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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Opens α
inst✝ : CompleteSpace α
u : ℕ → s.CompleteCopy
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < (1 / 2) ^ N
A : CauchySeq fun n ↦ ↑(u n)
x : α
xlim : Tendsto (fun n ↦ ↑(u n)) atTop (𝓝 x)
xs : x ∉ s
C : ℝ
hC : ∀ (n : ℕ), 1 / infDist (↑(u n)) (↑s)ᶜ < C
Cpos : 0 < C
y : α
⊢ y ∈ s ↔ 0 < infDist y (↑s)ᶜ

Tactic:
rw [← s.isOpen.isClosed_compl.notMem_iff_infDist_pos ⟨x, xs⟩]

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Opens α
inst✝ : CompleteSpace α
u : ℕ → s.CompleteCopy
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < (1 / 2) ^ N
A : CauchySeq fun n ↦ ↑(u n)
x : α
xlim : Tendsto (fun n ↦ ↑(u n)) atTop (𝓝 x)
xs : x ∉ s
C : ℝ
hC : ∀ (n : ℕ), 1 / infDist (↑(u n)) (↑s)ᶜ < C
Cpos : 0 < C
y : α
⊢ y ∈ s ↔ y ∉ (↑s)ᶜ
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: case h_nhds
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
x : (n : ℕ) → E n
u : Set ((n : ℕ) → E n)
hx : x ∈ u
u_open : IsOpen[Pi.topologicalSpace] u
U : (i : ℕ) → Set (E i)
F : Finset ℕ
xU : x ∈ (↑F).pi U
Uu : (↑F).pi U ⊆ u
n : ℕ
hn : n ∈ upperBounds ↑F
y : (n : ℕ) → E n
hy : y ∈ cylinder x (n + 1)
i : ℕ
⊢ i ∈ F → y i ∈ U i

Tactic:
hi

Goal state after:
case h_nhds
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
x : (n : ℕ) → E n
u : Set ((n : ℕ) → E n)
hx : x ∈ u
u_open : IsOpen[Pi.topologicalSpace] u
U : (i : ℕ) → Set (E i)
F : Finset ℕ
xU : x ∈ (↑F).pi U
Uu : (↑F).pi U ⊆ u
n : ℕ
hn : n ∈ upperBounds ↑F
y : (n : ℕ) → E n
hy : y ∈ cylinder x (n + 1)
i : ℕ
hi : i ∈ F
⊢ y i ∈ U i
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: X : Type u
Y : Type v
Z : Type w
inst✝³ : Nonempty Z
inst✝² : MetricSpace Z
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
Φ : Z → X
Ψ : Z → Y
hΦ : Isometry Φ
hΨ : Isometry Ψ
⊢ toGlueL hΦ hΨ ∘ Φ = toGlueR hΦ hΨ ∘ Ψ

Tactic:
let i : PseudoMetricSpace (X ⊕ Y) := gluePremetric hΦ hΨ

Goal state after:
X : Type u
Y : Type v
Z : Type w
inst✝³ : Nonempty Z
inst✝² : MetricSpace Z
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
Φ : Z → X
Ψ : Z → Y
hΦ : Isometry Φ
hΨ : Isometry Ψ
i : PseudoMetricSpace (X ⊕ Y) := gluePremetric hΦ hΨ
⊢ toGlueL hΦ hΨ ∘ Φ = toGlueR hΦ hΨ ∘ Ψ
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: E✝ : ℕ → Type u_1
inst✝² : (n : ℕ) → TopologicalSpace (E✝ n)
inst✝¹ : ∀ (n : ℕ), DiscreteTopology (E✝ n)
E : ℕ → Type u_2
inst✝ : (n : ℕ) → UniformSpace (E n)
h : ∀ (n : ℕ), uniformity (E n) = 𝓟 SetRel.id
this : ∀ (n : ℕ), DiscreteTopology (E n)
ε : ℝ
εpos : ε > 0
⊢ 1 / 2 < 1

Tactic:
norm_num

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

Goal state before:
pp: case h
β : Type u_1
α : Type u_2
A : List β → Set α
inst✝¹ : PseudoMetricSpace α
inst✝ : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : ℕ → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
umem : ∀ (n m : ℕ), n ≤ m → u m ∈ A (res x n)
ε : ℝ
ε_pos : ε > 0
n : ℕ
hn : ∀ y ∈ A (res x n), ∀ z ∈ A (res x n), dist y z < ε
⊢ ∀ m ≥ n, ∀ n_1 ≥ n, dist (u m) (u n_1) < ε

Tactic:
intro m₀

Goal state after:
case h
β : Type u_1
α : Type u_2
A : List β → Set α
inst✝¹ : PseudoMetricSpace α
inst✝ : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : ℕ → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
umem : ∀ (n m : ℕ), n ≤ m → u m ∈ A (res x n)
ε : ℝ
ε_pos : ε > 0
n : ℕ
hn : ∀ y ∈ A (res x n), ∀ z ∈ A (res x n), dist y z < ε
m₀ : ℕ
⊢ m₀ ≥ n → ∀ n_1 ≥ n, dist (u m₀) (u n_1) < ε
Module: Mathlib.Topology.MetricSpace.Perfect

Goal state before:
pp: case e'_4.ha₀
α : Type u_1
inst✝ : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
ε_pos : 0 < ε
x : α
xC : x ∈ C
this✝ : x ∈ eball x (ε / 2)
this : Perfect (closure (eball x (ε / 2) ∩ C)) ∧ (closure (eball x (ε / 2) ∩ C)).Nonempty
⊢ 2 ≠ 0

Tactic:
norm_num

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

Goal state before:
pp: case e'_4.ha
α : Type u_1
inst✝ : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
ε_pos : 0 < ε
x : α
xC : x ∈ C
this✝ : x ∈ eball x (ε / 2)
this : Perfect (closure (eball x (ε / 2) ∩ C)) ∧ (closure (eball x (ε / 2) ∩ C)).Nonempty
⊢ 2 ≠ ∞

Tactic:
norm_num

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

Goal state before:
pp: α : Type u_1
inst✝¹ : MetricSpace α
C : Set α
hC : Perfect C
hnonempty : C.Nonempty
inst✝ : CompleteSpace α
u : ℕ → ℝ≥0∞
upos' : ∀ (n : ℕ), u n ∈ Ioo 0 1
hu : Tendsto u atTop (nhds 0)
upos : ∀ (n : ℕ), 0 < u n
P : Type (max 0 u_1) := { E // Perfect E ∧ E.Nonempty }
C0 C1 : {C : Set α} → Perfect C → C.Nonempty → {ε : ℝ≥0∞} → 0 < ε → Set α
h0 :
  ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
    Perfect (C0 hC hnonempty hε) ∧
      (C0 hC hnonempty hε).Nonempty ∧ C0 hC hnonempty hε ⊆ C ∧ ediam (C0 hC hnonempty hε) ≤ ε
h1 :
  ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
    Perfect (C1 hC hnonempty hε) ∧
      (C1 hC hnonempty hε).Nonempty ∧ C1 hC hnonempty hε ⊆ C ∧ ediam (C1 hC hnonempty hε) ≤ ε
hdisj :
  ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
    Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)
l : List Bool
⊢ P

Tactic:
induction l with

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)
x y : (n : ℕ) → E n
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
H : longestPrefix x s < firstDiff x y
xs : x ∉ s
ys : y ∉ s
⊢ cylinder x (longestPrefix x s) = cylinder y (longestPrefix y s)

Tactic:
have l_eq : longestPrefix y s = longestPrefix x s := by
    rcases lt_trichotomy (longestPrefix y s) (longestPrefix x s) with (L | L | L)
    · have Ax : (s ∩ cylinder x (longestPrefix x s)).Nonempty :=
        inter_cylinder_longestPrefix_nonempty hs hne x
      have Z := disjoint_cylinder_of_longestPrefix_lt hs ys L
      rw [firstDiff_comm] at H
      rw [cylinder_eq_cylinder_of_le_firstDiff _ _ H.le] at Z
      exact (Ax.not_disjoint Z).elim
    · exact L
    · have Ay : (s ∩ cylinder y (longestPrefix y s)).Nonempty :=
        inter_cylinder_longestPrefix_nonempty hs hne y
      have A'y : (s ∩ cylinder y (longestPrefix x s).succ).Nonempty :=
        Ay.mono (inter_subset_inter_right s (cylinder_anti _ L))
      have Z := disjoint_cylinder_of_longestPrefix_lt hs xs (Nat.lt_succ_self _)
      rw [cylinder_eq_cylinder_of_le_firstDiff _ _ H] at Z
      exact (A'y.not_disjoint Z).elim

Goal state after:
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
x y : (n : ℕ) → E n
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
H : longestPrefix x s < firstDiff x y
xs : x ∉ s
ys : y ∉ s
l_eq : longestPrefix y s = longestPrefix x s
⊢ cylinder x (longestPrefix x s) = cylinder y (longestPrefix y s)
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Map

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
g : α → β
hg : MeasurableEmbedding g
f : β → ℝ≥0∞
f₀ : SimpleFunc α ℝ≥0∞
hf₀ : ⇑f₀ ≤ fun a ↦ f (g a)
⊢ ((f₀.extend g hg (const β 0)).comp g ⋯).lintegral μ ≤
    ⨆ g_1, ⨆ (_ : ⇑g_1 ≤ fun a ↦ f a), g_1.lintegral (Measure.map g μ)

Tactic:
← SimpleFunc.lintegral_map,

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
g : α → β
hg : MeasurableEmbedding g
f : β → ℝ≥0∞
f₀ : SimpleFunc α ℝ≥0∞
hf₀ : ⇑f₀ ≤ fun a ↦ f (g a)
⊢ (f₀.extend g hg (const β 0)).lintegral (Measure.map g μ) ≤
    ⨆ g_1, ⨆ (_ : ⇑g_1 ≤ fun a ↦ f a), g_1.lintegral (Measure.map g μ)
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: case mp
α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
⊢ AnalyticSet s → ∃ β h, ∃ (_ : PolishSpace β), ∃ f, Continuous[_, inst✝] f ∧ range f = s

Tactic:
intro h
    rw [AnalyticSet] at h
    rcases h with h | h
    · refine ⟨Empty, inferInstance, inferInstance, Empty.elim, continuous_bot, ?_⟩
      rw [h]
      exact range_eq_empty _
    · exact ⟨ℕ → ℕ, inferInstance, inferInstance, h⟩

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
hμs : μ s = 0
hμs_compl : μ sᶜ = 0
⊢ μ s + μ sᶜ = 0

Tactic:
hμs,

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
hμs : μ s = 0
hμs_compl : μ sᶜ = 0
⊢ 0 + μ sᶜ = 0
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
A : (s ∩ cylinder x (longestPrefix x s)).Nonempty
fx : f x = A.some
⊢ firstDiff x y ≤ firstDiff (f x) (f y)

Tactic:
have I : cylinder A.some (firstDiff x y) = cylinder x (firstDiff x y) := by
          rw [← mem_cylinder_iff_eq]
          apply cylinder_anti x _ A.some_mem.2
          apply firstDiff_le_longestPrefix hs xs ys

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
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) (f y)
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: case inl
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
⊢ cylinder x (longestPrefix x s) = cylinder y (longestPrefix y s)

pp: case inr
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 y s < firstDiff x y
⊢ cylinder x (longestPrefix x s) = cylinder y (longestPrefix y s)

Tactic:
· exact cylinder_longestPrefix_eq_of_longestPrefix_lt_firstDiff hs hne H xs ys

Goal state after:
case inr
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 y s < firstDiff x y
⊢ cylinder x (longestPrefix x s) = cylinder y (longestPrefix y s)
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, (∀ x ∈ s, f x = x) ∧ range f = s ∧ Continuous[Pi.topologicalSpace, Pi.topologicalSpace] f

Tactic:
rcases exists_lipschitz_retraction_of_isClosed hs hne with ⟨f, fs, frange, hf⟩
  exact ⟨f, fs, frange, hf.continuous⟩

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
⊢ 1 / 2 < 1

Tactic:
norm_num

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
y : α
⊢ ∃ a, g a = y

Tactic:
have : ∀ n : ℕ, ∃ j, y ∈ closedBall (u j) ((1 / 2) ^ n) := fun n ↦ by
      rcases hu.exists_dist_lt y (by simp : (0 : ℝ) < (1 / 2) ^ n) with ⟨j, hj⟩
      exact ⟨j, hj.le⟩

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
y : α
this : ∀ (n : ℕ), ∃ j, y ∈ closedBall (u j) ((1 / 2) ^ n)
⊢ ∃ a, g a = y
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)
ε : ℝ≥0∞
hε : 0 < ε
K : Finset ι
hK : ∑' (i : { j // j ∉ K }), 2⁻¹ ^ encode ↑i < ε / 2
δ : ℝ≥0∞
δpos : 0 < δ
hδ : δ * ↑K.card < ε / 2
x y : (i : ι) → F i
hxy : ∀ (x_1 : ι) (h : x_1 ∈ K), edist (x x_1) (y x_1) < δ
i : ι
hi : i ∈ K
⊢ edist (x i) (y i) ≤ δ

Tactic:
exact (hxy i hi).le

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) → PseudoEMetricSpace (F i)
i : ι
ε : ℝ≥0∞
hε₀ : 0 < ε
⊢ 0 < 2⁻¹

Tactic:
norm_num

Goal state after:
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
⊢ Summable fun i ↦ 2⁻¹ ^ encode i

Tactic:
simpa [one_div] using summable_geometric_two_encode

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

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

Tactic:
hi

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

Goal state before:
pp: case e'_4
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : T2Space α
inst✝¹ : MeasurableSpace α
inst✝ : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous[_, inst✝³] f
hg : Continuous[_, inst✝³] g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
I :
  ∀ (n : ℕ) (x y : ℕ → ℕ),
    ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →
      ∃ x' y',
        x' ∈ cylinder x n ∧
          y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))
A : Type := { p // ¬MeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }
F : A → A
hFn : ∀ (p : A), (↑(F p)).1 = (↑p).1 + 1
hFx : ∀ (p : A), (↑(F p)).2.1 ∈ cylinder (↑p).2.1 (↑p).1
hFy : ∀ (p : A), (↑(F p)).2.2 ∈ cylinder (↑p).2.2 (↑p).1
p0 : A := ⟨(0, fun x ↦ 0, fun x ↦ 0), ⋯⟩
p : ℕ → A := fun n ↦ F^[n] p0
prec : ∀ (n : ℕ), p (n + 1) = F (p n)
pn_fst : ∀ (n : ℕ), (↑(p n)).1 = n
Ix : ∀ (m n : ℕ), m + 1 ≤ n → (↑(p n)).2.1 m = (↑(p (m + 1))).2.1 m
Iy : ∀ (m n : ℕ), m + 1 ≤ n → (↑(p n)).2.2 m = (↑(p (m + 1))).2.2 m
x : ℕ → ℕ := fun n ↦ (↑(p (n + 1))).2.1 n
hx : x = fun n ↦ (↑(p (n + 1))).2.1 n
y : ℕ → ℕ := fun n ↦ (↑(p (n + 1))).2.2 n
hy : y = fun n ↦ (↑(p (n + 1))).2.2 n
n i : ℕ
⊢ i < n → y i = (↑(p n)).2.2 i

Tactic:
hi

Goal state after:
case e'_4
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : T2Space α
inst✝¹ : MeasurableSpace α
inst✝ : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous[_, inst✝³] f
hg : Continuous[_, inst✝³] g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
I :
  ∀ (n : ℕ) (x y : ℕ → ℕ),
    ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →
      ∃ x' y',
        x' ∈ cylinder x n ∧
          y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))
A : Type := { p // ¬MeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }
F : A → A
hFn : ∀ (p : A), (↑(F p)).1 = (↑p).1 + 1
hFx : ∀ (p : A), (↑(F p)).2.1 ∈ cylinder (↑p).2.1 (↑p).1
hFy : ∀ (p : A), (↑(F p)).2.2 ∈ cylinder (↑p).2.2 (↑p).1
p0 : A := ⟨(0, fun x ↦ 0, fun x ↦ 0), ⋯⟩
p : ℕ → A := fun n ↦ F^[n] p0
prec : ∀ (n : ℕ), p (n + 1) = F (p n)
pn_fst : ∀ (n : ℕ), (↑(p n)).1 = n
Ix : ∀ (m n : ℕ), m + 1 ≤ n → (↑(p n)).2.1 m = (↑(p (m + 1))).2.1 m
Iy : ∀ (m n : ℕ), m + 1 ≤ n → (↑(p n)).2.2 m = (↑(p (m + 1))).2.2 m
x : ℕ → ℕ := fun n ↦ (↑(p (n + 1))).2.1 n
hx : x = fun n ↦ (↑(p (n + 1))).2.1 n
y : ℕ → ℕ := fun n ↦ (↑(p (n + 1))).2.2 n
hy : y = fun n ↦ (↑(p (n + 1))).2.2 n
n i : ℕ
hi : i < n
⊢ y i = (↑(p n)).2.2 i
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
h : ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), ∀ t ≤ K, μ (closedBall x (t * ε)) ≤ ↑(Classical.choose ⋯) * μ (closedBall x ε)
⊢ ∃ 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:
rcases mem_nhdsGT_iff_exists_Ioc_subset.1 h with ⟨R, Rpos, hR⟩

Goal state after:
α : Type u_1
inst✝² : PseudoMetricSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsUnifLocDoublingMeasure μ
K : ℝ
h : ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), ∀ t ≤ K, μ (closedBall x (t * ε)) ≤ ↑(Classical.choose ⋯) * μ (closedBall x ε)
R : ℝ
Rpos : R ∈ Ioi 0
hR :
  Ioc 0 R ⊆ {x | (fun ε ↦ ∀ (x : α), ∀ t ≤ K, μ (closedBall x (t * ε)) ≤ ↑(Classical.choose ⋯) * μ (closedBall x ε)) x}
⊢ ∃ 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)
Module: Mathlib.MeasureTheory.Measure.Doubling

Goal state before:
pp: case inl
α : Type u_1
inst✝² : PseudoMetricSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsUnifLocDoublingMeasure μ
K : ℝ
h : ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), ∀ t ≤ K, μ (closedBall x (t * ε)) ≤ ↑(Classical.choose ⋯) * μ (closedBall x ε)
R : ℝ
Rpos : R ∈ Ioi 0
hR :
  Ioc 0 R ⊆ {x | (fun ε ↦ ∀ (x : α), ∀ t ≤ K, μ (closedBall x (t * ε)) ≤ ↑(Classical.choose ⋯) * μ (closedBall x ε)) x}
x : α
t r : ℝ
ht : t ∈ Ioc 0 K
hr : r ≤ R
rneg : r < 0
⊢ μ (closedBall x (t * r)) ≤ ↑(scalingConstantOf μ K) * μ (closedBall x r)

Tactic:
have : t * r < 0 := mul_neg_of_pos_of_neg ht.1 rneg
    simp only [closedBall_eq_empty.2 this, measure_empty, zero_le]

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:
norm_num

Goal state after: