mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.SimpleFuncDense

Goal state before:
pp: X : Type u_3
Y : Type u_4
α : Type u_5
inst✝¹⁰ : Zero α
inst✝⁹ : TopologicalSpace X
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : MeasurableSpace X
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : OpensMeasurableSpace X
inst✝⁴ : OpensMeasurableSpace Y
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetrizableSpace α
inst✝¹ : MeasurableSpace α
inst✝ : BorelSpace α
f : X × Y → α
hf : Continuous[instTopologicalSpaceProd, inst✝³] f
h'f : HasCompactSupport f
this : PseudoMetricSpace α := pseudoMetrizableSpacePseudoMetric α
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
g : ℕ → SimpleFunc (X × Y) α
hg : ∀ (n : ℕ) (x : X × Y), dist (f x) ((g n) x) < u n
x : X × Y
⊢ Tendsto (fun n ↦ (g n) x) atTop (𝓝 (f x))

Tactic:
tendsto_iff_dist_tendsto_zero

Goal state after:
X : Type u_3
Y : Type u_4
α : Type u_5
inst✝¹⁰ : Zero α
inst✝⁹ : TopologicalSpace X
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : MeasurableSpace X
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : OpensMeasurableSpace X
inst✝⁴ : OpensMeasurableSpace Y
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetrizableSpace α
inst✝¹ : MeasurableSpace α
inst✝ : BorelSpace α
f : X × Y → α
hf : Continuous[instTopologicalSpaceProd, inst✝³] f
h'f : HasCompactSupport f
this : PseudoMetricSpace α := pseudoMetrizableSpacePseudoMetric α
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
g : ℕ → SimpleFunc (X × Y) α
hg : ∀ (n : ℕ) (x : X × Y), dist (f x) ((g n) x) < u n
x : X × Y
⊢ Tendsto (fun b ↦ dist ((g b) x) (f x)) atTop (𝓝 0)
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable

Goal state before:
pp: α : Type u_1
m₀ : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetrizableSpace α
inst✝ : OpensMeasurableSpace α
s : Set α
h1 : IsSeparable s
h2 : μ sᶜ = 0
a✝ : Nontrivial α
a : α
h : StronglyMeasurable Subtype.val
x : α
⊢ (closure[inst✝²] s).piecewise id (fun x ↦ a) x = Function.extend Subtype.val Subtype.val (fun x ↦ a) x

Tactic:
by_cases hx : x ∈ closure s

Goal state after:
case pos
α : Type u_1
m₀ : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetrizableSpace α
inst✝ : OpensMeasurableSpace α
s : Set α
h1 : IsSeparable s
h2 : μ sᶜ = 0
a✝ : Nontrivial α
a : α
h : StronglyMeasurable Subtype.val
x : α
hx : x ∈ closure[inst✝²] s
⊢ (closure[inst✝²] s).piecewise id (fun x ↦ a) x = Function.extend Subtype.val Subtype.val (fun x ↦ a) x

case neg
α : Type u_1
m₀ : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetrizableSpace α
inst✝ : OpensMeasurableSpace α
s : Set α
h1 : IsSeparable s
h2 : μ sᶜ = 0
a✝ : Nontrivial α
a : α
h : StronglyMeasurable Subtype.val
x : α
hx : x ∉ closure[inst✝²] s
⊢ (closure[inst✝²] s).piecewise id (fun x ↦ a) x = Function.extend Subtype.val Subtype.val (fun x ↦ a) x
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
h : ⨆ φ, ⨆ (_ : ∀ (x : α), ↑(φ x) ≤ f x), (SimpleFunc.map ofNNReal φ).lintegral μ ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
this :
  ⨆ φ, ⨆ (_ : ∀ (x : α), ↑(φ x) ≤ f x), (SimpleFunc.map ofNNReal φ).lintegral μ <
    ⨆ i ∈ fun φ ↦ ∀ (x : α), ↑(φ x) ≤ f x, (SimpleFunc.map ofNNReal i).lintegral μ + ε
⊢ ∃ φ,
    (∀ (x : α), ↑(φ x) ≤ f x) ∧
      ∀ (ψ : α →ₛ ℝ≥0), (∀ (x : α), ↑(ψ x) ≤ f x) → (SimpleFunc.map ofNNReal (ψ - φ)).lintegral μ < ε

Tactic:
simp_rw [lt_iSup_iff, iSup_lt_iff, iSup_le_iff] at this

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
h : ⨆ φ, ⨆ (_ : ∀ (x : α), ↑(φ x) ≤ f x), (SimpleFunc.map ofNNReal φ).lintegral μ ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
this :
  ∃ i,
    ∃ (_ : i ∈ fun φ ↦ ∀ (x : α), ↑(φ x) ≤ f x),
      ∃ b < (SimpleFunc.map ofNNReal i).lintegral μ + ε,
        ∀ (i : α →ₛ ℝ≥0), (∀ (x : α), ↑(i x) ≤ f x) → (SimpleFunc.map ofNNReal i).lintegral μ ≤ b
⊢ ∃ φ,
    (∀ (x : α), ↑(φ x) ≤ f x) ∧
      ∀ (ψ : α →ₛ ℝ≥0), (∀ (x : α), ↑(ψ x) ≤ f x) → (SimpleFunc.map ofNNReal (ψ - φ)).lintegral μ < ε
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: case pos
α : Type u_1
m✝ : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), AEMeasurable (f n) μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n ↦ f n x
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' ↦ Monotone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
n m : ℕ
hnm : n ≤ m
x : α
hx : x ∈ aeSeqSet hf p
⊢ aeSeq hf p n x ≤ aeSeq hf p m x

Tactic:
exact aeSeq.prop_of_mem_aeSeqSet hf hx hnm

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' ↦ Directed LE.le f'
⊢ ∫⁻ (a : α), (⨆ i, f i) a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ

Tactic:
have hp : ∀ᵐ x ∂μ, p x fun i => f i x := by
    filter_upwards [] with x i j
    obtain ⟨z, hz₁, hz₂⟩ := h_directed i j
    exact ⟨z, hz₁ x, hz₂ x⟩

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' ↦ Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
⊢ ∫⁻ (a : α), (⨆ i, f i) a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
f : α → β
m m' : MeasurableSpace α
inst✝ : TopologicalSpace β
hf : StronglyMeasurable f
h_mono : m' ≤ m
⊢ StronglyMeasurable f

Tactic:
by
  let f_approx : ℕ → @SimpleFunc α m β := fun n =>
    @SimpleFunc.mk α m β
      (hf.approx n)
      (fun x => h_mono _ (SimpleFunc.measurableSet_fiber' _ x))
      (SimpleFunc.finite_range (hf.approx n))
  exact ⟨f_approx, hf.tendsto_approx⟩

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

Goal state before:
pp: case h_mono
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
i j : ℕ
h : i ≤ j
⊢ (fun n ↦ ⇑(eapprox f n) + ⇑(eapprox g n)) i ≤ (fun n ↦ ⇑(eapprox f n) + ⇑(eapprox g n)) j

Tactic:
a

Goal state after:
case h_mono
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
i j : ℕ
h : i ≤ j
a : α
⊢ (fun n ↦ ⇑(eapprox f n) + ⇑(eapprox g n)) i a ≤ (fun n ↦ ⇑(eapprox f n) + ⇑(eapprox g n)) j a
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : TopologicalSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
h : SecondCountableTopologyEither α β
f : α → β
hf : Continuous[inst✝³, inst✝¹] f
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
h✝ : SecondCountableTopology α
⊢ IsSeparable (range f)

Tactic:
exact isSeparable_range hf

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

Goal state before:
pp: case refine_2
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
g : α → ℝ≥0∞
hf_meas : ∀ (n : ℕ), Measurable (f n)
h_bound : ∀ (n : ℕ), f n ≤ᵐ[μ] g
h_fin : ∫⁻ (a : α), g a ∂μ ≠ ∞
n m : ℕ
hnm : n ≤ m
⊢ (fun n a ↦ ⨆ i, ⨆ (_ : i ≥ n), f i a) m ≤ (fun n a ↦ ⨆ i, ⨆ (_ : i ≥ n), f i a) n

Tactic:
a

Goal state after:
case refine_2
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
g : α → ℝ≥0∞
hf_meas : ∀ (n : ℕ), Measurable (f n)
h_bound : ∀ (n : ℕ), f n ≤ᵐ[μ] g
h_fin : ∫⁻ (a : α), g a ∂μ ≠ ∞
n m : ℕ
hnm : n ≤ m
a : α
⊢ (fun n a ↦ ⨆ i, ⨆ (_ : i ≥ n), f i a) m a ≤ (fun n a ↦ ⨆ i, ⨆ (_ : i ≥ n), f i a) n a
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
m : MeasurableSpace α
inst✝³ : TopologicalSpace β
inst✝² : PseudoMetrizableSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : PseudoMetrizableSpace γ
g : β → γ
f : α → β
hg : IsEmbedding g
this : PseudoMetricSpace γ := pseudoMetrizableSpacePseudoMetric γ
this✝³ : MeasurableSpace β := borel β
this✝² : BorelSpace β
this✝¹ : MeasurableSpace γ := borel γ
this✝ : BorelSpace γ
H : StronglyMeasurable fun x ↦ g (f x)
G : β → ↑(range g) := rangeFactorization g
⊢ IsClosed[instTopologicalSpaceSubtype] (range G)

Tactic:
rw [rangeFactorization_surjective.range_eq]
          exact isClosed_univ

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

Goal state before:
pp: case h_mono
α : Type u_1
m : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hf : Measurable f
i j : ℕ
h : i ≤ j
⊢ (fun n ↦ ⇑(const α r * eapprox f n)) i ≤ (fun n ↦ ⇑(const α r * eapprox f n)) j

Tactic:
a

Goal state after:
case h_mono
α : Type u_1
m : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hf : Measurable f
i j : ℕ
h : i ≤ j
a : α
⊢ (fun n ↦ ⇑(const α r * eapprox f n)) i a ≤ (fun n ↦ ⇑(const α r * eapprox f n)) j a
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f : α → ℝ≥0∞
hf : AEMeasurable f (μ.trim hm)
⊢ ∫⁻ (a : α), f a ∂μ.trim hm = ∫⁻ (a : α), f a ∂μ

Tactic:
lintegral_congr_ae (ae_eq_of_ae_eq_trim hf.ae_eq_mk),

Goal state after:
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f : α → ℝ≥0∞
hf : AEMeasurable f (μ.trim hm)
⊢ ∫⁻ (a : α), f a ∂μ.trim hm = ∫⁻ (a : α), AEMeasurable.mk f hf a ∂μ
Module: Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
s : Set α
inst✝ : IsFiniteMeasure ν
hs_subset : s ⊆ (μ.sigmaFiniteSetWRT' ν)ᶜ
hνs : ν s ≠ 0
this : ∀ t ⊆ (μ.sigmaFiniteSetWRT' ν)ᶜ, s ⊆ t → MeasurableSet t → μ t = ∞
t : Set α
⊢ s ⊆ t → MeasurableSet t → μ t = ∞

Tactic:
hts

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
s : Set α
inst✝ : IsFiniteMeasure ν
hs_subset : s ⊆ (μ.sigmaFiniteSetWRT' ν)ᶜ
hνs : ν s ≠ 0
this : ∀ t ⊆ (μ.sigmaFiniteSetWRT' ν)ᶜ, s ⊆ t → MeasurableSet t → μ t = ∞
t : Set α
hts : s ⊆ t
⊢ MeasurableSet t → μ t = ∞
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : Zero β
s : Set α
f : α → β
hs : MeasurableSet s
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
⊢ ∀ x ∉ s, ∀ (n : ℕ), ((fun n ↦ (hf.approx n).restrict s) n) x = 0

Tactic:
intro x hx n
    simp [SimpleFunc.coe_restrict, hs, hx]

Goal state after:
Module: Mathlib.Topology.Perfect

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : SecondCountableTopology α
hclosed : IsClosed[inst✝¹] C
hunc : ¬C.Countable
⊢ ∃ D, Perfect D ∧ D.Nonempty ∧ D ⊆ C

Tactic:
rcases exists_countable_union_perfect_of_isClosed hclosed with ⟨V, D, Vct, Dperf, VD⟩
  refine ⟨D, ⟨Dperf, ?_⟩⟩
  constructor
  · rw [nonempty_iff_ne_empty]
    by_contra h
    rw [h, union_empty] at VD
    rw [VD] at hunc
    contradiction
  rw [VD]
  exact subset_union_right

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: x : ↑I
⊢ Subtype.val ⁻¹' Ioi ↑x = Ioc x 1

Tactic:
preimage_subtype_val_Ioi

Goal state after:
x : ↑I
⊢ Ioi x = Ioc x 1
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : LinearOrder α
inst✝ : IsOrderedAddMonoid α
a b c✝ d✝ : α
h : a ≤ b
c d : α
hdc : 0 ≤ c - d
⊢ a - a ≤ c - d

Tactic:
sub_self

Goal state after:
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : LinearOrder α
inst✝ : IsOrderedAddMonoid α
a b c✝ d✝ : α
h : a ≤ b
c d : α
hdc : 0 ≤ c - d
⊢ 0 ≤ c - d
Module: Mathlib.Topology.Metrizable.CompletelyMetrizable

Goal state before:
pp: case refine_1
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
m : MetricSpace X :=
  { dist := fun x y ↦ if x = y then 0 else 1, dist_self := ⋯, dist_comm := ⋯, dist_triangle := ⋯, edist_dist := ⋯,
    uniformity_dist := ⋯, cobounded_sets := ⋯, eq_of_dist_eq_zero := ⋯ }
⊢ PseudoMetricSpace.toUniformSpace.toTopologicalSpace = inst✝¹

Tactic:
DiscreteTopology.eq_bot (α := X)

Goal state after:
case refine_1
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
m : MetricSpace X :=
  { dist := fun x y ↦ if x = y then 0 else 1, dist_self := ⋯, dist_comm := ⋯, dist_triangle := ⋯, edist_dist := ⋯,
    uniformity_dist := ⋯, cobounded_sets := ⋯, eq_of_dist_eq_zero := ⋯ }
⊢ PseudoMetricSpace.toUniformSpace.toTopologicalSpace = ⊥
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: case mp
E : ℕ → Type u_1
x y : (n : ℕ) → E n
hne : x ≠ y
i : ℕ
h : x ∈ cylinder y i
⊢ i ≤ firstDiff x y

Tactic:
by_contra!

Goal state after:
case mp
E : ℕ → Type u_1
x y : (n : ℕ) → E n
hne : x ≠ y
i : ℕ
h : x ∈ cylinder y i
this : firstDiff x y < i
⊢ False
Module: Mathlib.Topology.MetricSpace.Polish

Goal state before:
pp: α : Type u_1
ι : Type u_3
inst✝ : Countable ι
t : ι → TopologicalSpace α
ht₀ : ∃ i₀, ∀ (i : ι), t i ≤ t i₀
ht : ∀ (i : ι), PolishSpace α
⊢ PolishSpace α

Tactic:
by
  rcases ht₀ with ⟨i₀, hi₀⟩
  rcases CompletePseudometrizable.iInf ⟨t i₀, letI := t i₀; haveI := ht i₀; inferInstance, hi₀⟩
    fun i ↦
      letI := t i; haveI := ht i; letI := upgradeIsCompletelyMetrizable α
      ⟨inferInstance, inferInstance, inferInstance, rfl⟩
    with ⟨u, hcomp, hcount, htop⟩
  rw [← htop]
  have : @SecondCountableTopology α u.toTopologicalSpace :=
    htop.symm ▸ secondCountableTopology_iInf fun i ↦ letI := t i; (ht i).toSecondCountableTopology
  have : @T1Space α u.toTopologicalSpace :=
    htop.symm ▸ t1Space_antitone (iInf_le _ i₀) (by let := t i₀; have := ht i₀; infer_instance)
  infer_instance

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

Goal state before:
pp: case refine_4.refine_2
α : Type u_1
β : Type u_2
inst✝ : MetricSpace α
s : Opens α
t : Set s.CompleteCopy
h : ∀ x ∈ t, ∃ ε > 0, ∀ (y : s.CompleteCopy), dist x y < ε → y ∈ t
x : s.CompleteCopy
hx : x ∈ t
ε : ℝ
ε0 : ε > 0
hε : ∀ (y : s.CompleteCopy), dist ↑x ↑y + |(infDist (↑x) (↑s)ᶜ)⁻¹ - (infDist (↑y) (↑s)ᶜ)⁻¹| < ε → y ∈ t
this : Tendsto (fun y ↦ dist ↑x ↑y + |(infDist (↑x) (↑s)ᶜ)⁻¹ - (infDist (↑y) (↑s)ᶜ)⁻¹|) (𝓝 x) (𝓝 0)
⊢ t ∈ 𝓝 x

Tactic:
exact mem_of_superset (this <| gt_mem_nhds ε0) hε

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)
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)
l : List Bool
⊢ ∀ (a : Bool), D (a :: l) ⊆ D l

Tactic:
a

Goal state after:
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)
l : List Bool
a : Bool
⊢ D (a :: l) ⊆ D l
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: case refine_1.h₂
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 : (n : ℕ) → E n
hx : x ∈ s
⊢ x ∈ range f

Tactic:
← fs x hx

Goal state after:
case refine_1.h₂
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 : (n : ℕ) → E n
hx : x ∈ s
⊢ f x ∈ range f
Module: Mathlib.MeasureTheory.Measure.MutuallySingular

Goal state before:
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)

Tactic:
exact Disjoint.mono inter_subset_right inter_subset_right disjoint_compl_right

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

Goal state before:
pp: α : Type u_3
β : Type u_4
t : TopologicalSpace α
inst✝⁵ : PolishSpace α
inst✝⁴ : MeasurableSpace α
inst✝³ : BorelSpace α
tβ : TopologicalSpace β
inst✝² : MeasurableSpace β
inst✝¹ : OpensMeasurableSpace β
f : α → β
inst✝ : SecondCountableTopology ↑(range f)
hf : Measurable f
b : Set (Set ↑(range f))
b_count : b.Countable
hb : IsTopologicalBasis b
this : Countable ↑b
T : ↑b → TopologicalSpace α
Tt : ∀ (s : ↑b), T s ≤ t
Tpolish : ∀ (s : ↑b), PolishSpace α
h✝ : ∀ (s : ↑b), IsClosed[T s] (rangeFactorization f ⁻¹' ↑s)
Topen : ∀ (s : ↑b), IsOpen[T s] (rangeFactorization f ⁻¹' ↑s)
t' : TopologicalSpace α
t'T : ∀ (i : ↑b), t' ≤ T i
t't : t' ≤ t
t'_polish : PolishSpace α
⊢ Continuous[t', tβ] f

Tactic:
have : Continuous[t', _] (rangeFactorization f) :=
    hb.continuous_iff.2 fun s hs => t'T ⟨s, hs⟩ _ (Topen ⟨s, hs⟩)

Goal state after:
α : Type u_3
β : Type u_4
t : TopologicalSpace α
inst✝⁵ : PolishSpace α
inst✝⁴ : MeasurableSpace α
inst✝³ : BorelSpace α
tβ : TopologicalSpace β
inst✝² : MeasurableSpace β
inst✝¹ : OpensMeasurableSpace β
f : α → β
inst✝ : SecondCountableTopology ↑(range f)
hf : Measurable f
b : Set (Set ↑(range f))
b_count : b.Countable
hb : IsTopologicalBasis b
this✝ : Countable ↑b
T : ↑b → TopologicalSpace α
Tt : ∀ (s : ↑b), T s ≤ t
Tpolish : ∀ (s : ↑b), PolishSpace α
h✝ : ∀ (s : ↑b), IsClosed[T s] (rangeFactorization f ⁻¹' ↑s)
Topen : ∀ (s : ↑b), IsOpen[T s] (rangeFactorization f ⁻¹' ↑s)
t' : TopologicalSpace α
t'T : ∀ (i : ↑b), t' ≤ T i
t't : t' ≤ t
t'_polish : PolishSpace α
this : Continuous[t', instTopologicalSpaceSubtype] (rangeFactorization f)
⊢ Continuous[t', tβ] f
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: case succ
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)
n : ℕ
IH : (↑(p n)).1 = n
⊢ (↑(p (n + 1))).1 = n + 1

Tactic:
simp only [prec, hFn, IH]

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

Goal state before:
pp: α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
C : ℝ≥0∞
f : Set α → ℝ≥0∞
hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ∞ → f s ≤ C
h_F_lim : ∀ (S : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (S n)) → Monotone S → f (⋃ n, S n) ≤ ⨆ n, f (S n)
⊢ f univ ≤ C

Tactic:
let S := @spanningSets _ m (μ.trim hm) _
  have hS_mono : Monotone S := @monotone_spanningSets _ m (μ.trim hm) _
  have hS_meas : ∀ n, MeasurableSet[m] (S n) := @measurableSet_spanningSets _ m (μ.trim hm) _
  rw [← @iUnion_spanningSets _ m (μ.trim hm)]
  refine (h_F_lim S hS_meas hS_mono).trans ?_
  refine iSup_le fun n => hf (S n) (hS_meas n) ?_
  exact ((le_trim hm).trans_lt (@measure_spanningSets_lt_top _ m (μ.trim hm) _ n)).ne

Goal state after:
Module: Mathlib.GroupTheory.Complement

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

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

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OpensMeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : TopologicalSpace β
inst✝¹ : BorelSpace β
f : α ≃ₜ β
μ : Measure α
inst✝ : μ.OuterRegular
⊢ (map (⇑f) μ).OuterRegular

Tactic:
refine ⟨fun A hA r hr => ?_⟩

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OpensMeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : TopologicalSpace β
inst✝¹ : BorelSpace β
f : α ≃ₜ β
μ : Measure α
inst✝ : μ.OuterRegular
A : Set β
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > (map (⇑f) μ) A
⊢ ∃ U ⊇ A, IsOpen[inst✝²] U ∧ (map (⇑f) μ) U < r
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen[inst✝¹] (s n)
h'' : univ ⊆ ⋃ n, s n
⊢ μ.OuterRegular

Tactic:
refine ⟨fun A hA r hr => ?_⟩

Goal state after:
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen[inst✝¹] (s n)
h'' : univ ⊆ ⋃ n, s n
A : Set α
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > μ A
⊢ ∃ U ⊇ A, IsOpen[inst✝¹] U ∧ μ U < r