mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Countable

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
c : ℝ≥0
s : Set α
hs : MeasurableSet s
L : ℝ≥0∞
hL : L < ↑c * μ s
c_ne_zero : c ≠ 0
⊢ L < ↑c * μ s

pp: case h0
α : Type u_1
m : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
c : ℝ≥0
s : Set α
hs : MeasurableSet s
L : ℝ≥0∞
hL : L < ↑c * μ s
c_ne_zero : c ≠ 0
⊢ ↑c ≠ 0 ∨ L ≠ 0

pp: case ht
α : Type u_1
m : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
c : ℝ≥0
s : Set α
hs : MeasurableSet s
L : ℝ≥0∞
hL : L < ↑c * μ s
c_ne_zero : c ≠ 0
⊢ ↑c ≠ ∞ ∨ L ≠ ∞

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
c : ℝ≥0
s : Set α
hs : MeasurableSet s
L : ℝ≥0∞
hL : L < ↑c * μ s
c_ne_zero : c ≠ 0
⊢ L < ↑c * μ s

case h0
α : Type u_1
m : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
c : ℝ≥0
s : Set α
hs : MeasurableSet s
L : ℝ≥0∞
hL : L < ↑c * μ s
c_ne_zero : c ≠ 0
⊢ ↑c ≠ 0 ∨ L ≠ 0

case ht
α : Type u_1
m : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
c : ℝ≥0
s : Set α
hs : MeasurableSet s
L : ℝ≥0∞
hL : L < ↑c * μ s
c_ne_zero : c ≠ 0
⊢ ↑c ≠ ∞ ∨ L ≠ ∞
Module: Mathlib.MeasureTheory.Measure.GiryMonad

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_3
inst✝ : MeasurableSpace γ
m : Measure α
f : α → Measure β
g : β → Measure γ
hf : AEMeasurable f m
hg : AEMeasurable g (m.bind f)
s : Set γ
hs : MeasurableSet s
⊢ ∫⁻ (a : α), ∫⁻ (x : β), (g x) s ∂f a ∂m = ∫⁻ (a : α), ((f a).bind g) s ∂m

pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_3
inst✝ : MeasurableSpace γ
m : Measure α
f : α → Measure β
g : β → Measure γ
hf : AEMeasurable f m
hg : AEMeasurable g (m.bind f)
s : Set γ
hs : MeasurableSet s
⊢ AEMeasurable (fun a ↦ (f a).bind g) m

pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_3
inst✝ : MeasurableSpace γ
m : Measure α
f : α → Measure β
g : β → Measure γ
hf : AEMeasurable f m
hg : AEMeasurable g (m.bind f)
s : Set γ
hs : MeasurableSet s
⊢ AEMeasurable (fun a ↦ (g a) s) (m.bind f)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_3
inst✝ : MeasurableSpace γ
m : Measure α
f : α → Measure β
g : β → Measure γ
hf : AEMeasurable f m
hg : AEMeasurable g (m.bind f)
s : Set γ
hs : MeasurableSet s
⊢ ∫⁻ (a : α), ∫⁻ (x : β), (g x) s ∂f a ∂m = ∫⁻ (a : α), ((f a).bind g) s ∂m

α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_3
inst✝ : MeasurableSpace γ
m : Measure α
f : α → Measure β
g : β → Measure γ
hf : AEMeasurable f m
hg : AEMeasurable g (m.bind f)
s : Set γ
hs : MeasurableSet s
⊢ AEMeasurable (fun a ↦ (f a).bind g) m

α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_3
inst✝ : MeasurableSpace γ
m : Measure α
f : α → Measure β
g : β → Measure γ
hf : AEMeasurable f m
hg : AEMeasurable g (m.bind f)
s : Set γ
hs : MeasurableSet s
⊢ AEMeasurable (fun a ↦ (g a) s) (m.bind f)
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Countable

Goal state before:
pp: case const.refine_3
α : Type u_1
m : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
c : ℝ≥0
s : Set α
hs : MeasurableSet s
L : ℝ≥0∞
hL : L < ↑c * μ s
c_ne_zero : c ≠ 0
this : L / ↑c < μ s
t : Set α
ht : MeasurableSet t
ts : t ⊆ s
mlt : L / ↑c < μ t
t_top : μ t < ∞
⊢ L / ↑c < μ t

pp: case const.refine_3.h0
α : Type u_1
m : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
c : ℝ≥0
s : Set α
hs : MeasurableSet s
L : ℝ≥0∞
hL : L < ↑c * μ s
c_ne_zero : c ≠ 0
this : L / ↑c < μ s
t : Set α
ht : MeasurableSet t
ts : t ⊆ s
mlt : L / ↑c < μ t
t_top : μ t < ∞
⊢ ↑c ≠ 0 ∨ L ≠ 0

pp: case const.refine_3.ht
α : Type u_1
m : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
c : ℝ≥0
s : Set α
hs : MeasurableSet s
L : ℝ≥0∞
hL : L < ↑c * μ s
c_ne_zero : c ≠ 0
this : L / ↑c < μ s
t : Set α
ht : MeasurableSet t
ts : t ⊆ s
mlt : L / ↑c < μ t
t_top : μ t < ∞
⊢ ↑c ≠ ∞ ∨ L ≠ ∞

Tactic:
]

Goal state after:
case const.refine_3
α : Type u_1
m : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
c : ℝ≥0
s : Set α
hs : MeasurableSet s
L : ℝ≥0∞
hL : L < ↑c * μ s
c_ne_zero : c ≠ 0
this : L / ↑c < μ s
t : Set α
ht : MeasurableSet t
ts : t ⊆ s
mlt : L / ↑c < μ t
t_top : μ t < ∞
⊢ L / ↑c < μ t

case const.refine_3.h0
α : Type u_1
m : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
c : ℝ≥0
s : Set α
hs : MeasurableSet s
L : ℝ≥0∞
hL : L < ↑c * μ s
c_ne_zero : c ≠ 0
this : L / ↑c < μ s
t : Set α
ht : MeasurableSet t
ts : t ⊆ s
mlt : L / ↑c < μ t
t_top : μ t < ∞
⊢ ↑c ≠ 0 ∨ L ≠ 0

case const.refine_3.ht
α : Type u_1
m : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
c : ℝ≥0
s : Set α
hs : MeasurableSet s
L : ℝ≥0∞
hL : L < ↑c * μ s
c_ne_zero : c ≠ 0
this : L / ↑c < μ s
t : Set α
ht : MeasurableSet t
ts : t ⊆ s
mlt : L / ↑c < μ t
t_top : μ t < ∞
⊢ ↑c ≠ ∞ ∨ L ≠ ∞
Module: Mathlib.MeasureTheory.Measure.GiryMonad

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
m : Measure α
f : α → β
hf : Measurable f
⊢ (map (fun x ↦ dirac (f x)) m).join = (map (fun x ↦ dirac (f x)) m).join

pp: case hg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
m : Measure α
f : α → β
hf : Measurable f
⊢ Measurable dirac

pp: case hf
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
m : Measure α
f : α → β
hf : Measurable f
⊢ Measurable f

Tactic:
]

Goal state after:
case hg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
m : Measure α
f : α → β
hf : Measurable f
⊢ Measurable dirac

case hf
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
m : Measure α
f : α → β
hf : Measurable f
⊢ Measurable f
Module: Mathlib.MeasureTheory.Measure.GiryMonad

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure (Measure α)
⊢ μ.join = μ.join

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Countable

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
f₁ f₂ : α →ₛ ℝ≥0
a✝ : Disjoint (Function.support ⇑f₁) (Function.support ⇑f₂)
h₁ :
  ∀ {L : ℝ≥0∞},
    L < ∫⁻ (x : α), ↑(f₁ x) ∂μ → ∃ g, (∀ (x : α), g x ≤ f₁ x) ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ∞ ∧ L < ∫⁻ (x : α), ↑(g x) ∂μ
h₂ :
  ∀ {L : ℝ≥0∞},
    L < ∫⁻ (x : α), ↑(f₂ x) ∂μ → ∃ g, (∀ (x : α), g x ≤ f₂ x) ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ∞ ∧ L < ∫⁻ (x : α), ↑(g x) ∂μ
L : ℝ≥0∞
hL : L < ∫⁻ (x : α), ↑((f₁ + f₂) x) ∂μ
⊢ L < ∫⁻ (a : α), ↑(f₁ a) + ↑(f₂ a) ∂μ

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
f₁ f₂ : α →ₛ ℝ≥0
a✝ : Disjoint (Function.support ⇑f₁) (Function.support ⇑f₂)
h₁ :
  ∀ {L : ℝ≥0∞},
    L < ∫⁻ (x : α), ↑(f₁ x) ∂μ → ∃ g, (∀ (x : α), g x ≤ f₁ x) ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ∞ ∧ L < ∫⁻ (x : α), ↑(g x) ∂μ
h₂ :
  ∀ {L : ℝ≥0∞},
    L < ∫⁻ (x : α), ↑(f₂ x) ∂μ → ∃ g, (∀ (x : α), g x ≤ f₂ x) ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ∞ ∧ L < ∫⁻ (x : α), ↑(g x) ∂μ
L : ℝ≥0∞
hL : L < ∫⁻ (x : α), ↑((f₁ + f₂) x) ∂μ
⊢ L < ∫⁻ (a : α), ↑(f₁ a) + ↑(f₂ a) ∂μ
Module: Mathlib.MeasureTheory.Measure.GiryMonad

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
hf : Measurable f
μ : Measure (Measure α)
s : Set β
hs : MeasurableSet s
⊢ ∫⁻ (a : Measure α), (map f a) s ∂μ = ∫⁻ (μ : Measure α), μ (f ⁻¹' s) ∂μ

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
hf : Measurable f
μ : Measure (Measure α)
s : Set β
hs : MeasurableSet s
⊢ ∫⁻ (a : Measure α), (map f a) s ∂μ = ∫⁻ (μ : Measure α), μ (f ⁻¹' s) ∂μ
Module: Mathlib.MeasureTheory.Measure.GiryMonad

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure (Measure (Measure α))
⊢ μ.bind join = μ.bind fun a ↦ (id a).bind id

Tactic:
]

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure (Measure (Measure α))
⊢ μ.bind join = μ.bind fun a ↦ (id a).bind id
Module: Mathlib.MeasureTheory.Measure.OpenPos

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
m : MeasurableSpace X
μ : Measure X
inst✝ : μ.IsOpenPosMeasure
U : Set X
hU : IsOpen[inst✝¹] U
⊢ U = ∅ ↔ U = ∅

Tactic:
]

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
m : MeasurableSpace X
μ : Measure X
inst✝ : μ.IsOpenPosMeasure
F : Set X
hF : IsClosed[inst✝¹] F
h : F = univ
⊢ univ =ᵐ[μ] univ

Tactic:
]

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
m : MeasurableSpace X
μ : Measure X
inst✝ : μ.IsOpenPosMeasure
F : Set X
hF : IsClosed[inst✝¹] F
h : F = univ
⊢ F = univ

Tactic:
]

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
m : MeasurableSpace X
μ : Measure X
inst✝ : μ.IsOpenPosMeasure
F : Set X
hF : IsClosed[inst✝¹] F
h : F = univ
⊢ F = univ
Module: Mathlib.MeasureTheory.Measure.OpenPos

Goal state before:
pp: X : Type u_1
inst✝³ : TopologicalSpace X
m : MeasurableSpace X
μ : Measure X
inst✝² : μ.IsOpenPosMeasure
F : Set X
inst✝¹ : OpensMeasurableSpace X
inst✝ : IsFiniteMeasure μ
hF : IsClosed[inst✝³] F
⊢ F = univ ↔ F = univ

Tactic:
]

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

Goal state before:
pp: X : Type u_1
inst✝³ : TopologicalSpace X
m : MeasurableSpace X
μ : Measure X
inst✝² : μ.IsOpenPosMeasure
F : Set X
inst✝¹ : OpensMeasurableSpace X
inst✝ : IsProbabilityMeasure μ
hF : IsClosed[inst✝³] F
⊢ F = univ ↔ F = univ

Tactic:
]

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
m : MeasurableSpace X
μ : Measure X
inst✝ : μ.IsOpenPosMeasure
p : X → Prop
hp : ∀ᵐ (x : X) ∂μ, p x
⊢ interior {x | p x}ᶜ = ∅

Tactic:
]

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
m : MeasurableSpace X
μ : Measure X
inst✝ : μ.IsOpenPosMeasure
p : X → Prop
hp : ∀ᵐ (x : X) ∂μ, p x
⊢ interior {x | p x}ᶜ = ∅
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: E : ℕ → Type u_1
ι : Type u_2
inst✝¹ : Encodable ι
F : ι → Type u_3
inst✝ : (i : ι) → PseudoMetricSpace (F i)
x✝ y✝ : (i : ι) → F i
i : ι
x y : (i : ι) → F i
⊢ 0 ≤ ∑' (i : ι), min (2⁻¹ ^ encode i) (dist (x i) (y i))

Tactic:
]

Goal state after:
E : ℕ → Type u_1
ι : Type u_2
inst✝¹ : Encodable ι
F : ι → Type u_3
inst✝ : (i : ι) → PseudoMetricSpace (F i)
x✝ y✝ : (i : ι) → F i
i : ι
x y : (i : ι) → F i
⊢ 0 ≤ ∑' (i : ι), min (2⁻¹ ^ encode i) (dist (x i) (y i))
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: E : ℕ → Type u_1
ι : Type u_2
inst✝¹ : Encodable ι
F : ι → Type u_3
inst✝ : (i : ι) → PseudoMetricSpace (F i)
x✝ y✝ : (i : ι) → F i
i : ι
x y : (i : ι) → F i
⊢ ∑' (i : ι), min (2⁻¹ ^ encode i) (edist (x i) (y i)) =
    ∑' (n : ι), ENNReal.ofReal (min (2⁻¹ ^ encode n) (dist (x n) (y n)))

Tactic:
]

Goal state after:
E : ℕ → Type u_1
ι : Type u_2
inst✝¹ : Encodable ι
F : ι → Type u_3
inst✝ : (i : ι) → PseudoMetricSpace (F i)
x✝ y✝ : (i : ι) → F i
i : ι
x y : (i : ι) → F i
⊢ ∑' (i : ι), min (2⁻¹ ^ encode i) (edist (x i) (y i)) =
    ∑' (n : ι), ENNReal.ofReal (min (2⁻¹ ^ encode n) (dist (x n) (y n)))
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: ι : Type u_2
X : Type u_3
Y : ι → Type u_4
f : (i : ι) → X → Y i
inst✝² : Encodable ι
inst✝¹ : (i : ι) → PseudoMetricSpace (Y i)
inst✝ : TopologicalSpace X
continuous_f : ∀ (i : ι), Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f i)
⊢ Continuous[instTopologicalSpaceProd, _] fun x ↦ dist { ofPiNat := x.1 } { ofPiNat := x.2 }

Tactic:
]

Goal state after:
ι : Type u_2
X : Type u_3
Y : ι → Type u_4
f : (i : ι) → X → Y i
inst✝² : Encodable ι
inst✝¹ : (i : ι) → PseudoMetricSpace (Y i)
inst✝ : TopologicalSpace X
continuous_f : ∀ (i : ι), Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f i)
⊢ Continuous[instTopologicalSpaceProd, _] fun x ↦ dist { ofPiNat := x.1 } { ofPiNat := x.2 }
Module: Mathlib.MeasureTheory.Measure.OpenPos

Goal state before:
pp: X : Type u_1
inst✝⁵ : TopologicalSpace X
m : MeasurableSpace X
μ : Measure X
inst✝⁴ : μ.IsOpenPosMeasure
inst✝³ : OpensMeasurableSpace X
Z : Type u_3
inst✝² : TopologicalSpace Z
inst✝¹ : MeasurableSpace Z
inst✝ : BorelSpace Z
f : X → Z
hf : Continuous[inst✝⁵, inst✝²] f
hf_surj : Surjective f
U : Set Z
hUo : IsOpen[inst✝²] U
hUne : U.Nonempty
⊢ μ (f ⁻¹' U) ≠ 0

Tactic:
]

Goal state after:
X : Type u_1
inst✝⁵ : TopologicalSpace X
m : MeasurableSpace X
μ : Measure X
inst✝⁴ : μ.IsOpenPosMeasure
inst✝³ : OpensMeasurableSpace X
Z : Type u_3
inst✝² : TopologicalSpace Z
inst✝¹ : MeasurableSpace Z
inst✝ : BorelSpace Z
f : X → Z
hf : Continuous[inst✝⁵, inst✝²] f
hf_surj : Surjective f
U : Set Z
hUo : IsOpen[inst✝²] U
hUne : U.Nonempty
⊢ μ (f ⁻¹' U) ≠ 0
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: case e'_3
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : T2Space α
inst✝¹ : MeasurableSpace α
inst✝ : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous[_, inst✝³] f
hg : Continuous[_, inst✝³] g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
I :
  ∀ (n : ℕ) (x y : ℕ → ℕ),
    ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →
      ∃ x' y',
        x' ∈ cylinder x n ∧
          y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))
A : Type := { p // ¬MeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }
F : A → A
hFn : ∀ (p : A), (↑(F p)).1 = (↑p).1 + 1
hFx : ∀ (p : A), (↑(F p)).2.1 ∈ cylinder (↑p).2.1 (↑p).1
hFy : ∀ (p : A), (↑(F p)).2.2 ∈ cylinder (↑p).2.2 (↑p).1
p0 : A := ⟨(0, fun x ↦ 0, fun x ↦ 0), ⋯⟩
p : ℕ → A := fun n ↦ F^[n] p0
prec : ∀ (n : ℕ), p (n + 1) = F (p n)
pn_fst : ∀ (n : ℕ), (↑(p n)).1 = n
Ix : ∀ (m n : ℕ), m + 1 ≤ n → (↑(p n)).2.1 m = (↑(p (m + 1))).2.1 m
Iy : ∀ (m n : ℕ), m + 1 ≤ n → (↑(p n)).2.2 m = (↑(p (m + 1))).2.2 m
x : ℕ → ℕ := fun n ↦ (↑(p (n + 1))).2.1 n
hx : x = fun n ↦ (↑(p (n + 1))).2.1 n
y : ℕ → ℕ := fun n ↦ (↑(p (n + 1))).2.2 n
hy : y = fun n ↦ (↑(p (n + 1))).2.2 n
n : ℕ
⊢ ∀ i < n, x i = (↑(p n)).2.1 i

Tactic:
]

Goal state after:
case e'_3
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : T2Space α
inst✝¹ : MeasurableSpace α
inst✝ : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous[_, inst✝³] f
hg : Continuous[_, inst✝³] g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
I :
  ∀ (n : ℕ) (x y : ℕ → ℕ),
    ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →
      ∃ x' y',
        x' ∈ cylinder x n ∧
          y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))
A : Type := { p // ¬MeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }
F : A → A
hFn : ∀ (p : A), (↑(F p)).1 = (↑p).1 + 1
hFx : ∀ (p : A), (↑(F p)).2.1 ∈ cylinder (↑p).2.1 (↑p).1
hFy : ∀ (p : A), (↑(F p)).2.2 ∈ cylinder (↑p).2.2 (↑p).1
p0 : A := ⟨(0, fun x ↦ 0, fun x ↦ 0), ⋯⟩
p : ℕ → A := fun n ↦ F^[n] p0
prec : ∀ (n : ℕ), p (n + 1) = F (p n)
pn_fst : ∀ (n : ℕ), (↑(p n)).1 = n
Ix : ∀ (m n : ℕ), m + 1 ≤ n → (↑(p n)).2.1 m = (↑(p (m + 1))).2.1 m
Iy : ∀ (m n : ℕ), m + 1 ≤ n → (↑(p n)).2.2 m = (↑(p (m + 1))).2.2 m
x : ℕ → ℕ := fun n ↦ (↑(p (n + 1))).2.1 n
hx : x = fun n ↦ (↑(p (n + 1))).2.1 n
y : ℕ → ℕ := fun n ↦ (↑(p (n + 1))).2.2 n
hy : y = fun n ↦ (↑(p (n + 1))).2.2 n
n : ℕ
⊢ ∀ i < n, x i = (↑(p n)).2.1 i
Module: Mathlib.MeasureTheory.Measure.OpenPos

Goal state before:
pp: X : Type u_1
inst✝⁴ : TopologicalSpace X
m : MeasurableSpace X
inst✝³ : BorelSpace X
Z : Type u_3
inst✝² : TopologicalSpace Z
mZ : MeasurableSpace Z
inst✝¹ : BorelSpace Z
μ : Measure Z
inst✝ : μ.IsOpenPosMeasure
f : X → Z
hf : IsOpenEmbedding f
U : Set X
hU : IsOpen[inst✝⁴] U
Une : U.Nonempty
⊢ μ (f '' U) ≠ 0

Tactic:
]

Goal state after:
X : Type u_1
inst✝⁴ : TopologicalSpace X
m : MeasurableSpace X
inst✝³ : BorelSpace X
Z : Type u_3
inst✝² : TopologicalSpace Z
mZ : MeasurableSpace Z
inst✝¹ : BorelSpace Z
μ : Measure Z
inst✝ : μ.IsOpenPosMeasure
f : X → Z
hf : IsOpenEmbedding f
U : Set X
hU : IsOpen[inst✝⁴] U
Une : U.Nonempty
⊢ μ (f '' U) ≠ 0
Module: Mathlib.MeasureTheory.Measure.OpenPos

Goal state before:
pp: X : Type u_2
inst✝² : MetricSpace X
m : MeasurableSpace X
μ : Measure X
inst✝¹ : μ.IsOpenPosMeasure
inst✝ : NullSingletonClass μ
x : X
r : ℝ
h : r ≤ 0
⊢ 0 ≤ 0

Tactic:
]

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

Goal state before:
pp: ι : Type u_2
X : Type u_3
Y : ι → Type u_4
f : (i : ι) → X → Y i
inst✝³ : Encodable ι
inst✝² : (i : ι) → MetricSpace (Y i)
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
continuous_f : ∀ (i : ι), Continuous[inst✝¹, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f i)
separating_f : Pairwise fun x y ↦ ∃ i, f i x ≠ f i y
this : EMetricSpace (PiNatEmbed X Y f) := emetricSpace separating_f
⊢ Continuous[inst✝¹, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] toPiNat ∧ Bijective toPiNat

Tactic:
]

Goal state after:
ι : Type u_2
X : Type u_3
Y : ι → Type u_4
f : (i : ι) → X → Y i
inst✝³ : Encodable ι
inst✝² : (i : ι) → MetricSpace (Y i)
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
continuous_f : ∀ (i : ι), Continuous[inst✝¹, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f i)
separating_f : Pairwise fun x y ↦ ∃ i, f i x ≠ f i y
this : EMetricSpace (PiNatEmbed X Y f) := emetricSpace separating_f
⊢ Continuous[inst✝¹, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] toPiNat ∧ Bijective toPiNat
Module: Mathlib.MeasureTheory.Measure.OpenPos

Goal state before:
pp: X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : MeasurableSpace X
s : Set X
μ : Measure X
inst✝¹ : μ.IsOpenPosMeasure
inst✝ : T2Space X
h₂s : μ s = 0
K : ℕ → Set X
hcompact : ∀ (n : ℕ), IsCompact (K n)
hcover : ⋃ n, K n = s
h : ∀ (n : ℕ), IsNowhereDense (K n)
⊢ ∃ S, (∀ t ∈ S, IsNowhereDense t) ∧ S.Countable ∧ s ⊆ ⋃₀ S

Tactic:
]

Goal state after:
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : MeasurableSpace X
s : Set X
μ : Measure X
inst✝¹ : μ.IsOpenPosMeasure
inst✝ : T2Space X
h₂s : μ s = 0
K : ℕ → Set X
hcompact : ∀ (n : ℕ), IsCompact (K n)
hcover : ⋃ n, K n = s
h : ∀ (n : ℕ), IsNowhereDense (K n)
⊢ ∃ S, (∀ t ∈ S, IsNowhereDense t) ∧ S.Countable ∧ s ⊆ ⋃₀ S
Module: Mathlib.MeasureTheory.Measure.Doubling

Goal state before:
pp: α : Type u_1
inst✝² : PseudoMetricSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsUnifLocDoublingMeasure μ
K : ℝ
C : ℝ≥0 := doublingConstant μ
this : ∀ (n : ℕ), ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), μ (closedBall x (2 ^ n * ε)) ≤ ↑C ^ n * μ (closedBall x ε)
n : ℕ
hn : K < 2 ^ n
ε : ℝ
hε₀ : 0 < ε
hε : ∀ (x : α), μ (closedBall x (2 ^ n * ε)) ≤ ↑C ^ n * μ (closedBall x ε)
x : α
t : ℝ
ht : t ≤ K
⊢ μ (closedBall x (t * ε)) ≤ ↑(C ^ n) * μ (closedBall x ε)

Tactic:
]

Goal state after:
α : Type u_1
inst✝² : PseudoMetricSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsUnifLocDoublingMeasure μ
K : ℝ
C : ℝ≥0 := doublingConstant μ
this : ∀ (n : ℕ), ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), μ (closedBall x (2 ^ n * ε)) ≤ ↑C ^ n * μ (closedBall x ε)
n : ℕ
hn : K < 2 ^ n
ε : ℝ
hε₀ : 0 < ε
hε : ∀ (x : α), μ (closedBall x (2 ^ n * ε)) ≤ ↑C ^ n * μ (closedBall x ε)
x : α
t : ℝ
ht : t ≤ K
⊢ μ (closedBall x (t * ε)) ≤ ↑(C ^ n) * μ (closedBall x ε)
Module: Mathlib.MeasureTheory.Measure.Doubling

Goal state before:
pp: α : Type u_1
inst✝² : PseudoMetricSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsUnifLocDoublingMeasure μ
K : ℝ
C : ℝ≥0 := doublingConstant μ
this : ∀ (n : ℕ), ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), μ (closedBall x (2 ^ n * ε)) ≤ ↑C ^ n * μ (closedBall x ε)
n : ℕ
hn : K < 2 ^ n
ε : ℝ
hε₀ : 0 < ε
hε : ∀ (x : α), μ (closedBall x (2 ^ n * ε)) ≤ ↑C ^ n * μ (closedBall x ε)
x : α
t : ℝ
ht : t ≤ K
⊢ μ (closedBall x (2 ^ n * ε)) ≤ ↑C ^ n * μ (closedBall x ε)

Tactic:
]

Goal state after:
α : Type u_1
inst✝² : PseudoMetricSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsUnifLocDoublingMeasure μ
K : ℝ
C : ℝ≥0 := doublingConstant μ
this : ∀ (n : ℕ), ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), μ (closedBall x (2 ^ n * ε)) ≤ ↑C ^ n * μ (closedBall x ε)
n : ℕ
hn : K < 2 ^ n
ε : ℝ
hε₀ : 0 < ε
hε : ∀ (x : α), μ (closedBall x (2 ^ n * ε)) ≤ ↑C ^ n * μ (closedBall x ε)
x : α
t : ℝ
ht : t ≤ K
⊢ μ (closedBall x (2 ^ n * ε)) ≤ ↑C ^ n * μ (closedBall x ε)
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: case e'_3
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : T2Space α
inst✝¹ : MeasurableSpace α
inst✝ : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous[_, inst✝³] f
hg : Continuous[_, inst✝³] g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
I :
  ∀ (n : ℕ) (x y : ℕ → ℕ),
    ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →
      ∃ x' y',
        x' ∈ cylinder x n ∧
          y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))
A : Type := { p // ¬MeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }
F : A → A
hFn : ∀ (p : A), (↑(F p)).1 = (↑p).1 + 1
hFx : ∀ (p : A), (↑(F p)).2.1 ∈ cylinder (↑p).2.1 (↑p).1
hFy : ∀ (p : A), (↑(F p)).2.2 ∈ cylinder (↑p).2.2 (↑p).1
p0 : A := ⟨(0, fun x ↦ 0, fun x ↦ 0), ⋯⟩
p : ℕ → A := fun n ↦ F^[n] p0
prec : ∀ (n : ℕ), p (n + 1) = F (p n)
pn_fst : ∀ (n : ℕ), (↑(p n)).1 = n
Ix : ∀ (m n : ℕ), m + 1 ≤ n → (↑(p n)).2.1 m = (↑(p (m + 1))).2.1 m
Iy : ∀ (m n : ℕ), m + 1 ≤ n → (↑(p n)).2.2 m = (↑(p (m + 1))).2.2 m
x : ℕ → ℕ := fun n ↦ (↑(p (n + 1))).2.1 n
hx : x = fun n ↦ (↑(p (n + 1))).2.1 n
y : ℕ → ℕ := fun n ↦ (↑(p (n + 1))).2.2 n
hy : y = fun n ↦ (↑(p (n + 1))).2.2 n
n i : ℕ
hi : i < n
⊢ (fun n ↦ (↑(p (n + 1))).2.1 n) i = (↑(p n)).2.1 i

Tactic:
]

Goal state after:
case e'_3
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : T2Space α
inst✝¹ : MeasurableSpace α
inst✝ : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous[_, inst✝³] f
hg : Continuous[_, inst✝³] g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
I :
  ∀ (n : ℕ) (x y : ℕ → ℕ),
    ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →
      ∃ x' y',
        x' ∈ cylinder x n ∧
          y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))
A : Type := { p // ¬MeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }
F : A → A
hFn : ∀ (p : A), (↑(F p)).1 = (↑p).1 + 1
hFx : ∀ (p : A), (↑(F p)).2.1 ∈ cylinder (↑p).2.1 (↑p).1
hFy : ∀ (p : A), (↑(F p)).2.2 ∈ cylinder (↑p).2.2 (↑p).1
p0 : A := ⟨(0, fun x ↦ 0, fun x ↦ 0), ⋯⟩
p : ℕ → A := fun n ↦ F^[n] p0
prec : ∀ (n : ℕ), p (n + 1) = F (p n)
pn_fst : ∀ (n : ℕ), (↑(p n)).1 = n
Ix : ∀ (m n : ℕ), m + 1 ≤ n → (↑(p n)).2.1 m = (↑(p (m + 1))).2.1 m
Iy : ∀ (m n : ℕ), m + 1 ≤ n → (↑(p n)).2.2 m = (↑(p (m + 1))).2.2 m
x : ℕ → ℕ := fun n ↦ (↑(p (n + 1))).2.1 n
hx : x = fun n ↦ (↑(p (n + 1))).2.1 n
y : ℕ → ℕ := fun n ↦ (↑(p (n + 1))).2.2 n
hy : y = fun n ↦ (↑(p (n + 1))).2.2 n
n i : ℕ
hi : i < n
⊢ (fun n ↦ (↑(p (n + 1))).2.1 n) i = (↑(p n)).2.1 i