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

Goal state before:
pp: case neg
α : Type u_1
E : Type u_2
F' : Type u_4
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'
h_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T s = 0
f : α →ₛ E
hf : Integrable (⇑f) μ
x : E
x✝ : x ∈ f.range
hx0 : ¬x = 0
⊢ (T (⇑f ⁻¹' {x})) x = 0

Tactic:
h_zero (f ⁻¹' ({x} : Set E)) (measurableSet_fiber _ _)
      (measure_preimage_lt_top_of_integrable f hf hx0),

Goal state after:
case neg
α : Type u_1
E : Type u_2
F' : Type u_4
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'
h_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T s = 0
f : α →ₛ E
hf : Integrable (⇑f) μ
x : E
x✝ : x ∈ f.range
hx0 : ¬x = 0
⊢ 0 x = 0
Module: Mathlib.Analysis.Normed.Operator.Extend

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
Eₗ : Type u_4
F : Type u_5
Fₗ : Type u_6
inst✝¹³ : NormedField 𝕜
inst✝¹² : NormedField 𝕜₂
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : Module 𝕜 E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module 𝕜₂ F
inst✝⁷ : NormedAddCommGroup Eₗ
inst✝⁶ : NormedSpace 𝕜 Eₗ
inst✝⁵ : CompleteSpace Eₗ
inst✝⁴ : NormedAddCommGroup Fₗ
inst✝³ : NormedSpace 𝕜₂ Fₗ
inst✝² : CompleteSpace Fₗ
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₁ : 𝕜₂ →+* 𝕜
inst✝¹ : RingHomInvPair σ₁₂ σ₂₁
inst✝ : RingHomInvPair σ₂₁ σ₁₂
f : E ≃ₛₗ[σ₁₂] F
e₁ : E →ₗ[𝕜] Eₗ
e₂ : F →ₗ[𝕜₂] Fₗ
h_dense₁ : DenseRange ⇑e₁
h_dense₂ : DenseRange ⇑e₂
h_norm : ∀ (x : E), ‖e₂ (f x)‖ = ‖e₁ x‖
x : F
⊢ ‖e₁ (f.symm x)‖ = ‖e₂ x‖

Tactic:
simpa using (h_norm (f.symm x)).symm

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Extend

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
Eₗ : Type u_4
F : Type u_5
Fₗ : Type u_6
inst✝¹³ : NormedField 𝕜
inst✝¹² : NormedField 𝕜₂
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : Module 𝕜 E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module 𝕜₂ F
inst✝⁷ : NormedAddCommGroup Eₗ
inst✝⁶ : NormedSpace 𝕜 Eₗ
inst✝⁵ : CompleteSpace Eₗ
inst✝⁴ : NormedAddCommGroup Fₗ
inst✝³ : NormedSpace 𝕜₂ Fₗ
inst✝² : CompleteSpace Fₗ
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₁ : 𝕜₂ →+* 𝕜
inst✝¹ : RingHomInvPair σ₁₂ σ₂₁
inst✝ : RingHomInvPair σ₂₁ σ₁₂
f : E ≃ₛₗ[σ₁₂] F
e₁ : E →ₗ[𝕜] Eₗ
e₂ : F →ₗ[𝕜₂] Fₗ
h_dense₁ : DenseRange ⇑e₁
h_dense₂ : DenseRange ⇑e₂
h_norm : ∀ (x : E), ‖e₂ (f x)‖ = ‖e₁ x‖
x✝ x : F
⊢ ‖e₁ (f.symm x)‖ = ‖e₂ x‖

Tactic:
simpa using (h_norm (f.symm x)).symm

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:

Tactic:
(const α y).integral μ = ∑ z ∈ {y}, μ.real (const α y ⁻¹' {z}) • z :=
      integral_eq_sum_of_subset <| (filter_subset _ _).trans (range_const_subset _ _)
    _ = μ.real univ • y := by simp [Set.preimage]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedSpace ℝ E
T : Set α → E →L[ℝ] F
C : ℝ
hT_norm : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ‖T s‖ ≤ C * μ.real s
f : α →ₛ E
hf : Integrable (⇑f) μ
⊢ C * ∑ x ∈ f.range, μ.real (⇑f ⁻¹' {x}) * ‖x‖ = C * ∑ x ∈ f.range, μ.real (⇑f ⁻¹' {x}) • ‖x‖

Tactic:
smul_eq_mul

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝¹ : NormedAddCommGroup E
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedSpace ℝ E
ν : Measure α
f : α →ₛ E
hf : Integrable (⇑f) (μ + ν)
⊢ integral (μ + ν) f = integral μ f + integral ν f

Tactic:
simp_rw [integral_def]
  refine setToSimpleFunc_add_left'
    (weightedSMul μ) (weightedSMul ν) (weightedSMul (μ + ν)) (fun s _ hμνs => ?_) hf
  rw [lt_top_iff_ne_top, Measure.coe_add, Pi.add_apply, ENNReal.add_ne_top] at hμνs
  rw [weightedSMul_add_measure _ _ hμνs.1 hμνs.2]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝ : NormedAddCommGroup E
m : MeasurableSpace α
μ : Measure α
f : ↥(α →₁ₛ[μ] E)
⊢ ∑ x ∈ (toSimpleFunc f).range, μ.real (⇑(toSimpleFunc f) ⁻¹' {x}) * ‖x‖ =
    ∑ x ∈ (toSimpleFunc f).range, μ.real (⇑(toSimpleFunc f) ⁻¹' {x}) • ‖x‖

Tactic:
smul_eq_mul

Goal state after:
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: α : Type u_1
E : Type u_2
𝕜 : Type u_3
inst✝² : NormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
a b : α → 𝕜
u v : α → E
l : Filter α
hab : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : α) in l, ‖(a - b) x‖ ≤ c * ‖b x‖
huv : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : α) in l, ‖(u - v) x‖ ≤ c * ‖v x‖
φ : α → 𝕜
habφ : a =ᶠ[l] φ * b
this : ((fun x ↦ a x • u x) - fun x ↦ b x • v x) =ᶠ[l] fun x ↦ b x • (φ x • u x - v x)
C : ℝ
hC : C > 0
hCuv : ∀ᶠ (x : α) in l, ‖u x‖ ≤ C * ‖v x‖
hφ : ∀ ε > 0, ∀ᶠ (x : α) in l, ‖φ x - 1‖ < ε
c : ℝ
⊢ 0 < c → ∀ᶠ (x : α) in l, ‖φ x • u x - v x‖ ≤ c * ‖v x‖

Tactic:
hc

Goal state after:
α : Type u_1
E : Type u_2
𝕜 : Type u_3
inst✝² : NormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
a b : α → 𝕜
u v : α → E
l : Filter α
hab : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : α) in l, ‖(a - b) x‖ ≤ c * ‖b x‖
huv : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : α) in l, ‖(u - v) x‖ ≤ c * ‖v x‖
φ : α → 𝕜
habφ : a =ᶠ[l] φ * b
this : ((fun x ↦ a x • u x) - fun x ↦ b x • v x) =ᶠ[l] fun x ↦ b x • (φ x • u x - v x)
C : ℝ
hC : C > 0
hCuv : ∀ᶠ (x : α) in l, ‖u x‖ ≤ C * ‖v x‖
hφ : ∀ ε > 0, ∀ᶠ (x : α) in l, ‖φ x - 1‖ < ε
c : ℝ
hc : 0 < c
⊢ ∀ᶠ (x : α) in l, ‖φ x • u x - v x‖ ≤ c * ‖v x‖
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: α : Type u_1
β : Type u_3
inst✝ : NormedField β
u v : α → β
l : Filter α
huv : u ~[l] v
⊢ u⁻¹ ~[l] v⁻¹

Tactic:
rw [isEquivalent_iff_exists_eq_mul] at *
  rcases huv with ⟨φ, hφ, h⟩
  rw [← inv_one]
  refine ⟨fun x ↦ (φ x)⁻¹, Tendsto.inv₀ hφ (by simp), ?_⟩
  convert! h.fun_inv
  simp [mul_comm]

Goal state after:
Module: Mathlib.Analysis.Normed.Module.RCLike.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
r : ℝ
r_pos : 0 < r
c : ℝ
f : E →ₗ[𝕜] 𝕜
h : ∀ z ∈ sphere 0 r, ‖f z‖ ≤ c
z : E
z_zero : ¬z = 0
z₁ : E := (↑r * (↑‖z‖)⁻¹) • z
hz₁ : z₁ = (↑r * (↑‖z‖)⁻¹) • z
norm_f_z₁ : ‖f z₁‖ ≤ c
r_ne_zero : ↑r ≠ 0
⊢ f z = ↑‖z‖ / ↑r * (↑r * (↑‖z‖)⁻¹) • f z

Tactic:
smul_eq_mul

Goal state after:
𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
r : ℝ
r_pos : 0 < r
c : ℝ
f : E →ₗ[𝕜] 𝕜
h : ∀ z ∈ sphere 0 r, ‖f z‖ ≤ c
z : E
z_zero : ¬z = 0
z₁ : E := (↑r * (↑‖z‖)⁻¹) • z
hz₁ : z₁ = (↑r * (↑‖z‖)⁻¹) • z
norm_f_z₁ : ‖f z₁‖ ≤ c
r_ne_zero : ↑r ≠ 0
⊢ f z = ↑‖z‖ / ↑r * (↑r * (↑‖z‖)⁻¹ * f z)
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_6
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝⁵ : NormedRing 𝕜
inst✝⁴ : Module 𝕜 E
inst✝³ : Module 𝕜 F
inst✝² : IsBoundedSMul 𝕜 E
inst✝¹ : IsBoundedSMul 𝕜 F
inst✝ : CompleteSpace F
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c • x) = c • (T s) x
f : ↥(Lp E 1 μ)
⊢ (setToL1 hT) f = (setToL1' 𝕜 hT h_smul) f

Tactic:
by
  have h₁ : Dense (Set.range (coeToLp α E ℝ)) := simpleFunc.denseRange (μ := μ) one_ne_top
  apply Dense.induction (P := fun f : α →₁[μ] E ↦ (setToL1 hT) f = (setToL1' 𝕜 hT h_smul) f) h₁
  · intro f ⟨f', hf⟩
    simp [← hf]
  · exact isClosed_eq (setToL1 hT).continuous (setToL1' 𝕜 hT h_smul).continuous

Goal state after:
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
⊢ ↑(nndist (∫ (x : α), f x ∂μ) (∫ (x : α), f x ∂(μ + ν))) ≤ ∫⁻ (x : α), ‖f x‖ₑ ∂ν

Tactic:
rw [integral_add_measure h₁ h₂, nndist_comm, nndist_eq_nnnorm, add_sub_cancel_left]

Goal state after:
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
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
⊢ ∃ C, ∀ (x y : α), dist ((⇑f ⊔ ⇑g) x) ((⇑f ⊔ ⇑g) y) ≤ C

Tactic:
refine ⟨C₁ + C₂, fun x y ↦ ?_⟩

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 : 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
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
⊢ ∃ C, ∀ (x y : α), dist ((⇑f ⊓ ⇑g) x) ((⇑f ⊓ ⇑g) y) ≤ C

Tactic:
refine ⟨C₁ + C₂, fun x y ↦ ?_⟩

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 : 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
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
hc'0 : ¬c' = 0
f : ↥(Lp G 1 μ)
ε : ℝ
hε_pos : ε > 0
g : ↥(Lp G 1 μ)
hfg : (eLpNorm (↑↑g - ↑↑f) 1 μ).toReal < ε / 2 / c'.toReal
h_int : ∀ (f' : ↥(Lp G 1 μ)), Integrable (↑↑f') μ' :=
  fun f' ↦ Integrable.of_measure_le_smul hc' hμ'_le (L1.integrable_coeFn f')
this : eLpNorm (↑↑(Integrable.toL1 ↑↑g ⋯) - ↑↑(Integrable.toL1 ↑↑f ⋯)) 1 μ' = eLpNorm (↑↑g - ↑↑f) 1 μ'
h_eLpNorm_ne_top : eLpNorm (↑↑g - ↑↑f) 1 μ ≠ ∞
⊢ c' ^ (1 / 1).toReal • eLpNorm (↑↑g - ↑↑f) 1 μ = c' * eLpNorm (↑↑g - ↑↑f) 1 μ

Tactic:
smul_eq_mul

Goal state after:
α : Type u_1
G : Type u_5
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
hc'0 : ¬c' = 0
f : ↥(Lp G 1 μ)
ε : ℝ
hε_pos : ε > 0
g : ↥(Lp G 1 μ)
hfg : (eLpNorm (↑↑g - ↑↑f) 1 μ).toReal < ε / 2 / c'.toReal
h_int : ∀ (f' : ↥(Lp G 1 μ)), Integrable (↑↑f') μ' :=
  fun f' ↦ Integrable.of_measure_le_smul hc' hμ'_le (L1.integrable_coeFn f')
this : eLpNorm (↑↑(Integrable.toL1 ↑↑g ⋯) - ↑↑(Integrable.toL1 ↑↑f ⋯)) 1 μ' = eLpNorm (↑↑g - ↑↑f) 1 μ'
h_eLpNorm_ne_top : eLpNorm (↑↑g - ↑↑f) 1 μ ≠ ∞
⊢ c' ^ (1 / 1).toReal * eLpNorm (↑↑g - ↑↑f) 1 μ = c' * eLpNorm (↑↑g - ↑↑f) 1 μ
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: case pos
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
⊢ f =ᵐ[μ] 0

Tactic:
rw [hr] at hf

Goal state after:
case pos
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 ≤ ↑0
hr : r = 0
⊢ f =ᵐ[μ] 0
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
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ' T C'
f : α → E
hfμ : Integrable f μ
hF : CompleteSpace F
h_int : ∀ (g : α → E), Integrable g μ → Integrable g μ'
c : E
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ c' • μ s < ∞

Tactic:
smul_eq_mul

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
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ' T C'
f : α → E
hfμ : Integrable f μ
hF : CompleteSpace F
h_int : ∀ (g : α → E), Integrable g μ → Integrable g μ'
c : E
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ c' * μ s < ∞
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
hr : r = 0
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ 0
hnegf : ∫ (x : α), -f x ∂μ = 0
this : -f =ᵐ[μ] 0
ω : α
hω : (-f) ω = 0 ω
⊢ f ω = 0 ω

Tactic:
rwa [Pi.neg_apply, Pi.zero_apply, neg_eq_zero] at hω

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.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
h' : (𝓝[<] a).NeBot
b : α
hb : b ∈ Iio a
u : α
au : u < a
hu : Ioo u a ⊆ {x | f x ∈ s}
c : α
⊢ c ∈ Ioo u a → leftLim f c ∈ id s

Tactic:
hc

Goal state after:
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
b : α
hb : b ∈ Iio a
u : α
au : u < a
hu : Ioo u a ⊆ {x | f x ∈ s}
c : α
hc : c ∈ Ioo u a
⊢ leftLim f c ∈ id s
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))
h' : (𝓝[<] a).NeBot
b : α
hb : b ∈ Iio a
s : Set β
s_mem : s ∈ 𝓝 (leftLim f a)
s_closed : IsClosed[inst✝³] s
u : α
au : u < a
hu : Ioo u a ⊆ {x | f x ∈ s}
c : α
⊢ c ∈ Ioo u a → rightLim f c ∈ id s

Tactic:
hc

Goal state after:
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))
h' : (𝓝[<] a).NeBot
b : α
hb : b ∈ Iio a
s : Set β
s_mem : s ∈ 𝓝 (leftLim f a)
s_closed : IsClosed[inst✝³] s
u : α
au : u < a
hu : Ioo u a ⊆ {x | f x ∈ s}
c : α
hc : c ∈ Ioo u a
⊢ rightLim f c ∈ id s
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
⊢ ∀ i ∈ {x ∈ t.powerset | x.Nonempty}, Integrable ((⋂ i_1 ∈ i, s i_1).indicator fun a ↦ (-1) ^ (#i + 1) • f a) μ

pp: 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
⊢ ∫ (x : X), (⋃ i ∈ t, s i).indicator f x ∂μ =
    ∫ (a : X), ∑ i ∈ t.powerset with i.Nonempty, (⋂ i_1 ∈ i, s i_1).indicator (fun a ↦ (-1) ^ (#i + 1) • f a) a ∂μ

Tactic:
· intro u hu
    rw [integrable_indicator_iff (A u hu)]
    apply Integrable.smul
    simp only [Finset.mem_filter, Finset.mem_powerset] at hu
    rcases hu.2 with ⟨i, hi⟩
    exact (hf i (hu.1 hi)).mono (biInter_subset_of_mem hi) le_rfl

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
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
⊢ ∫ (x : X), (⋃ i ∈ t, s i).indicator f x ∂μ =
    ∫ (a : X), ∑ i ∈ t.powerset with i.Nonempty, (⋂ i_1 ∈ i, s i_1).indicator (fun a ↦ (-1) ^ (#i + 1) • f a) a ∂μ
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), NullMeasurableSet (s i) μ
h_mono : Monotone s
hne : atTop.NeBot
this : IsDirectedOrder ι
S : Set X := ⋃ i, s i
hfi : IntegrableOn f S μ
hfi' : (μ.withDensity fun x ↦ ↑‖f x‖₊) S < ∞
hSm : NullMeasurableSet S μ
hsub : ∀ {i : ι}, s i ⊆ S
ν : Measure X := μ.withDensity fun x ↦ ‖f x‖ₑ
hν : ν = μ.withDensity fun x ↦ ‖f x‖ₑ
ε : ℝ≥0
ε0 : 0 < ↑ε
⊢ ∀ᶠ (x : ι) in atTop, ∫ (x : X) in s x, f x ∂μ ∈ Metric.closedBall (∫ (x : X) in S, f x ∂μ) ↑ε

Tactic:
have : ∀ᶠ i in atTop, ν (s i) ∈ Icc (ν S - ε) (ν S + ε) :=
    tendsto_measure_iUnion_atTop h_mono (ENNReal.Icc_mem_nhds hfi'.ne (ENNReal.coe_pos.2 ε0).ne')

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), NullMeasurableSet (s i) μ
h_mono : Monotone s
hne : atTop.NeBot
this✝ : IsDirectedOrder ι
S : Set X := ⋃ i, s i
hfi : IntegrableOn f S μ
hfi' : (μ.withDensity fun x ↦ ↑‖f x‖₊) S < ∞
hSm : NullMeasurableSet S μ
hsub : ∀ {i : ι}, s i ⊆ S
ν : Measure X := μ.withDensity fun x ↦ ‖f x‖ₑ
hν : ν = μ.withDensity fun x ↦ ‖f x‖ₑ
ε : ℝ≥0
ε0 : 0 < ↑ε
this : ∀ᶠ (i : ι) in atTop, ν (s i) ∈ Icc (ν S - ↑ε) (ν S + ↑ε)
⊢ ∀ᶠ (x : ι) in atTop, ∫ (x : X) in s x, f x ∂μ ∈ Metric.closedBall (∫ (x : X) in S, f x ∂μ) ↑ε
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ ∑ i ∈ t, μ (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
⊢ μ K ≤ ∑' (i : ℕ), μ.innerContent (U i)

Tactic:
let L : ℕ → Compacts G := fun n => ⟨K' n, h1K' n⟩

Goal state after:
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ ∑ i ∈ t, μ (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n ↦ { carrier := K' n, isCompact' := ⋯ }
⊢ μ K ≤ ∑' (i : ℕ), μ.innerContent (U i)
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 : Set α
μ : Measure α
inst✝² : MeasurableConstSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
ν : Measure α
h : IsFundamentalDomain G s μ
hν : ν ≪ μ
⊢ (sum fun g ↦ ν.restrict (g • s)) = ν

Tactic:
by
  rw [← restrict_iUnion_ae (h.aedisjoint.mono fun i j h => hν h) fun g =>
      (h.nullMeasurableSet_smul g).mono_ac hν,
    restrict_congr_set (hν h.iUnion_smul_ae_eq), restrict_univ]

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

Goal state before:
pp: G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : MeasurableSpace G
inst✝¹ : R1Space G
inst✝ : BorelSpace G
H : μ.ContentRegular
K : Compacts G
ε : ℝ≥0
εpos : 0 < ε
a✝ : μ K < ∞
K' : Compacts G
K'_hyp : K.carrier ⊆ interior K'.carrier ∧ μ K' ≤ μ K + ↑ε
⊢ μ.outerMeasure (interior ↑K') ≤ μ K'

Tactic:
exact μ.outerMeasure_interior_compacts K'

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

Goal state before:
pp: R : Type u_1
inst✝⁶ : LinearOrder R
inst✝⁵ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁴ : OrderTopology R
inst✝³ : CompactIccSpace R
inst✝² : MeasurableSpace R
inst✝¹ : BorelSpace R
inst✝ : DenselyOrdered R
⊢ f.outer.trim = f.outer

Tactic:
refine le_antisymm (fun s => ?_) (OuterMeasure.le_trim _)
  rw [OuterMeasure.trim_eq_iInf]
  refine le_iInf fun t => le_iInf fun ht => ENNReal.le_of_forall_pos_le_add fun ε ε0 h => ?_
  rcases ENNReal.exists_pos_sum_of_countable (ENNReal.coe_pos.2 ε0).ne' ℕ with ⟨ε', ε'0, hε⟩
  grw [← hε]
  rw [← ENNReal.tsum_add]
  choose g hg using
    show ∀ i, ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ f.outer s ≤ f.length (t i) + ofReal (ε' i) by
      intro i
      rcases isEmpty_or_nonempty R with hR | hR
      · exact ⟨∅, by simp, MeasurableSet.empty, by simp⟩
      have hl :=
        ENNReal.lt_add_right ((ENNReal.le_tsum i).trans_lt h).ne (ENNReal.coe_pos.2 (ε'0 i)).ne'
      conv at hl =>
        lhs
        rw [length_eq]
      simp only [iInf_lt_iff] at hl
      rcases hl with ⟨a, b, h₁, h₂⟩
      rw [← f.outer_Ioc] at h₂
      rw [sdiff_subset_iff] at h₁
      refine ⟨_, h₁, measurableSet_botSet.union measurableSet_Ioc, le_of_lt ?_⟩
      calc f.outer (botSet ∪ Ioc a b)
      _ ≤ f.outer botSet + f.outer (Ioc a b) := measure_union_le _ _
      _ ≤ f.length botSet + f.outer (Ioc a b) := by gcongr; apply outer_le_length
      _ = 0 + f.outer (Ioc a b) := by
        simp only [← length_sdiff_botSet, sdiff_self, empty_sdiff, outer_Ioc, zero_add]
        simp [empty_sdiff]
      _ = f.outer (Ioc a b) := by simp
      _ < f.length (t i) + ofReal ↑(ε' i) := by simpa using h₂
  simp only [ofReal_coe_nnreal] at hg
  apply iInf_le_of_le (iUnion g) _
  apply iInf_le_of_le (ht.trans <| iUnion_mono fun i => (hg i).1) _
  apply iInf_le_of_le (MeasurableSet.iUnion fun i => (hg i).2.1) _
  exact le_trans (measure_iUnion_le _) (ENNReal.tsum_le_tsum fun i => (hg i).2.2)

Goal state after: