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

Goal state before:
pp: ι : Type u_1
X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝ : NormedAddCommGroup E
μ : ι → Measure X
f : X → E
hf : Integrable f (Measure.sum μ)
⊢ Summable fun i ↦ ∫ (x : X), ‖f x‖ ∂μ i

Tactic:
convert! ENNReal.summable_toReal (f := fun i ↦ ∫⁻ x, ‖f x‖ₑ ∂μ i) ?_ with i
  · rw [← integral_toReal ?_ (by simp)]
    · simp
    · exact (hf.aestronglyMeasurable.mono_measure (Measure.le_sum _ i)).enorm
  rw [← lintegral_sum_measure]
  exact hf.2.ne

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

Goal state before:
pp: case e'_3
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, Antitone fun i ↦ f i a
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), F a ≤ f i a
f' : ℕ → α → ℝ := fun i a ↦ -f i a
F' : α → ℝ := fun a ↦ -F a
this : ∀ᵐ (a : α) ∂μ, Tendsto (fun i ↦ f' i a) atTop (𝓝 (F' a))
a : α
ha_tendsto : Tendsto (fun i ↦ f' i a) atTop (𝓝 (F' a))
x✝ : ℕ
⊢ f x✝ a = -f' x✝ a

Tactic:
simp [f']

Goal state after:
Module: Mathlib.Topology.Algebra.Indicator

Goal state before:
pp: case inl
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
hf : ContinuousOn f (interior s)
x : α
h : x ∈ interior s
hs : interior s ∈ 𝓝 x
⊢ ContinuousAt (s.mulIndicator f) x

Tactic:
exact ContinuousAt.congr (hf.continuousAt hs) <| Filter.eventuallyEq_iff_exists_mem.mpr
      ⟨interior s, hs, Set.eqOn_mulIndicator.symm.mono interior_subset⟩

Goal state after:
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: 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
⊢ IsOpen {L | Injective ⇑L}

Tactic:
rw [isOpen_iff_eventually]

Goal state after:
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
⊢ ∀ x ∈ {L | Injective ⇑L}, ∀ᶠ (y : E →L[𝕜] F) in 𝓝 x, y ∈ {L | Injective ⇑L}
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
hE : CompleteSpace E
mα : MeasurableSpace α
f : α → E
hf : StronglyMeasurable f
a : α
s : Set α
hs : MeasurableSet s
inst✝ : Decidable (a ∈ s)
⊢ (∫ (x : α), f x ∂if a ∈ s then Measure.dirac a else 0) = if a ∈ s then f a else 0

Tactic:
split_ifs

Goal state after:
case pos
α : Type u_1
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
hE : CompleteSpace E
mα : MeasurableSpace α
f : α → E
hf : StronglyMeasurable f
a : α
s : Set α
hs : MeasurableSet s
inst✝ : Decidable (a ∈ s)
h✝ : a ∈ s
⊢ ∫ (x : α), f x ∂Measure.dirac a = f a

case neg
α : Type u_1
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
hE : CompleteSpace E
mα : MeasurableSpace α
f : α → E
hf : StronglyMeasurable f
a : α
s : Set α
hs : MeasurableSet s
inst✝ : Decidable (a ∈ s)
h✝ : a ∉ s
⊢ ∫ (x : α), f x ∂0 = 0
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSingletonClass α
f : α → E
a : α
s : Set α
inst✝ : Decidable (a ∈ s)
⊢ (∫ (x : α), f x ∂if a ∈ s then Measure.dirac a else 0) = if a ∈ s then f a else 0

Tactic:
split_ifs

Goal state after:
case pos
α : Type u_1
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSingletonClass α
f : α → E
a : α
s : Set α
inst✝ : Decidable (a ∈ s)
h✝ : a ∈ s
⊢ ∫ (x : α), f x ∂Measure.dirac a = f a

case neg
α : Type u_1
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSingletonClass α
f : α → E
a : α
s : Set α
inst✝ : Decidable (a ∈ s)
h✝ : a ∉ s
⊢ ∫ (x : α), f x ∂0 = 0
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: case refine_3
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.domRestrict (range ⇑f)ᶜ) x) ((h₂.domRestrict (range ⇑f)ᶜ) x) ≤ dist (extend f g₁ h₁) (extend f g₂ h₂)

Tactic:
calc
      dist (h₁ x) (h₂ x) = dist (extend f g₁ h₁ x) (extend f g₂ h₂ x) := by
        rw [extend_apply' x.coe_prop, extend_apply' x.coe_prop]
      _ ≤ _ := dist_coe_le_dist _

Goal state after:
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: case inr.intro
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type v
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CompleteSpace 𝕜
ι : Type u_1
inst✝ : Finite ι
i₀ : ι
ι' : Type (max 0 u_1) := { x // x ≠ i₀ }
val✝ : Fintype ι
⊢ IsOpen[Pi.topologicalSpace] {p | LinearIndependent 𝕜 fun i ↦ p ↑i -ᵥ p i₀}

Tactic:
have : Fintype ι' := Subtype.fintype _

Goal state after:
case inr.intro
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type v
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CompleteSpace 𝕜
ι : Type u_1
inst✝ : Finite ι
i₀ : ι
ι' : Type (max 0 u_1) := { x // x ≠ i₀ }
val✝ : Fintype ι
this : Fintype ι'
⊢ IsOpen[Pi.topologicalSpace] {p | LinearIndependent 𝕜 fun i ↦ p ↑i -ᵥ p i₀}
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
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
hfi : Integrable f μ
f_lp : ↥(Lp E 1 μ) := Integrable.toL1 f hfi
⊢ Tendsto (fun i ↦ setToFun μ T hT (fs i)) l (𝓝 (setToFun μ T hT f))

Tactic:
let F_lp i := if hFi : Integrable (fs i) μ then hFi.toL1 (fs i) else 0

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
hfi : Integrable f μ
f_lp : ↥(Lp E 1 μ) := Integrable.toL1 f hfi
F_lp : ι → ↥(Lp E 1 μ) := fun i ↦ if hFi : Integrable (fs i) μ then Integrable.toL1 (fs i) hFi else 0
⊢ Tendsto (fun i ↦ setToFun μ T hT (fs i)) l (𝓝 (setToFun μ T hT f))
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type w
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : CompleteSpace 𝕜
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : FiniteDimensional 𝕜 E
f : X → E →L[𝕜] F
s : Set X
x : X
⊢ ContinuousWithinAt f s x ↔ ∀ (y : E), ContinuousWithinAt (fun q ↦ (f q) y) s x

Tactic:
refine ⟨fun h y ↦ (apply 𝕜 F y).continuous.continuousAt.comp_continuousWithinAt h, fun h ↦ ?_⟩

Goal state after:
𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type w
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : CompleteSpace 𝕜
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : FiniteDimensional 𝕜 E
f : X → E →L[𝕜] F
s : Set X
x : X
h : ∀ (y : E), ContinuousWithinAt (fun q ↦ (f q) y) s x
⊢ ContinuousWithinAt f s x
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
hF : CompleteSpace F
hfi' : Integrable f μ'
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : SeparableSpace ↑(range f ∪ {0})
i : ℕ
⊢ ∑ x ∈ (SimpleFunc.approxOn f ⋯ (range f ∪ {0}) 0 ⋯ i).range,
      (T' (⇑(SimpleFunc.approxOn f ⋯ (range f ∪ {0}) 0 ⋯ i) ⁻¹' {x})) x =
    ∑ x ∈ ((SimpleFunc.approxOn f ⋯ (range f ∪ {0}) 0 ⋯ i).comp φ hφ).range,
      (T (⇑((SimpleFunc.approxOn f ⋯ (range f ∪ {0}) 0 ⋯ i).comp φ hφ) ⁻¹' {x})) x

Tactic:
simp only [union_singleton, SimpleFunc.measurableSet_preimage, h, ← preimage_comp,
    SimpleFunc.coe_comp]

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})
i : ℕ
⊢ ∑ x ∈ (SimpleFunc.approxOn f ⋯ (insert 0 (range f)) 0 ⋯ i).range,
      (T (⇑(SimpleFunc.approxOn f ⋯ (insert 0 (range f)) 0 ⋯ i) ∘ φ ⁻¹' {x})) x =
    ∑ x ∈ ((SimpleFunc.approxOn f ⋯ (insert 0 (range f)) 0 ⋯ i).comp φ hφ).range,
      (T (⇑(SimpleFunc.approxOn f ⋯ (insert 0 (range f)) 0 ⋯ i) ∘ φ ⁻¹' {x})) x
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : CompleteSpace 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : LocallyCompactSpace E
S : Submodule 𝕜 E
a✝ : Nontrivial E
⊢ ProperSpace ↥S

Tactic:
have : ProperSpace 𝕜 := .of_locallyCompact_module 𝕜 E

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : CompleteSpace 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : LocallyCompactSpace E
S : Submodule 𝕜 E
a✝ : Nontrivial E
this : ProperSpace 𝕜
⊢ ProperSpace ↥S
Module: Mathlib.Topology.ContinuousMap.Bounded.Normed

Goal state before:
pp: α : Type u
β : Type v
γ : Type w
inst✝¹ : TopologicalSpace α
R : Type u_1
inst✝ : SeminormedRing R
f : α →ᵇ R
n : ℕ
⊢ ∃ C, ∀ (x y : α), dist ((f.toContinuousMap ^ n).toFun x) ((f.toContinuousMap ^ n).toFun y) ≤ C

Tactic:
by simpa [coe_npowRec] using (npowRec n f).map_bounded'

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

Goal state before:
pp: case a
α : Type u_1
E : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : SeminormedAddCommGroup E
f : C(α, E)
inst✝ : Fintype α
⊢ ‖⇑f‖ ≤ ‖f‖

Tactic:
rw [pi_norm_le_iff_of_nonneg (by positivity)]
    exact f.norm_coe_le_norm

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

Goal state before:
pp: case neg
α : Type u_1
inst✝ : PseudoEMetricSpace α
δseq : ℕ → ℝ
E : Set α
x : α
x_mem_closure : x ∉ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
ε : ℝ
ε_pos : 0 < ε
ε_lt : ENNReal.ofReal ε < infEDist x E
N : ℕ
hN : ∀ (b : ℕ), N ≤ b → |δseq b| < ε
n : ℕ
n_large : n ≥ N
key : x ∉ thickening ε E
⊢ thickenedIndicatorAux ε E x ≤ 0

Tactic:
exact (thickenedIndicatorAux_zero ε_pos E key).le

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

Goal state before:
pp: case h.h₁
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x y : α
key : ∀ (x : α), thickenedIndicatorAux δ E x ≤ 1
⊢ ↑(thickenedIndicatorAux δ E x).toNNReal ≤ 1

Tactic:
· norm_cast
        exact (toNNReal_le_toNNReal (lt_of_le_of_lt (key _) one_lt_top).ne one_ne_top).mpr (key _)

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

Goal state before:
pp: case h₂
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x y : α
key : ∀ (x : α), thickenedIndicatorAux δ E x ≤ 1
⊢ ↑(thickenedIndicatorAux δ E y).toNNReal ≤ 1

Tactic:
· norm_cast
        exact (toNNReal_le_toNNReal (lt_of_le_of_lt (key _) one_lt_top).ne one_ne_top).mpr (key _)

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:
rw [← ENNReal.sub_mul (by simp [δ_pos]), mul_comm, edist_comm]

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
⊢ (↑δ.toNNReal)⁻¹ * (infEDist y E - infEDist x E) ≤ (↑δ.toNNReal)⁻¹ * edist y x
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
s : Finset δ
x y : (i : δ) → X i
f : ((i : δ) → X i) → ℝ≥0∞
h : ∀ i ∉ s, x i = y i
⊢ (∫⁻ (y : (i : ↥s) → X ↑i), f fun i ↦ if hi : i ∈ s then y ⟨i, hi⟩ else x i ∂Measure.pi fun i ↦ μ ↑i) =
    ∫⁻ (y_1 : (i : ↥s) → X ↑i), f fun i ↦ if hi : i ∈ s then y_1 ⟨i, hi⟩ else y i ∂Measure.pi fun i ↦ μ ↑i

Tactic:
rcongr

Goal state after:
case e_f.e_e
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
s : Finset δ
x y : (i : δ) → X i
f : ((i : δ) → X i) → ℝ≥0∞
h : ∀ i ∉ s, x i = y i
x✝² : (i : ↥s) → X ↑i
x✝¹ : δ
x✝ : x✝¹ ∉ s
⊢ x x✝¹ = y x✝¹
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : T3Space β
f : α → β
b : β
h : Tendsto f atTop (𝓝 b)
val✝ : OrderTop α
⊢ Tendsto (rightLim f) atTop (𝓝 b)

Tactic:
simp only [OrderTop.atTop_eq α] at h ⊢

Goal state after:
case inl
α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : T3Space β
f : α → β
b : β
val✝ : OrderTop α
h : Tendsto f (pure ⊤) (𝓝 b)
⊢ Tendsto (rightLim f) (pure ⊤) (𝓝 b)
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: case inr.refine_2
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
z : α
hz : z < x
⊢ f z ≤ f y

Tactic:
exact hf (hz.le.trans h)

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

Goal state before:
pp: case pos
R : Type u_1
inst✝² : LinearOrder R
inst✝¹ : TopologicalSpace R
inst✝ : OrderTopology R
a b : R
h : IsTop b
⊢ IsOpen[inst✝¹] (Ioc a b)

Tactic:
have : Ioc a b = Ioi a := Subset.antisymm (fun x hx ↦ hx.1) (fun x hx ↦ by exact ⟨hx, h _⟩)
    simp [this, isOpen_Ioi]

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

Goal state before:
pp: case inr.inr
R : Type u_1
inst✝² : LinearOrder R
inst✝¹ : TopologicalSpace R
f✝ : StieltjesFunction R
inst✝ : OrderTopology R
f : R → ℝ
hf : Monotone f
x : R
s : Set ℝ
hs : s ∈ 𝓝 (rightLim f x)
l u : ℝ
hlu : rightLim f x ∈ Ioo l u
lus : Ioo l u ⊆ s
y₀ : R
hy₀ : x < y₀
y : R
xy : x < y
h'y : Ioo x y ⊆ f ⁻¹' Ioo l u
z : R
hz : z ∈ Ico x y
h''z : x < z
h'z : (𝓝[>] z).NeBot
⊢ rightLim f z < u

Tactic:
obtain ⟨a, za, ay⟩ : ∃ a : R, z < a ∧ a < y := Filter.nonempty_of_mem (Ioo_mem_nhdsGT hz.2)
      calc
        rightLim f z ≤ f a := hf.rightLim_le za
        _ < u := (h'y ⟨hz.1.trans_lt za, ay⟩).2

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✝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), MeasurableSet (s i)
h_anti : Antitone s
hne : atTop.NeBot
this : IsDirectedOrder ι
i₀ : ι
hi₀ : IntegrableOn f (s i₀) μ
⊢ Tendsto (fun x ↦ ∫ (x : X) in s i₀, f x ∂μ - ∫ (x : X) in s i₀ \ s x, f x ∂μ) atTop
    (𝓝 (∫ (x : X) in s i₀, f x ∂μ - ∫ (x : X) in ⋃ i, s i₀ \ s i, f x ∂μ))

Tactic:
apply tendsto_const_nhds.sub

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), MeasurableSet (s i)
h_anti : Antitone s
hne : atTop.NeBot
this : IsDirectedOrder ι
i₀ : ι
hi₀ : IntegrableOn f (s i₀) μ
⊢ Tendsto (fun x ↦ ∫ (x : X) in s i₀ \ s x, f x ∂μ) atTop (𝓝 (∫ (x : X) in ⋃ i, s i₀ \ s i, f x ∂μ))
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
h3 : ∀ (g : G) {K : Compacts G}, μ (Compacts.map (fun x ↦ g * x) ⋯ K) = μ K
K : Compacts G
hK : μ K ≠ 0
U : Opens G
hU : (↑U).Nonempty
this✝ : (interior ↑U).Nonempty
s : Finset G
hs : K.carrier ⊆ ⋃ g ∈ s, (fun x ↦ g * x) ⁻¹' ↑U
this : ↑K ⊆ ↑(⨆ g ∈ s, (Opens.comap ↑(Homeomorph.mulLeft g)) U)
⊢ ∑ d ∈ s, μ.innerContent ((Opens.comap ↑(Homeomorph.mulLeft d)) U) ≤ ↑s.card * μ.innerContent U

Tactic:
simp only [μ.is_mul_left_invariant_innerContent h3, Finset.sum_const, nsmul_eq_mul, le_refl]

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

Goal state before:
pp: G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : R1Space G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
h3 : ∀ (g : G) {K : Compacts G}, μ (Compacts.map (fun x ↦ g * x) ⋯ K) = μ K
K : Compacts G
hK : μ K ≠ 0
U : Set G
h1U : IsOpen U
h2U : U.Nonempty
⊢ 0 < μ.outerMeasure U

Tactic:
by
  convert! μ.innerContent_pos_of_is_mul_left_invariant h3 K hK ⟨U, h1U⟩ h2U
  exact μ.outerMeasure_opens ⟨U, h1U⟩

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✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s : Set X
μ : Measure X
C : ℝ
hs : μ s < ∞
hC : ∀ᵐ (x : X) ∂μ, x ∈ s → ‖f x‖ ≤ C
hfm : AEStronglyMeasurable f (μ.restrict s)
a✝ : X
⊢ (a✝ ∈ s → ‖f a✝‖ ≤ C) →
    (a✝ ∈ s → f a✝ = AEStronglyMeasurable.mk f hfm a✝) → a✝ ∈ s → ‖AEStronglyMeasurable.mk f hfm a✝‖ ≤ C

Tactic:
h1

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 : Set X
μ : Measure X
C : ℝ
hs : μ s < ∞
hC : ∀ᵐ (x : X) ∂μ, x ∈ s → ‖f x‖ ≤ C
hfm : AEStronglyMeasurable f (μ.restrict s)
a✝ : X
h1 : a✝ ∈ s → ‖f a✝‖ ≤ C
⊢ (a✝ ∈ s → f a✝ = AEStronglyMeasurable.mk f hfm a✝) → a✝ ∈ s → ‖AEStronglyMeasurable.mk f hfm a✝‖ ≤ C
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 : Set X
μ : Measure X
C : ℝ
hs : μ s < ∞
hC : ∀ᵐ (x : X) ∂μ, x ∈ s → ‖f x‖ ≤ C
hfm : AEStronglyMeasurable f (μ.restrict s)
A : ∀ᵐ (x : X) ∂μ, x ∈ s → ‖AEStronglyMeasurable.mk f hfm x‖ ≤ C
B : MeasurableSet {x | ‖AEStronglyMeasurable.mk f hfm x‖ ≤ C}
a✝ : X
⊢ f a✝ = AEStronglyMeasurable.mk f hfm a✝ → ‖AEStronglyMeasurable.mk f hfm a✝‖ ≤ C → ‖f a✝‖ ≤ C

Tactic:
h1

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 : Set X
μ : Measure X
C : ℝ
hs : μ s < ∞
hC : ∀ᵐ (x : X) ∂μ, x ∈ s → ‖f x‖ ≤ C
hfm : AEStronglyMeasurable f (μ.restrict s)
A : ∀ᵐ (x : X) ∂μ, x ∈ s → ‖AEStronglyMeasurable.mk f hfm x‖ ≤ C
B : MeasurableSet {x | ‖AEStronglyMeasurable.mk f hfm x‖ ≤ C}
a✝ : X
h1 : f a✝ = AEStronglyMeasurable.mk f hfm a✝
⊢ ‖AEStronglyMeasurable.mk f hfm a✝‖ ≤ C → ‖f a✝‖ ≤ C
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 : Set X
μ : Measure X
C : ℝ
hs : μ s < ∞
hC : ∀ᵐ (x : X) ∂μ, x ∈ s → ‖f x‖ ≤ C
hfm : AEStronglyMeasurable f (μ.restrict s)
A : ∀ᵐ (x : X) ∂μ, x ∈ s → ‖AEStronglyMeasurable.mk f hfm x‖ ≤ C
B : MeasurableSet {x | ‖AEStronglyMeasurable.mk f hfm x‖ ≤ C}
a✝¹ : X
h1 : f a✝¹ = AEStronglyMeasurable.mk f hfm a✝¹
a✝ : ‖AEStronglyMeasurable.mk f hfm a✝¹‖ ≤ C
⊢ ‖f a✝¹‖ ≤ C

Tactic:
h1

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 : Set X
μ : Measure X
C : ℝ
hs : μ s < ∞
hC : ∀ᵐ (x : X) ∂μ, x ∈ s → ‖f x‖ ≤ C
hfm : AEStronglyMeasurable f (μ.restrict s)
A : ∀ᵐ (x : X) ∂μ, x ∈ s → ‖AEStronglyMeasurable.mk f hfm x‖ ≤ C
B : MeasurableSet {x | ‖AEStronglyMeasurable.mk f hfm x‖ ≤ C}
a✝¹ : X
h1 : f a✝¹ = AEStronglyMeasurable.mk f hfm a✝¹
a✝ : ‖AEStronglyMeasurable.mk f hfm a✝¹‖ ≤ C
⊢ ‖AEStronglyMeasurable.mk f hfm a✝¹‖ ≤ C
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : (interior U).Nonempty
⊢ prehaar (↑K₀) U ∈ haarProduct ↑K₀

Tactic:
by
    rintro ⟨K, hK⟩ _; rw [mem_Icc]; exact ⟨prehaar_nonneg K₀ _, prehaar_le_index K₀ _ hU⟩

Goal state after:
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
g : StieltjesFunction R
y : R
hfg : f.measure = g.measure
hy : ↑f y = ↑g y
x : R
hxy : y ≤ x
⊢ ↑f x = ↑g x

Tactic:
have hf := measure_Ioc f y x

Goal state after:
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
g : StieltjesFunction R
y : R
hfg : f.measure = g.measure
hy : ↑f y = ↑g y
x : R
hxy : y ≤ x
hf : f.measure (Ioc y x) = ofReal (↑f x - ↑f y)
⊢ ↑f x = ↑g x
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
ι : Type u_5
inst✝¹ : Countable ι
μ : Measure X
inst✝ : NormedAddCommGroup E
f : X → E
s : ι → Set X
hi : ∀ (i : ι), IntegrableOn f (s i) μ
h : Summable fun i ↦ ∫ (x : X) in s i, ‖f x‖ ∂μ
⊢ IntegrableOn f (iUnion s) μ

Tactic:
by
  refine ⟨AEStronglyMeasurable.iUnion fun i => (hi i).1, (lintegral_iUnion_le _ _).trans_lt ?_⟩
  have B := fun i => lintegral_coe_eq_integral (fun x : X => ‖f x‖₊) (hi i).norm
  simp_rw [enorm_eq_nnnorm, tsum_congr B]
  have S' : Summable fun i : ι =>
      (NNReal.mk (∫ x : X in s i, ‖f x‖₊ ∂μ) (integral_nonneg fun x => NNReal.coe_nonneg _)) := by
    rw [← NNReal.summable_coe]; exact h
  have S'' := ENNReal.tsum_coe_eq S'.hasSum
  simp_rw [ENNReal.coe_nnreal_eq, NNReal.coe_mk, coe_nnnorm] at S''
  convert! ENNReal.ofReal_lt_top

Goal state after: