mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.DiracProba

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompletelyRegularSpace X
K : Set X
K_closed : IsClosed[inst✝¹] K
x : X
x_notin_K : x ∉ K
g : X → ↑unitInterval
g_cont : Continuous[inst✝¹, _] g
gx_zero : g x = 0
g_one_on_K : EqOn g 1 K
g_bdd : ∀ (x y : X), dist (↑(g x)).toNNReal (↑(g y)).toNNReal ≤ 1
⊢ ∃ f, f x = 1 ∧ ∀ y ∈ K, f y = 0

Tactic:
set g' := BoundedContinuousFunction.mkOfBound
      ⟨fun x ↦ Real.toNNReal (g x), continuous_real_toNNReal.comp g_cont.subtype_val⟩ 1 g_bdd

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompletelyRegularSpace X
K : Set X
K_closed : IsClosed[inst✝¹] K
x : X
x_notin_K : x ∉ K
g : X → ↑unitInterval
g_cont : Continuous[inst✝¹, _] g
gx_zero : g x = 0
g_one_on_K : EqOn g 1 K
g_bdd : ∀ (x y : X), dist (↑(g x)).toNNReal (↑(g y)).toNNReal ≤ 1
g' : X →ᵇ ℝ≥0 := mkOfBound { toFun := fun x ↦ (↑(g x)).toNNReal, continuous_toFun := ⋯ } 1 g_bdd
⊢ ∃ f, f x = 1 ∧ ∀ y ∈ K, f y = 0
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : SecondCountableTopology E
inst✝² : CompleteSpace E
μ ν : Measure E
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : charFunDual μ = charFunDual ν
⊢ μ = ν

Tactic:
by
  refine ext_of_integral_char_eq continuous_probChar probChar_ne_one
    ?_ ?_ (fun L ↦ funext_iff.mp h L)
  · exact fun v hv ↦ DFunLike.ne_iff.mpr <| SeparatingDual.exists_ne_zero hv
  · exact isBoundedBilinearMap_apply.symm.continuous

Goal state after:
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 ≠ ∞
V : Set X
pV1 : V ⊇ K
pV2 : IsOpen[inst✝⁷] V
pV3 : ν V ≤ ν K + ↑ε
VltTop : ν V < ∞
f : X →C_c ℝ
pf1 : EqOn (⇑f) 1 K
pf2 : tsupport ⇑f ⊆ V
pf3 : ∀ (x : X), f x ∈ Icc 0 1
hfV : ∀ (x : X), f x ≤ V.indicator 1 x
hfK : ∀ (x : X), K.indicator 1 x ≤ f x
⊢ ∫ (x : X), K.indicator 1 x ∂μ ≤ ∫ (x : X), f x ∂μ

Tactic:
by
      refine integral_mono ?_ f.integrable hfK
      exact (continuousOn_const.integrableOn_compact hK).integrable_indicator hK.measurableSet

Goal state after:
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
a✝ b c d : E
γ₁ : Path a✝ b
γ₂ : Path c d
s : Set (↑I × ↑I)
t : Set E
ω : E → E →L[ℝ] F
dω : E → E →L[ℝ] E →L[ℝ] F
φ : (↑γ₁).Homotopy ↑γ₂
hs✝ : s.Countable
hφt : ∀ a ∈ Ioo 0 1, ∀ b_1 ∈ Ioo 0 1, φ (a, b_1) ∈ t
hω : ∀ a ∈ Ioo 0 1, ∀ b_1 ∈ Ioo 0 1, (a, b_1) ∉ s → HasFDerivWithinAt ω (dω (φ (a, b_1))) t (φ (a, b_1))
hωc : ContinuousOn ω (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t)
hdω_symm :
  ∀ a ∈ Ioo 0 1,
    ∀ b_1 ∈ Ioo 0 1,
      (a, b_1) ∉ s →
        ∀ u ∈ tangentConeAt ℝ t (φ (a, b_1)),
          ∀ v ∈ tangentConeAt ℝ t (φ (a, b_1)), ((dω (φ (a, b_1))) u) v = ((dω (φ (a, b_1))) v) u
U : Set (ℝ × ℝ) := Ioo 0 1 ×ˢ Ioo 0 1
hU✝ : U = Ioo 0 1 ×ˢ Ioo 0 1
hinterior : interior (Icc 0 1) = U
hunique : UniqueDiffOn ℝ (Icc 0 1)
hUopen : IsOpen U
hU_subset : U ⊆ Icc 0 1
hclosure : closure U = Icc 0 1
ψ : ℝ × ℝ → E := fun xy ↦ Set.IccExtend ⋯ (⇑(φ.extend xy.1)) xy.2
hcontdiff : ContDiffOn ℝ 2 ψ (Icc 0 1)
hψ : ψ = fun xy ↦ Set.IccExtend ⋯ (⇑(φ.extend xy.1)) xy.2
hψφ : ∀ (a b_1 : ↑I), ψ (↑a, ↑b_1) = φ (a, b_1)
hψ_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ψ
hψUt : MapsTo ψ U t
dψ : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fderivWithin ℝ ψ (Icc 0 1)
s' : Set (ℝ × ℝ) := Prod.map Subtype.val Subtype.val '' s
hs' : s' = Prod.map Subtype.val Subtype.val '' s
hmem_s' : ∀ (x y : ↑I), (↑x, ↑y) ∈ s' ↔ (x, y) ∈ s
hs'c : s'.Countable
hdψ : ∀ a ∈ U, HasFDerivAt ψ (dψ a) a
d2ψ : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ →L[ℝ] E := fderivWithin ℝ dψ (Icc 0 1)
hd2ψ : ∀ a ∈ U, HasFDerivAt dψ (d2ψ a) a
hd2ψ_symm : ∀ a ∈ Icc 0 1, ∀ (x y : ℝ × ℝ), ((d2ψ a) x) y = ((d2ψ a) y) x
η : ℝ × ℝ → ℝ × ℝ →L[ℝ] F := fun a ↦ ω (ψ a) ∘SL dψ a
f : ℝ × ℝ → F := fun a ↦ (η a) (0, 1)
hf : ∀ a ∈ Icc 0 1, f a = (ω (ψ a)) (derivWithin (ψ ∘ fun x ↦ (a.1, x)) I a.2)
g : ℝ × ℝ → F := fun a ↦ -(η a) (1, 0)
hg : ∀ a ∈ Icc 0 1, g a = (ω (ψ a)) (-derivWithin (ψ ∘ fun x ↦ (x, a.2)) I a.1)
dη : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ →L[ℝ] F :=
  fun a ↦ (ContinuousLinearMap.compL ℝ (ℝ × ℝ) E F) (ω (ψ a)) ∘SL d2ψ a + (dω (ψ a)).bilinearComp (dψ a) (dψ a)
hdη : ∀ a ∈ U \ s', HasFDerivAt η (dη a) a
hdη_symm : ∀ a ∈ U \ s', ∀ (u v : ℝ × ℝ), ((dη a) u) v = ((dη a) v) u
f' : ℝ × ℝ → ℝ × ℝ →L[ℝ] F := fun a ↦ (ContinuousLinearMap.apply ℝ F) (0, 1) ∘SL dη a
hf' : ∀ a ∈ U \ s', HasFDerivAt f (f' a) a
g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] F := fun a ↦ -(ContinuousLinearMap.apply ℝ F) (1, 0) ∘SL dη a
hg' : ∀ a ∈ U \ s', HasFDerivAt g (g' a) a
a : ℝ × ℝ
hU : a ∈ Ioo 0.1 1.1 ×ˢ Ioo 0.2 1.2
hs : a ∉ s'
⊢ (f' a) (1, 0) + (g' a) (0, 1) = 0 a

Tactic:
simp [f', g', hdη_symm a ⟨hU, hs⟩ (0, 1)]

Goal state after:
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✝¹⁵ : Group A
inst✝¹⁴ : Group B
inst✝¹³ : Group C
inst✝¹² : TopologicalSpace A
inst✝¹¹ : TopologicalSpace B
inst✝¹⁰ : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst✝⁹ : IsTopologicalGroup A
inst✝⁸ : IsTopologicalGroup B
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : MeasurableSpace A
inst✝⁵ : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst✝⁴ : NormedSpace ℝ E
inst✝³ : IsTopologicalGroup C
inst✝² : LocallyCompactSpace B
inst✝¹ : MeasurableSpace C
inst✝ : BorelSpace C
μC : Measure C
hμC : μC.IsHaarMeasure
f g : CompactlySupportedContinuousMap B E
⊢ ∫ (c : C), ((H.pushforward μA) (f + g)) c ∂μC =
    ∫ (c : C), ((H.pushforward μA) f) c ∂μC + ∫ (c : C), ((H.pushforward μA) g) c ∂μC

Tactic:
map_add

Goal state after:
A : Type u_1
B : Type u_2
C : Type u_3
E : Type u_4
inst✝¹⁵ : Group A
inst✝¹⁴ : Group B
inst✝¹³ : Group C
inst✝¹² : TopologicalSpace A
inst✝¹¹ : TopologicalSpace B
inst✝¹⁰ : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst✝⁹ : IsTopologicalGroup A
inst✝⁸ : IsTopologicalGroup B
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : MeasurableSpace A
inst✝⁵ : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst✝⁴ : NormedSpace ℝ E
inst✝³ : IsTopologicalGroup C
inst✝² : LocallyCompactSpace B
inst✝¹ : MeasurableSpace C
inst✝ : BorelSpace C
μC : Measure C
hμC : μC.IsHaarMeasure
f g : CompactlySupportedContinuousMap B E
⊢ ∫ (c : C), ((H.pushforward μA) f + (H.pushforward μA) g) c ∂μC =
    ∫ (c : C), ((H.pushforward μA) f) c ∂μC + ∫ (c : C), ((H.pushforward μA) g) c ∂μC
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

Goal state before:
pp: case e_f
μ : Measure ℝ
r : ℝ
inst✝ : IsFiniteMeasure μ
hr : 0 < r
h_int : Integrable (Function.uncurry fun x y ↦ cexp (↑x * ↑y * I)) ((volume.restrict (Set.uIoc (-r) r)).prod μ)
y : ℝ
⊢ (if r * y = 0 then 2 * ↑r else ↑y⁻¹ * ∫ (x : ℝ) in -(y * r)..y * r, cexp (↑x * I)) = 2 * ↑r * ↑(sinc (r * y))

Tactic:
rw [integral_exp_mul_I_eq_sinc]

Goal state after:
case e_f
μ : Measure ℝ
r : ℝ
inst✝ : IsFiniteMeasure μ
hr : 0 < r
h_int : Integrable (Function.uncurry fun x y ↦ cexp (↑x * ↑y * I)) ((volume.restrict (Set.uIoc (-r) r)).prod μ)
y : ℝ
⊢ (if r * y = 0 then 2 * ↑r else ↑y⁻¹ * (2 * ↑(y * r) * ↑(sinc (y * r)))) = 2 * ↑r * ↑(sinc (r * y))
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

Goal state before:
pp: case e_s
μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
x : ℝ
⊢ r < |x| ↔ 1 < r⁻¹ * |x|

Tactic:
lt_inv_mul_iff₀ (by positivity),

Goal state after:
case e_s
μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
x : ℝ
⊢ r < |x| ↔ r * 1 < |x|
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)
A : ∀ (n : ℕ), IsCompact ((partialSups K) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∈ f
n : ℕ
ν : FiniteMeasure E
hν : ν ∈ {ρ | ρ.mass ≤ C ∧ ρ ((partialSups K) n)ᶜ = 0}
ν_lim : Tendsto (fun ρ ↦ ρ.restrict (disjointed K n)) (↑f) (𝓝 ν)
ν' : Measure E
ν'_reg : ν'.InnerRegular
ν'_fin : IsFiniteMeasure ν'
ν'K : ν' ((partialSups K) n)ᶜ = 0
hν' : ∀ (g : E →ᵇ ℝ), ∫ (x : E), g x ∂↑ν = ∫ (x : E), g x ∂ν'
μ : FiniteMeasure E := ⟨ν', ν'_fin⟩
⊢ μ ((partialSups K) n)ᶜ = 0

Tactic:
by simp [μ, ν'K]

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

Goal state before:
pp: 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)‼

Tactic:
suffices √π / (↑(2 * k + 1)‼ * √π / 2 ^ (k + 1)) = 2 ^ (k + 1) / ↑(2 * k + 1)‼ by
    simpa [add_div, add_right_comm, -one_div, Gamma_nat_add_one_add_half]

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 / (↑(2 * k + 1)‼ * √π / 2 ^ (k + 1)) = 2 ^ (k + 1) / ↑(2 * k + 1)‼
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare

Goal state before:
pp: 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 := NormedSpace.restrictScalars ℝ 𝕜 E
⊢ ∃ g, ∀ a ∈ s, HasDerivWithinAt g (f a) s a

Tactic:
apply hs.exists_forall_hasFDerivWithinAt_of_hasFDerivWithinAt_symmetric

Goal state after:
case hω
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 := ⋯
⊢ ∀ x ∈ s, HasFDerivWithinAt (fun a ↦ ContinuousLinearMap.toSpanSingleton 𝕜 (f a)) (?m.54 x) s x

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, ((?m.54 a) x) y = ((?m.54 a) y) x

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 := ⋯
⊢ 𝕜 → 𝕜 →L[ℝ] 𝕜 →L[𝕜] E
Module: Mathlib.MeasureTheory.Measure.ResolventTransform

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : HereditarilyLindelofSpace 𝕜
inst✝⁵ : CompleteSpace 𝕜
inst✝⁴ : MeasurableSpace 𝕜
inst✝³ : BorelSpace 𝕜
inst✝² : RCLike A
inst✝¹ : NormedAlgebra 𝕜 A
μ : Measure 𝕜
inst✝ : IsFiniteMeasure μ
a : A
ha : a ∉ ⇑(algebraMap 𝕜 A) '' μ.support
h : μ.support.Nonempty
this : 0 < infDist a (⇑(algebraMap 𝕜 A) '' μ.support)
s : Set A := ball a (infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2)
hs_z : s ∈ 𝓝 a
hs_μ : s ⊆ (⇑(algebraMap 𝕜 A) '' μ.support)ᶜ
resolvent_meas : ∀ᶠ (w : A) in 𝓝 a, AEStronglyMeasurable (resolvent w) μ
x : 𝕜
hx : x ∈ μ.support
w : A
hw : w ∈ s
⊢ ‖((algebraMap 𝕜 A) x - w)⁻¹ ^ 2‖ ≤ (infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2)⁻¹ ^ 2

Tactic:
norm_pow,

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : HereditarilyLindelofSpace 𝕜
inst✝⁵ : CompleteSpace 𝕜
inst✝⁴ : MeasurableSpace 𝕜
inst✝³ : BorelSpace 𝕜
inst✝² : RCLike A
inst✝¹ : NormedAlgebra 𝕜 A
μ : Measure 𝕜
inst✝ : IsFiniteMeasure μ
a : A
ha : a ∉ ⇑(algebraMap 𝕜 A) '' μ.support
h : μ.support.Nonempty
this : 0 < infDist a (⇑(algebraMap 𝕜 A) '' μ.support)
s : Set A := ball a (infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2)
hs_z : s ∈ 𝓝 a
hs_μ : s ⊆ (⇑(algebraMap 𝕜 A) '' μ.support)ᶜ
resolvent_meas : ∀ᶠ (w : A) in 𝓝 a, AEStronglyMeasurable (resolvent w) μ
x : 𝕜
hx : x ∈ μ.support
w : A
hw : w ∈ s
⊢ ‖((algebraMap 𝕜 A) x - w)⁻¹‖ ^ 2 ≤ (infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2)⁻¹ ^ 2
Module: Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsZeroOneMeasure μ
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
hμs : μ s = 1
hμt : μ t = 1
h✝ : μ (s ∩ t) = 0
⊢ μ (s ∩ t)ᶜ = μ (sᶜ ∪ tᶜ)

Tactic:
simp [compl_inter]

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

Goal state before:
pp: E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
S : Set (ProbabilityMeasure E)
hS : IsTightMeasureSet {x | ∃ μ ∈ S, ↑μ = x}
u : ℕ → ℝ≥0
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
K : ℕ → Set E
K_comp : ∀ (n : ℕ), IsCompact (K n)
hK : ∀ (n : ℕ), ∀ μ ∈ S, μ (K n)ᶜ ≤ u n
K' : ℕ → Set E := fun n ↦ ⋃ i ∈ Iic n, K i
⊢ IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)

Tactic:
have h'K : IsCompact {μ : ProbabilityMeasure E | ∀ n, μ (K' n)ᶜ ≤ u n} := by
    apply isCompact_setOfPred_probabilityMeasure_mass_eq_compl_isCompact_le u_lim
    · exact fun n ↦ (finite_Iic n).isCompact_biUnion (fun i hi ↦ K_comp i)
    · right
      simp only [Monotone, mem_Iic, iUnion_subset_iff, K']
      intro a b hab i hi
      apply subset_biUnion_of_mem
      exact hi.trans hab

Goal state after:
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
S : Set (ProbabilityMeasure E)
hS : IsTightMeasureSet {x | ∃ μ ∈ S, ↑μ = x}
u : ℕ → ℝ≥0
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
K : ℕ → Set E
K_comp : ∀ (n : ℕ), IsCompact (K n)
hK : ∀ (n : ℕ), ∀ μ ∈ S, μ (K n)ᶜ ≤ u n
K' : ℕ → Set E := fun n ↦ ⋃ i ∈ Iic n, K i
h'K : IsCompact {μ | ∀ (n : ℕ), μ (K' n)ᶜ ≤ u n}
⊢ IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
Module: Mathlib.MeasureTheory.SpecificCodomains.WithLp

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
q : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ q)
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f : X → WithLp q (E × F)
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
hf : Integrable f μ
| (∫ (x : X), f x ∂μ).ofLp.1 = ∫ (x : X), (f x).fst ∂μ

Tactic:
enter [1, 1]; change WithLp.prodContinuousLinearEquiv q ℝ E F _

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
q : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ q)
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f : X → WithLp q (E × F)
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
hf : Integrable f μ
| (WithLp.prodContinuousLinearEquiv q ℝ E F) (∫ (x : X), f x ∂μ)
Module: Mathlib.MeasureTheory.SpecificCodomains.WithLp

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
q : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ q)
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f : X → WithLp q (E × F)
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace E
hf : Integrable f μ
| (∫ (x : X), f x ∂μ).ofLp.2 = ∫ (x : X), (f x).snd ∂μ

Tactic:
enter [1, 1]; change WithLp.prodContinuousLinearEquiv q ℝ E F _

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
q : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ q)
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f : X → WithLp q (E × F)
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace E
hf : Integrable f μ
| (WithLp.prodContinuousLinearEquiv q ℝ E F) (∫ (x : X), f x ∂μ)
Module: Mathlib.MeasureTheory.Measure.SeparableMeasure

Goal state before:
pp: case refine_2.refine_1
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
⊢ ‖indicatorConstLp p ms ⋯ (a - b)‖ ≤ ε / 3

Tactic:
norm_indicatorConstLp p_ne_zero p_ne_top.elim

Goal state after:
case refine_2.refine_1
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
⊢ ‖a - b‖ * μ.real s ^ (1 / p.toReal) ≤ ε / 3
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

Goal state before:
pp: X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝² : TopologicalSpace V
inst✝¹ : ENormedAddCommMonoid V
inst✝ : T2Space V
μ : VectorMeasure X V
s : Set X
hs : MeasurableSet s
P : Finset (Set X)
hP₁ : ∀ t ∈ P, t ⊆ s
hP₂ : (↑P).PairwiseDisjoint id
Q : Finpartition (P.sup id) := ⋯
defQ : Q = Finpartition.ofPairwiseDisjoint P hP₂
Q' : Finpartition ((filter MeasurableSet Q.parts).sup id) := ⋯
defQ' : Q' = Q.ofSubset ⋯ ⋯
hQ' : ∀ t ∈ Q'.parts, t ⊆ s
⊢ ∀ p ∈ (Q'.extendOfLE ⋯).parts, MeasurableSet p

Tactic:
intro p

Goal state after:
X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝² : TopologicalSpace V
inst✝¹ : ENormedAddCommMonoid V
inst✝ : T2Space V
μ : VectorMeasure X V
s : Set X
hs : MeasurableSet s
P : Finset (Set X)
hP₁ : ∀ t ∈ P, t ⊆ s
hP₂ : (↑P).PairwiseDisjoint id
Q : Finpartition (P.sup id) := Finpartition.ofPairwiseDisjoint P hP₂
defQ : Q = Finpartition.ofPairwiseDisjoint P hP₂
Q' : Finpartition ((filter MeasurableSet Q.parts).sup id) := Q.ofSubset ⋯ ⋯
defQ' : Q' = Q.ofSubset ⋯ ⋯
hQ' : ∀ t ∈ Q'.parts, t ⊆ s
p : Set X
⊢ p ∈ (Q'.extendOfLE ⋯).parts → MeasurableSet p
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
inst✝ : IsFiniteMeasure μ
s : Set X
hs : MeasurableSet s
⊢ ‖μ.toSignedMeasure s‖ₑ ≤ μ s

Tactic:
simp [hs, Measure.real, Real.enorm_eq_ofReal]

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
inst✝ : IsFiniteMeasure μ
s : Set X
hs : MeasurableSet s
⊢ μ s ≤ ‖μ.toSignedMeasure s‖ₑ

Tactic:
simp [hs, Measure.real, Real.enorm_eq_ofReal]

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : SignedMeasure X
s : Set X
hs : MeasurableSet s
a : ℝ≥0∞
ha : a < (variation μ) s
P : Finset (Set X)
Ps : ∀ t ∈ P, t ⊆ s
P_disj : (↑P).PairwiseDisjoint id
P_meas : ∀ t ∈ P, MeasurableSet t
hP : a < ∑ p ∈ P, ‖μ p‖ₑ
I : ∑ p ∈ P with 0 ≤ μ p, ‖μ p‖ₑ = ‖μ (⋃ p ∈ {p ∈ P | 0 ≤ μ p}, p)‖ₑ
J : ∑ p ∈ P with ¬0 ≤ μ p, ‖μ p‖ₑ = ‖μ (⋃ p ∈ {p ∈ P | ¬0 ≤ μ p}, p)‖ₑ
⊢ ∃ t ⊆ s, MeasurableSet t ∧ a < 2 * ‖μ t‖ₑ

Tactic:
simp_rw [two_mul]

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : SignedMeasure X
s : Set X
hs : MeasurableSet s
a : ℝ≥0∞
ha : a < (variation μ) s
P : Finset (Set X)
Ps : ∀ t ∈ P, t ⊆ s
P_disj : (↑P).PairwiseDisjoint id
P_meas : ∀ t ∈ P, MeasurableSet t
hP : a < ∑ p ∈ P, ‖μ p‖ₑ
I : ∑ p ∈ P with 0 ≤ μ p, ‖μ p‖ₑ = ‖μ (⋃ p ∈ {p ∈ P | 0 ≤ μ p}, p)‖ₑ
J : ∑ p ∈ P with ¬0 ≤ μ p, ‖μ p‖ₑ = ‖μ (⋃ p ∈ {p ∈ P | ¬0 ≤ μ p}, p)‖ₑ
⊢ ∃ t ⊆ s, MeasurableSet t ∧ a < ‖μ t‖ₑ + ‖μ t‖ₑ
Module: Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Goal state before:
pp: case insert
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedAddCommGroup G
μ : VectorMeasure X F
f : X → E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
ι : Type u_7
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✝, μ.IntegrableOn f (s i)) →
        ∫ᵛ (x : X) in ⋃ i ∈ s✝, s i, f x ∂[B; μ] = ∑ i ∈ s✝, ∫ᵛ (x : X) in s i, f x ∂[B; μ]
hs : ∀ i ∈ insert a✝ s✝, MeasurableSet (s i)
h's : (↑(insert a✝ s✝)).Pairwise (Disjoint on s)
hf : ∀ i ∈ insert a✝ s✝, μ.IntegrableOn f (s i)
⊢ ∫ᵛ (x : X) in ⋃ i ∈ insert a✝ s✝, s i, f x ∂[B; μ] = ∑ i ∈ insert a✝ s✝, ∫ᵛ (x : X) in s i, f x ∂[B; μ]

Tactic:
simp only [Finset.coe_insert, Finset.forall_mem_insert, Set.pairwise_insert,
      Finset.set_biUnion_insert] at hs hf h's ⊢

Goal state after:
case insert
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedAddCommGroup G
μ : VectorMeasure X F
f : X → E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
ι : Type u_7
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✝, μ.IntegrableOn f (s i)) →
        ∫ᵛ (x : X) in ⋃ i ∈ s✝, s i, f x ∂[B; μ] = ∑ i ∈ s✝, ∫ᵛ (x : X) in s i, f x ∂[B; μ]
hs : MeasurableSet (s a✝) ∧ ∀ x ∈ s✝, MeasurableSet (s x)
hf : μ.IntegrableOn f (s a✝) ∧ ∀ x ∈ s✝, μ.IntegrableOn f (s x)
h's : (↑s✝).Pairwise (Disjoint on s) ∧ ∀ b ∈ ↑s✝, a✝ ≠ b → (Disjoint on s) a✝ b ∧ (Disjoint on s) b a✝
⊢ ∫ᵛ (x : X) in s a✝ ∪ ⋃ x ∈ s✝, s x, f x ∂[B; μ] = ∑ i ∈ insert a✝ s✝, ∫ᵛ (x : X) in s i, f x ∂[B; μ]
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.JordanSub

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
s : Set X
μ ν : Measure X
hs : IsHahnDecomposition μ ν s
⊢ (μ - ν) s = 0

Tactic:
by
  rw [← restrict_eq_zero, restrict_sub_eq_restrict_sub_restrict hs.measurableSet]
  exact sub_eq_zero_of_le hs.le_on

Goal state after: