awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution
Goal:
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst_1 : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst_2 : SeminormedAddCommGroup E
inst_3 : SecondCountableTopology E
inst_4 : BorelSpace E
X : ι → Ω'' → E
inst_5 : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution (Ω := fun (i : ι) ↦ Ω'') (m := fun (i : ι) ↦ m'') (mE := mE) X l Z (fun (x : ι) ↦ μ'') μ'
hXY : TendstoInMeasure μ'' (Y - X) l 0
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : Nonempty E
x₀ : E := hE.some
F : E → ℝ
M : ℝ
hF_bounded : ∀ (x y : E), dist (F x) (F y) ≤ M
L : NNReal
hF_lip : LipschitzWith L F
hF_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] F
hL : 0 < L
ε : ℝ
hε : ε > 0
n : ι
h_int_Y : Integrable (ε := ℝ) (fun (x : Ω'') ↦ F (Y n x)) μ''
h_int_X : Integrable (ε := ℝ) (fun (x : Ω'') ↦ F (X n x)) μ''
h_int_sub : Integrable (ε := ℝ) (fun (a : Ω'') ↦ ‖F (Y n a) - F (X n a)‖) μ''
⊢ ∀ (x : Ω''),
    Membership.mem (γ := Set Ω'') {x : Ω'' | ‖Y n x - X n x‖ < ε / 2} x → ‖F (Y n x) - F (X n x)‖ ≤ (↑L : ℝ) * (ε / 2)

Tactic:
exact fun x hx ↦ hF_lip.norm_sub_le_of_le hx.le

Premises used: LipschitzWith.norm_sub_le_of_le
-- Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution
Goal:
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst_1 : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst_2 : SeminormedAddCommGroup E
inst_3 : SecondCountableTopology E
inst_4 : BorelSpace E
X : ι → Ω'' → E
inst_5 : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution (Ω := fun (i : ι) ↦ Ω'') (m := fun (i : ι) ↦ m'') (mE := mE) X l Z (fun (x : ι) ↦ μ'') μ'
hXY : TendstoInMeasure μ'' (Y - X) l 0
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : Nonempty E
x₀ : E := hE.some
F : E → ℝ
M : ℝ
hF_bounded : ∀ (x y : E), dist (F x) (F y) ≤ M
L : NNReal
hF_lip : LipschitzWith L F
hF_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] F
hL : 0 < L
ε : ℝ
hε : ε > 0
n : ι
h_int_Y : Integrable (ε := ℝ) (fun (x : Ω'') ↦ F (Y n x)) μ''
h_int_X : Integrable (ε := ℝ) (fun (x : Ω'') ↦ F (X n x)) μ''
h_int_sub : Integrable (ε := ℝ) (fun (a : Ω'') ↦ ‖F (Y n a) - F (X n a)‖) μ''
⊢ LE.le (α := ℝ) (∫ (a : Ω'') in {x : Ω'' | ε / 2 ≤ ‖Y n x - X n x‖}, ‖F (Y n a) - F (X n a)‖ ∂μ'')
    (∫ (a : Ω'') in {x : Ω'' | ε / 2 ≤ ‖Y n x - X n x‖}, M ∂μ'')

Tactic:
setIntegral_mono h_int_sub.integrableOn integrableOn_const fun a ↦ ?_

Premises used: MeasureTheory.integrableOn_const, MeasureTheory.Integrable.integrableOn, MeasureTheory.setIntegral_mono
-- Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution
Goal:
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst_1 : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst_2 : SeminormedAddCommGroup E
inst_3 : SecondCountableTopology E
inst_4 : BorelSpace E
X : ι → Ω'' → E
inst_5 : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution (Ω := fun (i : ι) ↦ Ω'') (m := fun (i : ι) ↦ m'') (mE := mE) X l Z (fun (x : ι) ↦ μ'') μ'
hXY : TendstoInMeasure μ'' (Y - X) l 0
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : Nonempty E
x₀ : E := hE.some
F : E → ℝ
M : ℝ
hF_bounded : ∀ (x y : E), dist (F x) (F y) ≤ M
L : NNReal
hF_lip : LipschitzWith L F
hF_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] F
hL : 0 < L
ε : ℝ
hε : ε > 0
n : ι
h_int_Y : Integrable (ε := ℝ) (fun (x : Ω'') ↦ F (Y n x)) μ''
h_int_X : Integrable (ε := ℝ) (fun (x : Ω'') ↦ F (X n x)) μ''
h_int_sub : Integrable (ε := ℝ) (fun (a : Ω'') ↦ ‖F (Y n a) - F (X n a)‖) μ''
h₂ : ∀ (a : Ω''), ‖F (Y n a) - F (X n a)‖ ≤ M
⊢ LE.le (α := ℝ) (∫ (a : Ω'') in {x : Ω'' | ε / 2 ≤ ‖Y n x - X n x‖}, ‖F (Y n a) - F (X n a)‖ ∂μ'')
    (∫ (a : Ω'') in {x : Ω'' | ε / 2 ≤ ‖Y n x - X n x‖}, M ∂μ'')

Tactic:
refine setIntegral_mono h_int_sub.integrableOn integrableOn_const fun a ↦ ?_

Premises used: MeasureTheory.integrableOn_const, MeasureTheory.Integrable.integrableOn, MeasureTheory.setIntegral_mono
-- Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution
Goal:
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst_1 : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst_2 : SeminormedAddCommGroup E
inst_3 : SecondCountableTopology E
inst_4 : BorelSpace E
X : ι → Ω'' → E
inst_5 : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution (Ω := fun (i : ι) ↦ Ω'') (m := fun (i : ι) ↦ m'') (mE := mE) X l Z (fun (x : ι) ↦ μ'') μ'
hXY : TendstoInMeasure μ'' (Y - X) l 0
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : Nonempty E
x₀ : E := hE.some
F : E → ℝ
M : ℝ
hF_bounded : ∀ (x y : E), dist (F x) (F y) ≤ M
L : NNReal
hF_lip : LipschitzWith L F
hF_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] F
hL : 0 < L
ε : ℝ
hε : ε > 0
n : ι
h_int_Y : Integrable (ε := ℝ) (fun (x : Ω'') ↦ F (Y n x)) μ''
h_int_X : Integrable (ε := ℝ) (fun (x : Ω'') ↦ F (X n x)) μ''
h_int_sub : Integrable (ε := ℝ) (fun (a : Ω'') ↦ ‖F (Y n a) - F (X n a)‖) μ''
a : Ω''
⊢ dist (F (Y n a)) (F (X n a)) ≤ M

Tactic:
hF_bounded _
            _
-- Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution
Goal:
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst_1 : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst_2 : SeminormedAddCommGroup E
inst_3 : SecondCountableTopology E
inst_4 : BorelSpace E
X : ι → Ω'' → E
inst_5 : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution (Ω := fun (i : ι) ↦ Ω'') (m := fun (i : ι) ↦ m'') (mE := mE) X l Z (fun (x : ι) ↦ μ'') μ'
hXY : TendstoInMeasure μ'' (Y - X) l 0
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : Nonempty E
x₀ : E := hE.some
F : E → ℝ
M : ℝ
hF_bounded : ∀ (x y : E), dist (F x) (F y) ≤ M
L : NNReal
hF_lip : LipschitzWith L F
hF_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] F
hL : 0 < L
ε : ℝ
hε : ε > 0
n : ι
h_int_Y : Integrable (ε := ℝ) (fun (x : Ω'') ↦ F (Y n x)) μ''
h_int_X : Integrable (ε := ℝ) (fun (x : Ω'') ↦ F (X n x)) μ''
h_int_sub : Integrable (ε := ℝ) (fun (a : Ω'') ↦ ‖F (Y n a) - F (X n a)‖) μ''
a : Ω''
⊢ dist (F (Y n a)) (F (X n a)) ≤ M

Tactic:
convert!
          hF_bounded _
            _
-- Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution
Goal:
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst_1 : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst_2 : SeminormedAddCommGroup E
inst_3 : SecondCountableTopology E
inst_4 : BorelSpace E
X : ι → Ω'' → E
inst_5 : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution (Ω := fun (i : ι) ↦ Ω'') (m := fun (i : ι) ↦ m'') (mE := mE) X l Z (fun (x : ι) ↦ μ'') μ'
hXY : TendstoInMeasure μ'' (Y - X) l 0
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : Nonempty E
x₀ : E := hE.some
F : E → ℝ
M : ℝ
hF_bounded : ∀ (x y : E), dist (F x) (F y) ≤ M
L : NNReal
hF_lip : LipschitzWith L F
hF_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] F
hL : 0 < L
ε : ℝ
hε : ε > 0
n : ι
h_int_Y : Integrable (ε := ℝ) (fun (x : Ω'') ↦ F (Y n x)) μ''
h_int_X : Integrable (ε := ℝ) (fun (x : Ω'') ↦ F (X n x)) μ''
h_int_sub : Integrable (ε := ℝ) (fun (a : Ω'') ↦ ‖F (Y n a) - F (X n a)‖) μ''
simp :
  μ''.real {x : Ω'' | ‖Y n x - X n x‖ < ε / 2} * ((↑L : ℝ) * (ε / 2)) +
      μ''.real {x : Ω'' | ε / 2 ≤ ‖Y n x - X n x‖} * M =
    (↑L : ℝ) * (ε / 2) * μ''.real {x : Ω'' | ‖Y n x - X n x‖ < ε / 2} + M * μ''.real {x : Ω'' | ε / 2 ≤ ‖Y n x - X n x‖}
⊢ HAdd.hAdd (α := ℝ) (∫ (a : Ω'') in {x : Ω'' | ‖Y n x - X n x‖ < ε / 2}, (↑L : ℝ) * (ε / 2) ∂μ'')
      (∫ (a : Ω'') in {x : Ω'' | ε / 2 ≤ ‖Y n x - X n x‖}, M ∂μ'') =
    (↑L : ℝ) * (ε / 2) * μ''.real {x : Ω'' | ‖Y n x - X n x‖ < ε / 2} + M * μ''.real {ω : Ω'' | ε / 2 ≤ ‖Y n ω - X n ω‖}

Tactic:
simp only [integral_const, MeasurableSet.univ, measureReal_restrict_apply, Set.univ_inter,
        smul_eq_mul]

Premises used: Set.univ_inter, MeasureTheory.integral_const, smul_eq_mul, MeasureTheory.measureReal_restrict_apply, MeasurableSet.univ
-- Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution
Goal:
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst_1 : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst_2 : SeminormedAddCommGroup E
inst_3 : SecondCountableTopology E
inst_4 : BorelSpace E
X : ι → Ω'' → E
inst_5 : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution (Ω := fun (i : ι) ↦ Ω'') (m := fun (i : ι) ↦ m'') (mE := mE) X l Z (fun (x : ι) ↦ μ'') μ'
hXY : TendstoInMeasure μ'' (Y - X) l 0
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : Nonempty E
x₀ : E := hE.some
F : E → ℝ
M : ℝ
hF_bounded : ∀ (x y : E), dist (F x) (F y) ≤ M
L : NNReal
hF_lip : LipschitzWith L F
hF_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] F
hL : 0 < L
ε : ℝ
hε : ε > 0
n : ι
h_int_Y : Integrable (ε := ℝ) (fun (x : Ω'') ↦ F (Y n x)) μ''
h_int_X : Integrable (ε := ℝ) (fun (x : Ω'') ↦ F (X n x)) μ''
h_int_sub : Integrable (ε := ℝ) (fun (a : Ω'') ↦ ‖F (Y n a) - F (X n a)‖) μ''
⊢ μ''.real {x : Ω'' | ‖Y n x - X n x‖ < ε / 2} * ((↑L : ℝ) * (ε / 2)) +
      μ''.real {x : Ω'' | ε / 2 ≤ ‖Y n x - X n x‖} * M =
    (↑L : ℝ) * (ε / 2) * μ''.real {x : Ω'' | ‖Y n x - X n x‖ < ε / 2} + M * μ''.real {x : Ω'' | ε / 2 ≤ ‖Y n x - X n x‖}

Tactic:
ring
-- Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution
Goal:
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst_1 : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst_2 : SeminormedAddCommGroup E
inst_3 : SecondCountableTopology E
inst_4 : BorelSpace E
X : ι → Ω'' → E
inst_5 : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution (Ω := fun (i : ι) ↦ Ω'') (m := fun (i : ι) ↦ m'') (mE := mE) X l Z (fun (x : ι) ↦ μ'') μ'
hXY : TendstoInMeasure μ'' (Y - X) l 0
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : Nonempty E
x₀ : E := hE.some
F : E → ℝ
M : ℝ
hF_bounded : ∀ (x y : E), dist (F x) (F y) ≤ M
L : NNReal
hF_lip : LipschitzWith L F
hF_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] F
hL : 0 < L
ε : ℝ
hε : ε > 0
n : ι
h_int_Y : Integrable (ε := ℝ) (fun (x : Ω'') ↦ F (Y n x)) μ''
h_int_X : Integrable (ε := ℝ) (fun (x : Ω'') ↦ F (X n x)) μ''
h_int_sub : Integrable (ε := ℝ) (fun (a : Ω'') ↦ ‖F (Y n a) - F (X n a)‖) μ''
rw :
  (↑L : ℝ) * (ε / 2 * μ''.real {x : Ω'' | ‖Y n x - X n x‖ < ε / 2}) + M * μ''.real {ω : Ω'' | ε / 2 ≤ ‖Y n ω - X n ω‖} ≤
    (↑L : ℝ) * (ε / 2) + M * μ''.real {ω : Ω'' | ε / 2 ≤ ‖Y n ω - X n ω‖}
⊢ (↑L : ℝ) * (ε / 2) * μ''.real {x : Ω'' | ‖Y n x - X n x‖ < ε / 2} + M * μ''.real {ω : Ω'' | ε / 2 ≤ ‖Y n ω - X n ω‖} ≤
    (↑L : ℝ) * (ε / 2) + M * μ''.real {ω : Ω'' | ε / 2 ≤ ‖Y n ω - X n ω‖}

Tactic:
rw [mul_assoc]

Premises used: mul_assoc
-- Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution
Goal:
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst_1 : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst_2 : SeminormedAddCommGroup E
inst_3 : SecondCountableTopology E
inst_4 : BorelSpace E
X : ι → Ω'' → E
inst_5 : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution (Ω := fun (i : ι) ↦ Ω'') (m := fun (i : ι) ↦ m'') (mE := mE) X l Z (fun (x : ι) ↦ μ'') μ'
hXY : TendstoInMeasure μ'' (Y - X) l 0
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : Nonempty E
x₀ : E := hE.some
F : E → ℝ
M : ℝ
hF_bounded : ∀ (x y : E), dist (F x) (F y) ≤ M
L : NNReal
hF_lip : LipschitzWith L F
hF_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] F
hL : 0 < L
ε : ℝ
hε : ε > 0
n : ι
h_int_Y : Integrable (ε := ℝ) (fun (x : Ω'') ↦ F (Y n x)) μ''
h_int_X : Integrable (ε := ℝ) (fun (x : Ω'') ↦ F (X n x)) μ''
h_int_sub : Integrable (ε := ℝ) (fun (a : Ω'') ↦ ‖F (Y n a) - F (X n a)‖) μ''
bc : ε / 2 * μ''.real {x : Ω'' | ‖Y n x - X n x‖ < ε / 2} ≤ ε / 2
⊢ (↑L : ℝ) * (ε / 2 * μ''.real {x : Ω'' | ‖Y n x - X n x‖ < ε / 2}) + M * μ''.real {ω : Ω'' | ε / 2 ≤ ‖Y n ω - X n ω‖} ≤
    (↑L : ℝ) * (ε / 2) + M * μ''.real {ω : Ω'' | ε / 2 ≤ ‖Y n ω - X n ω‖}

Tactic:
gcongr
-- Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution
Goal:
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst_1 : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst_2 : SeminormedAddCommGroup E
inst_3 : SecondCountableTopology E
inst_4 : BorelSpace E
X : ι → Ω'' → E
inst_5 : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution (Ω := fun (i : ι) ↦ Ω'') (m := fun (i : ι) ↦ m'') (mE := mE) X l Z (fun (x : ι) ↦ μ'') μ'
hXY : TendstoInMeasure μ'' (Y - X) l 0
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : Nonempty E
x₀ : E := hE.some
F : E → ℝ
M : ℝ
hF_bounded : ∀ (x y : E), dist (F x) (F y) ≤ M
L : NNReal
hF_lip : LipschitzWith L F
hF_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] F
hL : 0 < L
ε : ℝ
hε : ε > 0
n : ι
h_int_Y : Integrable (ε := ℝ) (fun (x : Ω'') ↦ F (Y n x)) μ''
h_int_X : Integrable (ε := ℝ) (fun (x : Ω'') ↦ F (X n x)) μ''
h_int_sub : Integrable (ε := ℝ) (fun (a : Ω'') ↦ ‖F (Y n a) - F (X n a)‖) μ''
h₁ : 0 ≤ ε / 2
bc : ε / 2 * 1 ≤ ε / 2
bc₁ bc₂ : ε / 2 ≤ ε / 2
⊢ ε / 2 * μ''.real {x : Ω'' | ‖Y n x - X n x‖ < ε / 2} ≤ ε / 2

Tactic:
grw [measureReal_le_one, mul_one]

Premises used: MeasureTheory.measureReal_le_one, mul_one
-- Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution
Goal:
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst_1 : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst_2 : SeminormedAddCommGroup E
inst_3 : SecondCountableTopology E
inst_4 : BorelSpace E
X : ι → Ω'' → E
inst_5 : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution (Ω := fun (i : ι) ↦ Ω'') (m := fun (i : ι) ↦ m'') (mE := mE) X l Z (fun (x : ι) ↦ μ'') μ'
hXY : TendstoInMeasure μ'' (Y - X) l 0
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : Nonempty E
x₀ : E := hE.some
F : E → ℝ
M : ℝ
hF_bounded : ∀ (x y : E), dist (F x) (F y) ≤ M
L : NNReal
hF_lip : LipschitzWith L F
hF_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] F
hL : 0 < L
ε : ℝ
hε : ε > 0
h_le :
  ∀ (n : ι),
    |HSub.hSub (α := ℝ) (@integral _ _ _ _ mE (Measure.map (Y n) μ'') fun (ω : E) ↦ F ω)
          (@integral _ _ _ _ mE (Measure.map Z μ') fun (ω : E) ↦ F ω)| ≤
      (↑L : ℝ) * (ε / 2) + M * μ''.real {ω : Ω'' | ε / 2 ≤ ‖Y n ω - X n ω‖} +
        |HSub.hSub (α := ℝ) (@integral _ _ _ _ mE (Measure.map (X n) μ'') fun (ω : E) ↦ F ω)
            (@integral _ _ _ _ mE (Measure.map Z μ') fun (ω : E) ↦ F ω)|
this :
  Tendsto
    (fun (n : ι) ↦
      (↑L : ℝ) * (ε / 2) + M * μ''.real {ω : Ω'' | ε / 2 ≤ ‖Y n ω - X n ω‖} +
        |HSub.hSub (α := ℝ) (@integral _ _ _ _ mE (Measure.map (X n) μ'') fun (ω : E) ↦ F ω)
            (@integral _ _ _ _ mE (Measure.map Z μ') fun (ω : E) ↦ F ω)|)
    l (𝓝 ((↑L : ℝ) * ε / 2 + M * 0 + 0))
⊢ Tendsto
    (fun (n : ι) ↦
      (↑L : ℝ) * (ε / 2) + M * μ''.real {ω : Ω'' | ε / 2 ≤ ‖Y n ω - X n ω‖} +
        |HSub.hSub (α := ℝ) (@integral _ _ _ _ mE (Measure.map (X n) μ'') fun (ω : E) ↦ F ω)
            (@integral _ _ _ _ mE (Measure.map Z μ') fun (ω : E) ↦ F ω)|)
    l (𝓝 ((↑L : ℝ) * ε / 2))

Tactic:
simpa
-- Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution
Goal:
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst_1 : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst_2 : SeminormedAddCommGroup E
inst_3 : SecondCountableTopology E
inst_4 : BorelSpace E
X : ι → Ω'' → E
inst_5 : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution (Ω := fun (i : ι) ↦ Ω'') (m := fun (i : ι) ↦ m'') (mE := mE) X l Z (fun (x : ι) ↦ μ'') μ'
hXY : TendstoInMeasure μ'' (Y - X) l 0
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : Nonempty E
x₀ : E := hE.some
F : E → ℝ
M : ℝ
hF_bounded : ∀ (x y : E), dist (F x) (F y) ≤ M
L : NNReal
hF_lip : LipschitzWith L F
hF_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] F
hL : 0 < L
ε : ℝ
hε : ε > 0
h_le :
  ∀ (n : ι),
    |HSub.hSub (α := ℝ) (@integral _ _ _ _ mE (Measure.map (Y n) μ'') fun (ω : E) ↦ F ω)
          (@integral _ _ _ _ mE (Measure.map Z μ') fun (ω : E) ↦ F ω)| ≤
      (↑L : ℝ) * (ε / 2) + M * μ''.real {ω : Ω'' | ε / 2 ≤ ‖Y n ω - X n ω‖} +
        |HSub.hSub (α := ℝ) (@integral _ _ _ _ mE (Measure.map (X n) μ'') fun (ω : E) ↦ F ω)
            (@integral _ _ _ _ mE (Measure.map Z μ') fun (ω : E) ↦ F ω)|
suffices :
  Tendsto
    (fun (n : ι) ↦
      (↑L : ℝ) * (ε / 2) + M * μ''.real {ω : Ω'' | ε / 2 ≤ ‖Y n ω - X n ω‖} +
        |HSub.hSub (α := ℝ) (@integral _ _ _ _ mE (Measure.map (X n) μ'') fun (ω : E) ↦ F ω)
            (@integral _ _ _ _ mE (Measure.map Z μ') fun (ω : E) ↦ F ω)|)
    l (𝓝 ((↑L : ℝ) * ε / 2 + M * 0 + 0))
⊢ Tendsto
    (fun (n : ι) ↦
      (↑L : ℝ) * (ε / 2) + M * μ''.real {ω : Ω'' | ε / 2 ≤ ‖Y n ω - X n ω‖} +
        |HSub.hSub (α := ℝ) (@integral _ _ _ _ mE (Measure.map (X n) μ'') fun (ω : E) ↦ F ω)
            (@integral _ _ _ _ mE (Measure.map Z μ') fun (ω : E) ↦ F ω)|)
    l (𝓝 ((↑L : ℝ) * ε / 2))

Tactic:
suffices Tendsto (fun n ↦ L * (ε / 2) + M * μ''.real {ω | ε / 2 ≤ ‖Y n ω - X n ω‖}
        + |∫ ω, F ω ∂(μ''.map (X n)) - ∫ ω, F ω ∂(μ'.map Z)|) l (𝓝 (L * ε / 2 + M * 0 + 0)) by
      simpa
-- Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution
Goal:
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst_1 : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst_2 : SeminormedAddCommGroup E
inst_3 : SecondCountableTopology E
inst_4 : BorelSpace E
X : ι → Ω'' → E
inst_5 : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution (Ω := fun (i : ι) ↦ Ω'') (m := fun (i : ι) ↦ m'') (mE := mE) X l Z (fun (x : ι) ↦ μ'') μ'
hXY : TendstoInMeasure μ'' (Y - X) l 0
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : Nonempty E
x₀ : E := hE.some
F : E → ℝ
M : ℝ
hF_bounded : ∀ (x y : E), dist (F x) (F y) ≤ M
L : NNReal
hF_lip : LipschitzWith L F
hF_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] F
hL : 0 < L
ε : ℝ
hε : ε > 0
h_le :
  ∀ (n : ι),
    |HSub.hSub (α := ℝ) (@integral _ _ _ _ mE (Measure.map (Y n) μ'') fun (ω : E) ↦ F ω)
          (@integral _ _ _ _ mE (Measure.map Z μ') fun (ω : E) ↦ F ω)| ≤
      (↑L : ℝ) * (ε / 2) + M * μ''.real {ω : Ω'' | ε / 2 ≤ ‖Y n ω - X n ω‖} +
        |HSub.hSub (α := ℝ) (@integral _ _ _ _ mE (Measure.map (X n) μ'') fun (ω : E) ↦ F ω)
            (@integral _ _ _ _ mE (Measure.map Z μ') fun (ω : E) ↦ F ω)|
⊢ Tendsto
    (fun (n : ι) ↦
      (↑L : ℝ) * (ε / 2) + M * μ''.real {ω : Ω'' | ε / 2 ≤ ‖Y n ω - X n ω‖} +
        |HSub.hSub (α := ℝ) (@integral _ _ _ _ mE (Measure.map (X n) μ'') fun (ω : E) ↦ F ω)
            (@integral _ _ _ _ mE (Measure.map Z μ') fun (ω : E) ↦ F ω)|)
    l (𝓝 ((↑L : ℝ) * ε / 2 + M * 0 + 0))

Tactic:
(Tendsto.add ?_ (Tendsto.const_mul _ ?_)).add ?_

Premises used: Filter.Tendsto.const_mul, Filter.Tendsto.add
-- Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution
Goal:
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst_1 : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst_2 : SeminormedAddCommGroup E
inst_3 : SecondCountableTopology E
inst_4 : BorelSpace E
X : ι → Ω'' → E
inst_5 : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution (Ω := fun (i : ι) ↦ Ω'') (m := fun (i : ι) ↦ m'') (mE := mE) X l Z (fun (x : ι) ↦ μ'') μ'
hXY : TendstoInMeasure μ'' (Y - X) l 0
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : Nonempty E
x₀ : E := hE.some
F : E → ℝ
M : ℝ
hF_bounded : ∀ (x y : E), dist (F x) (F y) ≤ M
L : NNReal
hF_lip : LipschitzWith L F
hF_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] F
hL : 0 < L
ε : ℝ
hε : ε > 0
h_le :
  ∀ (n : ι),
    |HSub.hSub (α := ℝ) (@integral _ _ _ _ mE (Measure.map (Y n) μ'') fun (ω : E) ↦ F ω)
          (@integral _ _ _ _ mE (Measure.map Z μ') fun (ω : E) ↦ F ω)| ≤
      (↑L : ℝ) * (ε / 2) + M * μ''.real {ω : Ω'' | ε / 2 ≤ ‖Y n ω - X n ω‖} +
        |HSub.hSub (α := ℝ) (@integral _ _ _ _ mE (Measure.map (X n) μ'') fun (ω : E) ↦ F ω)
            (@integral _ _ _ _ mE (Measure.map Z μ') fun (ω : E) ↦ F ω)|
refine_1 : Tendsto (fun (n : ι) ↦ (↑L : ℝ) * (ε / 2)) l (𝓝 ((↑L : ℝ) * ε / 2))
refine_2 : Tendsto (β := ℝ) (fun (n : ι) ↦ μ''.real {ω : Ω'' | ε / 2 ≤ ‖Y n ω - X n ω‖}) l (𝓝 0)
refine_3 :
  Tendsto (β := ℝ)
    (fun (n : ι) ↦
      |HSub.hSub (α := ℝ) (@integral _ _ _ _ mE (Measure.map (X n) μ'') fun (ω : E) ↦ F ω)
          (@integral _ _ _ _ mE (Measure.map Z μ') fun (ω : E) ↦ F ω)|)
    l (𝓝 0)
⊢ Tendsto
    (fun (n : ι) ↦
      (↑L : ℝ) * (ε / 2) + M * μ''.real {ω : Ω'' | ε / 2 ≤ ‖Y n ω - X n ω‖} +
        |HSub.hSub (α := ℝ) (@integral _ _ _ _ mE (Measure.map (X n) μ'') fun (ω : E) ↦ F ω)
            (@integral _ _ _ _ mE (Measure.map Z μ') fun (ω : E) ↦ F ω)|)
    l (𝓝 ((↑L : ℝ) * ε / 2 + M * 0 + 0))

Tactic:
refine (Tendsto.add ?_ (Tendsto.const_mul _ ?_)).add ?_

Premises used: Filter.Tendsto.const_mul, Filter.Tendsto.add