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

Goal state before:
pp: case pos
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
inst✝¹ : MeasurableSpace δ
f✝ g : α → β
μ✝ ν μ : Measure α
f : α → β
inst✝ : SFinite μ
H : AEMeasurable f μ
⊢ SFinite (map f μ)

pp: case neg
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
inst✝¹ : MeasurableSpace δ
f✝ g : α → β
μ✝ ν μ : Measure α
f : α → β
inst✝ : SFinite μ
H : ¬AEMeasurable f μ
⊢ SFinite (map f μ)

Tactic:
·

Goal state after:
case pos
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
inst✝¹ : MeasurableSpace δ
f✝ g : α → β
μ✝ ν μ : Measure α
f : α → β
inst✝ : SFinite μ
H : AEMeasurable f μ
⊢ SFinite (map f μ)
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : Preorder α
s : Set α
hsc : s.Countable
hst : ∀ (x : α), ∃ y ∈ s, x ≤ y
hdir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s
⊢ μ (⋃ i ∈ s, Iic i) = μ univ

pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : Preorder α
s : Set α
hsc : s.Countable
hst : ∀ (x : α), ∃ y ∈ s, x ≤ y
hdir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s
⊢ DirectedOn ((fun x1 x2 ↦ x1 ⊆ x2) on Iic) s

Tactic:
·

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : Preorder α
s : Set α
hsc : s.Countable
hst : ∀ (x : α), ∃ y ∈ s, x ≤ y
hdir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s
⊢ μ (⋃ i ∈ s, Iic i) = μ univ
Module: Mathlib.MeasureTheory.Measure.AEMeasurable

Goal state before:
pp: case neg
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
inst✝¹ : MeasurableSpace δ
f✝ g : α → β
μ✝ ν μ : Measure α
f : α → β
inst✝ : SFinite μ
H : ¬AEMeasurable f μ
⊢ SFinite (map f μ)

Tactic:
·

Goal state after:
case neg
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
inst✝¹ : MeasurableSpace δ
f✝ g : α → β
μ✝ ν μ : Measure α
f : α → β
inst✝ : SFinite μ
H : ¬AEMeasurable f μ
⊢ SFinite (map f μ)
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : Preorder α
s : Set α
hsc : s.Countable
hst : ∀ (x : α), ∃ y ∈ s, x ≤ y
hdir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s
⊢ DirectedOn ((fun x1 x2 ↦ x1 ⊆ x2) on Iic) s

Tactic:
·

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : Preorder α
s : Set α
hsc : s.Countable
hst : ∀ (x : α), ∃ y ∈ s, x ≤ y
hdir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s
⊢ DirectedOn ((fun x1 x2 ↦ x1 ⊆ x2) on Iic) s
Module: Mathlib.Topology.Separation.GDelta

Goal state before:
pp: case inl
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_open : ∀ t ∈ ∅, IsOpen[inst✝¹] t
T_count : ∅.Countable
T_int : t = ⋂₀ ∅
⊢ HasSeparatingCover s t

pp: case inr
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
⊢ HasSeparatingCover s t

Tactic:
·

Goal state after:
case inl
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_open : ∀ t ∈ ∅, IsOpen[inst✝¹] t
T_count : ∅.Countable
T_int : t = ⋂₀ ∅
⊢ HasSeparatingCover s t
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)
⊢ s ⊆ ⋃ n, (closure[inst✝¹] (g' n))ᶜ

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), 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), IsOpen[inst✝¹] (closure[inst✝¹] (g' n))ᶜ ∧ Disjoint (closure[inst✝¹] (closure[inst✝¹] (g' n))ᶜ) t

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)
⊢ s ⊆ ⋃ n, (closure[inst✝¹] (g' n))ᶜ
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), 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), IsOpen[inst✝¹] (closure[inst✝¹] (g' n))ᶜ ∧ Disjoint (closure[inst✝¹] (closure[inst✝¹] (g' n))ᶜ) t

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), 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), IsOpen[inst✝¹] (closure[inst✝¹] (g' n))ᶜ ∧ Disjoint (closure[inst✝¹] (closure[inst✝¹] (g' n))ᶜ) 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:
by 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:
by simpa only [inv_inv] using h.fun_inv

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

Goal state before:
pp: case ofNat
α : Type u_1
G : Type u
inst✝³ : DivInvMonoid G
inst✝² : MeasurableSpace G
inst✝¹ : MeasurableMul₂ G
inst✝ : MeasurableInv G
n : ℕ
⊢ Measurable fun x ↦ (x, Int.ofNat n).1 ^ (x, Int.ofNat n).2

pp: case negSucc
α : Type u_1
G : Type u
inst✝³ : DivInvMonoid G
inst✝² : MeasurableSpace G
inst✝¹ : MeasurableMul₂ G
inst✝ : MeasurableInv G
n : ℕ
⊢ Measurable fun x ↦ (x, Int.negSucc n).1 ^ (x, Int.negSucc n).2

Tactic:
·

Goal state after:
case ofNat
α : Type u_1
G : Type u
inst✝³ : DivInvMonoid G
inst✝² : MeasurableSpace G
inst✝¹ : MeasurableMul₂ G
inst✝ : MeasurableInv G
n : ℕ
⊢ Measurable fun x ↦ (x, Int.ofNat n).1 ^ (x, Int.ofNat n).2
Module: Mathlib.MeasureTheory.Group.Arithmetic

Goal state before:
pp: case negSucc
α : Type u_1
G : Type u
inst✝³ : DivInvMonoid G
inst✝² : MeasurableSpace G
inst✝¹ : MeasurableMul₂ G
inst✝ : MeasurableInv G
n : ℕ
⊢ Measurable fun x ↦ (x, Int.negSucc n).1 ^ (x, Int.negSucc n).2

Tactic:
·

Goal state after:
case negSucc
α : Type u_1
G : Type u
inst✝³ : DivInvMonoid G
inst✝² : MeasurableSpace G
inst✝¹ : MeasurableMul₂ G
inst✝ : MeasurableInv G
n : ℕ
⊢ Measurable fun x ↦ (x, Int.negSucc n).1 ^ (x, Int.negSucc n).2
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Basic

Goal state before:
pp: case inl
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 → s.Nonempty → s ≠ univ → MeasurableSet (f ⁻¹' s)
s : Set γ
hs : IsClosed[inst✝³] s
h1 : s = ∅
⊢ MeasurableSet (f ⁻¹' s)

pp: case inr
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 → s.Nonempty → s ≠ univ → MeasurableSet (f ⁻¹' s)
s : Set γ
hs : IsClosed[inst✝³] s
h1 : s.Nonempty
⊢ MeasurableSet (f ⁻¹' s)

Tactic:
·

Goal state after:
case inl
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 → s.Nonempty → s ≠ univ → MeasurableSet (f ⁻¹' s)
s : Set γ
hs : IsClosed[inst✝³] s
h1 : s = ∅
⊢ MeasurableSet (f ⁻¹' s)
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Basic

Goal state before:
pp: case pos
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 → s.Nonempty → s ≠ univ → MeasurableSet (f ⁻¹' s)
s : Set γ
hs : IsClosed[inst✝³] s
h1 : s.Nonempty
h2 : s = univ
⊢ MeasurableSet (f ⁻¹' s)

pp: case neg
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 → s.Nonempty → s ≠ univ → MeasurableSet (f ⁻¹' s)
s : Set γ
hs : IsClosed[inst✝³] s
h1 : s.Nonempty
h2 : ¬s = univ
⊢ MeasurableSet (f ⁻¹' s)

Tactic:
·

Goal state after:
case pos
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 → s.Nonempty → s ≠ univ → MeasurableSet (f ⁻¹' s)
s : Set γ
hs : IsClosed[inst✝³] s
h1 : s.Nonempty
h2 : s = univ
⊢ MeasurableSet (f ⁻¹' s)
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
⊢ borel α ≤ MeasurableSpace.generateFrom (range Iio)

pp: case refine_2
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
⊢ ∀ t ∈ range Iio, MeasurableSet t

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
⊢ borel α ≤ MeasurableSpace.generateFrom (range Iio)
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case refine_1.inl
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
a : α
⊢ MeasurableSet (Ioi a)

pp: case refine_1.inr
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
a : α
⊢ MeasurableSet (Iio a)

Tactic:
·

Goal state after:
case refine_1.inl
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
a : α
⊢ MeasurableSet (Ioi a)
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case refine_1.inl.inl
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
a b : α
hb : a ⋖ b
⊢ MeasurableSet (Ioi a)

pp: case refine_1.inl.inr
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
a : α
hcovBy : ¬∃ b, a ⋖ b
⊢ MeasurableSet (Ioi a)

Tactic:
·

Goal state after:
case refine_1.inl.inl
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
a b : α
hb : a ⋖ b
⊢ MeasurableSet (Ioi a)
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✝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 : Subsingleton ι
inst✝² : (i : ι) → TopologicalSpace (X i)
inst✝¹ : (i : ι) → MeasurableSpace (X i)
inst✝ : ∀ (i : ι), OpensMeasurableSpace (X i)
h : IsEmpty ι
⊢ borel ((i : ι) → X i) ≤ MeasurableSpace.pi

pp: case inr
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 : Subsingleton ι
inst✝² : (i : ι) → TopologicalSpace (X i)
inst✝¹ : (i : ι) → MeasurableSpace (X i)
inst✝ : ∀ (i : ι), OpensMeasurableSpace (X i)
h : Nonempty ι
⊢ borel ((i : ι) → X i) ≤ MeasurableSpace.pi

Tactic:
·

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✝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 : Subsingleton ι
inst✝² : (i : ι) → TopologicalSpace (X i)
inst✝¹ : (i : ι) → MeasurableSpace (X i)
inst✝ : ∀ (i : ι), OpensMeasurableSpace (X i)
h : IsEmpty ι
⊢ borel ((i : ι) → X i) ≤ MeasurableSpace.pi
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case refine_1.inl.inr
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
a : α
hcovBy : ¬∃ b, a ⋖ b
⊢ MeasurableSet (Ioi a)

Tactic:
·

Goal state after:
case refine_1.inl.inr
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
a : α
hcovBy : ¬∃ b, a ⋖ b
⊢ MeasurableSet (Ioi a)
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case refine_1.inr
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
a : α
⊢ MeasurableSet (Iio a)

Tactic:
·

Goal state after:
case refine_1.inr
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
a : α
⊢ MeasurableSet (Iio a)
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case refine_2
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
⊢ ∀ t ∈ range Iio, MeasurableSet t

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
⊢ ∀ t ∈ range Iio, MeasurableSet t
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 (range Ioi) ≤ MeasurableSpace.generateFrom (range Iic)

pp: case refine_2
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
⊢ MeasurableSpace.generateFrom (range Iic) ≤ MeasurableSpace.generateFrom (range Ioi)

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 (range Ioi) ≤ MeasurableSpace.generateFrom (range Iic)
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case refine_2
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
⊢ MeasurableSpace.generateFrom (range Iic) ≤ MeasurableSpace.generateFrom (range Ioi)

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
⊢ MeasurableSpace.generateFrom (range Iic) ≤ MeasurableSpace.generateFrom (range Ioi)
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
⊢ MeasurableSet s

pp: case inr
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
hβ : SecondCountableTopology β
⊢ MeasurableSet s

Tactic:
·

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
⊢ MeasurableSet s
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Basic

Goal state before:
pp: case h₁
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 := ⋯
A : ∀ (a : Set α), IsOpen[inst✝⁹] (F a)
⊢ s ⊆ ⋃ a ∈ countableBasis α, a ×ˢ F a

pp: case h₂
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 := ⋯
A : ∀ (a : Set α), IsOpen[inst✝⁹] (F a)
⊢ ⋃ a ∈ countableBasis α, a ×ˢ F a ⊆ s

Tactic:
·

Goal state after:
case h₁
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 := ⋯
A : ∀ (a : Set α), IsOpen[inst✝⁹] (F a)
⊢ s ⊆ ⋃ a ∈ countableBasis α, a ×ˢ F a