mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Separation.CompletelyRegular

Goal state before:
pp: case a
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompletelyRegularSpace X
x : X
U : Set X
hxU : x ∈ U
hU : IsOpen[inst✝¹] U
f : X → ↑I
hf : Continuous[inst✝¹, _] f
efx : f x = 0
hfU : f ⁻¹' {1}ᶜ ⊆ U
⊢ ∃ i, stoneCechUnit x ∈ i ∧ IsOpen i ∧ stoneCechUnit ⁻¹' i ⊆ U

Tactic:
← stoneCechExtend_extends hf,

Goal state after:
case a
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompletelyRegularSpace X
x : X
U : Set X
hxU : x ∈ U
hU : IsOpen[inst✝¹] U
f : X → ↑I
hf : Continuous[inst✝¹, _] f
efx : f x = 0
hfU : stoneCechExtend hf ∘ stoneCechUnit ⁻¹' {1}ᶜ ⊆ U
⊢ ∃ i, stoneCechUnit x ∈ i ∧ IsOpen i ∧ stoneCechUnit ⁻¹' i ⊆ U
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: case calc_5
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
Λ : (X →C_c ℝ) →ₚ[ℝ] ℝ
inst✝ : LocallyCompactSpace X
f : X →C_c ℝ
μ : Measure X := rieszMeasure Λ
K : Set X := tsupport ⇑f
ε : ℝ
hε : 0 < ε
a b : ℝ
hab : a < b ∧ range ⇑f ⊆ Ioo a b
N : ℕ
hN : 0 < N
ε' : ℝ := (b - a) / ↑N
hε' : 0 < ε' ∧ ε' * (2 * μ.real K + |a| + b + ε') ≤ ε
E : Fin N → Set X
hE :
  tsupport ⇑f = ⋃ j, E j ∧
    univ.PairwiseDisjoint E ∧
      (∀ (n : Fin N), ∀ x ∈ E n, a + ε' * ↑↑n < f x ∧ f x ≤ a + ε' * (↑↑n + 1)) ∧ ∀ (n : Fin N), MeasurableSet (E n)
y : Fin N → ℝ := fun n ↦ a + ε' * (↑↑n + 1)
hE' : ∀ (n : Fin N), μ (E n) ≠ ∞
V : Fin N → Opens X
hV : ∀ (n : Fin N), E n ⊆ ↑(V n) ∧ (∀ x ∈ V n, f x < y n + ε') ∧ μ ↑(V n) ≤ μ (E n) + ENNReal.ofReal (ε' / ↑N)
g : Fin N → X →C_c ℝ
hg :
  (∀ (n : Fin N), tsupport ⇑(g n) ⊆ (V n).carrier) ∧
    EqOn (∑ n, ⇑(g n)) 1 (tsupport f.toFun) ∧
      (∀ (n : Fin N) (x : X), (g n) x ∈ Icc 0 1) ∧ ∀ (n : Fin N), HasCompactSupport ⇑(g n)
⊢ ∑ n, (|a| + y n + ε') * (μ.real (E n) + ε' / ↑N) - |a| * μ.real K =
    ∑ n, (y n - ε') * μ.real (E n) + 2 * ε' * μ.real K + ε' / ↑N * ∑ n, (|a| + y n + ε')

Tactic:
have (n : Fin N) : (|a| + y n + ε') * μ.real (E n) =
        (|a| + 2 * ε') * μ.real (E n) + (y n - ε') * μ.real (E n) := by linarith

Goal state after:
case calc_5
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
Λ : (X →C_c ℝ) →ₚ[ℝ] ℝ
inst✝ : LocallyCompactSpace X
f : X →C_c ℝ
μ : Measure X := rieszMeasure Λ
K : Set X := tsupport ⇑f
ε : ℝ
hε : 0 < ε
a b : ℝ
hab : a < b ∧ range ⇑f ⊆ Ioo a b
N : ℕ
hN : 0 < N
ε' : ℝ := (b - a) / ↑N
hε' : 0 < ε' ∧ ε' * (2 * μ.real K + |a| + b + ε') ≤ ε
E : Fin N → Set X
hE :
  tsupport ⇑f = ⋃ j, E j ∧
    univ.PairwiseDisjoint E ∧
      (∀ (n : Fin N), ∀ x ∈ E n, a + ε' * ↑↑n < f x ∧ f x ≤ a + ε' * (↑↑n + 1)) ∧ ∀ (n : Fin N), MeasurableSet (E n)
y : Fin N → ℝ := fun n ↦ a + ε' * (↑↑n + 1)
hE' : ∀ (n : Fin N), μ (E n) ≠ ∞
V : Fin N → Opens X
hV : ∀ (n : Fin N), E n ⊆ ↑(V n) ∧ (∀ x ∈ V n, f x < y n + ε') ∧ μ ↑(V n) ≤ μ (E n) + ENNReal.ofReal (ε' / ↑N)
g : Fin N → X →C_c ℝ
hg :
  (∀ (n : Fin N), tsupport ⇑(g n) ⊆ (V n).carrier) ∧
    EqOn (∑ n, ⇑(g n)) 1 (tsupport f.toFun) ∧
      (∀ (n : Fin N) (x : X), (g n) x ∈ Icc 0 1) ∧ ∀ (n : Fin N), HasCompactSupport ⇑(g n)
this : ∀ (n : Fin N), (|a| + y n + ε') * μ.real (E n) = (|a| + 2 * ε') * μ.real (E n) + (y n - ε') * μ.real (E n)
⊢ ∑ n, (|a| + y n + ε') * (μ.real (E n) + ε' / ↑N) - |a| * μ.real K =
    ∑ n, (y n - ε') * μ.real (E n) + 2 * ε' * μ.real K + ε' / ↑N * ∑ n, (|a| + y n + ε')
Module: Mathlib.MeasureTheory.Measure.FiniteMeasurePi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝² : Fintype ι
inst✝¹ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → FiniteMeasure (α i)
β : ι → Type u_3
inst✝ : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → α i → β i
f_mble : ∀ (i : ι), AEMeasurable (f i) ↑(μ i)
⊢ ↑((pi μ).map fun x i ↦ f i (x i)) = ↑(pi fun i ↦ (μ i).map (f i))

Tactic:
simp only [val_eq_toMeasure, toMeasure_map, toMeasure_pi]

Goal state after:
ι : Type u_1
α : ι → Type u_2
inst✝² : Fintype ι
inst✝¹ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → FiniteMeasure (α i)
β : ι → Type u_3
inst✝ : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → α i → β i
f_mble : ∀ (i : ι), AEMeasurable (f i) ↑(μ i)
⊢ Measure.map (fun x i ↦ f i (x i)) (Measure.pi fun i ↦ ↑(μ i)) = Measure.pi fun i ↦ Measure.map (f i) ↑(μ i)
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: X : Type u_1
inst✝⁷ : TopologicalSpace X
inst✝⁶ : T2Space X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
μ ν : Measure X
inst✝³ : LocallyCompactSpace X
inst✝² : ν.OuterRegular
inst✝¹ : IsFiniteMeasureOnCompacts ν
inst✝ : IsFiniteMeasureOnCompacts μ
hμν : ∀ (f : X →C_c ℝ), ∫ (x : X), f x ∂μ ≤ ∫ (x : X), f x ∂ν
K : Set X
hK : IsCompact K
ε : NNReal
hε : 0 < ε
hν : ν K < ∞
hνK : ν K ≠ ∞
hμK : μ K ≠ ∞
⊢ μ K ≤ ν K + ↑ε

Tactic:
obtain ⟨V, pV1, pV2, pV3⟩ : ∃ V ⊇ K, IsOpen V ∧ ν V ≤ ν K + ε :=
    exists_isOpen_le_add K ν (ne_of_gt (ENNReal.coe_lt_coe.mpr hε))

Goal state after:
X : Type u_1
inst✝⁷ : TopologicalSpace X
inst✝⁶ : T2Space X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
μ ν : Measure X
inst✝³ : LocallyCompactSpace X
inst✝² : ν.OuterRegular
inst✝¹ : IsFiniteMeasureOnCompacts ν
inst✝ : IsFiniteMeasureOnCompacts μ
hμν : ∀ (f : X →C_c ℝ), ∫ (x : X), f x ∂μ ≤ ∫ (x : X), f x ∂ν
K : Set X
hK : IsCompact K
ε : NNReal
hε : 0 < ε
hν : ν K < ∞
hνK : ν K ≠ ∞
hμK : μ K ≠ ∞
V : Set X
pV1 : V ⊇ K
pV2 : IsOpen[inst✝⁷] V
pV3 : ν V ≤ ν K + ↑ε
⊢ μ K ≤ ν K + ↑ε
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)
ν : LevyProkhorov (ProbabilityMeasure Ω)
hμs : Tendsto μs atTop (𝓝 ν)
P : ProbabilityMeasure Ω := ν.toMeasure
Ps : ℕ → ProbabilityMeasure Ω := toMeasure ∘ μs
f✝ f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
f_zero : ¬‖f‖ = 0
norm_f_pos : 0 < ‖f‖
δ : ℝ
δ_pos : 0 < δ
⊢ IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) atTop fun i ↦ ∫ (x : Ω), f x ∂↑(Ps i)

Tactic:
simp only [IsCoboundedUnder, IsCobounded, eventually_map, eventually_atTop,
               forall_exists_index]

Goal state after:
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)
ν : LevyProkhorov (ProbabilityMeasure Ω)
hμs : Tendsto μs atTop (𝓝 ν)
P : ProbabilityMeasure Ω := ν.toMeasure
Ps : ℕ → ProbabilityMeasure Ω := toMeasure ∘ μs
f✝ f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
f_zero : ¬‖f‖ = 0
norm_f_pos : 0 < ‖f‖
δ : ℝ
δ_pos : 0 < δ
⊢ ∃ b, ∀ (a : ℝ) (x : ℕ), (∀ (b : ℕ), x ≤ b → ∫ (x : Ω), f x ∂↑(Ps b) ≤ a) → b ≤ a
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

Goal state before:

Tactic:
∫ t in -r..r, charFun μ t
  _ = ∫ x in -r..r, ∫ y, cexp (x * y * I) ∂μ := by simp_rw [charFun_apply_real]
  _ = ∫ y, ∫ x in -r..r, cexp (x * y * I) ∂volume ∂μ := by
    rw [intervalIntegral_integral_swap h_int]
  _ = ∫ y, if r * y = 0 then 2 * (r : ℂ)
      else y⁻¹ * ∫ x in -(y * r)..y * r, cexp (x * I) ∂volume ∂μ := by
    congr with y
    by_cases hy : y = 0
    · simp [hy, two_mul]
    simp only [mul_eq_zero, hr.ne', hy, or_self, ↓reduceIte, ofReal_inv]
    have h := intervalIntegral.integral_deriv_smul_comp (E := ℂ) (a := -r) (b := r)
      (f := fun x ↦ y * x) (f' := fun _ ↦ y) (g := fun x ↦ cexp (x * I)) ?_ (by fun_prop)
      (by fun_prop)
    swap
    · intro x hx
      simp_rw [mul_comm y]
      exact hasDerivAt_mul_const _
    simp only [Function.comp_apply, ofReal_mul, real_smul, intervalIntegral.integral_const_mul,
      mul_neg] at h
    rw [← h, ← mul_assoc]
    norm_cast
    simp [mul_comm _ y, mul_inv_cancel₀ hy]
  _ = ∫ x, 2 * (r : ℂ) * sinc (r * x) ∂μ := by
    congr with y
    rw [integral_exp_mul_I_eq_sinc]
    split_ifs with hry
    · simp [hry]
    have hy : y ≠ 0 := fun hy ↦ hry (by simp [hy])
    norm_cast
    field_simp
  _ = 2 * r * ∫ x, sinc (r * x) ∂μ := by
    norm_cast
    rw [← integral_const_mul]

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

Goal state before:
pp: case refine_2
ι : Type u_1
κ : Type u_2
X : ι → Type u_5
Y : κ → Type u_6
mX : (i : ι) → MeasurableSpace (X i)
inst✝⁸ : (i : ι) → TopologicalSpace (X i)
inst✝⁷ : ∀ (i : ι), BorelSpace (X i)
inst✝⁶ : ∀ (i : ι), HasOuterApproxClosed (X i)
mY : (j : κ) → MeasurableSpace (Y j)
inst✝⁵ : (j : κ) → TopologicalSpace (Y j)
inst✝⁴ : ∀ (j : κ), BorelSpace (Y j)
inst✝³ : ∀ (j : κ), HasOuterApproxClosed (Y j)
inst✝² : Fintype ι
inst✝¹ : Fintype κ
μ ν : Measure (((i : ι) → X i) × ((j : κ) → Y j))
inst✝ : IsFiniteMeasure μ
h :
  ∀ (f : (i : ι) → X i →ᵇ ℝ≥0) (g : (j : κ) → Y j →ᵇ ℝ≥0),
    ∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑(∏ i, (f i) (p.1 i)) * ↑(∏ j, (g j) (p.2 j)) ∂μ =
      ∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑(∏ i, (f i) (p.1 i)) * ↑(∏ j, (g j) (p.2 j)) ∂ν
hμν : μ univ = ν univ
this : IsFiniteMeasure ν
π : Set (Set (((i : ι) → X i) × ((j : κ) → Y j))) :=
  image2 (fun s t ↦ s ×ˢ t) (univ.pi '' univ.pi fun x ↦ {s | IsClosed[inst✝⁸ x] s})
    (univ.pi '' univ.pi fun x ↦ {t | IsClosed[inst✝⁵ x] t})
s₁ : (i : ι) → Set (X i)
hs₁ : s₁ ∈ univ.pi fun x ↦ {s | IsClosed[inst✝⁸ x] s}
t₁ : (i : κ) → Set (Y i)
ht₁ : t₁ ∈ univ.pi fun x ↦ {t | IsClosed[inst✝⁵ x] t}
s₂ : (i : ι) → Set (X i)
hs₂ : s₂ ∈ univ.pi fun x ↦ {s | IsClosed[inst✝⁸ x] s}
t₂ : (i : κ) → Set (Y i)
ht₂ : t₂ ∈ univ.pi fun x ↦ {t | IsClosed[inst✝⁵ x] t}
⊢ (fun j ↦ t₁ j ∩ t₂ j) ∈ univ.pi fun x ↦ {t | IsClosed[inst✝⁵ x] t}

Tactic:
simp only [Set.mem_pi, mem_univ, mem_ofPred_eq, forall_const] at ht₁ ht₂ ⊢

Goal state after:
case refine_2
ι : Type u_1
κ : Type u_2
X : ι → Type u_5
Y : κ → Type u_6
mX : (i : ι) → MeasurableSpace (X i)
inst✝⁸ : (i : ι) → TopologicalSpace (X i)
inst✝⁷ : ∀ (i : ι), BorelSpace (X i)
inst✝⁶ : ∀ (i : ι), HasOuterApproxClosed (X i)
mY : (j : κ) → MeasurableSpace (Y j)
inst✝⁵ : (j : κ) → TopologicalSpace (Y j)
inst✝⁴ : ∀ (j : κ), BorelSpace (Y j)
inst✝³ : ∀ (j : κ), HasOuterApproxClosed (Y j)
inst✝² : Fintype ι
inst✝¹ : Fintype κ
μ ν : Measure (((i : ι) → X i) × ((j : κ) → Y j))
inst✝ : IsFiniteMeasure μ
h :
  ∀ (f : (i : ι) → X i →ᵇ ℝ≥0) (g : (j : κ) → Y j →ᵇ ℝ≥0),
    ∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑(∏ i, (f i) (p.1 i)) * ↑(∏ j, (g j) (p.2 j)) ∂μ =
      ∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑(∏ i, (f i) (p.1 i)) * ↑(∏ j, (g j) (p.2 j)) ∂ν
hμν : μ univ = ν univ
this : IsFiniteMeasure ν
π : Set (Set (((i : ι) → X i) × ((j : κ) → Y j))) :=
  image2 (fun s t ↦ s ×ˢ t) (univ.pi '' univ.pi fun x ↦ {s | IsClosed[inst✝⁸ x] s})
    (univ.pi '' univ.pi fun x ↦ {t | IsClosed[inst✝⁵ x] t})
s₁ : (i : ι) → Set (X i)
hs₁ : s₁ ∈ univ.pi fun x ↦ {s | IsClosed[inst✝⁸ x] s}
t₁ : (i : κ) → Set (Y i)
s₂ : (i : ι) → Set (X i)
hs₂ : s₂ ∈ univ.pi fun x ↦ {s | IsClosed[inst✝⁸ x] s}
t₂ : (i : κ) → Set (Y i)
ht₁ : ∀ (i : κ), IsClosed[inst✝⁵ i] (t₁ i)
ht₂ : ∀ (i : κ), IsClosed[inst✝⁵ i] (t₂ i)
⊢ ∀ (i : κ), IsClosed[inst✝⁵ i] (t₁ i ∩ t₂ i)
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: case h₂
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
C : ℝ≥0
K : Set E
hK : IsCompact K
f : ↑K → E := ⋯
hf : IsClosedEmbedding f
rf : range f = K
F : FiniteMeasure ↑K → FiniteMeasure E := ⋯
T : Set (FiniteMeasure ↑K) := ⋯
⊢ F '' T ⊆ {μ | μ.mass ≤ C ∧ μ Kᶜ = 0}

Tactic:
simp only [null_iff_toMeasure_null, image_subset_iff, preimage_ofPred_eq, toMeasure_map,
        ofPred_subset_ofPred, F, T]
      intro μ hμ
      rw [Measure.map_apply hf.continuous.measurable hK.measurableSet.compl]
      refine ⟨(mass_map_le _ _).trans hμ, by simp [f]⟩

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
k : ℕ
hk : finrank ℝ E = 2 * k + 1
x : E
r : ℝ
this : Nontrivial E
⊢ ENNReal.ofReal r ^ (2 * k + 1) * ENNReal.ofReal (π ^ k * (√π / Gamma (↑(2 * k + 1) / 2 + 1))) =
    ENNReal.ofReal r ^ (2 * k + 1) * ENNReal.ofReal (π ^ k * (2 ^ (k + 1) / ↑(2 * k + 1)‼))

Tactic:
congr 3

Goal state after:
case e_a.e_a
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
k : ℕ
hk : finrank ℝ E = 2 * k + 1
x : E
r : ℝ
this : Nontrivial E
⊢ √π / Gamma (↑(2 * k + 1) / 2 + 1) = 2 ^ (k + 1) / ↑(2 * k + 1)‼
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: x : EuclideanSpace ℝ (Fin 3)
r : ℝ
⊢ volume (ball x r) = ENNReal.ofReal r ^ 3 * ENNReal.ofReal (π * 4 / 3)

Tactic:
volume_ball_fin_three x

Goal state after:
x : EuclideanSpace ℝ (Fin 3)
r : ℝ
⊢ ENNReal.ofReal r ^ 3 * ENNReal.ofReal (π * 4 / 3) = ENNReal.ofReal r ^ 3 * ENNReal.ofReal (π * 4 / 3)
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), ‖y‖ = 1 → Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
y : E
hy : ¬y = 0
⊢ ‖y‖⁻¹ * ‖y‖ = 1

Tactic:
inv_mul_cancel₀ (by positivity)

Goal state after:
E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), ‖y‖ = 1 → Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
y : E
hy : ¬y = 0
⊢ 1 = 1
Module: Mathlib.MeasureTheory.Measure.MeasuredSets

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
s t : MeasuredSets μ
⊢ μ.real ↑s - μ.real ↑t ≤ dist s t

Tactic:
dist_edist,

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
s t : MeasuredSets μ
⊢ μ.real ↑s - μ.real ↑t ≤ (edist s t).toReal
Module: Mathlib.MeasureTheory.Measure.PreVariation

Goal state before:
pp: X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
s : Set X
hs : MeasurableSet s
P : Finpartition ⟨s, hs⟩
⊢ ∑ p ∈ P.parts, f ↑p ≤ preVariationFun f s

Tactic:
by
  simpa [preVariationFun, hs, le_iSup_iff] using fun _ a ↦ a P

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

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : MeasurableSpace X
μ : Measure X
inst✝ : HereditarilyLindelofSpace X
s : Set X
hμ : 0 < μ s
⊢ (s ∩ μ.support).Nonempty

Tactic:
by
  rw [← Set.not_disjoint_iff_nonempty_inter]
  contrapose! hμ
  exact μ.mono hμ.subset_compl_right |>.trans <| by simp

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : MeasurableSpace 𝕜
inst✝² : CompleteSpace 𝕜
inst✝¹ : NormedDivisionRing A
inst✝ : NormedAlgebra 𝕜 A
μ : Measure 𝕜
a : A
hz : a ∉ ⇑(algebraMap 𝕜 A) '' μ.support
x : 𝕜
hx : x ∈ μ.support
⊢ ‖resolvent a x‖ ≤ (infDist a (⇑(algebraMap 𝕜 A) '' μ.support))⁻¹

Tactic:
have : 0 < infDist a (algebraMap 𝕜 A '' μ.support) := by
    refine (IsClosed.notMem_iff_infDist_pos ?_ ((Set.nonempty_of_mem hx).image _)).mp hz
    refine (Topology.IsClosedEmbedding.isClosed_iff_image_isClosed ?_).mp isClosed_support
    exact (algebraMap_isometry 𝕜 A).isClosedEmbedding

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : MeasurableSpace 𝕜
inst✝² : CompleteSpace 𝕜
inst✝¹ : NormedDivisionRing A
inst✝ : NormedAlgebra 𝕜 A
μ : Measure 𝕜
a : A
hz : a ∉ ⇑(algebraMap 𝕜 A) '' μ.support
x : 𝕜
hx : x ∈ μ.support
this : 0 < infDist a (⇑(algebraMap 𝕜 A) '' μ.support)
⊢ ‖resolvent a x‖ ≤ (infDist a (⇑(algebraMap 𝕜 A) '' μ.support))⁻¹
Module: Mathlib.MeasureTheory.Measure.ResolventTransform

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : MeasurableSpace 𝕜
inst✝² : CompleteSpace 𝕜
inst✝¹ : NormedDivisionRing A
inst✝ : NormedAlgebra 𝕜 A
μ : Measure 𝕜
a : A
hz : a ∉ ⇑(algebraMap 𝕜 A) '' μ.support
x : 𝕜
hx : x ∈ μ.support
⊢ ‖resolvent a x‖ ≤ (infDist a (⇑(algebraMap 𝕜 A) '' μ.support))⁻¹

Tactic:
have : 0 < infDist a (algebraMap 𝕜 A '' μ.support) := by
    refine (IsClosed.notMem_iff_infDist_pos ?_ ((Set.nonempty_of_mem hx).image _)).mp hz
    refine (Topology.IsClosedEmbedding.isClosed_iff_image_isClosed ?_).mp isClosed_support
    exact (algebraMap_isometry 𝕜 A).isClosedEmbedding
  have : infDist a (algebraMap 𝕜 A '' μ.support) ≤ ‖(algebraMap 𝕜 A) x - a‖ := by
    grw [infDist_le_dist_of_mem (y := (algebraMap 𝕜 A) x), ← dist_eq_norm, dist_comm]
    simp [hx]
  grw [resolvent, Ring.inverse_eq_inv', norm_inv, inv_le_inv₀ (by linarith) (by positivity), this]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : HereditarilyLindelofSpace 𝕜
inst✝⁴ : CompleteSpace 𝕜
inst✝³ : MeasurableSpace 𝕜
inst✝² : BorelSpace 𝕜
inst✝¹ : NormedAlgebra 𝕜 ℂ
μ : Measure 𝕜
inst✝ : IsFiniteMeasure μ
⊢ DifferentiableOn ℂ (resolventTransform μ) (⇑(algebraMap 𝕜 ℂ) '' μ.support)ᶜ

Tactic:
intro z hz

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : HereditarilyLindelofSpace 𝕜
inst✝⁴ : CompleteSpace 𝕜
inst✝³ : MeasurableSpace 𝕜
inst✝² : BorelSpace 𝕜
inst✝¹ : NormedAlgebra 𝕜 ℂ
μ : Measure 𝕜
inst✝ : IsFiniteMeasure μ
z : ℂ
hz : z ∈ (⇑(algebraMap 𝕜 ℂ) '' μ.support)ᶜ
⊢ DifferentiableWithinAt ℂ (resolventTransform μ) (⇑(algebraMap 𝕜 ℂ) '' μ.support)ᶜ z
Module: Mathlib.MeasureTheory.Order.UpperLower

Goal state before:
pp: case h
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior sᶜ
H : Tendsto (fun r ↦ volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
⊢ closure s ∩ closedBall x (ε n) ⊆ closedBall x (ε n) \ closedBall (f (ε n) ⋯) (ε n / 4)

Tactic:
sdiff_eq_compl_inter

Goal state after:
case h
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior sᶜ
H : Tendsto (fun r ↦ volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
⊢ closure s ∩ closedBall x (ε n) ⊆ (closedBall (f (ε n) ⋯) (ε n / 4))ᶜ ∩ closedBall x (ε n)
Module: Mathlib.MeasureTheory.OuterMeasure.OfAddContent

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
m_top : ∀ s ∉ C, m s = ∞
hs : s ∈ C
t : Set α
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_subset : t ⊆ ⋃ i, f i
A : ℕ → Finset (Set α) := fun i ↦ hC.disjointOfDiff ⋯ ⋯
h_diff_eq_sUnion : ∀ (i : ℕ), f i \ s = ⋃₀ ↑(A i)
h_m_eq : ∀ (i : ℕ), m (f i) = m (f i ∩ s) + ∑ u ∈ A i, m u
⊢ (OuterMeasure.ofFunction ⇑m ⋯) (t ∩ s) + (OuterMeasure.ofFunction ⇑m ⋯) (t \ s) ≤ ∑' (i : ℕ), m (f i)

Tactic:
simp_rw [h_m_eq]

Goal state after:
α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
m_top : ∀ s ∉ C, m s = ∞
hs : s ∈ C
t : Set α
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_subset : t ⊆ ⋃ i, f i
A : ℕ → Finset (Set α) := fun i ↦ hC.disjointOfDiff ⋯ ⋯
h_diff_eq_sUnion : ∀ (i : ℕ), f i \ s = ⋃₀ ↑(A i)
h_m_eq : ∀ (i : ℕ), m (f i) = m (f i ∩ s) + ∑ u ∈ A i, m u
⊢ (OuterMeasure.ofFunction ⇑m ⋯) (t ∩ s) + (OuterMeasure.ofFunction ⇑m ⋯) (t \ s) ≤
    ∑' (i : ℕ), (m (f i ∩ s) + ∑ u ∈ A i, m u)
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Defs

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
V : Type u_2
inst✝² : TopologicalSpace V
inst✝¹ : ENormedAddCommMonoid V
inst✝ : T2Space V
μ : VectorMeasure X V
s : ℕ → { t // MeasurableSet t }
hs : Pairwise (Function.onFun Disjoint (Subtype.val ∘ s))
hmeas : ∀ (i : ℕ), MeasurableSet ↑(s i)
⊢ (fun x ↦ ‖μ x‖ₑ) (⋃ i, ↑(s i)) ≤ ∑' (i : ℕ), (fun x ↦ ‖μ x‖ₑ) ↑(s i)

Tactic:
simpa [VectorMeasure.of_disjoint_iUnion hmeas hs] using enorm_tsum_le_tsum_enorm

Goal state after:
Module: Mathlib.MeasureTheory.Order.UpperLower

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
hs : s.OrdConnected
⊢ volume (frontier s) = 0

Tactic:
rw [← hs.upperClosure_inter_lowerClosure]
  exact measure_mono_null (frontier_inter_subset _ _) <| measure_union_null
    (measure_inter_null_of_null_left _ (UpperSet.upper _).null_frontier)
    (measure_inter_null_of_null_right _ (LowerSet.lower _).null_frontier)

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

Goal state before:
pp: X : Type u_1
E : Type u_2
m : MeasurableSpace X
inst✝² : NormedAddCommGroup E
μ : Measure X
p : ℝ≥0∞
one_le_p : Fact (1 ≤ p)
p_ne_top : Fact (p ≠ ∞)
𝒜✝ : Set (Set X)
inst✝¹ : IsSeparable μ
inst✝ : SeparableSpace E
𝒜 : Set (Set X)
count_𝒜 : 𝒜.Countable
h𝒜 : μ.MeasureDense 𝒜
𝒜₀ : Set (Set X) := {s | s ∈ 𝒜 ∧ μ s ≠ ∞}
h𝒜₀ : μ.MeasureDense 𝒜₀
count_𝒜₀ : 𝒜₀.Countable
p_ne_zero : p ≠ 0
u : Set E
countable_u : u.Countable
dense_u : Dense u
key : (n : ℕ) → (Fin n → ↑u) → (Fin n → ↑𝒜₀) → ↥(Lp E p μ) := fun n d s ↦ ∑ i, indicatorConstLp p ⋯ ⋯ ↑(d i)
D : Set ↥(Lp E p μ) := {s | ∃ n d t, s = key n d t}
a : E
s : Set X
ms : MeasurableSet s
hμs✝ : μ s < ∞
hμs : μ s ≠ ∞
ε : ℝ
ε_pos : 0 < ε
μs_pow_nonneg : 0 ≤ μ.real s ^ (1 / p.toReal)
approx_a_pos : 0 < ε / (3 * (1 + μ.real s ^ (1 / p.toReal)))
b : E
b_mem : b ∈ u
hb : ‖a - b‖ < ε / (3 * (1 + μ.real s ^ (1 / p.toReal)))
t : Set X
ht : t ∈ 𝒜₀
hμt : μ t ≠ ∞
hst : ‖indicatorConstLp p ms hμs b - indicatorConstLp p ⋯ hμt b‖ < ε / 3
mt : MeasurableSet t
⊢ ε / (3 * (1 + μ.real s ^ (1 / p.toReal))) * μ.real s ^ (1 / p.toReal) ≤ ε / 3

Tactic:
rw [← mul_one (ε / 3), div_mul_eq_div_mul_one_div, mul_assoc, one_div_mul_eq_div]

Goal state after:
X : Type u_1
E : Type u_2
m : MeasurableSpace X
inst✝² : NormedAddCommGroup E
μ : Measure X
p : ℝ≥0∞
one_le_p : Fact (1 ≤ p)
p_ne_top : Fact (p ≠ ∞)
𝒜✝ : Set (Set X)
inst✝¹ : IsSeparable μ
inst✝ : SeparableSpace E
𝒜 : Set (Set X)
count_𝒜 : 𝒜.Countable
h𝒜 : μ.MeasureDense 𝒜
𝒜₀ : Set (Set X) := {s | s ∈ 𝒜 ∧ μ s ≠ ∞}
h𝒜₀ : μ.MeasureDense 𝒜₀
count_𝒜₀ : 𝒜₀.Countable
p_ne_zero : p ≠ 0
u : Set E
countable_u : u.Countable
dense_u : Dense u
key : (n : ℕ) → (Fin n → ↑u) → (Fin n → ↑𝒜₀) → ↥(Lp E p μ) := fun n d s ↦ ∑ i, indicatorConstLp p ⋯ ⋯ ↑(d i)
D : Set ↥(Lp E p μ) := {s | ∃ n d t, s = key n d t}
a : E
s : Set X
ms : MeasurableSet s
hμs✝ : μ s < ∞
hμs : μ s ≠ ∞
ε : ℝ
ε_pos : 0 < ε
μs_pow_nonneg : 0 ≤ μ.real s ^ (1 / p.toReal)
approx_a_pos : 0 < ε / (3 * (1 + μ.real s ^ (1 / p.toReal)))
b : E
b_mem : b ∈ u
hb : ‖a - b‖ < ε / (3 * (1 + μ.real s ^ (1 / p.toReal)))
t : Set X
ht : t ∈ 𝒜₀
hμt : μ t ≠ ∞
hst : ‖indicatorConstLp p ms hμs b - indicatorConstLp p ⋯ hμt b‖ < ε / 3
mt : MeasurableSet t
⊢ ε / 3 * (μ.real s ^ (1 / p.toReal) / (1 + μ.real s ^ (1 / p.toReal))) ≤ ε / 3 * 1
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

Goal state before:
pp: X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝ : NormedAddCommGroup V
μ ν : VectorMeasure X V
⊢ (μ - ν).variation ≤ μ.variation + ν.variation

Tactic:
grw [sub_eq_add_neg, variation_add_le, variation_neg]

Goal state after: