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

Goal state before:
pp: W : Type u_1
inst✝¹¹ : AddCommGroup W
inst✝¹⁰ : Module ℝ W
inst✝⁹ : TopologicalSpace W
e : AddChar ℝ Circle
V : Type u_2
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module ℝ V
inst✝⁶ : PseudoEMetricSpace V
inst✝⁵ : MeasurableSpace V
inst✝⁴ : BorelSpace V
inst✝³ : CompleteSpace V
inst✝² : SecondCountableTopology V
L : V →ₗ[ℝ] W →ₗ[ℝ] ℝ
he : Continuous ⇑e
he' : e ≠ 1
hL' : ∀ (v : V), v ≠ 0 → L v ≠ 0
hL : Continuous fun p ↦ (L p.1) p.2
P P' : Measure V
inst✝¹ : IsFiniteMeasure P
inst✝ : IsFiniteMeasure P'
h : ∀ (w : W), ∫ (v : V), (char he hL w) v ∂P = ∫ (v : V), (char he hL w) v ∂P'
g✝ : V →ᵇ ℂ
w : AddMonoidAlgebra ℂ W
hw : ⇑g✝ = fun x ↦ w.coeff.sum fun a z ↦ z * ↑(e ((L x) a))
hsum :
  ∀ (P : Measure V) [IsFiniteMeasure P],
    ∫ (v : V), w.coeff.sum fun a z ↦ z * ↑(e ((L v) a)) ∂P = w.coeff.sum fun a z ↦ ∫ (v : V), z * ↑(e ((L v) a)) ∂P
⊢ (w.coeff.sum fun a z ↦ ∫ (v : V), z * ↑(e ((L v) a)) ∂P) =
    ∫ (x : V), (fun x ↦ w.coeff.sum fun a z ↦ z * ↑(e ((L x) a))) x ∂P'

Tactic:
hsum P'

Goal state after:
W : Type u_1
inst✝¹¹ : AddCommGroup W
inst✝¹⁰ : Module ℝ W
inst✝⁹ : TopologicalSpace W
e : AddChar ℝ Circle
V : Type u_2
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module ℝ V
inst✝⁶ : PseudoEMetricSpace V
inst✝⁵ : MeasurableSpace V
inst✝⁴ : BorelSpace V
inst✝³ : CompleteSpace V
inst✝² : SecondCountableTopology V
L : V →ₗ[ℝ] W →ₗ[ℝ] ℝ
he : Continuous ⇑e
he' : e ≠ 1
hL' : ∀ (v : V), v ≠ 0 → L v ≠ 0
hL : Continuous fun p ↦ (L p.1) p.2
P P' : Measure V
inst✝¹ : IsFiniteMeasure P
inst✝ : IsFiniteMeasure P'
h : ∀ (w : W), ∫ (v : V), (char he hL w) v ∂P = ∫ (v : V), (char he hL w) v ∂P'
g✝ : V →ᵇ ℂ
w : AddMonoidAlgebra ℂ W
hw : ⇑g✝ = fun x ↦ w.coeff.sum fun a z ↦ z * ↑(e ((L x) a))
hsum :
  ∀ (P : Measure V) [IsFiniteMeasure P],
    ∫ (v : V), w.coeff.sum fun a z ↦ z * ↑(e ((L v) a)) ∂P = w.coeff.sum fun a z ↦ ∫ (v : V), z * ↑(e ((L v) a)) ∂P
⊢ (w.coeff.sum fun a z ↦ ∫ (v : V), z * ↑(e ((L v) a)) ∂P) = w.coeff.sum fun a z ↦ ∫ (v : V), z * ↑(e ((L v) a)) ∂P'
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: W : Type u_1
inst✝¹¹ : AddCommGroup W
inst✝¹⁰ : Module ℝ W
inst✝⁹ : TopologicalSpace W
e : AddChar ℝ Circle
V : Type u_2
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module ℝ V
inst✝⁶ : PseudoEMetricSpace V
inst✝⁵ : MeasurableSpace V
inst✝⁴ : BorelSpace V
inst✝³ : CompleteSpace V
inst✝² : SecondCountableTopology V
L : V →ₗ[ℝ] W →ₗ[ℝ] ℝ
he : Continuous ⇑e
he' : e ≠ 1
hL' : ∀ (v : V), v ≠ 0 → L v ≠ 0
hL : Continuous fun p ↦ (L p.1) p.2
P P' : Measure V
inst✝¹ : IsFiniteMeasure P
inst✝ : IsFiniteMeasure P'
h : ∀ (w : W), ∫ (v : V), (char he hL w) v ∂P = ∫ (v : V), (char he hL w) v ∂P'
g✝ : V →ᵇ ℂ
w : AddMonoidAlgebra ℂ W
hw : ⇑g✝ = fun x ↦ w.coeff.sum fun a z ↦ z * ↑(e ((L x) a))
hsum :
  ∀ (P : Measure V) [IsFiniteMeasure P],
    ∫ (v : V), w.coeff.sum fun a z ↦ z * ↑(e ((L v) a)) ∂P = w.coeff.sum fun a z ↦ ∫ (v : V), z * ↑(e ((L v) a)) ∂P
⊢ (w.coeff.sum fun a z ↦ ∫ (v : V), z * ↑(e ((L v) a)) ∂P) = w.coeff.sum fun a z ↦ ∫ (v : V), z * ↑(e ((L v) a)) ∂P'

Tactic:
apply Finset.sum_congr rfl fun i _ => ?_

Goal state after:
W : Type u_1
inst✝¹¹ : AddCommGroup W
inst✝¹⁰ : Module ℝ W
inst✝⁹ : TopologicalSpace W
e : AddChar ℝ Circle
V : Type u_2
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module ℝ V
inst✝⁶ : PseudoEMetricSpace V
inst✝⁵ : MeasurableSpace V
inst✝⁴ : BorelSpace V
inst✝³ : CompleteSpace V
inst✝² : SecondCountableTopology V
L : V →ₗ[ℝ] W →ₗ[ℝ] ℝ
he : Continuous ⇑e
he' : e ≠ 1
hL' : ∀ (v : V), v ≠ 0 → L v ≠ 0
hL : Continuous fun p ↦ (L p.1) p.2
P P' : Measure V
inst✝¹ : IsFiniteMeasure P
inst✝ : IsFiniteMeasure P'
h : ∀ (w : W), ∫ (v : V), (char he hL w) v ∂P = ∫ (v : V), (char he hL w) v ∂P'
g✝ : V →ᵇ ℂ
w : AddMonoidAlgebra ℂ W
hw : ⇑g✝ = fun x ↦ w.coeff.sum fun a z ↦ z * ↑(e ((L x) a))
hsum :
  ∀ (P : Measure V) [IsFiniteMeasure P],
    ∫ (v : V), w.coeff.sum fun a z ↦ z * ↑(e ((L v) a)) ∂P = w.coeff.sum fun a z ↦ ∫ (v : V), z * ↑(e ((L v) a)) ∂P
i : W
x✝ : i ∈ w.coeff.support
⊢ (fun a z ↦ ∫ (v : V), z * ↑(e ((L v) a)) ∂P) i (w.coeff i) =
    (fun a z ↦ ∫ (v : V), z * ↑(e ((L v) a)) ∂P') i (w.coeff i)
Module: Mathlib.Topology.Separation.CompletelyRegular

Goal state before:
pp: X : Type u
inst✝² : TopologicalSpace X
Y : Type v
inst✝¹ : TopologicalSpace Y
inst✝ : CompletelyRegularSpace Y
f : X → Y
hf : IsInducing f
x : X
K : Set Y
hK : IsClosed[inst✝¹] K
hxK : f x ∉ K
g : Y → ↑I
hcf : Continuous[inst✝¹, _] g
egfx : g (f x) = 0
hgK : EqOn g 1 K
⊢ EqOn (g ∘ f) (1 ∘ f) (f ⁻¹' K)

Tactic:
exact hgK.comp_right <| mapsTo_preimage _ _

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

Goal state before:
pp: X : Type u_1
inst✝² : MeasurableSpace X
inst✝¹ : TopologicalSpace X
inst✝ : OpensMeasurableSpace X
⊢ Continuous[inst✝¹, ProbabilityMeasure.instTopologicalSpace] fun x ↦ diracProba x

Tactic:
continuous_iff_continuousAt

Goal state after:
X : Type u_1
inst✝² : MeasurableSpace X
inst✝¹ : TopologicalSpace X
inst✝ : OpensMeasurableSpace X
⊢ ∀ (x : X), ContinuousAt (fun x ↦ diracProba x) x
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: case pos
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : PseudoEMetricSpace Ω
ε₁ ε₂ : ℝ≥0∞
μ ν : Measure Ω
h_le : ε₁ ≤ ε₂
B : Set Ω
hε₁ : μ B ≤ ν (thickening ε₁.toReal B) + ε₁
ε_top : ε₂ = ∞
⊢ μ B ≤ ν (thickening ε₂.toReal B) + ε₂

Tactic:
simp only [ε_top, toReal_top,
                add_top, le_top]

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

Goal state before:
pp: case h_bound
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
f : Ω →ᵇ ℝ
A : Set ℝ
A_finmeas : volume A ≠ ∞
μ : ProbabilityMeasure Ω
⊢ ∀ᶠ (n : ℝ) in 𝓝[>] 0, ∀ᵐ (a : ℝ) ∂volume.restrict A, ‖↑(μ (thickening n {a_1 | a ≤ f a_1}))‖ ≤ 1 a

Tactic:
apply Eventually.of_forall (fun i ↦ ?_)
    apply Eventually.of_forall (fun t ↦ ?_)
    simp only [Real.norm_eq_abs, NNReal.abs_eq, Pi.one_apply]
    exact ENNReal.toReal_mono one_ne_top prob_le_one

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

Goal state before:
pp: α : Type u_1
inst✝⁹ : MeasurableSpace α
β : Type u_2
inst✝⁸ : MeasurableSpace β
inst✝⁷ : TopologicalSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : SecondCountableTopology β
inst✝³ : PseudoMetrizableSpace α
inst✝² : PseudoMetrizableSpace β
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace β
⊢ Continuous[instTopologicalSpaceProd, instTopologicalSpace] fun μ ↦ μ.1.prod μ.2

Tactic:
refine continuous_iff_continuousAt.2 (fun μ ↦ ?_)
  /- It suffices to check the convergence along elements of a π-system containing arbitrarily
  small neighborhoods of any point, by `tendsto_probabilityMeasure_of_tendsto_of_mem`.
  We take as a π-system the sets of the form `a ×ˢ b` where `a` and `b` have null frontier. -/
  let S : Set (Set (α × β)) := {t | ∃ (a : Set α) (b : Set β),
    MeasurableSet a ∧ μ.1 (frontier a) = 0 ∧ MeasurableSet b ∧ μ.2 (frontier b) = 0
    ∧ t = a ×ˢ b}
  have : IsPiSystem S := by
    rintro - ⟨a, b, ameas, ha, bmeas, hb, rfl⟩ - ⟨a', b', a'meas, ha', b'meas, hb', rfl⟩ -
    refine ⟨a ∩ a', b ∩ b', ameas.inter a'meas, ?_, bmeas.inter b'meas, ?_, prod_inter_prod⟩
    · rw [null_iff_toMeasure_null] at ha ha' ⊢
      exact null_frontier_inter ha ha'
    · rw [null_iff_toMeasure_null] at hb hb' ⊢
      exact null_frontier_inter hb hb'
  apply this.tendsto_probabilityMeasure_of_tendsto_of_mem
  · rintro s ⟨a, b, ameas, -, bmeas, -, rfl⟩
    exact ameas.prod bmeas
  · let : PseudoMetricSpace α := TopologicalSpace.pseudoMetrizableSpacePseudoMetric α
    let : PseudoMetricSpace β := TopologicalSpace.pseudoMetrizableSpacePseudoMetric β
    intro u u_open x xu
    obtain ⟨ε, εpos, hε⟩ : ∃ ε > 0, ball x ε ⊆ u := Metric.isOpen_iff.1 u_open x xu
    rcases exists_null_frontier_thickening (μ.1 : Measure α) {x.1} εpos with ⟨r, hr, μr⟩
    rcases exists_null_frontier_thickening (μ.2 : Measure β) {x.2} εpos with ⟨r', hr', μr'⟩
    simp only [thickening_singleton] at μr μr'
    refine ⟨ball x.1 r ×ˢ ball x.2 r', ⟨ball x.1 r, ball x.2 r', measurableSet_ball,
      by simp [coeFn_def, μr], measurableSet_ball, by simp [coeFn_def, μr'], rfl⟩, ?_, ?_⟩
    · exact (isOpen_ball.prod isOpen_ball).mem_nhds (by simp [hr.1, hr'.1])
    · calc ball x.1 r ×ˢ ball x.2 r'
      _ ⊆ ball x.1 ε ×ˢ ball x.2 ε := by gcongr; exacts [hr.2.le, hr'.2.le]
      _ ⊆ _ := by rwa [ball_prod_same]
  · rintro s ⟨a, b, ameas, ha, bmeas, hb, rfl⟩
    simp only [prod_prod]
    apply Filter.Tendsto.mul
    · exact tendsto_measure_of_null_frontier_of_tendsto tendsto_id.fst_nhds ha
    · exact tendsto_measure_of_null_frontier_of_tendsto tendsto_id.snd_nhds hb

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✝¹³ : Group A
inst✝¹² : Group B
inst✝¹¹ : Group C
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : TopologicalSpace B
inst✝⁸ : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst✝⁷ : IsTopologicalGroup A
inst✝⁶ : IsTopologicalGroup B
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace A
inst✝³ : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst✝² : NormedSpace ℝ E
inst✝¹ : IsTopologicalGroup C
inst✝ : LocallyCompactSpace B
f : CompactlySupportedContinuousMap B E
this : UniformSpace B := IsTopologicalGroup.rightUniformSpace B
b : B
ε : ℝ
hε : ε > 0
U₀ : Set B
hU₀ : IsCompact U₀
hb : U₀ ∈ nhds b
K : Set B
hK : IsCompact K
hf₀ : ∀ x ∉ K, f x = 0
S : Set A := ⇑φ ⁻¹' (U₀⁻¹ * K)
hSc : IsCompact S
⊢ ∃ δ > 0, ENNReal.ofReal δ * μA S < ENNReal.ofReal ε

Tactic:
← ENNReal.ofReal_toReal hSc.measure_ne_top,

Goal state after:
A : Type u_1
B : Type u_2
C : Type u_3
E : Type u_4
inst✝¹³ : Group A
inst✝¹² : Group B
inst✝¹¹ : Group C
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : TopologicalSpace B
inst✝⁸ : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst✝⁷ : IsTopologicalGroup A
inst✝⁶ : IsTopologicalGroup B
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace A
inst✝³ : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst✝² : NormedSpace ℝ E
inst✝¹ : IsTopologicalGroup C
inst✝ : LocallyCompactSpace B
f : CompactlySupportedContinuousMap B E
this : UniformSpace B := IsTopologicalGroup.rightUniformSpace B
b : B
ε : ℝ
hε : ε > 0
U₀ : Set B
hU₀ : IsCompact U₀
hb : U₀ ∈ nhds b
K : Set B
hK : IsCompact K
hf₀ : ∀ x ∉ K, f x = 0
S : Set A := ⇑φ ⁻¹' (U₀⁻¹ * K)
hSc : IsCompact S
⊢ ∃ δ > 0, ENNReal.ofReal δ * ENNReal.ofReal (μA S).toReal < ENNReal.ofReal 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
⊢ ∃ b, ∀ (a : ℝ) (x : ℕ), (∀ (b : ℕ), x ≤ b → ∫ (x : Ω), f x ∂↑(Ps b) ≤ a) → b ≤ a

Tactic:
refine ⟨0, fun a i hia ↦ le_trans (integral_nonneg f_nn) (hia i le_rfl)⟩

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

Goal state before:
pp: Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
⊢ Continuous[ProbabilityMeasure.instTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ofMeasure

Tactic:
continuous_iff_continuousAt

Goal state after:
Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
⊢ ∀ (x : ProbabilityMeasure Ω), ContinuousAt ofMeasure x
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✝¹⁸ : Group A
inst✝¹⁷ : Group B
inst✝¹⁶ : Group C
inst✝¹⁵ : TopologicalSpace A
inst✝¹⁴ : TopologicalSpace B
inst✝¹³ : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst✝¹² : IsTopologicalGroup A
inst✝¹¹ : IsTopologicalGroup B
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace A
inst✝⁸ : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : IsTopologicalGroup C
inst✝⁵ : LocallyCompactSpace B
inst✝⁴ : MeasurableSpace C
inst✝³ : BorelSpace C
μC : Measure C
hμC : μC.IsHaarMeasure
inst✝² : T2Space B
inst✝¹ : MeasurableSpace B
inst✝ : BorelSpace B
U : Set B
hU : IsOpen U
b : B
hb✝ : b ∈ U
K : Set B
hK : IsCompact K
hb : b ∈ interior K
hKU : K ⊆ U
f : C(B, ℝ)
hf1 : Set.EqOn (⇑f) 1 K
hf2 : IsCompact (tsupport ⇑f)
hf3 : tsupport ⇑f ⊆ U
hf4 : ∀ (x : B), f x ∈ Set.Icc 0 1
⊢ (H.inducedMeasure μA μC) U ≠ 0

Tactic:
have hf0 : 0 ≤ H.pushforward μA ⟨f, hf2⟩ := by
      rw [← map_zero (H.pushforward μA)]
      apply pushforward_mono
      exact fun x ↦ (hf4 x).1

Goal state after:
A : Type u_1
B : Type u_2
C : Type u_3
E : Type u_4
inst✝¹⁸ : Group A
inst✝¹⁷ : Group B
inst✝¹⁶ : Group C
inst✝¹⁵ : TopologicalSpace A
inst✝¹⁴ : TopologicalSpace B
inst✝¹³ : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst✝¹² : IsTopologicalGroup A
inst✝¹¹ : IsTopologicalGroup B
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace A
inst✝⁸ : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : IsTopologicalGroup C
inst✝⁵ : LocallyCompactSpace B
inst✝⁴ : MeasurableSpace C
inst✝³ : BorelSpace C
μC : Measure C
hμC : μC.IsHaarMeasure
inst✝² : T2Space B
inst✝¹ : MeasurableSpace B
inst✝ : BorelSpace B
U : Set B
hU : IsOpen U
b : B
hb✝ : b ∈ U
K : Set B
hK : IsCompact K
hb : b ∈ interior K
hKU : K ⊆ U
f : C(B, ℝ)
hf1 : Set.EqOn (⇑f) 1 K
hf2 : IsCompact (tsupport ⇑f)
hf3 : tsupport ⇑f ⊆ U
hf4 : ∀ (x : B), f x ∈ Set.Icc 0 1
hf0 : 0 ≤ (H.pushforward μA) { toContinuousMap := f, hasCompactSupport' := hf2 }
⊢ (H.inducedMeasure μA μC) U ≠ 0
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: case h₁.hf
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 := Subtype.val
hf : IsClosedEmbedding f
rf : range f = K
F : FiniteMeasure ↑K → FiniteMeasure E := fun μ ↦ μ.map f
T : Set (FiniteMeasure ↑K) := {μ | μ.mass ≤ C}
μ : FiniteMeasure E
hμ : μ ∈ {μ | μ.mass ≤ C ∧ μ Kᶜ = 0}
s : Set E
hs : MeasurableSet s
⊢ ∀ (s : Set { x // x ∈ K }), MeasurableSet s → MeasurableSet (Subtype.val '' s)

Tactic:
exact fun t ht ↦ hf.measurableEmbedding.measurableSet_image' ht

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
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
ρ : FiniteMeasure E
hρ : ρ.mass ≤ C ∧ ∀ (n : ℕ), ρ (K n)ᶜ ≤ u n
⊢ (ρ.restrict (disjointed K n)).mass ≤ C ∧ ↑(ρ.restrict (disjointed K n)) ((partialSups K) n)ᶜ = 0

Tactic:
simp only [restrict_mass, restrict_measure_eq,
        Measure.restrict_apply (A n).measurableSet.compl]

Goal state after:
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
ρ : FiniteMeasure E
hρ : ρ.mass ≤ C ∧ ∀ (n : ℕ), ρ (K n)ᶜ ≤ u n
⊢ ρ (disjointed K n) ≤ C ∧ ↑ρ (((partialSups K) n)ᶜ ∩ disjointed K n) = 0
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
a b c : E
s : Set E
ω : E → E →L[𝕜] F
dω : E → E →L[ℝ] E →L[𝕜] F
hs : Convex ℝ s
hω : ∀ x ∈ s, HasFDerivWithinAt ω (dω x) s x
hdω : ∀ a ∈ s, ∀ x ∈ tangentConeAt ℝ s a, ∀ y ∈ tangentConeAt ℝ s a, ((dω a) x) y = ((dω a) y) x
ha : a ∈ s
hb : b ∈ s
hc : c ∈ s
φ : (↑(Path.segment a b)).Homotopy ↑(Path.segment a c) :=
  ContinuousMap.Homotopy.affine ↑(Path.segment a b) ↑(Path.segment a c)
⊢ range ⇑φ ⊆ s

Tactic:
range_subset_iff

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
a b c : E
s : Set E
ω : E → E →L[𝕜] F
dω : E → E →L[ℝ] E →L[𝕜] F
hs : Convex ℝ s
hω : ∀ x ∈ s, HasFDerivWithinAt ω (dω x) s x
hdω : ∀ a ∈ s, ∀ x ∈ tangentConeAt ℝ s a, ∀ y ∈ tangentConeAt ℝ s a, ((dω a) x) y = ((dω a) y) x
ha : a ∈ s
hb : b ∈ s
hc : c ∈ s
φ : (↑(Path.segment a b)).Homotopy ↑(Path.segment a c) :=
  ContinuousMap.Homotopy.affine ↑(Path.segment a b) ↑(Path.segment a c)
⊢ ∀ (y : ↑I × ↑I), φ y ∈ s
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
⊢ volume (ball x r) = ENNReal.ofReal r ^ finrank ℝ E * ENNReal.ofReal (π ^ k * 2 ^ (k + 1) / ↑(finrank ℝ E)‼)

Tactic:
rw [volume_ball, hk, pow_succ (√π), pow_mul, sq_sqrt pi_nonneg, mul_div_assoc, mul_div_assoc]

Goal state after:
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)‼))
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)
⊢ IsTightMeasureSet (Set.range μ) ↔
    ∀ (y : E), Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)

Tactic:
refine ⟨fun h z ↦ ?_, isTightMeasureSet_range_of_tendsto_limsup_inner 𝕜⟩

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 : IsTightMeasureSet (Set.range μ)
z : E
⊢ Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪z, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
Module: Mathlib.MeasureTheory.Measure.LevyConvergence

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun n ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
h_le_4 : ∀ (n : ℕ) (r : ℝ), 0 < r → 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖ ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ),
    0 < r → (μ n).real {x | r < |⟪z, x⟫|} ≤ 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖
h_limsup_le :
  ∀ (r : ℝ),
    0 < r →
      limsup (fun n ↦ (μ n).real {x | r < |⟪z, x⟫|}) atTop ≤ 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z)‖
ε : ℝ
hε : ε > 0
hf0 : f 0 = 1
hf_tendsto : ∀ (ε : ℝ), 0 < ε → ∃ δ, 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε
δ : ℝ
hδ : 0 < δ
hδ_lt : ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε / 4
r : ℝ
hrδ : 4 * δ⁻¹ ≤ r
hr : 0 < r
hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹
h_le_Ioc : ∀ x ∈ Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹), ‖1 - f (x • z)‖ ≤ ε / 4
⊢ 2⁻¹ * r * ‖∫ (t : ℝ) in -(2 * r⁻¹)..2 * r⁻¹, 1 - f (t • z)‖ ≤
    2⁻¹ * r * ∫ (t : ℝ) in -(2 * r⁻¹)..2 * r⁻¹, ‖1 - f (t • z)‖

Tactic:
grw [intervalIntegral.norm_integral_le_integral_norm hr']

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

Goal state before:
pp: X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
h𝒜 : μ.MeasureDense 𝒜
⊢ 𝒜.Nonempty

Tactic:
by
  rcases h𝒜.approx ∅ MeasurableSet.empty (by simp) 1 (by simp) with ⟨t, ht, -⟩
  exact ⟨t, ht⟩

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

Goal state before:
pp: X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
inst✝ : IsFiniteMeasure μ
h𝒜 : IsSetAlgebra 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
f : ℕ → Set X
hs✝ : ∀ (n : ℕ), MeasurableSet (f n)
hf : ∀ (n : ℕ), MeasurableSet (f n) ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, μ.real (f n ∆ t) < ε
ε : ℝ
ε_pos : 0 < ε
this : Filter.Tendsto (fun n ↦ (μ (accumulate f n)).toReal) Filter.atTop (nhds (μ (⋃ i, f i)).toReal)
N : ℕ
hN : ∀ n ≥ N, dist (μ (accumulate f n)).toReal (μ (⋃ i, f i)).toReal < ε / 2
g : ℕ → Set X
g_mem : ∀ (n : ℕ), g n ∈ 𝒜
hg : ∀ (n : ℕ), μ.real (f n ∆ g n) < ε / (2 * (↑N + 1))
⊢ (↑N + 1) * (ε / (2 * (↑N + 1))) ≤ ε / 2

Tactic:
div_mul_eq_div_mul_one_div,

Goal state after:
X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
inst✝ : IsFiniteMeasure μ
h𝒜 : IsSetAlgebra 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
f : ℕ → Set X
hs✝ : ∀ (n : ℕ), MeasurableSet (f n)
hf : ∀ (n : ℕ), MeasurableSet (f n) ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, μ.real (f n ∆ t) < ε
ε : ℝ
ε_pos : 0 < ε
this : Filter.Tendsto (fun n ↦ (μ (accumulate f n)).toReal) Filter.atTop (nhds (μ (⋃ i, f i)).toReal)
N : ℕ
hN : ∀ n ≥ N, dist (μ (accumulate f n)).toReal (μ (⋃ i, f i)).toReal < ε / 2
g : ℕ → Set X
g_mem : ∀ (n : ℕ), g n ∈ 𝒜
hg : ∀ (n : ℕ), μ.real (f n ∆ g n) < ε / (2 * (↑N + 1))
⊢ (↑N + 1) * (ε / 2 * (1 / (↑N + 1))) ≤ ε / 2
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: case h
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 := ⋯
⊢ NormalSpace E ∨ Monotone K'

Tactic:
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: