mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Group.Arithmetic

Goal state before:
pp: α✝ : Type u_1
M✝ : Type u_2
X : Type u_3
α : Type u_4
β : Type u_5
inst✝⁷ : MeasurableSpace X
inst✝⁶ : SMul M✝ X
m : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
f : α → M✝
g : α → X
inst✝⁵ : MeasurableSpace M✝
inst✝⁴ : MeasurableSMul M✝ X
M : Type u_6
inst✝³ : SubNegMonoid M
inst✝² : MeasurableSpace M
inst✝¹ : MeasurableAdd₂ M
inst✝ : MeasurableNeg M
⊢ Measurable fun p ↦ p.2 • p.1

Tactic:
refine measurable_from_prod_countable_left fun n => ?_

Goal state after:
α✝ : Type u_1
M✝ : Type u_2
X : Type u_3
α : Type u_4
β : Type u_5
inst✝⁷ : MeasurableSpace X
inst✝⁶ : SMul M✝ X
m : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
f : α → M✝
g : α → X
inst✝⁵ : MeasurableSpace M✝
inst✝⁴ : MeasurableSMul M✝ X
M : Type u_6
inst✝³ : SubNegMonoid M
inst✝² : MeasurableSpace M
inst✝¹ : MeasurableAdd₂ M
inst✝ : MeasurableNeg M
n : ℤ
⊢ Measurable fun x ↦ (x, n).2 • (x, n).1
Module: Mathlib.MeasureTheory.Group.Arithmetic

Goal state before:
pp: M : Type u_2
α : Type u_4
inst✝² : CommMonoid M
inst✝¹ : MeasurableSpace M
inst✝ : MeasurableMul₂ M
m : MeasurableSpace α
l : Multiset (α → M)
hl : ∀ f ∈ l, Measurable f
⊢ Measurable l.prod

Tactic:
by
  rcases l with ⟨l⟩
  simpa using l.measurable_prod (by simpa using hl)

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
h : SecondCountableTopologyEither α β
s : Set (α × β)
hs : IsOpen[instTopologicalSpaceProd] s
⊢ MeasurableSet s

Tactic:
rcases h.out with hα | hβ

Goal state after:
case inl
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
h : SecondCountableTopologyEither α β
s : Set (α × β)
hs : IsOpen[instTopologicalSpaceProd] s
hα : SecondCountableTopology α
⊢ MeasurableSet s

case inr
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
h : SecondCountableTopologyEither α β
s : Set (α × β)
hs : IsOpen[instTopologicalSpaceProd] s
hβ : SecondCountableTopology β
⊢ MeasurableSet s
Module: Mathlib.Topology.MetricSpace.HausdorffDistance

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
x y : α
s : Set α
⊢ ⨅ z ∈ s, edist y z + edist x y = (⨅ z ∈ s, edist y z) + edist x y

Tactic:
by simp only [ENNReal.iInf_add]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
mβ : TopologicalSpace β
inst✝² : MeasurableSpace β
inst✝¹ : BorelSpace β
μ : Measure α
inst✝ : IsFiniteMeasureOnCompacts μ
f : α ≃ₜ β
K : Set β
hK : IsCompact K
⊢ μ (⇑f.toMeasurableEquiv ⁻¹' K) < ∞

Tactic:
exact IsCompact.measure_lt_top (f.isCompact_preimage.2 hK)

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_5
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
⊢ borel α = MeasurableSpace.generateFrom {S | ∃ l u, l < u ∧ Ico l u = S}

Tactic:
by
  simpa only [exists_prop, mem_univ, true_and] using
    (@dense_univ α _).borel_eq_generateFrom_Ico_mem_aux (fun _ _ => mem_univ _) fun _ _ _ _ =>
      mem_univ _

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
s : Set α
x y : α
⊢ (infEDist x s).toReal ≤ (infEDist y s).toReal + (edist x y).toReal

Tactic:
refine ENNReal.toReal_le_add' infEDist_le_infEDist_add_edist ?_ (flip absurd (edist_ne_top _ _))

Goal state after:
α : Type u
inst✝ : PseudoMetricSpace α
s : Set α
x y : α
⊢ infEDist y s = ∞ → infEDist x s = ∞
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_5
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
hμν : μ univ = ν univ
h : ∀ ⦃a b : α⦄, a < b → μ (Ioc a b) = ν (Ioc a b)
⊢ μ = ν

Tactic:
refine @ext_of_Ico_finite αᵒᵈ _ _ _ _ _ ‹_› μ ν _ hμν fun a b hab => ?_

Goal state after:
α : Type u_5
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
hμν : μ univ = ν univ
h : ∀ ⦃a b : α⦄, a < b → μ (Ioc a b) = ν (Ioc a b)
a b : αᵒᵈ
hab : a < b
⊢ μ (Ico a b) = ν (Ico a b)
Module: Mathlib.Topology.MetricSpace.HausdorffDistance

Goal state before:
pp: α : Type u
inst✝¹ : PseudoMetricSpace α
s : Set α
inst✝ : ProperSpace α
h : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
x z : α
hz : z ∈ s
t : Set α := s ∩ closedBall x (dist z x)
htc : IsCompact t
htne : t.Nonempty
y : α
hyd : infDist x t = dist x y
hys : y ∈ s
⊢ ∃ y ∈ s, infDist x t = dist x y

Tactic:
exact ⟨y, hys, hyd⟩

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

Goal state before:
pp: case inr
α : Type u
inst✝ : PseudoMetricSpace α
s : Set α
h : s.Nonempty
⊢ hausdorffDist s ∅ = 0

Tactic:
simp [hausdorffDist, hausdorffEDist_empty h]

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_5
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
h : ∀ (a : α), μ (Iic a) = ν (Iic a)
⊢ μ = ν

Tactic:
by
  refine ext_of_Ioc_finite μ ν ?_ fun a b hlt => ?_
  · rcases exists_countable_dense_bot_top α with ⟨s, hsc, hsd, -, hst⟩
    have : DirectedOn (· ≤ ·) s := directedOn_iff_directed.2 (Subtype.mono_coe _).directed_le
    simp only [← biSup_measure_Iic hsc (hsd.exists_ge' hst) this, h]
  rw [← Iic_sdiff_Iic, measure_sdiff (Iic_subset_Iic.2 hlt.le) nullMeasurableSet_Iic,
    measure_sdiff (Iic_subset_Iic.2 hlt.le) nullMeasurableSet_Iic, h a, h b]
  · rw [← h a]
    finiteness
  · finiteness

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Metric

Goal state before:
pp: α : Type u_5
m : MeasurableSpace α
inst✝¹ : CountablyGenerated α
inst✝ : MeasurableSpace.SeparatesPoints α
s : Set (ℕ → Bool)
f : α ≃ᵐ ↑s
this : TopologicalSpace α := induced (⇑f) inferInstance
F : α ≃ₜ ↑s := f.toHomeomorphOfIsInducing ⋯
⊢ ∃ x, SecondCountableTopology α ∧ T4Space α ∧ BorelSpace α

Tactic:
exact ⟨inferInstance, F.secondCountableTopology, F.symm.t4Space,
    f.measurableEmbedding.borelSpace F.isInducing⟩

Goal state after:
Module: Mathlib.MeasureTheory.Function.SimpleFunc

Goal state before:
pp: case e_f
α : Type u_1
β : Type u_2
inst✝⁷ : MeasurableSpace α
inst✝⁶ : SemilatticeSup β
inst✝⁵ : OrderBot β
inst✝⁴ : Zero β
inst✝³ : TopologicalSpace β
inst✝² : OrderClosedTopology β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
i : ℕ → β
f : α → β
n : ℕ
a : α
hf : Measurable f
k : ℕ
⊢ {a | i k ≤ f a}.indicator (⇑(const α (i k))) a = if i k ≤ f a then i k else 0

pp: case e_f.hs
α : Type u_1
β : Type u_2
inst✝⁷ : MeasurableSpace α
inst✝⁶ : SemilatticeSup β
inst✝⁵ : OrderBot β
inst✝⁴ : Zero β
inst✝³ : TopologicalSpace β
inst✝² : OrderClosedTopology β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
i : ℕ → β
f : α → β
n : ℕ
a : α
hf : Measurable f
k : ℕ
⊢ MeasurableSet {a | i k ≤ f a}

Tactic:
· simp only [coe_const, mem_ofPred_eq, indicator_apply, Function.const_apply]

Goal state after:
case e_f.hs
α : Type u_1
β : Type u_2
inst✝⁷ : MeasurableSpace α
inst✝⁶ : SemilatticeSup β
inst✝⁵ : OrderBot β
inst✝⁴ : Zero β
inst✝³ : TopologicalSpace β
inst✝² : OrderClosedTopology β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
i : ℕ → β
f : α → β
n : ℕ
a : α
hf : Measurable f
k : ℕ
⊢ MeasurableSet {a | i k ≤ f a}
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case pos
α : Type u_1
δ : Type u_4
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
mδ : MeasurableSpace δ
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_5
ι' : Type u_6
f : ι → δ → α
v : Filter ι
hf : ∀ (i : ι), Measurable (f i)
p : ι' → Prop
s : ι' → Set ι
hv : v.HasCountableBasis p s
hs : ∀ (j : ι'), (s j).Countable
this : Countable (Subtype p)
hp : Nonempty (Subtype p)
H : ∃ j, s ↑j = ∅
⊢ Measurable fun x ↦ sSup univ

Tactic:
measurable_const

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Real

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ι : Type u_5
f : ι → α → ℝ≥0
g : α → ℝ≥0
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
hf : ∀ (i : ι), Measurable (f i)
lim : Tendsto f u (𝓝 g)
⊢ Measurable g

Tactic:
simp_rw [← measurable_coe_nnreal_ennreal_iff] at hf ⊢
  refine ENNReal.measurable_of_tendsto' u hf ?_
  rw [tendsto_pi_nhds] at lim ⊢
  exact fun x => (ENNReal.continuous_coe.tendsto (g x)).comp (lim x)

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case hd
α : Type u_5
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : μ s = μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' Ioi 0)
⊢ Disjoint (s ∩ f ⁻¹' {∞}) (s ∩ f ⁻¹' Ioo 0 ∞)

pp: case h
α : Type u_5
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : μ s = μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' Ioi 0)
⊢ MeasurableSet (s ∩ f ⁻¹' Ioo 0 ∞)

Tactic:
· refine disjoint_left.2 fun x hx h'x => ?_
      have : f x < ∞ := h'x.2.2
      exact lt_irrefl _ (this.trans_le (le_of_eq hx.2.symm))

Goal state after:
case h
α : Type u_5
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : μ s = μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' Ioi 0)
⊢ MeasurableSet (s ∩ f ⁻¹' Ioo 0 ∞)
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_5
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : μ s = μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' Ioi 0)
B : μ (s ∩ f ⁻¹' Ioi 0) = μ (s ∩ f ⁻¹' {∞}) + μ (s ∩ f ⁻¹' Ioo 0 ∞)
C : μ (s ∩ f ⁻¹' Ioo 0 ∞) = ∑' (n : ℤ), μ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))
⊢ μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' Ioi 0) =
    μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' {∞}) + ∑' (n : ℤ), μ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))

Tactic:
B,

Goal state after:
α : Type u_5
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : μ s = μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' Ioi 0)
B : μ (s ∩ f ⁻¹' Ioi 0) = μ (s ∩ f ⁻¹' {∞}) + μ (s ∩ f ⁻¹' Ioo 0 ∞)
C : μ (s ∩ f ⁻¹' Ioo 0 ∞) = ∑' (n : ℤ), μ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))
⊢ μ (s ∩ f ⁻¹' {0}) + (μ (s ∩ f ⁻¹' {∞}) + μ (s ∩ f ⁻¹' Ioo 0 ∞)) =
    μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' {∞}) + ∑' (n : ℤ), μ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
m₀ : MeasurableSpace α
μ : Measure α
inst✝¹ : PseudoMetrizableSpace β
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 : AEStronglyMeasurable (fun x ↦ g (f x)) μ
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.Function.StronglyMeasurable.AEStronglyMeasurable

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁶ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
f : α → β
G₀ : Type u_7
inst✝⁵ : GroupWithZero G₀
inst✝⁴ : MulAction G₀ β
inst✝³ : TopologicalSpace G₀
inst✝² : ContinuousInv₀ G₀
inst✝¹ : MetrizableSpace G₀
inst✝ : ContinuousSMul G₀ β
c : α → G₀
hc : AEStronglyMeasurable c μ
hc0 : ∀ᵐ (x : α) ∂μ, c x ≠ 0
⊢ AEStronglyMeasurable (fun x ↦ c x • f x) μ ↔ AEStronglyMeasurable f μ

Tactic:
by
  refine ⟨fun h => (hc.fun_inv₀.fun_smul h).congr ?_, fun h => hc.fun_smul h⟩
  filter_upwards [hc0] with x hx
  simp [hx]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
hf : ∀ (n : ℕ), AEMeasurable (f n) μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n ↦ f n x
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))
this : Monotone fun n ↦ ∫⁻ (x : α), f n x ∂μ
a✝ : α
⊢ (Monotone fun n ↦ f n a✝) → Tendsto (fun n ↦ f n a✝) atTop (𝓝 (F a✝)) → F a✝ = ⨆ n, f n a✝

Tactic:
hx_mono

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
hf : ∀ (n : ℕ), AEMeasurable (f n) μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n ↦ f n x
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))
this : Monotone fun n ↦ ∫⁻ (x : α), f n x ∂μ
a✝ : α
hx_mono : Monotone fun n ↦ f n a✝
⊢ Tendsto (fun n ↦ f n a✝) atTop (𝓝 (F a✝)) → F a✝ = ⨆ n, f n a✝
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Markov

Goal state before:
pp: case h₁
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hle : f ≤ᵐ[μ] g
hg : AEMeasurable g μ
ε : ℝ≥0∞
φ : α → ℝ≥0∞
hφm : Measurable φ
hφ_le : φ ≤ f
hφ_eq : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), φ a ∂μ
x : α
⊢ φ x ≤ f x

Tactic:
exact hφ_le _

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

Goal state before:
pp: case neg
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ≥0∞
hf : ∀ (a : α), f a ≤ 1
h'f : ∀ a ∈ sᶜ, f a = 0
x : α
hx : x ∉ s
⊢ f x ≤ s.indicator 1 x

Tactic:
· simpa [hx] using h'f x hx

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hμ : μ ≠ 0
hg : AEMeasurable g μ
hfi : ∫⁻ (x : α), f x ∂μ ≠ ∞
h : ∀ᵐ (x : α) ∂μ, f x < g x
⊢ ∫⁻ (x : α), f x ∂μ < ∫⁻ (x : α), g x ∂μ

Tactic:
rw [Ne, ← Measure.measure_univ_eq_zero] at hμ

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hμ : ¬μ univ = 0
hg : AEMeasurable g μ
hfi : ∫⁻ (x : α), f x ∂μ ≠ ∞
h : ∀ᵐ (x : α) ∂μ, f x < g x
⊢ ∫⁻ (x : α), f x ∂μ < ∫⁻ (x : α), g x ∂μ
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
f f' : α → β
h✝ : f =ᵐ[μ] f'
g : β → ℝ≥0∞
a : α
h : f a = f' a
⊢ (fun a ↦ g (f a)) a = (fun a ↦ g (f' a)) a

Tactic:
by dsimp only; rw [h]

Goal state after:
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_5
E : Type u_6
x✝ : MeasurableSpace α
inst✝⁶ : AddGroup E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : ContinuousAdd E
inst✝¹ : ContinuousNeg E
inst✝ : PseudoMetrizableSpace E
g f : α → E
hg : Measurable g
hf : StronglyMeasurable f
⊢ Measurable (g - f)

Tactic:
sub_eq_add_neg

Goal state after:
α : Type u_5
E : Type u_6
x✝ : MeasurableSpace α
inst✝⁶ : AddGroup E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : ContinuousAdd E
inst✝¹ : ContinuousNeg E
inst✝ : PseudoMetrizableSpace E
g f : α → E
hg : Measurable g
hf : StronglyMeasurable f
⊢ Measurable (g + -f)
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hf : AEMeasurable f μ
A : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), AEMeasurable.mk f hf a ∂μ
B : ∫⁻ (a : α), r * f a ∂μ = ∫⁻ (a : α), r * AEMeasurable.mk f hf a ∂μ
⊢ ∫⁻ (a : α), r * f a ∂μ = r * ∫⁻ (a : α), AEMeasurable.mk f hf a ∂μ

Tactic:
B,

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hf : AEMeasurable f μ
A : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), AEMeasurable.mk f hf a ∂μ
B : ∫⁻ (a : α), r * f a ∂μ = ∫⁻ (a : α), r * AEMeasurable.mk f hf a ∂μ
⊢ ∫⁻ (a : α), r * AEMeasurable.mk f hf a ∂μ = r * ∫⁻ (a : α), AEMeasurable.mk f hf a ∂μ
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
f : α → β
inst✝¹ : SeminormedAddCommGroup β
m m0 : MeasurableSpace α
hm : m ≤ m0
hf : StronglyMeasurable f
μ : Measure α
inst✝ : SigmaFinite (μ.trim hm)
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n) ∧ (μ.trim hm) (s n) < ∞ ∧ ∀ x ∈ s n, ‖f x‖₊ ≤ ↑n
hs_univ : ⋃ i, s i = univ
⊢ ∃ s, (∀ (n : ℕ), MeasurableSet (s n) ∧ μ (s n) < ∞ ∧ ∀ x ∈ s n, ‖f x‖ ≤ ↑n) ∧ ⋃ i, s i = univ

Tactic:
refine ⟨s, fun n ↦ ⟨(hs n).1, (le_trim hm).trans_lt (hs n).2.1, fun x hx ↦ ?_⟩, hs_univ⟩

Goal state after:
α : Type u_1
β : Type u_2
f : α → β
inst✝¹ : SeminormedAddCommGroup β
m m0 : MeasurableSpace α
hm : m ≤ m0
hf : StronglyMeasurable f
μ : Measure α
inst✝ : SigmaFinite (μ.trim hm)
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n) ∧ (μ.trim hm) (s n) < ∞ ∧ ∀ x ∈ s n, ‖f x‖₊ ≤ ↑n
hs_univ : ⋃ i, s i = univ
n : ℕ
x : α
hx : x ∈ s n
⊢ ‖f x‖ ≤ ↑n
Module: Mathlib.Topology.Perfect

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
⊢ PerfectSpace X ↔ ∀ (x : X), (𝓝[≠] x).NeBot

Tactic:
simp [perfectSpace_def, Preperfect, AccPt]

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

Goal state before:
pp: α : Type u_2
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ∞
g : ℕ → α → ℝ≥0∞
gmeas : ∀ (n : ℕ), Measurable (g n)
gf : ∀ (n : ℕ), g n ≤ f n
hg : ∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g n a ∂μ
g' : ℕ → α → ℝ≥0∞ := ⋯
M : ∀ (n : ℕ), Measurable (g' n)
I : ∀ (n : ℕ) (x : α), g n x ≤ g' n x
I' : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), g' n x ≤ f n x
Int_eq : ∀ (n : ℕ), ∫⁻ (x : α), g' n x ∂μ = ∫⁻ (x : α), f n x ∂μ
⊢ ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), g' i a ≤ F a

Tactic:
filter_upwards [h_bound, I'] with x h'x hx n using (hx n).trans (h'x n)

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

Goal state before:
pp: case mp.inl.inr
X : Type u
Y : Type v
Z : Type w
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
Φ : Z → X
Ψ : Z → Y
ε : ℝ
hε : 0 < ε
s : Set ((X ⊕ Y) × (X ⊕ Y))
δX : ℝ
δX0 : δX > 0
hX : ∀ ⦃a b : X⦄, dist a b < δX → Prod.map Sum.inl Sum.inl (a, b) ∈ s
δY : ℝ
δY0 : δY > 0
hY : ∀ ⦃a b : Y⦄, dist a b < δY → Prod.map Sum.inr Sum.inr (a, b) ∈ s
a : X
b : Y
h :
  (glueDist Φ Ψ ε (Sum.inl a) (Sum.inr b) < δX ∧ glueDist Φ Ψ ε (Sum.inl a) (Sum.inr b) < δY) ∧
    glueDist Φ Ψ ε (Sum.inl a) (Sum.inr b) < ε
⊢ (Sum.inl a, Sum.inr b) ∈ s

pp: case mp.inr.inl
X : Type u
Y : Type v
Z : Type w
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
Φ : Z → X
Ψ : Z → Y
ε : ℝ
hε : 0 < ε
s : Set ((X ⊕ Y) × (X ⊕ Y))
δX : ℝ
δX0 : δX > 0
hX : ∀ ⦃a b : X⦄, dist a b < δX → Prod.map Sum.inl Sum.inl (a, b) ∈ s
δY : ℝ
δY0 : δY > 0
hY : ∀ ⦃a b : Y⦄, dist a b < δY → Prod.map Sum.inr Sum.inr (a, b) ∈ s
a : Y
b : X
h :
  (glueDist Φ Ψ ε (Sum.inr a) (Sum.inl b) < δX ∧ glueDist Φ Ψ ε (Sum.inr a) (Sum.inl b) < δY) ∧
    glueDist Φ Ψ ε (Sum.inr a) (Sum.inl b) < ε
⊢ (Sum.inr a, Sum.inl b) ∈ s

pp: case mp.inr.inr
X : Type u
Y : Type v
Z : Type w
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
Φ : Z → X
Ψ : Z → Y
ε : ℝ
hε : 0 < ε
s : Set ((X ⊕ Y) × (X ⊕ Y))
δX : ℝ
δX0 : δX > 0
hX : ∀ ⦃a b : X⦄, dist a b < δX → Prod.map Sum.inl Sum.inl (a, b) ∈ s
δY : ℝ
δY0 : δY > 0
hY : ∀ ⦃a b : Y⦄, dist a b < δY → Prod.map Sum.inr Sum.inr (a, b) ∈ s
a b : Y
h :
  (glueDist Φ Ψ ε (Sum.inr a) (Sum.inr b) < δX ∧ glueDist Φ Ψ ε (Sum.inr a) (Sum.inr b) < δY) ∧
    glueDist Φ Ψ ε (Sum.inr a) (Sum.inr b) < ε
⊢ (Sum.inr a, Sum.inr b) ∈ s

Tactic:
· exact absurd h.2 (le_glueDist_inl_inr _ _ _ _ _).not_gt

Goal state after:
case mp.inr.inl
X : Type u
Y : Type v
Z : Type w
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
Φ : Z → X
Ψ : Z → Y
ε : ℝ
hε : 0 < ε
s : Set ((X ⊕ Y) × (X ⊕ Y))
δX : ℝ
δX0 : δX > 0
hX : ∀ ⦃a b : X⦄, dist a b < δX → Prod.map Sum.inl Sum.inl (a, b) ∈ s
δY : ℝ
δY0 : δY > 0
hY : ∀ ⦃a b : Y⦄, dist a b < δY → Prod.map Sum.inr Sum.inr (a, b) ∈ s
a : Y
b : X
h :
  (glueDist Φ Ψ ε (Sum.inr a) (Sum.inl b) < δX ∧ glueDist Φ Ψ ε (Sum.inr a) (Sum.inl b) < δY) ∧
    glueDist Φ Ψ ε (Sum.inr a) (Sum.inl b) < ε
⊢ (Sum.inr a, Sum.inl b) ∈ s

case mp.inr.inr
X : Type u
Y : Type v
Z : Type w
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
Φ : Z → X
Ψ : Z → Y
ε : ℝ
hε : 0 < ε
s : Set ((X ⊕ Y) × (X ⊕ Y))
δX : ℝ
δX0 : δX > 0
hX : ∀ ⦃a b : X⦄, dist a b < δX → Prod.map Sum.inl Sum.inl (a, b) ∈ s
δY : ℝ
δY0 : δY > 0
hY : ∀ ⦃a b : Y⦄, dist a b < δY → Prod.map Sum.inr Sum.inr (a, b) ∈ s
a b : Y
h :
  (glueDist Φ Ψ ε (Sum.inr a) (Sum.inr b) < δX ∧ glueDist Φ Ψ ε (Sum.inr a) (Sum.inr b) < δY) ∧
    glueDist Φ Ψ ε (Sum.inr a) (Sum.inr b) < ε
⊢ (Sum.inr a, Sum.inr b) ∈ s
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: E : ℕ → Type u_1
x y : (n : ℕ) → E n
⊢ 0 ≤ dist x y

Tactic:
by
  rcases eq_or_ne x y with (rfl | h)
  · simp [dist]
  · simp [dist, h]

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

Goal state before:
pp: E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
x : (n : ℕ) → E n
hx : x ∉ s
⊢ ∃ n, Disjoint s (cylinder x n)

Tactic:
rcases eq_empty_or_nonempty s with (rfl | hne)

Goal state after:
case inl
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
x : (n : ℕ) → E n
hs : IsClosed[Pi.topologicalSpace] ∅
hx : x ∉ ∅
⊢ ∃ n, Disjoint ∅ (cylinder x n)

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
x : (n : ℕ) → E n
hx : x ∉ s
hne : s.Nonempty
⊢ ∃ n, Disjoint s (cylinder x n)
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: X : ℕ → Type u
inst✝ : (n : ℕ) → MetricSpace (X n)
hs : ∀ (n : ℕ), TopologicalSpace.SeparableSpace (X n)
f : (n : ℕ) → X n → X (n + 1)
I : ∀ (n : ℕ), Isometry (f n)
hsX : (n : ℕ) → Set (X n)
hcX : ∀ (n : ℕ), (hsX n).Countable
hdX : ∀ (n : ℕ), Dense (hsX n)
s : Set (InductiveLimit I) := ⋃ i, toInductiveLimit I i '' hsX i
⊢ TopologicalSpace.SeparableSpace (InductiveLimit I)

Tactic:
refine ⟨s, countable_iUnion (fun n => (hcX n).image _), ?_⟩

Goal state after:
X : ℕ → Type u
inst✝ : (n : ℕ) → MetricSpace (X n)
hs : ∀ (n : ℕ), TopologicalSpace.SeparableSpace (X n)
f : (n : ℕ) → X n → X (n + 1)
I : ∀ (n : ℕ), Isometry (f n)
hsX : (n : ℕ) → Set (X n)
hcX : ∀ (n : ℕ), (hsX n).Countable
hdX : ∀ (n : ℕ), Dense (hsX n)
s : Set (InductiveLimit I) := ⋃ i, toInductiveLimit I i '' hsX i
⊢ Dense s