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

Goal state before:
pp: γ : Type u_3
β : Type u_4
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : PolishSpace γ
inst✝³ : TopologicalSpace β
inst✝² : T2Space β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
f : γ → β
f_cont : Continuous[inst✝⁵, inst✝³] f
f_inj : Injective f
this✝¹ : UpgradedIsCompletelyMetrizableSpace γ := upgradeIsCompletelyMetrizable γ
b : Set (Set γ)
b_count : b.Countable
b_nonempty : ∅ ∉ b
hb : IsTopologicalBasis b
this✝ : Encodable ↑b
A : Type (max 0 u_3) := { p // Disjoint ↑p.1 ↑p.2 }
q : A → Set β
hq1 : ∀ (p : A), f '' ↑(↑p).1 ⊆ q p
hq2 : ∀ (p : A), Disjoint (f '' ↑(↑p).2) (q p)
q_meas : ∀ (p : A), MeasurableSet (q p)
E : ↑b → Set β := fun s ↦ closure[inst✝³] (f '' ↑s) ∩ ⋂ t, ⋂ (ht : Disjoint ↑s ↑t), q ⟨(s, t), ht⟩ \ q ⟨(t, s), ⋯⟩
u : ℕ → ℝ
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
F : ℕ → Set β := fun n ↦ ⋃ s, ⋃ (_ : Bornology.IsBounded ↑s ∧ diam ↑s ≤ u n), E s
this : range f = ⋂ n, F n
E_meas : ∀ (s : ↑b), MeasurableSet (E s)
⊢ MeasurableSet (range f)

Tactic:
have F_meas : ∀ n, MeasurableSet (F n) := by
      intro n
      refine MeasurableSet.iUnion fun s => ?_
      exact MeasurableSet.iUnion fun _ => E_meas _

Goal state after:
γ : Type u_3
β : Type u_4
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : PolishSpace γ
inst✝³ : TopologicalSpace β
inst✝² : T2Space β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
f : γ → β
f_cont : Continuous[inst✝⁵, inst✝³] f
f_inj : Injective f
this✝¹ : UpgradedIsCompletelyMetrizableSpace γ := upgradeIsCompletelyMetrizable γ
b : Set (Set γ)
b_count : b.Countable
b_nonempty : ∅ ∉ b
hb : IsTopologicalBasis b
this✝ : Encodable ↑b
A : Type (max 0 u_3) := { p // Disjoint ↑p.1 ↑p.2 }
q : A → Set β
hq1 : ∀ (p : A), f '' ↑(↑p).1 ⊆ q p
hq2 : ∀ (p : A), Disjoint (f '' ↑(↑p).2) (q p)
q_meas : ∀ (p : A), MeasurableSet (q p)
E : ↑b → Set β := fun s ↦ closure[inst✝³] (f '' ↑s) ∩ ⋂ t, ⋂ (ht : Disjoint ↑s ↑t), q ⟨(s, t), ht⟩ \ q ⟨(t, s), ⋯⟩
u : ℕ → ℝ
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
F : ℕ → Set β := fun n ↦ ⋃ s, ⋃ (_ : Bornology.IsBounded ↑s ∧ diam ↑s ≤ u n), E s
this : range f = ⋂ n, F n
E_meas : ∀ (s : ↑b), MeasurableSet (E s)
F_meas : ∀ (n : ℕ), MeasurableSet (F n)
⊢ MeasurableSet (range f)
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: case h₁
γ : Type u_3
β : Type u_4
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : PolishSpace γ
inst✝³ : TopologicalSpace β
inst✝² : T2Space β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
f : γ → β
f_cont : Continuous[inst✝⁵, inst✝³] f
f_inj : Injective f
this✝ : UpgradedIsCompletelyMetrizableSpace γ := upgradeIsCompletelyMetrizable γ
b : Set (Set γ)
b_count : b.Countable
b_nonempty : ∅ ∉ b
hb : IsTopologicalBasis b
this : Encodable ↑b
A : Type (max 0 u_3) := { p // Disjoint ↑p.1 ↑p.2 }
q : A → Set β
hq1 : ∀ (p : A), f '' ↑(↑p).1 ⊆ q p
hq2 : ∀ (p : A), Disjoint (f '' ↑(↑p).2) (q p)
q_meas : ∀ (p : A), MeasurableSet (q p)
E : ↑b → Set β := fun s ↦ closure[inst✝³] (f '' ↑s) ∩ ⋂ t, ⋂ (ht : Disjoint ↑s ↑t), q ⟨(s, t), ht⟩ \ q ⟨(t, s), ⋯⟩
u : ℕ → ℝ
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
F : ℕ → Set β := fun n ↦ ⋃ s, ⋃ (_ : Bornology.IsBounded ↑s ∧ diam ↑s ≤ u n), E s
y : γ
⊢ f y ∈ ⋂ n, F n

Tactic:
refine mem_iInter.2 fun n => ?_

Goal state after:
case h₁
γ : Type u_3
β : Type u_4
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : PolishSpace γ
inst✝³ : TopologicalSpace β
inst✝² : T2Space β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
f : γ → β
f_cont : Continuous[inst✝⁵, inst✝³] f
f_inj : Injective f
this✝ : UpgradedIsCompletelyMetrizableSpace γ := upgradeIsCompletelyMetrizable γ
b : Set (Set γ)
b_count : b.Countable
b_nonempty : ∅ ∉ b
hb : IsTopologicalBasis b
this : Encodable ↑b
A : Type (max 0 u_3) := { p // Disjoint ↑p.1 ↑p.2 }
q : A → Set β
hq1 : ∀ (p : A), f '' ↑(↑p).1 ⊆ q p
hq2 : ∀ (p : A), Disjoint (f '' ↑(↑p).2) (q p)
q_meas : ∀ (p : A), MeasurableSet (q p)
E : ↑b → Set β := fun s ↦ closure[inst✝³] (f '' ↑s) ∩ ⋂ t, ⋂ (ht : Disjoint ↑s ↑t), q ⟨(s, t), ht⟩ \ q ⟨(t, s), ⋯⟩
u : ℕ → ℝ
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
F : ℕ → Set β := fun n ↦ ⋃ s, ⋃ (_ : Bornology.IsBounded ↑s ∧ diam ↑s ≤ u n), E s
y : γ
n : ℕ
⊢ f y ∈ F n
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : Set α → Prop
h : μ.InnerRegularWRT p fun s ↦ MeasurableSet s ∧ μ s ≠ ∞
A s : Set α
s_meas : MeasurableSet s
hs : (μ.restrict A) s ≠ ∞
r : ℝ≥0∞
hr : r < (μ.restrict A) s
⊢ ∃ K ⊆ s, p K ∧ r < (μ.restrict A) K

Tactic:
rw [restrict_apply s_meas] at hs

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : Set α → Prop
h : μ.InnerRegularWRT p fun s ↦ MeasurableSet s ∧ μ s ≠ ∞
A s : Set α
s_meas : MeasurableSet s
hs : μ (s ∩ A) ≠ ∞
r : ℝ≥0∞
hr : r < (μ.restrict A) s
⊢ ∃ K ⊆ s, p K ∧ r < (μ.restrict A) K
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_1
inst✝² : MeasurableSpace α
μ : Measure α
p : Set α → Prop
inst✝¹ : TopologicalSpace α
inst✝ : μ.OuterRegular
H : μ.InnerRegularWRT p IsOpen[inst✝¹]
hd : ∀ ⦃s U : Set α⦄, p s → IsOpen[inst✝¹] U → p (s \ U)
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
r : ℝ≥0∞
hr : r < μ s
h0 : p ∅
⊢ ∃ ε, ε ≠ 0 ∧ ε + ε ≤ μ s ∧ r = μ s - (ε + ε)

Tactic:
use (μ s - r) / 2
    simp [*, hr.le, ENNReal.add_halves, ENNReal.sub_sub_cancel, tsub_eq_zero_iff_le]

Goal state after:
Module: Mathlib.MeasureTheory.Group.Action

Goal state before:
pp: G : Type u
α : Type w
m : MeasurableSpace α
inst✝⁵ : Group G
inst✝⁴ : MulAction G α
μ : Measure α
inst✝³ : SMulInvariantMeasure G α μ
inst✝² : TopologicalSpace α
inst✝¹ : ContinuousConstSMul G α
inst✝ : MulAction.IsMinimal G α
U : Set α
hU : IsOpen[inst✝²] U
hne : U.Nonempty
hμU : μ U ≠ ∞
x : α
g : G
hg : g • x ∈ U
⊢ μ ((fun x ↦ g • x) ⁻¹' U) ≠ ∞

Tactic:
by rwa [measure_preimage_smul]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: case compl
α : Type u_1
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
H✝ : μ.InnerRegularWRT IsClosed[inst✝²] IsOpen[inst✝²]
hfin : ∀ {s : Set α}, μ s ≠ ∞
s : Set α
hs : MeasurableSet s
H : ∀ (ε : ℝ≥0∞), ε ≠ 0 → ∃ F ⊆ s, ∃ U ⊇ s, IsClosed[inst✝²] F ∧ IsOpen[inst✝²] U ∧ μ s ≤ μ F + ε ∧ μ U ≤ μ s + ε
ε : ℝ≥0∞
hε : ε ≠ 0
F : Set α
hFs : F ⊆ s
U : Set α
hsU : U ⊇ s
hFc : IsClosed[inst✝²] F
hUo : IsOpen[inst✝²] U
hF : μ s ≤ μ F + ε
hU : μ U ≤ μ s + ε
⊢ μ sᶜ ≤ μ Uᶜ + ε ∧ μ Fᶜ ≤ μ sᶜ + ε

Tactic:
simp only [measure_compl_le_add_iff, *, hUo.measurableSet, hFc.measurableSet, true_and]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_4
β : Type u_5
γ : Type u_6
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
f : α × β → γ
μ : Measure α
ν : Measure β
τ : Measure γ
hf : Measurable f
inst✝¹ : SFinite μ
inst✝ : SFinite ν
h2f : ∀ᵐ (y : β) ∂ν, QuasiMeasurePreserving (fun x ↦ f (x, y)) μ τ
⊢ QuasiMeasurePreserving f (μ.prod ν) τ

Tactic:
rw [← prod_swap]

Goal state after:
α : Type u_4
β : Type u_5
γ : Type u_6
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
f : α × β → γ
μ : Measure α
ν : Measure β
τ : Measure γ
hf : Measurable f
inst✝¹ : SFinite μ
inst✝ : SFinite ν
h2f : ∀ᵐ (y : β) ∂ν, QuasiMeasurePreserving (fun x ↦ f (x, y)) μ τ
⊢ QuasiMeasurePreserving f (map Prod.swap (ν.prod μ)) τ
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: G : Type u_1
inst✝³ : MeasurableSpace G
inst✝² : Group G
inst✝¹ : MeasurableMul G
μ : Measure G
inst✝ : μ.IsMulLeftInvariant
t : G
p : G → Prop
⊢ (∀ᵐ (x : G) ∂μ, p (t * x)) ↔ ∀ᵐ (x : G) ∂μ, p x

Tactic:
conv_rhs => rw [Filter.Eventually, ← map_mul_left_ae μ t]
  rfl

Goal state after:
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: G : Type u_1
H : Type u_2
inst✝⁵ : MeasurableSpace G
inst✝⁴ : MeasurableSpace H
inst✝³ : DivisionMonoid G
inst✝² : MeasurableMul G
inst✝¹ : MeasurableInv G
μ : Measure G
inst✝ : μ.IsMulRightInvariant
g : G
⊢ map (fun x ↦ g * x) (map inv μ) = map inv (map (fun x ↦ x * g⁻¹) μ)

Tactic:
map_map (measurable_const_mul g) measurable_inv,

Goal state after:
G : Type u_1
H : Type u_2
inst✝⁵ : MeasurableSpace G
inst✝⁴ : MeasurableSpace H
inst✝³ : DivisionMonoid G
inst✝² : MeasurableMul G
inst✝¹ : MeasurableInv G
μ : Measure G
inst✝ : μ.IsMulRightInvariant
g : G
⊢ map ((fun x ↦ g * x) ∘ inv) μ = map inv (map (fun x ↦ x * g⁻¹) μ)
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: case mp
G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul G
H : Type u_3
inst✝² : Group H
mH : MeasurableSpace H
inst✝¹ : MeasurableMul H
μ : Measure H
inst✝ : μ.IsMulLeftInvariant
f : G →* H
hf : MeasurableEmbedding ⇑f
g : G
s : Set G
hs : MeasurableSet s
x✝ : H
⊢ x✝ ∈ ⇑f '' (fun x ↦ g * x) ⁻¹' s → x✝ ∈ (fun x ↦ f g * x) ⁻¹' ⇑f '' s

Tactic:
rintro ⟨y, hy, rfl⟩

Goal state after:
case mp
G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul G
H : Type u_3
inst✝² : Group H
mH : MeasurableSpace H
inst✝¹ : MeasurableMul H
μ : Measure H
inst✝ : μ.IsMulLeftInvariant
f : G →* H
hf : MeasurableEmbedding ⇑f
g : G
s : Set G
hs : MeasurableSet s
y : G
hy : y ∈ (fun x ↦ g * x) ⁻¹' s
⊢ f y ∈ (fun x ↦ f g * x) ⁻¹' ⇑f '' s
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : SFinite ν
inst✝ : SFinite μ
s : Set α
t : Set β
f : α × β → ℝ≥0∞
hf : AEMeasurable f ((μ.prod ν).restrict (s ×ˢ t))
⊢ ∫⁻ (z : β × α) in t ×ˢ s, f z.swap ∂ν.prod μ = ∫⁻ (y : β) in t, ∫⁻ (x : α) in s, f (x, y) ∂μ ∂ν

Tactic:
setLIntegral_prod

Goal state after:
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : SFinite ν
inst✝ : SFinite μ
s : Set α
t : Set β
f : α × β → ℝ≥0∞
hf : AEMeasurable f ((μ.prod ν).restrict (s ×ˢ t))
⊢ ∫⁻ (x : β) in t, ∫⁻ (y : α) in s, f (x, y).swap ∂μ ∂ν = ∫⁻ (y : β) in t, ∫⁻ (x : α) in s, f (x, y) ∂μ ∂ν

case hf
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : SFinite ν
inst✝ : SFinite μ
s : Set α
t : Set β
f : α × β → ℝ≥0∞
hf : AEMeasurable f ((μ.prod ν).restrict (s ×ˢ t))
⊢ AEMeasurable (fun z ↦ f z.swap) ((ν.prod μ).restrict (t ×ˢ s))
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: case mp
G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul G
H : Type u_3
inst✝² : Group H
mH : MeasurableSpace H
inst✝¹ : MeasurableMul H
μ : Measure H
inst✝ : μ.IsMulRightInvariant
f : G →* H
hf : MeasurableEmbedding ⇑f
g : G
s : Set G
hs : MeasurableSet s
x✝ : H
⊢ x✝ ∈ ⇑f '' (fun x ↦ x * g) ⁻¹' s → x✝ ∈ (fun x ↦ x * f g) ⁻¹' ⇑f '' s

Tactic:
rintro ⟨y, hy, rfl⟩

Goal state after:
case mp
G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul G
H : Type u_3
inst✝² : Group H
mH : MeasurableSpace H
inst✝¹ : MeasurableMul H
μ : Measure H
inst✝ : μ.IsMulRightInvariant
f : G →* H
hf : MeasurableEmbedding ⇑f
g : G
s : Set G
hs : MeasurableSet s
y : G
hy : y ∈ (fun x ↦ x * g) ⁻¹' s
⊢ f y ∈ (fun x ↦ x * f g) ⁻¹' ⇑f '' s
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : SFinite ν
ρ : Measure (α × β)
inst✝ : IsZeroOrProbabilityMeasure ρ
h : ρ = 0
⊢ IsZeroOrProbabilityMeasure ρ.fst

Tactic:
simp only [h, fst_zero]

Goal state after:
case inl
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : SFinite ν
ρ : Measure (α × β)
inst✝ : IsZeroOrProbabilityMeasure ρ
h : ρ = 0
⊢ IsZeroOrProbabilityMeasure 0
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: case mp
G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul G
H : Type u_3
inst✝² : Group H
mH : MeasurableSpace H
inst✝¹ : MeasurableMul H
μ : Measure H
inst✝ : μ.IsMulRightInvariant
f : G →* H
hf : MeasurableEmbedding ⇑f
g : G
s : Set G
hs : MeasurableSet s
x✝ : H
⊢ x✝ ∈ ⇑f '' (fun x ↦ x * g) ⁻¹' s → x✝ ∈ (fun x ↦ x * f g) ⁻¹' ⇑f '' s

pp: case mpr
G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul G
H : Type u_3
inst✝² : Group H
mH : MeasurableSpace H
inst✝¹ : MeasurableMul H
μ : Measure H
inst✝ : μ.IsMulRightInvariant
f : G →* H
hf : MeasurableEmbedding ⇑f
g : G
s : Set G
hs : MeasurableSet s
x✝ : H
⊢ x✝ ∈ (fun x ↦ x * f g) ⁻¹' ⇑f '' s → x✝ ∈ ⇑f '' (fun x ↦ x * g) ⁻¹' s

Tactic:
· rintro ⟨y, hy, rfl⟩
        exact ⟨y * g, hy, by simp⟩

Goal state after:
case mpr
G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul G
H : Type u_3
inst✝² : Group H
mH : MeasurableSpace H
inst✝¹ : MeasurableMul H
μ : Measure H
inst✝ : μ.IsMulRightInvariant
f : G →* H
hf : MeasurableEmbedding ⇑f
g : G
s : Set G
hs : MeasurableSet s
x✝ : H
⊢ x✝ ∈ (fun x ↦ x * f g) ⁻¹' ⇑f '' s → x✝ ∈ ⇑f '' (fun x ↦ x * g) ⁻¹' s
Module: Mathlib.MeasureTheory.Group.Prod

Goal state before:
pp: G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul₂ G
μ : Measure G
inst✝² : SFinite μ
inst✝¹ : MeasurableInv G
inst✝ : μ.IsMulLeftInvariant
⊢ ae μ ≤ (ae μ)⁻¹

Tactic:
← inv_inv (ae μ)

Goal state after:
G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul₂ G
μ : Measure G
inst✝² : SFinite μ
inst✝¹ : MeasurableInv G
inst✝ : μ.IsMulLeftInvariant
⊢ (ae μ)⁻¹⁻¹ ≤ (ae μ)⁻¹
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hf : Measurable f
⊢ μ.withDensity (r • f) = r • μ.withDensity f

Tactic:
by
  refine Measure.ext fun s hs => ?_
  rw [withDensity_apply _ hs, Measure.coe_smul, Pi.smul_apply, withDensity_apply _ hs,
    smul_eq_mul, ← lintegral_const_mul r hf]
  simp only [Pi.smul_apply, smul_eq_mul]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : TopologicalSpace α
h : μ.InnerRegularCompactLTTop
c : ℝ≥0∞
hc : ¬c = 0
h'c : c = ∞
s : Set α
hs : MeasurableSet s ∧ (c • μ) s ≠ ∞
r : ℝ≥0∞
hr : r < (c • μ) s
⊢ ∃ K ⊆ s, IsCompact K ∧ r < (c • μ) K

Tactic:
by_cases h's : μ s = 0

Goal state after:
case pos
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : TopologicalSpace α
h : μ.InnerRegularCompactLTTop
c : ℝ≥0∞
hc : ¬c = 0
h'c : c = ∞
s : Set α
hs : MeasurableSet s ∧ (c • μ) s ≠ ∞
r : ℝ≥0∞
hr : r < (c • μ) s
h's : μ s = 0
⊢ ∃ K ⊆ s, IsCompact K ∧ r < (c • μ) K

case neg
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : TopologicalSpace α
h : μ.InnerRegularCompactLTTop
c : ℝ≥0∞
hc : ¬c = 0
h'c : c = ∞
s : Set α
hs : MeasurableSet s ∧ (c • μ) s ≠ ∞
r : ℝ≥0∞
hr : r < (c • μ) s
h's : ¬μ s = 0
⊢ ∃ K ⊆ s, IsCompact K ∧ r < (c • μ) K
Module: Mathlib.MeasureTheory.Group.Prod

Goal state before:
pp: G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
s : Set G
inst✝⁴ : MeasurableInv G
μ' ν' : Measure G
inst✝³ : SigmaFinite μ'
inst✝² : SigmaFinite ν'
inst✝¹ : μ'.IsMulLeftInvariant
inst✝ : ν'.IsMulLeftInvariant
h2s : ν' s ≠ 0
h3s : ν' s ≠ ∞
⊢ ∀ᵐ (x : G) ∂μ', ν' ((fun y ↦ y * x) ⁻¹' s) < ∞

Tactic:
refine (ae_measure_preimage_mul_right_lt_top ν' ν' h3s).filter_mono ?_
  refine (absolutelyContinuous_of_isMulLeftInvariant μ' ν' ?_).ae_le
  refine mt ?_ h2s
  intro hν
  rw [hν, Measure.coe_zero, Pi.zero_apply]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: case mp.ht
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0
hf : AEMeasurable f μ
g : α → ℝ≥0∞
f' : α → ℝ≥0
hf'_m : Measurable f'
hf'_ae : f =ᵐ[μ] f'
g' : α → ℝ≥0∞
g'meas : Measurable g'
hg' : ∀ᵐ (x : α) ∂μ.withDensity fun x ↦ ↑(f x), g x = g' x
A : MeasurableSet {x | f' x ≠ 0}
⊢ ∀ᵐ (x : α) ∂μ.restrict {x | f' x ≠ 0}, (fun x ↦ ↑(f x) * g x) x = (fun x ↦ ↑(f' x) * g' x) x

Tactic:
ae_withDensity_iff' hf.coe_nnreal_ennreal

Goal state after:
case mp.ht
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0
hf : AEMeasurable f μ
g : α → ℝ≥0∞
f' : α → ℝ≥0
hf'_m : Measurable f'
hf'_ae : f =ᵐ[μ] f'
g' : α → ℝ≥0∞
g'meas : Measurable g'
hg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = g' x
A : MeasurableSet {x | f' x ≠ 0}
⊢ ∀ᵐ (x : α) ∂μ.restrict {x | f' x ≠ 0}, (fun x ↦ ↑(f x) * g x) x = (fun x ↦ ↑(f' x) * g' x) x
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: case mpr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0
hf : AEMeasurable f μ
g : α → ℝ≥0∞
f' : α → ℝ≥0
hf'_m : Measurable f'
hf'_ae : f =ᵐ[μ] f'
g' : α → ℝ≥0∞
g'meas : Measurable g'
hg' : (fun x ↦ ↑(f x) * g x) =ᵐ[μ] g'
⊢ ∀ᵐ (x : α) ∂μ.withDensity fun x ↦ ↑(f x), g x = (↑(f' x))⁻¹ * g' x

Tactic:
ae_withDensity_iff' hf.coe_nnreal_ennreal

Goal state after:
case mpr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0
hf : AEMeasurable f μ
g : α → ℝ≥0∞
f' : α → ℝ≥0
hf'_m : Measurable f'
hf'_ae : f =ᵐ[μ] f'
g' : α → ℝ≥0∞
g'meas : Measurable g'
hg' : (fun x ↦ ↑(f x) * g x) =ᵐ[μ] g'
⊢ ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = (↑(f' x))⁻¹ * g' x
Module: Mathlib.Algebra.QuadraticDiscriminant

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
a b c : R
inst✝¹ : NeZero 2
inst✝ : NoZeroDivisors R
ha : a ≠ 0
x : R
h : discrim a b c = (2 * a * x + b) ^ 2
⊢ a * (x * x) + b * x + c = 0

Tactic:
rw [discrim] at h

Goal state after:
R : Type u_1
inst✝² : CommRing R
a b c : R
inst✝¹ : NeZero 2
inst✝ : NoZeroDivisors R
ha : a ≠ 0
x : R
h : b ^ 2 - 4 * a * c = (2 * a * x + b) ^ 2
⊢ a * (x * x) + b * x + c = 0
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: case add
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
h_mf : Measurable f
g h : α → ℝ≥0∞
a✝¹ : Disjoint (Function.support g) (Function.support h)
h_mea_g : Measurable g
a✝ : Measurable h
h_ind_g : ∫⁻ (a : α), g a ∂μ.withDensity f = ∫⁻ (a : α), (f * g) a ∂μ
⊢ ∫⁻ (a : α), h a ∂μ.withDensity f = ∫⁻ (a : α), (f * h) a ∂μ →
    ∫⁻ (a : α), (g + h) a ∂μ.withDensity f = ∫⁻ (a : α), (f * (g + h)) a ∂μ

Tactic:
h_ind_h

Goal state after:
case add
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
h_mf : Measurable f
g h : α → ℝ≥0∞
a✝¹ : Disjoint (Function.support g) (Function.support h)
h_mea_g : Measurable g
a✝ : Measurable h
h_ind_g : ∫⁻ (a : α), g a ∂μ.withDensity f = ∫⁻ (a : α), (f * g) a ∂μ
h_ind_h : ∫⁻ (a : α), h a ∂μ.withDensity f = ∫⁻ (a : α), (f * h) a ∂μ
⊢ ∫⁻ (a : α), (g + h) a ∂μ.withDensity f = ∫⁻ (a : α), (f * (g + h)) a ∂μ
Module: Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
a b : ℝ
X : α → ℝ
h : ∀ᵐ (ω : α) ∂μ, X ω ∈ Icc a b
⊢ ∀ᵐ (a_1 : α) ∂μ, ‖X a_1‖ ≤ max ‖a‖ ‖b‖

Tactic:
filter_upwards [h.mono fun ω h ↦ h.1, h.mono fun ω h ↦ h.2] with ω using abs_le_max_abs_abs

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
inst✝⁴ : PseudoEMetricSpace β
μ : Measure α
inst✝³ : SemilatticeSup ι
inst✝² : Nonempty ι
inst✝¹ : Countable ι
f : ι → α → β
g : α → β
inst✝ : IsFiniteMeasure μ
hf : ∀ (n : ι), Measurable fun a ↦ edist (f n a) (g a)
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ε : ℝ
hε : 0 < ε
t : Set α
left✝ : t ⊆ univ
ht : MeasurableSet t
htendsto : μ t ≤ ENNReal.ofReal ε ∧ TendstoUniformlyOn f g atTop (univ \ t)
⊢ ∃ t, MeasurableSet t ∧ μ t ≤ ENNReal.ofReal ε ∧ TendstoUniformlyOn f g atTop tᶜ

Tactic:
refine ⟨_, ht, ?_⟩

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
inst✝⁴ : PseudoEMetricSpace β
μ : Measure α
inst✝³ : SemilatticeSup ι
inst✝² : Nonempty ι
inst✝¹ : Countable ι
f : ι → α → β
g : α → β
inst✝ : IsFiniteMeasure μ
hf : ∀ (n : ι), Measurable fun a ↦ edist (f n a) (g a)
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ε : ℝ
hε : 0 < ε
t : Set α
left✝ : t ⊆ univ
ht : MeasurableSet t
htendsto : μ t ≤ ENNReal.ofReal ε ∧ TendstoUniformlyOn f g atTop (univ \ t)
⊢ μ t ≤ ENNReal.ofReal ε ∧ TendstoUniformlyOn f g atTop tᶜ
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
hf : ∀ᵐ (x : α) ∂μ, f x < ∞
g : α → ℝ≥0∞
⊢ ∫⁻ (a : α), g a ∂μ.withDensity f = ∫⁻ (a : α), (f * g) a ∂μ

Tactic:
refine le_antisymm (lintegral_withDensity_le_lintegral_mul μ f_meas g) ?_

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
hf : ∀ᵐ (x : α) ∂μ, f x < ∞
g : α → ℝ≥0∞
⊢ ∫⁻ (a : α), (f * g) a ∂μ ≤ ∫⁻ (a : α), g a ∂μ.withDensity f
Module: Mathlib.Analysis.CStarAlgebra.Basic

Goal state before:
pp: E : Type u_2
inst✝² : NonUnitalNormedRing E
inst✝¹ : StarRing E
inst✝ : CStarRing E
x : E
⊢ ‖x⋆⋆ * x⋆‖ = ‖x‖ * ‖x‖

Tactic:
simp only [norm_star_mul_self, norm_star]

Goal state after:
Module: Mathlib.LinearAlgebra.Complex.Module

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : SMul R ℝ
inst✝¹ : SMul Rᵐᵒᵖ ℝ
inst✝ : IsCentralScalar R ℝ
r : R
x : ℂ
⊢ MulOpposite.op r • x = r • x

Tactic:
ext <;> simp [smul_re, smul_im, op_smul_eq_smul]

Goal state after:
Module: Mathlib.Analysis.Complex.Basic

Goal state before:
pp: n : ℤ
⊢ ‖↑n‖₊ = ‖n‖₊

Tactic:
ext; exact norm_intCast n

Goal state after:
Module: Mathlib.Analysis.Complex.Basic

Goal state before:
pp: z : ℂ
hz : z ≠ 0
⊢ z ∈ slitPlane ∨ -z ∈ slitPlane

Tactic:
rw [mem_slitPlane_iff, mem_slitPlane_iff]
  rw [ne_eq, Complex.ext_iff] at hz
  push Not at hz
  simp_all only [ne_eq, zero_re, zero_im, neg_re, Left.neg_pos_iff, neg_im, neg_eq_zero]
  by_contra! contra
  exact hz (le_antisymm contra.1.1 contra.2.1) contra.1.2

Goal state after:
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
z w : K
⊢ normSq (z + w) = normSq z + normSq w + 2 * re (z * (starRingEnd K) w)

Tactic:
simp only [normSq_apply, map_add, rclike_simps]

Goal state after:
K : Type u_1
inst✝ : RCLike K
z w : K
⊢ (re z + re w) * (re z + re w) + (im z + im w) * (im z + im w) =
    re z * re z + im z * im z + (re w * re w + im w * im w) + 2 * (re z * re w - im z * -im w)
Module: Mathlib.LinearAlgebra.Complex.Module

Goal state before:
pp: A : Type u_1
inst✝⁵ : NonUnitalNonAssocRing A
inst✝⁴ : StarRing A
inst✝³ : Module ℂ A
inst✝² : IsScalarTower ℂ A A
inst✝¹ : SMulCommClass ℂ A A
inst✝ : StarModule ℂ A
a : A
a_eq : a = ↑(ℜ a) + I • ↑(ℑ a)
⊢ star (↑(ℜ a) + I • ↑(ℑ a)) * (↑(ℜ a) + I • ↑(ℑ a)) - (↑(ℜ a) + I • ↑(ℑ a)) * star (↑(ℜ a) + I • ↑(ℑ a)) =
    2 • I • (↑(ℜ a) * ↑(ℑ a) - ↑(ℑ a) * ↑(ℜ a))

Tactic:
by
      simp [mul_add, add_mul, mul_smul_comm, smul_mul_assoc, smul_smul]
      module

Goal state after:
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
x : K
⊢ ‖↑(2 * re x)‖ ^ 2 = re ↑(2 * re x) ^ 2

Tactic:
norm_ofReal,

Goal state after:
K : Type u_1
inst✝ : RCLike K
x : K
⊢ |2 * re x| ^ 2 = re ↑(2 * re x) ^ 2
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
z : K
r : ℤ
x : K
⊢ ‖↑↑r‖ * ‖x‖ = ‖r‖ * ‖x‖

Tactic:
norm_ofReal,

Goal state after:
K : Type u_1
E : Type u_2
inst✝ : RCLike K
z : K
r : ℤ
x : K
⊢ |↑r| * ‖x‖ = ‖r‖ * ‖x‖
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
z : K
⊢ z ≤ 0 ↔ ∃ x ≤ 0, ↑x = z

Tactic:
nonpos_iff (K := K),

Goal state after:
K : Type u_1
inst✝ : RCLike K
z : K
⊢ re z ≤ 0 ∧ im z = 0 ↔ ∃ x ≤ 0, ↑x = z
Module: Mathlib.Topology.OpenPartialHomeomorph.Continuity

Goal state before:
pp: X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : OpenPartialHomeomorph X Y
x : X
p : Y → Prop
hx : x ∈ e.source
⊢ (∀ᶠ (y : Y) in 𝓝 (↑e x), p y) ↔ ∀ᶠ (b : Y) in map (↑e) (𝓝 x), p b

Tactic:
rw [e.map_nhds_eq hx]

Goal state after: