mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: case refine_4
α : Type u_1
mα : MeasurableSpace α
inst✝ : CountablyGenerated α
s : Set α
hs : MeasurableSet s
t : ℕ → Set α
ht : ∀ (n : ℕ), MeasurableSet (t n)
⊢ (∀ (n : ℕ), ∃ q, t n = ⋃ p, if q p then countablyGeneratedAtom α p else ∅) →
    ∃ q, ⋃ i, t i = ⋃ p, if q p then countablyGeneratedAtom α p else ∅

Tactic:
h

Goal state after:
case refine_4
α : Type u_1
mα : MeasurableSpace α
inst✝ : CountablyGenerated α
s : Set α
hs : MeasurableSet s
t : ℕ → Set α
ht : ∀ (n : ℕ), MeasurableSet (t n)
h : ∀ (n : ℕ), ∃ q, t n = ⋃ p, if q p then countablyGeneratedAtom α p else ∅
⊢ ∃ q, ⋃ i, t i = ⋃ p, if q p then countablyGeneratedAtom α p else ∅
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
t : Set α
x✝ : μ (t ∩ s) + ν (tᶜ ∩ s) ∈ {m | ∃ t, m = μ (t ∩ s) + ν (tᶜ ∩ s)}
t' : ℕ → Set α := fun n ↦ if n = 0 then t ∩ s else if n = 1 then tᶜ ∩ s else ∅
ht' : t' = fun n ↦ if n = 0 then t ∩ s else if n = 1 then tᶜ ∩ s else ∅
n : ℕ
hn₁ : ¬n = 1
hn₀ : ¬n = 0
⊢ min (μ (t' n)) (ν (t' n)) = 0

Tactic:
simp only [ht', if_neg hn₀, if_neg hn₁, measure_empty, le_refl, inf_of_le_left]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: case intro.refine_2
α : Type u_1
β : Type u_2
ι✝ : Type u_3
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
μ✝ ν : Measure α
s t : Set α
a : α
ι : Type u_4
inst✝¹ : Finite ι
μ : ι → Measure α
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
val✝ : Fintype ι
this : ∀ (n : ℕ), MeasurableSet (⋂ i, spanningSets (μ i) n)
⊢ ∀ (i : ι), Monotone (spanningSets (μ i))

Tactic:
exact fun i => monotone_spanningSets (μ i)

Goal state after:
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: case inl
β : Type u_2
α : Type u_3
γ : Type u_4
inst✝ : MeasurableSpace γ
h : Countable (α × β)
⊢ CountableOrCountablyGenerated (β × α) γ

Tactic:
refine ⟨Or.inl countable_prod_swap⟩

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

Goal state before:
pp: α : Type u_2
δ : Type u_5
m0 : MeasurableSpace α
inst✝ : MeasurableSpace δ
μ : Measure α
f : α → δ
hf : AEMeasurable f μ
s : Set δ
hs : MeasurableSet s
⊢ map f (μ.restrict (AEMeasurable.mk f hf ⁻¹' s)) = map f (μ.restrict (f ⁻¹' s))

Tactic:
apply congr_arg
      ext1 t ht
      simp only [ht, Measure.restrict_apply]
      apply measure_congr
      apply (EventuallyEq.refl _ _).inter (hf.ae_eq_mk.symm.preimage s)

Goal state after:
Module: Mathlib.MeasureTheory.Group.Arithmetic

Goal state before:
pp: α : Type u_1
G : Type u_2
inst✝³ : MeasurableSpace G
inst✝² : DivInvMonoid G
inst✝¹ : MeasurableMul G
inst✝ : MeasurableInv G
c : G
⊢ Measurable fun x ↦ c / x

Tactic:
convert! measurable_inv.const_mul c using 1

Goal state after:
α : Type u_1
G : Type u_2
inst✝³ : MeasurableSpace G
inst✝² : DivInvMonoid G
inst✝¹ : MeasurableMul G
inst✝ : MeasurableInv G
c : G
⊢ (fun x ↦ c / x) = fun x ↦ c * x⁻¹
Module: Mathlib.Topology.MetricSpace.HausdorffDistance

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
x : α
s : Set α
h : infEDist x s = 0
ε : ℝ≥0∞
εpos : ε > 0
⊢ infEDist x s < ε

Tactic:
h

Goal state after:
α : Type u
inst✝ : PseudoEMetricSpace α
x : α
s : Set α
h : infEDist x s = 0
ε : ℝ≥0∞
εpos : ε > 0
⊢ 0 < ε
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case neg
α : Type u_5
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → x ∈ s
S : Set (Set α) := {S | ∃ l ∈ s, ∃ u ∈ s, l ≤ u ∧ Icc l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : t.Countable
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
b : α
ba : b < a
hb : Ioo b a = ∅
hbs : b ∈ s
⊢ MeasurableSet (⋃ l ∈ t, ⋃ (_ : l ≤ b), Icc l b)

Tactic:
· refine .biUnion hc fun x hx => MeasurableSet.iUnion fun hlt => ?_
      exact .basic _ ⟨x, hts hx, b, hbs, hlt, mem_singleton _⟩

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

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : PseudoEMetricSpace α
inst✝ : PseudoEMetricSpace β
s₁ t₁ : Set α
s₂ t₂ : Set β
x✝¹ : ℝ≥0∞
x✝ : max (hausdorffEDist s₁ t₁) (hausdorffEDist s₂ t₂) ≤ x✝¹
⊢ hausdorffEDist (s₁ ×ˢ s₂) (t₁ ×ˢ t₂) ≤ x✝¹

Tactic:
simp_all only [hausdorffEDist, infEDist_prod, max_le_iff, iSup_le_iff, mem_prod, true_and,
    implies_true]

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 ∧ Ioc l u = S}

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

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

Goal state before:
pp: case a
α : Type u_5
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
u v : α
this✝¹ : MeasurableSpace α := borel α
this✝ : BorelSpace α
⊢ MeasurableSet (Ioc u v)

Tactic:
exact measurableSet_Ioc

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
δ : ℝ
hδ : 0 < δ
h : Disjoint (thickening δ s) (thickening δ t)
⊢ ∃ δ, 0 < δ ∧ Disjoint (cthickening δ s) (cthickening δ t)

Tactic:
refine ⟨δ / 2, half_pos hδ, h.mono ?_ ?_⟩ <;>
    exact cthickening_subset_thickening' hδ (half_lt_self hδ) _

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

Goal state before:
pp: case refine_3
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
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → μ (Ico a b) ≠ ∞
h : ∀ ⦃a b : α⦄, a < b → μ (Ico a b) = ν (Ico a b)
s : Set α
hsc : s.Countable
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
this : {x | ∃ i, ∃ (_ : i ∈ s), ∃ i_2, ∃ (_ : i_2 ∈ s) (_ : i < i_2), x = Ico i i_2}.Countable
⊢ ∀ s_1 ∈ {x | ∃ i, ∃ (_ : i ∈ s), ∃ i_2, ∃ (_ : i_2 ∈ s) (_ : i < i_2), x = Ico i i_2}, μ s_1 ≠ ∞

Tactic:
rintro _ ⟨l, -, u, -, hlt, rfl⟩

Goal state after:
case refine_3
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
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → μ (Ico a b) ≠ ∞
h : ∀ ⦃a b : α⦄, a < b → μ (Ico a b) = ν (Ico a b)
s : Set α
hsc : s.Countable
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
this : {x | ∃ i, ∃ (_ : i ∈ s), ∃ i_2, ∃ (_ : i_2 ∈ s) (_ : i < i_2), x = Ico i i_2}.Countable
l u : α
hlt : l < u
⊢ μ (Ico l u) ≠ ∞
Module: Mathlib.Topology.MetricSpace.HausdorffDistance

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
s t : Set α
⊢ hausdorffDist s t = hausdorffDist t s

Tactic:
by
  simp [hausdorffDist, hausdorffEDist_comm]

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

Goal state before:
pp: case refine_3
α : Type u_5
inst✝⁴ : TopologicalSpace α
m : MeasurableSpace α
inst✝³ : SecondCountableTopology α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : BorelSpace α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a ≤ b → μ (Icc a b) ≠ ∞
h : ∀ ⦃a b : α⦄, a ≤ b → μ (Icc a b) = ν (Icc a b)
s : Set α
hsc : s.Countable
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
hst : ∀ (x : α), IsTop x → x ∈ s
this : {x | ∃ i, ∃ (_ : i ∈ s), ∃ i_2, ∃ (_ : i_2 ∈ s) (_ : i ≤ i_2), x = Icc i i_2}.Countable
⊢ ∀ s_1 ∈ {x | ∃ i, ∃ (_ : i ∈ s), ∃ i_2, ∃ (_ : i_2 ∈ s) (_ : i ≤ i_2), x = Icc i i_2}, μ s_1 ≠ ∞

Tactic:
rintro _ ⟨l, -, u, -, hlt, rfl⟩

Goal state after:
case refine_3
α : Type u_5
inst✝⁴ : TopologicalSpace α
m : MeasurableSpace α
inst✝³ : SecondCountableTopology α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : BorelSpace α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a ≤ b → μ (Icc a b) ≠ ∞
h : ∀ ⦃a b : α⦄, a ≤ b → μ (Icc a b) = ν (Icc a b)
s : Set α
hsc : s.Countable
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
hst : ∀ (x : α), IsTop x → x ∈ s
this : {x | ∃ i, ∃ (_ : i ∈ s), ∃ i_2, ∃ (_ : i_2 ∈ s) (_ : i ≤ i_2), x = Icc i i_2}.Countable
l u : α
hlt : l ≤ u
⊢ μ (Icc l u) ≠ ∞
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
⊢ ∃ x, SecondCountableTopology α ∧ T4Space α ∧ BorelSpace α

Tactic:
rcases measurableEquiv_nat_bool_of_countablyGenerated α with ⟨s, ⟨f⟩⟩
  let := induced f inferInstance
  let F := f.toEquiv.toHomeomorphOfIsInducing <| .induced _
  exact ⟨inferInstance, F.secondCountableTopology, F.symm.t4Space,
    f.measurableEmbedding.borelSpace F.isInducing⟩

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.NullSingletonClass

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s✝ : Set α
inst✝ : NullSingletonClass μ
s : Set α
x : α
⊢ (μ.restrict s) {x} = 0

Tactic:
obtain ⟨t, hxt, ht1, ht2⟩ := exists_measurable_superset_of_null (measure_singleton x : μ {x} = 0)

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s✝ : Set α
inst✝ : NullSingletonClass μ
s : Set α
x : α
t : Set α
hxt : {x} ⊆ t
ht1 : MeasurableSet t
ht2 : μ t = 0
⊢ (μ.restrict s) {x} = 0
Module: Mathlib.MeasureTheory.Function.SimpleFunc

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : CompleteLattice β
inst✝³ : OrderClosedTopology β
inst✝² : Zero β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
i : ℕ → β
f : α → β
a : α
hf : Measurable f
h_zero : 0 = ⊥
k : ℕ
hk : i k ≤ f a
⊢ i k ≤ (Finset.range (k + 1)).sup fun k ↦ if i k ≤ f a then i k else 0

Tactic:
have : k ∈ Finset.range (k + 1) := Finset.mem_range.2 (Nat.lt_succ_self _)

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : CompleteLattice β
inst✝³ : OrderClosedTopology β
inst✝² : Zero β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
i : ℕ → β
f : α → β
a : α
hf : Measurable f
h_zero : 0 = ⊥
k : ℕ
hk : i k ≤ f a
this : k ∈ Finset.range (k + 1)
⊢ i k ≤ (Finset.range (k + 1)).sup fun k ↦ if i k ≤ f a then i k else 0
Module: Mathlib.MeasureTheory.Function.SimpleFunc

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : CompleteLattice β
inst✝³ : OrderClosedTopology β
inst✝² : Zero β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
i : ℕ → β
f : α → β
a : α
hf : Measurable f
h_zero : 0 = ⊥
k : ℕ
hk : i k ≤ f a
⊢ i k ≤ ⨆ n, (approx i f n) a

Tactic:
· refine le_iSup_of_le (k + 1) ?_
    rw [approx_apply a hf]
    have : k ∈ Finset.range (k + 1) := Finset.mem_range.2 (Nat.lt_succ_self _)
    refine le_trans (le_of_eq ?_) (Finset.le_sup this)
    rw [if_pos hk]

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

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ≥0∞
hf : Measurable f
a : α
⊢ ⨆ n, (eapprox f n) a = f a

Tactic:
rw [eapprox, iSup_approx_apply ennrealRatEmbed f a hf rfl]

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ≥0∞
hf : Measurable f
a : α
⊢ ⨆ k, ⨆ (_ : ennrealRatEmbed k ≤ f a), ennrealRatEmbed k = f a
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: 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
μ : Measure δ
s : Set ι
f : ι → δ → α
hs : s.Countable
hf : ∀ i ∈ s, AEMeasurable (f i) μ
g : ι → δ → α := fun i ↦ if hi : i ∈ s then mk (f i) ⋯ else fun _b ↦ sSup ∅
i : ι
hi : i ∈ s
⊢ Measurable (g i)

Tactic:
simpa [g, hi] using! (hf i hi).measurable_mk

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : PseudoEMetricSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : MeasurableSpace β
f : β → α
hf : Measurable f
s : Set α
y₀ : α
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
x : β
n : ℕ
⊢ edist y₀ (f x) + edist ((approxOn f hf s y₀ h₀ n) x) (f x) ≤ edist y₀ (f x) + edist y₀ (f x)

Tactic:
edist_approxOn_le hf h₀ x n

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : PseudoEMetricSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : MeasurableSpace β
f : β → α
hf : Measurable f
s : Set α
y₀ : α
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
x : β
n : ℕ
⊢ edist y₀ (f x) + edist y₀ (f x) ≤ edist y₀ (f x) + edist y₀ (f x)
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∞
hf : ∀ (n : ℕ), Measurable (f n)
h_mono : Monotone f
c : ℝ≥0 → ℝ≥0∞ := ofNNReal
F : α → ℝ≥0∞ := fun a ↦ ⨆ n, f n a
s : α →ₛ ℝ≥0
hsf : ∀ (x : α), ↑(s x) ≤ ⨆ n, f n x
r✝ : ℝ≥0
right✝ ha✝ : ↑r✝ < 1
ha : r✝ < 1
rs : α →ₛ ℝ≥0 := SimpleFunc.map (fun a ↦ r✝ * a) s
eq_rs : SimpleFunc.map c rs = const α ↑r✝ * SimpleFunc.map c s
eq : ∀ (p : ℝ≥0∞), ⇑(SimpleFunc.map c rs) ⁻¹' {p} = ⋃ n, ⇑(SimpleFunc.map c rs) ⁻¹' {p} ∩ {a | p ≤ f n a}
r : ℝ≥0∞
i j : ℕ
⊢ i ≤ j →
    (fun n ↦ ⇑(SimpleFunc.map c rs) ⁻¹' {r} ∩ {a | r ≤ f n a}) i ⊆
      (fun n ↦ ⇑(SimpleFunc.map c rs) ⁻¹' {r} ∩ {a | r ≤ f n a}) j

Tactic:
h

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), Measurable (f n)
h_mono : Monotone f
c : ℝ≥0 → ℝ≥0∞ := ofNNReal
F : α → ℝ≥0∞ := fun a ↦ ⨆ n, f n a
s : α →ₛ ℝ≥0
hsf : ∀ (x : α), ↑(s x) ≤ ⨆ n, f n x
r✝ : ℝ≥0
right✝ ha✝ : ↑r✝ < 1
ha : r✝ < 1
rs : α →ₛ ℝ≥0 := SimpleFunc.map (fun a ↦ r✝ * a) s
eq_rs : SimpleFunc.map c rs = const α ↑r✝ * SimpleFunc.map c s
eq : ∀ (p : ℝ≥0∞), ⇑(SimpleFunc.map c rs) ⁻¹' {p} = ⋃ n, ⇑(SimpleFunc.map c rs) ⁻¹' {p} ∩ {a | p ≤ f n a}
r : ℝ≥0∞
i j : ℕ
h : i ≤ j
⊢ (fun n ↦ ⇑(SimpleFunc.map c rs) ⁻¹' {r} ∩ {a | r ≤ f n a}) i ⊆
    (fun n ↦ ⇑(SimpleFunc.map c rs) ⁻¹' {r} ∩ {a | r ≤ f n a}) j
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: case neg
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : SeminormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
c : ℝ
hf : StronglyMeasurable f
hc : 0 ≤ c
n : ℕ
x : α
h0 : ¬‖(hf.approx n) x‖ = 0
⊢ ‖min 1 (c / ‖(hf.approx n) x‖)‖ * ‖(hf.approx n) x‖ ≤ c

Tactic:
rcases le_total ‖hf.approx n x‖ c with h | h

Goal state after:
case neg.inl
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : SeminormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
c : ℝ
hf : StronglyMeasurable f
hc : 0 ≤ c
n : ℕ
x : α
h0 : ¬‖(hf.approx n) x‖ = 0
h : ‖(hf.approx n) x‖ ≤ c
⊢ ‖min 1 (c / ‖(hf.approx n) x‖)‖ * ‖(hf.approx n) x‖ ≤ c

case neg.inr
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : SeminormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
c : ℝ
hf : StronglyMeasurable f
hc : 0 ≤ c
n : ℕ
x : α
h0 : ¬‖(hf.approx n) x‖ = 0
h : c ≤ ‖(hf.approx n) x‖
⊢ ‖min 1 (c / ‖(hf.approx n) x‖)‖ * ‖(hf.approx n) x‖ ≤ c
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∞
hfg : f ≤ᵐ[μ] g
hf : ∫⁻ (x : α), f x ∂μ ≠ ∞
hg : AEMeasurable g μ
hgf : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), f x ∂μ
n : ℕ
this : (↑n)⁻¹ * μ {x | f x + (↑n)⁻¹ ≤ g x} ≤ 0
⊢ μ {a | f a + (↑n)⁻¹ ≤ g a} = 0

Tactic:
nonpos_iff_eq_zero,

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hfg : f ≤ᵐ[μ] g
hf : ∫⁻ (x : α), f x ∂μ ≠ ∞
hg : AEMeasurable g μ
hgf : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), f x ∂μ
n : ℕ
this : (↑n)⁻¹ * μ {x | f x + (↑n)⁻¹ ≤ g x} = 0
⊢ μ {a | f a + (↑n)⁻¹ ≤ g a} = 0
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
⊢ g (f a) = g (f' a)

Tactic:
h

Goal state after:
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
⊢ g (f' a) = g (f' a)
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
h : μ = ν
x✝¹ : α → ℝ≥0∞
x✝ : Measurable x✝¹
⊢ ∫⁻ (a : α), x✝¹ a ∂μ = ∫⁻ (a : α), x✝¹ a ∂ν

Tactic:
h

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
h : μ = ν
x✝¹ : α → ℝ≥0∞
x✝ : Measurable x✝¹
⊢ ∫⁻ (a : α), x✝¹ a ∂ν = ∫⁻ (a : α), x✝¹ a ∂ν
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

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

Tactic:
h

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

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

Tactic:
h

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

Goal state before:
pp: case e_f.hf
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
a : α
⊢ Monotone fun n ↦ (eapprox f n) a

Tactic:
intro i j h
        exact monotone_eapprox _ h a

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

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

Tactic:
h

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

Goal state before:
pp: case e_f.hg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
a : α
⊢ Monotone fun n ↦ (eapprox g n) a

Tactic:
intro i j h
        exact monotone_eapprox _ h a

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
⊢ i ≤ j → (fun n ↦ ⇑(eapprox f n) + ⇑(eapprox g n)) i ≤ (fun n ↦ ⇑(eapprox f n) + ⇑(eapprox g n)) j

Tactic:
h

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
⊢ (fun n ↦ ⇑(eapprox f n) + ⇑(eapprox g n)) i ≤ (fun n ↦ ⇑(eapprox f n) + ⇑(eapprox g n)) j
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
⊢ ∫⁻ (a : α), ⨆ n, (⇑(eapprox f n) + ⇑(eapprox g n)) a ∂μ = ⨆ n, (eapprox f n).lintegral μ + (eapprox g n).lintegral μ

Tactic:
rw [lintegral_iSup]
      · congr
        funext n
        rw [← SimpleFunc.add_lintegral, ← SimpleFunc.lintegral_eq_lintegral]
        simp only [Pi.add_apply, SimpleFunc.coe_add]
      · fun_prop
      · intro i j h a
        dsimp
        gcongr <;> exact monotone_eapprox _ h _

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

Goal state before:
pp: case refine_1
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
i j : ℕ
⊢ i ≤ j → (fun n ↦ (eapprox f n).lintegral μ) i ≤ (fun n ↦ (eapprox f n).lintegral μ) j

Tactic:
h

Goal state after:
case refine_1
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).lintegral μ) i ≤ (fun n ↦ (eapprox f n).lintegral μ) j
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: case refine_2
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
i j : ℕ
⊢ i ≤ j → (fun n ↦ (eapprox g n).lintegral μ) i ≤ (fun n ↦ (eapprox g n).lintegral μ) j

Tactic:
h

Goal state after:
case refine_2
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 g n).lintegral μ) i ≤ (fun n ↦ (eapprox g n).lintegral μ) j
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
⊢ i ≤ j → (fun n ↦ ⇑(const α r * eapprox f n)) i ≤ (fun n ↦ ⇑(const α r * eapprox f n)) j

Tactic:
h

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
⊢ (fun n ↦ ⇑(const α r * eapprox f n)) i ≤ (fun n ↦ ⇑(const α r * eapprox f n)) j
Module: Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure ν
⊢ Tendsto (fun n ↦ ν (μ.sigmaFiniteSetGE ν n)) atTop
    (𝓝 (⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s))

Tactic:
refine tendsto_of_tendsto_of_tendsto_of_le_of_le ?_
    tendsto_const_nhds (measure_sigmaFiniteSetGE_ge μ ν) (measure_sigmaFiniteSetGE_le μ ν)
  nth_rewrite 2 [← tsub_zero (⨆ (s) (_ : MeasurableSet s) (_ : SigmaFinite (μ.restrict s)), ν s)]
  refine ENNReal.Tendsto.sub tendsto_const_nhds ?_ (Or.inr ENNReal.zero_ne_top)
  simp only [one_div]
  exact ENNReal.tendsto_inv_nat_nhds_zero

Goal state after: