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

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : MeasurableSingletonClass α
a : α
s : Set α
h : a ∉ s
f : α → ℝ≥0∞
⊢ a ∉ s

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : MeasurableSingletonClass α
a : α
s : Set α
h : a ∉ s
f : α → ℝ≥0∞
⊢ a ∉ s
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Countable

Goal state before:
pp: α : Type u_1
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : MeasurableSingletonClass α
inst✝ : Fintype α
f : α → ℝ≥0∞
⊢ ∫⁻ (x : α), f x ∂μ = ∫⁻ (x : α), f x ∂μ

Tactic:
]

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
m : Measure (Measure α)
s : Set α
hs : s ∈ ae m.join
t : Set α
hst : sᶜ ⊆ t
htm : MeasurableSet t
ht : (fun μ ↦ μ t) =ᵐ[m] 0
⊢ s ∈ (ae m).bind ae

Tactic:
]

Goal state after:
α : Type u_1
mα : MeasurableSpace α
m : Measure (Measure α)
s : Set α
hs : s ∈ ae m.join
t : Set α
hst : sᶜ ⊆ t
htm : MeasurableSet t
ht : (fun μ ↦ μ t) =ᵐ[m] 0
⊢ s ∈ (ae m).bind ae
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: case succ
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
m n : ℕ
hmn : m + 1 ≤ n
IH : (↑(p n)).2.1 m = (↑(p (m + 1))).2.1 m
I : (↑(F (p n))).2.1 m = (↑(p n)).2.1 m
⊢ (↑(p (m + 1))).2.1 m = (↑(p (m + 1))).2.1 m

Tactic:
]

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
m : Measure (Measure α)
s : Set α
hs : s ∈ ae m.join
t : Set α
hst : sᶜ ⊆ t
htm : MeasurableSet t
ht : (fun μ ↦ μ t) =ᵐ[m] 0
⊢ {a | s ∈ ae a} ∈ ae m

Tactic:
]

Goal state after:
α : Type u_1
mα : MeasurableSpace α
m : Measure (Measure α)
s : Set α
hs : s ∈ ae m.join
t : Set α
hst : sᶜ ⊆ t
htm : MeasurableSet t
ht : (fun μ ↦ μ t) =ᵐ[m] 0
⊢ {a | s ∈ ae a} ∈ ae m
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: α : Type u_2
inst✝³ : MetricSpace α
inst✝² : CompleteSpace α
inst✝¹ : SecondCountableTopology α
inst✝ : Nonempty α
this✝ : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x | (⋂ n, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty}
g : ↑s → α := fun x ↦ Set.Nonempty.some ⋯
A : ∀ (x : ↑s) (n : ℕ), dist (g x) (u (↑x n)) ≤ (1 / 2) ^ n
g_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
g_surj : Surjective g
x : ℕ → ℕ
hx : ClusterPt x (𝓟 s)
this : Tendsto (fun n ↦ 2 * (1 / 2) ^ n) atTop (𝓝 0)
⊢ Tendsto (fun n ↦ diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (𝓝 0)

Tactic:
]

Goal state after:
α : Type u_2
inst✝³ : MetricSpace α
inst✝² : CompleteSpace α
inst✝¹ : SecondCountableTopology α
inst✝ : Nonempty α
this✝ : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x | (⋂ n, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty}
g : ↑s → α := fun x ↦ Set.Nonempty.some ⋯
A : ∀ (x : ↑s) (n : ℕ), dist (g x) (u (↑x n)) ≤ (1 / 2) ^ n
g_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
g_surj : Surjective g
x : ℕ → ℕ
hx : ClusterPt x (𝓟 s)
this : Tendsto (fun n ↦ 2 * (1 / 2) ^ n) atTop (𝓝 0)
⊢ Tendsto (fun n ↦ diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (𝓝 0)
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Countable

Goal state before:
pp: case inr
α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
f : α → ℝ≥0∞
this :
  ∀ (μ : Measure α) [SFinite μ],
    IsFiniteMeasure μ → ∃ g, Measurable g ∧ g ≤ f ∧ ∀ (s : Set α), ∫⁻ (a : α) in s, f a ∂μ ≤ ∫⁻ (a : α) in s, g a ∂μ
h : ¬IsFiniteMeasure μ
g : ℕ → α → ℝ≥0∞
hgm : ∀ (n : ℕ), Measurable (g n)
hgle : ∀ (n : ℕ), g n ≤ f
hgint : ∀ (n : ℕ) (s : Set α), ∫⁻ (a : α) in s, f a ∂sfiniteSeq μ n ≤ ∫⁻ (a : α) in s, g n a ∂sfiniteSeq μ n
s : Set α
⊢ ∑' (i : ℕ), ∫⁻ (a : α) in s, f a ∂sfiniteSeq μ i ≤ ∑' (i : ℕ), ∫⁻ (a : α) in s, ⨆ n, g n a ∂sfiniteSeq μ i

Tactic:
]

Goal state after:
case inr
α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
f : α → ℝ≥0∞
this :
  ∀ (μ : Measure α) [SFinite μ],
    IsFiniteMeasure μ → ∃ g, Measurable g ∧ g ≤ f ∧ ∀ (s : Set α), ∫⁻ (a : α) in s, f a ∂μ ≤ ∫⁻ (a : α) in s, g a ∂μ
h : ¬IsFiniteMeasure μ
g : ℕ → α → ℝ≥0∞
hgm : ∀ (n : ℕ), Measurable (g n)
hgle : ∀ (n : ℕ), g n ≤ f
hgint : ∀ (n : ℕ) (s : Set α), ∫⁻ (a : α) in s, f a ∂sfiniteSeq μ n ≤ ∫⁻ (a : α) in s, g n a ∂sfiniteSeq μ n
s : Set α
⊢ ∑' (i : ℕ), ∫⁻ (a : α) in s, f a ∂sfiniteSeq μ i ≤ ∑' (i : ℕ), ∫⁻ (a : α) in s, ⨆ n, g n a ∂sfiniteSeq μ i
Module: Mathlib.MeasureTheory.Measure.GiryMonad

Goal state before:
pp: case inr
α : Type u_1
mα : MeasurableSpace α
m : Measure (Measure α)
f : α → ℝ≥0∞
this :
  ∀ {f : α → ℝ≥0∞},
    AEMeasurable f m.join → Measurable f → ∫⁻ (x : α), f x ∂m.join = ∫⁻ (μ : Measure α), ∫⁻ (x : α), f x ∂μ ∂m
hfm : ¬Measurable f
g : α → ℝ≥0∞
hgm : Measurable g
hfg : f =ᵐ[m.join] g
⊢ ∫⁻ (μ : Measure α), ∫⁻ (x : α), g x ∂μ ∂m = ∫⁻ (μ : Measure α), ∫⁻ (x : α), f x ∂μ ∂m

Tactic:
]

Goal state after:
case inr
α : Type u_1
mα : MeasurableSpace α
m : Measure (Measure α)
f : α → ℝ≥0∞
this :
  ∀ {f : α → ℝ≥0∞},
    AEMeasurable f m.join → Measurable f → ∫⁻ (x : α), f x ∂m.join = ∫⁻ (μ : Measure α), ∫⁻ (x : α), f x ∂μ ∂m
hfm : ¬Measurable f
g : α → ℝ≥0∞
hgm : Measurable g
hfg : f =ᵐ[m.join] g
⊢ ∫⁻ (μ : Measure α), ∫⁻ (x : α), g x ∂μ ∂m = ∫⁻ (μ : Measure α), ∫⁻ (x : α), f x ∂μ ∂m
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: α : Type u_2
inst✝³ : MetricSpace α
inst✝² : CompleteSpace α
inst✝¹ : SecondCountableTopology α
inst✝ : Nonempty α
this✝ : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x | (⋂ n, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty}
g : ↑s → α := fun x ↦ Set.Nonempty.some ⋯
A : ∀ (x : ↑s) (n : ℕ), dist (g x) (u (↑x n)) ≤ (1 / 2) ^ n
g_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
g_surj : Surjective g
x : ℕ → ℕ
hx : ClusterPt x (𝓟 s)
L : Tendsto (fun n ↦ diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (𝓝 0)
N : ℕ
y : ℕ → ℕ
hxy : y ∈ ball x ((1 / 2) ^ N)
ys : y ∈ s
n : ℕ
hn : n ≤ N
this : x n = y n
⊢ closedBall (u (y n)) ((1 / 2) ^ n) = closedBall (u (y n)) ((1 / 2) ^ n)

Tactic:
]

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

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
μ✝ : Measure α
f : α → ℝ≥0∞
μ : Measure α
inst✝ : SFinite μ
h : IsFiniteMeasure μ
g : ℕ → α → ℝ≥0∞
hgm : ∀ (n : ℕ), Measurable (g n)
hgf : ∀ (n : ℕ), g n ≤ f
hgle : ∀ (n : ℕ), g n ≤ ↑n
hgint : ∀ (n : ℕ), ∫⁻ (a : α), min (f a) ↑n ∂μ = ∫⁻ (a : α), g n a ∂μ
φ : α → ℝ≥0∞ := fun x ↦ ⨆ n, g n x
hφm : Measurable φ
hφle : φ ≤ f
s : Set α
⊢ ⨆ φ, ⨆ (_ : ∀ (x : α), ↑(φ x) ≤ f x), (SimpleFunc.map ofNNReal φ).lintegral (μ.restrict s) ≤ ∫⁻ (a : α) in s, φ a ∂μ

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : MeasurableSpace α
μ✝ : Measure α
f : α → ℝ≥0∞
μ : Measure α
inst✝ : SFinite μ
h : IsFiniteMeasure μ
g : ℕ → α → ℝ≥0∞
hgm : ∀ (n : ℕ), Measurable (g n)
hgf : ∀ (n : ℕ), g n ≤ f
hgle : ∀ (n : ℕ), g n ≤ ↑n
hgint : ∀ (n : ℕ), ∫⁻ (a : α), min (f a) ↑n ∂μ = ∫⁻ (a : α), g n a ∂μ
φ : α → ℝ≥0∞ := fun x ↦ ⨆ n, g n x
hφm : Measurable φ
hφle : φ ≤ f
s : Set α
⊢ ⨆ φ, ⨆ (_ : ∀ (x : α), ↑(φ x) ≤ f x), (SimpleFunc.map ofNNReal φ).lintegral (μ.restrict s) ≤ ∫⁻ (a : α) in s, φ a ∂μ
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Countable

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
μ✝ : Measure α
f : α → ℝ≥0∞
μ : Measure α
inst✝ : SFinite μ
h : IsFiniteMeasure μ
g : ℕ → α → ℝ≥0∞
hgm : ∀ (n : ℕ), Measurable (g n)
hgf : ∀ (n : ℕ), g n ≤ f
hgle : ∀ (n : ℕ), g n ≤ ↑n
hgint : ∀ (n : ℕ), ∫⁻ (a : α), min (f a) ↑n ∂μ = ∫⁻ (a : α), g n a ∂μ
φ : α → ℝ≥0∞ := fun x ↦ ⨆ n, g n x
hφm : Measurable φ
hφle : φ ≤ f
s : Set α
ψ : SimpleFunc α ℝ≥0
hψ : ∀ (x : α), ↑(ψ x) ≤ f x
C : ℝ≥0
hC : C ∈ upperBounds ↑ψ.range
⊢ ∃ n, ∀ (x : α), ψ x ≤ ↑n

Tactic:
exact ⟨⌈C⌉₊, fun x ↦ (hC <| ψ.mem_range_self x).trans (Nat.le_ceil _)⟩

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

Goal state before:
pp: α : Type u_2
inst✝³ : MetricSpace α
inst✝² : CompleteSpace α
inst✝¹ : SecondCountableTopology α
inst✝ : Nonempty α
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x | (⋂ n, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty}
g : ↑s → α := fun x ↦ Set.Nonempty.some ⋯
A : ∀ (x : ↑s) (n : ℕ), dist (g x) (u (↑x n)) ≤ (1 / 2) ^ n
g_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
g_surj : Surjective g
x : ℕ → ℕ
hx : ClusterPt x (𝓟 s)
L : Tendsto (fun n ↦ diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (𝓝 0)
N : ℕ
y : ℕ → ℕ
hxy : y ∈ ball x ((1 / 2) ^ N)
ys : y ∈ s
E : ⋂ n, ⋂ (_ : n ≤ N), closedBall (u (x n)) ((1 / 2) ^ n) = ⋂ n, ⋂ (_ : n ≤ N), closedBall (u (y n)) ((1 / 2) ^ n)
⊢ (⋂ n, ⋂ (_ : n ≤ N), closedBall (u (y n)) ((1 / 2) ^ n)).Nonempty

Tactic:
]

Goal state after:
α : Type u_2
inst✝³ : MetricSpace α
inst✝² : CompleteSpace α
inst✝¹ : SecondCountableTopology α
inst✝ : Nonempty α
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x | (⋂ n, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty}
g : ↑s → α := fun x ↦ Set.Nonempty.some ⋯
A : ∀ (x : ↑s) (n : ℕ), dist (g x) (u (↑x n)) ≤ (1 / 2) ^ n
g_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
g_surj : Surjective g
x : ℕ → ℕ
hx : ClusterPt x (𝓟 s)
L : Tendsto (fun n ↦ diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (𝓝 0)
N : ℕ
y : ℕ → ℕ
hxy : y ∈ ball x ((1 / 2) ^ N)
ys : y ∈ s
E : ⋂ n, ⋂ (_ : n ≤ N), closedBall (u (x n)) ((1 / 2) ^ n) = ⋂ n, ⋂ (_ : n ≤ N), closedBall (u (y n)) ((1 / 2) ^ n)
⊢ (⋂ n, ⋂ (_ : n ≤ N), closedBall (u (y n)) ((1 / 2) ^ n)).Nonempty
Module: Mathlib.MeasureTheory.Measure.GiryMonad

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
m : Measure (Measure α)
f✝ : α → ℝ≥0∞
hfm : Measurable f✝
s : ℕ → Finset ℝ≥0∞
f : ℕ → ℝ≥0∞ → Measure α → ℝ≥0∞
hf : ∀ (n : ℕ) (r : ℝ≥0∞), Measurable (f n r)
hm : Monotone fun n μ ↦ ∑ r ∈ s n, r * f n r μ
⊢ ⨆ n, ∑ r ∈ s n, r * ∫⁻ (μ : Measure α), f n r μ ∂m = ⨆ n, ∫⁻ (a : Measure α), ∑ r ∈ s n, r * f n r a ∂m

pp: α : Type u_1
mα : MeasurableSpace α
m : Measure (Measure α)
f✝ : α → ℝ≥0∞
hfm : Measurable f✝
s : ℕ → Finset ℝ≥0∞
f : ℕ → ℝ≥0∞ → Measure α → ℝ≥0∞
hf : ∀ (n : ℕ) (r : ℝ≥0∞), Measurable (f n r)
hm : Monotone fun n μ ↦ ∑ r ∈ s n, r * f n r μ
⊢ ∀ (n : ℕ), Measurable fun μ ↦ ∑ r ∈ s n, r * f n r μ

Tactic:
]

Goal state after:
α : Type u_1
mα : MeasurableSpace α
m : Measure (Measure α)
f✝ : α → ℝ≥0∞
hfm : Measurable f✝
s : ℕ → Finset ℝ≥0∞
f : ℕ → ℝ≥0∞ → Measure α → ℝ≥0∞
hf : ∀ (n : ℕ) (r : ℝ≥0∞), Measurable (f n r)
hm : Monotone fun n μ ↦ ∑ r ∈ s n, r * f n r μ
⊢ ⨆ n, ∑ r ∈ s n, r * ∫⁻ (μ : Measure α), f n r μ ∂m = ⨆ n, ∫⁻ (a : Measure α), ∑ r ∈ s n, r * f n r a ∂m

α : Type u_1
mα : MeasurableSpace α
m : Measure (Measure α)
f✝ : α → ℝ≥0∞
hfm : Measurable f✝
s : ℕ → Finset ℝ≥0∞
f : ℕ → ℝ≥0∞ → Measure α → ℝ≥0∞
hf : ∀ (n : ℕ) (r : ℝ≥0∞), Measurable (f n r)
hm : Monotone fun n μ ↦ ∑ r ∈ s n, r * f n r μ
⊢ ∀ (n : ℕ), Measurable fun μ ↦ ∑ r ∈ s n, r * f n r μ
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: 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
m n : ℕ
hmn : m + 1 ≤ n
IH : (↑(p n)).2.2 m = (↑(p (m + 1))).2.2 m
⊢ m < n

Tactic:
]

Goal state after:
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
m n : ℕ
hmn : m + 1 ≤ n
IH : (↑(p n)).2.2 m = (↑(p (m + 1))).2.2 m
⊢ m < n
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: ι : Type u_2
inst✝¹ : Encodable ι
F : ι → Type u_3
inst✝ : (i : ι) → EDist (F i)
x y : (i : ι) → F i
⊢ ∑' (i : ι), min (2⁻¹ ^ encode i) (edist (x i) (y i)) ≤ ∑' (i : ?m.10), 2⁻¹ ^ encode i

Tactic:
]

Goal state after:
ι : Type u_2
inst✝¹ : Encodable ι
F : ι → Type u_3
inst✝ : (i : ι) → EDist (F i)
x y : (i : ι) → F i
⊢ ∑' (i : ι), min (2⁻¹ ^ encode i) (edist (x i) (y i)) ≤ ∑' (i : ?m.10), 2⁻¹ ^ encode i
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Countable

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
μ✝ : Measure α
f : α → ℝ≥0∞
μ : Measure α
inst✝ : SFinite μ
h : IsFiniteMeasure μ
g : ℕ → α → ℝ≥0∞
hgm : ∀ (n : ℕ), Measurable (g n)
hgf : ∀ (n : ℕ), g n ≤ f
hgle : ∀ (n : ℕ), g n ≤ ↑n
hgint : ∀ (n : ℕ), ∫⁻ (a : α), min (f a) ↑n ∂μ = ∫⁻ (a : α), g n a ∂μ
φ : α → ℝ≥0∞ := fun x ↦ ⨆ n, g n x
hφm : Measurable φ
hφle : φ ≤ f
s : Set α
ψ : SimpleFunc α ℝ≥0
hψ : ∀ (x : α), ↑(ψ x) ≤ f x
n : ℕ
hn : ∀ (x : α), ψ x ≤ ↑n
this : ∫⁻ (a : α) in (toMeasurable μ s)ᶜ, min (f a) ↑n ∂μ ≠ ∞
hsm : MeasurableSet (toMeasurable μ s)
⊢ ∫⁻ (x : α) in toMeasurable μ s, g n x ∂μ + ∫⁻ (x : α) in (toMeasurable μ s)ᶜ, g n x ∂μ ≤
    ∫⁻ (a : α) in toMeasurable μ s, g n a ∂μ + ∫⁻ (a : α) in (toMeasurable μ s)ᶜ, min (f a) ↑n ∂μ

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : MeasurableSpace α
μ✝ : Measure α
f : α → ℝ≥0∞
μ : Measure α
inst✝ : SFinite μ
h : IsFiniteMeasure μ
g : ℕ → α → ℝ≥0∞
hgm : ∀ (n : ℕ), Measurable (g n)
hgf : ∀ (n : ℕ), g n ≤ f
hgle : ∀ (n : ℕ), g n ≤ ↑n
hgint : ∀ (n : ℕ), ∫⁻ (a : α), min (f a) ↑n ∂μ = ∫⁻ (a : α), g n a ∂μ
φ : α → ℝ≥0∞ := fun x ↦ ⨆ n, g n x
hφm : Measurable φ
hφle : φ ≤ f
s : Set α
ψ : SimpleFunc α ℝ≥0
hψ : ∀ (x : α), ↑(ψ x) ≤ f x
n : ℕ
hn : ∀ (x : α), ψ x ≤ ↑n
this : ∫⁻ (a : α) in (toMeasurable μ s)ᶜ, min (f a) ↑n ∂μ ≠ ∞
hsm : MeasurableSet (toMeasurable μ s)
⊢ ∫⁻ (x : α) in toMeasurable μ s, g n x ∂μ + ∫⁻ (x : α) in (toMeasurable μ s)ᶜ, g n x ∂μ ≤
    ∫⁻ (a : α) in toMeasurable μ s, g n a ∂μ + ∫⁻ (a : α) in (toMeasurable μ s)ᶜ, min (f a) ↑n ∂μ
Module: Mathlib.MeasureTheory.Measure.GiryMonad

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
m : Measure (Measure α)
f✝ : α → ℝ≥0∞
hfm : Measurable f✝
s : ℕ → Finset ℝ≥0∞
f : ℕ → ℝ≥0∞ → Measure α → ℝ≥0∞
hf : ∀ (n : ℕ) (r : ℝ≥0∞), Measurable (f n r)
hm : Monotone fun n μ ↦ ∑ r ∈ s n, r * f n r μ
n : ℕ
⊢ ∑ r ∈ s n, r * ∫⁻ (μ : Measure α), f n r μ ∂m = ∑ b ∈ s n, ∫⁻ (a : Measure α), b * f n b a ∂m

pp: α : Type u_1
mα : MeasurableSpace α
m : Measure (Measure α)
f✝ : α → ℝ≥0∞
hfm : Measurable f✝
s : ℕ → Finset ℝ≥0∞
f : ℕ → ℝ≥0∞ → Measure α → ℝ≥0∞
hf : ∀ (n : ℕ) (r : ℝ≥0∞), Measurable (f n r)
hm : Monotone fun n μ ↦ ∑ r ∈ s n, r * f n r μ
n : ℕ
⊢ ∀ b ∈ s n, Measurable fun a ↦ b * f n b a

Tactic:
]

Goal state after:
α : Type u_1
mα : MeasurableSpace α
m : Measure (Measure α)
f✝ : α → ℝ≥0∞
hfm : Measurable f✝
s : ℕ → Finset ℝ≥0∞
f : ℕ → ℝ≥0∞ → Measure α → ℝ≥0∞
hf : ∀ (n : ℕ) (r : ℝ≥0∞), Measurable (f n r)
hm : Monotone fun n μ ↦ ∑ r ∈ s n, r * f n r μ
n : ℕ
⊢ ∑ r ∈ s n, r * ∫⁻ (μ : Measure α), f n r μ ∂m = ∑ b ∈ s n, ∫⁻ (a : Measure α), b * f n b a ∂m

α : Type u_1
mα : MeasurableSpace α
m : Measure (Measure α)
f✝ : α → ℝ≥0∞
hfm : Measurable f✝
s : ℕ → Finset ℝ≥0∞
f : ℕ → ℝ≥0∞ → Measure α → ℝ≥0∞
hf : ∀ (n : ℕ) (r : ℝ≥0∞), Measurable (f n r)
hm : Monotone fun n μ ↦ ∑ r ∈ s n, r * f n r μ
n : ℕ
⊢ ∀ b ∈ s n, Measurable fun a ↦ b * f n b a
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: case h
E : ℕ → Type u_1
ι : Type u_2
inst✝¹ : Encodable ι
F : ι → Type u_3
inst✝ : (i : ι) → PseudoEMetricSpace (F i)
x y z : (i : ι) → F i
n : ι
⊢ min (2⁻¹ ^ encode n) (edist (x n) (y n)) + min (2⁻¹ ^ encode n) (edist (y n) (z n)) ≤
    min (2⁻¹ ^ encode n) (edist (x n) (y n)) + min (2⁻¹ ^ encode n) (edist (y n) (z n))

Tactic:
]

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
f : α → ℝ≥0∞
m : Measure (Measure α)
g : α → ℝ≥0∞
hgm : Measurable g
hgf : g ≤ f
hfg_int : ∫⁻ (a : α), f a ∂m.join = ∫⁻ (a : α), g a ∂m.join
⊢ ∫⁻ (μ : Measure α), ∫⁻ (x : α), g x ∂μ ∂m ≤ ∫⁻ (μ : Measure α), ∫⁻ (x : α), f x ∂μ ∂m

Tactic:
]

Goal state after:
α : Type u_1
mα : MeasurableSpace α
f : α → ℝ≥0∞
m : Measure (Measure α)
g : α → ℝ≥0∞
hgm : Measurable g
hgf : g ≤ f
hfg_int : ∫⁻ (a : α), f a ∂m.join = ∫⁻ (a : α), g a ∂m.join
⊢ ∫⁻ (μ : Measure α), ∫⁻ (x : α), g x ∂μ ∂m ≤ ∫⁻ (μ : Measure α), ∫⁻ (x : α), f x ∂μ ∂m
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: case succ
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
m n : ℕ
hmn : m + 1 ≤ n
IH : (↑(p n)).2.2 m = (↑(p (m + 1))).2.2 m
I : (↑(F (p n))).2.2 m = (↑(p n)).2.2 m
⊢ (↑(p (m + 1))).2.2 m = (↑(p (m + 1))).2.2 m

Tactic:
]

Goal state after:
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 → Measure β
s : Set β
hs : MeasurableSet s
hf : AEMeasurable f m
⊢ ∫⁻ (a : α), (f a) s ∂m = ∫⁻ (a : α), (f a) s ∂m

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
C : ℝ≥0∞
f : Set α → ℝ≥0∞
hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ∞ → f s ≤ C
h_F_lim : ∀ (S : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (S n)) → Monotone S → f (⋃ n, S n) ≤ ⨆ n, f (S n)
S : ℕ → Set α := spanningSets (μ.trim hm)
hS_mono : Monotone S
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
⊢ f (⋃ i, spanningSets (μ.trim hm) i) ≤ C

Tactic:
]

Goal state after:
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
C : ℝ≥0∞
f : Set α → ℝ≥0∞
hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ∞ → f s ≤ C
h_F_lim : ∀ (S : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (S n)) → Monotone S → f (⋃ n, S n) ≤ ⨆ n, f (S n)
S : ℕ → Set α := spanningSets (μ.trim hm)
hS_mono : Monotone S
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
⊢ f (⋃ i, spanningSets (μ.trim hm) i) ≤ C
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 → Measure β
s : Set β
hs : MeasurableSet s
⊢ ∫⁻ (μ : Measure β), μ s ∂map f m ≤ ∫⁻ (a : α), (f a) s ∂m

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
m : Measure α
f : α → Measure β
s : Set β
hs : MeasurableSet s
⊢ ∫⁻ (μ : Measure β), μ s ∂map f m ≤ ∫⁻ (a : α), (f a) s ∂m
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Countable

Goal state before:
pp: α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
C : ℝ≥0∞
f : α → ℝ≥0∞
hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ∞ → ∫⁻ (x : α) in s, f x ∂μ ≤ C
this : ∫⁻ (x : α) in univ, f x ∂μ = ∫⁻ (x : α), f x ∂μ
⊢ ∫⁻ (x : α) in univ, f x ∂μ ≤ C

Tactic:
]

Goal state after:
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
C : ℝ≥0∞
f : α → ℝ≥0∞
hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ∞ → ∫⁻ (x : α) in s, f x ∂μ ≤ C
this : ∫⁻ (x : α) in univ, f x ∂μ = ∫⁻ (x : α), f x ∂μ
⊢ ∫⁻ (x : α) in univ, f x ∂μ ≤ C
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Countable

Goal state before:
pp: α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
C : ℝ≥0∞
f : α → ℝ≥0∞
hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ∞ → ∫⁻ (x : α) in s, f x ∂μ ≤ C
this : ∫⁻ (x : α) in univ, f x ∂μ = ∫⁻ (x : α), f x ∂μ
S : ℕ → Set α
x✝ : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
⊢ ⨆ i, ∫⁻ (x : α) in S i, f x ∂μ ≤ ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ

pp: case hd
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
C : ℝ≥0∞
f : α → ℝ≥0∞
hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ∞ → ∫⁻ (x : α) in s, f x ∂μ ≤ C
this : ∫⁻ (x : α) in univ, f x ∂μ = ∫⁻ (x : α), f x ∂μ
S : ℕ → Set α
x✝ : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
⊢ Directed (fun x1 x2 ↦ x1 ⊆ x2) S

Tactic:
]

Goal state after:
case hd
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
C : ℝ≥0∞
f : α → ℝ≥0∞
hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ∞ → ∫⁻ (x : α) in s, f x ∂μ ≤ C
this : ∫⁻ (x : α) in univ, f x ∂μ = ∫⁻ (x : α), f x ∂μ
S : ℕ → Set α
x✝ : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
⊢ Directed (fun x1 x2 ↦ x1 ⊆ x2) S
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Countable

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
C : ℝ≥0∞
f : α → ℝ≥0∞
hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ∞ → ∫⁻ (x : α) in s, f x ∂μ ≤ C
⊢ SigmaFinite μ

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
C : ℝ≥0∞
f : α → ℝ≥0∞
hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ∞ → ∫⁻ (x : α) in s, f x ∂μ ≤ C
⊢ SigmaFinite μ