mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
mβ : MeasurableSpace β
μ✝ ν ν₁ ν₂ : Measure α
s t : Set α
inst✝¹ : TopologicalSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
c : ℝ≥0
x : α
o : Set α
xo : x ∈ o
o_open : IsOpen[inst✝¹] o
μo : μ o < ∞
⊢ ↑c < ∞

Tactic:
simp

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

Goal state before:
pp: case empty
α : Type u_2
ι : Type u_6
m0 : MeasurableSpace α
μ ν : Measure α
t : ι → Set α
⊢ μ.restrict (⋃ i ∈ ∅, t i) = ν.restrict (⋃ i ∈ ∅, t i) ↔ ∀ i ∈ ∅, μ.restrict (t i) = ν.restrict (t i)

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : ∀ s ∈ C, MeasurableSet s
m : ℝ≥0∞ := ⨆ D ∈ {D | D ⊆ C ∧ D.Countable}, μ (⋃₀ D)
hm : ⊥ < m
u : ℕ → ℝ≥0∞
u_mem : ∀ (n : ℕ), u n ∈ Ioo 0 m
u_lim : Tendsto u atTop (𝓝 m)
D : ℕ → Set (Set α)
D_mem : ∀ (n : ℕ), D n ∈ {D | D ⊆ C ∧ D.Countable}
huD : ∀ (n : ℕ), u n < μ (⋃₀ D n)
⊢ ⋃ n, D n ∈ {D | D ⊆ C ∧ D.Countable}

Tactic:
simp

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : ∀ s ∈ C, MeasurableSet s
m : ℝ≥0∞ := ⨆ D ∈ {D | D ⊆ C ∧ D.Countable}, μ (⋃₀ D)
hm : ⊥ < m
u : ℕ → ℝ≥0∞
u_mem : ∀ (n : ℕ), u n ∈ Ioo 0 m
u_lim : Tendsto u atTop (𝓝 m)
D : ℕ → Set (Set α)
D_mem : ∀ (n : ℕ), D n ∈ {D | D ⊆ C ∧ D.Countable}
huD : ∀ (n : ℕ), u n < μ (⋃₀ D n)
⊢ (∀ (i : ℕ), D i ⊆ C) ∧ ∀ (i : ℕ), (D i).Countable
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
β : Type u_5
inst✝³ : TopologicalSpace β
inst✝² : T1Space β
inst✝¹ : SecondCountableTopology β
inst✝ : Nonempty β
f : α → β
m : OuterMeasure β := (OuterMeasure.map f) μ.toOuterMeasure
h : ∀ (b : β), m {b}ᶜ ≠ 0
⊢ ∃ a b, a ≠ b ∧ (∀ s ∈ 𝓝 a, 0 < μ (f ⁻¹' s)) ∧ ∀ t ∈ 𝓝 b, 0 < μ (f ⁻¹' t)

Tactic:
inhabit β

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
β : Type u_5
inst✝³ : TopologicalSpace β
inst✝² : T1Space β
inst✝¹ : SecondCountableTopology β
inst✝ : Nonempty β
f : α → β
m : OuterMeasure β := (OuterMeasure.map f) μ.toOuterMeasure
h : ∀ (b : β), m {b}ᶜ ≠ 0
inhabited_h : Inhabited β
⊢ ∃ a b, a ≠ b ∧ (∀ s ∈ 𝓝 a, 0 < μ (f ⁻¹' s)) ∧ ∀ t ∈ 𝓝 b, 0 < μ (f ⁻¹' t)
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: case empty
α : Type u_5
m₀ : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hμν : ∀ s ∈ C, μ s = ν s
m : MeasurableSpace α
h : m ≤ m₀
hA : m = generateFrom C
hC : IsPiSystem C
h_univ : μ univ = ν univ
s : Set α
this : IsFiniteMeasure ν
⊢ μ ∅ = ν ∅

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : ∀ s ∈ C, MeasurableSet s
m : ℝ≥0∞ := ⨆ D ∈ {D | D ⊆ C ∧ D.Countable}, μ (⋃₀ D)
D : Set (Set α)
D_mem : D ∈ {D | D ⊆ C ∧ D.Countable}
hD : μ (⋃₀ D) = m
s : Set α
hs : s ∈ C
⊢ s ∪ ⋃₀ D = ⋃₀ (D ∪ {s})

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: case h
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ : Measure α
s : Set α
h : IsCompact s
hμ : ∀ x ∈ s, μ.FiniteAtFilter (𝓝 x)
⊢ ∅ ⊇ ∅ ∧ IsOpen[inst✝¹] ∅ ∧ μ ∅ < ∞

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: case ht
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : ∀ s ∈ C, MeasurableSet s
m : ℝ≥0∞ := ⋯
D : Set (Set α)
D_mem : D ∈ {D | D ⊆ C ∧ D.Countable}
hD : μ (⋃₀ D) = m
s : Set α
hs : s ∈ C
⊢ μ (s ∪ ⋃₀ D) ≠ ∞

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ : Measure α
s : Set α
h : IsCompact s
hμ : ∀ x ∈ s, μ.FiniteAtFilter (𝓝[s] x)
⊢ μ ∅ < ∞

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
h✝ : IsEmpty α
x✝ : ℕ
⊢ ∅ ∈ {K | IsOpen[inst✝²] K}

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
h✝ : IsEmpty α
x✝ : ℕ
⊢ μ ∅ < ∞

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
h✝ : Nonempty α
inhabited_h : Inhabited α
S : Set (Set α) := ⋯
T : Set (Set α)
T_count : T.Countable
TS : T ⊆ S
hT : ⋃₀ T = univ
T_ne : T.Nonempty
f : ℕ → Set α
hf : T = range f
n : ℕ
⊢ f n ∈ T

Tactic:
hf

Goal state after:
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
h✝ : Nonempty α
inhabited_h : Inhabited α
S : Set (Set α) := {s | IsOpen[inst✝²] s ∧ μ s < ∞}
T : Set (Set α)
T_count : T.Countable
TS : T ⊆ S
hT : ⋃₀ T = univ
T_ne : T.Nonempty
f : ℕ → Set α
hf : T = range f
n : ℕ
⊢ f n ∈ range f
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
ms : MeasurableSpace α
h : ms ≤ OuterMeasure.caratheodory 0
⊢ OuterMeasure.toMeasure 0 h = 0

Tactic:
simp

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
⊢ ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), MeasurableSet (f i)) → Pairwise (Disjoint on f) → 0 = ∑' (i : ℕ), 0

Tactic:
simp

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁴ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝³ : Semiring R
inst✝² : Module R ℝ≥0∞
inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝ : Module.IsTorsionFree R ℝ≥0∞
⊢ ∀ (r : R) (m : Measure α), ⇑(r • m) = r • ⇑m

Tactic:
simp

Goal state after:
Module: Mathlib.Data.Set.MemPartition

Goal state before:
pp: case zero
α : Type u_1
f : ℕ → Set α
⊢ ⋃₀ memPartition f 0 = univ

Tactic:
simp

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

Goal state before:
pp: α : Type u_2
ι : Type u_6
m0 : MeasurableSpace α
μ : Measure α
inst✝ : Countable ι
s : ι → Set α
p : α → Prop
⊢ (∀ᵐ (x : α) ∂μ.restrict (⋃ i, s i), p x) ↔ ∀ (i : ι), ∀ᵐ (x : α) ∂μ.restrict (s i), p x

Tactic:
simp

Goal state after:
Module: Mathlib.Data.Set.MemPartition

Goal state before:
pp: case zero
α : Type u_1
f : ℕ → Set α
⊢ (memPartition f 0).Finite

Tactic:
simp

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

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
p : α → Prop
⊢ (∀ᵐ (x : α) ∂μ.restrict (s ∪ t), p x) ↔ (∀ᵐ (x : α) ∂μ.restrict s, p x) ∧ ∀ᵐ (x : α) ∂μ.restrict t, p x

Tactic:
simp

Goal state after:
Module: Mathlib.Data.Set.MemPartition

Goal state before:
pp: case succ
α : Type u_1
f : ℕ → Set α
n : ℕ
ih : (memPartition f n).Finite
this : Finite ↑(memPartition f n)
u : Set α
x✝ : u ∈ memPartition f n
⊢ ({a | a = u ∩ f n} ∪ {a | a = u \ f n}).Finite

Tactic:
simp

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ₁ μ₂ : Measure α
h : ∀ (s : Set α), MeasurableSet s → s.Nonempty → μ₁ s ≤ μ₂ s
hs : MeasurableSet ∅
⊢ μ₁ ∅ ≤ μ₂ ∅

Tactic:
simp

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

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : NullMeasurableSet s μ
t : Set α
h : NullMeasurableSet (t ∩ s) μ
⊢ μ (t \ s ∩ s) = 0

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : Countable α
s : Set α
hs : MeasurableSet s
x : α
hx : x ∈ s
⊢ x ∈ measurableAtom x

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : Countable α
s : Set α
hs : MeasurableSet s
this : s = ⋃ y ∈ s, measurableAtom y
x : α
_hx : x ∈ s
⊢ measurableAtom x ∈ ⋃ y, {measurableAtom y}

Tactic:
simp

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

Goal state before:
pp: case compl
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
t : Set ↑s
hs : NullMeasurableSet s μ
t' : Set ↑s
ht✝ : MeasurableSet t'
ht' : NullMeasurableSet (Subtype.val '' t') μ
⊢ NullMeasurableSet (Subtype.val '' t'ᶜ) μ

Tactic:
simp only [← range_sdiff_image Subtype.coe_injective, Subtype.range_coe_subtype, ofPred_mem_eq]

Goal state after:
case compl
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
t : Set ↑s
hs : NullMeasurableSet s μ
t' : Set ↑s
ht✝ : MeasurableSet t'
ht' : NullMeasurableSet (Subtype.val '' t') μ
⊢ NullMeasurableSet (s \ (fun a ↦ ↑a) '' t') μ
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
S : Set (Set α)
hc : S.Countable
hμ : ∀ s ∈ S, μ s < ∞
hU : ⋃₀ S = univ
⊢ μ ∅ < ∞

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
inst✝ : SigmaFinite μ
i : ℕ
s : Set α
hs : MeasurableSet s
h : ∀ (s : Set α), μ s + ν₁ s = μ s + ν₂ s
⊢ μ (s ∩ spanningSets μ i) + (ν₁.restrict (spanningSets μ i)) s =
    μ (s ∩ spanningSets μ i) + (ν₂.restrict (spanningSets μ i)) s

Tactic:
simp [hs, h]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : Countable α
hμ : ∀ (a : α), μ {a} < ∞
h✝ : Nonempty α
f : ℕ → α
hf : Surjective f
⊢ ∀ (i : ℕ), {f i} ∈ univ

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
inst✝ : CountablyGenerated α
s : Set α
hs : MeasurableSet s
⊢ ∅ = ⋃ p, if (fun x ↦ False) p then countablyGeneratedAtom α p else ∅

Tactic:
simp

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

Goal state before:
pp: case e_6
α : Type u_2
β : Type u_3
x✝¹ : MeasurableSpace α
x✝ : MeasurableSpace β
f : α → β
g : β → α
μ : Measure α
hf : Measurable f
hg : MeasurableEmbedding g
hμg : ∀ᵐ (a : α) ∂μ, a ∈ range g
hfg : ∀ (a : β), f (g a) = a
s : Set β
hs : MeasurableSet s
⊢ f ⁻¹' s \ {x | (fun a ↦ a ∈ range g) x}ᶜ = g '' s

Tactic:
simp

Goal state after:
case e_6
α : Type u_2
β : Type u_3
x✝¹ : MeasurableSpace α
x✝ : MeasurableSpace β
f : α → β
g : β → α
μ : Measure α
hf : Measurable f
hg : MeasurableEmbedding g
hμg : ∀ᵐ (a : α) ∂μ, a ∈ range g
hfg : ∀ (a : β), f (g a) = a
s : Set β
hs : MeasurableSet s
⊢ f ⁻¹' s ∩ {x | ∃ y, g y = x} = g '' s
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace β
inst✝¹ : Nonempty β
inst✝ : SeparatesPoints β
f : α → β
hf : Measurable f
h : ∀ (a₁ a₂ : α), f a₁ = f a₂
h' : IsEmpty α
⊢ ∀ (x : α), f x = Classical.ofNonempty

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : MeasurableSpace α
h : CountablySeparated α
s : Set α
⊢ CountablySeparated ↑s

Tactic:
by
  rw [CountablySeparated.subtype_iff]
  exact h.countably_separated.mono (fun s ↦ id) <| subset_univ _

Goal state after:
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: case inr
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : Countable α
inst✝ : SeparatesPoints α
x : α
s : (y : α) → x ≠ y → Set α
hsm : ∀ (y : α) (h : x ≠ y), MeasurableSet (s y h)
hxs : ∀ (y : α) (h : x ≠ y), x ∈ s y h
hys : ∀ (y : α) (h : x ≠ y), y ∉ s y h
y : α
h : x ≠ y
⊢ y ∈ {x} ↔ y ∈ ⋂ b, ⋂ (b_1 : x ≠ b), s b b_1

Tactic:
simp only [mem_singleton_iff, h.symm, false_iff, mem_iInter, not_forall]
      exact ⟨y, h, hys y h⟩

Goal state after:
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
b : Set (Set α)
inst✝ : SeparatesPoints α
hbc : b.Countable
hb : inst✝¹ = generateFrom b
a✝ : α
⊢ (a✝ ∈ univ → ∀ y ∈ univ, (∀ s ∈ b, a✝ ∈ s ↔ y ∈ s) → a✝ = y) ↔ ∀ (y : α), (∀ s ∈ b, a✝ ∈ s ↔ y ∈ s) → a✝ = y

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_1
t : ℕ → Set α
n : ℕ
s : Set α
hs : s ∈ memPartition t n
⊢ s \ t n = s ∩ t n ∨ s \ t n = s \ t n

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_1
t : ℕ → Set α
n : ℕ
s : Set α
hs : s ∈ memPartition t n
⊢ s ∩ t n = s ∩ t n ∨ s ∩ t n = s \ t n

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_1
t : ℕ → Set α
n : ℕ
s u : Set α
hu : u ∈ memPartition t n
ht✝ : MeasurableSet u
⊢ u = ⋃₀ ↑{u}

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
f : α → β
hf : MeasurableEmbedding f
inst✝ : SigmaFinite μ
⊢ ∀ (i : ℕ), f '' spanningSets μ i ∪ (range f)ᶜ ∈ univ

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_1
t : ℕ → Set α
n : ℕ
s : Set α
⊢ ↑∅ ⊆ memPartition t n

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_1
t : ℕ → Set α
n : ℕ
s : Set α
⊢ ∅ = ⋃₀ ↑∅

Tactic:
simp

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

Goal state before:
pp: ι : Type u_1
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
f : α → β
inst✝ : Countable ι
μ : ι → Measure α
h : ∀ (i : ι), AEMeasurable f (μ i)
a✝ : Nontrivial β
⊢ AEMeasurable f (sum μ)

Tactic:
inhabit β

Goal state after:
ι : Type u_1
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
f : α → β
inst✝ : Countable ι
μ : ι → Measure α
h : ∀ (i : ι), AEMeasurable f (μ i)
a✝ : Nontrivial β
inhabited_h : Inhabited β
⊢ AEMeasurable f (sum μ)
Module: Mathlib.MeasureTheory.Order.Lattice

Goal state before:
pp: α : Type u_2
m : MeasurableSpace α
δ : Type u_3
inst✝² : MeasurableSpace δ
inst✝¹ : SemilatticeSup α
inst✝ : MeasurableSup₂ α
f : ℕ → δ → α
n : ℕ
hf : ∀ k ≤ n, Measurable (f k)
x : δ
⊢ ((range (n + 1)).sup' ⋯ fun k ↦ f k x) = (range (n + 1)).sup' ⋯ f x

Tactic:
simp

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

Goal state before:
pp: α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
f : α → β
hc : MeasurableSpace.CountablyGenerated β
h : NullMeasurable f μ
a✝ : Nontrivial β
⊢ AEMeasurable f μ

Tactic:
inhabit β

Goal state after:
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
f : α → β
hc : MeasurableSpace.CountablyGenerated β
h : NullMeasurable f μ
a✝ : Nontrivial β
inhabited_h : Inhabited β
⊢ AEMeasurable f μ
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: case pos
α : Type u_2
ι : Type u_6
x✝ : MeasurableSpace α
μ : Measure α
s : ι → Set α
M : ℕ
hs_meas : ∀ (i : ι), MeasurableSet (s i)
hs : ∀ (y : α), {i | y ∈ s i}.encard ≤ ↑M
t : Set α
ht : MeasurableSet t
F : Finset ι
P : Finset ι → Set α := fun C ↦ (⋂ i ∈ C, s i) ∩ ⋂ i ∈ F \ C, (s i)ᶜ
Cs : Finset (Finset ι) := F.powerset \ {∅}
G : ι → Set (Finset ι) := fun i ↦ {C | C ∈ F.powerset ∧ i ∈ C}
P_meas : ∀ (C : Finset ι), MeasurableSet (P C)
P_cover : ∀ {i : ι}, i ∈ F → s i ⊆ ⋃ C ∈ G i, P C
iUnion_P : ⋃ C ∈ Cs, P C ⊆ ⋃ i, s i
P_subset_s : ∀ {i : ι} {C : Finset ι}, i ∈ C → P C ⊆ s i
mem_C : ∀ {i : ι}, i ∈ F → ∀ {C : Finset ι} {x : α}, x ∈ P C → x ∈ s i → i ∈ C
C_subset_C : ∀ {C₁ C₂ : Finset ι}, C₁ ∈ Cs → ∀ {x : α}, x ∈ P C₁ ∩ P C₂ → C₁ ⊆ C₂
C : Finset ι
hC : C ∈ Cs
hPC : P C = ∅
⊢ #({a ∈ F | a ∈ C}) • (μ.restrict (P C)) t ≤ M • (μ.restrict (P C)) t

pp: case neg
α : Type u_2
ι : Type u_6
x✝ : MeasurableSpace α
μ : Measure α
s : ι → Set α
M : ℕ
hs_meas : ∀ (i : ι), MeasurableSet (s i)
hs : ∀ (y : α), {i | y ∈ s i}.encard ≤ ↑M
t : Set α
ht : MeasurableSet t
F : Finset ι
P : Finset ι → Set α := fun C ↦ (⋂ i ∈ C, s i) ∩ ⋂ i ∈ F \ C, (s i)ᶜ
Cs : Finset (Finset ι) := F.powerset \ {∅}
G : ι → Set (Finset ι) := fun i ↦ {C | C ∈ F.powerset ∧ i ∈ C}
P_meas : ∀ (C : Finset ι), MeasurableSet (P C)
P_cover : ∀ {i : ι}, i ∈ F → s i ⊆ ⋃ C ∈ G i, P C
iUnion_P : ⋃ C ∈ Cs, P C ⊆ ⋃ i, s i
P_subset_s : ∀ {i : ι} {C : Finset ι}, i ∈ C → P C ⊆ s i
mem_C : ∀ {i : ι}, i ∈ F → ∀ {C : Finset ι} {x : α}, x ∈ P C → x ∈ s i → i ∈ C
C_subset_C : ∀ {C₁ C₂ : Finset ι}, C₁ ∈ Cs → ∀ {x : α}, x ∈ P C₁ ∩ P C₂ → C₁ ⊆ C₂
C : Finset ι
hC : C ∈ Cs
hPC : ¬P C = ∅
⊢ #({a ∈ F | a ∈ C}) • (μ.restrict (P C)) t ≤ M • (μ.restrict (P C)) t

Tactic:
· simp [hPC]

Goal state after:
case neg
α : Type u_2
ι : Type u_6
x✝ : MeasurableSpace α
μ : Measure α
s : ι → Set α
M : ℕ
hs_meas : ∀ (i : ι), MeasurableSet (s i)
hs : ∀ (y : α), {i | y ∈ s i}.encard ≤ ↑M
t : Set α
ht : MeasurableSet t
F : Finset ι
P : Finset ι → Set α := fun C ↦ (⋂ i ∈ C, s i) ∩ ⋂ i ∈ F \ C, (s i)ᶜ
Cs : Finset (Finset ι) := F.powerset \ {∅}
G : ι → Set (Finset ι) := fun i ↦ {C | C ∈ F.powerset ∧ i ∈ C}
P_meas : ∀ (C : Finset ι), MeasurableSet (P C)
P_cover : ∀ {i : ι}, i ∈ F → s i ⊆ ⋃ C ∈ G i, P C
iUnion_P : ⋃ C ∈ Cs, P C ⊆ ⋃ i, s i
P_subset_s : ∀ {i : ι} {C : Finset ι}, i ∈ C → P C ⊆ s i
mem_C : ∀ {i : ι}, i ∈ F → ∀ {C : Finset ι} {x : α}, x ∈ P C → x ∈ s i → i ∈ C
C_subset_C : ∀ {C₁ C₂ : Finset ι}, C₁ ∈ Cs → ∀ {x : α}, x ∈ P C₁ ∩ P C₂ → C₁ ⊆ C₂
C : Finset ι
hC : C ∈ Cs
hPC : ¬P C = ∅
⊢ #({a ∈ F | a ∈ C}) • (μ.restrict (P C)) t ≤ M • (μ.restrict (P C)) t
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0✝ : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
m0 : MeasurableSpace α
μ : Measure α
⊢ μ ∅ < ∞

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Basic

Goal state before:
pp: α✝ : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
α : Type u_6
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
inst✝ : SecondCountableTopology α
⊢ CountablyGenerated α

Tactic:
obtain ⟨b, bct, -, hb⟩ := exists_countable_basis α
  refine ⟨⟨b, bct, ?_⟩⟩
  borelize α
  exact hb.borel_eq_generateFrom

Goal state after:
Module: Mathlib.Topology.MetricSpace.HausdorffDistance

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
s : Set α
⊢ 1 ≠ ∞

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Basic

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
h : SecondCountableTopologyEither α β
s : Set (α × β)
hs : IsOpen[instTopologicalSpaceProd] s
hα : SecondCountableTopology α
F : Set α → Set β := fun a ↦ {y | ∃ b, IsOpen[inst✝⁹] b ∧ y ∈ b ∧ a ×ˢ b ⊆ s}
A : ∀ (a : Set α), IsOpen[inst✝⁹] (F a)
⊢ MeasurableSet s

Tactic:
have : s = ⋃ a ∈ countableBasis α, a ×ˢ F a := by
      apply Subset.antisymm
      · rintro ⟨y1, y2⟩ hy
        rcases isOpen_prod_iff.1 hs y1 y2 hy with ⟨u, v, u_open, v_open, yu, yv, huv⟩
        obtain ⟨a, ha, ya, au⟩ : ∃ a ∈ countableBasis α, y1 ∈ a ∧ a ⊆ u :=
          IsTopologicalBasis.exists_subset_of_mem_open (isBasis_countableBasis α) yu u_open
        simp only [mem_iUnion, mem_prod, exists_and_left, exists_prop]
        exact ⟨a, ya, ha, v, v_open, yv, (Set.prod_mono_left au).trans huv⟩
      · rintro ⟨y1, y2⟩ hy
        simp only [mem_iUnion, mem_prod, exists_and_left, exists_prop] at hy
        rcases hy with ⟨a, ya, -, b, -, yb, hb⟩
        exact hb (mem_prod.2 ⟨ya, yb⟩)

Goal state after:
case inl
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
h : SecondCountableTopologyEither α β
s : Set (α × β)
hs : IsOpen[instTopologicalSpaceProd] s
hα : SecondCountableTopology α
F : Set α → Set β := fun a ↦ {y | ∃ b, IsOpen[inst✝⁹] b ∧ y ∈ b ∧ a ×ˢ b ⊆ s}
A : ∀ (a : Set α), IsOpen[inst✝⁹] (F a)
this : s = ⋃ a ∈ countableBasis α, a ×ˢ F a
⊢ MeasurableSet s
Module: Mathlib.Topology.MetricSpace.HausdorffDistance

Goal state before:
pp: case h
α : Type u
inst✝ : PseudoEMetricSpace α
t : Set α
ht : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] t
hs : IsCompact ∅
hst : Disjoint ∅ t
⊢ 0 < 1 ∧ ∀ x ∈ ∅, ∀ y ∈ t, ↑1 < edist x y

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_5
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : OrderTopology α
inst✝² : SecondCountableTopology α
inst✝¹ : DenselyOrdered α
inst✝ : NoMinOrder α
s : Set α
hd : Dense s
⊢ ∀ (x : α), IsBot x → x ∈ s

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Basic

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
inst✝³ : TopologicalSpace α
mα : MeasurableSpace α
inst✝² : BorelSpace α
mβ : TopologicalSpace β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
⊢ MeasurableSpace.comap Prod.fst (borel α) ≤ borel (α × β)

pp: case refine_2
α : Type u_1
β : Type u_2
inst✝³ : TopologicalSpace α
mα : MeasurableSpace α
inst✝² : BorelSpace α
mβ : TopologicalSpace β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
⊢ MeasurableSpace.comap Prod.snd (borel β) ≤ borel (α × β)

Tactic:
· exact comap_le_iff_le_map.mpr continuous_fst.borel_measurable

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
inst✝³ : TopologicalSpace α
mα : MeasurableSpace α
inst✝² : BorelSpace α
mβ : TopologicalSpace β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
⊢ MeasurableSpace.comap Prod.snd (borel β) ≤ borel (α × β)
Module: Mathlib.Topology.MetricSpace.HausdorffDistance

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
s t u : Set α
x : α
xu : x ∈ u
⊢ hausdorffEDist u t + hausdorffEDist t s = hausdorffEDist s t + hausdorffEDist t u

Tactic:
simp [hausdorffEDist_comm, add_comm]

Goal state after: