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
n : ℤ
⊢ Measurable fun x ↦ (x, n).2 • (x, n).1

Tactic:
cases n with
    | ofNat n =>
      simp only [Int.ofNat_eq_natCast, natCast_zsmul]
      exact measurable_const_smul _
    | negSucc n =>
      simp only [negSucc_zsmul]
      exact (measurable_const_smul _).neg

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

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
x y : α
s : Set α
⊢ infEDist x s ≤ edist x y + infEDist y s

Tactic:
rw [add_comm]
  exact infEDist_le_infEDist_add_edist

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
s : Set α
x : α
⊢ infDist x s = ⨅ y, dist x ↑y

Tactic:
rw [infDist, infEDist, iInf_subtype', ENNReal.toReal_iInf]
  · simp only [dist_edist]
  · finiteness

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

Goal state before:
pp: ε : ℝ
α : Type u_2
inst✝ : PseudoMetricSpace α
s : Set α
hε : 0 ≤ ε
⊢ diam (cthickening ε s) ≤ diam s + 2 * ε

Tactic:
by
  lift ε to ℝ≥0 using hε
  refine (toReal_le_add' (ediam_cthickening_le _) ?_ ?_).trans_eq ?_
  · exact fun h ↦ top_unique <| h ▸ ediam_mono (self_subset_cthickening _)
  · simp [mul_eq_top]
  · simp [diam]

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

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
s t : Set α
hst : Disjoint s t
hs : IsCompact s
ht : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] t
r : ℝ≥0
hr : 0 < r
h : ∀ x ∈ s, ∀ y ∈ t, ↑r < edist x y
z x : α
hx : x ∈ s
hzx : edist z x < ↑(r / 2)
y : α
hy : y ∈ t
hzy : edist z y < ↑(r / 2)
⊢ z ∈ ⊥

Tactic:
refine (h x hx y hy).not_ge ?_

Goal state after:
α : Type u
inst✝ : PseudoEMetricSpace α
s t : Set α
hst : Disjoint s t
hs : IsCompact s
ht : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] t
r : ℝ≥0
hr : 0 < r
h : ∀ x ∈ s, ∀ y ∈ t, ↑r < edist x y
z x : α
hx : x ∈ s
hzx : edist z x < ↑(r / 2)
y : α
hy : y ∈ t
hzy : edist z y < ↑(r / 2)
⊢ edist x y ≤ ↑r
Module: Mathlib.Topology.MetricSpace.Thickening

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
ε δ : ℝ
s : Set α
⊢ thickening ε (thickening δ s) ⊆ thickening (ε + δ) s

Tactic:
obtain hε | hε := le_total ε 0
  · simp only [thickening_of_nonpos hε, empty_subset]
  obtain hδ | hδ := le_total δ 0
  · simp only [thickening_of_nonpos hδ, thickening_empty, empty_subset]
  intro x
  simp_rw [mem_thickening_iff_exists_edist_lt, ENNReal.ofReal_add hε hδ]
  exact fun ⟨y, ⟨z, hz, hy⟩, hx⟩ =>
    ⟨z, hz, (edist_triangle _ _ _).trans_lt <| ENNReal.add_lt_add hx hy⟩

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

Goal state before:
pp: α : Type u_1
inst✝² : PseudoEMetricSpace α
inst✝¹ : MeasurableSpace α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : Set α
hs : ∃ R > 0, μ (cthickening R s) ≠ ∞
A :
  Tendsto (fun r ↦ μ (cthickening r s)) (𝓝[>] 0)
    (𝓝 (μ (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s)))
a✝ : ℝ
⊢ a✝ ∈ Iic 0 → μ (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s) = μ (cthickening a✝ s)

Tactic:
hr

Goal state after:
α : Type u_1
inst✝² : PseudoEMetricSpace α
inst✝¹ : MeasurableSpace α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : Set α
hs : ∃ R > 0, μ (cthickening R s) ≠ ∞
A :
  Tendsto (fun r ↦ μ (cthickening r s)) (𝓝[>] 0)
    (𝓝 (μ (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s)))
a✝ : ℝ
hr : a✝ ∈ Iic 0
⊢ μ (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s) = μ (cthickening a✝ s)
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Metric

Goal state before:
pp: α : Type u_1
inst✝² : PseudoEMetricSpace α
inst✝¹ : MeasurableSpace α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : Set α
hs : ∃ R > 0, μ (cthickening R s) ≠ ∞
A :
  Tendsto (fun r ↦ μ (cthickening r s)) (𝓝[>] 0)
    (𝓝 (μ (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s)))
⊢ Tendsto (fun r ↦ μ (cthickening r s)) (𝓝 0) (𝓝 (μ (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s)))

Tactic:
have B : Tendsto (fun r => μ (cthickening r s)) (𝓝[Iic 0] 0) (𝓝 (μ (closure s))) := by
    apply Tendsto.congr' _ tendsto_const_nhds
    filter_upwards [self_mem_nhdsWithin (α := ℝ)] with _ hr
    rw [cthickening_of_nonpos hr]

Goal state after:
α : Type u_1
inst✝² : PseudoEMetricSpace α
inst✝¹ : MeasurableSpace α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : Set α
hs : ∃ R > 0, μ (cthickening R s) ≠ ∞
A :
  Tendsto (fun r ↦ μ (cthickening r s)) (𝓝[>] 0)
    (𝓝 (μ (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s)))
B :
  Tendsto (fun r ↦ μ (cthickening r s)) (𝓝[≤] 0)
    (𝓝 (μ (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s)))
⊢ Tendsto (fun r ↦ μ (cthickening r s)) (𝓝 0) (𝓝 (μ (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s)))
Module: Mathlib.MeasureTheory.Function.SimpleFunc

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : Zero β
f : α →ₛ β
s : Set α
hs : MeasurableSet s
⊢ ⇑(if hs : MeasurableSet s then piecewise s hs f 0 else 0) = s.indicator ⇑f

Tactic:
dif_pos hs,

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : Zero β
f : α →ₛ β
s : Set α
hs : MeasurableSet s
⊢ ⇑(piecewise s hs f 0) = s.indicator ⇑f
Module: Mathlib.MeasureTheory.MeasurableSpace.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
C : Set (Set α)
D : Set (Set β)
hC : generateFrom C = inst✝¹
hD : generateFrom D = inst✝
h2C : IsCountablySpanning C
h2D : IsCountablySpanning D
⊢ generateFrom (image2 (fun x1 x2 ↦ x1 ×ˢ x2) C D) = Prod.instMeasurableSpace

Tactic:
← hC,

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
C : Set (Set α)
D : Set (Set β)
hC : generateFrom C = inst✝¹
hD : generateFrom D = inst✝
h2C : IsCountablySpanning C
h2D : IsCountablySpanning D
⊢ generateFrom (image2 (fun x1 x2 ↦ x1 ×ˢ x2) C D) = Prod.instMeasurableSpace
Module: Mathlib.MeasureTheory.Measure.Typeclasses.NullSingletonClass

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : NullSingletonClass μ
a : α
⊢ μ.restrict {a} = 0

Tactic:
by
  simp only [measure_singleton, Measure.restrict_eq_zero]

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

Goal state before:
pp: case neg
δ : Type u_4
mδ : MeasurableSpace δ
α : Type u_5
mα : MeasurableSpace α
inst✝ : ConditionallyCompleteLattice α
p : Prop
f : δ → α
hf : Measurable f
h : ¬p
⊢ Measurable fun b ↦ sSup ∅

Tactic:
· exact measurable_const

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

Goal state before:
pp: case neg
δ : Type u_4
mδ : MeasurableSpace δ
α : Type u_5
mα : MeasurableSpace α
inst✝ : ConditionallyCompleteLattice α
p : Prop
f : δ → α
hf : Measurable f
h : ¬p
⊢ Measurable fun b ↦ sInf ∅

Tactic:
· exact measurable_const

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_5
inst✝ : MeasurableSpace β
μ' : Measure β
f : α →ₛ ℝ≥0∞
g : β →ₛ ℝ≥0∞
m' : α → β
eq : ∀ (a : α), f a = g (m' a)
h : ∀ (s : Set β), MeasurableSet s → μ' s = μ (m' ⁻¹' s)
y : ℝ≥0∞
⊢ μ {x | g (m' x) ∈ {y}} = μ' {x | g x ∈ {y}}

Tactic:
exact (h (g ⁻¹' {y}) (g.measurableSet_preimage _)).symm

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : Zero β
f : α →ₛ β
x : α
⊢ x ∈ support ⇑f ↔ x ∈ ⋃ y ∈ {y ∈ f.range | y ≠ 0}, ⇑f ⁻¹' {y}

Tactic:
simp only [mem_support, Set.mem_preimage, mem_filter, mem_range_self, true_and, exists_prop,
      mem_iUnion, mem_singleton_iff, exists_eq_right']

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

Goal state before:
pp: case mpr
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : Zero β
μ : Measure α
f : α →ₛ β
H : ∀ (y : β), y ≠ 0 → μ (⇑f ⁻¹' {y}) < ∞
⊢ μ (⋃ y ∈ {y ∈ f.range | y ≠ 0}, ⇑f ⁻¹' {y}) < ∞

Tactic:
exact measure_biUnion_lt_top (finite_toSet _) fun y hy ↦ H y (mem_filter.1 hy).2

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

Goal state before:
pp: α : Type u_1
inst✝³ : MeasurableSpace α
β : Type u_3
inst✝² : MeasurableSpace β
inst✝¹ : PseudoMetricSpace β
inst✝ : BorelSpace β
μ : Measure α
g : α → β
hf : ∀ ε > 0, ∃ f, AEMeasurable f μ ∧ ∀ᵐ (x : α) ∂μ, dist (f x) (g x) ≤ ε
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
⊢ AEMeasurable g μ

Tactic:
choose f Hf using fun n : ℕ => hf (u n) (u_pos n)

Goal state after:
α : Type u_1
inst✝³ : MeasurableSpace α
β : Type u_3
inst✝² : MeasurableSpace β
inst✝¹ : PseudoMetricSpace β
inst✝ : BorelSpace β
μ : Measure α
g : α → β
hf : ∀ ε > 0, ∃ f, AEMeasurable f μ ∧ ∀ᵐ (x : α) ∂μ, dist (f x) (g x) ≤ ε
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
f : ℕ → α → β
Hf : ∀ (n : ℕ), AEMeasurable (f n) μ ∧ ∀ᵐ (x : α) ∂μ, dist (f n x) (g x) ≤ u n
⊢ AEMeasurable g μ
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
⊢ μ s = μ (s ∩ f ⁻¹' ({0} ∪ Ioi 0))

Tactic:
singleton_union,

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
⊢ μ s = μ (s ∩ f ⁻¹' insert 0 (Ioi 0))
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
ι : Type u_5
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
f : ι → α → β
g : α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) u (𝓝 (g x))
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
v : ℕ → ι
hv : Tendsto v atTop u
t : ℕ → Set β
t_sep : ∀ (n : ℕ), IsSeparable (t n)
ht : ∀ (n : ℕ), f (v n) ⁻¹' t n ∈ ae μ
x : α
hx : ∀ (i : ℕ), f (v i) x ∈ t i
⊢ Tendsto (fun n ↦ f n x) u (𝓝 (g x)) → g x ∈ closure[inst✝³] (⋃ i, t i)

Tactic:
h'x

Goal state after:
α : Type u_1
β : Type u_2
inst✝³ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
ι : Type u_5
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
f : ι → α → β
g : α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) u (𝓝 (g x))
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
v : ℕ → ι
hv : Tendsto v atTop u
t : ℕ → Set β
t_sep : ∀ (n : ℕ), IsSeparable (t n)
ht : ∀ (n : ℕ), f (v n) ⁻¹' t n ∈ ae μ
x : α
hx : ∀ (i : ℕ), f (v i) x ∈ t i
h'x : Tendsto (fun n ↦ f n x) u (𝓝 (g x))
⊢ g x ∈ closure[inst✝³] (⋃ i, t i)
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
inst✝ : PseudoMetrizableSpace β
f : ℕ → α → β
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n ↦ f n x) atTop (𝓝 l)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
g : α → β
left✝ : Measurable g
hg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
Hg : AEStronglyMeasurable g μ
x : α
hx : Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
⊢ g x = AEStronglyMeasurable.mk g Hg x → Tendsto (fun n ↦ f n x) atTop (𝓝 (AEStronglyMeasurable.mk g Hg x))

Tactic:
h'x

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
inst✝ : PseudoMetrizableSpace β
f : ℕ → α → β
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n ↦ f n x) atTop (𝓝 l)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
g : α → β
left✝ : Measurable g
hg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
Hg : AEStronglyMeasurable g μ
x : α
hx : Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
h'x : g x = AEStronglyMeasurable.mk g Hg x
⊢ Tendsto (fun n ↦ f n x) atTop (𝓝 (AEStronglyMeasurable.mk g Hg x))
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
inst✝ : PseudoMetrizableSpace β
f : ℕ → α → β
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n ↦ f n x) atTop (𝓝 l)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
g : α → β
left✝ : Measurable g
hg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
Hg : AEStronglyMeasurable g μ
x : α
hx : Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
h'x : g x = AEStronglyMeasurable.mk g Hg x
⊢ Tendsto (fun n ↦ f n x) atTop (𝓝 (AEStronglyMeasurable.mk g Hg x))

Tactic:
h'x

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
inst✝ : PseudoMetrizableSpace β
f : ℕ → α → β
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n ↦ f n x) atTop (𝓝 l)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
g : α → β
left✝ : Measurable g
hg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
Hg : AEStronglyMeasurable g μ
x : α
hx : Tendsto (fun n ↦ f n x) atTop (𝓝 (AEStronglyMeasurable.mk g Hg x))
h'x : g x = AEStronglyMeasurable.mk g Hg x
⊢ Tendsto (fun n ↦ f n x) atTop (𝓝 (AEStronglyMeasurable.mk g Hg x))
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: case refine_2
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), Measurable (f b)
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
val✝ : Encodable β
h✝ : Nonempty β
inhabited_h : Inhabited β
this : ∀ (a : α), ⨆ b, f b a = ⨆ n, f (Directed.sequence f h_directed n) a
b : β
⊢ ∫⁻ (a : α), f b a ∂μ ≤ ⨆ n, ∫⁻ (a : α), f (Directed.sequence f h_directed n) a ∂μ

Tactic:
exact le_iSup_of_le (encode b + 1) (lintegral_mono <| h_directed.le_sequence b)

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

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
⊢ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), g x - f x ∂μ + ∫⁻ (x : α), f x ∂μ

Tactic:
by_cases hfi : ∫⁻ x, f x ∂μ = ∞

Goal state after:
case pos
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hfi : ∫⁻ (x : α), f x ∂μ = ∞
⊢ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), g x - f x ∂μ + ∫⁻ (x : α), f x ∂μ

case neg
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hfi : ¬∫⁻ (x : α), f x ∂μ = ∞
⊢ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), g x - f x ∂μ + ∫⁻ (x : α), f x ∂μ
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
M : Type u_5
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : ContinuousMul M
m : MeasurableSpace α
n : MeasurableSpace β
ι : Type u_6
f : ι → α → β → M
g : α → β
s : Finset ι
hf : ∀ i ∈ s, StronglyMeasurable ↿(f i)
hg : Measurable g
⊢ StronglyMeasurable fun a ↦ (∏ i ∈ s, f i a) (g a)

Tactic:
by
  simp only [Finset.prod_apply]; fun_prop

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

Goal state before:
pp: α : Type u_1
β : Type u_2
f : α → β
mα : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : TopologicalSpace β
inst✝² : PseudoMetrizableSpace β
inst✝¹ : SecondCountableTopology β
inst✝ : OpensMeasurableSpace β
hf : Measurable f
⊢ StronglyMeasurable f

Tactic:
by
  let := pseudoMetrizableSpacePseudoMetric β
  nontriviality β; inhabit β
  exact ⟨SimpleFunc.approxOn f hf Set.univ default (Set.mem_univ _), fun x ↦
    SimpleFunc.tendsto_approxOn hf (Set.mem_univ _) (by simp)⟩

Goal state after:
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 : MeasurableSet s
hs_subset : s ⊆ (μ.sigmaFiniteSetWRT' ν)ᶜ
hνs : ν s ≠ 0
⊢ μ s = ∞

Tactic:
by
  suffices ¬ SigmaFinite (μ.restrict s) by
    by_contra h
    have h_lt_top : Fact (μ s < ∞) := ⟨Ne.lt_top h⟩
    exact this inferInstance
  intro hsσ
  have h_lt : ν (μ.sigmaFiniteSetWRT' ν) < ν (μ.sigmaFiniteSetWRT' ν ∪ s) := by
    rw [measure_union _ hs]
    · exact ENNReal.lt_add_right (measure_ne_top _ _) hνs
    · exact disjoint_compl_right.mono_right hs_subset
  have h_le : ν (μ.sigmaFiniteSetWRT' ν ∪ s) ≤ ν (μ.sigmaFiniteSetWRT' ν) := by
    conv_rhs => rw [measure_sigmaFiniteSetWRT']
    refine (le_iSup
      (f := fun (_ : SigmaFinite (μ.restrict (μ.sigmaFiniteSetWRT' ν ∪ s))) ↦ _) ?_).trans ?_
    · have := sigmaFinite_restrict_sigmaFiniteSetWRT' μ ν
      infer_instance
    · exact le_iSup₂ (f := fun s _ ↦ ⨆ (_ : SigmaFinite (μ.restrict _)), ν s)
        (μ.sigmaFiniteSetWRT' ν ∪ s) (measurableSet_sigmaFiniteSetWRT'.union hs)
  exact h_lt.not_ge h_le

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∞ := fun t ↦ Nat.rec (motive := fun x ↦ α → ℝ≥0∞) (g 0) (fun n I x ↦ max (g (n + 1) x) (I x)) t
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
x : α
⊢ (∀ (i : ℕ), f i x ≤ F x) → (∀ (n : ℕ), g' n x ≤ f n x) → ∀ (i : ℕ), g' i x ≤ F x

Tactic:
h'x

Goal state after:
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∞ := fun t ↦ Nat.rec (motive := fun x ↦ α → ℝ≥0∞) (g 0) (fun n I x ↦ max (g (n + 1) x) (I x)) t
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
x : α
h'x : ∀ (i : ℕ), f i x ≤ F x
⊢ (∀ (n : ℕ), g' n x ≤ f n x) → ∀ (i : ℕ), g' i x ≤ F x
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∞ := fun t ↦ Nat.rec (motive := fun x ↦ α → ℝ≥0∞) (g 0) (fun n I x ↦ max (g (n + 1) x) (I x)) t
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
this : ∀ᵐ (a : α) ∂μ, Tendsto (fun i ↦ g' i a) atTop (𝓝 (F a))
x : α
hx : Tendsto (fun i ↦ g' i x) atTop (𝓝 (F x))
⊢ (∀ (n : ℕ), g' n x ≤ f n x) → (∀ (i : ℕ), f i x ≤ F x) → Tendsto (fun i ↦ f i x) atTop (𝓝 (F x))

Tactic:
h'x

Goal state after:
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∞ := fun t ↦ Nat.rec (motive := fun x ↦ α → ℝ≥0∞) (g 0) (fun n I x ↦ max (g (n + 1) x) (I x)) t
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
this : ∀ᵐ (a : α) ∂μ, Tendsto (fun i ↦ g' i a) atTop (𝓝 (F a))
x : α
hx : Tendsto (fun i ↦ g' i x) atTop (𝓝 (F x))
h'x : ∀ (n : ℕ), g' n x ≤ f n x
⊢ (∀ (i : ℕ), f i x ≤ F x) → Tendsto (fun i ↦ f i x) atTop (𝓝 (F x))
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: case neg
a b : ℝ
x y z : ↑(Icc a b)
s t : ↑unitInterval
hs : ↑s = 1
ht : ¬↑t = 1
this : 1 - ↑t ≠ 0
⊢ (1 - 1) * ↑x + 1 * ((1 - ↑t) * ↑y + ↑t * ↑z) =
    (1 - 1 * ↑t) * ((1 - 1 * (1 - ↑t) / (1 - 1 * ↑t)) * ↑x + 1 * (1 - ↑t) / (1 - 1 * ↑t) * ↑y) + 1 * ↑t * ↑z

Tactic:
field_simp

Goal state after:
case neg
a b : ℝ
x y z : ↑(Icc a b)
s t : ↑unitInterval
hs : ↑s = 1
ht : ¬↑t = 1
this : 1 - ↑t ≠ 0
⊢ (1 - 1) * ↑x + ((1 - ↑t) * ↑y + ↑t * ↑z) = (1 - ↑t) * ((1 - 1) * ↑x + ↑y) + ↑t * ↑z
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: case neg
a b : ℝ
x y z : ↑(Icc a b)
s t : ↑unitInterval
hs : ¬↑s = 1
ht : ¬↑t = 1
this : 1 - ↑s * ↑t ≠ 0
⊢ (1 - ↑s) * ↑x + ↑s * ((1 - ↑t) * ↑y + ↑t * ↑z) =
    (1 - ↑s * ↑t) * ((1 - ↑s * (1 - ↑t) / (1 - ↑s * ↑t)) * ↑x + ↑s * (1 - ↑t) / (1 - ↑s * ↑t) * ↑y) + ↑s * ↑t * ↑z

Tactic:
field_simp

Goal state after:
case neg
a b : ℝ
x y z : ↑(Icc a b)
s t : ↑unitInterval
hs : ¬↑s = 1
ht : ¬↑t = 1
this : 1 - ↑s * ↑t ≠ 0
⊢ (1 - ↑s) * ↑x + ↑s * ((1 - ↑t) * ↑y + ↑t * ↑z) =
    ↑x * (1 - ↑s * ↑t - ↑s * (1 - ↑t)) + ↑s * (1 - ↑t) * ↑y + ↑s * ↑t * ↑z
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: case mp.h₁
E : ℕ → Type u_1
x y : (n : ℕ) → E n
n : ℕ
hy : y ∈ cylinder x n
z : (n : ℕ) → E n
hz : z ∈ cylinder y n
i : ℕ
hi : i < n
⊢ z i = x i

Tactic:
← hy i hi

Goal state after:
case mp.h₁
E : ℕ → Type u_1
x y : (n : ℕ) → E n
n : ℕ
hy : y ∈ cylinder x n
z : (n : ℕ) → E n
hz : z ∈ cylinder y n
i : ℕ
hi : i < n
⊢ z i = y i
Module: Mathlib.Topology.Metrizable.CompletelyMetrizable

Goal state before:
pp: 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 := ⋯ }
x y : X
hy : dist y x < 1
⊢ y = x

Tactic:
by_contra

Goal state after:
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 := ⋯ }
x y : X
hy : dist y x < 1
this : ¬y = x
⊢ False