mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Separation.GDelta

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
s t : Set X
inst✝ : NormalSpace X
st_dis : Disjoint s t
t_cl : IsClosed[inst✝¹] t
T : Set (Set X)
T_open : ∀ t ∈ T, IsOpen[inst✝¹] t
T_count : T.Countable
T_int : t = ⋂₀ T
T_nonempty : T.Nonempty
g : ℕ → ↑T
g_surj : Surjective g
g' : ℕ → Set X
g'_open : ∀ (n : ℕ), IsOpen[inst✝¹] (g' n)
clt_sub_g' : ∀ (n : ℕ), t ⊆ g' n
clg'_sub_g : ∀ (n : ℕ), closure[inst✝¹] (g' n) ⊆ ↑(g n)
⊢ ⋂ i, closure[inst✝¹] (g' i) ⊆ ⋂₀ T

Tactic:
]

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
s t : Set X
inst✝ : NormalSpace X
st_dis : Disjoint s t
t_cl : IsClosed[inst✝¹] t
T : Set (Set X)
T_open : ∀ t ∈ T, IsOpen[inst✝¹] t
T_count : T.Countable
T_int : t = ⋂₀ T
T_nonempty : T.Nonempty
g : ℕ → ↑T
g_surj : Surjective g
g' : ℕ → Set X
g'_open : ∀ (n : ℕ), IsOpen[inst✝¹] (g' n)
clt_sub_g' : ∀ (n : ℕ), t ⊆ g' n
clg'_sub_g : ∀ (n : ℕ), closure[inst✝¹] (g' n) ⊆ ↑(g n)
⊢ ⋂ i, closure[inst✝¹] (g' i) ⊆ ⋂₀ T
Module: Mathlib.Topology.GDelta.MetrizableSpace

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : PseudoMetrizableSpace X
s : Set X
hs : IsClosed s
this✝ : UniformSpace X := pseudoMetrizableSpaceUniformity X
this : (uniformity X).IsCountablyGenerated
U : ℕ → SetRel X X
hUo : ∀ (i : ℕ), U i ∈ uniformity X ∧ IsOpen (U i)
hU : (uniformity X).HasBasis (fun x ↦ True) fun i ↦ id (U i)
⊢ IsGδ (⋂ i, ⋂ (_ : True), ⋃ x ∈ s, UniformSpace.ball x (id (U i)))

Tactic:
]

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : PseudoMetrizableSpace X
s : Set X
hs : IsClosed s
this✝ : UniformSpace X := pseudoMetrizableSpaceUniformity X
this : (uniformity X).IsCountablyGenerated
U : ℕ → SetRel X X
hUo : ∀ (i : ℕ), U i ∈ uniformity X ∧ IsOpen (U i)
hU : (uniformity X).HasBasis (fun x ↦ True) fun i ↦ id (U i)
⊢ IsGδ (⋂ i, ⋂ (_ : True), ⋃ x ∈ s, UniformSpace.ball x (id (U i)))
Module: Mathlib.Topology.Separation.GDelta

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
s t✝ : Set X
inst✝ : NormalSpace X
st_dis : Disjoint s t✝
t_cl : IsClosed[inst✝¹] t✝
T : Set (Set X)
T_open : ∀ t ∈ T, IsOpen[inst✝¹] t
T_count : T.Countable
T_int : t✝ = ⋂₀ T
T_nonempty : T.Nonempty
g : ℕ → ↑T
g_surj : Surjective g
g' : ℕ → Set X
g'_open : ∀ (n : ℕ), IsOpen[inst✝¹] (g' n)
clt_sub_g' : ∀ (n : ℕ), t✝ ⊆ g' n
clg'_sub_g : ∀ (n : ℕ), closure[inst✝¹] (g' n) ⊆ ↑(g n)
t : Set X
tinT : t ∈ T
n : ℕ
gn : g n = ⟨t, tinT⟩
⊢ ↑⟨t, tinT⟩ ⊆ t

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Separation.GDelta

Goal state before:
pp: case h.left
X : Type u_1
inst✝¹ : TopologicalSpace X
s t : Set X
inst✝ : NormalSpace X
st_dis : Disjoint s t
t_cl : IsClosed[inst✝¹] t
T : Set (Set X)
T_open : ∀ t ∈ T, IsOpen[inst✝¹] t
T_count : T.Countable
T_int : t = ⋂₀ T
T_nonempty : T.Nonempty
g : ℕ → ↑T
g_surj : Surjective g
g' : ℕ → Set X
g'_open : ∀ (n : ℕ), IsOpen[inst✝¹] (g' n)
clt_sub_g' : ∀ (n : ℕ), t ⊆ g' n
clg'_sub_g : ∀ (n : ℕ), closure[inst✝¹] (g' n) ⊆ ↑(g n)
clg'_int : t = ⋂ i, closure[inst✝¹] (g' i)
⊢ t ⊆ sᶜ

Tactic:
]

Goal state after:
case h.left
X : Type u_1
inst✝¹ : TopologicalSpace X
s t : Set X
inst✝ : NormalSpace X
st_dis : Disjoint s t
t_cl : IsClosed[inst✝¹] t
T : Set (Set X)
T_open : ∀ t ∈ T, IsOpen[inst✝¹] t
T_count : T.Countable
T_int : t = ⋂₀ T
T_nonempty : T.Nonempty
g : ℕ → ↑T
g_surj : Surjective g
g' : ℕ → Set X
g'_open : ∀ (n : ℕ), IsOpen[inst✝¹] (g' n)
clt_sub_g' : ∀ (n : ℕ), t ⊆ g' n
clg'_sub_g : ∀ (n : ℕ), closure[inst✝¹] (g' n) ⊆ ↑(g n)
clg'_int : t = ⋂ i, closure[inst✝¹] (g' i)
⊢ t ⊆ sᶜ
Module: Mathlib.Topology.Separation.GDelta

Goal state before:
pp: case h.right
X : Type u_1
inst✝¹ : TopologicalSpace X
s t : Set X
inst✝ : NormalSpace X
st_dis : Disjoint s t
t_cl : IsClosed[inst✝¹] t
T : Set (Set X)
T_open : ∀ t ∈ T, IsOpen[inst✝¹] t
T_count : T.Countable
T_int : t = ⋂₀ T
T_nonempty : T.Nonempty
g : ℕ → ↑T
g_surj : Surjective g
g' : ℕ → Set X
g'_open : ∀ (n : ℕ), IsOpen[inst✝¹] (g' n)
clt_sub_g' : ∀ (n : ℕ), closure[inst✝¹] t ⊆ g' n
clg'_sub_g : ∀ (n : ℕ), closure[inst✝¹] (g' n) ⊆ ↑(g n)
clg'_int : t = ⋂ i, closure[inst✝¹] (g' i)
n : ℕ
⊢ t ⊆ interior (closure[inst✝¹] (g' n))

Tactic:
]

Goal state after:
case h.right
X : Type u_1
inst✝¹ : TopologicalSpace X
s t : Set X
inst✝ : NormalSpace X
st_dis : Disjoint s t
t_cl : IsClosed[inst✝¹] t
T : Set (Set X)
T_open : ∀ t ∈ T, IsOpen[inst✝¹] t
T_count : T.Countable
T_int : t = ⋂₀ T
T_nonempty : T.Nonempty
g : ℕ → ↑T
g_surj : Surjective g
g' : ℕ → Set X
g'_open : ∀ (n : ℕ), IsOpen[inst✝¹] (g' n)
clt_sub_g' : ∀ (n : ℕ), closure[inst✝¹] t ⊆ g' n
clg'_sub_g : ∀ (n : ℕ), closure[inst✝¹] (g' n) ⊆ ↑(g n)
clg'_int : t = ⋂ i, closure[inst✝¹] (g' i)
n : ℕ
⊢ t ⊆ interior (closure[inst✝¹] (g' n))
Module: Mathlib.Topology.Separation.GDelta

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PerfectlyNormalSpace X
x y : X
hxy : x ⤳ y
⊢ ∀ (s : Set X), IsClosed[inst✝²] s → y ∈ s → x ∈ s

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PerfectlyNormalSpace X
x y : X
hxy : x ⤳ y
⊢ ∀ (s : Set X), IsClosed[inst✝²] s → y ∈ s → x ∈ s
Module: Mathlib.Topology.Separation.GDelta

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PerfectlyNormalSpace X
x y : X
hxy : x ⤳ y
Ts : Set (Set X)
hoTs : ∀ t ∈ Ts, IsOpen[inst✝²] t
hyK : ∀ t ∈ Ts, y ∈ t
⊢ ∀ t ∈ Ts, x ∈ t

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PerfectlyNormalSpace X
x y : X
hxy : x ⤳ y
Ts : Set (Set X)
hoTs : ∀ t ∈ Ts, IsOpen[inst✝²] t
hyK : ∀ t ∈ Ts, y ∈ t
⊢ ∀ t ∈ Ts, x ∈ t
Module: Mathlib.MeasureTheory.Group.Arithmetic

Goal state before:
pp: α : Type u_3
m : MeasurableSpace α
G : Type u_4
inst✝² : InvolutiveInv G
inst✝¹ : MeasurableSpace G
inst✝ : MeasurableInv G
f : α → G
h : Measurable fun x ↦ (f x)⁻¹
⊢ Measurable f

Tactic:
simpa only [inv_inv] using h.fun_inv

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

Goal state before:
pp: α : Type u_3
m : MeasurableSpace α
μ : Measure α
G : Type u_4
inst✝² : InvolutiveInv G
inst✝¹ : MeasurableSpace G
inst✝ : MeasurableInv G
f : α → G
h : AEMeasurable (fun x ↦ (f x)⁻¹) μ
⊢ AEMeasurable f μ

Tactic:
simpa only [inv_inv] using h.fun_inv

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

Goal state before:
pp: α : Type u_1
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u : Set α
hu : TopologicalSpace.IsOpen u
⊢ MeasurableSet u

Tactic:
]

Goal state after:
α : Type u_1
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u : Set α
hu : TopologicalSpace.IsOpen u
⊢ MeasurableSet u
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Basic

Goal state before:
pp: α : Type u_1
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u : Set α
f : Set (Set α)
hf : ∀ s_1 ∈ f, GenerateOpen s s_1
ih : ∀ s_1 ∈ f, MeasurableSet s_1
⊢ ∀ s_1 ∈ f, IsOpen[TopologicalSpace.generateFrom s] s_1

Tactic:
]

Goal state after:
α : Type u_1
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u : Set α
f : Set (Set α)
hf : ∀ s_1 ∈ f, GenerateOpen s s_1
ih : ∀ s_1 ∈ f, MeasurableSet s_1
⊢ ∀ s_1 ∈ f, IsOpen[TopologicalSpace.generateFrom s] s_1
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Basic

Goal state before:
pp: case sUnion
α : Type u_1
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u : Set α
f : Set (Set α)
hf : ∀ s_1 ∈ f, GenerateOpen s s_1
ih : ∀ s_1 ∈ f, MeasurableSet s_1
v : Set (Set α)
hv : v.Countable
vf : v ⊆ f
vu : ⋃₀ v = ⋃₀ f
⊢ MeasurableSet (⋃₀ v)

Tactic:
]

Goal state after:
case sUnion
α : Type u_1
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u : Set α
f : Set (Set α)
hf : ∀ s_1 ∈ f, GenerateOpen s s_1
ih : ∀ s_1 ∈ f, MeasurableSet s_1
v : Set (Set α)
hv : v.Countable
vf : v ⊆ f
vu : ⋃₀ v = ⋃₀ f
⊢ MeasurableSet (⋃₀ v)
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Basic

Goal state before:
pp: α : Type u_1
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u : Set α
hu : u ∈ s
⊢ TopologicalSpace.IsOpen u

Tactic:
]

Goal state after:
α : Type u_1
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u : Set α
hu : u ∈ s
⊢ TopologicalSpace.IsOpen u
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✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
h : OpensMeasurableSpace α
p : α → Prop
⊢ MeasurableSpace.comap val (borel α) ≤ instMeasurableSpace

Tactic:
]

Goal state after:
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✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
h : OpensMeasurableSpace α
p : α → Prop
⊢ MeasurableSpace.comap val (borel α) ≤ instMeasurableSpace
Module: Mathlib.MeasureTheory.Group.Arithmetic

Goal state before:
pp: case cons
M : Type u_2
α : Type u_3
inst✝² : Monoid M
inst✝¹ : MeasurableSpace M
inst✝ : MeasurableMul₂ M
m : MeasurableSpace α
f : α → M
l : List (α → M)
ihl : (∀ f ∈ l, Measurable f) → Measurable l.prod
hl : Measurable f ∧ ∀ x ∈ l, Measurable x
⊢ Measurable (f :: l).prod

Tactic:
]

Goal state after:
case cons
M : Type u_2
α : Type u_3
inst✝² : Monoid M
inst✝¹ : MeasurableSpace M
inst✝ : MeasurableMul₂ M
m : MeasurableSpace α
f : α → M
l : List (α → M)
ihl : (∀ f ∈ l, Measurable f) → Measurable l.prod
hl : Measurable f ∧ ∀ x ∈ l, Measurable x
⊢ Measurable (f :: l).prod
Module: Mathlib.MeasureTheory.Group.Arithmetic

Goal state before:
pp: case cons
M : Type u_2
α : Type u_3
inst✝² : Monoid M
inst✝¹ : MeasurableSpace M
inst✝ : MeasurableMul₂ M
m : MeasurableSpace α
f : α → M
l : List (α → M)
ihl : (∀ f ∈ l, Measurable f) → Measurable l.prod
hl : Measurable f ∧ ∀ x ∈ l, Measurable x
⊢ Measurable (f * l.prod)

Tactic:
]

Goal state after:
case cons
M : Type u_2
α : Type u_3
inst✝² : Monoid M
inst✝¹ : MeasurableSpace M
inst✝ : MeasurableMul₂ M
m : MeasurableSpace α
f : α → M
l : List (α → M)
ihl : (∀ f ∈ l, Measurable f) → Measurable l.prod
hl : Measurable f ∧ ∀ x ∈ l, Measurable x
⊢ Measurable (f * l.prod)
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
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
s : Set α
h : HasCountableSeparatingOn α IsOpen[inst✝¹²] s
⊢ HasCountableSeparatingOn α MeasurableSet s

Tactic:
]

Goal state after:
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
s : Set α
h : HasCountableSeparatingOn α IsOpen[inst✝¹²] s
⊢ HasCountableSeparatingOn α MeasurableSet s
Module: Mathlib.MeasureTheory.Group.Arithmetic

Goal state before:
pp: case cons
M : Type u_2
α : Type u_3
inst✝² : Monoid M
inst✝¹ : MeasurableSpace M
inst✝ : MeasurableMul₂ M
m : MeasurableSpace α
μ : Measure α
f : α → M
l : List (α → M)
ihl : (∀ f ∈ l, AEMeasurable f μ) → AEMeasurable l.prod μ
hl : AEMeasurable f μ ∧ ∀ x ∈ l, AEMeasurable x μ
⊢ AEMeasurable (f :: l).prod μ

Tactic:
]

Goal state after:
case cons
M : Type u_2
α : Type u_3
inst✝² : Monoid M
inst✝¹ : MeasurableSpace M
inst✝ : MeasurableMul₂ M
m : MeasurableSpace α
μ : Measure α
f : α → M
l : List (α → M)
ihl : (∀ f ∈ l, AEMeasurable f μ) → AEMeasurable l.prod μ
hl : AEMeasurable f μ ∧ ∀ x ∈ l, AEMeasurable x μ
⊢ AEMeasurable (f :: l).prod μ
Module: Mathlib.MeasureTheory.Group.Arithmetic

Goal state before:
pp: case cons
M : Type u_2
α : Type u_3
inst✝² : Monoid M
inst✝¹ : MeasurableSpace M
inst✝ : MeasurableMul₂ M
m : MeasurableSpace α
μ : Measure α
f : α → M
l : List (α → M)
ihl : (∀ f ∈ l, AEMeasurable f μ) → AEMeasurable l.prod μ
hl : AEMeasurable f μ ∧ ∀ x ∈ l, AEMeasurable x μ
⊢ AEMeasurable (f * l.prod) μ

Tactic:
]

Goal state after:
case cons
M : Type u_2
α : Type u_3
inst✝² : Monoid M
inst✝¹ : MeasurableSpace M
inst✝ : MeasurableMul₂ M
m : MeasurableSpace α
μ : Measure α
f : α → M
l : List (α → M)
ihl : (∀ f ∈ l, AEMeasurable f μ) → AEMeasurable l.prod μ
hl : AEMeasurable f μ ∧ ∀ x ∈ l, AEMeasurable x μ
⊢ AEMeasurable (f * l.prod) μ
Module: Mathlib.MeasureTheory.Group.Arithmetic

Goal state before:
pp: M : Type u_2
α : Type u_3
inst✝² : Monoid M
inst✝¹ : MeasurableSpace M
inst✝ : MeasurableMul₂ M
m : MeasurableSpace α
μ : Measure α
l : List (α → M)
hl : ∀ f ∈ l, AEMeasurable f μ
⊢ AEMeasurable l.prod μ

Tactic:
by
  induction l with
  | nil => exact aemeasurable_one
  | cons f l ihl =>
    rw [List.forall_mem_cons] at hl
    rw [List.prod_cons]
    exact hl.1.mul (ihl hl.2)

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

Goal state before:
pp: γ : Type u_3
inst✝³ : TopologicalSpace γ
inst✝² : MeasurableSpace γ
inst✝¹ : BorelSpace γ
inst✝ : R1Space γ
K s : Set γ
hK : IsCompact K
hs : MeasurableSet s
hKs : K ⊆ s
⊢ ∀ i ∈ K, {y | Inseparable i y} ⊆ s

Tactic:
]

Goal state after:
γ : Type u_3
inst✝³ : TopologicalSpace γ
inst✝² : MeasurableSpace γ
inst✝¹ : BorelSpace γ
inst✝ : R1Space γ
K s : Set γ
hK : IsCompact K
hs : MeasurableSet s
hKs : K ⊆ s
⊢ ∀ i ∈ K, {y | Inseparable i y} ⊆ s
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Basic

Goal state before:

Tactic:
μ (closure K) ≤ μ (toMeasurable μ K) := measure_mono <|
      hK.closure_subset_measurableSet (measurableSet_toMeasurable ..) (subset_toMeasurable ..)
    _ = μ K := measure_toMeasurable ..

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

Goal state before:
pp: γ : Type u_3
δ : Type u_5
inst✝³ : TopologicalSpace γ
inst✝² : MeasurableSpace γ
inst✝¹ : BorelSpace γ
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsOpen[inst✝³] s → MeasurableSet (f ⁻¹' s)
⊢ Measurable f

Tactic:
]

Goal state after:
γ : Type u_3
δ : Type u_5
inst✝³ : TopologicalSpace γ
inst✝² : MeasurableSpace γ
inst✝¹ : BorelSpace γ
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsOpen[inst✝³] s → MeasurableSet (f ⁻¹' s)
⊢ Measurable f
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Basic

Goal state before:
pp: γ : Type u_3
δ : Type u_5
inst✝³ : TopologicalSpace γ
inst✝² : MeasurableSpace γ
inst✝¹ : BorelSpace γ
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsClosed[inst✝³] s → MeasurableSet (f ⁻¹' s)
s : Set γ
hs : IsOpen[inst✝³] s
⊢ MeasurableSet (f ⁻¹' sᶜ)

Tactic:
]

Goal state after:
γ : Type u_3
δ : Type u_5
inst✝³ : TopologicalSpace γ
inst✝² : MeasurableSpace γ
inst✝¹ : BorelSpace γ
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsClosed[inst✝³] s → MeasurableSet (f ⁻¹' s)
s : Set γ
hs : IsOpen[inst✝³] s
⊢ MeasurableSet (f ⁻¹' sᶜ)
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Basic

Goal state before:
pp: γ : Type u_3
δ : Type u_5
inst✝³ : TopologicalSpace γ
inst✝² : MeasurableSpace γ
inst✝¹ : BorelSpace γ
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsClosed[inst✝³] s → MeasurableSet (f ⁻¹' s)
s : Set γ
hs : IsOpen[inst✝³] s
⊢ IsOpen[inst✝³] s

Tactic:
]

Goal state after:
γ : Type u_3
δ : Type u_5
inst✝³ : TopologicalSpace γ
inst✝² : MeasurableSpace γ
inst✝¹ : BorelSpace γ
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsClosed[inst✝³] s → MeasurableSet (f ⁻¹' s)
s : Set γ
hs : IsOpen[inst✝³] s
⊢ IsOpen[inst✝³] s
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Basic

Goal state before:
pp: γ : Type u_3
δ : Type u_5
inst✝³ : TopologicalSpace γ
inst✝² : MeasurableSpace γ
inst✝¹ : BorelSpace γ
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsClosed[inst✝³] s → MeasurableSet (f ⁻¹' s)
s : Set γ
hs : IsOpen[inst✝³] s
⊢ IsOpen[inst✝³] s

Tactic:
exact hs

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
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
a : α
⊢ (⨅ x, 𝓟 ↑x).IsMeasurablyGenerated

Tactic:
]

Goal state after:
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
a : α
⊢ (⨅ x, 𝓟 ↑x).IsMeasurablyGenerated
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
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : OpensMeasurableSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : OpensMeasurableSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : TopologicalSpace γ₂
inst✝³ : MeasurableSpace γ₂
inst✝² : BorelSpace γ₂
inst✝¹ : MeasurableSpace δ
inst✝ : T0Space α
⊢ ∀ (x y : α), (∀ (s : Set α), MeasurableSet s → (x ∈ s ↔ y ∈ s)) → x = y

Tactic:
]

Goal state after:
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✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : OpensMeasurableSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : OpensMeasurableSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : TopologicalSpace γ₂
inst✝³ : MeasurableSpace γ₂
inst✝² : BorelSpace γ₂
inst✝¹ : MeasurableSpace δ
inst✝ : T0Space α
⊢ ∀ (x y : α), (∀ (s : Set α), MeasurableSet s → (x ∈ s ↔ y ∈ s)) → x = y
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
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : OpensMeasurableSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : OpensMeasurableSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : TopologicalSpace γ₂
inst✝³ : MeasurableSpace γ₂
inst✝² : BorelSpace γ₂
inst✝¹ : MeasurableSpace δ
inst✝ : T0Space α
x y : α
hxy : ∀ (s : Set α), MeasurableSet s → (x ∈ s ↔ y ∈ s)
⊢ ∀ (s : Set α), IsOpen[inst✝¹³] s → (x ∈ s ↔ y ∈ s)

Tactic:
]

Goal state after:
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✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : OpensMeasurableSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : OpensMeasurableSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : TopologicalSpace γ₂
inst✝³ : MeasurableSpace γ₂
inst✝² : BorelSpace γ₂
inst✝¹ : MeasurableSpace δ
inst✝ : T0Space α
x y : α
hxy : ∀ (s : Set α), MeasurableSet s → (x ∈ s ↔ y ∈ s)
⊢ ∀ (s : Set α), IsOpen[inst✝¹³] s → (x ∈ s ↔ y ∈ s)
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
inst✝¹⁶ : TopologicalSpace α
inst✝¹⁵ : MeasurableSpace α
inst✝¹⁴ : OpensMeasurableSpace α
inst✝¹³ : TopologicalSpace β
inst✝¹² : MeasurableSpace β
inst✝¹¹ : OpensMeasurableSpace β
inst✝¹⁰ : TopologicalSpace γ
inst✝⁹ : MeasurableSpace γ
inst✝⁸ : BorelSpace γ
inst✝⁷ : TopologicalSpace γ₂
inst✝⁶ : MeasurableSpace γ₂
inst✝⁵ : BorelSpace γ₂
inst✝⁴ : MeasurableSpace δ
ι : Type u_6
X : ι → Type u_7
inst✝³ : Countable ι
t' : (i : ι) → TopologicalSpace (X i)
inst✝² : (i : ι) → MeasurableSpace (X i)
inst✝¹ : ∀ (i : ι), SecondCountableTopology (X i)
inst✝ : ∀ (i : ι), OpensMeasurableSpace (X i)
this :
  topologicalSpace = TopologicalSpace.generateFrom {t | ∃ s i, (∀ a ∈ i, s a ∈ countableBasis (X a)) ∧ t = (↑i).pi s}
⊢ MeasurableSpace.generateFrom {t | ∃ s i, (∀ a ∈ i, s a ∈ countableBasis (X a)) ∧ t = (↑i).pi s} ≤ MeasurableSpace.pi

Tactic:
]

Goal state after:
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✝¹⁶ : TopologicalSpace α
inst✝¹⁵ : MeasurableSpace α
inst✝¹⁴ : OpensMeasurableSpace α
inst✝¹³ : TopologicalSpace β
inst✝¹² : MeasurableSpace β
inst✝¹¹ : OpensMeasurableSpace β
inst✝¹⁰ : TopologicalSpace γ
inst✝⁹ : MeasurableSpace γ
inst✝⁸ : BorelSpace γ
inst✝⁷ : TopologicalSpace γ₂
inst✝⁶ : MeasurableSpace γ₂
inst✝⁵ : BorelSpace γ₂
inst✝⁴ : MeasurableSpace δ
ι : Type u_6
X : ι → Type u_7
inst✝³ : Countable ι
t' : (i : ι) → TopologicalSpace (X i)
inst✝² : (i : ι) → MeasurableSpace (X i)
inst✝¹ : ∀ (i : ι), SecondCountableTopology (X i)
inst✝ : ∀ (i : ι), OpensMeasurableSpace (X i)
this :
  topologicalSpace = TopologicalSpace.generateFrom {t | ∃ s i, (∀ a ∈ i, s a ∈ countableBasis (X a)) ∧ t = (↑i).pi s}
⊢ MeasurableSpace.generateFrom {t | ∃ s i, (∀ a ∈ i, s a ∈ countableBasis (X a)) ∧ t = (↑i).pi s} ≤ MeasurableSpace.pi
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case refine_1
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
⊢ MeasurableSpace.generateFrom {s | ∃ a, s = Ioi a ∨ s = Iio a} ≤ MeasurableSpace.generateFrom (range Iio)

Tactic:
]

Goal state after:
case refine_1
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
⊢ MeasurableSpace.generateFrom {s | ∃ a, s = Ioi a ∨ s = Iio a} ≤ MeasurableSpace.generateFrom (range Iio)