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

Goal state before:
pp: β : Type u_2
inst✝¹ : MeasurableSpace β
p : ℝ≥0∞
E : Type u_7
inst✝ : NormedAddCommGroup E
f : β → E
μ : Measure β
hf : MemLp f p μ
hp_ne_top : p ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
f' : β → E := AEStronglyMeasurable.mk f ⋯
g : β →ₛ E
hg : eLpNorm (f' - ⇑g) p μ < ε
g_mem : MemLp (⇑g) p μ
this : eLpNorm (f - ⇑g) p μ = eLpNorm (f' - ⇑g) p μ
⊢ eLpNorm (f - ⇑g) p μ < ε

Tactic:
by rwa [this]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: case pos
α : Type u_1
F : Type u_3
F' : Type u_4
G : Type u_5
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
T : Set α → F →L[ℝ] F'
h_add : FinMeasAdditive μ T
f : α →ₛ G
hf : Integrable (⇑f) μ
g : G → F
hg : g 0 = 0
T_empty : T ∅ = 0
hfp : ∀ x ∈ f.range, x ≠ 0 → μ (⇑f ⁻¹' {x}) ≠ ∞
a : α
hb : f a ∈ f.range
h0 : g (f a) = 0
⊢ (T (⇑(map g f) ⁻¹' {g (f a)})) (g (f a)) = ∑ j ∈ f.range with g j = g (f a), (T (⇑f ⁻¹' {j})) (g j)

Tactic:
h0

Goal state after:
case pos
α : Type u_1
F : Type u_3
F' : Type u_4
G : Type u_5
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
T : Set α → F →L[ℝ] F'
h_add : FinMeasAdditive μ T
f : α →ₛ G
hf : Integrable (⇑f) μ
g : G → F
hg : g 0 = 0
T_empty : T ∅ = 0
hfp : ∀ x ∈ f.range, x ≠ 0 → μ (⇑f ⁻¹' {x}) ≠ ∞
a : α
hb : f a ∈ f.range
h0 : g (f a) = 0
⊢ (T (⇑(map g f) ⁻¹' {0})) 0 = ∑ x ∈ f.range with g x = 0, (T (⇑f ⁻¹' {x})) (g x)
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
hp_pos : p ≠ 0
hp_ne_top : p ≠ ∞
P : ↥(simpleFunc E p μ) → Prop
indicatorConst : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), P (simpleFunc.indicatorConst p hs ⋯ c)
add :
  ∀ ⦃f g : α →ₛ E⦄ (hf : MemLp (⇑f) p μ) (hg : MemLp (⇑g) p μ),
    Disjoint (support ⇑f) (support ⇑g) → P (f.toLp hf) → P (g.toLp hg) → P (f.toLp hf + g.toLp hg)
f : ↥(simpleFunc E p μ)
⊢ P f

Tactic:
suffices ∀ f : α →ₛ E, ∀ hf : MemLp f p μ, P (toLp f hf) by
    rw [← toLp_toSimpleFunc f]
    apply this
  clear f
  apply SimpleFunc.induction
  · intro c s hs hf
    by_cases hc : c = 0
    · convert! indicatorConst 0 MeasurableSet.empty (by simp) using 1
      ext1
      simp [hc]
    exact indicatorConst c hs
      (SimpleFunc.measure_lt_top_of_memLp_indicator hp_pos hp_ne_top hc hs hf)
  · intro f g hfg hf hg hfg'
    obtain ⟨hf', hg'⟩ : MemLp f p μ ∧ MemLp g p μ :=
      (memLp_add_of_disjoint hfg f.stronglyMeasurable g.stronglyMeasurable).mp hfg'
    exact add hf' hg' hfg (hf hf') (hg hg')

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: α : Type u_1
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : DecidablePred fun x ↦ x ≠ 0
m : MeasurableSpace α
f : α →ₛ F
μ : Measure α
⊢ integral μ f = ∑ x ∈ f.range with x ≠ 0, μ.real (⇑f ⁻¹' {x}) • x

Tactic:
integral_def,

Goal state after:
α : Type u_1
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : DecidablePred fun x ↦ x ≠ 0
m : MeasurableSpace α
f : α →ₛ F
μ : Measure α
⊢ setToSimpleFunc (weightedSMul μ) f = ∑ x ∈ f.range with x ≠ 0, μ.real (⇑f ⁻¹' {x}) • x
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: case pos
α : Type u_1
E : Type u_2
inst✝ : NormedAddCommGroup E
m : MeasurableSpace α
μ : Measure α
f : α →ₛ E
g : E → ℝ≥0∞
hf : Integrable (⇑f) μ
hg0 : g 0 = 0
ht : ∀ (b : E), g b ≠ ∞
hf' : f.FinMeasSupp μ
a : E
a0 : a = 0
⊢ g 0 * μ (⇑f ⁻¹' {0}) ≠ ∞

Tactic:
hg0,

Goal state after:
case pos
α : Type u_1
E : Type u_2
inst✝ : NormedAddCommGroup E
m : MeasurableSpace α
μ : Measure α
f : α →ₛ E
g : E → ℝ≥0∞
hf : Integrable (⇑f) μ
hg0 : g 0 = 0
ht : ∀ (b : E), g b ≠ ∞
hf' : f.FinMeasSupp μ
a : E
a0 : a = 0
⊢ 0 * μ (⇑f ⁻¹' {0}) ≠ ∞
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: case pos
α : Type u_1
E : Type u_2
inst✝ : NormedAddCommGroup E
m : MeasurableSpace α
μ : Measure α
f : α →ₛ E
g : E → ℝ≥0∞
hf : Integrable (⇑f) μ
hg0 : g 0 = 0
ht : ∀ (b : E), g b ≠ ∞
hf' : f.FinMeasSupp μ
a : E
a0 : a = 0
⊢ g a * μ (⇑f ⁻¹' {a}) ≠ ∞

pp: case neg
α : Type u_1
E : Type u_2
inst✝ : NormedAddCommGroup E
m : MeasurableSpace α
μ : Measure α
f : α →ₛ E
g : E → ℝ≥0∞
hf : Integrable (⇑f) μ
hg0 : g 0 = 0
ht : ∀ (b : E), g b ≠ ∞
hf' : f.FinMeasSupp μ
a : E
a0 : ¬a = 0
⊢ g a * μ (⇑f ⁻¹' {a}) ≠ ∞

Tactic:
· rw [a0, hg0, zero_mul]; exact WithTop.zero_ne_top

Goal state after:
case neg
α : Type u_1
E : Type u_2
inst✝ : NormedAddCommGroup E
m : MeasurableSpace α
μ : Measure α
f : α →ₛ E
g : E → ℝ≥0∞
hf : Integrable (⇑f) μ
hg0 : g 0 = 0
ht : ∀ (b : E), g b ≠ ∞
hf' : f.FinMeasSupp μ
a : E
a0 : ¬a = 0
⊢ g a * μ (⇑f ⁻¹' {a}) ≠ ∞
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
f : α → G
⊢ ‖setToFun μ (weightedSMul μ) ⋯ f‖ ≤ (∫⁻ (a : α), ENNReal.ofReal ‖f a‖ ∂μ).toReal

Tactic:
exact (norm_setToFun_le_toReal _ (by simp)).trans (by simp)

Goal state after:
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : NormedAddCommGroup β
u : α → β
l : Filter α
h : u =O[l] 0
⊢ u ~[l] 0

Tactic:
rw [isEquivalent_zero_iff_eventually_zero, eventuallyEq_iff_exists_mem]

Goal state after:
α : Type u_1
β : Type u_2
inst✝ : NormedAddCommGroup β
u : α → β
l : Filter α
h : u =O[l] 0
⊢ ∃ s ∈ l, Set.EqOn u 0 s
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: case neg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfm : AEStronglyMeasurable f μ
hfi : ¬Integrable f μ
⊢ 0 = (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal

Tactic:
simp_rw [Integrable, hfm, hasFiniteIntegral_iff_norm, lt_top_iff_ne_top, Ne, true_and,
      Classical.not_not] at hfi

Goal state after:
case neg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfm : AEStronglyMeasurable f μ
hfi : ∫⁻ (a : α), ENNReal.ofReal ‖f a‖ ∂μ = ∞
⊢ 0 = (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : NormedField β
u v : α → β
l : Filter α
φ : α → β
hφ : Tendsto φ l (𝓝 1)
h : u =ᶠ[l] φ * v
| 𝓝 0

Tactic:
← sub_self (1 : β)

Goal state after:
α : Type u_1
β : Type u_2
inst✝ : NormedField β
u v : α → β
l : Filter α
φ : α → β
hφ : Tendsto φ l (𝓝 1)
h : u =ᶠ[l] φ * v
| 𝓝 (1 - 1)
Module: Mathlib.Analysis.Normed.Module.RieszLemma

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
F : Subspace 𝕜 E
hFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F
r : ℝ
hr : r < 1
x : E
hx : x ∉ F
d : ℝ := infDist x ↑F
hFn : (↑F).Nonempty
⊢ ∃ x₀ ∉ F, ∀ y ∈ F, r * ‖x₀‖ ≤ ‖x₀ - y‖

Tactic:
have hdp : 0 < d :=
    lt_of_le_of_ne Metric.infDist_nonneg fun heq =>
      hx ((hFc.mem_iff_infDist_zero hFn).2 heq.symm)

Goal state after:
𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
F : Subspace 𝕜 E
hFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F
r : ℝ
hr : r < 1
x : E
hx : x ∉ F
d : ℝ := infDist x ↑F
hFn : (↑F).Nonempty
hdp : 0 < d
⊢ ∃ x₀ ∉ F, ∀ y ∈ F, r * ‖x₀‖ ≤ ‖x₀ - y‖
Module: Mathlib.Analysis.Normed.Module.RieszLemma

Goal state before:
pp: 𝕜 : Type u_4
inst✝² : RCLike 𝕜
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
F : Subspace 𝕜 E
hFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F
hF : ∃ x, x ∉ F
r : ℝ
hr : r < 1
x₀ : E
hx₀ : x₀ ∉ F
h : ∀ y ∈ F, r * ‖x₀‖ ≤ ‖x₀ - y‖
hx₀' : x₀ ≠ 0
y : E
hy : y ∈ F
⊢ ‖(↑‖x₀‖)⁻¹‖ * ‖x₀ - ↑‖x₀‖ • y‖ = ‖x₀‖⁻¹ * ‖x₀ - ↑‖x₀‖ • y‖

Tactic:
norm_inv,

Goal state after:
𝕜 : Type u_4
inst✝² : RCLike 𝕜
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
F : Subspace 𝕜 E
hFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F
hF : ∃ x, x ∉ F
r : ℝ
hr : r < 1
x₀ : E
hx₀ : x₀ ∉ F
h : ∀ y ∈ F, r * ‖x₀‖ ≤ ‖x₀ - y‖
hx₀' : x₀ ≠ 0
y : E
hy : y ∈ F
⊢ ‖↑‖x₀‖‖⁻¹ * ‖x₀ - ↑‖x₀‖ • y‖ = ‖x₀‖⁻¹ * ‖x₀ - ↑‖x₀‖ • y‖
Module: Mathlib.Topology.Instances.Matrix

Goal state before:
pp: X : Type u_1
l : Type u_3
R : Type u_8
m' : l → Type u_9
n' : l → Type u_10
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace R
inst✝¹ : Zero R
inst✝ : DecidableEq l
A : X → (i : l) → Matrix (m' i) (n' i) R
hA : Continuous A
x✝¹ : (i : l) × m' i
x✝ : (i : l) × n' i
i₁ : l
i₂ : m' i₁
j₁ : l
j₂ : n' j₁
⊢ Continuous fun a ↦ blockDiagonal' (A a) ⟨i₁, i₂⟩ ⟨j₁, j₂⟩

Tactic:
by
    dsimp only [blockDiagonal'_apply']
    split_ifs with h
    · subst h
      exact ((continuous_apply i₁).comp hA).matrix_elem i₂ j₂
    · exact continuous_const

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case pos
α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
c : ℝ
f : α → E
hF : CompleteSpace F
hf : Integrable f μ
⊢ setToFun μ (fun s ↦ c • T s) ⋯ f = c • setToFun μ T hT f

Tactic:
simp_rw [setToFun_eq _ hf, L1.setToL1_smul_left hT c]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.SumMeasure

Goal state before:
pp: X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝⁴ : NormedAddCommGroup E
f : X → E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSingletonClass X
μ : Measure X
inst✝ : Countable X
hf : Integrable f (Measure.sum fun a ↦ μ {a} • Measure.dirac a)
⊢ ∫ (x : X), f x ∂μ = ∑' (x : X), μ.real {x} • f x

Tactic:
← Measure.sum_smul_dirac μ,

Goal state after:
X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝⁴ : NormedAddCommGroup E
f : X → E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSingletonClass X
μ : Measure X
inst✝ : Countable X
hf : Integrable f (Measure.sum fun a ↦ μ {a} • Measure.dirac a)
⊢ (∫ (x : X), f x ∂Measure.sum fun a ↦ μ {a} • Measure.dirac a) =
    ∑' (x : X), (Measure.sum fun a ↦ μ {a} • Measure.dirac a).real {x} • f x
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case pos
α : Type u_1
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
G'' : Type u_8
inst✝⁴ : NormedAddCommGroup G''
inst✝³ : PartialOrder G''
inst✝² : IsOrderedAddMonoid G''
inst✝¹ : NormedSpace ℝ G''
inst✝ : OrderClosedTopology G''
T T' : Set α → E →L[ℝ] G''
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
hTT' : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∀ (x : E), (T s) x ≤ (T' s) x
f : α → E
hG'' : CompleteSpace G''
hf : Integrable f μ
⊢ setToFun μ T hT f ≤ setToFun μ T' hT' f

Tactic:
simp_rw [setToFun_eq _ hf]; exact L1.setToL1_mono_left' hT hT' hTT' _

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : PseudoMetricSpace β
f g : α →ᵇ β
⊢ dist f g = ⨆ x, dist (f x) (g x)

Tactic:
cases isEmpty_or_nonempty α
  · rw [iSup_of_empty', Real.sSup_empty, dist_zero_of_empty]
  refine (dist_le_iff_of_nonempty.mpr <| le_ciSup ?_).antisymm (ciSup_le dist_coe_le_dist)
  exact dist_set_exists.imp fun C hC => forall_mem_range.2 hC.2

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

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x : α
⊢ (thickenedIndicator δ_pos E) x ≤ 1

Tactic:
rw [thickenedIndicator.coeFn_eq_comp]
  simpa using (toNNReal_le_toNNReal (by finiteness) one_ne_top).mpr
    (thickenedIndicatorAux_le_one δ E x)

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Marginal

Goal state before:
pp: δ : Type u_1
X : δ → Type u_3
inst✝² : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝¹ : DecidableEq δ
s t : Finset δ
inst✝ : ∀ (i : δ), SigmaFinite (μ i)
f : ((i : δ) → X i) → ℝ≥0∞
hf : Measurable f
hst : Disjoint s t
x : (i : δ) → X i
e : ((i : ↥s) → X ↑i) × ((i : ↥t) → X ↑i) ≃ᵐ ((i : ↥(s ∪ t)) → X ↑i) := MeasurableEquiv.piFinsetUnion X hst
⊢ (∫⁻ (y : (i : ↥s) → X ↑i),
      ∫⁻ (z : (j : ↥t) → X ↑j),
        f (updateFinset x (s ∪ t) (e (y, z))) ∂Measure.pi fun j ↦ μ ↑j ∂Measure.pi fun i ↦ μ ↑i) =
    ∫⁻ (y : (i : ↥s) → X ↑i),
      ∫⁻ (y_1 : (i : ↥t) → X ↑i),
        f (updateFinset (updateFinset x s y) t y_1) ∂Measure.pi fun j ↦ μ ↑j ∂Measure.pi fun i ↦ μ ↑i

Tactic:
updateFinset_updateFinset hst

Goal state after:
δ : Type u_1
X : δ → Type u_3
inst✝² : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝¹ : DecidableEq δ
s t : Finset δ
inst✝ : ∀ (i : δ), SigmaFinite (μ i)
f : ((i : δ) → X i) → ℝ≥0∞
hf : Measurable f
hst : Disjoint s t
x : (i : δ) → X i
e : ((i : ↥s) → X ↑i) × ((i : ↥t) → X ↑i) ≃ᵐ ((i : ↥(s ∪ t)) → X ↑i) := MeasurableEquiv.piFinsetUnion X hst
⊢ (∫⁻ (y : (i : ↥s) → X ↑i),
      ∫⁻ (z : (j : ↥t) → X ↑j),
        f (updateFinset x (s ∪ t) (e (y, z))) ∂Measure.pi fun j ↦ μ ↑j ∂Measure.pi fun i ↦ μ ↑i) =
    ∫⁻ (y : (i : ↥s) → X ↑i),
      ∫⁻ (y_1 : (i : ↥t) → X ↑i),
        f (updateFinset x (s ∪ t) ((piFinsetUnion X hst) (y, y_1))) ∂Measure.pi fun j ↦ μ ↑j ∂Measure.pi fun i ↦ μ ↑i
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
fs : ℕ → α → E
f : α → E
bound : α → ℝ
fs_measurable : ∀ (n : ℕ), AEStronglyMeasurable (fs n) μ
bound_integrable : Integrable bound μ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ fs n a) atTop (𝓝 (f a))
hF : CompleteSpace F
f_measurable : AEStronglyMeasurable f μ
fs_int : ∀ (n : ℕ), Integrable (fs n) μ
f_int : Integrable f μ
lintegral_norm_tendsto_zero : Tendsto (fun n ↦ (∫⁻ (a : α), ENNReal.ofReal ‖fs n a - f a‖ ∂μ).toReal) atTop (𝓝 0)
n : ℕ
⊢ (fun a ↦ ‖↑↑(Integrable.toL1 (fs n) ⋯ - Integrable.toL1 f f_int) a‖ₑ) =ᵐ[μ] fun a ↦ ENNReal.ofReal ‖fs n a - f a‖

Tactic:
rw [← Integrable.toL1_sub]

Goal state after:
α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
fs : ℕ → α → E
f : α → E
bound : α → ℝ
fs_measurable : ∀ (n : ℕ), AEStronglyMeasurable (fs n) μ
bound_integrable : Integrable bound μ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ fs n a) atTop (𝓝 (f a))
hF : CompleteSpace F
f_measurable : AEStronglyMeasurable f μ
fs_int : ∀ (n : ℕ), Integrable (fs n) μ
f_int : Integrable f μ
lintegral_norm_tendsto_zero : Tendsto (fun n ↦ (∫⁻ (a : α), ENNReal.ofReal ‖fs n a - f a‖ ∂μ).toReal) atTop (𝓝 0)
n : ℕ
⊢ (fun a ↦ ‖↑↑(Integrable.toL1 (fs n - f) ⋯) a‖ₑ) =ᵐ[μ] fun a ↦ ENNReal.ofReal ‖fs n a - f a‖
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
Y : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
μ : Measure X
mY : MeasurableSpace Y
ν : Measure Y
f : X → Y → E
s : Set X
hs : MeasurableSet s
⊢ ∫ (x : X), ∫ (y : Y), s.indicator (fun x ↦ f x y) x ∂ν ∂μ = ∫ (x : X) in s, ∫ (y : Y), f x y ∂ν ∂μ

Tactic:
← integral_indicator hs,

Goal state after:
X : Type u_1
Y : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
μ : Measure X
mY : MeasurableSpace Y
ν : Measure Y
f : X → Y → E
s : Set X
hs : MeasurableSet s
⊢ ∫ (x : X), ∫ (y : Y), s.indicator (fun x ↦ f x y) x ∂ν ∂μ = ∫ (x : X), s.indicator (fun x ↦ ∫ (y : Y), f x y ∂ν) x ∂μ
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁴ : LinearOrder R
inst✝³ : TopologicalSpace R
f : StieltjesFunction R
inst✝² : OrderTopology R
inst✝¹ : CompactIccSpace R
inst✝ : DenselyOrdered R
a b : R
hab : a < b
s : ℕ → Set R
hs : Ioc a b ⊆ ⋃ i, s i
ε : ℝ≥0
εpos : 0 < ε
h : ∑' (i : ℕ), f.length (s i) < ∞
δ : ℝ≥0 := ε / 2
δpos : 0 < ↑δ
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑δ
A : ContinuousWithinAt (fun r ↦ ↑f r - ↑f a) (Ioi a) a
⊢ ↑f a - ↑f a < ↑δ

Tactic:
sub_self,

Goal state after:
R : Type u_1
inst✝⁴ : LinearOrder R
inst✝³ : TopologicalSpace R
f : StieltjesFunction R
inst✝² : OrderTopology R
inst✝¹ : CompactIccSpace R
inst✝ : DenselyOrdered R
a b : R
hab : a < b
s : ℕ → Set R
hs : Ioc a b ⊆ ⋃ i, s i
ε : ℝ≥0
εpos : 0 < ε
h : ∑' (i : ℕ), f.length (s i) < ∞
δ : ℝ≥0 := ε / 2
δpos : 0 < ↑δ
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑δ
A : ContinuousWithinAt (fun r ↦ ↑f r - ↑f a) (Ioi a) a
⊢ 0 < ↑δ
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s t : Set X
μ : Measure X
ht : NullMeasurableSet t μ
hts : s ⊆ t
h't : ∀ᵐ (x : X) ∂μ, x ∈ t \ s → f x = 0
h : IntegrableOn f t μ
f' : X → E := AEStronglyMeasurable.mk f ⋯
x : X
hx : x ∈ t \ s → f x = 0
h'x : x ∈ t → f x = AEStronglyMeasurable.mk f ⋯ x
h''x : x ∈ t \ s
⊢ AEStronglyMeasurable.mk f ⋯ x = 0

Tactic:
← h'x h''x.1,

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s t : Set X
μ : Measure X
ht : NullMeasurableSet t μ
hts : s ⊆ t
h't : ∀ᵐ (x : X) ∂μ, x ∈ t \ s → f x = 0
h : IntegrableOn f t μ
f' : X → E := AEStronglyMeasurable.mk f ⋯
x : X
hx : x ∈ t \ s → f x = 0
h'x : x ∈ t → f x = AEStronglyMeasurable.mk f ⋯ x
h''x : x ∈ t \ s
⊢ f x = 0
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inr
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
a b : R
hab : b < a
⊢ f.measure (Icc a b) = ofReal (↑f b - leftLim (↑f) a)

Tactic:
simp only [hab, measure_empty, Icc_eq_empty, not_le]
    symm
    simp [ENNReal.ofReal_eq_zero, f.mono.le_leftLim hab]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : MeasurableSet s
hf : StronglyMeasurable f
hfi : Integrable f μ
⊢ ∫ (x : X) in s, f x ∂μ ≤ ∫ (x : X) in {y | 0 ≤ f y}, f x ∂μ

Tactic:
← integral_indicator hs,

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : MeasurableSet s
hf : StronglyMeasurable f
hfi : Integrable f μ
⊢ ∫ (x : X), s.indicator f x ∂μ ≤ ∫ (x : X) in {y | 0 ≤ f y}, f x ∂μ
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : MeasurableSet s
hf : StronglyMeasurable f
hfi : Integrable f μ
⊢ ∫ (x : X) in {y | f y ≤ 0}, f x ∂μ ≤ ∫ (x : X) in s, f x ∂μ

Tactic:
← integral_indicator hs,

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : MeasurableSet s
hf : StronglyMeasurable f
hfi : Integrable f μ
⊢ ∫ (x : X) in {y | f y ≤ 0}, f x ∂μ ≤ ∫ (x : X), s.indicator f x ∂μ
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
f_int : Integrable f μ
f_nonneg : 0 ≤ᵐ[μ] f
s : Set X
hs : ∀ x ∈ s, 1 ≤ f x
⊢ μ s ≤ ENNReal.ofReal (∫ (x : X), f x ∂μ)

Tactic:
rw [ofReal_integral_eq_lintegral_ofReal f_int f_nonneg]

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
f_int : Integrable f μ
f_nonneg : 0 ≤ᵐ[μ] f
s : Set X
hs : ∀ x ∈ s, 1 ≤ f x
⊢ μ s ≤ ∫⁻ (x : X), ENNReal.ofReal (f x) ∂μ
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
C : ℝ
h : ∀ (x : R), |↑f x| ≤ C
h✝ : Nonempty R
u : ℝ
hu : Tendsto (↑f) atTop (𝓝 u)
⊢ ∃ l, Tendsto (↑f) atBot (𝓝 l)

Tactic:
rcases tendsto_atBot_of_monotone f.mono with H | H

Goal state after:
case inl
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
C : ℝ
h : ∀ (x : R), |↑f x| ≤ C
h✝ : Nonempty R
u : ℝ
hu : Tendsto (↑f) atTop (𝓝 u)
H : Tendsto (↑f) atBot atBot
⊢ ∃ l, Tendsto (↑f) atBot (𝓝 l)

case inr
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
C : ℝ
h : ∀ (x : R), |↑f x| ≤ C
h✝ : Nonempty R
u : ℝ
hu : Tendsto (↑f) atTop (𝓝 u)
H : ∃ l, Tendsto (↑f) atBot (𝓝 l)
⊢ ∃ l, Tendsto (↑f) atBot (𝓝 l)
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
g : StieltjesFunction R
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : R
hf : ↑f x - l = ↑g x - l
⊢ ↑f x = ↑g x

pp: case hp
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
g : StieltjesFunction R
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : R
hf : ofReal (↑g x - l) = ofReal (↑f x - l)
⊢ 0 ≤ ↑g x - l

pp: case hq
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
g : StieltjesFunction R
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : R
hf : ofReal (↑g x - l) = ofReal (↑f x - l)
⊢ 0 ≤ ↑f x - l

Tactic:
· simpa using hf

Goal state after:
case hp
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
g : StieltjesFunction R
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : R
hf : ofReal (↑g x - l) = ofReal (↑f x - l)
⊢ 0 ≤ ↑g x - l

case hq
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
g : StieltjesFunction R
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : R
hf : ofReal (↑g x - l) = ofReal (↑f x - l)
⊢ 0 ≤ ↑f x - l