mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: case refine_1
α : Type u_1
inst✝³ : PseudoMetricSpace α
E' : Type u_2
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : FiniteDimensional ℝ E'
s : Set α
f : α → E'
K : ℝ≥0
hf : LipschitzOnWith K f s
ι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')
A : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ := (Basis.ofVectorSpace ℝ E').equivFun.toContinuousLinearEquiv
LA : LipschitzWith ‖↑A‖₊ ⇑A
L : LipschitzOnWith (‖↑A‖₊ * K) (⇑A ∘ f) s
g : α → ι → ℝ
hg : LipschitzWith (‖↑A‖₊ * K) g
gs : EqOn (⇑A ∘ f) g s
⊢ LipschitzWith (lipschitzExtensionConstant E' * K) (⇑A.symm ∘ g)

pp: case refine_2
α : Type u_1
inst✝³ : PseudoMetricSpace α
E' : Type u_2
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : FiniteDimensional ℝ E'
s : Set α
f : α → E'
K : ℝ≥0
hf : LipschitzOnWith K f s
ι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')
A : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ := (Basis.ofVectorSpace ℝ E').equivFun.toContinuousLinearEquiv
LA : LipschitzWith ‖↑A‖₊ ⇑A
L : LipschitzOnWith (‖↑A‖₊ * K) (⇑A ∘ f) s
g : α → ι → ℝ
hg : LipschitzWith (‖↑A‖₊ * K) g
gs : EqOn (⇑A ∘ f) g s
⊢ EqOn f (⇑A.symm ∘ g) s

Tactic:
· have LAsymm : LipschitzWith ‖A.symm.toContinuousLinearMap‖₊ A.symm := by
      apply A.symm.lipschitz
    apply (LAsymm.comp hg).weaken
    rw [lipschitzExtensionConstant, ← mul_assoc]
    exact mul_le_mul' (le_max_left _ _) le_rfl

Goal state after:
case refine_2
α : Type u_1
inst✝³ : PseudoMetricSpace α
E' : Type u_2
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : FiniteDimensional ℝ E'
s : Set α
f : α → E'
K : ℝ≥0
hf : LipschitzOnWith K f s
ι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')
A : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ := (Basis.ofVectorSpace ℝ E').equivFun.toContinuousLinearEquiv
LA : LipschitzWith ‖↑A‖₊ ⇑A
L : LipschitzOnWith (‖↑A‖₊ * K) (⇑A ∘ f) s
g : α → ι → ℝ
hg : LipschitzWith (‖↑A‖₊ * K) g
gs : EqOn (⇑A ∘ f) g s
⊢ EqOn f (⇑A.symm ∘ g) s
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetricSpace β
δ : Type u_2
inst✝¹ : TopologicalSpace δ
inst✝ : DiscreteTopology δ
f : α ↪ δ
g₁ g₂ : α →ᵇ β
h₁ h₂ : δ →ᵇ β
x : ↑(range ⇑f)ᶜ
⊢ dist (h₁ ↑x) (h₂ ↑x) = dist ((extend f g₁ h₁) ↑x) ((extend f g₂ h₂) ↑x)

Tactic:
by
        rw [extend_apply' x.coe_prop, extend_apply' x.coe_prop]

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

Goal state before:
pp: case inl
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
ι : Type u_7
f : α → E
hf : AEStronglyMeasurable f μ
fs : ι → α → E
hfsi : ∀ᶠ (i : ι) in ⊥, Integrable (fs i) μ
hfs : Tendsto (fun i ↦ ∫⁻ (x : α), ‖fs i x - f x‖ₑ ∂μ) ⊥ (𝓝 0)
⊢ Tendsto (fun i ↦ setToFun μ T hT (fs i)) ⊥ (𝓝 (setToFun μ T hT f))

pp: case inr
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
ι : Type u_7
f : α → E
hf : AEStronglyMeasurable f μ
fs : ι → α → E
l : Filter ι
hfsi : ∀ᶠ (i : ι) in l, Integrable (fs i) μ
hfs : Tendsto (fun i ↦ ∫⁻ (x : α), ‖fs i x - f x‖ₑ ∂μ) l (𝓝 0)
hl : l.NeBot
⊢ Tendsto (fun i ↦ setToFun μ T hT (fs i)) l (𝓝 (setToFun μ T hT f))

Tactic:
· simp

Goal state after:
case inr
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
ι : Type u_7
f : α → E
hf : AEStronglyMeasurable f μ
fs : ι → α → E
l : Filter ι
hfsi : ∀ᶠ (i : ι) in l, Integrable (fs i) μ
hfs : Tendsto (fun i ↦ ∫⁻ (x : α), ‖fs i x - f x‖ₑ ∂μ) l (𝓝 0)
hl : l.NeBot
⊢ Tendsto (fun i ↦ setToFun μ T hT (fs i)) l (𝓝 (setToFun μ T hT f))
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: case hp_ne_zero
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_6
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : p.HolderConjugate q
hf : MemLp f (ENNReal.ofReal p) μ
hg : MemLp g (ENNReal.ofReal q) μ
h_left : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ * ‖g a‖) ∂μ = ∫⁻ (a : α), ((fun x ↦ ‖f x‖ₑ) * fun x ↦ ‖g x‖ₑ) a ∂μ
h_right_f : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ ^ p) ∂μ = ∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ
h_right_g : ∫⁻ (a : α), ENNReal.ofReal (‖g a‖ ^ q) ∂μ = ∫⁻ (a : α), ‖g a‖ₑ ^ q ∂μ
⊢ 0 < p

Tactic:
exact hpq.pos

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

Goal state before:
pp: case hf
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})
i : ℕ
⊢ Integrable (⇑(SimpleFunc.approxOn (f ∘ φ) ⋯ (range f ∪ {0}) 0 ⋯ i)) μ

Tactic:
apply SimpleFunc.integrable_approxOn _ hf (by simp) (by simp)

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_6
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : p.HolderConjugate q
hf : MemLp f (ENNReal.ofReal p) μ
hg : MemLp g (ENNReal.ofReal q) μ
h_left : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ * ‖g a‖) ∂μ = ∫⁻ (a : α), ((fun x ↦ ‖f x‖ₑ) * fun x ↦ ‖g x‖ₑ) a ∂μ
h_right_f : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ ^ p) ∂μ = ∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ
h_right_g : ∫⁻ (a : α), ENNReal.ofReal (‖g a‖ ^ q) ∂μ = ∫⁻ (a : α), ‖g a‖ₑ ^ q ∂μ
⊢ (∫⁻ (a : α), ‖g a‖ₑ ^ q ∂μ) ^ (1 / q) =
    (∫⁻ (x : α), ‖g x‖ₑ ^ (ENNReal.ofReal q).toReal ∂μ) ^ (1 / (ENNReal.ofReal q).toReal)

Tactic:
rw [ENNReal.toReal_ofReal hpq.symm.nonneg]

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

Goal state before:
pp: α : Type u
inst✝² : TopologicalSpace α
R : Type u_1
inst✝¹ : NonUnitalSeminormedRing R
inst✝ : IsCancelMulZero R
ι : Type u_2
f : ι → α →ᵇ R
h : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)
j : ι
s : Finset ι
hj : j ∉ s
ih : ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊
i : ι
hi : i ∈ s
⊢ j ≠ i

Tactic:
by grind

Goal state after:
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 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
⊢ setToFun μ' T' hT' f = setToFun μ T hT (f ∘ φ)

Tactic:
by
  by_cases hF : CompleteSpace F; swap
  · simp [setToFun, hF]
  have hfi' : Integrable f μ' :=
    ((integrable_map_measure hfm.aestronglyMeasurable hφ.aemeasurable).2 hf).mono_measure hμ'
  borelize E
  have : SeparableSpace (range f ∪ {0} : Set E) := hfm.separableSpace_range_union_singleton
  refine tendsto_nhds_unique
    (tendsto_setToFun_approxOn_of_measurable_of_range_subset
      hT' hfm.measurable hfi' _ Subset.rfl) ?_
  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
  ext i : 1
  rw [setToFun_simpleFunc _ _ (SimpleFunc.integrable_approxOn_range _ hfi' _),
    setToFun_simpleFunc, SimpleFunc.approxOn_comp hfm.measurable hφ]; swap
  · apply SimpleFunc.integrable_approxOn _ hf (by simp) (by simp)
  simp only [union_singleton, SimpleFunc.measurableSet_preimage, h, ← preimage_comp,
    SimpleFunc.coe_comp]
  refine (Finset.sum_subset (SimpleFunc.range_comp_subset_range _ hφ) fun y _ hy => ?_).symm
  rw [SimpleFunc.mem_range, ← Set.preimage_singleton_eq_empty, SimpleFunc.coe_comp] at hy
  simp [hy, hT.1.map_empty_eq_zero]

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
R : Type u_4
inst✝¹ : NonUnitalSeminormedRing R
inst✝ : IsCancelMulZero R
ι : Type u_5
f : ι → C(α, R)
h : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)
j : ι
s : Finset ι
hj : j ∉ s
ih : ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊
i : ι
hi : i ∈ s
⊢ j ≠ i

Tactic:
by grind

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : SeminormedAddCommGroup β
inst✝² : StarAddMonoid β
inst✝¹ : NormedStarGroup β
inst✝ : CompactSpace α
f : C(α, β)
⊢ ‖star (mkOfCompact f)‖ ≤ ‖f‖

Tactic:
norm_star,

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : SeminormedAddCommGroup β
inst✝² : StarAddMonoid β
inst✝¹ : NormedStarGroup β
inst✝ : CompactSpace α
f : C(α, β)
⊢ ‖mkOfCompact f‖ ≤ ‖f‖
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: case neg.hnc
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
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ ↑r
hr : ¬r = 0
hμ : ¬IsFiniteMeasure μ
this : μ univ = ∞
⊢ ¬2 = 0

Tactic:
· simp

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 y : α
h : infEDist x E ≤ infEDist y E
h_le : infEDist y E * (↑δ.toNNReal)⁻¹ ≤ 1
⊢ infEDist y E * (↑δ.toNNReal)⁻¹ - infEDist x E * (↑δ.toNNReal)⁻¹ ≤ (↑δ.toNNReal)⁻¹ * edist x y

Tactic:
← ENNReal.sub_mul (by simp [δ_pos]),

Goal state after:
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x y : α
h : infEDist x E ≤ infEDist y E
h_le : infEDist y E * (↑δ.toNNReal)⁻¹ ≤ 1
⊢ (infEDist y E - infEDist x E) * (↑δ.toNNReal)⁻¹ ≤ (↑δ.toNNReal)⁻¹ * edist x y
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: case pos
α : Type u_1
inst✝¹ : PseudoEMetricSpace α
β : Type u_2
inst✝ : One β
f : α → β
E : Set α
x : α
x_mem_closure : x ∈ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
⊢ ∀ᶠ (δ : ℝ) in 𝓝[>] 0,
    (thickening δ E).mulIndicator f x =
      (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).mulIndicator f x

Tactic:
filter_upwards [self_mem_nhdsWithin] with δ δ_pos

Goal state after:
α : Type u_1
inst✝¹ : PseudoEMetricSpace α
β : Type u_2
inst✝ : One β
f : α → β
E : Set α
x : α
x_mem_closure : x ∈ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
δ : ℝ
δ_pos : δ ∈ Ioi 0
⊢ (thickening δ E).mulIndicator f x = (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).mulIndicator f x
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
hfint : Integrable f μ
hfint' : 0 ≤ ∫ (x : α), f x ∂μ
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ r
⊢ ∀ᵐ (ω : α) ∂μ, f ω ≤ r ∨ f ω ≤ 0

Tactic:
filter_upwards [hf] with ω hω using Or.inl hω

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

Goal state before:
pp: case h₁
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
f : α → E
hT : DominatedFinMeasAdditive μ T C
hC : 0 ≤ C
hF : CompleteSpace F
hf : Integrable f μ
⊢ C ≤ ↑(NNReal.mk C hC)

pp: case h₂
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
f : α → E
hT : DominatedFinMeasAdditive μ T C
hC : 0 ≤ C
hF : CompleteSpace F
hf : Integrable f μ
⊢ ‖Integrable.toL1 f hf‖ ≤ (∫⁻ (a : α), ENNReal.ofReal ‖f a‖ ∂μ).toReal

Tactic:
· simp

Goal state after:
case h₂
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
f : α → E
hT : DominatedFinMeasAdditive μ T C
hC : 0 ≤ C
hF : CompleteSpace F
hf : Integrable f μ
⊢ ‖Integrable.toL1 f hf‖ ≤ (∫⁻ (a : α), ENNReal.ofReal ‖f a‖ ∂μ).toReal
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: case inr.refine_1
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x ≤ y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : (𝓝[<] x).NeBot
⊢ (f '' Iio x).Nonempty

Tactic:
simp only [image_nonempty]
    exact (forall_mem_nonempty_iff_neBot.2 h') _ self_mem_nhdsWithin

Goal state after:
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x < y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
⊢ f x ≤ leftLim f y

Tactic:
rcases eq_or_neBot (𝓝[<] y) with h' | h'

Goal state after:
case inl
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x < y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : 𝓝[<] y = ⊥
⊢ f x ≤ leftLim f y

case inr
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x < y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : (𝓝[<] y).NeBot
⊢ f x ≤ leftLim f y
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F✝ : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F✝
inst✝¹ : NormedSpace ℝ F✝
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F✝
C : ℝ
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
inst✝ : Countable ι
F : ι → α → E
f : α → E
bound : ι → α → ℝ
hF_meas : ∀ (n : ι), AEStronglyMeasurable (F n) μ
h_bound : ∀ (n : ι), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound n a
bound_summable : ∀ᵐ (a : α) ∂μ, Summable fun n ↦ bound n a
bound_integrable : Integrable (fun a ↦ ∑' (n : ι), bound n a) μ
h_lim : ∀ᵐ (a : α) ∂μ, HasSum (fun n ↦ F n a) (f a)
hb_nonneg : ∀ᵐ (a : α) ∂μ, ∀ (n : ι), 0 ≤ bound n a
hb_le_tsum : ∀ (n : ι), bound n ≤ᵐ[μ] fun a ↦ ∑' (n : ι), bound n a
hF_integrable : ∀ (n : ι), Integrable (F n) μ
s : Finset ι
a : α
⊢ (∀ (i : ι), ‖F i a‖ ≤ bound i a) →
    (∀ (n : ι), 0 ≤ bound n a) → (Summable fun n ↦ bound n a) → ‖∑ i ∈ s, F i a‖ ≤ ∑' (n : ι), bound n a

Tactic:
hFa

Goal state after:
α : Type u_1
E : Type u_2
F✝ : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F✝
inst✝¹ : NormedSpace ℝ F✝
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F✝
C : ℝ
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
inst✝ : Countable ι
F : ι → α → E
f : α → E
bound : ι → α → ℝ
hF_meas : ∀ (n : ι), AEStronglyMeasurable (F n) μ
h_bound : ∀ (n : ι), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound n a
bound_summable : ∀ᵐ (a : α) ∂μ, Summable fun n ↦ bound n a
bound_integrable : Integrable (fun a ↦ ∑' (n : ι), bound n a) μ
h_lim : ∀ᵐ (a : α) ∂μ, HasSum (fun n ↦ F n a) (f a)
hb_nonneg : ∀ᵐ (a : α) ∂μ, ∀ (n : ι), 0 ≤ bound n a
hb_le_tsum : ∀ (n : ι), bound n ≤ᵐ[μ] fun a ↦ ∑' (n : ι), bound n a
hF_integrable : ∀ (n : ι), Integrable (F n) μ
s : Finset ι
a : α
hFa : ∀ (i : ι), ‖F i a‖ ≤ bound i a
⊢ (∀ (n : ι), 0 ≤ bound n a) → (Summable fun n ↦ bound n a) → ‖∑ i ∈ s, F i a‖ ≤ ∑' (n : ι), bound n a
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝ : Fintype ι
s : ι → Set X
hs : ∀ (i : ι), MeasurableSet (s i)
h's : Pairwise (Disjoint on s)
hf : ∀ (i : ι), IntegrableOn f (s i) μ
x✝ : ι
⊢ s x✝ = ⋃ (_ : x✝ ∈ Finset.univ), s x✝

pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝ : Fintype ι
s : ι → Set X
hs : ∀ (i : ι), MeasurableSet (s i)
h's : Pairwise (Disjoint on s)
hf : ∀ (i : ι), IntegrableOn f (s i) μ
⊢ (↑Finset.univ).Pairwise (Disjoint on s)

Tactic:
· simp

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝ : Fintype ι
s : ι → Set X
hs : ∀ (i : ι), MeasurableSet (s i)
h's : Pairwise (Disjoint on s)
hf : ∀ (i : ι), IntegrableOn f (s i) μ
⊢ (↑Finset.univ).Pairwise (Disjoint on s)
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case pos.refine_1.refine_2
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
β : Type u_7
mβ : MeasurableSpace β
inst✝ : SFinite μ
hT : DominatedFinMeasAdditive μ T C
h'T : ∀ (s : Set (β × α)), MeasurableSet s → StronglyMeasurable fun x ↦ T (Prod.mk x ⁻¹' s)
f : β → α → E
hf : StronglyMeasurable (Function.uncurry f)
hF : CompleteSpace F
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
this✝ : SeparableSpace ↑(range (Function.uncurry f) ∪ {0})
s : ℕ → SimpleFunc (β × α) E := SimpleFunc.approxOn (Function.uncurry f) ⋯ (range (Function.uncurry f) ∪ {0}) 0 ⋯
s' : ℕ → β → SimpleFunc α E := fun n x ↦ (s n).comp (Prod.mk x) ⋯
f' : ℕ → β → F := fun n ↦ {x | Integrable (f x) μ}.indicator fun x ↦ SimpleFunc.setToSimpleFunc T (s' n x)
hf' : ∀ (n : ℕ), StronglyMeasurable (f' n)
x : β
hfx : Integrable (f x) μ
this : ∀ (n : ℕ), Integrable (⇑(s' n x)) μ
n : ℕ
y : α
⊢ 0 ∈ range (Function.uncurry f) ∪ {0}

Tactic:
· simp

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

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : X → E
μ : Measure X
inst✝ : IsFiniteMeasure μ
C : ℝ
hf : ∀ᵐ (x : X) ∂μ, ‖f x‖ ≤ C
s : Set X
hs : MeasurableSet s
hf1 : Integrable f μ
⊢ ∫ (x : X), f x ∂μ - ∫ (x : X) in s, f x ∂μ = ∫ (x : X) in sᶜ, f x ∂μ

Tactic:
sub_eq_iff_eq_add,

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : X → E
μ : Measure X
inst✝ : IsFiniteMeasure μ
C : ℝ
hf : ∀ᵐ (x : X) ∂μ, ‖f x‖ ≤ C
s : Set X
hs : MeasurableSet s
hf1 : Integrable f μ
⊢ ∫ (x : X), f x ∂μ = ∫ (x : X) in sᶜ, f x ∂μ + ∫ (x : X) in s, f x ∂μ
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
μ : Measure X
f : X → E
s : Set X
inst✝¹ : ClosedIciTopology E
inst✝ : CompleteSpace E
c : E
hs : MeasurableSet s
hμs : μ s ≠ ∞
hf : ∀ x ∈ s, c ≤ f x
hfint : IntegrableOn (fun x ↦ f x) s μ
⊢ μ.real s • c ≤ ∫ (x : X) in s, f x ∂μ

Tactic:
← setIntegral_const c

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
μ : Measure X
f : X → E
s : Set X
inst✝¹ : ClosedIciTopology E
inst✝ : CompleteSpace E
c : E
hs : MeasurableSet s
hμs : μ s ≠ ∞
hf : ∀ x ∈ s, c ≤ f x
hfint : IntegrableOn (fun x ↦ f x) s μ
⊢ ∫ (x : X) in s, c ∂μ ≤ ∫ (x : X) in s, f x ∂μ
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
⊢ ⟪x - y, x - y⟫ = ⟪x, x⟫ - ⟪x, y⟫ - ⟪y, x⟫ + ⟪y, y⟫

Tactic:
simp only [inner_sub_left, inner_sub_right]; ring

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 : ∀ x ∈ s, f x ≤ 1
h's : ∀ x ∈ sᶜ, f x ≤ 0
⊢ ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ μ s

Tactic:
by_cases H : Integrable f μ; swap
  · simp [integral_undef H]
  let g x := max (f x) 0
  have g_int : Integrable g μ := H.pos_part
  have : ENNReal.ofReal (∫ x, f x ∂μ) ≤ ENNReal.ofReal (∫ x, g x ∂μ) := by
    apply ENNReal.ofReal_le_ofReal
    exact integral_mono H g_int (fun x ↦ le_max_left _ _)
  apply this.trans
  rw [ofReal_integral_eq_lintegral_ofReal g_int (Eventually.of_forall (fun x ↦ le_max_right _ _))]
  apply lintegral_le_meas
  · intro x
    apply ENNReal.ofReal_le_of_le_toReal
    by_cases H : x ∈ s
    · simpa [g] using hs x H
    · apply le_trans _ zero_le_one
      simpa [g] using h's x H
  · intro x hx
    simpa [g] using h's x hx

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

Goal state before:
pp: R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace 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
a b : R
hab : a ≤ b
⊢ c ≤ ↑(StieltjesFunction.const R c) a

pp: R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace 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
a b : R
hab : a ≤ b
⊢ ContinuousWithinAt (↑(StieltjesFunction.const R c)) (Iic a) a

Tactic:
· simp

Goal state after:
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace 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
a b : R
hab : a ≤ b
⊢ ContinuousWithinAt (↑(StieltjesFunction.const R c)) (Iic a) a
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure X
f g : ↥(Lp E p μ)
s : Set X
⊢ MemLp.toLp ↑↑(f + g) ⋯ = MemLp.toLp ↑↑f ⋯ + MemLp.toLp ↑↑g ⋯

Tactic:
ext1
  refine (ae_restrict_of_ae (Lp.coeFn_add f g)).mp ?_
  refine
    (Lp.coeFn_add (MemLp.toLp f ((Lp.memLp f).restrict s))
          (MemLp.toLp g ((Lp.memLp g).restrict s))).mp ?_
  refine (MemLp.coeFn_toLp ((Lp.memLp f).restrict s)).mp ?_
  refine (MemLp.coeFn_toLp ((Lp.memLp g).restrict s)).mp ?_
  refine (MemLp.coeFn_toLp ((Lp.memLp (f + g)).restrict s)).mono fun x hx1 hx2 hx3 hx4 hx5 => ?_
  rw [hx4, hx1, Pi.add_apply, hx2, hx3, hx5, Pi.add_apply]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁸ : RCLike 𝕜
inst✝⁷ : SeminormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
𝕝 : Type u_4
inst✝⁵ : CommSemiring 𝕝
inst✝⁴ : StarRing 𝕝
inst✝³ : Algebra 𝕝 𝕜
inst✝² : Module 𝕝 E
inst✝¹ : IsScalarTower 𝕝 𝕜 E
inst✝ : StarModule 𝕝 𝕜
x y : E
r : 𝕝
⊢ star (star r • ⟪y, x⟫) = r • ⟪x, y⟫

Tactic:
star_smul,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁸ : RCLike 𝕜
inst✝⁷ : SeminormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
𝕝 : Type u_4
inst✝⁵ : CommSemiring 𝕝
inst✝⁴ : StarRing 𝕝
inst✝³ : Algebra 𝕝 𝕜
inst✝² : Module 𝕝 E
inst✝¹ : IsScalarTower 𝕝 𝕜 E
inst✝ : StarModule 𝕝 𝕜
x y : E
r : 𝕝
⊢ star (star r) • star ⟪y, x⟫ = r • ⟪x, y⟫