mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: case neg
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x ↦ if x ∈ s then x else ⋯.some
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)
xs : x ∉ s
⊢ firstDiff x y ≤ firstDiff (f x) (f y)

Tactic:
·

Goal state after:
case neg
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x ↦ if x ∈ s then x else ⋯.some
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)
xs : x ∉ s
⊢ firstDiff x y ≤ firstDiff (f x) (f y)
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: case h₁
α : Type u_1
ι : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : Countable ι
inst✝ : T2Space α
s : ι → Set α
hs : ∀ (n : ι), AnalyticSet (s n)
i₀ : ι
β : ι → Type
hβ : (n : ι) → TopologicalSpace (β n)
h'β : ∀ (n : ι), PolishSpace (β n)
f : (n : ι) → β n → α
f_cont : ∀ (n : ι), Continuous[hβ n, inst✝²] (f n)
f_range : ∀ (n : ι), range (f n) = s n
γ : Type u_2 := ⋯
t : Set γ := ⋯
t_closed : IsClosed[Pi.topologicalSpace] t
this : PolishSpace ↑t
F : ↑t → α := ⋯
F_cont : Continuous[instTopologicalSpaceSubtype, inst✝²] F
⊢ range F ⊆ ⋂ n, s n

pp: case h₂
α : Type u_1
ι : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : Countable ι
inst✝ : T2Space α
s : ι → Set α
hs : ∀ (n : ι), AnalyticSet (s n)
i₀ : ι
β : ι → Type
hβ : (n : ι) → TopologicalSpace (β n)
h'β : ∀ (n : ι), PolishSpace (β n)
f : (n : ι) → β n → α
f_cont : ∀ (n : ι), Continuous[hβ n, inst✝²] (f n)
f_range : ∀ (n : ι), range (f n) = s n
γ : Type u_2 := ⋯
t : Set γ := ⋯
t_closed : IsClosed[Pi.topologicalSpace] t
this : PolishSpace ↑t
F : ↑t → α := ⋯
F_cont : Continuous[instTopologicalSpaceSubtype, inst✝²] F
⊢ ⋂ n, s n ⊆ range F

Tactic:
·

Goal state after:
case h₁
α : Type u_1
ι : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : Countable ι
inst✝ : T2Space α
s : ι → Set α
hs : ∀ (n : ι), AnalyticSet (s n)
i₀ : ι
β : ι → Type
hβ : (n : ι) → TopologicalSpace (β n)
h'β : ∀ (n : ι), PolishSpace (β n)
f : (n : ι) → β n → α
f_cont : ∀ (n : ι), Continuous[hβ n, inst✝²] (f n)
f_range : ∀ (n : ι), range (f n) = s n
γ : Type u_2 := ⋯
t : Set γ := ⋯
t_closed : IsClosed[Pi.topologicalSpace] t
this : PolishSpace ↑t
F : ↑t → α := ⋯
F_cont : Continuous[instTopologicalSpaceSubtype, inst✝²] F
⊢ range F ⊆ ⋂ n, s n
Module: Mathlib.MeasureTheory.Measure.MutuallySingular

Goal state before:
pp: case hd
α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
h : μ ≪ ν₁ + ν₂
h_ms : μ ⟂ₘ ν₂
s : Set α
hs : MeasurableSet s
hs_zero : ν₁ s = 0
t : Set α := h_ms.nullSet
ht : MeasurableSet t
htμ : μ t = 0
htν₂ : ν₂ tᶜ = 0
⊢ Disjoint (s ∩ t) (s ∩ tᶜ)

pp: case h
α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
h : μ ≪ ν₁ + ν₂
h_ms : μ ⟂ₘ ν₂
s : Set α
hs : MeasurableSet s
hs_zero : ν₁ s = 0
t : Set α := h_ms.nullSet
ht : MeasurableSet t
htμ : μ t = 0
htν₂ : ν₂ tᶜ = 0
⊢ MeasurableSet (s ∩ tᶜ)

Tactic:
·

Goal state after:
case hd
α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
h : μ ≪ ν₁ + ν₂
h_ms : μ ⟂ₘ ν₂
s : Set α
hs : MeasurableSet s
hs_zero : ν₁ s = 0
t : Set α := h_ms.nullSet
ht : MeasurableSet t
htμ : μ t = 0
htν₂ : ν₂ tᶜ = 0
⊢ Disjoint (s ∩ t) (s ∩ tᶜ)
Module: Mathlib.MeasureTheory.Measure.MutuallySingular

Goal state before:
pp: case h
α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
h : μ ≪ ν₁ + ν₂
h_ms : μ ⟂ₘ ν₂
s : Set α
hs : MeasurableSet s
hs_zero : ν₁ s = 0
t : Set α := h_ms.nullSet
ht : MeasurableSet t
htμ : μ t = 0
htν₂ : ν₂ tᶜ = 0
⊢ MeasurableSet (s ∩ tᶜ)

Tactic:
·

Goal state after:
case h
α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
h : μ ≪ ν₁ + ν₂
h_ms : μ ⟂ₘ ν₂
s : Set α
hs : MeasurableSet s
hs_zero : ν₁ s = 0
t : Set α := h_ms.nullSet
ht : MeasurableSet t
htμ : μ t = 0
htν₂ : ν₂ tᶜ = 0
⊢ MeasurableSet (s ∩ tᶜ)
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: case pos
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x ↦ if x ∈ s then x else ⋯.some
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)
xs : x ∉ s
ys : y ∈ s
⊢ firstDiff x y ≤ firstDiff (f x) (f y)

pp: case neg
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x ↦ if x ∈ s then x else ⋯.some
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)
xs : x ∉ s
ys : y ∉ s
⊢ firstDiff x y ≤ firstDiff (f x) (f y)

Tactic:
·

Goal state after:
case pos
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x ↦ if x ∈ s then x else ⋯.some
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)
xs : x ∉ s
ys : y ∈ s
⊢ firstDiff x y ≤ firstDiff (f x) (f y)
Module: Mathlib.MeasureTheory.Measure.MutuallySingular

Goal state before:
pp: case refine_1
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
β : Type u_2
x✝ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
hμν : μ ⟂ₘ ν
⊢ (map f μ) (f '' hμν.nullSet) = 0

pp: case refine_2
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
β : Type u_2
x✝ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
hμν : μ ⟂ₘ ν
⊢ (map f ν) (f '' hμν.nullSet)ᶜ = 0

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
β : Type u_2
x✝ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
hμν : μ ⟂ₘ ν
⊢ (map f μ) (f '' hμν.nullSet) = 0
Module: Mathlib.MeasureTheory.Measure.MutuallySingular

Goal state before:
pp: case refine_2
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
β : Type u_2
x✝ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
hμν : μ ⟂ₘ ν
⊢ (map f ν) (f '' hμν.nullSet)ᶜ = 0

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
β : Type u_2
x✝ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
hμν : μ ⟂ₘ ν
⊢ (map f ν) (f '' hμν.nullSet)ᶜ = 0
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: case h₂
α : Type u_1
ι : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : Countable ι
inst✝ : T2Space α
s : ι → Set α
hs : ∀ (n : ι), AnalyticSet (s n)
i₀ : ι
β : ι → Type
hβ : (n : ι) → TopologicalSpace (β n)
h'β : ∀ (n : ι), PolishSpace (β n)
f : (n : ι) → β n → α
f_cont : ∀ (n : ι), Continuous[hβ n, inst✝²] (f n)
f_range : ∀ (n : ι), range (f n) = s n
γ : Type u_2 := ⋯
t : Set γ := ⋯
t_closed : IsClosed[Pi.topologicalSpace] t
this : PolishSpace ↑t
F : ↑t → α := ⋯
F_cont : Continuous[instTopologicalSpaceSubtype, inst✝²] F
⊢ ⋂ n, s n ⊆ range F

Tactic:
·

Goal state after:
case h₂
α : Type u_1
ι : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : Countable ι
inst✝ : T2Space α
s : ι → Set α
hs : ∀ (n : ι), AnalyticSet (s n)
i₀ : ι
β : ι → Type
hβ : (n : ι) → TopologicalSpace (β n)
h'β : ∀ (n : ι), PolishSpace (β n)
f : (n : ι) → β n → α
f_cont : ∀ (n : ι), Continuous[hβ n, inst✝²] (f n)
f_range : ∀ (n : ι), range (f n) = s n
γ : Type u_2 := ⋯
t : Set γ := ⋯
t_closed : IsClosed[Pi.topologicalSpace] t
this : PolishSpace ↑t
F : ↑t → α := ⋯
F_cont : Continuous[instTopologicalSpaceSubtype, inst✝²] F
⊢ ⋂ n, s n ⊆ range F
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Probability

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
s : Set α
⊢ IsZeroOrProbabilityMeasure ((0 univ)⁻¹ • 0)

pp: case inr
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s : Set α
h : NeZero μ
⊢ IsZeroOrProbabilityMeasure ((μ univ)⁻¹ • μ)

Tactic:
·

Goal state after:
case inl
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
s : Set α
⊢ IsZeroOrProbabilityMeasure ((0 univ)⁻¹ • 0)
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Probability

Goal state before:
pp: case inr.inl
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s : Set α
h✝ : NeZero μ
h : μ univ = ∞
⊢ IsZeroOrProbabilityMeasure ((μ univ)⁻¹ • μ)

pp: case inr.inr
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s : Set α
h✝ : NeZero μ
h : μ univ < ∞
⊢ IsZeroOrProbabilityMeasure ((μ univ)⁻¹ • μ)

Tactic:
·

Goal state after:
case inr.inl
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s : Set α
h✝ : NeZero μ
h : μ univ = ∞
⊢ IsZeroOrProbabilityMeasure ((μ univ)⁻¹ • μ)
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Probability

Goal state before:
pp: case inl
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsZeroOrProbabilityMeasure μ
h : μ univ = 0
⊢ μ = 0 ∨ IsProbabilityMeasure μ

pp: case inr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsZeroOrProbabilityMeasure μ
h : μ univ = 1
⊢ μ = 0 ∨ IsProbabilityMeasure μ

Tactic:
·

Goal state after:
case inl
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsZeroOrProbabilityMeasure μ
h : μ univ = 0
⊢ μ = 0 ∨ IsProbabilityMeasure μ
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Probability

Goal state before:
pp: case inr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsZeroOrProbabilityMeasure μ
h : μ univ = 1
⊢ μ = 0 ∨ IsProbabilityMeasure μ

Tactic:
·

Goal state after:
case inr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsZeroOrProbabilityMeasure μ
h : μ univ = 1
⊢ μ = 0 ∨ IsProbabilityMeasure μ
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Probability

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s : Set α
inst✝ : IsZeroOrProbabilityMeasure μ
p : α → Prop
f✝ : β → α
f : α → β
hf : AEMeasurable f μ
⊢ IsZeroOrProbabilityMeasure (Measure.map f μ)

pp: case neg
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s : Set α
inst✝ : IsZeroOrProbabilityMeasure μ
p : α → Prop
f✝ : β → α
f : α → β
hf : ¬AEMeasurable f μ
⊢ IsZeroOrProbabilityMeasure (Measure.map f μ)

Tactic:
·

Goal state after:
case pos
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s : Set α
inst✝ : IsZeroOrProbabilityMeasure μ
p : α → Prop
f✝ : β → α
f : α → β
hf : AEMeasurable f μ
⊢ IsZeroOrProbabilityMeasure (Measure.map f μ)
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Probability

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s : Set α
inst✝ : IsZeroOrProbabilityMeasure μ
p : α → Prop
f✝ : β → α
f : α → β
hf : ¬AEMeasurable f μ
⊢ IsZeroOrProbabilityMeasure (Measure.map f μ)

Tactic:
·

Goal state after:
case neg
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s : Set α
inst✝ : IsZeroOrProbabilityMeasure μ
p : α → Prop
f✝ : β → α
f : α → β
hf : ¬AEMeasurable f μ
⊢ IsZeroOrProbabilityMeasure (Measure.map f μ)
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Probability

Goal state before:
pp: case inl
α : Type u_1
m0 : MeasurableSpace α
s : Set α
p : ℝ≥0∞
s_mble : MeasurableSet s
inst✝ : IsZeroOrProbabilityMeasure 0
hμs : p < 0 s
⊢ 0 sᶜ < 1 - p

pp: case inr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : IsZeroOrProbabilityMeasure μ
p : ℝ≥0∞
hμs : p < μ s
s_mble : MeasurableSet s
h : IsProbabilityMeasure μ
⊢ μ sᶜ < 1 - p

Tactic:
·

Goal state after:
case inl
α : Type u_1
m0 : MeasurableSpace α
s : Set α
p : ℝ≥0∞
s_mble : MeasurableSet s
inst✝ : IsZeroOrProbabilityMeasure 0
hμs : p < 0 s
⊢ 0 sᶜ < 1 - p
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Probability

Goal state before:
pp: case inr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : IsZeroOrProbabilityMeasure μ
p : ℝ≥0∞
hμs : p < μ s
s_mble : MeasurableSet s
h : IsProbabilityMeasure μ
⊢ μ sᶜ < 1 - p

Tactic:
·

Goal state after:
case inr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : IsZeroOrProbabilityMeasure μ
p : ℝ≥0∞
hμs : p < μ s
s_mble : MeasurableSet s
h : IsProbabilityMeasure μ
⊢ μ sᶜ < 1 - p
Module: Mathlib.MeasureTheory.Measure.MutuallySingular

Goal state before:
pp: case refine_1
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
h : Disjoint μ ν
ε : ℝ≥0
hε : 0 < ε
h₁ : sInf {m | ∃ t, m = μ t + ν tᶜ} = 0
t : ℕ → Set α
ht₂ : ∀ (n : ℕ), μ (t n) + ν (t n)ᶜ < ↑ε * (1 / 2) ^ n
⊢ μ (⋂ n, t n) = 0

pp: case refine_2
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
h : Disjoint μ ν
ε : ℝ≥0
hε : 0 < ε
h₁ : sInf {m | ∃ t, m = μ t + ν tᶜ} = 0
t : ℕ → Set α
ht₂ : ∀ (n : ℕ), μ (t n) + ν (t n)ᶜ < ↑ε * (1 / 2) ^ n
⊢ ν (⋂ n, t n)ᶜ ≤ 2 * ↑ε

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
h : Disjoint μ ν
ε : ℝ≥0
hε : 0 < ε
h₁ : sInf {m | ∃ t, m = μ t + ν tᶜ} = 0
t : ℕ → Set α
ht₂ : ∀ (n : ℕ), μ (t n) + ν (t n)ᶜ < ↑ε * (1 / 2) ^ n
⊢ μ (⋂ n, t n) = 0
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Probability

Goal state before:
pp: case inl
α : Type u_1
m0 : MeasurableSpace α
s : Set α
p : ℝ≥0∞
s_mble : MeasurableSet s
inst✝ : IsZeroOrProbabilityMeasure 0
hμs : p ≤ 0 s
⊢ 0 sᶜ ≤ 1 - p

pp: case inr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : IsZeroOrProbabilityMeasure μ
p : ℝ≥0∞
hμs : p ≤ μ s
s_mble : MeasurableSet s
h : IsProbabilityMeasure μ
⊢ μ sᶜ ≤ 1 - p

Tactic:
·

Goal state after:
case inl
α : Type u_1
m0 : MeasurableSpace α
s : Set α
p : ℝ≥0∞
s_mble : MeasurableSet s
inst✝ : IsZeroOrProbabilityMeasure 0
hμs : p ≤ 0 s
⊢ 0 sᶜ ≤ 1 - p
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Probability

Goal state before:
pp: case inr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : IsZeroOrProbabilityMeasure μ
p : ℝ≥0∞
hμs : p ≤ μ s
s_mble : MeasurableSet s
h : IsProbabilityMeasure μ
⊢ μ sᶜ ≤ 1 - p

Tactic:
·

Goal state after:
case inr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : IsZeroOrProbabilityMeasure μ
p : ℝ≥0∞
hμs : p ≤ μ s
s_mble : MeasurableSet s
h : IsProbabilityMeasure μ
⊢ μ sᶜ ≤ 1 - p
Module: Mathlib.MeasureTheory.Measure.MutuallySingular

Goal state before:
pp: case refine_2
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
h : Disjoint μ ν
ε : ℝ≥0
hε : 0 < ε
h₁ : sInf {m | ∃ t, m = μ t + ν tᶜ} = 0
t : ℕ → Set α
ht₂ : ∀ (n : ℕ), μ (t n) + ν (t n)ᶜ < ↑ε * (1 / 2) ^ n
⊢ ν (⋂ n, t n)ᶜ ≤ 2 * ↑ε

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
h : Disjoint μ ν
ε : ℝ≥0
hε : 0 < ε
h₁ : sInf {m | ∃ t, m = μ t + ν tᶜ} = 0
t : ℕ → Set α
ht₂ : ∀ (n : ℕ), μ (t n) + ν (t n)ᶜ < ↑ε * (1 / 2) ^ n
⊢ ν (⋂ n, t n)ᶜ ≤ 2 * ↑ε
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: case neg
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x ↦ if x ∈ s then x else ⋯.some
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)
xs : x ∉ s
ys : y ∉ s
⊢ firstDiff x y ≤ firstDiff (f x) (f y)

Tactic:
·

Goal state after:
case neg
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x ↦ if x ∈ s then x else ⋯.some
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)
xs : x ∉ s
ys : y ∉ s
⊢ firstDiff x y ≤ firstDiff (f x) (f y)
Module: Mathlib.MeasureTheory.Measure.Dirac

Goal state before:
pp: case pos
α : Type u_1
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSingletonClass α
a : α
s : Set α
h : a ∈ s
⊢ (dirac a) s = s.indicator 1 a

pp: case neg
α : Type u_1
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSingletonClass α
a : α
s : Set α
h : a ∉ s
⊢ (dirac a) s = s.indicator 1 a

Tactic:
·

Goal state after:
case pos
α : Type u_1
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSingletonClass α
a : α
s : Set α
h : a ∈ s
⊢ (dirac a) s = s.indicator 1 a
Module: Mathlib.MeasureTheory.Measure.MutuallySingular

Goal state before:
pp: case refine_1
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
h : Disjoint μ ν
s : ℕ → Set α
hs₂ : ∀ (n : ℕ), μ (s n) = 0
hs₃ : ∀ (n : ℕ), ν (s n)ᶜ ≤ (1 / 2) ^ n
⊢ ν (⋃ n, s n)ᶜ = 0

pp: case refine_2
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
h : Disjoint μ ν
s : ℕ → Set α
hs₂ : ∀ (n : ℕ), μ (s n) = 0
hs₃ : ∀ (n : ℕ), ν (s n)ᶜ ≤ (1 / 2) ^ n
⊢ univ ⊆ (⋃ i, s i) ∪ (⋃ n, s n)ᶜ

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
h : Disjoint μ ν
s : ℕ → Set α
hs₂ : ∀ (n : ℕ), μ (s n) = 0
hs₃ : ∀ (n : ℕ), ν (s n)ᶜ ≤ (1 / 2) ^ n
⊢ ν (⋃ n, s n)ᶜ = 0
Module: Mathlib.MeasureTheory.Measure.Dirac

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
c : β
s : Set β
hs : MeasurableSet s
hsc : c ∈ s
⊢ μ (if c ∈ s then univ else ∅) = μ univ * s.indicator 1 c

pp: case neg
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
c : β
s : Set β
hs : MeasurableSet s
hsc : c ∉ s
⊢ μ (if c ∈ s then univ else ∅) = μ univ * s.indicator 1 c

Tactic:
·

Goal state after:
case pos
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
c : β
s : Set β
hs : MeasurableSet s
hsc : c ∈ s
⊢ μ (if c ∈ s then univ else ∅) = μ univ * s.indicator 1 c
Module: Mathlib.MeasureTheory.Measure.Dirac

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
c : β
s : Set β
hs : MeasurableSet s
hsc : c ∉ s
⊢ μ (if c ∈ s then univ else ∅) = μ univ * s.indicator 1 c

Tactic:
·

Goal state after:
case neg
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
c : β
s : Set β
hs : MeasurableSet s
hsc : c ∉ s
⊢ μ (if c ∈ s then univ else ∅) = μ univ * s.indicator 1 c
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: case pos
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x ↦ if x ∈ s then x else ⋯.some
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)
xs : x ∉ s
ys : y ∉ s
Ax : (s ∩ cylinder x (longestPrefix x s)).Nonempty
fx : f x = Ax.some
Ay : (s ∩ cylinder y (longestPrefix y s)).Nonempty
fy : f y = Ay.some
H : longestPrefix x s < firstDiff x y ∨ longestPrefix y s < firstDiff x y
⊢ firstDiff x y ≤ firstDiff (f x) (f y)

pp: case neg
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x ↦ if x ∈ s then x else ⋯.some
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)
xs : x ∉ s
ys : y ∉ s
Ax : (s ∩ cylinder x (longestPrefix x s)).Nonempty
fx : f x = Ax.some
Ay : (s ∩ cylinder y (longestPrefix y s)).Nonempty
fy : f y = Ay.some
H : firstDiff x y ≤ longestPrefix x s ∧ firstDiff x y ≤ longestPrefix y s
⊢ firstDiff x y ≤ firstDiff (f x) (f y)

Tactic:
·

Goal state after:
case pos
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x ↦ if x ∈ s then x else ⋯.some
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)
xs : x ∉ s
ys : y ∉ s
Ax : (s ∩ cylinder x (longestPrefix x s)).Nonempty
fx : f x = Ax.some
Ay : (s ∩ cylinder y (longestPrefix y s)).Nonempty
fy : f y = Ay.some
H : longestPrefix x s < firstDiff x y ∨ longestPrefix y s < firstDiff x y
⊢ firstDiff x y ≤ firstDiff (f x) (f y)
Module: Mathlib.MeasureTheory.Measure.Dirac

Goal state before:
pp: case pos
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
a : α
s : Set α
hs : MeasurableSet s
ha : a ∈ s
⊢ (μ.restrict {a}) s = (μ {a} • dirac a) s

pp: case neg
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
a : α
s : Set α
hs : MeasurableSet s
ha : a ∉ s
⊢ (μ.restrict {a}) s = (μ {a} • dirac a) s

Tactic:
·

Goal state after:
case pos
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
a : α
s : Set α
hs : MeasurableSet s
ha : a ∈ s
⊢ (μ.restrict {a}) s = (μ {a} • dirac a) s
Module: Mathlib.MeasureTheory.Measure.Dirac

Goal state before:
pp: case neg
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
a : α
s : Set α
hs : MeasurableSet s
ha : a ∉ s
⊢ (μ.restrict {a}) s = (μ {a} • dirac a) s

Tactic:
·

Goal state after:
case neg
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
a : α
s : Set α
hs : MeasurableSet s
ha : a ∉ s
⊢ (μ.restrict {a}) s = (μ {a} • dirac a) s
Module: Mathlib.MeasureTheory.Measure.MutuallySingular

Goal state before:
pp: case refine_2
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
h : Disjoint μ ν
s : ℕ → Set α
hs₂ : ∀ (n : ℕ), μ (s n) = 0
hs₃ : ∀ (n : ℕ), ν (s n)ᶜ ≤ (1 / 2) ^ n
⊢ univ ⊆ (⋃ i, s i) ∪ (⋃ n, s n)ᶜ

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
h : Disjoint μ ν
s : ℕ → Set α
hs₂ : ∀ (n : ℕ), μ (s n) = 0
hs₃ : ∀ (n : ℕ), ν (s n)ᶜ ≤ (1 / 2) ^ n
⊢ univ ⊆ (⋃ i, s i) ∪ (⋃ n, s n)ᶜ