mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable 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
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
f' : ℕ → α → ℝ≥0∞ := fun n a ↦ ENNReal.ofReal (f n a - f 0 a)
F' : α → ℝ≥0∞ := fun a ↦ ENNReal.ofReal (F a - f 0 a)
hf'_int_eq : ∀ (i : ℕ), ∫⁻ (a : α), f' i a ∂μ = ENNReal.ofReal (∫ (a : α), f i a ∂μ - ∫ (a : α), f 0 a ∂μ)
hF'_int_eq : ∫⁻ (a : α), F' a ∂μ = ENNReal.ofReal (∫ (a : α), F a ∂μ - ∫ (a : α), f 0 a ∂μ)
h_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f' i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F' a ∂μ))
h_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 : ∀ᵐ (a : α) ∂μ, Tendsto (fun i ↦ f' i a) atTop (𝓝 (F' a))
a : α
ha : Tendsto (fun i ↦ f' i a) atTop (𝓝 (F' a))
ha_mono : Monotone fun i ↦ f i a
ha_bound : ∀ (i : ℕ), f i a ≤ F a
h1 : (fun i ↦ f i a) = fun i ↦ (f' i a).toReal + f 0 a
h2 : F a = (F' a).toReal + f 0 a
⊢ Tendsto (fun i ↦ f i a) atTop (𝓝 (F a))

Tactic:
h1,

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable 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
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
f' : ℕ → α → ℝ≥0∞ := fun n a ↦ ENNReal.ofReal (f n a - f 0 a)
F' : α → ℝ≥0∞ := fun a ↦ ENNReal.ofReal (F a - f 0 a)
hf'_int_eq : ∀ (i : ℕ), ∫⁻ (a : α), f' i a ∂μ = ENNReal.ofReal (∫ (a : α), f i a ∂μ - ∫ (a : α), f 0 a ∂μ)
hF'_int_eq : ∫⁻ (a : α), F' a ∂μ = ENNReal.ofReal (∫ (a : α), F a ∂μ - ∫ (a : α), f 0 a ∂μ)
h_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f' i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F' a ∂μ))
h_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 : ∀ᵐ (a : α) ∂μ, Tendsto (fun i ↦ f' i a) atTop (𝓝 (F' a))
a : α
ha : Tendsto (fun i ↦ f' i a) atTop (𝓝 (F' a))
ha_mono : Monotone fun i ↦ f i a
ha_bound : ∀ (i : ℕ), f i a ≤ F a
h1 : (fun i ↦ f i a) = fun i ↦ (f' i a).toReal + f 0 a
h2 : F a = (F' a).toReal + f 0 a
⊢ Tendsto (fun i ↦ (f' i a).toReal + f 0 a) atTop (𝓝 (F a))
Module: Mathlib.Topology.Algebra.Indicator

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → β
s : Set α
inst✝ : One β
hs : ∀ a ∈ frontier s, f a = 1
hf : ContinuousOn f (closure[inst✝²] s)
⊢ Continuous[inst✝², inst✝¹] (s.mulIndicator f)

Tactic:
by
  classical exact continuous_piecewise hs hf continuousOn_const

Goal state after:
Module: Mathlib.Topology.Bornology.BoundedOperation

Goal state before:
pp: R : Type u_2
inst✝² : Bornology R
inst✝¹ : Monoid R
inst✝ : BoundedMul R
s : Set R
s_bdd : Bornology.IsBounded s
n : ℕ
hn : Bornology.IsBounded ((fun x ↦ x ^ n) '' s)
x y : R
y_in_s : y ∈ s
ypow_eq_x : y ^ (n + 1) = x
⊢ x ∈ (fun x ↦ x ^ n) '' s * s

Tactic:
← ypow_eq_x,

Goal state after:
R : Type u_2
inst✝² : Bornology R
inst✝¹ : Monoid R
inst✝ : BoundedMul R
s : Set R
s_bdd : Bornology.IsBounded s
n : ℕ
hn : Bornology.IsBounded ((fun x ↦ x ^ n) '' s)
x y : R
y_in_s : y ∈ s
ypow_eq_x : y ^ (n + 1) = x
⊢ y ^ (n + 1) ∈ (fun x ↦ x ^ n) '' s * s
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: case pos
𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
E : Type v
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace 𝕜
s : Finset E
b : Basis (↥s) 𝕜 E
this : FiniteDimensional 𝕜 E
⊢ Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun f ↦ ((LinearMap.toMatrix b b) ↑f).det

Tactic:
refine Continuous.matrix_det ?_

Goal state after:
case pos
𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
E : Type v
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace 𝕜
s : Finset E
b : Basis (↥s) 𝕜 E
this : FiniteDimensional 𝕜 E
⊢ Continuous[_, instTopologicalSpaceMatrix] fun f ↦ (LinearMap.toMatrix b b) ↑f
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: case mp
𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type w
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace 𝕜
inst✝ : FiniteDimensional 𝕜 E
f : E →ₗ[𝕜] F
⊢ f.ker = ⊥ → ∃ K > 0, AntilipschitzWith K ⇑f

Tactic:
exact f.exists_antilipschitzWith

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

Goal state before:
pp: case pos
α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝⁷ : NormedAddCommGroup G''
inst✝⁶ : PartialOrder G''
inst✝⁵ : IsOrderedAddMonoid G''
inst✝⁴ : NormedSpace ℝ G''
inst✝³ : NormedAddCommGroup G'
inst✝² : PartialOrder G'
inst✝¹ : NormedSpace ℝ G'
inst✝ : ClosedIciTopology G''
T : Set α → G' →L[ℝ] G''
C : ℝ
hT : DominatedFinMeasAdditive μ T C
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f : α → G'
hf : 0 ≤ᵐ[μ] f
hG'' : CompleteSpace G''
hfi : Integrable f μ
⊢ 0 ≤ (L1.setToL1 hT) (Integrable.toL1 f hfi)

Tactic:
exact L1.setToL1_nonneg hT hT_nonneg hf

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

Goal state before:
pp: F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetricSpace β
inst✝¹ : PseudoMetricSpace γ
f✝ g : α →ᵇ β
x : α
C : ℝ
inst✝ : CompleteSpace β
f : ℕ → α →ᵇ β
hf : CauchySeq f
⊢ ∃ a, Tendsto f atTop (𝓝 a)

Tactic:
rcases cauchySeq_iff_le_tendsto_0.1 hf with ⟨b, b0, b_bound, b_lim⟩
    have f_bdd := fun x n m N hn hm => le_trans (dist_coe_le_dist x) (b_bound n m N hn hm)
    have fx_cau : ∀ x, CauchySeq fun n => f n x :=
      fun x => cauchySeq_iff_le_tendsto_0.2 ⟨b, b0, f_bdd x, b_lim⟩
    choose F hF using fun x => cauchySeq_tendsto_of_complete (fx_cau x)
    /- `F : α → β`, `hF : ∀ (x : α), Tendsto (fun n ↦ ↑(f n) x) atTop (𝓝 (F x))`
      `F` is the desired limit function. Check that it is uniformly approximated by `f N`. -/
    have fF_bdd : ∀ x N, dist (f N x) (F x) ≤ b N :=
      fun x N => le_of_tendsto (tendsto_const_nhds.dist (hF x))
        (Filter.eventually_atTop.2 ⟨N, fun n hn => f_bdd x N n N (le_refl N) hn⟩)
    refine ⟨⟨⟨F, ?_⟩, ?_⟩, ?_⟩
    · -- Check that `F` is continuous, as a uniform limit of continuous functions
      have : TendstoUniformly (fun n x => f n x) F atTop := by
        refine Metric.tendstoUniformly_iff.2 fun ε ε0 => ?_
        refine ((tendsto_order.1 b_lim).2 ε ε0).mono fun n hn x => ?_
        rw [dist_comm]
        exact lt_of_le_of_lt (fF_bdd x n) hn
      exact this.continuous (Frequently.of_forall fun N => (f N).continuous)
    · -- Check that `F` is bounded
      rcases (f 0).bounded with ⟨C, hC⟩
      refine ⟨C + (b 0 + b 0), fun x y => ?_⟩
      calc
        dist (F x) (F y) ≤ dist (f 0 x) (f 0 y) + (dist (f 0 x) (F x) + dist (f 0 y) (F y)) :=
          dist_triangle4_left _ _ _ _
        _ ≤ C + (b 0 + b 0) := add_le_add (hC x y) (add_le_add (fF_bdd x 0) (fF_bdd y 0))
    · -- Check that `F` is close to `f N` in distance terms
      refine tendsto_iff_dist_tendsto_zero.2 (squeeze_zero (fun _ => dist_nonneg) ?_ b_lim)
      exact fun N => (dist_le (b0 _)).2 fun x => fF_bdd x N

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 C' : ℝ
hT : DominatedFinMeasAdditive μ T C
β : Type u_7
x✝ : MeasurableSpace β
μ' : Measure β
φ : α → β
T' : Set β → E →L[ℝ] F
hT' : DominatedFinMeasAdditive μ' T' C'
f : β → E
hf : Integrable (f ∘ φ) μ
hfm : StronglyMeasurable f
hφ : Measurable φ
hμ' : μ' ≤ Measure.map φ μ
h : ∀ (s : Set β) (x : E), MeasurableSet s → (T' s) x = (T (φ ⁻¹' s)) x
hF : CompleteSpace F
hfi' : Integrable f μ'
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : SeparableSpace ↑(range f ∪ {0})
⊢ Tendsto (fun n ↦ setToFun μ' T' hT' ⇑(SimpleFunc.approxOn f ⋯ (range f ∪ {0}) 0 ⋯ n)) atTop
    (𝓝 (setToFun μ T hT (f ∘ φ)))

Tactic:
convert tendsto_setToFun_approxOn_of_measurable_of_range_subset
    hT (hfm.measurable.comp hφ) hf (range f ∪ {0})
    (union_subset_union_left {0} (range_comp_subset_range φ f)) using 1

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 C' : ℝ
hT : DominatedFinMeasAdditive μ T C
β : Type u_7
x✝ : MeasurableSpace β
μ' : Measure β
φ : α → β
T' : Set β → E →L[ℝ] F
hT' : DominatedFinMeasAdditive μ' T' C'
f : β → E
hf : Integrable (f ∘ φ) μ
hfm : StronglyMeasurable f
hφ : Measurable φ
hμ' : μ' ≤ Measure.map φ μ
h : ∀ (s : Set β) (x : E), MeasurableSet s → (T' s) x = (T (φ ⁻¹' s)) x
hF : CompleteSpace F
hfi' : Integrable f μ'
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : SeparableSpace ↑(range f ∪ {0})
⊢ (fun n ↦ setToFun μ' T' hT' ⇑(SimpleFunc.approxOn f ⋯ (range f ∪ {0}) 0 ⋯ n)) = fun n ↦
    setToFun μ T hT ⇑(SimpleFunc.approxOn (f ∘ φ) ⋯ (range f ∪ {0}) 0 ⋯ n)
Module: Mathlib.Topology.ContinuousMap.Bounded.Normed

Goal state before:
pp: α : Type u
β : Type v
γ : Type w
𝕜 : Type u_1
inst✝⁵ : TopologicalSpace α
inst✝⁴ : SeminormedAddCommGroup β
f✝ g✝¹ : α →ᵇ β
x : α
C : ℝ
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 β
inst✝¹ : SeminormedAddCommGroup γ
inst✝ : NormedSpace 𝕜 γ
g✝ : β →L[𝕜] γ
f g : α →ᵇ β
⊢ ofNormedAddCommGroup (⇑g✝ ∘ ⇑(f + g)) ⋯ (‖g✝‖ * ‖f + g‖) ⋯ =
    ofNormedAddCommGroup (⇑g✝ ∘ ⇑f) ⋯ (‖g✝‖ * ‖f‖) ⋯ + ofNormedAddCommGroup (⇑g✝ ∘ ⇑g) ⋯ (‖g✝‖ * ‖g‖) ⋯

Tactic:
by ext; simp

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

Goal state before:
pp: α : Type u
β : Type v
γ : Type w
𝕜 : Type u_1
inst✝⁵ : TopologicalSpace α
inst✝⁴ : SeminormedAddCommGroup β
f✝ g✝ : α →ᵇ β
x : α
C : ℝ
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 β
inst✝¹ : SeminormedAddCommGroup γ
inst✝ : NormedSpace 𝕜 γ
g : β →L[𝕜] γ
c : 𝕜
f : α →ᵇ β
⊢ ofNormedAddCommGroup (⇑g ∘ ⇑(c • f)) ⋯ (‖g‖ * ‖c • f‖) ⋯ =
    (RingHom.id 𝕜) c • ofNormedAddCommGroup (⇑g ∘ ⇑f) ⋯ (‖g‖ * ‖f‖) ⋯

Tactic:
by ext; simp

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : p.HolderConjugate q
f g : α → ℝ
hf_nonneg : 0 ≤ᵐ[μ] f
hg_nonneg : 0 ≤ᵐ[μ] g
hf : MemLp f (ENNReal.ofReal p) μ
hg : MemLp g (ENNReal.ofReal q) μ
h_left : ∫ (a : α), f a * g a ∂μ = ∫ (a : α), ‖f a‖ * ‖g a‖ ∂μ
h_right_f : ∫ (a : α), f a ^ p ∂μ = ∫ (a : α), ‖f a‖ ^ p ∂μ
⊢ (fun a ↦ g a ^ q) =ᵐ[μ] fun a ↦ ‖g a‖ ^ q

Tactic:
filter_upwards [hg_nonneg] with x hxg

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : p.HolderConjugate q
f g : α → ℝ
hf_nonneg : 0 ≤ᵐ[μ] f
hg_nonneg : 0 ≤ᵐ[μ] g
hf : MemLp f (ENNReal.ofReal p) μ
hg : MemLp g (ENNReal.ofReal q) μ
h_left : ∫ (a : α), f a * g a ∂μ = ∫ (a : α), ‖f a‖ * ‖g a‖ ∂μ
h_right_f : ∫ (a : α), f a ^ p ∂μ = ∫ (a : α), ‖f a‖ ^ p ∂μ
x : α
hxg : 0 x ≤ g x
⊢ g x ^ q = ‖g x‖ ^ q
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
hc'0 : c' = 0
⊢ μ' = 0

Tactic:
rw [← Measure.nonpos_iff_eq_zero']

Goal state after:
α : Type u_1
G : Type u_5
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
hc'0 : c' = 0
⊢ μ' ≤ 0
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
r : ℝ≥0
hfint : Integrable f μ
hfint' : 0 ≤ ∫ (x : α), f x ∂μ
hr : r = 0
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ 0
hnegf : ∫ (x : α), -f x ∂μ = 0
ω : α
hω : f ω ≤ 0
⊢ 0 ω ≤ (-f) ω

Tactic:
rwa [Pi.zero_apply, Pi.neg_apply, Right.nonneg_neg_iff]

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

Goal state before:
pp: case pos.h_closed
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 C' : ℝ
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ' T C'
f : α → E
hfμ : Integrable f μ
hF : CompleteSpace F
h_int : ∀ (g : α → E), Integrable g μ → Integrable g μ'
this : (fun f ↦ setToFun μ' T hT' ↑↑f) = fun f ↦ setToFun μ' T hT' ↑↑(Integrable.toL1 ↑↑f ⋯)
⊢ Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun f ↦ setToFun μ' T hT' ↑↑(Integrable.toL1 ↑↑f ⋯)

Tactic:
exact (continuous_setToFun hT').comp (continuous_L1_toL1 c' hc' hμ'_le)

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

Goal state before:
pp: δ : Type u_1
X : δ → Type u_3
inst✝¹ : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝ : DecidableEq δ
f : ((i : δ) → X i) → ℝ≥0∞
i : δ
⊢ ∫⋯∫⁻_{i}, f ∂μ = fun x ↦ ∫⁻ (xᵢ : X i), f (update x i xᵢ) ∂μ i

Tactic:
by
  let α : Type _ := ({i} : Finset δ)
  let e := (MeasurableEquiv.piUnique fun j : α ↦ X j).symm
  ext1 x
  calc (∫⋯∫⁻_{i}, f ∂μ) x
      = ∫⁻ (y : X (default : α)), f (updateFinset x {i} (e y)) ∂μ (default : α) := by
        simp_rw [lmarginal,
          measurePreserving_piUnique (fun j : ({i} : Finset δ) ↦ μ j) |>.symm _
            |>.lintegral_map_equiv, e, α]
    _ = ∫⁻ xᵢ, f (Function.update x i xᵢ) ∂μ i := by simp [update_eq_updateFinset]; rfl

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

Goal state before:
pp: case e'_3
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
this :
  Tendsto (fun n ↦ (L1.setToL1 hT) (Integrable.toL1 (fs n) ⋯)) atTop (𝓝 ((L1.setToL1 hT) (Integrable.toL1 f f_int)))
n : ℕ
⊢ setToFun μ T hT (fs n) = (L1.setToL1 hT) (Integrable.toL1 (fs n) ⋯)

pp: case e'_5
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
this :
  Tendsto (fun n ↦ (L1.setToL1 hT) (Integrable.toL1 (fs n) ⋯)) atTop (𝓝 ((L1.setToL1 hT) (Integrable.toL1 f f_int)))
⊢ setToFun μ T hT f = (L1.setToL1 hT) (Integrable.toL1 f f_int)

Tactic:
· exact setToFun_eq hT (fs_int n)

Goal state after:
case e'_5
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
this :
  Tendsto (fun n ↦ (L1.setToL1 hT) (Integrable.toL1 (fs n) ⋯)) atTop (𝓝 ((L1.setToL1 hT) (Integrable.toL1 f f_int)))
⊢ setToFun μ T hT f = (L1.setToL1 hT) (Integrable.toL1 f f_int)
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
μ : Measure X
ι : Type u_5
t : Finset ι
s : ι → Set X
hs : ∀ i ∈ t, MeasurableSet (s i)
hf : ∀ i ∈ t, IntegrableOn f (s i) μ
A : ∀ u ∈ {x ∈ t.powerset | x.Nonempty}, MeasurableSet (⋂ i ∈ u, s i)
x : Finset ι
hx : x ∈ {x ∈ t.powerset | x.Nonempty}
⊢ ∫ (a : X) in ⋂ i ∈ x, s i, (-1) ^ (#x + 1) • f a ∂μ =
    ∫ (a : X), (⋂ i ∈ x, s i).indicator (fun a ↦ (-1) ^ (#x + 1) • f a) a ∂μ

Tactic:
← integral_indicator (A x hx)

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
μ : Measure X
ι : Type u_5
t : Finset ι
s : ι → Set X
hs : ∀ i ∈ t, MeasurableSet (s i)
hf : ∀ i ∈ t, IntegrableOn f (s i) μ
A : ∀ u ∈ {x ∈ t.powerset | x.Nonempty}, MeasurableSet (⋂ i ∈ u, s i)
x : Finset ι
hx : x ∈ {x ∈ t.powerset | x.Nonempty}
⊢ ∫ (x_1 : X), (⋂ i ∈ x, s i).indicator (fun a ↦ (-1) ^ (#x + 1) • f a) x_1 ∂μ =
    ∫ (a : X), (⋂ i ∈ x, s i).indicator (fun a ↦ (-1) ^ (#x + 1) • f a) a ∂μ
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
inst✝¹ : CompleteSpace E
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
inst✝ : Countable ι
f : ι → α → E
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hf' : ∑' (i : ι), ∫⁻ (a : α), ‖f i a‖ₑ ∂μ ≠ ∞
hF : CompleteSpace F
hf'' : ∀ (i : ι), AEMeasurable (fun x ↦ ‖f i x‖ₑ) μ
hhh : ∀ᵐ (a : α) ∂μ, Summable fun n ↦ ↑‖f n a‖₊
this : ∫⁻ (a : α), ∑' (n : ι), ‖f n a‖ₑ ∂μ < ∞
⊢ (fun a ↦ ‖∑' (n : ι), ‖f n a‖‖ₑ) =ᵐ[μ] fun a ↦ ∑' (n : ι), ‖f n a‖ₑ

Tactic:
← coe_nnnorm,

Goal state after:
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
inst✝¹ : CompleteSpace E
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
inst✝ : Countable ι
f : ι → α → E
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hf' : ∑' (i : ι), ∫⁻ (a : α), ‖f i a‖ₑ ∂μ ≠ ∞
hF : CompleteSpace F
hf'' : ∀ (i : ι), AEMeasurable (fun x ↦ ‖f i x‖ₑ) μ
hhh : ∀ᵐ (a : α) ∂μ, Summable fun n ↦ ↑‖f n a‖₊
this : ∫⁻ (a : α), ∑' (n : ι), ‖f n a‖ₑ ∂μ < ∞
⊢ (fun a ↦ ‖∑' (n : ι), ↑‖f n a‖₊‖ₑ) =ᵐ[μ] fun a ↦ ∑' (n : ι), ‖f n a‖ₑ
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set X
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
h_mono : Monotone s
hne : atTop.NeBot
this✝ : IsDirectedOrder ι
S : Set X := ⋃ i, s i
hfi : IntegrableOn f S μ
hfi' : (μ.withDensity fun x ↦ ↑‖f x‖₊) S < ∞
hSm : NullMeasurableSet S μ
hsub : ∀ {i : ι}, s i ⊆ S
ν : Measure X := μ.withDensity fun x ↦ ‖f x‖ₑ
hν : ν = μ.withDensity fun x ↦ ‖f x‖ₑ
ε : ℝ≥0
ε0 : 0 < ↑ε
this : ∀ᶠ (i : ι) in atTop, ν (s i) ∈ Icc (ν S - ↑ε) (ν S + ↑ε)
i : ι
hi : ν (s i) ∈ Icc (ν S - ↑ε) (ν S + ↑ε)
⊢ ‖∫ (x : X) in S \ s i, f x ∂μ‖ ≤ ↑ε

Tactic:
← coe_nnnorm,

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set X
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
h_mono : Monotone s
hne : atTop.NeBot
this✝ : IsDirectedOrder ι
S : Set X := ⋃ i, s i
hfi : IntegrableOn f S μ
hfi' : (μ.withDensity fun x ↦ ↑‖f x‖₊) S < ∞
hSm : NullMeasurableSet S μ
hsub : ∀ {i : ι}, s i ⊆ S
ν : Measure X := μ.withDensity fun x ↦ ‖f x‖ₑ
hν : ν = μ.withDensity fun x ↦ ‖f x‖ₑ
ε : ℝ≥0
ε0 : 0 < ↑ε
this : ∀ᶠ (i : ι) in atTop, ν (s i) ∈ Icc (ν S - ↑ε) (ν S + ↑ε)
i : ι
hi : ν (s i) ∈ Icc (ν S - ↑ε) (ν S + ↑ε)
⊢ ↑‖∫ (x : X) in S \ s i, f x ∂μ‖₊ ≤ ↑ε
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
X : Type u_7
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
hT : DominatedFinMeasAdditive μ T C
fs : X → α → E
bound : α → ℝ
s : Set X
hfs_meas : ∀ x ∈ s, AEStronglyMeasurable (fs x) μ
h_bound : ∀ x ∈ s, ∀ᵐ (a : α) ∂μ, ‖fs x a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_cont : ∀ᵐ (a : α) ∂μ, ContinuousOn (fun x ↦ fs x a) s
⊢ ContinuousOn (fun x ↦ setToFun μ T hT (fs x)) s

Tactic:
by
  intro x hx
  refine continuousWithinAt_setToFun_of_dominated hT ?_ ?_ bound_integrable ?_
  · filter_upwards [self_mem_nhdsWithin] with x hx using hfs_meas x hx
  · filter_upwards [self_mem_nhdsWithin] with x hx using h_bound x hx
  · filter_upwards [h_cont] with a ha using ha x hx

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

Goal state before:
pp: case e'_3
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ ∑ i ∈ t, μ (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n ↦ { carrier := K' n, isCompact' := ⋯ }
⊢ K = t.sup L

pp: case convert_2
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ ∑ i ∈ t, μ (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n ↦ { carrier := K' n, isCompact' := ⋯ }
⊢ ∑ i ∈ t, μ (L i) ≤ ∑' (i : ℕ), μ.innerContent (U i)

Tactic:
· ext1
    rw [Compacts.coe_finset_sup, Finset.sup_eq_iSup]
    exact h3K'

Goal state after:
case convert_2
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ ∑ i ∈ t, μ (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n ↦ { carrier := K' n, isCompact' := ⋯ }
⊢ ∑ i ∈ t, μ (L i) ≤ ∑' (i : ℕ), μ.innerContent (U i)
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case refine_1
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 : R
ha : ¬IsBot a
b : R
hb : b < a
u : ℕ → R
u_mono : StrictMono u
u_lim : Tendsto u atTop (𝓝 a)
u_lt_a : ∀ (n : ℕ), u n < a
A : {a} = ⋂ n, Ioc (u n) a
m n : ℕ
hmn : m ≤ n
⊢ Ioc (u n) a ⊆ Ioc (u m) a

Tactic:
exact Ioc_subset_Ioc_left (u_mono.monotone hmn)

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

Goal state before:
pp: 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
a b : R
hab : a ≤ b
⊢ f.measure (Icc a b) = ofReal (↑f b - leftLim (↑f) a)

Tactic:
have A : Disjoint {a} (Ioc a b) := by simp
    simp [← Icc_union_Ioc_eq_Icc le_rfl hab, -singleton_union, ← ENNReal.ofReal_add,
      f.mono.leftLim_le, measure_union A measurableSet_Ioc, f.mono hab]

Goal state after: