mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case refine_2
α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝⁸ : NormedAddCommGroup G''
inst✝⁷ : PartialOrder G''
inst✝⁶ : IsOrderedAddMonoid G''
inst✝⁵ : NormedSpace ℝ G''
inst✝⁴ : CompleteSpace G''
inst✝³ : NormedAddCommGroup G'
inst✝² : PartialOrder G'
inst✝¹ : NormedSpace ℝ G'
inst✝ : ClosedIciTopology G''
T : Set α → G' →L[ℝ] G''
C : ℝ
hT : DominatedFinMeasAdditive μ T C
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f : ↥(Lp G' 1 μ)
hf : 0 ≤ f
g✝ : { g // 0 ≤ g }
g : { g // 0 ≤ g }
this : ↑(coeSimpleFuncNonnegToLpNonneg 1 μ G' g) = ↑↑g
⊢ 0 ≤ (setToL1SCLM α G' μ hT) ↑g

Tactic:
exact setToL1S_nonneg (fun s => hT.eq_zero_of_measure_zero) hT.1 hT_nonneg g.2

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

Goal state before:
pp: ι : Type u_1
X : Type u_2
E : Type u_3
inst✝¹ : Countable ι
mX : MeasurableSpace X
inst✝ : NormedAddCommGroup E
μ : ι → Measure X
f : X → E
hf : ∀ (i : ι), Integrable f (μ i)
h : Summable fun i ↦ ∫ (x : X), ‖f x‖ ∂μ i
i : ι
⊢ ∫⁻ (a : X), ‖f a‖ₑ ∂μ i = ∫⁻ (x : X), ENNReal.ofReal ‖f x‖ ∂μ i

Tactic:
ofReal_norm

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

Goal state before:
pp: case pos
α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
hF : CompleteSpace F
⊢ (L1.setToL1 hT) (Integrable.toL1 0 ⋯) = 0

Tactic:
Integrable.toL1_zero,

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 : ℝ
hT : DominatedFinMeasAdditive μ T C
hF : CompleteSpace F
⊢ (L1.setToL1 hT) 0 = 0
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
H : Type u_6
inst✝ : NormedAddCommGroup H
f : α → H
p : ℝ≥0∞
hp1 : p ≠ 0
hp2 : p ≠ ∞
hf : MemLp f p μ
⊢ ∫⁻ (a : α), ENNReal.ofReal ‖f a‖ ^ p.toReal ∂μ = ∫⁻ (a : α), ‖f a‖ₑ ^ p.toReal ∂μ

Tactic:
ofReal_norm

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

Goal state before:
pp: F✝ : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetricSpace β
inst✝¹ : PseudoMetricSpace γ
f✝ g : α →ᵇ β
x : α
C : ℝ
inst✝ : CompleteSpace β
f : ℕ → α →ᵇ β
hf : CauchySeq f
b : ℕ → ℝ
b0 : ∀ (n : ℕ), 0 ≤ b n
b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N
b_lim : Tendsto b atTop (𝓝 0)
f_bdd : ∀ (x : α) (n m N : ℕ), N ≤ n → N ≤ m → dist ((f n) x) ((f m) x) ≤ b N
fx_cau : ∀ (x : α), CauchySeq fun n ↦ (f n) x
F : α → β
hF : ∀ (x : α), Tendsto (fun n ↦ (f n) x) atTop (𝓝 (F x))
fF_bdd : ∀ (x : α) (N : ℕ), dist ((f N) x) (F x) ≤ b N
⊢ ∃ a, Tendsto f atTop (𝓝 a)

Tactic:
refine ⟨⟨⟨F, ?_⟩, ?_⟩, ?_⟩

Goal state after:
case refine_1
F✝ : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetricSpace β
inst✝¹ : PseudoMetricSpace γ
f✝ g : α →ᵇ β
x : α
C : ℝ
inst✝ : CompleteSpace β
f : ℕ → α →ᵇ β
hf : CauchySeq f
b : ℕ → ℝ
b0 : ∀ (n : ℕ), 0 ≤ b n
b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N
b_lim : Tendsto b atTop (𝓝 0)
f_bdd : ∀ (x : α) (n m N : ℕ), N ≤ n → N ≤ m → dist ((f n) x) ((f m) x) ≤ b N
fx_cau : ∀ (x : α), CauchySeq fun n ↦ (f n) x
F : α → β
hF : ∀ (x : α), Tendsto (fun n ↦ (f n) x) atTop (𝓝 (F x))
fF_bdd : ∀ (x : α) (N : ℕ), dist ((f N) x) (F x) ≤ b N
⊢ Continuous[inst✝³, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] F

case refine_2
F✝ : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetricSpace β
inst✝¹ : PseudoMetricSpace γ
f✝ g : α →ᵇ β
x : α
C : ℝ
inst✝ : CompleteSpace β
f : ℕ → α →ᵇ β
hf : CauchySeq f
b : ℕ → ℝ
b0 : ∀ (n : ℕ), 0 ≤ b n
b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N
b_lim : Tendsto b atTop (𝓝 0)
f_bdd : ∀ (x : α) (n m N : ℕ), N ≤ n → N ≤ m → dist ((f n) x) ((f m) x) ≤ b N
fx_cau : ∀ (x : α), CauchySeq fun n ↦ (f n) x
F : α → β
hF : ∀ (x : α), Tendsto (fun n ↦ (f n) x) atTop (𝓝 (F x))
fF_bdd : ∀ (x : α) (N : ℕ), dist ((f N) x) (F x) ≤ b N
⊢ ∃ C,
    ∀ (x y : α),
      dist ({ toFun := F, continuous_toFun := ?refine_1 }.toFun x)
          ({ toFun := F, continuous_toFun := ?refine_1 }.toFun y) ≤
        C

case refine_3
F✝ : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetricSpace β
inst✝¹ : PseudoMetricSpace γ
f✝ g : α →ᵇ β
x : α
C : ℝ
inst✝ : CompleteSpace β
f : ℕ → α →ᵇ β
hf : CauchySeq f
b : ℕ → ℝ
b0 : ∀ (n : ℕ), 0 ≤ b n
b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N
b_lim : Tendsto b atTop (𝓝 0)
f_bdd : ∀ (x : α) (n m N : ℕ), N ≤ n → N ≤ m → dist ((f n) x) ((f m) x) ≤ b N
fx_cau : ∀ (x : α), CauchySeq fun n ↦ (f n) x
F : α → β
hF : ∀ (x : α), Tendsto (fun n ↦ (f n) x) atTop (𝓝 (F x))
fF_bdd : ∀ (x : α) (N : ℕ), dist ((f N) x) (F x) ≤ b N
⊢ Tendsto f atTop (𝓝 { toFun := F, continuous_toFun := ?refine_1, map_bounded' := ?refine_2 })
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_L1 : Tendsto F_lp l (𝓝 f_lp)
this : Tendsto (fun i ↦ setToFun μ T hT ↑↑(F_lp i)) l (𝓝 (setToFun μ T hT f))
i : ι
hi : Integrable (fs i) μ
⊢ setToFun μ T hT ↑↑(F_lp i) = setToFun μ T hT (fs i)

Tactic:
suffices h_ae_eq : F_lp i =ᵐ[μ] fs i from setToFun_congr_ae hT h_ae_eq

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_L1 : Tendsto F_lp l (𝓝 f_lp)
this : Tendsto (fun i ↦ setToFun μ T hT ↑↑(F_lp i)) l (𝓝 (setToFun μ T hT f))
i : ι
hi : Integrable (fs i) μ
⊢ ↑↑(F_lp i) =ᵐ[μ] fs i
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: F : Type u_1
α : Type u
β : Type v
γ : Type w
𝕜 : Type u_2
inst✝⁶ : PseudoMetricSpace 𝕜
inst✝⁵ : TopologicalSpace α
inst✝⁴ : PseudoMetricSpace β
inst✝³ : Zero 𝕜
inst✝² : Zero β
inst✝¹ : SMul 𝕜 β
inst✝ : IsBoundedSMul 𝕜 β
c : 𝕜
f₁ f₂ : α →ᵇ β
⊢ dist (c • f₁) (c • f₂) ≤ dist c 0 * dist f₁ f₂

Tactic:
dist_le (mul_nonneg dist_nonneg dist_nonneg)

Goal state after:
F : Type u_1
α : Type u
β : Type v
γ : Type w
𝕜 : Type u_2
inst✝⁶ : PseudoMetricSpace 𝕜
inst✝⁵ : TopologicalSpace α
inst✝⁴ : PseudoMetricSpace β
inst✝³ : Zero 𝕜
inst✝² : Zero β
inst✝¹ : SMul 𝕜 β
inst✝ : IsBoundedSMul 𝕜 β
c : 𝕜
f₁ f₂ : α →ᵇ β
⊢ ∀ (x : α), dist ((c • f₁) x) ((c • f₂) x) ≤ dist c 0 * dist f₁ f₂
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
f g : α →ᵇ R
h : f * g = 0
⊢ ‖f + g‖ = max ‖f‖ ‖g‖

Tactic:
by
  have hfg : ∀ x, f x = 0 ∨ g x = 0 := by simpa [DFunLike.ext_iff, mul_eq_zero] using h
  have hfg' x : ‖(f + g) x‖ = max ‖f x‖ ‖g x‖ := by obtain (h | h) := hfg x <;> simp [h]
  have key (c : ℝ) (hc : 0 ≤ c) : ‖f + g‖ ≤ c ↔ max ‖f‖ ‖g‖ ≤ c := by
    simp_rw [norm_le hc, hfg', max_le_iff, norm_le hc, forall_and]
  exact le_antisymm (by rw [key]; positivity) (by rw [← key]; positivity)

Goal state after:
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 < q

Tactic:
exact hpq.symm.pos

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

Goal state before:
pp: α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedAddCommGroup β
inst✝² : Lattice β
inst✝¹ : HasSolidNorm β
inst✝ : IsOrderedAddMonoid β
f g : α →ᵇ β
C₁ : ℝ
hf : ∀ (x y : α), dist (f x) (f y) ≤ C₁
C₂ : ℝ
hg : ∀ (x y : α), dist (g x) (g y) ≤ C₂
x y : α
⊢ dist ((⇑f ⊔ ⇑g) x) ((⇑f ⊔ ⇑g) y) ≤ C₁ + C₂

Tactic:
simp_rw [dist_eq_norm_sub] at hf hg ⊢

Goal state after:
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedAddCommGroup β
inst✝² : Lattice β
inst✝¹ : HasSolidNorm β
inst✝ : IsOrderedAddMonoid β
f g : α →ᵇ β
C₁ C₂ : ℝ
x y : α
hf : ∀ (x y : α), ‖f x - f y‖ ≤ C₁
hg : ∀ (x y : α), ‖g x - g y‖ ≤ C₂
⊢ ‖(⇑f ⊔ ⇑g) x - (⇑f ⊔ ⇑g) y‖ ≤ C₁ + C₂
Module: Mathlib.Topology.ContinuousMap.Bounded.Normed

Goal state before:
pp: α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedAddCommGroup β
inst✝² : Lattice β
inst✝¹ : HasSolidNorm β
inst✝ : IsOrderedAddMonoid β
f g : α →ᵇ β
C₁ : ℝ
hf : ∀ (x y : α), dist (f x) (f y) ≤ C₁
C₂ : ℝ
hg : ∀ (x y : α), dist (g x) (g y) ≤ C₂
x y : α
⊢ dist ((⇑f ⊓ ⇑g) x) ((⇑f ⊓ ⇑g) y) ≤ C₁ + C₂

Tactic:
simp_rw [dist_eq_norm_sub] at hf hg ⊢

Goal state after:
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedAddCommGroup β
inst✝² : Lattice β
inst✝¹ : HasSolidNorm β
inst✝ : IsOrderedAddMonoid β
f g : α →ᵇ β
C₁ C₂ : ℝ
x y : α
hf : ∀ (x y : α), ‖f x - f y‖ ≤ C₁
hg : ∀ (x y : α), ‖g x - g y‖ ≤ C₂
⊢ ‖(⇑f ⊓ ⇑g) x - (⇑f ⊓ ⇑g) y‖ ≤ C₁ + C₂
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: case inr
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : Nontrivial E
x : E
K : Set E
hK : IsCompact K
hx : x ∈ K
hx' : x ∈ frontier K
⊢ ∃ y ∈ frontier K, infDist x Kᶜ = dist x y

Tactic:
refine ⟨x, hx', ?_⟩
    rw [frontier_eq_closure_inter_closure] at hx'
    rw [Metric.infDist_zero_of_mem_closure hx'.2, dist_self]

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

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
m : MeasurableSpace α
hE : CompleteSpace E
μ : Measure α
f : α → E
hf : StronglyMeasurable f
a : α
⊢ ∫ (a : α) in {a}, f a ∂μ = μ.real {a} • f a

Tactic:
simp only [Measure.restrict_singleton, integral_smul_measure, integral_dirac' f a hf,
    measureReal_def]

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

Goal state before:

Tactic:
1 - infEDist x E * (↑δ.toNNReal)⁻¹ - (1 - infEDist y E * (↑δ.toNNReal)⁻¹)
    _ ≤ infEDist y E * (↑δ.toNNReal)⁻¹ - infEDist x E * (↑δ.toNNReal)⁻¹ := by
      rw [ENNReal.sub_sub_sub_cancel_left (by finiteness) h_le]
    _ ≤ (↑δ.toNNReal)⁻¹ * edist x y := by
      rw [← ENNReal.sub_mul (by simp [δ_pos]), mul_comm, edist_comm]
      gcongr
      simp only [tsub_le_iff_right]
      exact infEDist_le_edist_add_infEDist

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
x : (i : δ) → X i
⊢ (∫⋯∫⁻_insert i s, f ∂μ) x = ∫⁻ (xᵢ : X i), (∫⋯∫⁻_s, f ∂μ) (update x i xᵢ) ∂μ i

Tactic:
rw [Finset.insert_eq, lmarginal_union μ f hf (Finset.disjoint_singleton_left.mpr hi),
    lmarginal_singleton]

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

Goal state before:
pp: case hμ
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s t : Set X
μ : Measure X
ht : NullMeasurableSet t μ
hfs : IntegrableOn f s μ
⊢ Integrable f (μ.restrict (s ∩ t))

pp: case hν
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s t : Set X
μ : Measure X
ht : NullMeasurableSet t μ
hfs : IntegrableOn f s μ
⊢ Integrable f (μ.restrict (s \ t))

Tactic:
· exact Integrable.mono_measure hfs (Measure.restrict_mono inter_subset_left le_rfl)

Goal state after:
case hν
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s t : Set X
μ : Measure X
ht : NullMeasurableSet t μ
hfs : IntegrableOn f s μ
⊢ Integrable f (μ.restrict (s \ t))
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case insert
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
s : ι → Set X
a✝ : ι
s✝ : Finset ι
hat : a✝ ∉ s✝
IH :
  (∀ i ∈ s✝, MeasurableSet (s i)) →
    (↑s✝).Pairwise (Disjoint on s) →
      (∀ i ∈ s✝, IntegrableOn f (s i) μ) → ∫ (x : X) in ⋃ i ∈ s✝, s i, f x ∂μ = ∑ i ∈ s✝, ∫ (x : X) in s i, f x ∂μ
hs : MeasurableSet (s a✝) ∧ ∀ x ∈ s✝, MeasurableSet (s x)
hf : IntegrableOn f (s a✝) μ ∧ ∀ x ∈ s✝, IntegrableOn f (s x) μ
h's : (↑s✝).Pairwise (Disjoint on s) ∧ ∀ b ∈ ↑s✝, a✝ ≠ b → (Disjoint on s) a✝ b ∧ (Disjoint on s) b a✝
⊢ ∫ (x : X) in s a✝, f x ∂μ + ∫ (x : X) in ⋃ i ∈ s✝, s i, f x ∂μ =
    ∫ (x : X) in s a✝, f x ∂μ + ∑ x ∈ s✝, ∫ (x : X) in s x, f x ∂μ

Tactic:
IH hs.2 h's.1 hf.2

Goal state after:
case insert
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
s : ι → Set X
a✝ : ι
s✝ : Finset ι
hat : a✝ ∉ s✝
IH :
  (∀ i ∈ s✝, MeasurableSet (s i)) →
    (↑s✝).Pairwise (Disjoint on s) →
      (∀ i ∈ s✝, IntegrableOn f (s i) μ) → ∫ (x : X) in ⋃ i ∈ s✝, s i, f x ∂μ = ∑ i ∈ s✝, ∫ (x : X) in s i, f x ∂μ
hs : MeasurableSet (s a✝) ∧ ∀ x ∈ s✝, MeasurableSet (s x)
hf : IntegrableOn f (s a✝) μ ∧ ∀ x ∈ s✝, IntegrableOn f (s x) μ
h's : (↑s✝).Pairwise (Disjoint on s) ∧ ∀ b ∈ ↑s✝, a✝ ≠ b → (Disjoint on s) a✝ b ∧ (Disjoint on s) b a✝
⊢ ∫ (x : X) in s a✝, f x ∂μ + ∑ i ∈ s✝, ∫ (x : X) in s i, f x ∂μ =
    ∫ (x : X) in s a✝, f x ∂μ + ∑ x ∈ s✝, ∫ (x : X) in s x, f x ∂μ
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝ : TopologicalSpace G
μ : Content G
f : G ≃ₜ G
h : ∀ ⦃K : Compacts G⦄, μ (Compacts.map ⇑f ⋯ K) = μ K
U : Opens G
K : Compacts G
hK : ↑K ⊆ ↑((Opens.comap ↑f) U)
⊢ μ ({ toFun := Compacts.map ⇑f ⋯, invFun := Compacts.map ⇑f.symm ⋯, left_inv := ⋯, right_inv := ⋯ } K) = μ K

Tactic:
apply h

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

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G α
inst✝² : MeasurableSpace α
s : Set α
μ : Measure α
inst✝¹ : Countable G
inst✝ : SMulInvariantMeasure G α μ
hμ : μ ≠ 0
h : IsFundamentalDomain G s μ
⊢ μ s ≠ 0

Tactic:
have hc := measure_univ_pos.mpr hμ
  contrapose! hc
  rw [← measure_congr h.iUnion_smul_ae_eq]
  refine le_trans (measure_iUnion_le _) ?_
  simp_rw [measure_smul, hc, tsum_zero, le_refl]

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
K V : Set G
hK : IsCompact K
hV : (interior V).Nonempty
⊢ ∃ n, n ∈ Finset.card '' {t | K ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V}

Tactic:
by
  rcases compact_covered_by_mul_left_translates hK hV with ⟨t, ht⟩; exact ⟨t.card, t, ht, rfl⟩

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

Goal state before:
pp: case neg
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
hx : ∀ (x : R), ¬IsBot x
⊢ f.measure botSet = 0

Tactic:
· simp [botSet, 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
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) atTop atTop
x : R
this : Nonempty R
⊢ f.measure (Ioi x) = ∞

Tactic:
refine ENNReal.eq_top_of_forall_nnreal_le fun r ↦ ?_

Goal state after:
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) atTop atTop
x : R
this : Nonempty R
r : ℝ≥0
⊢ ↑r ≤ f.measure (Ioi x)
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: 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
⊢ f.measure (Iic x) = ∞

Tactic:
refine ENNReal.eq_top_of_forall_nnreal_le fun r ↦ ?_

Goal state after:
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
r : ℝ≥0
⊢ ↑r ≤ f.measure (Iic x)
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 : F
⊢ x = 0 → normSq x = 0

Tactic:
by
  rintro rfl
  simp [normSq, inner_self_of_eq_zero]

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

Goal state before:
pp: 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
x : R
⊢ f.measure.FiniteAtFilter (𝓝 x)

Tactic:
obtain ⟨b, c, -, h, -⟩ : ∃ b c, x ∈ Icc b c ∧ Icc b c ∈ 𝓝 x ∧ Icc b c ⊆ univ :=
    exists_Icc_mem_subset_of_mem_nhds (by simp)

Goal state after:
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
x b c : R
h : Icc b c ∈ 𝓝 x
⊢ f.measure.FiniteAtFilter (𝓝 x)
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case pos.hf
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
H : Integrable f μ
g : X → ℝ := fun x ↦ max (f x) 0
g_int : Integrable g μ
this : ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ ENNReal.ofReal (∫ (x : X), g x ∂μ)
x : X
⊢ ENNReal.ofReal (g x) ≤ 1

Tactic:
apply ENNReal.ofReal_le_of_le_toReal

Goal state after:
case pos.hf
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
H : Integrable f μ
g : X → ℝ := ⋯
g_int : Integrable g μ
this : ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ ENNReal.ofReal (∫ (x : X), g x ∂μ)
x : X
⊢ g x ≤ ENNReal.toReal 1
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)
⊢ Summable fun a ↦ ↑(NNReal.mk (∫ (x : X) in s a, ↑‖f x‖₊ ∂μ) ⋯)

Tactic:
exact h

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

Goal state before:
pp: F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
⊢ (innerₗ F).flip = innerₗ F

Tactic:
ext v w
  exact real_inner_comm v w

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

Goal state before:
pp: F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
⊢ ⟪x, y⟫_ℝ * ⟪x, y⟫_ℝ ≤ ‖⟪x, y⟫_ℝ‖ * ‖⟪y, x⟫_ℝ‖

Tactic:
by
      rw [real_inner_comm y, ← norm_mul]
      exact le_abs_self _

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
h : ∀ (v : E), ⟪x, v⟫ = ⟪y, v⟫
⊢ x = y

Tactic:
by
  rw [← sub_eq_zero, ← @inner_self_eq_zero 𝕜, inner_sub_left, sub_eq_zero, h (x - y)]

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

Goal state before:
pp: Y : Type u_2
E : Type u_3
X : Type u_5
inst✝⁹ : TopologicalSpace X
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : MeasurableSpace Y
inst✝⁶ : OpensMeasurableSpace Y
μ : Measure Y
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FirstCountableTopology X
inst✝² : LocallyCompactSpace X
inst✝¹ : SecondCountableTopologyEither Y E
inst✝ : IsLocallyFiniteMeasure μ
f : X → Y → E
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (uncurry f)
s : Set Y
hs : IsCompact s
x₀ : X
U : Set X
U_cpct : IsCompact U
U_nhds : U ∈ 𝓝 x₀
M : ℝ
hM : M ∈ upperBounds ((fun x ↦ ‖uncurry f x‖) '' U ×ˢ s)
x : X
x_in : x ∈ U
⊢ MeasurableSet {x_1 | ‖f x x_1‖ ≤ M}

Tactic:
· exact (isClosed_le (by fun_prop) (by fun_prop)).measurableSet

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
⊢ (haarMeasure K₀).IsMulLeftInvariant

Tactic:
rw [← forall_measure_preimage_mul_iff]
  intro g A hA
  rw [haarMeasure_apply hA, haarMeasure_apply (measurable_const_mul g hA)]
  -- Porting note: Here was `congr 1`, but `to_additive` failed to generate a theorem.
  refine congr_arg (fun x : ℝ≥0∞ => x / (haarContent K₀).measure K₀) ?_
  apply Content.is_mul_left_invariant_outerMeasure
  apply is_left_invariant_haarContent

Goal state after: