mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
μ : Measure Ω'
X : Ω → ℝ
Y : Ω' ≃ᵐ Ω
⊢ Var[X; Measure.map (⇑Y) μ] = Var[X ∘ ⇑Y; μ]

Tactic:
variance,

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
μ : Measure Ω'
X : Ω → ℝ
Y : Ω' ≃ᵐ Ω
⊢ eVar[X; Measure.map (⇑Y) μ].toReal = eVar[X ∘ ⇑Y; μ].toReal
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: case neg
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
X : Ω → ℝ
hm : AEStronglyMeasurable X μ
hX : ¬MemLp X 2 μ
⊢ Var[X; μ] ≤ ∫ (x : Ω), (X ^ 2) x ∂μ

Tactic:
variance,

Goal state after:
case neg
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
X : Ω → ℝ
hm : AEStronglyMeasurable X μ
hX : ¬MemLp X 2 μ
⊢ eVar[X; μ].toReal ≤ ∫ (x : Ω), (X ^ 2) x ∂μ
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
ν : Measure β
f : α → ℝ≥0∞
g : β → ℝ≥0∞
h : IdentDistrib f g μ ν
⊢ ∫⁻ (x : α), f x ∂μ = ∫⁻ (x : β), g x ∂ν

Tactic:
change ∫⁻ x, id (f x) ∂μ = ∫⁻ x, id (g x) ∂ν
  rw [← lintegral_map' aemeasurable_id h.aemeasurable_fst, ←
    lintegral_map' aemeasurable_id h.aemeasurable_snd, h.map_eq]

Goal state after:
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
inst✝² : NormedAddCommGroup γ
inst✝¹ : NormedSpace ℝ γ
inst✝ : BorelSpace γ
h : IdentDistrib f g μ ν
hf : AEStronglyMeasurable f μ
A : AEStronglyMeasurable id (Measure.map f μ)
⊢ ∫ (x : α), id (f x) ∂μ = ∫ (x : β), id (g x) ∂ν

Tactic:
← integral_map h.aemeasurable_fst A

Goal state after:
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
inst✝² : NormedAddCommGroup γ
inst✝¹ : NormedSpace ℝ γ
inst✝ : BorelSpace γ
h : IdentDistrib f g μ ν
hf : AEStronglyMeasurable f μ
A : AEStronglyMeasurable id (Measure.map f μ)
⊢ ∫ (y : γ), id y ∂Measure.map f μ = ∫ (x : β), id (g x) ∂ν
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: case e'_3
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
ν : Measure β
f : α → ℝ
g : β → ℝ
h : IdentDistrib f g μ ν
⊢ eVar[g; ν] = ∫⁻ (x : β), ↑‖g x - ∫ (x : α), f x ∂μ‖₊ ^ 2 ∂ν

Tactic:
rw [h.integral_eq]

Goal state after:
case e'_3
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
ν : Measure β
f : α → ℝ
g : β → ℝ
h : IdentDistrib f g μ ν
⊢ eVar[g; ν] = ∫⁻ (x : β), ↑‖g x - ∫ (x : β), g x ∂ν‖₊ ^ 2 ∂ν
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
ν : Measure β
f : α → ℝ
g : β → ℝ
h : IdentDistrib f g μ ν
⊢ Var[f; μ] = Var[g; ν]

Tactic:
variance,

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
ν : Measure β
f : α → ℝ
g : β → ℝ
h : IdentDistrib f g μ ν
⊢ eVar[f; μ].toReal = Var[g; ν]
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
ν : Measure β
f : α → ℝ
g : β → ℝ
h : IdentDistrib f g μ ν
⊢ eVar[f; μ].toReal = Var[g; ν]

Tactic:
h.evariance_eq

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
ν : Measure β
f : α → ℝ
g : β → ℝ
h : IdentDistrib f g μ ν
⊢ eVar[g; ν].toReal = Var[g; ν]
Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution

Goal state before:
pp: ι : Type u_1
E : Type u_2
Ω' : Type u_3
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁴ : IsProbabilityMeasure μ'
mE : MeasurableSpace E
Z : Ω' → E
l : Filter ι
inst✝³ : PseudoEMetricSpace E
inst✝² : BorelSpace E
inst✝¹ : l.IsCountablyGenerated
inst✝ : l.NeBot
X : ι → Ω' → E
h : TendstoInMeasure μ' X l Z
hX : ∀ (i : ι), AEMeasurable (X i) μ'
hZ : AEMeasurable Z μ'
⊢ TendstoInDistribution X l Z (fun x ↦ μ') μ'

Tactic:
refine ⟨hX, hZ, ?_⟩

Goal state after:
ι : Type u_1
E : Type u_2
Ω' : Type u_3
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁴ : IsProbabilityMeasure μ'
mE : MeasurableSpace E
Z : Ω' → E
l : Filter ι
inst✝³ : PseudoEMetricSpace E
inst✝² : BorelSpace E
inst✝¹ : l.IsCountablyGenerated
inst✝ : l.NeBot
X : ι → Ω' → E
h : TendstoInMeasure μ' X l Z
hX : ∀ (i : ι), AEMeasurable (X i) μ'
hZ : AEMeasurable Z μ'
⊢ Tendsto (fun n ↦ ⟨Measure.map (X n) μ', ⋯⟩) l (𝓝 ⟨Measure.map Z μ', ⋯⟩)
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: α : Type u_1
Ω : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : Type u_8
m : MeasurableSpace β
inst✝¹ : Div β
inst✝ : MeasurableDiv₂ β
f : ι → Ω → β
hf_indep : iIndepFun f κ μ
hf_meas : ∀ (i : ι), AEMeasurable (f i) (⇑κ ∘ₘ μ)
i j k : ι
hik : i ≠ k
hjk : j ≠ k
⊢ IndepFun (f i / f j) (f k) κ μ

Tactic:
have : IndepFun (fun ω => (f i ω, f j ω)) (f k) κ μ :=
    hf_indep.indepFun_prodMk₀ hf_meas i j k hik hjk
  simpa using! this.comp (measurable_fst.div measurable_snd) measurable_id

Goal state after:
Module: Mathlib.MeasureTheory.Function.Floor

Goal state before:
pp: R : Type u_2
inst✝⁶ : Semiring R
inst✝⁵ : LinearOrder R
inst✝⁴ : FloorSemiring R
inst✝³ : TopologicalSpace R
inst✝² : OrderTopology R
inst✝¹ : MeasurableSpace R
inst✝ : OpensMeasurableSpace R
n : R
⊢ MeasurableSet (ceil ⁻¹' {⌈n⌉₊})

Tactic:
by
    rcases eq_or_ne ⌈n⌉₊ 0 with h | h <;> simp_all [Nat.preimage_ceil_of_ne_zero, -ceil_eq_zero]

Goal state after:
Module: Mathlib.MeasureTheory.Function.Intersectivity

Goal state before:
pp: α : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ μ (s n)
M : (α → ℝ) → Set α := fun f ↦ {x | eLpNormEssSup f μ < ↑‖f x‖₊}
N : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)
hN₀ : μ N = 0
hN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)
⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)

Tactic:
let f (n : ℕ) : α → ℝ≥0∞ := (↑(n + 1) : ℝ≥0∞)⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1

Goal state after:
α : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ μ (s n)
M : (α → ℝ) → Set α := fun f ↦ {x | eLpNormEssSup f μ < ↑‖f x‖₊}
N : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)
hN₀ : μ N = 0
hN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)
f : ℕ → α → ℝ≥0∞ := fun n ↦ (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1
⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)
Module: Mathlib.MeasureTheory.Group.ModularCharacter

Goal state before:
pp: G : Type u_1
inst✝³ : TopologicalSpace G
inst✝² : Group G
inst✝¹ : IsTopologicalGroup G
inst✝ : LocallyCompactSpace G
g h : G
this✝¹ : MeasurableSpace G := ⋯
this✝ : BorelSpace G
mul_g_meas : Measurable fun x ↦ x * g
mul_h_meas : Measurable fun x ↦ x * h
ν : Measure G := ⋯
⊢ modularCharacterFun g * modularCharacterFun h = modularCharacterFun (g * h)

Tactic:
calc
    modularCharacterFun g * modularCharacterFun h =
      modularCharacterFun h * modularCharacterFun g := mul_comm _ _
    _ = haarScalarFactor (map (· * h) (map (· * g) ν)) (map (· * g) ν) *
      modularCharacterFun g := by
      rw [modularCharacterFun_eq_haarScalarFactor (map (· * g) ν) _]
    _ = haarScalarFactor (map (· * h) (map (· * g) ν)) (map (· * g) ν) *
      haarScalarFactor (map (· * g) ν) ν := rfl
    _ = haarScalarFactor (map (· * (g * h)) ν) ν := by simp only [map_map mul_h_meas mul_g_meas,
      comp_mul_right, ← haarScalarFactor_eq_mul]

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

Goal state before:
pp: E : Type u_1
inst✝⁸ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : Nontrivial E
inst✝² : μ.IsAddHaarMeasure
L : AddSubgroup E
inst✝¹ : Countable ↥L
inst✝ : DiscreteTopology ↥L
fund : IsAddFundamentalDomain (↥L) F μ
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h_cpt : IsCompact s
h : μ F * 2 ^ finrank ℝ E ≤ μ s
h_mes : μ s ≠ 0
h_nemp : s.Nonempty
u : ℕ → ℝ≥0 := ⋯
K : ConvexBody E := ⋯
S : ℕ → ConvexBody E := ⋯
Z : ℕ → Set E := ⋯
h_zero : 0 ∈ K
⊢ ∃ U, IsOpen U ∧ U ∩ ↑L = {0}

Tactic:
exact isOpen_inter_eq_singleton_of_mem_discrete ⟨inferInstance⟩ (zero_mem L)

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

Goal state before:
pp: α : Type u_2
inst✝ : TopologicalSpace α
f : α →C_c ℝ
a : ℝ
ha : a ≤ 0
x : α
⊢ ↑((a • f).nnrealPart x) = ↑(((-a).toNNReal • (-f).nnrealPart) x)

Tactic:
simp only [nnrealPart_apply, coe_smul, Pi.smul_apply, smul_eq_mul, Real.coe_toNNReal', coe_neg,
    Pi.neg_apply, NNReal.coe_mul]

Goal state after:
α : Type u_2
inst✝ : TopologicalSpace α
f : α →C_c ℝ
a : ℝ
ha : a ≤ 0
x : α
⊢ max (a * f x) 0 = max (-a) 0 * max (-f x) 0
Module: Mathlib.Order.Interval.Set.Union

Goal state before:
pp: X : Type u_1
inst✝ : LinearOrder X
N : ℕ
a : ℕ → X
⊢ Ico (a 0) (a N) ⊆ ⋃ i ∈ Finset.range N, Ico (a i) (a (i + 1))

Tactic:
induction N with
  | zero => simp
  | succ N ih => calc
    _ ⊆ Ico (a 0) (a N) ∪ Ico (a N) (a (N + 1)) := Ico_subset_Ico_union_Ico
    _ ⊆ _ := by simpa [Finset.range_add_one] using
                  union_subset_union_right (Ico (a N) (a (N + 1))) ih

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

Goal state before:
pp: μ : Measure ℝ
t : ℝ
⊢ charFun μ t = ∫ (x : ℝ), cexp (↑t * ↑x * I) ∂μ

Tactic:
by simp [charFun_apply]

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

Goal state before:
pp: E : Type u_2
mE : MeasurableSpace E
t : E
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
⊢ charFun 0 t = 0

Tactic:
by simp [charFun_apply]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: 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 := ⋯
K : Set X := ⋯
ε : ℝ
hε : 0 < ε
a b : ℝ
hab : a < b ∧ range ⇑f ⊆ Ioo a b
N : ℕ
hN : 0 < N
ε' : ℝ := ⋯
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 := ⋯
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 : Fin N
hn : n ∈ Finset.univ
⊢ μ (E n) + ENNReal.ofReal (ε' / ↑N) < ∞

Tactic:
rw [WithTop.add_lt_top]

Goal state after:
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 : Fin N
hn : n ∈ Finset.univ
⊢ μ (E n) < ⊤ ∧ ENNReal.ofReal (ε' / ↑N) < ⊤
Module: Mathlib.MeasureTheory.Measure.DiracProba

Goal state before:
pp: case h
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
U : Set X
U_nhds : U ∈ 𝓝 x
hU : ∃ᶠ (x : X) in L, x ∉ U
Uint_nhds : interior U ∈ 𝓝 x
f : X →ᵇ ℝ≥0
fx_eq_one : f x = 1
f_vanishes_outside : ∀ y ∈ (interior U)ᶜ, f y = 0
⊢ ¬Tendsto (fun i ↦ ↑(f i)) L (𝓝 ↑1)

Tactic:
apply not_tendsto_iff_exists_frequently_notMem.mpr

Goal state after:
case h
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
U : Set X
U_nhds : U ∈ 𝓝 x
hU : ∃ᶠ (x : X) in L, x ∉ U
Uint_nhds : interior U ∈ 𝓝 x
f : X →ᵇ ℝ≥0
fx_eq_one : f x = 1
f_vanishes_outside : ∀ y ∈ (interior U)ᶜ, f y = 0
⊢ ∃ s ∈ 𝓝 ↑1, ∃ᶠ (x : X) in L, ↑(f x) ∉ s
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace ℝ F
mF : MeasurableSpace F
μ : Measure E
ν : Measure F
inst✝⁸ : BorelSpace E
inst✝⁷ : SecondCountableTopology E
inst✝⁶ : BorelSpace F
inst✝⁵ : SecondCountableTopology F
inst✝⁴ : CompleteSpace E
inst✝³ : CompleteSpace F
ξ : Measure (E × F)
inst✝² : IsFiniteMeasure μ
inst✝¹ : IsFiniteMeasure ν
inst✝ : IsFiniteMeasure ξ
h :
  ∀ (L : StrongDual ℝ (E × F)),
    charFunDual ξ L =
      charFunDual μ (L ∘SL ContinuousLinearMap.inl ℝ E F) * charFunDual ν (L ∘SL ContinuousLinearMap.inr ℝ E F)
⊢ ξ = μ.prod ν

Tactic:
refine Measure.ext_of_charFunDual <| funext fun t ↦ ?_
    rw [h, charFunDual_prod]

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

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : PseudoEMetricSpace Ω
μ : Measure Ω
⊢ levyProkhorovEDist μ μ ≤ 0

Tactic:
← csInf_Ioo zero_lt_top

Goal state after:
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : PseudoEMetricSpace Ω
μ : Measure Ω
⊢ levyProkhorovEDist μ μ ≤ sInf (Ioo 0 ∞)
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: case hf
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 := ⋯
K : Set X := ⋯
ε : ℝ
hε : 0 < ε
a b : ℝ
hab : a < b ∧ range ⇑f ⊆ Ioo a b
N : ℕ
hN : 0 < N
ε' : ℝ := ⋯
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 := ⋯
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 : Fin N
⊢ ∀ x ∈ E n, y n - ε' ≤ f x

Tactic:
intro x hx
        dsimp [y]; linarith [(hE.2.2.1 n x hx).1]

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

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝⁵ : Fintype ι
inst✝⁴ : (i : ι) → MeasurableSpace (α i)
inst✝³ : (i : ι) → TopologicalSpace (α i)
inst✝² : ∀ (i : ι), SecondCountableTopology (α i)
inst✝¹ : ∀ (i : ι), PseudoMetrizableSpace (α i)
inst✝ : ∀ (i : ι), OpensMeasurableSpace (α i)
μ : (i : ι) → ProbabilityMeasure (α i)
S : Set (Set ((i : ι) → α i)) :=
  {t | ∃ s, t = univ.pi s ∧ (∀ (i : ι), MeasurableSet (s i)) ∧ ∀ (i : ι), (μ i) (frontier (s i)) = 0}
this✝ : IsPiSystem S
this : (i : ι) → PseudoMetricSpace (α i) := fun i ↦ pseudoMetrizableSpacePseudoMetric (α i)
u : Set ((i : ι) → α i)
u_open : IsOpen[Pi.topologicalSpace] u
x : (i : ι) → α i
xu : x ∈ u
ε : ℝ
εpos : ε > 0
hε : ball x ε ⊆ u
r : ι → ℝ
rpos : ∀ (i : ι), r i ∈ Ioo 0 ε
hr : ∀ (i : ι), ↑(μ i) (frontier (thickening (r i) {x i})) = 0
⊢ (univ.pi fun i ↦ ball (x i) ε) ⊆ u

Tactic:
← ball_pi _ εpos

Goal state after:
ι : Type u_1
α : ι → Type u_2
inst✝⁵ : Fintype ι
inst✝⁴ : (i : ι) → MeasurableSpace (α i)
inst✝³ : (i : ι) → TopologicalSpace (α i)
inst✝² : ∀ (i : ι), SecondCountableTopology (α i)
inst✝¹ : ∀ (i : ι), PseudoMetrizableSpace (α i)
inst✝ : ∀ (i : ι), OpensMeasurableSpace (α i)
μ : (i : ι) → ProbabilityMeasure (α i)
S : Set (Set ((i : ι) → α i)) :=
  {t | ∃ s, t = univ.pi s ∧ (∀ (i : ι), MeasurableSet (s i)) ∧ ∀ (i : ι), (μ i) (frontier (s i)) = 0}
this✝ : IsPiSystem S
this : (i : ι) → PseudoMetricSpace (α i) := fun i ↦ pseudoMetrizableSpacePseudoMetric (α i)
u : Set ((i : ι) → α i)
u_open : IsOpen[Pi.topologicalSpace] u
x : (i : ι) → α i
xu : x ∈ u
ε : ℝ
εpos : ε > 0
hε : ball x ε ⊆ u
r : ι → ℝ
rpos : ∀ (i : ι), r i ∈ Ioo 0 ε
hr : ∀ (i : ι), ↑(μ i) (frontier (thickening (r i) {x i})) = 0
⊢ ball x ε ⊆ u
Module: Mathlib.MeasureTheory.Measure.Haar.Extension

Goal state before:
pp: A : Type u_1
B : Type u_2
C : Type u_3
E : Type u_4
inst✝¹¹ : Group A
inst✝¹⁰ : Group B
inst✝⁹ : Group C
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst✝⁵ : IsTopologicalGroup A
inst✝⁴ : IsTopologicalGroup B
inst✝³ : NormedAddCommGroup E
inst✝² : MeasurableSpace A
inst✝¹ : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst✝ : NormedSpace ℝ E
f : CompactlySupportedContinuousMap B E
b : B
⊢ ψ ((Function.invFun (⇑ψ) (ψ b))⁻¹ * b) = 1

Tactic:
simp [Function.apply_invFun_apply]

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

Goal state before:
pp: case pos
Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n)
Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3
Es_cover : ⋃ n, Es n = univ
Es_disjoint : Pairwise fun n m ↦ Disjoint (Es n) (Es m)
N : ℕ
hN : ↑P (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)
Js_finite : {J | J ⊆ Iio N}.Finite
Gs : Set (Set Ω) := (fun J ↦ thickening (ε / 3) (⋃ j ∈ J, Es j)) '' {J | J ⊆ Iio N}
Gs_open :
  ∀ (J : Set ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (thickening (ε / 3) (⋃ j ∈ J, Es j))
mem_nhds_P :
  ∀ (G : Set Ω),
    IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] G → {Q | ↑P G < ↑Q G + ENNReal.ofReal (ε / 3)} ∈ 𝓝 P
Q : ProbabilityMeasure Ω
hQ :
  ∀ i ⊆ Iio N,
    ↑P (⋃ i_2 ∈ i, thickening (ε / 3) (Es i_2)) < ↑Q (⋃ i_2 ∈ i, thickening (ε / 3) (Es i_2)) + ENNReal.ofReal (ε / 3)
δ : ℝ
B : Set Ω
δ_gt : 2 * (ε / 3) < δ
x✝ : MeasurableSet B
JB : Set ℕ := {i | (B ∩ Es i).Nonempty ∧ i ∈ Iio N}
ω : Ω
ω_in_B : ω ∈ B
i : ℕ
hi : ω ∈ Es i
i_small : i ∈ Iio N
⊢ i ∈ JB

Tactic:
simp only [mem_Iio, mem_ofPred_eq, JB]

Goal state after:
case pos
Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n)
Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3
Es_cover : ⋃ n, Es n = univ
Es_disjoint : Pairwise fun n m ↦ Disjoint (Es n) (Es m)
N : ℕ
hN : ↑P (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)
Js_finite : {J | J ⊆ Iio N}.Finite
Gs : Set (Set Ω) := (fun J ↦ thickening (ε / 3) (⋃ j ∈ J, Es j)) '' {J | J ⊆ Iio N}
Gs_open :
  ∀ (J : Set ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (thickening (ε / 3) (⋃ j ∈ J, Es j))
mem_nhds_P :
  ∀ (G : Set Ω),
    IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] G → {Q | ↑P G < ↑Q G + ENNReal.ofReal (ε / 3)} ∈ 𝓝 P
Q : ProbabilityMeasure Ω
hQ :
  ∀ i ⊆ Iio N,
    ↑P (⋃ i_2 ∈ i, thickening (ε / 3) (Es i_2)) < ↑Q (⋃ i_2 ∈ i, thickening (ε / 3) (Es i_2)) + ENNReal.ofReal (ε / 3)
δ : ℝ
B : Set Ω
δ_gt : 2 * (ε / 3) < δ
x✝ : MeasurableSet B
JB : Set ℕ := {i | (B ∩ Es i).Nonempty ∧ i ∈ Iio N}
ω : Ω
ω_in_B : ω ∈ B
i : ℕ
hi : ω ∈ Es i
i_small : i ∈ Iio N
⊢ (B ∩ Es i).Nonempty ∧ i < N
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: case inl
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : ∀ (x : ι → ℂ), 0 ≤ ∑ i, ‖x i‖ ^ p
h₂ : ∀ (x : ι → ℂ), 0 ≤ (∑ i, ‖x i‖ ^ p) ^ (1 / p)
hr : r ≤ 0
⊢ volume {x | (∑ i, ‖x i‖ ^ p) ^ (1 / p) < r} =
    ENNReal.ofReal r ^ (2 * card ι) *
      ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * ↑(card ι) / p + 1))

pp: case inr
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : ∀ (x : ι → ℂ), 0 ≤ ∑ i, ‖x i‖ ^ p
h₂ : ∀ (x : ι → ℂ), 0 ≤ (∑ i, ‖x i‖ ^ p) ^ (1 / p)
hr : 0 < r
⊢ volume {x | (∑ i, ‖x i‖ ^ p) ^ (1 / p) < r} =
    ENNReal.ofReal r ^ (2 * card ι) *
      ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * ↑(card ι) / p + 1))

Tactic:
· have : {x : ι → ℂ | (∑ i, ‖x i‖ ^ p) ^ (1 / p) < r} = ∅ := by
      ext x
      refine ⟨fun hx => ?_, fun hx => hx.elim⟩
      exact not_le.mpr (lt_of_lt_of_le (Set.mem_ofPred.mp hx) hr) (h₂ x)
    rw [this, measure_empty, ← zero_eq_ofReal.mpr hr, zero_pow Fin.pos'.ne', zero_mul]

Goal state after:
case inr
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : ∀ (x : ι → ℂ), 0 ≤ ∑ i, ‖x i‖ ^ p
h₂ : ∀ (x : ι → ℂ), 0 ≤ (∑ i, ‖x i‖ ^ p) ^ (1 / p)
hr : 0 < r
⊢ volume {x | (∑ i, ‖x i‖ ^ p) ^ (1 / p) < r} =
    ENNReal.ofReal r ^ (2 * card ι) *
      ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * ↑(card ι) / p + 1))
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare

Goal state before:
pp: case hdω
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace E
f : 𝕜 → E
s : Set 𝕜
hs : Convex ℝ s
hf : DifferentiableOn 𝕜 f s
this : NormedSpace ℝ E := ⋯
⊢ ∀ a ∈ s,
    ∀ x ∈ tangentConeAt ℝ s a,
      ∀ y ∈ tangentConeAt ℝ s a,
        ((ContinuousLinearMap.restrictScalars ℝ
                (ContinuousLinearMap.toSpanSingleton 𝕜
                  (((ContinuousLinearMap.smulRightL 𝕜 𝕜 E) 1) (derivWithin f s a))))
              x)
            y =
          ((ContinuousLinearMap.restrictScalars ℝ
                (ContinuousLinearMap.toSpanSingleton 𝕜
                  (((ContinuousLinearMap.smulRightL 𝕜 𝕜 E) 1) (derivWithin f s a))))
              y)
            x

Tactic:
rintro a ha x - y -
    simpa using smul_comm ..

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : IsSetRing C
h'C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0
h : mα = generateFrom C
s✝ : Set α
hs✝ : MeasurableSet s✝
ε✝ : ℝ≥0∞
hε : 0 < ε✝
s : Set α
hs : MeasurableSet s
h's : ∀ (ε : ℝ≥0∞), 0 < ε → ∃ t ∈ C, μ (t ∆ s) < ε
ε : ℝ≥0∞
εpos : 0 < ε
t : Set α
tC : t ∈ C
ht : μ (t ∆ s) < ε / 2
D : Set (Set α)
D_count : D.Countable
DC : D ⊆ C
Dne : D.Nonempty
f : ℕ → Set α
hD : μ (⋂ n, (accumulate f n)ᶜ) = 0
hf : D = range f
fC : ∀ (n : ℕ), accumulate f n ∈ C
this : ⋃₀ D = ⋃ n, accumulate f n
i j : ℕ
hij : i ≤ j
⊢ (accumulate f j)ᶜ ⊆ (accumulate f i)ᶜ

Tactic:
by simpa using monotone_accumulate hij

Goal state after: