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

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
ι : Type u_6
f : α → G
hfi : AEStronglyMeasurable f μ
F : ι → α → G
l : Filter ι
hFi : ∀ᶠ (i : ι) in l, Integrable (F i) μ
hF : Tendsto (fun i ↦ eLpNorm (F i - f) 1 μ) l (𝓝 0)
s : Set α
⊢ Tendsto (fun i ↦ ∫⁻ (x : α), ‖F i x - f x‖ₑ ∂μ) l (𝓝 0)

Tactic:
eLpNorm_one_eq_lintegral_enorm,

Goal state after:
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
ι : Type u_6
f : α → G
hfi : AEStronglyMeasurable f μ
F : ι → α → G
l : Filter ι
hFi : ∀ᶠ (i : ι) in l, Integrable (F i) μ
s : Set α
hF : Tendsto (fun i ↦ ∫⁻ (x : α), ‖(F i - f) x‖ₑ ∂μ) l (𝓝 0)
⊢ Tendsto (fun i ↦ ∫⁻ (x : α), ‖F i x - f x‖ₑ ∂μ) l (𝓝 0)
Module: Mathlib.Analysis.Normed.Module.RieszLemma

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
F : Subspace 𝕜 E
hFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F
hF : ∃ x, x ∉ F
⊢ ∃ x₀, ‖x₀‖ ≤ R ∧ ∀ y ∈ F, 1 ≤ ‖x₀ - y‖

Tactic:
have Rpos : 0 < R := (norm_nonneg _).trans_lt hR

Goal state after:
𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
F : Subspace 𝕜 E
hFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F
hF : ∃ x, x ∉ F
Rpos : 0 < R
⊢ ∃ x₀, ‖x₀‖ ≤ R ∧ ∀ y ∈ F, 1 ≤ ‖x₀ - y‖
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: case refine_2
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
⊢ ∫⁻ (a : α), F' a ∂μ ≠ ∞

Tactic:
exact ((lintegral_ofReal_le_lintegral_enorm _).trans_lt (hF_int.sub (hf_int 0)).2).ne

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

Goal state before:
pp: R : Type u_1
inst✝ : NonUnitalSeminormedRing R
s t : Set R
hs : ∃ C, ∀ ⦃x : R⦄, x ∈ s → ∀ ⦃y : R⦄, y ∈ s → dist x y ≤ C
ht : ∃ C, ∀ ⦃x : R⦄, x ∈ t → ∀ ⦃y : R⦄, y ∈ t → dist x y ≤ C
Af : ℝ
hAf : s ⊆ Metric.closedBall 0 Af
Ag : ℝ
hAg : t ⊆ Metric.closedBall 0 Ag
z : R
hz : z ∈ s * t
w : R
hw : w ∈ s * t
x₁ : R
hx₁ : x₁ ∈ s
y₁ : R
hy₁ : y₁ ∈ t
z_eq : x₁ * y₁ = z
x₂ : R
hx₂ : x₂ ∈ s
y₂ : R
hy₂ : y₂ ∈ t
w_eq : x₂ * y₂ = w
hAf' : 0 ≤ Af
x y : R
x_in_s : x ∈ s
y_in_t : y ∈ t
⊢ ‖y‖ ≤ Ag

Tactic:
· exact mem_closedBall_zero_iff.mp (hAg y_in_t)

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

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → G
C : ℝ
h : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C
⊢ μ.real univ • C = C * μ.real univ

Tactic:
smul_eq_mul,

Goal state after:
α : Type u_1
G : Type u_5
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → G
C : ℝ
h : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C
⊢ μ.real univ * C = C * μ.real univ
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
m : MeasurableSpace α
μ ν : Measure α
f : α → G
h₁ : Integrable f μ
h₂ : Integrable f ν
⊢ ↑‖∫ (x : α), f x ∂ν‖₊ ≤ ∫⁻ (x : α), ‖f x‖ₑ ∂ν

Tactic:
exact enorm_integral_le_lintegral_enorm _

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

Goal state before:
pp: case pos
α : Type u_1
G : Type u_5
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
β : Type u_6
inst✝ : MeasurableSpace β
φ : α → β
hφ : Measurable φ
f : β → G
hfm : StronglyMeasurable f
hfi : Integrable f (Measure.map φ μ)
⊢ ∀ (s : Set β) (x : G), MeasurableSet s → (weightedSMul (Measure.map φ μ) s) x = (weightedSMul μ (φ ⁻¹' s)) x

Tactic:
intro s x hs

Goal state after:
case pos
α : Type u_1
G : Type u_5
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
β : Type u_6
inst✝ : MeasurableSpace β
φ : α → β
hφ : Measurable φ
f : β → G
hfm : StronglyMeasurable f
hfi : Integrable f (Measure.map φ μ)
s : Set β
x : G
hs : MeasurableSet s
⊢ (weightedSMul (Measure.map φ μ) s) x = (weightedSMul μ (φ ⁻¹' s)) x
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: case refine_1.inl
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 : α
⊢ dist ((extend f g₁ h₁) (f x)) ((extend f g₂ h₂) (f x)) ≤
    max (dist g₁ g₂) (dist (h₁.domRestrict (range ⇑f)ᶜ) (h₂.domRestrict (range ⇑f)ᶜ))

Tactic:
simp only [extend_apply]

Goal state after:
case refine_1.inl
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 : α
⊢ dist (g₁ x) (g₂ x) ≤ max (dist g₁ g₂) (dist (h₁.domRestrict (range ⇑f)ᶜ) (h₂.domRestrict (range ⇑f)ᶜ))
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
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 n ↦ eLpNorm (↑↑(F_lp n) - ↑↑f_lp) 1 μ) l (𝓝 0)

Tactic:
eLpNorm_one_eq_lintegral_enorm,

Goal state after:
α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
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 n ↦ ∫⁻ (x : α), ‖(↑↑(F_lp n) - ↑↑f_lp) x‖ₑ ∂μ) l (𝓝 0)
Module: Mathlib.Topology.ContinuousMap.Bounded.Normed

Goal state before:
pp: α : Type u
inst✝¹ : TopologicalSpace α
R : Type u_1
inst✝ : SeminormedRing R
f : α →ᵇ R
n : ℕ
⊢ ⇑(npowRec n f) * ⇑f = ⇑f ^ n * ⇑f

Tactic:
coe_npowRec f n

Goal state after:
α : Type u
inst✝¹ : TopologicalSpace α
R : Type u_1
inst✝ : SeminormedRing R
f : α →ᵇ R
n : ℕ
⊢ ⇑f ^ n * ⇑f = ⇑f ^ n * ⇑f
Module: Mathlib.Topology.ContinuousMap.Bounded.Normed

Goal state before:
pp: α : Type u
γ : Type w
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : TopologicalSpace α
inst✝¹ : NormedRing γ
inst✝ : NormedAlgebra 𝕜 γ
k : 𝕜
a : α
⊢ ((algebraMap 𝕜 (α →ᵇ γ)) k) a = k • 1

Tactic:
simp only [Algebra.algebraMap_eq_smul_one, coe_smul, coe_one, Pi.one_apply]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : SeminormedAddCommGroup E
x y : C(α, E)
⊢ dist (mkOfCompact x) (mkOfCompact y) = ‖mkOfCompact (-x + y)‖

Tactic:
dist_eq_norm_neg_add,

Goal state after:
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : SeminormedAddCommGroup E
x y : C(α, E)
⊢ ‖-mkOfCompact x + mkOfCompact y‖ = ‖mkOfCompact (-x + y)‖
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
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ ↑r
hr : r = 0
this : f =ᵐ[μ] 0
⊢ eLpNorm f 1 μ ≤ 2 * μ univ * ↑r

Tactic:
rw [eLpNorm_congr_ae this, eLpNorm_zero, hr, ENNReal.coe_zero, mul_zero]

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

Goal state before:
pp: δ : Type u_1
X : δ → Type u_3
inst✝² : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝¹ : DecidableEq δ
s : Finset δ
inst✝ : ∀ (i : δ), SigmaFinite (μ i)
f : ((i : δ) → X i) → ℝ≥0∞
hf : Measurable f
i : δ
hi : i ∈ s
⊢ ∫⋯∫⁻_s, f ∂μ = ∫⋯∫⁻_s.erase i, fun x ↦ ∫⁻ (xᵢ : X i), f (update x i xᵢ) ∂μ i ∂μ

Tactic:
by
  simpa [insert_erase hi] using lmarginal_insert' _ hf (notMem_erase i s)

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

Goal state before:
pp: case inr
δ : Type u_1
X : δ → Type u_3
inst✝³ : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝² : DecidableEq δ
inst✝¹ : ∀ (i : δ), SigmaFinite (μ i)
inst✝ : Fintype δ
s : Finset δ
f g : ((i : δ) → X i) → ℝ≥0∞
hf : Measurable f
hg : Measurable g
hfg : ∫⋯∫⁻_s, f ∂μ = ∫⋯∫⁻_s, g ∂μ
x : (i : δ) → X i
⊢ ∫⁻ (x : (i : δ) → X i), f x ∂Measure.pi μ = ∫⁻ (x : (i : δ) → X i), g x ∂Measure.pi μ

Tactic:
lintegral_eq_lmarginal_univ x,

Goal state after:
case inr
δ : Type u_1
X : δ → Type u_3
inst✝³ : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝² : DecidableEq δ
inst✝¹ : ∀ (i : δ), SigmaFinite (μ i)
inst✝ : Fintype δ
s : Finset δ
f g : ((i : δ) → X i) → ℝ≥0∞
hf : Measurable f
hg : Measurable g
hfg : ∫⋯∫⁻_s, f ∂μ = ∫⋯∫⁻_s, g ∂μ
x : (i : δ) → X i
⊢ (∫⋯∫⁻_Finset.univ, f ∂μ) x = (∫⋯∫⁻_Finset.univ, g ∂μ) x
Module: Mathlib.MeasureTheory.Integral.Marginal

Goal state before:
pp: case inr
δ : Type u_1
X : δ → Type u_3
inst✝³ : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝² : DecidableEq δ
inst✝¹ : ∀ (i : δ), SigmaFinite (μ i)
inst✝ : Fintype δ
s : Finset δ
f g : ((i : δ) → X i) → ℝ≥0∞
hf : Measurable f
hg : Measurable g
hfg : ∫⋯∫⁻_s, f ∂μ ≤ ∫⋯∫⁻_s, g ∂μ
x : (i : δ) → X i
⊢ ∫⁻ (x : (i : δ) → X i), f x ∂Measure.pi μ ≤ ∫⁻ (x : (i : δ) → X i), g x ∂Measure.pi μ

Tactic:
lintegral_eq_lmarginal_univ x,

Goal state after:
case inr
δ : Type u_1
X : δ → Type u_3
inst✝³ : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝² : DecidableEq δ
inst✝¹ : ∀ (i : δ), SigmaFinite (μ i)
inst✝ : Fintype δ
s : Finset δ
f g : ((i : δ) → X i) → ℝ≥0∞
hf : Measurable f
hg : Measurable g
hfg : ∫⋯∫⁻_s, f ∂μ ≤ ∫⋯∫⁻_s, g ∂μ
x : (i : δ) → X i
⊢ (∫⋯∫⁻_Finset.univ, f ∂μ) x ≤ (∫⋯∫⁻_Finset.univ, g ∂μ) x
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁴ : LinearOrder R
inst✝³ : TopologicalSpace R
inst✝² : OrderTopology R
inst✝¹ : DenselyOrdered R
inst✝ : NoMaxOrder R
f : StieltjesFunction R
x : R
this : rightLim (↑f) x = ⨅ r, ↑f ↑r
⊢ ⨅ r, ↑f ↑r = ↑f x

Tactic:
← this,

Goal state after:
R : Type u_1
inst✝⁴ : LinearOrder R
inst✝³ : TopologicalSpace R
inst✝² : OrderTopology R
inst✝¹ : DenselyOrdered R
inst✝ : NoMaxOrder R
f : StieltjesFunction R
x : R
this : rightLim (↑f) x = ⨅ r, ↑f ↑r
⊢ rightLim (↑f) x = ↑f x
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
hs : NullMeasurableSet s μ
hfi : Integrable f μ
this : ∫ (x : X) in s ∪ sᶜ, f x ∂μ = ∫ (x : X) in s, f x ∂μ + ∫ (x : X) in sᶜ, f x ∂μ
⊢ ∫ (x : X) in s, f x ∂μ + ∫ (x : X) in sᶜ, f x ∂μ = ∫ (x : X), f x ∂μ

Tactic:
← this,

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
hs : NullMeasurableSet s μ
hfi : Integrable f μ
this : ∫ (x : X) in s ∪ sᶜ, f x ∂μ = ∫ (x : X) in s, f x ∂μ + ∫ (x : X) in sᶜ, f x ∂μ
⊢ ∫ (x : X) in s ∪ sᶜ, f x ∂μ = ∫ (x : X), f x ∂μ
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
H : Type u_2
α : Type u_3
β : Type u_4
inst✝⁵ : Group G
inst✝⁴ : Group H
inst✝³ : MulAction G α
inst✝² : MeasurableSpace α
inst✝¹ : MulAction H β
inst✝ : MeasurableSpace β
s : Set α
μ : Measure α
ν : Measure β
h : IsFundamentalDomain G s μ
f : β → α
hf : QuasiMeasurePreserving f ν μ
e : G → H
he : Bijective e
hef : ∀ (g : G), Semiconj f (fun x ↦ e g • x) fun x ↦ g • x
x : β
x✝ : ∃ g, g • f x ∈ s
g : G
hg : g • f x ∈ s
⊢ e g • x ∈ f ⁻¹' s

Tactic:
by rwa [mem_preimage, hef g x]

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

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝⁵ : Group G
inst✝⁴ : MulAction G α
inst✝³ : MeasurableSpace α
s t : Set α
μ : Measure α
inst✝² : MeasurableConstSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
hs : IsFundamentalDomain G s μ
ht : IsFundamentalDomain G t μ
f : α → ℝ≥0∞
hf : ∀ (g : G) (x : α), f (g • x) = f x
⊢ ∑' (g : G), ∫⁻ (x : α) in s ∩ g • t, f x ∂μ = ∑' (g : G), ∫⁻ (x : α) in g • t ∩ s, f (g⁻¹ • x) ∂μ

Tactic:
simp only [hf, inter_comm]

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
⊢ ∀ᵐ (x : X) ∂μ, x ∈ univ \ s → f x = 0

Tactic:
filter_upwards [h] with x hx h'x using hx h'x.2

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

Goal state before:
pp: G : Type u_1
α : Type u_3
E : Type u_5
inst✝⁷ : Group G
inst✝⁶ : MulAction G α
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
s t : Set α
μ : Measure α
inst✝³ : MeasurableConstSMul G α
inst✝² : SMulInvariantMeasure G α μ
inst✝¹ : Countable G
inst✝ : NormedSpace ℝ E
hs : IsFundamentalDomain G s μ
ht : IsFundamentalDomain G t μ
f : α → E
hf : ∀ (g : G) (x : α), f (g • x) = f x
hfs : IntegrableOn f s μ
hft : IntegrableOn f t μ
⊢ ∑' (g : G), ∫ (x : α) in s ∩ g • t, f x ∂μ = ∑' (g : G), ∫ (x : α) in g • t ∩ s, f (g⁻¹ • x) ∂μ

Tactic:
simp only [hf, inter_comm]

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
hf : Tendsto (↑f) atBot atBot
x : R
this : Nonempty R
val✝ : OrderBot R
⊢ f.measure (Iio x) = ∞

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
hf : Tendsto (↑f) atBot atBot
x : R
this : Nonempty R
val✝ : NoBotOrder R
⊢ f.measure (Iio x) = ∞

Tactic:
· rw [atBot_eq_pure_of_isBot isBot_bot] at hf
    simpa using (tendsto_pure_left.1 hf) _ (Iio_mem_atBot (f ⊥))

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
hf : Tendsto (↑f) atBot atBot
x : R
this : Nonempty R
val✝ : NoBotOrder R
⊢ f.measure (Iio x) = ∞
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
g : X → ℝ
hf : ∀ x ∈ s, 0 ≤ f x
h : ∀ x ∈ s, f x ≤ g x
hg : IntegrableOn g s μ
h'f : ¬AEStronglyMeasurable f (μ.restrict s)
⊢ NullMeasurableSet {x | 0 x ≤ g x} (μ.restrict s)

Tactic:
exact nullMeasurableSet_le aemeasurable_const hg.aemeasurable

Goal state after:
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
B : ∀ (i : ι), ∫⁻ (a : X) in s i, ↑‖f a‖₊ ∂μ = ENNReal.ofReal (∫ (a : X) in s i, ↑‖f a‖₊ ∂μ)
S' : Summable fun i ↦ NNReal.mk (∫ (x : X) in s i, ↑‖f x‖₊ ∂μ) ⋯
S'' :
  ∑' (a : ι), ↑(NNReal.mk (∫ (x : X) in s a, ↑‖f x‖₊ ∂μ) ⋯) = ↑(∑' (b : ι), NNReal.mk (∫ (x : X) in s b, ↑‖f x‖₊ ∂μ) ⋯)
⊢ ∑' (b : ι), ENNReal.ofReal (∫ (a : X) in s b, ↑‖f a‖₊ ∂μ) < ∞

Tactic:
simp_rw [ENNReal.coe_nnreal_eq, NNReal.coe_mk, coe_nnnorm] at S''

Goal state after:
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
B : ∀ (i : ι), ∫⁻ (a : X) in s i, ↑‖f a‖₊ ∂μ = ENNReal.ofReal (∫ (a : X) in s i, ↑‖f a‖₊ ∂μ)
S' : Summable fun i ↦ NNReal.mk (∫ (x : X) in s i, ↑‖f x‖₊ ∂μ) ⋯
S'' :
  ∑' (a : ι), ENNReal.ofReal (∫ (x : X) in s a, ‖f x‖ ∂μ) =
    ENNReal.ofReal ↑(∑' (b : ι), NNReal.mk (∫ (x : X) in s b, ‖f x‖ ∂μ) ⋯)
⊢ ∑' (b : ι), ENNReal.ofReal (∫ (a : X) in s b, ↑‖f a‖₊ ∂μ) < ∞
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
μ : Measure X
inst✝ : NormedSpace ℝ E
s : Set X
this : Fact (1 ≤ 1)
f : ↥(Lp E 1 μ)
⊢ ∫ (x : X) in s, ↑↑f x ∂μ = (integral (μ.restrict s) ∘ fun f ↦ ↑↑((LpToLpRestrictCLM X E ℝ μ 1 s) f)) f

Tactic:
rw [Function.comp_apply, integral_congr_ae (LpToLpRestrictCLM_coeFn ℝ s f)]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
r : 𝕜
x : F
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
rw [RCLike.conj_mul, ← ofReal_pow, re_ofReal_mul, sqrt_mul, ← ofReal_normSq_eq_inner_self,
      ofReal_re]

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
r : 𝕜
x : F
⊢ √(‖r‖ ^ 2) * √(normSq x) ≤ ‖r‖ * ‖x‖

case hx
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
r : 𝕜
x : F
⊢ 0 ≤ ‖r‖ ^ 2
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
c : 𝕜
x : F
⊢ ‖c • x‖ = ‖c‖ * ‖x‖

Tactic:
let : NormedSpace 𝕜 F := toNormedSpace
    exact _root_.norm_smul c x

Goal state after:
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
cd : Core 𝕜 F
c : 𝕜
x : F
⊢ ‖c • x‖ = ‖c‖ * ‖x‖

Tactic:
let : NormedSpace 𝕜 F := toNormedSpace
    exact _root_.norm_smul c x

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
pp: G : Type u_1
inst✝⁴ : Group G
inst✝³ : TopologicalSpace G
inst✝² : IsTopologicalGroup G
inst✝¹ : MeasurableSpace G
inst✝ : BorelSpace G
K₀ : PositiveCompacts G
this : LocallyCompactSpace G
⊢ (haarMeasure K₀) ↑K₀ = 1

Tactic:
simp only [haarMeasure, coe_smul, Pi.smul_apply, smul_eq_mul]

Goal state after:
G : Type u_1
inst✝⁴ : Group G
inst✝³ : TopologicalSpace G
inst✝² : IsTopologicalGroup G
inst✝¹ : MeasurableSpace G
inst✝ : BorelSpace G
K₀ : PositiveCompacts G
this : LocallyCompactSpace G
⊢ ((haarContent K₀).measure ↑K₀)⁻¹ * (haarContent K₀).measure ↑K₀ = 1
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
hx : x ≠ 0
hy : y ≠ 0
R : ℝ
⊢ √((R ^ 2 / (‖x‖ * ‖y‖)) ^ 2 * ‖x - y‖ ^ 2) = R ^ 2 / (‖x‖ * ‖y‖) * dist x y

Tactic:
rw [sqrt_mul, sqrt_sq, sqrt_sq, dist_eq_norm] <;> positivity

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

Goal state before:
pp: case mp
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
h₀ : x ≠ 0
h₀' : ¬↑‖x‖ = 0
h : ⟪x, y⟫ = ↑‖x‖ * ↑‖y‖
this : x = 0 ∨ y = (⟪x, y⟫ / ⟪x, x⟫) • x
⊢ (↑‖(↑‖x‖ * ↑‖y‖ / ⟪x, x⟫) • x‖ / ↑‖x‖) • x = (↑‖x‖ * ↑‖y‖ / ⟪x, x⟫) • x

Tactic:
simp [norm_smul, mul_div_cancel_right₀ _ h₀']

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
h : Orthonormal 𝕜 v
i : ι
this : ↑‖v i‖₊ = 1
⊢ ‖v i‖₊ = 1

Tactic:
norm_cast at this

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝ : RCLike 𝕜
x y z : 𝕜
⊢ y * (starRingEnd 𝕜) (z • x) = (starRingEnd 𝕜) z * (y * (starRingEnd 𝕜) x)

Tactic:
smul_eq_mul,

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝ : RCLike 𝕜
x y z : 𝕜
⊢ y * (starRingEnd 𝕜) (z * x) = (starRingEnd 𝕜) z * (y * (starRingEnd 𝕜) x)
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
hv : Orthonormal 𝕜 v
i : ι
⊢ v i ≠ 0

Tactic:
refine ne_of_apply_ne norm ?_

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
hv : Orthonormal 𝕜 v
i : ι
⊢ ‖v i‖ ≠ ‖0‖
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
hv : Orthonormal 𝕜 v
i : ι
⊢ v i ≠ 0

Tactic:
by
  refine ne_of_apply_ne norm ?_
  rw [hv.1 i, norm_zero]
  simp

Goal state after:
Module: Mathlib.Topology.Baire.CompleteMetrizable

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : IsCompletelyPseudoMetrizableSpace X
x✝ : UpgradedIsCompletelyPseudoMetrizableSpace X := upgradeIsCompletelyPseudoMetrizable X
f : ℕ → Set X
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n ↦ 1 / 2 ^ n
⊢ Dense (⋂ n, f n)

Tactic:
have Bpos : ∀ n, 0 < B n := fun n ↦
    ENNReal.div_pos one_ne_zero <| by finiteness

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : IsCompletelyPseudoMetrizableSpace X
x✝ : UpgradedIsCompletelyPseudoMetrizableSpace X := upgradeIsCompletelyPseudoMetrizable X
f : ℕ → Set X
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n ↦ 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
⊢ Dense (⋂ n, f n)