mathlib-initiative/mathlib-tactics huggingface dataset:
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) → PseudoMetricSpace (F i)
x y : (i : ι) → F i
i : ι
h : dist x y < 2⁻¹ ^ encode i
⊢ dist (x i) (y i) ≤ dist x y

Tactic:
by

Goal state after:
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 ≤ dist x y

Tactic:
by

Goal state after:
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✝¹ y✝ : (i : ι) → F i
i : ι
x y : (i : ι) → F i
x✝ : ι
⊢ 0 ≤ min (2⁻¹ ^ encode x✝) (dist (x x✝) (y x✝))

Tactic:
by

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
m : MeasurableSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : T2Space Y
μ : Measure X
inst✝ : μ.IsOpenPosMeasure
U : Set X
f g : X → Y
hU : IsOpen[inst✝³] U
hf : ContinuousOn f U
hg : ContinuousOn g U
h : μ {a | a ∈ U ∧ ¬f a = g a} = 0
⊢ IsOpen[inst✝³] (U ∩ {a | f a ≠ g a})

Tactic:
by

Goal state after:
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
⊢ edist x y = ENNReal.ofReal (dist x y)

Tactic:
by

Goal state after:
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) → MetricSpace (F i)
⊢ ∀ (x y : (i : ι) → F i), 0 ≤ dist x y

Tactic:
by

Goal state after:
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) → MetricSpace (F i)
⊢ ∀ (x y : (i : ι) → F i), edist x y = ENNReal.ofReal (dist x y)

Tactic:
by

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
x y : PiNatEmbed X Y f
hxy : x.ofPiNat = y.ofPiNat
⊢ x = y

Tactic:
by

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
m : MeasurableSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : T2Space Y
μ : Measure X
inst✝ : μ.IsOpenPosMeasure
U : Set X
f g : X → Y
h : f =ᵐ[μ.restrict U] g
hU : IsOpen[inst✝³] U
hf : ContinuousOn f U
hg : ContinuousOn g U
⊢ EqOn f g U

Tactic:
by

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
separating_f : Pairwise fun x y ↦ ∃ i, f i x ≠ f i y
⊢ Injective (embed X Y f)

Tactic:
by

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✝² : Encodable ι
inst✝¹ : (i : ι) → PseudoMetricSpace (Y i)
inst✝ : TopologicalSpace X
continuous_f : ∀ (i : ι), Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f i)
⊢ ∀ (i : ι), Continuous[instTopologicalSpaceProd, _] fun x ↦ min (2⁻¹ ^ encode i) (dist (f i x.1) (f i x.2))

Tactic:
by

Goal state after:
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
⊢ (map f μ).IsOpenPosMeasure

Tactic:
by

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✝² : Encodable ι
inst✝¹ : (i : ι) → PseudoMetricSpace (Y i)
inst✝ : TopologicalSpace X
continuous_f : ∀ (i : ι), Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f i)
⊢ ∀ (n : ι) (x : X × X), ‖min (2⁻¹ ^ encode n) (dist (f n x.1) (f n x.2))‖ ≤ (1 / 2) ^ encode n

Tactic:
by

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✝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[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] toPiNat

Tactic:
by

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

Goal state before:
pp: E : ℕ → Type u_1
ι : Type u_2
X : Type u_3
Y : ι → Type u_4
f : (i : ι) → X → Y i
inst✝¹ : Encodable ι
inst✝ : (i : ι) → MetricSpace (Y i)
separating_f : Pairwise fun x y ↦ ∃ i, f i x ≠ f i y
x y : PiNatEmbed X Y f
⊢ edist x y ≠ ∞

Tactic:
by

Goal state after:
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
⊢ (comap f μ) U ≠ 0

Tactic:
by

Goal state after:
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
⊢ 0 < μ (closedBall x r) ↔ 0 < r

Tactic:
by

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
⊢ IsHomeomorph toPiNat

Tactic:
isHomeomorph_iff_continuous_bijective

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
n : ℕ
⊢ IsNowhereDense (K n)

Tactic:
by

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
⊢ IsHomeomorph toPiNat

Tactic:
by

Goal state after:
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
h₁s : IsSigmaCompact s
h₂s : μ s = 0
⊢ IsMeagre s

Tactic:
by

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

Goal state before:
pp: X : Type u_3
inst✝¹ : MetricSpace X
inst✝ : SeparableSpace X
n : ℕ
⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] (distDenseSeq X n)

Tactic:
by

Goal state after:
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)
⊢ ∃ C, ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), ∀ t ≤ K, μ (closedBall x (t * ε)) ≤ ↑C * μ (closedBall x ε)

Tactic:
by

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

Goal state before:
pp: X : Type u_3
inst✝¹ : MetricSpace X
inst✝ : SeparableSpace X
x : X
C : Set X
hxC : C ∈ 𝓝 x
ε : ℝ := min (infDist x (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ)) 1
hC : (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ).Nonempty
this : Nonempty X
⊢ ε / 2 > 0

Tactic:
by

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

Goal state before:
pp: X : Type u_3
inst✝¹ : MetricSpace X
inst✝ : SeparableSpace X
x : X
C : Set X
hxC : C ∈ 𝓝 x
ε : ℝ := min (infDist x (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ)) 1
hC : (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ).Nonempty
this : Nonempty X
n : ℕ
hn : dist x (denseSeq X n) < ε / 2
y : X
hy : y ∈ distDenseSeq X n ⁻¹' ball 0 (ε / 2)
⊢ dist y (denseSeq X n) < ε / 2

Tactic:
by

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

Goal state before:
pp: X : Type u_3
inst✝¹ : MetricSpace X
inst✝ : SeparableSpace X
x : X
C : Set X
hxC : C ∈ 𝓝 x
ε : ℝ := min (infDist x (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ)) 1
hC : (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ).Nonempty
this : Nonempty X
n : ℕ
hn : dist x (denseSeq X n) < ε / 2
y : X
hy : dist y (denseSeq X n) < ε / 2
⊢ ε / 2 + ε / 2 ≤ infDist x (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ)

Tactic:
by

Goal state after:
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, ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), ∀ t ≤ K, μ (closedBall x (t * ε)) ≤ ↑C * μ (closedBall x ε)

Tactic:
by

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

Goal state before:
pp: X : Type u_3
inst✝¹ : MetricSpace X
inst✝ : SeparableSpace X
x : X
C : Set X
hxC : C ∈ 𝓝 x
⊢ ∃ n, C ∈ comap (distDenseSeq X n) (𝓝 (distDenseSeq X n x))

Tactic:
by

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

Goal state before:
pp: X : Type u_3
inst✝¹ : MetricSpace X
inst✝ : SeparableSpace X
x y : X
hxy : x ≠ y
n : ℕ
hn : {y}ᶜ ∈ comap (distDenseSeq X n) (𝓝 (distDenseSeq X n x))
e : distDenseSeq X n x = distDenseSeq X n y
⊢ False

Tactic:
by

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

Goal state before:
pp: X : Type u_3
inst✝¹ : MetricSpace X
inst✝ : SeparableSpace X
x y : X
hxy : x ≠ y
⊢ ∃ n, distDenseSeq X n x ≠ distDenseSeq X n y

Tactic:
by

Goal state after:
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
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), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)

Tactic:
by

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

Goal state before:
pp: X : Type u_3
inst✝¹ : MetricSpace X
inst✝ : SeparableSpace X
⊢ Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ofPiNat

Tactic:
by

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

Goal state before:
pp: X : Type u_3
inst✝¹ : MetricSpace X
inst✝ : SeparableSpace X
⊢ ∃ F, IsEmbedding F

Tactic:
by

Goal state after:
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
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
M : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)
⊢ ∃ u v, IsOpen[inst✝³] u ∧ IsOpen[inst✝³] v ∧ f x ∈ u ∧ g y ∈ v ∧ Disjoint u v

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
ν : Measure β
inst✝ : IsFiniteMeasure ν
s : Set (α × β)
f : ℕ → Set (α × β)
hfd : Pairwise (Disjoint on f)
hfm : ∀ (i : ℕ), MeasurableSet (f i)
ihf : ∀ (i : ℕ), Measurable fun x ↦ ν (Prod.mk x ⁻¹' f i)
a : α
⊢ ν (Prod.mk a ⁻¹' ⋃ i, f i) = ∑' (i : ℕ), ν (Prod.mk a ⁻¹' f i)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
ν : Measure β
inst✝ : IsFiniteMeasure ν
s : Set (α × β)
hs : MeasurableSet s
⊢ Measurable fun x ↦ ν (Prod.mk x ⁻¹' s)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
s : Set (α × β)
hs : MeasurableSet s
⊢ Measurable fun x ↦ ν (Prod.mk x ⁻¹' s)

Tactic:
by

Goal state after:
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
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
M : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)
u v : Set α
u_open : IsOpen[inst✝³] u
v_open : IsOpen[inst✝³] v
xu : f x ∈ u
yv : g y ∈ v
huv : Disjoint u v
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
⊢ ∃ εx > 0, ball x εx ⊆ f ⁻¹' u

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
⊢ Measurable fun x ↦ map (Prod.mk x) ν

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
inst✝ : SFinite μ
⊢ Measurable fun y ↦ map (fun x ↦ (x, y)) μ

Tactic:
by

Goal state after:
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
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
M : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)
u v : Set α
u_open : IsOpen[inst✝³] u
v_open : IsOpen[inst✝³] v
xu : f x ∈ u
yv : g y ∈ v
huv : Disjoint u v
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
εx : ℝ
εxpos : εx > 0
hεx : ball x εx ⊆ f ⁻¹' u
⊢ ∃ εy > 0, ball y εy ⊆ g ⁻¹' v

Tactic:
by

Goal state after:
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
⊢ ∃ R,
    0 < R ∧
      ∀ (x : α) (t r : ℝ),
        t ∈ Ioc 0 K → r ≤ R → μ (closedBall x (t * r)) ≤ ↑(scalingConstantOf μ K) * μ (closedBall x r)

Tactic:
by

Goal state after:
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
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
M : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)
u v : Set α
u_open : IsOpen[inst✝³] u
v_open : IsOpen[inst✝³] v
xu : f x ∈ u
yv : g y ∈ v
huv : Disjoint u v
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
εx : ℝ
εxpos : εx > 0
hεx : ball x εx ⊆ f ⁻¹' u
εy : ℝ
εypos : εy > 0
hεy : ball y εy ⊆ g ⁻¹' v
⊢ 1 / 2 < 1

Tactic:
by

Goal state after:
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
⊢ ∀ᶠ (r : ℝ) in 𝓝[>] 0, ∀ (x : α), μ (closedBall x (K * r)) ≤ ↑(scalingConstantOf μ K) * μ (closedBall x r)

Tactic:
by

Goal state after:
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
hK : 0 < K
⊢ ∀ᶠ (r : ℝ) in 𝓝[>] 0, ∀ (x : α), μ (closedBall x r) ≤ ↑(scalingConstantOf μ K⁻¹) * μ (closedBall x (K * r))

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
m : ∀ {α : Type ?u.42} {β : Type ?u.41} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x)
c : ℝ≥0∞
s : Set (α × β)
hs : MeasurableSet s
this : Measurable fun x ↦ c * ν (Prod.mk x ⁻¹' s)
⊢ Measurable fun x ↦ ∫⁻ (y : β), (Prod.mk x ⁻¹' s).indicator ((fun x ↦ c) ∘ Prod.mk x) y ∂ν

Tactic:
by

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

Goal state before:
pp: M : Type u_1
inst✝² : Monoid M
inst✝¹ : MeasurableSpace M
inst✝ : MeasurableMul₂ M
μ ν : Measure M
f : M → ℝ≥0∞
hf : Measurable f
⊢ ∫⁻ (z : M), f z ∂μ ∗ₘ ν = ∫⁻ (z : M × M), f (z.1 * z.2) ∂μ.prod ν

Tactic:
by

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

Goal state before:
pp: M : Type u_1
inst✝³ : Monoid M
inst✝² : MeasurableSpace M
inst✝¹ : MeasurableMul₂ M
μ ν : Measure M
inst✝ : SFinite ν
f : M → ℝ≥0∞
hf : Measurable f
⊢ AEMeasurable (fun z ↦ f (z.1 * z.2)) (μ.prod ν)

Tactic:
by

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

Goal state before:
pp: M : Type u_1
inst✝³ : Monoid M
inst✝² : MeasurableSpace M
inst✝¹ : MeasurableMul₂ M
μ ν : Measure M
inst✝ : SFinite ν
f : M → ℝ≥0∞
hf : Measurable f
⊢ ∫⁻ (z : M), f z ∂μ ∗ₘ ν = ∫⁻ (x : M), ∫⁻ (y : M), f (x * y) ∂ν ∂μ

Tactic:
by

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

Goal state before:
pp: M : Type u_1
inst✝³ : Monoid M
inst✝² : MeasurableSpace M
inst✝¹ : MeasurableMul₂ M
x : M
μ : Measure M
inst✝ : SFinite μ
⊢ Measurable fun x ↦ x.1 * x.2

Tactic:
by

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

Goal state before:
pp: M : Type u_1
inst✝³ : Monoid M
inst✝² : MeasurableSpace M
inst✝¹ : MeasurableMul₂ M
x : M
μ : Measure M
inst✝ : SFinite μ
⊢ Measurable (Prod.mk x)

Tactic:
by

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

Goal state before:
pp: M : Type u_1
inst✝³ : Monoid M
inst✝² : MeasurableSpace M
inst✝¹ : MeasurableMul₂ M
x : M
μ : Measure M
inst✝ : SFinite μ
⊢ dirac x ∗ₘ μ = map (fun y ↦ x * y) μ

Tactic:
by

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

Goal state before:
pp: M : Type u_1
inst✝³ : Monoid M
inst✝² : MeasurableSpace M
inst✝¹ : MeasurableMul₂ M
μ : Measure M
inst✝ : SFinite μ
x : M
⊢ Measurable fun x ↦ x.1 * x.2

Tactic:
by

Goal state after: