g : α → E
hf : AEMeasurable f μ
hfg : Integrable (f • g) μ
⊢ μ.withDensityᵥ (f • g) = (μ.withDensity fun x ↦ ↑(f x)).withDensityᵥ g

Tactic:
ext s hs

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
g : α → E
hf : AEMeasurable f μ
hfg : Integrable (f • g) μ
s : Set α
hs : MeasurableSet s
⊢ (μ.withDensityᵥ (f • g)) s = ((μ.withDensity fun x ↦ ↑(f x)).withDensityᵥ g) s
Module: Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm

Goal state before:
pp: case hfm
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f : α → E
hp₀ : p ≠ 0
hp : p ≠ ∞
hf : AEStronglyMeasurable f μ
⊢ AEMeasurable (fun x ↦ ‖f x‖ₑ ^ p.toReal) μ

pp: case hf
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f : α → E
hp₀ : p ≠ 0
hp : p ≠ ∞
hf : AEStronglyMeasurable f μ
⊢ ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ^ p.toReal < ∞

Tactic:
· simp_rw [← ofReal_norm]
    borelize E
    fun_prop

Goal state after:
case hf
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f : α → E
hp₀ : p ≠ 0
hp : p ≠ ∞
hf : AEStronglyMeasurable f μ
⊢ ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ^ p.toReal < ∞
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ∞
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg : MemLp g p μ
hui : UnifIntegrable f p μ
hfg : TendstoInMeasure μ f atTop g
ns : ℕ → ℕ
hns : Tendsto ns atTop atTop
ms : ℕ → ℕ
left✝ : StrictMono ms
hms' : ∀ᵐ (x : α) ∂μ, Tendsto (fun i ↦ f (ns (ms i)) x) atTop (𝓝 (g x))
⊢ ∃ ms, Tendsto (fun n ↦ eLpNorm (f (ns (ms n)) - g) p μ) atTop (𝓝 0)

Tactic:
exact ⟨ms,
    tendsto_Lp_finite_of_tendsto_ae hp hp' (fun _ => hf _) hg (fun ε hε =>
      let ⟨δ, hδ, hδ'⟩ := hui hε
      ⟨δ, hδ, fun i s hs hμs => hδ' _ s hs hμs⟩)
      hms'⟩

Goal state after:
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ p.toReal)
hμs' : μ s = 0
⊢ eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε

Tactic:
rw [(eLpNorm_eq_zero_iff ((hf i).indicator hs).aestronglyMeasurable hpzero).2
        (indicator_meas_zero hμs')]

Goal state after:
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ p.toReal)
hμs' : μ s = 0
⊢ 0 ≤ ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.Measure.HasOuterApproxClosed

Goal state before:
pp: case h.refine_3
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace.PseudoMetrizableSpace X
this : PseudoMetricSpace X := TopologicalSpace.pseudoMetrizableSpacePseudoMetric X
F : Set X
hF : IsClosed[inst✝¹] F
⊢ Tendsto (fun n x ↦ ((fun n ↦ thickenedIndicator ⋯ F) n) x) atTop (𝓝 (F.indicator fun x ↦ 1))

Tactic:
have key := thickenedIndicator_tendsto_indicator_closure
              (δseq := fun (n : ℕ) ↦ (1 : ℝ) / (n + 1))
              (fun _ ↦ Nat.one_div_pos_of_nat) tendsto_one_div_add_atTop_nhds_zero_nat F
    rw [tendsto_pi_nhds] at *
    intro x
    nth_rw 2 [← IsClosed.closure_eq hF]
    exact key x

Goal state after:
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ p.toReal)
hμs' : ¬μ s = 0
⊢ ∀ (a : α), ‖{x | ‖f i x‖₊ < C}.indicator (f i) a‖ ≤ ↑C

Tactic:
simp_rw [norm_indicator_eq_indicator_norm]

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ p.toReal)
hμs' : ¬μ s = 0
⊢ ∀ (a : α), {x | ‖f i x‖₊ < C}.indicator (fun a ↦ ‖f i a‖) a ≤ ↑C