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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
s : Set α
hs : IsCompact s
hf' : ContinuousOn f s
ε : (x : α) → x ∈ s → ℝ
hε : ∀ (x : α) (a : x ∈ s), ε x a > 0
K : (x : α) → x ∈ s → ℝ≥0
hf : ∀ (x : α) (a : x ∈ s), LipschitzOnWith (K x a) f (ball x (ε x a) ∩ s)
K' : (x : α) → x ∈ s → ℝ≥0
hK' : ∀ (x : α) (hx : x ∈ s), ∀ y ∈ s.diff (ball x (ε x hx)), edist (f x) (f y) ≤ ↑(K' x hx) * edist x y
t : Finset ↑s
ht : s ⊆ ⋃ x ∈ t, s ∩ ball (↑x) (ε ↑x ⋯ / 2)
x : α
hx : x ∈ s
y : α
hy : y ∈ s
z : ↑s
hz : z ∈ t
hx' : x ∈ s ∩ ball (↑z) (ε ↑z ⋯ / 2)
hy' : y ∉ ball (↑z) (ε ↑z ⋯)
⊢ ↑(K ↑z ⋯) * edist (↑z) x + ↑(K' ↑z ⋯) * edist (↑z) y ≤
    ↑(K ↑z ⋯) * edist (↑z) x + ↑(K' ↑z ⋯) * (edist x ↑z + edist x y)

Tactic:
gcongr; exact edist_triangle_left z.1 y x

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

Goal state before:
pp: ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
i : ι
x : E i
j : ι
y : E j
z : E i
hij : i ≠ j
⊢ dist x ⋯.some + 0 + 0 + (0 + 0 + dist ⋯.some z) ≤
    dist x ⋯.some + 1 + dist ⋯.some y + (dist y ⋯.some + 1 + dist ⋯.some z)

Tactic:
apply_rules [add_le_add, le_rfl, dist_nonneg, zero_le_one]

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

Goal state before:
pp: β : Type u_1
α : Type u_2
A : List β → Set α
inst✝ : PseudoMetricSpace α
ε : ℝ
ε_pos : 0 < ε
x : ℕ → β
hA : Tendsto (fun n ↦ Metric.ediam (A (res x n))) atTop (𝓝 0)
⊢ ∃ n, ∀ y ∈ A (res x n), ∀ z ∈ A (res x n), dist y z < ε

Tactic:
rw [ENNReal.tendsto_atTop_zero] at hA

Goal state after:
β : Type u_1
α : Type u_2
A : List β → Set α
inst✝ : PseudoMetricSpace α
ε : ℝ
ε_pos : 0 < ε
x : ℕ → β
hA : ∀ ε > 0, ∃ N, ∀ n ≥ N, Metric.ediam (A (res x n)) ≤ ε
⊢ ∃ n, ∀ y ∈ A (res x n), ∀ z ∈ A (res x n), dist y z < ε
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: case a.hV
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
n : ℕ
hn : (1 / 2) ^ n < ε
⊢ ∀ (i : ↑↑(Finset.range n)), {p | p.1 ↑i = p.2 ↑i} ∈ 𝓟 {a | a.1 ↑i = a.2 ↑i}

Tactic:
simp only [mem_principal, ofPred_subset_ofPred, imp_self, imp_true_iff]

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)
this : CauchySeq u
y : α
hy : Tendsto u atTop (𝓝 y)
n : ℕ
⊢ ∀ᶠ (x_1 : ℕ) in atTop, u x_1 ∈ A (x n :: res x n)

Tactic:
eventually_atTop

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)
this : CauchySeq u
y : α
hy : Tendsto u atTop (𝓝 y)
n : ℕ
⊢ ∃ a, ∀ (b : ℕ), a ≤ b → u b ∈ A (x n :: res x n)
Module: Mathlib.Topology.MetricSpace.Perfect

Goal state before:
pp: case hfh
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) := ⋯
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)
DP : List Bool → P := ⋯
D : List Bool → Set α := ⋯
hanti : ClosureAntitone D
x : ℕ → Bool
⊢ ∀ᶠ (b : ℕ) in atTop, ediam (D (PiNat.res x b)) ≤ u b

Tactic:
eventually_atTop

Goal state after:
case hfh
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)
DP : List Bool → P :=
  fun l ↦
    List.rec ⟨C, ⋯⟩
      (fun a l ih ↦ Bool.casesOn (motive := fun t ↦ a = t → P) a (fun h ↦ ⟨C0 ⋯ ⋯ ⋯, ⋯⟩) (fun h ↦ ⟨C1 ⋯ ⋯ ⋯, ⋯⟩) ⋯) l
D : List Bool → Set α := fun l ↦ ↑(DP l)
hanti : ClosureAntitone D
x : ℕ → Bool
⊢ ∃ a, ∀ (b : ℕ), a ≤ b → ediam (D (PiNat.res x b)) ≤ u b
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: case mpr
α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
β : Type
h : TopologicalSpace β
h' : PolishSpace β
f : β → α
f_cont : Continuous[h, inst✝] f
f_range : range f = s
⊢ AnalyticSet s

Tactic:
← f_range

Goal state after:
case mpr
α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
β : Type
h : TopologicalSpace β
h' : PolishSpace β
f : β → α
f_cont : Continuous[h, inst✝] f
f_range : range f = s
⊢ AnalyticSet (range f)
Module: Mathlib.MeasureTheory.Measure.MutuallySingular

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

Tactic:
by
  refine ⟨?_, fun h ↦ by (rw [h]; exact zero_left)⟩
  rintro ⟨s, hs, hμs, hμs_compl⟩
  suffices μ Set.univ = 0 by rwa [measure_univ_eq_zero] at this
  rw [← Set.union_compl_self s, measure_union disjoint_compl_right hs.compl, hμs, hμs_compl,
    add_zero]

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
s : Set α
inst✝ : IsZeroOrProbabilityMeasure μ
p : α → Prop
f✝ : β → α
f : α → β
⊢ IsZeroOrProbabilityMeasure (Measure.map f μ)

Tactic:
by_cases hf : AEMeasurable f μ

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)

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.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
⊢ Disjoint μ ν

Tactic:
intro ξ

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
⊢ ξ ≤ μ → ξ ≤ ν → ξ ≤ ⊥
Module: Mathlib.MeasureTheory.Measure.Dirac

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
s : Set α
a : α
hs : MeasurableSet s
⊢ s ∈ ae (dirac a) ↔ a ∈ s

Tactic:
by_cases a ∈ s <;> simp [mem_ae_iff, dirac_apply', hs.compl, *]

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

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
s : Set α
a : α
hs : MeasurableSet s
inst✝ : Decidable (a ∈ s)
⊢ (dirac a).restrict s = if a ∈ s then dirac a else 0

Tactic:
split_ifs with has
  · apply restrict_eq_self_of_ae_mem
    rw [ae_dirac_iff] <;> assumption
  · rw [restrict_eq_zero, dirac_apply' _ hs, indicator_of_notMem has]

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

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
a : α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
inst✝ : Decidable (a ∈ s)
⊢ ∫⁻ (x : α) in s, f x ∂dirac a = if a ∈ s then f a else 0

Tactic:
restrict_dirac' hs

Goal state after:
α : Type u_1
inst✝¹ : MeasurableSpace α
a : α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
inst✝ : Decidable (a ∈ s)
⊢ (∫⁻ (x : α), f x ∂if a ∈ s then dirac a else 0) = if a ∈ s then f a else 0
Module: Mathlib.MeasureTheory.Measure.GiryMonad

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
s : Set α
hs : MeasurableSet s
⊢ Measurable fun b ↦ (dirac b) s

Tactic:
dirac_apply' _ hs

Goal state after:
α : Type u_1
mα : MeasurableSpace α
s : Set α
hs : MeasurableSet s
⊢ Measurable fun b ↦ s.indicator 1 b
Module: Mathlib.MeasureTheory.Measure.GiryMonad

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
m : Measure (Measure α)
f : α → ℝ≥0∞
hfm : Measurable f
this :
  ∀ (s : ℕ → Finset ℝ≥0∞) (f : ℕ → ℝ≥0∞ → Measure α → ℝ≥0∞),
    (∀ (n : ℕ) (r : ℝ≥0∞), Measurable (f n r)) →
      (Monotone fun n μ ↦ ∑ r ∈ s n, r * f n r μ) →
        ⨆ n, ∑ r ∈ s n, r * ∫⁻ (μ : Measure α), f n r μ ∂m = ∫⁻ (μ : Measure α), ⨆ n, ∑ r ∈ s n, r * f n r μ ∂m
⊢ ⨆ n, ∑ x ∈ (SimpleFunc.eapprox f n).range, x * ∫⁻ (μ : Measure α), μ (⇑(SimpleFunc.eapprox f n) ⁻¹' {x}) ∂m =
    ∫⁻ (μ : Measure α), ⨆ n, ∑ x ∈ (SimpleFunc.eapprox f n).range, x * μ (⇑(SimpleFunc.eapprox f n) ⁻¹' {x}) ∂m

Tactic:
refine
      this (fun n => SimpleFunc.range (SimpleFunc.eapprox f n))
        (fun n r μ => μ (SimpleFunc.eapprox f n ⁻¹' {r})) ?_ ?_

Goal state after:
case refine_1
α : Type u_1
mα : MeasurableSpace α
m : Measure (Measure α)
f : α → ℝ≥0∞
hfm : Measurable f
this :
  ∀ (s : ℕ → Finset ℝ≥0∞) (f : ℕ → ℝ≥0∞ → Measure α → ℝ≥0∞),
    (∀ (n : ℕ) (r : ℝ≥0∞), Measurable (f n r)) →
      (Monotone fun n μ ↦ ∑ r ∈ s n, r * f n r μ) →
        ⨆ n, ∑ r ∈ s n, r * ∫⁻ (μ : Measure α), f n r μ ∂m = ∫⁻ (μ : Measure α), ⨆ n, ∑ r ∈ s n, r * f n r μ ∂m
⊢ ∀ (n : ℕ) (r : ℝ≥0∞), Measurable fun μ ↦ μ (⇑(SimpleFunc.eapprox f n) ⁻¹' {r})

case refine_2
α : Type u_1
mα : MeasurableSpace α
m : Measure (Measure α)
f : α → ℝ≥0∞
hfm : Measurable f
this :
  ∀ (s : ℕ → Finset ℝ≥0∞) (f : ℕ → ℝ≥0∞ → Measure α → ℝ≥0∞),
    (∀ (n : ℕ) (r : ℝ≥0∞), Measurable (f n r)) →
      (Monotone fun n μ ↦ ∑ r ∈ s n, r * f n r μ) →
        ⨆ n, ∑ r ∈ s n, r * ∫⁻ (μ : Measure α), f n r μ ∂m = ∫⁻ (μ : Measure α), ⨆ n, ∑ r ∈ s n, r * f n r μ ∂m
⊢ Monotone fun n μ ↦ ∑ r ∈ (SimpleFunc.eapprox f n).range, r * μ (⇑(SimpleFunc.eapprox f n) ⁻¹' {r})
Module: Mathlib.Topology.MetricSpace.PiNat

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

Tactic:
rw [edist_eq_tsum]

Goal state after:
ι : Type u_2
inst✝¹ : Encodable ι
F : ι → Type u_3
inst✝ : (i : ι) → EDist (F i)
x y : (i : ι) → F i
⊢ ∑' (i : ι), min (2⁻¹ ^ encode i) (edist (x i) (y i)) ≤ ∑' (i : ?m.10), 2⁻¹ ^ encode i
Module: Mathlib.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)
x : (i : ι) → F i
⊢ edist x x = 0

Tactic:
by simp [edist_eq_tsum]

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

Goal state before:
pp: case left.hV
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
i : ι
hi : i ∈ ↑K
⊢ {p | edist (p.1 ↑⟨i, hi⟩) (p.2 ↑⟨i, hi⟩) < δ} ∈
    ⨅ i_1, ⨅ (_ : 0 < i_1), 𝓟 {a | edist (a.1 ↑⟨i, hi⟩) (a.2 ↑⟨i, hi⟩) < i_1}

Tactic:
refine mem_iInf_of_mem δ (mem_iInf_of_mem δpos ?_)

Goal state after:
case left.hV
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
i : ι
hi : i ∈ ↑K
⊢ {p | edist (p.1 ↑⟨i, hi⟩) (p.2 ↑⟨i, hi⟩) < δ} ∈ 𝓟 {a | edist (a.1 ↑⟨i, hi⟩) (a.2 ↑⟨i, hi⟩) < δ}
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
hi : i < n
⊢ x i = (↑(p n)).2.1 i

Tactic:
rw [hx]

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
⊢ (fun n ↦ (↑(p (n + 1))).2.1 n) i = (↑(p n)).2.1 i
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: case basic
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)
s : Set α
hs : s ∈ {s | MeasurableSet s}
t : Set β
⊢ Measurable fun x ↦ ν (Prod.mk x ⁻¹' (fun x1 x2 ↦ x1 ×ˢ x2) s t)

Tactic:
simpa only [mk_preimage_prod_right_eq_if, measure_if]
      using measurable_const.indicator hs

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
⊢ False

Tactic:
let : MetricSpace (ℕ → ℕ) := metricSpaceNatNat

Goal state after:
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : T2Space α
inst✝¹ : MeasurableSpace α
inst✝ : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous[_, inst✝³] f
hg : Continuous[_, inst✝³] g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
I :
  ∀ (n : ℕ) (x y : ℕ → ℕ),
    ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →
      ∃ x' y',
        x' ∈ cylinder x n ∧
          y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))
A : Type := { p // ¬MeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }
F : A → A
hFn : ∀ (p : A), (↑(F p)).1 = (↑p).1 + 1
hFx : ∀ (p : A), (↑(F p)).2.1 ∈ cylinder (↑p).2.1 (↑p).1
hFy : ∀ (p : A), (↑(F p)).2.2 ∈ cylinder (↑p).2.2 (↑p).1
p0 : A := ⟨(0, fun x ↦ 0, fun x ↦ 0), ⋯⟩
p : ℕ → A := fun n ↦ F^[n] p0
prec : ∀ (n : ℕ), p (n + 1) = F (p n)
pn_fst : ∀ (n : ℕ), (↑(p n)).1 = n
Ix : ∀ (m n : ℕ), m + 1 ≤ n → (↑(p n)).2.1 m = (↑(p (m + 1))).2.1 m
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
⊢ False
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
ι : Type u_4
m : ι → Measure α
μ : Measure β
inst✝ : SFinite μ
⊢ (sum m).prod μ = sum fun i ↦ (m i).prod μ

Tactic:
ext s hs
  simp only [prod_apply hs, lintegral_sum_measure, hs, sum_apply]

Goal state after:
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: case mpr
G : Type u_1
inst✝ : Group G
H : Subgroup G
T : Set G
hHT : IsComplement (↑H) T
g₁ g₂ : G
⊢ g₂ * g₁⁻¹ ∈ H → (hHT.equiv g₁).2 = (hHT.equiv g₂).2

Tactic:
intro h
    apply (isComplement_iff_existsUnique_mul_inv_mem.1 hHT g₁).unique
    · -- This used to be `simp [...]` before https://github.com/leanprover/lean4/pull/2644
      rw [equiv_snd_eq_inv_mul]; simp
    · rw [SetLike.mem_coe, ← mul_mem_cancel_left h]
      -- This used to be `simp [...]` before https://github.com/leanprover/lean4/pull/2644
      rw [equiv_snd_eq_inv_mul, mul_assoc]; simp

Goal state after:
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H : Subgroup G
T : Set G
hHT : IsComplement (↑H) T
h : ↥H
g : G
⊢ hHT.equiv (↑h * g) = (h * (hHT.equiv g).1, (hHT.equiv g).2)

Tactic:
have : (hHT.equiv (h * g)).2 = (hHT.equiv g).2 := hHT.equiv_snd_eq_iff_rightCosetEquivalence.2 ?_

Goal state after:
case refine_2
G : Type u_1
inst✝ : Group G
H : Subgroup G
T : Set G
hHT : IsComplement (↑H) T
h : ↥H
g : G
this : (hHT.equiv (↑h * g)).2 = (hHT.equiv g).2
⊢ hHT.equiv (↑h * g) = (h * (hHT.equiv g).1, (hHT.equiv g).2)

case refine_1
G : Type u_1
inst✝ : Group G
H : Subgroup G
T : Set G
hHT : IsComplement (↑H) T
h : ↥H
g : G
⊢ RightCosetEquivalence (↑H) (↑h * g) g
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝² : SFinite ν
inst✝¹ : SFinite μ
μ' : Measure α
inst✝ : SFinite μ'
⊢ (sum (sfiniteSeq μ) + μ').prod ν = (sum (sfiniteSeq μ)).prod ν + μ'.prod ν

Tactic:
← sum_sfiniteSeq μ',

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝² : SFinite ν
inst✝¹ : SFinite μ
μ' : Measure α
inst✝ : SFinite μ'
⊢ (sum (sfiniteSeq μ) + sum (sfiniteSeq μ')).prod ν = (sum (sfiniteSeq μ)).prod ν + (sum (sfiniteSeq μ')).prod ν
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: case map_eq.inr
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
δ : Type u_4
inst✝² : MeasurableSpace δ
μa : Measure α
μb : Measure β
μc : Measure γ
μd : Measure δ
inst✝¹ : SFinite μa
inst✝ : SFinite μc
f : α → β
hf : MeasurePreserving f μa μb
g : α → γ → δ
hgm : Measurable (uncurry g)
hg : ∀ᵐ (a : α) ∂μa, map (g a) μc = μd
this : Measurable fun p ↦ (f p.1, g p.1 p.2)
h✝ : NeZero μa
⊢ map (fun p ↦ (f p.1, g p.1 p.2)) (μa.prod μc) = μb.prod μd

Tactic:
have sf : SFinite μd := by
    obtain ⟨a, ha⟩ : ∃ a, map (g a) μc = μd := hg.exists
    rw [← ha]
    infer_instance

Goal state after:
case map_eq.inr
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
δ : Type u_4
inst✝² : MeasurableSpace δ
μa : Measure α
μb : Measure β
μc : Measure γ
μd : Measure δ
inst✝¹ : SFinite μa
inst✝ : SFinite μc
f : α → β
hf : MeasurePreserving f μa μb
g : α → γ → δ
hgm : Measurable (uncurry g)
hg : ∀ᵐ (a : α) ∂μa, map (g a) μc = μd
this : Measurable fun p ↦ (f p.1, g p.1 p.2)
h✝ : NeZero μa
sf : SFinite μd
⊢ map (fun p ↦ (f p.1, g p.1 p.2)) (μa.prod μc) = μb.prod μd
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
μ : Measure α
ν : Measure β
τ : Measure γ
ω : Type u_4
mω : MeasurableSpace ω
υ : Measure ω
inst✝² : SFinite μ
inst✝¹ : SFinite τ
inst✝ : SFinite υ
f : α → β
g : γ → ω
hf : QuasiMeasurePreserving f μ ν
hg : QuasiMeasurePreserving g τ υ
⊢ map (Prod.map f g) (μ.prod τ) ≪ ν.prod υ

Tactic:
rw [← map_prod_map _ _ (by fun_prop) (by fun_prop)]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
μ : Measure α
ν : Measure β
τ : Measure γ
ω : Type u_4
mω : MeasurableSpace ω
υ : Measure ω
inst✝² : SFinite μ
inst✝¹ : SFinite τ
inst✝ : SFinite υ
f : α → β
g : γ → ω
hf : QuasiMeasurePreserving f μ ν
hg : QuasiMeasurePreserving g τ υ
⊢ (map f μ).prod (map g τ) ≪ ν.prod υ
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul G
H : Type u_3
inst✝² : Group H
mH : MeasurableSpace H
inst✝¹ : MeasurableMul H
μ : Measure H
inst✝ : μ.IsMulLeftInvariant
f : G →* H
hf : MeasurableEmbedding ⇑f
g : G
s : Set G
hs : MeasurableSet s
x✝ : H
y : G
yins : y ∈ s
hy : f y = (fun x ↦ f g * x) x✝
⊢ f (g⁻¹ * y) = x✝

Tactic:
simp [hy]

Goal state after: