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

Goal state before:
pp: case cons
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
ι : Type u_6
m : MeasurableSpace α
f : α → G
μ : ι → Measure α
a✝ : ι
s✝ : Finset ι
h : a✝ ∉ s✝
ih : (∀ i ∈ s✝, Integrable f (μ i)) → ∫ (a : α), f a ∂∑ i ∈ s✝, μ i = ∑ i ∈ s✝, ∫ (a : α), f a ∂μ i
hf : Integrable f (μ a✝) ∧ ∀ x ∈ s✝, Integrable f (μ x)
⊢ ∫ (a : α), f a ∂∑ i ∈ Finset.cons a✝ s✝ h, μ i = ∑ i ∈ Finset.cons a✝ s✝ h, ∫ (a : α), f a ∂μ i

Tactic:
Finset.sum_cons,

Goal state after:
case cons
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
ι : Type u_6
m : MeasurableSpace α
f : α → G
μ : ι → Measure α
a✝ : ι
s✝ : Finset ι
h : a✝ ∉ s✝
ih : (∀ i ∈ s✝, Integrable f (μ i)) → ∫ (a : α), f a ∂∑ i ∈ s✝, μ i = ∑ i ∈ s✝, ∫ (a : α), f a ∂μ i
hf : Integrable f (μ a✝) ∧ ∀ x ∈ s✝, Integrable f (μ x)
⊢ ∫ (a : α), f a ∂(μ a✝ + ∑ x ∈ s✝, μ x) = ∑ i ∈ Finset.cons a✝ s✝ h, ∫ (a : α), f a ∂μ i
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: case cons
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
ι : Type u_6
m : MeasurableSpace α
f : α → G
μ : ι → Measure α
a✝ : ι
s✝ : Finset ι
h : a✝ ∉ s✝
ih : (∀ i ∈ s✝, Integrable f (μ i)) → ∫ (a : α), f a ∂∑ i ∈ s✝, μ i = ∑ i ∈ s✝, ∫ (a : α), f a ∂μ i
hf : Integrable f (μ a✝) ∧ ∀ x ∈ s✝, Integrable f (μ x)
⊢ ∫ (a : α), f a ∂(μ a✝ + ∑ x ∈ s✝, μ x) = ∑ i ∈ Finset.cons a✝ s✝ h, ∫ (a : α), f a ∂μ i

Tactic:
Finset.sum_cons,

Goal state after:
case cons
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
ι : Type u_6
m : MeasurableSpace α
f : α → G
μ : ι → Measure α
a✝ : ι
s✝ : Finset ι
h : a✝ ∉ s✝
ih : (∀ i ∈ s✝, Integrable f (μ i)) → ∫ (a : α), f a ∂∑ i ∈ s✝, μ i = ∑ i ∈ s✝, ∫ (a : α), f a ∂μ i
hf : Integrable f (μ a✝) ∧ ∀ x ∈ s✝, Integrable f (μ x)
⊢ ∫ (a : α), f a ∂(μ a✝ + ∑ x ∈ s✝, μ x) = ∫ (a : α), f a ∂μ a✝ + ∑ x ∈ s✝, ∫ (a : α), f a ∂μ x
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
inst✝ : CompleteSpace F
hT : DominatedFinMeasAdditive μ T C
f : SimpleFunc α E
hf : Integrable (⇑f) μ
h'f : MemLp (⇑f) 1 μ
g : ↥(α →₁ₛ[μ] E) := f.toLp h'f
A : ⇑f =ᵐ[μ] ↑↑↑g
⊢ (L1.setToL1 hT) ↑g = ∑ x ∈ f.range, (T (⇑f ⁻¹' {x})) x

Tactic:
L1.setToL1_eq_setToL1SCLM

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
inst✝ : CompleteSpace F
hT : DominatedFinMeasAdditive μ T C
f : SimpleFunc α E
hf : Integrable (⇑f) μ
h'f : MemLp (⇑f) 1 μ
g : ↥(α →₁ₛ[μ] E) := f.toLp h'f
A : ⇑f =ᵐ[μ] ↑↑↑g
⊢ (L1.SimpleFunc.setToL1SCLM α E μ hT) g = ∑ x ∈ f.range, (T (⇑f ⁻¹' {x})) x
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
⊢ setToFun μ' T' hT' f = setToFun μ T hT (f ∘ φ)

Tactic:
have hfi' : Integrable f μ' :=
    ((integrable_map_measure hfm.aestronglyMeasurable hφ.aemeasurable).2 hf).mono_measure hμ'

Goal state after:
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'
⊢ setToFun μ' T' hT' f = setToFun μ T hT (f ∘ φ)
Module: Mathlib.Topology.ContinuousMap.Bounded.Normed

Goal state before:
pp: α : Type u
inst✝ : TopologicalSpace α
f : α →ᵇ ℝ
x : α
⊢ f x = max (f x) 0 - max (-f x) 0

Tactic:
simp only [max_zero_sub_max_neg_zero_eq_self]

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

Goal state before:
pp: case inr
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
E : Set α
x y : α
h : infEDist x E ≤ infEDist y E
hle : 1 ≤ infEDist x E / ENNReal.ofReal δ
⊢ 1 - infEDist y E / ENNReal.ofReal δ ≤ 0

Tactic:
tsub_eq_zero_of_le

Goal state after:
case inr
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
E : Set α
x y : α
h : infEDist x E ≤ infEDist y E
hle : 1 ≤ infEDist x E / ENNReal.ofReal δ
⊢ 0 ≤ 0

case inr
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
E : Set α
x y : α
h : infEDist x E ≤ infEDist y E
hle : 1 ≤ infEDist x E / ENNReal.ofReal δ
⊢ 1 ≤ infEDist y E / ENNReal.ofReal δ
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δseq : ℕ → ℝ
δseq_lim : Tendsto δseq atTop (𝓝 0)
E : Set α
⊢ Tendsto (fun n ↦ thickenedIndicatorAux (δseq n) E) atTop
    (𝓝 ((closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).indicator fun x ↦ 1))

Tactic:
by
  rw [tendsto_pi_nhds]
  intro x
  by_cases x_mem_closure : x ∈ closure E
  · simp_rw [thickenedIndicatorAux_one_of_mem_closure _ E x_mem_closure]
    rw [show (indicator (closure E) fun _ => (1 : ℝ≥0∞)) x = 1 by
        simp only [x_mem_closure, indicator_of_mem]]
    exact tendsto_const_nhds
  · rw [show (closure E).indicator (fun _ => (1 : ℝ≥0∞)) x = 0 by
        simp only [x_mem_closure, indicator_of_notMem, not_false_iff]]
    rcases exists_real_pos_lt_infEDist_of_notMem_closure x_mem_closure with ⟨ε, ⟨ε_pos, ε_lt⟩⟩
    rw [Metric.tendsto_nhds] at δseq_lim
    specialize δseq_lim ε ε_pos
    simp only [dist_zero_right, Real.norm_eq_abs, eventually_atTop] at δseq_lim
    rcases δseq_lim with ⟨N, hN⟩
    apply tendsto_atTop_of_eventually_const (i₀ := N)
    intro n n_large
    have key : x ∉ thickening ε E := by simpa only [thickening, mem_ofPred_eq, not_lt] using ε_lt.le
    refine le_antisymm ?_ bot_le
    apply (thickenedIndicatorAux_mono (lt_of_abs_lt (hN n n_large)).le E x).trans
    exact (thickenedIndicatorAux_zero ε_pos E key).le

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
inst✝ : ∀ (i : δ), SigmaFinite (μ i)
f : ((i : δ) → X i) → ℝ≥0∞
hf : Measurable f
s t : Finset δ
hst : Disjoint s t
⊢ ∫⋯∫⁻_s ∪ t, f ∂μ = ∫⋯∫⁻_t, ∫⋯∫⁻_s, f ∂μ ∂μ

Tactic:
Finset.union_comm,

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
inst✝ : ∀ (i : δ), SigmaFinite (μ i)
f : ((i : δ) → X i) → ℝ≥0∞
hf : Measurable f
s t : Finset δ
hst : Disjoint s t
⊢ ∫⋯∫⁻_t ∪ s, f ∂μ = ∫⋯∫⁻_t, ∫⋯∫⁻_s, f ∂μ ∂μ
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 : Finset δ
inst✝ : ∀ (i : δ), SigmaFinite (μ i)
f : ((i : δ) → X i) → ℝ≥0∞
hf : Measurable f
i : δ
hi : i ∉ s
⊢ ∫⋯∫⁻_{i} ∪ s, f ∂μ = ∫⋯∫⁻_s, fun x ↦ ∫⁻ (xᵢ : X i), f (update x i xᵢ) ∂μ i ∂μ

Tactic:
Finset.union_comm,

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 : Finset δ
inst✝ : ∀ (i : δ), SigmaFinite (μ i)
f : ((i : δ) → X i) → ℝ≥0∞
hf : Measurable f
i : δ
hi : i ∉ s
⊢ ∫⋯∫⁻_s ∪ {i}, f ∂μ = ∫⋯∫⁻_s, fun x ↦ ∫⁻ (xᵢ : X i), f (update x i xᵢ) ∂μ i ∂μ
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: 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✝ : T2Space β
f : α → β
a : α
hf : ContinuousWithinAt f (Iic a) a
⊢ leftLim f a = f a

Tactic:
rcases eq_or_neBot (𝓝[<] a) with h' | 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✝ : T2Space β
f : α → β
a : α
hf : ContinuousWithinAt f (Iic a) a
h' : 𝓝[<] a = ⊥
⊢ leftLim f a = f a

case inr
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✝ : T2Space β
f : α → β
a : α
hf : ContinuousWithinAt f (Iic a) a
h' : (𝓝[<] a).NeBot
⊢ leftLim f a = f a
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : LinearOrder α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace α
h'α : OrderTopology α
f : α → β
a : α
h : ∃ y, Tendsto f (𝓝[<] a) (𝓝 y)
⊢ Tendsto f (𝓝[<] a) (𝓝 (leftLim f a))

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

Goal state after:
case inl
α : Type u_1
β : Type u_2
inst✝² : LinearOrder α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace α
h'α : OrderTopology α
f : α → β
a : α
h : ∃ y, Tendsto f (𝓝[<] a) (𝓝 y)
h' : 𝓝[<] a = ⊥
⊢ Tendsto f (𝓝[<] a) (𝓝 (leftLim f a))

case inr
α : Type u_1
β : Type u_2
inst✝² : LinearOrder α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace α
h'α : OrderTopology α
f : α → β
a : α
h : ∃ y, Tendsto f (𝓝[<] a) (𝓝 y)
h' : (𝓝[<] a).NeBot
⊢ Tendsto f (𝓝[<] a) (𝓝 (leftLim f a))
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → β
a : α
A : (𝓝 (f a) ⊓ map f (𝓝[≤] a)).NeBot
⊢ MapClusterPt (leftLim f a) (𝓝[≤] a) f

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

Goal state after:
case inl
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → β
a : α
A : (𝓝 (f a) ⊓ map f (𝓝[≤] a)).NeBot
h' : 𝓝[<] a = ⊥
⊢ MapClusterPt (leftLim f a) (𝓝[≤] a) f

case inr
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → β
a : α
A : (𝓝 (f a) ⊓ map f (𝓝[≤] a)).NeBot
h' : (𝓝[<] a).NeBot
⊢ MapClusterPt (leftLim f a) (𝓝[≤] a) f
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: 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
a : α
h : Tendsto f (𝓝[<] a) (𝓝 (leftLim f a))
this : 𝓝[≤] a = 𝓝[<] a ⊔ pure a
s : Set β
s_mem : s ∈ 𝓝 (leftLim f a)
s_closed : IsClosed[inst✝³] s
⊢ ∀ᶠ (x : α) in 𝓝[<] a, leftLim f x ∈ id s

Tactic:
rcases eq_or_neBot (𝓝[<] a) with h' | 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
a : α
h : Tendsto f (𝓝[<] a) (𝓝 (leftLim f a))
this : 𝓝[≤] a = 𝓝[<] a ⊔ pure a
s : Set β
s_mem : s ∈ 𝓝 (leftLim f a)
s_closed : IsClosed[inst✝³] s
h' : 𝓝[<] a = ⊥
⊢ ∀ᶠ (x : α) in 𝓝[<] a, leftLim f x ∈ id s

case inr
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
a : α
h : Tendsto f (𝓝[<] a) (𝓝 (leftLim f a))
this : 𝓝[≤] a = 𝓝[<] a ⊔ pure a
s : Set β
s_mem : s ∈ 𝓝 (leftLim f a)
s_closed : IsClosed[inst✝³] s
h' : (𝓝[<] a).NeBot
⊢ ∀ᶠ (x : α) in 𝓝[<] a, leftLim f x ∈ id s
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
l : Filter ι
inst✝ : l.IsCountablyGenerated
fs : ι → α → E
f : α → E
bound : α → ℝ
hfs_meas : ∀ᶠ (n : ι) in l, AEStronglyMeasurable (fs n) μ
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ fs n a) l (𝓝 (f a))
x : ℕ → ι
xl : Tendsto x atTop l
hxl : ∀ s ∈ l, ∃ a, ∀ b ≥ a, x b ∈ s
h✝ : {x | (fun n ↦ AEStronglyMeasurable (fs n) μ) x} ∩ {x | (fun n ↦ ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) x} ∈ l
k : ℕ
h : ∀ b ≥ k, x b ∈ {x | (fun n ↦ AEStronglyMeasurable (fs n) μ) x} ∩ {x | (fun n ↦ ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) x}
⊢ Tendsto ((fun n ↦ setToFun μ T hT (fs n)) ∘ x) atTop (𝓝 (setToFun μ T hT f))

Tactic:
rw [← tendsto_add_atTop_iff_nat k]

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
l : Filter ι
inst✝ : l.IsCountablyGenerated
fs : ι → α → E
f : α → E
bound : α → ℝ
hfs_meas : ∀ᶠ (n : ι) in l, AEStronglyMeasurable (fs n) μ
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ fs n a) l (𝓝 (f a))
x : ℕ → ι
xl : Tendsto x atTop l
hxl : ∀ s ∈ l, ∃ a, ∀ b ≥ a, x b ∈ s
h✝ : {x | (fun n ↦ AEStronglyMeasurable (fs n) μ) x} ∩ {x | (fun n ↦ ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) x} ∈ l
k : ℕ
h : ∀ b ≥ k, x b ∈ {x | (fun n ↦ AEStronglyMeasurable (fs n) μ) x} ∩ {x | (fun n ↦ ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) x}
⊢ Tendsto (fun n ↦ ((fun n ↦ setToFun μ T hT (fs n)) ∘ x) (n + k)) atTop (𝓝 (setToFun μ T hT f))
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
f : α → β
hf : Monotone f
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
x : α
⊢ ∀ᶠ (x_1 : α) in 𝓝[<] x, f x_1 ∈ Iic (leftLim f x)

Tactic:
filter_upwards [@self_mem_nhdsWithin _ _ x (Iio x)] with y hy using hf.le_leftLim hy

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

Goal state before:
pp: f : StieltjesFunction ℝ
x : ℝ
⊢ ⨅ r, ↑f ↑↑r = ↑f x

Tactic:
rw [← iInf_Ioi_eq f x]

Goal state after:
f : StieltjesFunction ℝ
x : ℝ
⊢ ⨅ r, ↑f ↑↑r = ⨅ r, ↑f ↑r
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case hf
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)
this :
  ∑ x ∈ t.powerset with x.Nonempty, ∫ (a : X) in ⋂ i ∈ x, s i, (-1) ^ (#x + 1) • f a ∂μ =
    ∑ x ∈ t.powerset with x.Nonempty, ∫ (a : X), (⋂ i ∈ x, s i).indicator (fun a ↦ (-1) ^ (#x + 1) • f a) a ∂μ
u : Finset ι
hu : u ∈ {x ∈ t.powerset | x.Nonempty}
⊢ Integrable ((⋂ i ∈ u, s i).indicator fun a ↦ (-1) ^ (#u + 1) • f a) μ

Tactic:
rw [integrable_indicator_iff (A u hu)]

Goal state after:
case hf
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)
this :
  ∑ x ∈ t.powerset with x.Nonempty, ∫ (a : X) in ⋂ i ∈ x, s i, (-1) ^ (#x + 1) • f a ∂μ =
    ∑ x ∈ t.powerset with x.Nonempty, ∫ (a : X), (⋂ i ∈ x, s i).indicator (fun a ↦ (-1) ^ (#x + 1) • f a) a ∂μ
u : Finset ι
hu : u ∈ {x ∈ t.powerset | x.Nonempty}
⊢ IntegrableOn (fun a ↦ (-1) ^ (#u + 1) • f a) (⋂ i ∈ u, s i) 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
⊢ 0 < ↑δ

Tactic:
NNReal.coe_pos,

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.Measure.Content

Goal state before:
pp: G : Type w
inst✝ : TopologicalSpace G
⊢ Function.Injective fun μ s ↦ ↑(μ.toFun s)

Tactic:
rintro ⟨μ, _, _⟩ ⟨v, _, _⟩ h

Goal state after:
G : Type w
inst✝ : TopologicalSpace G
μ : Compacts G → ℝ≥0
mono'✝¹ : ∀ (K₁ K₂ : Compacts G), ↑K₁ ⊆ ↑K₂ → μ K₁ ≤ μ K₂
sup_disjoint'✝¹ : ∀ (K₁ K₂ : Compacts G), Disjoint ↑K₁ ↑K₂ → IsClosed ↑K₁ → IsClosed ↑K₂ → μ (K₁ ⊔ K₂) = μ K₁ + μ K₂
sup_le'✝¹ : ∀ (K₁ K₂ : Compacts G), μ (K₁ ⊔ K₂) ≤ μ K₁ + μ K₂
v : Compacts G → ℝ≥0
mono'✝ : ∀ (K₁ K₂ : Compacts G), ↑K₁ ⊆ ↑K₂ → v K₁ ≤ v K₂
sup_disjoint'✝ : ∀ (K₁ K₂ : Compacts G), Disjoint ↑K₁ ↑K₂ → IsClosed ↑K₁ → IsClosed ↑K₂ → v (K₁ ⊔ K₂) = v K₁ + v K₂
sup_le'✝ : ∀ (K₁ K₂ : Compacts G), v (K₁ ⊔ K₂) ≤ v K₁ + v K₂
h :
  (fun μ s ↦ ↑(μ.toFun s)) { toFun := μ, mono' := mono'✝¹, sup_disjoint' := sup_disjoint'✝¹, sup_le' := sup_le'✝¹ } =
    (fun μ s ↦ ↑(μ.toFun s)) { toFun := v, mono' := mono'✝, sup_disjoint' := sup_disjoint'✝, sup_le' := sup_le'✝ }
⊢ { toFun := μ, mono' := mono'✝¹, sup_disjoint' := sup_disjoint'✝¹, sup_le' := sup_le'✝¹ } =
    { toFun := v, mono' := mono'✝, sup_disjoint' := sup_disjoint'✝, sup_le' := sup_le'✝ }
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' : R
ha' : ↑f a' - ↑f a < ↑δ
aa' : a < a'
this : Nonempty R
i : ℕ
p q' : R
spq : s i \ botSet ⊆ Ioc p q'
hq' : ofReal (↑f q' - ↑f p) < f.length (s i) + ↑(ε' i)
x : R
hx : x ∈ Icc a' b
h'x : x ∈ s i
⊢ x ∈ Ioc p q'

Tactic:
apply spq

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 := ⋯
δpos : 0 < ↑δ
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑δ
a' : R
ha' : ↑f a' - ↑f a < ↑δ
aa' : a < a'
this : Nonempty R
i : ℕ
p q' : R
spq : s i \ botSet ⊆ Ioc p q'
hq' : ofReal (↑f q' - ↑f p) < f.length (s i) + ↑(ε' i)
x : R
hx : x ∈ Icc a' b
h'x : x ∈ s i
⊢ x ∈ s i \ botSet
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inl.inl
R : Type u_1
inst✝¹ : LinearOrder R
inst✝ : TopologicalSpace R
f : StieltjesFunction R
c : R
t : Set R
this : Nonempty R
a b : R
h : t \ botSet ⊆ Ioc a b
hac : a ≤ c
hbc : b ≤ c
⊢ f.length (Ioc a b ∩ Ioi c) + f.length (Ioc a b \ Ioi c) ≤ ofReal (↑f b - ↑f a)

Tactic:
simp only [Ioc_inter_Ioi, f.length_Ioc, hac, hbc, le_refl, Ioc_eq_empty,
      max_eq_right, min_eq_left, Ioc_sdiff_Ioi, f.length_empty, zero_add, not_lt]

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
h : ∀ᵐ (x : X) ∂μ, x ∉ s → f x = 0
⊢ ∫ (x : X) in s, f x ∂μ = ∫ (x : X), f x ∂μ

Tactic:
symm
  nth_rw 1 [← setIntegral_univ]
  apply setIntegral_eq_of_subset_of_ae_sdiff_eq_zero nullMeasurableSet_univ (subset_univ _)
  filter_upwards [h] with x hx h'x using hx h'x.2

Goal state after:
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
K : Compacts G
V : Set G
hV : (interior V).Nonempty
⊢ index (↑K) V ≤ index ↑K ↑K₀ * index (↑K₀) V

Tactic:
classical
  obtain ⟨s, h1s, h2s⟩ := index_elim K.isCompact K₀.interior_nonempty
  obtain ⟨t, h1t, h2t⟩ := index_elim K₀.isCompact hV
  rw [← h2s, ← h2t, mul_comm]
  refine le_trans ?_ Finset.card_mul_le
  apply Nat.sInf_le; refine ⟨_, ?_, rfl⟩; rw [mem_ofPred_eq]; refine Subset.trans h1s ?_
  apply iUnion₂_subset; intro g₁ hg₁; rw [preimage_subset_iff]; intro g₂ hg₂
  have := h1t hg₂
  rcases this with ⟨_, ⟨g₃, rfl⟩, A, ⟨hg₃, rfl⟩, h2V⟩; rw [mem_preimage, ← mul_assoc] at h2V
  exact mem_biUnion (Finset.mul_mem_mul hg₃ hg₁) h2V

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

Goal state before:
pp: case hA
G : Type u_1
α : Type u_3
inst✝⁵ : Group G
inst✝⁴ : MulAction G α
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : Countable G
s t : Set α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : MeasurableConstSMul G α
fund_dom_s : IsFundamentalDomain G s μ
fund_dom_t : IsFundamentalDomain G t μ
U : Set (Quotient α_mod_G)
meas_U : MeasurableSet U
g : G
x : α
⊢ x ∈ (fun x ↦ g • x) ⁻¹' Quotient.mk α_mod_G ⁻¹' U ↔ x ∈ Quotient.mk α_mod_G ⁻¹' U

Tactic:
have : Quotient.mk α_mod_G (g • x) = Quotient.mk α_mod_G x := by
      apply Quotient.sound
      use g

Goal state after:
case hA
G : Type u_1
α : Type u_3
inst✝⁵ : Group G
inst✝⁴ : MulAction G α
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : Countable G
s t : Set α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : MeasurableConstSMul G α
fund_dom_s : IsFundamentalDomain G s μ
fund_dom_t : IsFundamentalDomain G t μ
U : Set (Quotient α_mod_G)
meas_U : MeasurableSet U
g : G
x : α
this : ⟦g • x⟧ = ⟦x⟧
⊢ x ∈ (fun x ↦ g • x) ⁻¹' Quotient.mk α_mod_G ⁻¹' U ↔ x ∈ Quotient.mk α_mod_G ⁻¹' U
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
R : ℝ
f : X → ℝ
hR : 0 ≤ R
hfint : IntegrableOn f {x | R < f x} μ
hμ : μ {x | R < f x} ≠ 0
this : IntegrableOn (fun x ↦ R) {x | R < f x} μ
⊢ 0 < μ ((support fun a ↦ f a - R) ∩ {x | R < f x})

pp: case hf
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
R : ℝ
f : X → ℝ
hR : 0 ≤ R
hfint : IntegrableOn f {x | R < f x} μ
hμ : μ {x | R < f x} ≠ 0
this : IntegrableOn (fun x ↦ R) {x | R < f x} μ
⊢ 0 ≤ᵐ[μ.restrict {x | R < f x}] fun a ↦ f a - R

pp: case hfi
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
R : ℝ
f : X → ℝ
hR : 0 ≤ R
hfint : IntegrableOn f {x | R < f x} μ
hμ : μ {x | R < f x} ≠ 0
this : IntegrableOn (fun x ↦ R) {x | R < f x} μ
⊢ IntegrableOn (fun a ↦ f a - R) {x | R < f x} μ

Tactic:
· rw [← pos_iff_ne_zero] at hμ
    rwa [Set.inter_eq_self_of_subset_right]
    exact fun x hx => Ne.symm (ne_of_lt <| sub_pos.2 hx)

Goal state after:
case hf
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
R : ℝ
f : X → ℝ
hR : 0 ≤ R
hfint : IntegrableOn f {x | R < f x} μ
hμ : μ {x | R < f x} ≠ 0
this : IntegrableOn (fun x ↦ R) {x | R < f x} μ
⊢ 0 ≤ᵐ[μ.restrict {x | R < f x}] fun a ↦ f a - R

case hfi
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
R : ℝ
f : X → ℝ
hR : 0 ≤ R
hfint : IntegrableOn f {x | R < f x} μ
hμ : μ {x | R < f x} ≠ 0
this : IntegrableOn (fun x ↦ R) {x | R < f x} μ
⊢ IntegrableOn (fun a ↦ f a - R) {x | R < f x} μ