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 : x ≠ f y
I2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)
xs : x ∈ s
ys : y ∉ s
A : (s ∩ cylinder y (longestPrefix y s)).Nonempty
fy : f y = A.some
I : firstDiff x y ≤ firstDiff x (f y)
⊢ firstDiff x y ≤ firstDiff 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 : x ≠ f y
I2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)
xs : x ∈ s
ys : y ∉ s
A : (s ∩ cylinder y (longestPrefix y s)).Nonempty
fy : f y = A.some
I : firstDiff x y ≤ firstDiff x (f y)
⊢ firstDiff x y ≤ firstDiff x (f y)
Module: Mathlib.MeasureTheory.Measure.MutuallySingular

Goal state before:
pp: 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
⊢ μ (s ∩ tᶜ) = μ (s ∩ tᶜ)

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)

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.MeasureTheory.Measure.MutuallySingular

Goal state before:
pp: 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
this : μ s = μ (s ∩ tᶜ)
⊢ μ (s ∩ tᶜ) = 0

Tactic:
]

Goal state after:
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
this : μ s = μ (s ∩ tᶜ)
⊢ μ (s ∩ tᶜ) = 0
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
⊢ 0 = 0

Tactic:
]

Goal state after:
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
⊢ 0 = 0

Tactic:
]

Goal state after:
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 := (n : ι) → β n
t : Set γ := ⋂ n, {x | f n (x n) = f i₀ (x i₀)}
t_closed : IsClosed[Pi.topologicalSpace] t
this✝ : PolishSpace ↑t
F : ↑t → α := fun x ↦ f i₀ (↑x i₀)
F_cont : Continuous[instTopologicalSpaceSubtype, inst✝²] F
x : ↑t
n : ι
this : f n (↑x n) = F x
⊢ f n (↑x n) ∈ range (f n)

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 := (n : ι) → β n
t : Set γ := ⋂ n, {x | f n (x n) = f i₀ (x i₀)}
t_closed : IsClosed[Pi.topologicalSpace] t
this✝ : PolishSpace ↑t
F : ↑t → α := fun x ↦ f i₀ (↑x i₀)
F_cont : Continuous[instTopologicalSpaceSubtype, inst✝²] F
x : ↑t
n : ι
this : f n (↑x n) = F x
⊢ f n (↑x n) ∈ range (f n)
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: 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
A : (s ∩ cylinder x (longestPrefix x s)).Nonempty
fx : f x = A.some
⊢ A.some ∈ cylinder x (firstDiff x y)

Tactic:
]

Goal state after:
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
A : (s ∩ cylinder x (longestPrefix x s)).Nonempty
fx : f x = A.some
⊢ A.some ∈ cylinder x (firstDiff x y)
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Probability

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsProbabilityMeasure μ
maybe_empty : IsEmpty α
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Probability

Goal state before:
pp: α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
f : α → β
inst✝ : IsProbabilityMeasure (map f μ)
hf : AEMeasurable f μ
⊢ (map f μ) univ = 1

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
f : α → β
inst✝ : IsProbabilityMeasure (map f μ)
hf : AEMeasurable f μ
⊢ (map f μ) univ = 1
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Probability

Goal state before:
pp: α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s : Set α
inst✝ : IsProbabilityMeasure μ
p : α → Prop
f✝ : β → α
f : β ≃ᵐ α
⊢ IsProbabilityMeasure (Measure.map (⇑f.symm) μ)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s : Set α
inst✝ : IsProbabilityMeasure μ
p : α → Prop
f✝ : β → α
f : β ≃ᵐ α
⊢ IsProbabilityMeasure (Measure.map (⇑f.symm) μ)
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Probability

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : IsProbabilityMeasure μ
h : NullMeasurableSet s μ
⊢ 1 - μ s = 1 - μ s

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Probability

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : IsProbabilityMeasure μ
hs : NullMeasurableSet s μ
⊢ μ s = 1 ↔ μ s = 1

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Probability

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : IsProbabilityMeasure μ
hs : NullMeasurableSet s μ
⊢ μ s = 0 ↔ μ s = 0

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Probability

Goal state before:
pp: α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
inst✝ : IsProbabilityMeasure μ
f : β → α
hf : Injective f
hf' : ∀ᵐ (a : α) ∂μ, a ∈ range f
hf'' : ∀ (s : Set β), MeasurableSet s → MeasurableSet (f '' s)
⊢ f '' univ ∈ ae μ

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
inst✝ : IsProbabilityMeasure μ
f : β → α
hf : Injective f
hf' : ∀ᵐ (a : α) ∂μ, a ∈ range f
hf'' : ∀ (s : Set β), MeasurableSet s → MeasurableSet (f '' s)
⊢ f '' univ ∈ ae μ
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 ≠ y
I2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)
xs : x ∉ s
ys : y ∈ s
A : (s ∩ cylinder x (longestPrefix x s)).Nonempty
fx : f x = A.some
I : cylinder A.some (firstDiff x y) = cylinder x (firstDiff x y)
⊢ firstDiff x y ≤ firstDiff (f x) 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 ≠ y
I2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)
xs : x ∉ s
ys : y ∈ s
A : (s ∩ cylinder x (longestPrefix x s)).Nonempty
fx : f x = A.some
I : cylinder A.some (firstDiff x y) = cylinder x (firstDiff x y)
⊢ firstDiff x y ≤ firstDiff (f x) y
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
⊢ 1 - μ s < 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
⊢ 1 - μ s < 1 - p
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 ≠ y
I2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)
xs : x ∉ s
ys : y ∈ s
A : (s ∩ cylinder x (longestPrefix x s)).Nonempty
fx : f x = A.some
I : firstDiff x y ≤ firstDiff (f x) y
⊢ firstDiff x y ≤ firstDiff (f x) 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 ≠ y
I2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)
xs : x ∉ s
ys : y ∈ s
A : (s ∩ cylinder x (longestPrefix x s)).Nonempty
fx : f x = A.some
I : firstDiff x y ≤ firstDiff (f x) y
⊢ firstDiff x y ≤ firstDiff (f x) y
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 (⋃ i, (t i)ᶜ) ≤ ∑' (n : ℕ), ↑ε * (1 / 2) ^ n

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 (⋃ i, (t i)ᶜ) ≤ ∑' (n : ℕ), ↑ε * (1 / 2) ^ n
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: 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 := (n : ι) → β n
t : Set γ := ⋂ n, {x | f n (x n) = f i₀ (x i₀)}
t_closed : IsClosed[Pi.topologicalSpace] t
this : PolishSpace ↑t
F : ↑t → α := fun x ↦ f i₀ (↑x i₀)
F_cont : Continuous[instTopologicalSpaceSubtype, inst✝²] F
y : α
hy : y ∈ ⋂ n, s n
n : ι
⊢ y ∈ s n

Tactic:
]

Goal state after:
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 := (n : ι) → β n
t : Set γ := ⋂ n, {x | f n (x n) = f i₀ (x i₀)}
t_closed : IsClosed[Pi.topologicalSpace] t
this : PolishSpace ↑t
F : ↑t → α := fun x ↦ f i₀ (↑x i₀)
F_cont : Continuous[instTopologicalSpaceSubtype, inst✝²] F
y : α
hy : y ∈ ⋂ n, s n
n : ι
⊢ y ∈ s n
Module: Mathlib.MeasureTheory.Measure.Dirac

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
s : Set α
a : α
⊢ (if a ∈ toMeasurable (dirac a) s then 1 a else 0) = 0 ∨ (if a ∈ toMeasurable (dirac a) s then 1 a else 0) = 1

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
s : Set α
a : α
⊢ (if a ∈ toMeasurable (dirac a) s then 1 a else 0) = 0 ∨ (if a ∈ toMeasurable (dirac a) s then 1 a else 0) = 1
Module: Mathlib.MeasureTheory.Measure.Dirac

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
s : Set α
a : α
h : a ∈ s
this : ∀ (t : Set α), a ∈ t → t.indicator 1 a = 1
⊢ (dirac a) s ≤ (dirac a) univ

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
s : Set α
a : α
h : a ∈ s
this : ∀ (t : Set α), a ∈ t → t.indicator 1 a = 1
⊢ (dirac a) s ≤ (dirac a) univ
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
⊢ 1 = 1

Tactic:
]

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

Goal state before:
pp: 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
⊢ f y = Ay.some

Tactic:
simp_rw [f, if_neg ys]

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

Goal state before:
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 ≤ 0

Tactic:
]

Goal state after:
case neg
α : Type u_1
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSingletonClass α
a : α
s : Set α
h : a ∉ s
⊢ (dirac a) s ≤ 0
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 (⋂ i, (s i)ᶜ) = 0

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 (⋂ i, (s i)ᶜ) = 0
Module: Mathlib.MeasureTheory.Measure.Dirac

Goal state before:
pp: 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
⊢ μ (if c ∈ s then univ else ∅) = μ univ * s.indicator 1 c

Tactic:
]

Goal state after:
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
⊢ μ (if c ∈ s then univ else ∅) = μ univ * s.indicator 1 c
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 univ = μ univ

Tactic:
]

Goal state after:
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
⊢ 0 = 0

Tactic:
]

Goal state after:
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
n : ℕ
⊢ ν (⋂ i, (s i)ᶜ) ≤ (1 / 2) ^ n

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
n : ℕ
⊢ ν (⋂ i, (s i)ᶜ) ≤ (1 / 2) ^ n
Module: Mathlib.MeasureTheory.Measure.Dirac

Goal state before:
pp: α : Type u_1
inst✝³ : MeasurableSpace α
inst✝² : Countable α
μ1 μ2 : Measure α
inst✝¹ : SigmaFinite μ1
inst✝ : SigmaFinite μ2
⊢ (∀ (a : α), μ1 {a} = μ2 {a}) ↔ ∀ (x : α), μ1.real {x} = μ2.real {x}

Tactic:
]

Goal state after:
α : Type u_1
inst✝³ : MeasurableSpace α
inst✝² : Countable α
μ1 μ2 : Measure α
inst✝¹ : SigmaFinite μ1
inst✝ : SigmaFinite μ2
⊢ (∀ (a : α), μ1 {a} = μ2 {a}) ↔ ∀ (x : α), μ1.real {x} = μ2.real {x}
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 ⊆ univ

Tactic:
]

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

Goal state before:
pp: α : Type u_1
inst✝³ : MeasurableSpace α
inst✝² : Countable α
μ1 μ2 : Measure α
inst✝¹ : SigmaFinite μ1
inst✝ : SigmaFinite μ2
x : α
⊢ μ1 {x} = μ2 {x} ↔ μ1 {x} = μ2 {x} ∨ μ1 {x} = 0 ∧ μ2 {x} = ∞ ∨ μ1 {x} = ∞ ∧ μ2 {x} = 0

Tactic:
]

Goal state after:
α : Type u_1
inst✝³ : MeasurableSpace α
inst✝² : Countable α
μ1 μ2 : Measure α
inst✝¹ : SigmaFinite μ1
inst✝ : SigmaFinite μ2
x : α
⊢ μ1 {x} = μ2 {x} ↔ μ1 {x} = μ2 {x} ∨ μ1 {x} = 0 ∧ μ2 {x} = ∞ ∨ μ1 {x} = ∞ ∧ μ2 {x} = 0
Module: Mathlib.MeasureTheory.Measure.MutuallySingular

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
h : μ ⟂ₘ ν
ξ : Measure α
⊢ ξ = ⊥ ↔ ξ = 0

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
h : μ ⟂ₘ ν
ξ : Measure α
⊢ ξ = ⊥ ↔ ξ = 0
Module: Mathlib.MeasureTheory.Measure.Dirac

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
inst✝ : Countable α
μ : Measure α
measurableAtoms : Set (Set α) := measurableAtom '' univ
s : ↑measurableAtoms
y : α
left✝ : y ∈ univ
hy : measurableAtom y = ↑s
⊢ (measurableAtom y).Nonempty

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : MeasurableSpace α
inst✝ : Countable α
μ : Measure α
measurableAtoms : Set (Set α) := measurableAtom '' univ
s : ↑measurableAtoms
y : α
left✝ : y ∈ univ
hy : measurableAtom y = ↑s
⊢ (measurableAtom y).Nonempty
Module: Mathlib.MeasureTheory.Measure.Dirac

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
inst✝ : Countable α
μ : Measure α
measurableAtoms : Set (Set α) := measurableAtom '' univ
h_nonempty : ∀ (s : ↑measurableAtoms), (↑s).Nonempty
points : ↑measurableAtoms → α := fun s ↦ ⋯.some
h_points_mem : ∀ (s : ↑measurableAtoms), points s ∈ ↑s
x : α
⊢ μ (measurableAtom x) = ∑' (i : ↑(range points)), (μ (measurableAtom ↑i) • dirac ↑i) (measurableAtom x)

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : MeasurableSpace α
inst✝ : Countable α
μ : Measure α
measurableAtoms : Set (Set α) := measurableAtom '' univ
h_nonempty : ∀ (s : ↑measurableAtoms), (↑s).Nonempty
points : ↑measurableAtoms → α := fun s ↦ ⋯.some
h_points_mem : ∀ (s : ↑measurableAtoms), points s ∈ ↑s
x : α
⊢ μ (measurableAtom x) = ∑' (i : ↑(range points)), (μ (measurableAtom ↑i) • dirac ↑i) (measurableAtom x)
Module: Mathlib.MeasureTheory.Measure.MutuallySingular

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
h : μ ⟂ₘ ν
h_bot_iff : ∀ (ξ : Measure α), ξ ≤ ⊥ ↔ ξ = 0
ξ : Measure α
hξμ : ξ ≤ μ
hξν : ξ ≤ ν
⊢ ξ = 0

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
h : μ ⟂ₘ ν
h_bot_iff : ∀ (ξ : Measure α), ξ ≤ ⊥ ↔ ξ = 0
ξ : Measure α
hξμ : ξ ≤ μ
hξν : ξ ≤ ν
⊢ ξ = 0
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: 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 := (n : ι) → β n
t : Set γ := ⋂ n, {x | f n (x n) = f i₀ (x i₀)}
t_closed : IsClosed[Pi.topologicalSpace] t
this : PolishSpace ↑t
F : ↑t → α := fun x ↦ f i₀ (↑x i₀)
F_cont : Continuous[instTopologicalSpaceSubtype, inst✝²] F
F_range : range F = ⋂ n, s n
⊢ AnalyticSet (range F)

Tactic:
]

Goal state after:
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 := (n : ι) → β n
t : Set γ := ⋂ n, {x | f n (x n) = f i₀ (x i₀)}
t_closed : IsClosed[Pi.topologicalSpace] t
this : PolishSpace ↑t
F : ↑t → α := fun x ↦ f i₀ (↑x i₀)
F_cont : Continuous[instTopologicalSpaceSubtype, inst✝²] F
F_range : range F = ⋂ n, s n
⊢ AnalyticSet (range F)
Module: Mathlib.MeasureTheory.Measure.MutuallySingular

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
h : μ ⟂ₘ ν
h_bot_iff : ∀ (ξ : Measure α), ξ ≤ ⊥ ↔ ξ = 0
ξ : Measure α
hξμ : ξ ≤ μ
hξν : ξ ≤ ν
s : Set α
hs : MeasurableSet s
⊢ ξ (s ∩ h.nullSet) = 0 ∧ ξ (s ∩ h.nullSetᶜ) = 0

pp: case hd
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
h : μ ⟂ₘ ν
h_bot_iff : ∀ (ξ : Measure α), ξ ≤ ⊥ ↔ ξ = 0
ξ : Measure α
hξμ : ξ ≤ μ
hξν : ξ ≤ ν
s : Set α
hs : MeasurableSet s
⊢ Disjoint (s ∩ h.nullSet) (s ∩ h.nullSetᶜ)

pp: case h
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
h : μ ⟂ₘ ν
h_bot_iff : ∀ (ξ : Measure α), ξ ≤ ⊥ ↔ ξ = 0
ξ : Measure α
hξμ : ξ ≤ μ
hξν : ξ ≤ ν
s : Set α
hs : MeasurableSet s
⊢ MeasurableSet (s ∩ h.nullSetᶜ)

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
h : μ ⟂ₘ ν
h_bot_iff : ∀ (ξ : Measure α), ξ ≤ ⊥ ↔ ξ = 0
ξ : Measure α
hξμ : ξ ≤ μ
hξν : ξ ≤ ν
s : Set α
hs : MeasurableSet s
⊢ ξ (s ∩ h.nullSet) = 0 ∧ ξ (s ∩ h.nullSetᶜ) = 0

case hd
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
h : μ ⟂ₘ ν
h_bot_iff : ∀ (ξ : Measure α), ξ ≤ ⊥ ↔ ξ = 0
ξ : Measure α
hξμ : ξ ≤ μ
hξν : ξ ≤ ν
s : Set α
hs : MeasurableSet s
⊢ Disjoint (s ∩ h.nullSet) (s ∩ h.nullSetᶜ)

case h
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
h : μ ⟂ₘ ν
h_bot_iff : ∀ (ξ : Measure α), ξ ≤ ⊥ ↔ ξ = 0
ξ : Measure α
hξμ : ξ ≤ μ
hξν : ξ ≤ ν
s : Set α
hs : MeasurableSet s
⊢ MeasurableSet (s ∩ h.nullSetᶜ)