mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Independence.Basic

Goal state before:
pp: case refine_1
Ω : Type u_10
Ω' : Type u_11
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
μ : Measure Ω
ν : Measure Ω'
inst✝³ : IsProbabilityMeasure μ
inst✝² : IsProbabilityMeasure ν
𝓧 : Type u_12
𝓨 : Type u_13
inst✝¹ : MeasurableSpace 𝓧
inst✝ : MeasurableSpace 𝓨
X : Ω → 𝓧
Y : Ω' → 𝓨
mX : AEMeasurable X μ
mY : AEMeasurable Y ν
this : (fun ω ↦ AEMeasurable.mk X mX ω.1) ⟂ᵢ[μ.prod ν] fun ω ↦ AEMeasurable.mk Y mY ω.2
⊢ (fun ω ↦ AEMeasurable.mk X mX ω.1) =ᵐ[μ.prod ν] fun ω ↦ X ω.1

Tactic:
rw [← Function.comp_def, ← Function.comp_def]

Goal state after:
case refine_1
Ω : Type u_10
Ω' : Type u_11
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
μ : Measure Ω
ν : Measure Ω'
inst✝³ : IsProbabilityMeasure μ
inst✝² : IsProbabilityMeasure ν
𝓧 : Type u_12
𝓨 : Type u_13
inst✝¹ : MeasurableSpace 𝓧
inst✝ : MeasurableSpace 𝓨
X : Ω → 𝓧
Y : Ω' → 𝓨
mX : AEMeasurable X μ
mY : AEMeasurable Y ν
this : (fun ω ↦ AEMeasurable.mk X mX ω.1) ⟂ᵢ[μ.prod ν] fun ω ↦ AEMeasurable.mk Y mY ω.2
⊢ AEMeasurable.mk X mX ∘ Prod.fst =ᵐ[μ.prod ν] X ∘ Prod.fst
Module: Mathlib.Probability.Independence.Basic

Goal state before:
pp: case refine_2
Ω : Type u_10
Ω' : Type u_11
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
μ : Measure Ω
ν : Measure Ω'
inst✝³ : IsProbabilityMeasure μ
inst✝² : IsProbabilityMeasure ν
𝓧 : Type u_12
𝓨 : Type u_13
inst✝¹ : MeasurableSpace 𝓧
inst✝ : MeasurableSpace 𝓨
X : Ω → 𝓧
Y : Ω' → 𝓨
mX : AEMeasurable X μ
mY : AEMeasurable Y ν
this : (fun ω ↦ AEMeasurable.mk X mX ω.1) ⟂ᵢ[μ.prod ν] fun ω ↦ AEMeasurable.mk Y mY ω.2
⊢ (fun ω ↦ AEMeasurable.mk Y mY ω.2) =ᵐ[μ.prod ν] fun ω ↦ Y ω.2

Tactic:
rw [← Function.comp_def, ← Function.comp_def]

Goal state after:
case refine_2
Ω : Type u_10
Ω' : Type u_11
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
μ : Measure Ω
ν : Measure Ω'
inst✝³ : IsProbabilityMeasure μ
inst✝² : IsProbabilityMeasure ν
𝓧 : Type u_12
𝓨 : Type u_13
inst✝¹ : MeasurableSpace 𝓧
inst✝ : MeasurableSpace 𝓨
X : Ω → 𝓧
Y : Ω' → 𝓨
mX : AEMeasurable X μ
mY : AEMeasurable Y ν
this : (fun ω ↦ AEMeasurable.mk X mX ω.1) ⟂ᵢ[μ.prod ν] fun ω ↦ AEMeasurable.mk Y mY ω.2
⊢ AEMeasurable.mk Y mY ∘ Prod.snd =ᵐ[μ.prod ν] Y ∘ Prod.snd
Module: Mathlib.Probability.Density

Goal state before:
pp: Ω : Type u_1
E : Type u_2
inst✝ : MeasurableSpace E
m : MeasurableSpace Ω
X : Ω → E
ℙ : Measure Ω
μ : Measure E
hX : HasPDF X ℙ μ
s : Set E
hs : MeasurableSet s
⊢ (map X ℙ) s = ∫⁻ (x : E) in s, pdf X ℙ μ x ∂μ

Tactic:
rw [← withDensity_apply _ hs, map_eq_withDensity_pdf X ℙ μ]

Goal state after:
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: case inr
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
m : ι → MeasurableSpace Ω
h_le : ∀ (i : ι), m i ≤ _mΩ
π : ι → Set (Set Ω)
h_pi : ∀ (n : ι), IsPiSystem (π n)
h_generate : ∀ (i : ι), m i = generateFrom (π i)
h_ind : iIndepSets π κ μ
hμ : μ ≠ 0
η : Kernel α Ω
η_eq : ⇑κ =ᵐ[μ] ⇑η
hη : IsMarkovKernel η
⊢ Kernel.iIndep m η μ

Tactic:
intro s f

Goal state after:
case inr
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
m : ι → MeasurableSpace Ω
h_le : ∀ (i : ι), m i ≤ _mΩ
π : ι → Set (Set Ω)
h_pi : ∀ (n : ι), IsPiSystem (π n)
h_generate : ∀ (i : ι), m i = generateFrom (π i)
h_ind : iIndepSets π κ μ
hμ : μ ≠ 0
η : Kernel α Ω
η_eq : ⇑κ =ᵐ[μ] ⇑η
hη : IsMarkovKernel η
s : Finset ι
f : ι → Set Ω
⊢ (∀ i ∈ s, f i ∈ (fun x ↦ {s | MeasurableSet s}) i) → ∀ᵐ (a : α) ∂μ, (η a) (⋂ i ∈ s, f i) = ∏ i ∈ s, (η a) (f i)
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
s t : Set Ω
m0 : MeasurableSpace Ω
hs_meas : MeasurableSet s
ht_meas : MeasurableSet t
κ : Kernel α Ω
μ : Measure α
inst✝ : IsZeroOrMarkovKernel κ
h : IndepSets {s} {t} κ μ
u : Set Ω
hu : u ∈ {s}
⊢ MeasurableSet u

Tactic:
by rwa [Set.mem_singleton_iff.mp hu]

Goal state after:
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
s t : Set Ω
m0 : MeasurableSpace Ω
hs_meas : MeasurableSet s
ht_meas : MeasurableSet t
κ : Kernel α Ω
μ : Measure α
inst✝ : IsZeroOrMarkovKernel κ
h : IndepSets {s} {t} κ μ
u : Set Ω
hu : u ∈ {t}
⊢ MeasurableSet u

Tactic:
by rwa [Set.mem_singleton_iff.mp hu]

Goal state after:
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hXm : AEStronglyMeasurable X μ
hX : ¬MemLp X 2 μ
h : eVar[X; μ] < ∞
this : MemLp (fun ω ↦ X ω - ∫ (x : Ω), X x ∂μ) 2 μ
ω : Ω
⊢ X ω = ((fun ω ↦ X ω - ∫ (x : Ω), X x ∂μ) + fun x ↦ ∫ (x : Ω), X x ∂μ) ω

Tactic:
rw [Pi.add_apply, sub_add_cancel]

Goal state after:
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
ι : Type u_2
s : Finset ι
X : ι → Ω → ℝ
inst✝ : IsFiniteMeasure μ
hX : ∀ i ∈ s, MemLp (X i) 2 μ
⊢ AEMeasurable (∑ i ∈ s, X i) μ

Tactic:
exact (memLp_finsetSum' _ (by simpa)).aemeasurable

Goal state after:
Module: Mathlib.Probability.Moments.Covariance

Goal state before:
pp: case insert.hX
Ω : Type u_1
mΩ : MeasurableSpace Ω
Y : Ω → ℝ
μ : Measure Ω
ι : Type u_2
X : ι → Ω → ℝ
s✝ : Finset ι
inst✝ : IsFiniteMeasure μ
hY : MemLp Y 2 μ
i : ι
s : Finset ι
hi : i ∉ s
h_ind : (∀ i ∈ s, MemLp (X i) 2 μ) → cov[∑ i ∈ s, X i, Y; μ] = ∑ i ∈ s, cov[X i, Y; μ]
hX : ∀ i_1 ∈ insert i s, MemLp (X i_1) 2 μ
⊢ MemLp (X i) 2 μ

pp: case insert.hY
Ω : Type u_1
mΩ : MeasurableSpace Ω
Y : Ω → ℝ
μ : Measure Ω
ι : Type u_2
X : ι → Ω → ℝ
s✝ : Finset ι
inst✝ : IsFiniteMeasure μ
hY : MemLp Y 2 μ
i : ι
s : Finset ι
hi : i ∉ s
h_ind : (∀ i ∈ s, MemLp (X i) 2 μ) → cov[∑ i ∈ s, X i, Y; μ] = ∑ i ∈ s, cov[X i, Y; μ]
hX : ∀ i_1 ∈ insert i s, MemLp (X i_1) 2 μ
⊢ MemLp (∑ x ∈ s, X x) 2 μ

pp: case insert.hZ
Ω : Type u_1
mΩ : MeasurableSpace Ω
Y : Ω → ℝ
μ : Measure Ω
ι : Type u_2
X : ι → Ω → ℝ
s✝ : Finset ι
inst✝ : IsFiniteMeasure μ
hY : MemLp Y 2 μ
i : ι
s : Finset ι
hi : i ∉ s
h_ind : (∀ i ∈ s, MemLp (X i) 2 μ) → cov[∑ i ∈ s, X i, Y; μ] = ∑ i ∈ s, cov[X i, Y; μ]
hX : ∀ i_1 ∈ insert i s, MemLp (X i_1) 2 μ
⊢ MemLp Y 2 μ

Tactic:
· exact hX i (by simp)

Goal state after:
case insert.hY
Ω : Type u_1
mΩ : MeasurableSpace Ω
Y : Ω → ℝ
μ : Measure Ω
ι : Type u_2
X : ι → Ω → ℝ
s✝ : Finset ι
inst✝ : IsFiniteMeasure μ
hY : MemLp Y 2 μ
i : ι
s : Finset ι
hi : i ∉ s
h_ind : (∀ i ∈ s, MemLp (X i) 2 μ) → cov[∑ i ∈ s, X i, Y; μ] = ∑ i ∈ s, cov[X i, Y; μ]
hX : ∀ i_1 ∈ insert i s, MemLp (X i_1) 2 μ
⊢ MemLp (∑ x ∈ s, X x) 2 μ

case insert.hZ
Ω : Type u_1
mΩ : MeasurableSpace Ω
Y : Ω → ℝ
μ : Measure Ω
ι : Type u_2
X : ι → Ω → ℝ
s✝ : Finset ι
inst✝ : IsFiniteMeasure μ
hY : MemLp Y 2 μ
i : ι
s : Finset ι
hi : i ∉ s
h_ind : (∀ i ∈ s, MemLp (X i) 2 μ) → cov[∑ i ∈ s, X i, Y; μ] = ∑ i ∈ s, cov[X i, Y; μ]
hX : ∀ i_1 ∈ insert i s, MemLp (X i_1) 2 μ
⊢ MemLp Y 2 μ
Module: Mathlib.Probability.Moments.Covariance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
Y : Ω → ℝ
μ : Measure Ω
ι : Type u_2
X : ι → Ω → ℝ
s : Finset ι
inst✝ : IsFiniteMeasure μ
hX : ∀ i ∈ s, MemLp (X i) 2 μ
hY : MemLp Y 2 μ
⊢ cov[Y, fun ω ↦ ∑ i ∈ s, X i ω; μ] = ∑ i ∈ s, cov[Y, X i; μ]

Tactic:
convert! covariance_sum_right' hX hY

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
Y : Ω → ℝ
μ : Measure Ω
ι : Type u_2
X : ι → Ω → ℝ
s : Finset ι
inst✝ : IsFiniteMeasure μ
hX : ∀ i ∈ s, MemLp (X i) 2 μ
hY : MemLp Y 2 μ
x✝ : Ω
⊢ ∑ i ∈ s, X i x✝ = (∑ i ∈ s, X i) x✝
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: ι : Type u_2
inst✝¹ : Fintype ι
Ω : ι → Type u_3
mΩ : (i : ι) → MeasurableSpace (Ω i)
μ : (i : ι) → Measure (Ω i)
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μ i)
X : (i : ι) → Ω i → ℝ
h : ∀ (i : ι), MemLp (X i) 2 (μ i)
⊢ Var[∑ i, fun ω ↦ X i (ω i); Measure.pi μ] = ∑ i, Var[X i; μ i]

Tactic:
rw [IndepFun.variance_sum]
  · congr with i
    change Var[(X i) ∘ (fun ω ↦ ω i); Measure.pi μ] = _
    rw [← variance_map, (measurePreserving_eval _ i).map_eq]
    · rw [(measurePreserving_eval _ i).map_eq]
      exact (h i).aestronglyMeasurable.aemeasurable
    · exact Measurable.aemeasurable (by fun_prop)
  · exact fun i _ ↦ (h i).comp_measurePreserving (measurePreserving_eval _ i)
  · exact fun i _ j _ hij ↦
      (iIndepFun_pi fun i ↦ (h i).aestronglyMeasurable.aemeasurable).indepFun hij

Goal state after:
Module: Mathlib.Probability.HasLaw

Goal state before:
pp: case hX
Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
X : Ω → 𝓧
μ : Measure 𝓧
P : Measure Ω
hX : HasLaw X μ P
f g : 𝓧 → ℝ
hf : AEMeasurable f μ
hg : AEMeasurable g μ
⊢ AEStronglyMeasurable f (map X P)

Tactic:
rw [hX.map_eq]
    exact hf.aestronglyMeasurable

Goal state after:
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: case neg.e_a
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
inst✝¹ : NormedAddCommGroup γ
inst✝ : OpensMeasurableSpace γ
h : IdentDistrib f g μ ν
p : ℝ≥0∞
h0 : ¬p = 0
h_top : ¬p = ∞
⊢ IdentDistrib (fun x ↦ ‖f x‖ₑ ^ p.toReal) (fun x ↦ ‖g x‖ₑ ^ p.toReal) μ ν

Tactic:
exact h.comp (Measurable.pow_const (measurable_coe_nnreal_ennreal.comp measurable_nnnorm)
    p.toReal)

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

Goal state before:
pp: case inr
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁵ : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst✝⁴ : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst✝³ : SeminormedAddCommGroup E
inst✝² : SecondCountableTopology E
inst✝¹ : BorelSpace E
X : ι → Ω'' → E
inst✝ : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution X l Z (fun x ↦ 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
⊢ Tendsto (fun n ↦ ∫ (ω : E), F ω ∂Measure.map (Y n) μ'') l (𝓝 (∫ (ω : E), F ω ∂Measure.map Z μ'))

Tactic:
have hF_cont : Continuous F := hF_lip.continuous

Goal state after:
case inr
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁵ : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst✝⁴ : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst✝³ : SeminormedAddCommGroup E
inst✝² : SecondCountableTopology E
inst✝¹ : BorelSpace E
X : ι → Ω'' → E
inst✝ : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution X l Z (fun x ↦ 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
⊢ Tendsto (fun n ↦ ∫ (ω : E), F ω ∂Measure.map (Y n) μ'') l (𝓝 (∫ (ω : E), F ω ∂Measure.map Z μ'))
Module: Mathlib.MeasureTheory.Integral.CircleTransform

Goal state before:
pp: case hg.hg.hf
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
hR : 0 < R
f : ℂ → E
z w : ℂ
hf : ContinuousOn f (sphere z R)
hw : w ∈ ball z R
⊢ Continuous fun θ ↦ (circleMap z R θ - w)⁻¹

pp: case hg.hg.hg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
hR : 0 < R
f : ℂ → E
z w : ℂ
hf : ContinuousOn f (sphere z R)
hw : w ∈ ball z R
⊢ Continuous fun θ ↦ f (circleMap z R θ)

Tactic:
· exact continuous_circleMap_inv hw

Goal state after:
case hg.hg.hg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
hR : 0 < R
f : ℂ → E
z w : ℂ
hf : ContinuousOn f (sphere z R)
hw : w ∈ ball z R
⊢ Continuous fun θ ↦ f (circleMap z R θ)
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
x : α
inst✝ : Zero β
⊢ HasCompactSupport ⇑0

Tactic:
simp [HasCompactSupport, tsupport]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Gamma

Goal state before:
pp: p q : ℝ
hp : 0 < p
hq : -1 < q
⊢ ∫ (x : ℝ) in Ioi 0, x ^ q * rexp (-x ^ p) = 1 / p * Gamma ((q + 1) / p)

Tactic:
calc
    _ = ∫ (x : ℝ) in Ioi 0, (1 / p * x ^ (1 / p - 1)) • ((x ^ (1 / p)) ^ q * exp (-x)) := by
      rw [← integral_comp_rpow_Ioi _ (one_div_ne_zero (ne_of_gt hp)),
        abs_eq_self.mpr (le_of_lt (one_div_pos.mpr hp))]
      refine setIntegral_congr_fun measurableSet_Ioi (fun _ hx => ?_)
      rw [← rpow_mul (le_of_lt hx) _ p, one_div_mul_cancel (ne_of_gt hp), rpow_one]
    _ = ∫ (x : ℝ) in Ioi 0, 1 / p * exp (-x) * x ^ (1 / p - 1 + q / p) := by
      simp_rw [smul_eq_mul, mul_assoc]
      refine setIntegral_congr_fun measurableSet_Ioi (fun _ hx => ?_)
      rw [← rpow_mul (le_of_lt hx), div_mul_eq_mul_div, one_mul, rpow_add hx]
      ring_nf
    _ = (1 / p) * Gamma ((q + 1) / p) := by
      rw [Gamma_eq_integral (div_pos (neg_lt_iff_pos_add.mp hq) hp)]
      simp_rw [show 1 / p - 1 + q / p = (q + 1) / p - 1 by ring, ← integral_const_mul,
        ← mul_assoc]

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_1 ∈ Ioo 0 1, ∀ b_1 ∈ Ioo 0 1, φ (a_1, b_1) ∈ t
hω : ∀ a_1 ∈ Ioo 0 1, ∀ b_1 ∈ Ioo 0 1, (a_1, b_1) ∉ s → HasFDerivWithinAt ω (dω (φ (a_1, b_1))) t (φ (a_1, b_1))
hωc : ContinuousOn ω (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t)
hdω_symm :
  ∀ a_1 ∈ Ioo 0 1,
    ∀ b_1 ∈ Ioo 0 1,
      (a_1, b_1) ∉ s →
        ∀ u ∈ tangentConeAt ℝ t (φ (a_1, b_1)),
          ∀ v ∈ tangentConeAt ℝ t (φ (a_1, b_1)), ((dω (φ (a_1, b_1))) u) v = ((dω (φ (a_1, 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_1 b_1 : ↑I), ψ (↑a_1, ↑b_1) = φ (a_1, b_1)
hψ_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ψ
⊢ ∫ᶜ (x : E) in γ₁, ω x + ∫ᶜ (x : E) in φ.evalAt 1, ω x = ∫ᶜ (x : E) in γ₂, ω x + ∫ᶜ (x : E) in φ.evalAt 0, ω x

Tactic:
have hψUt : MapsTo ψ U t := by
    rintro ⟨a, b⟩ ⟨ha, hb⟩
    lift a to I using Ioo_subset_Icc_self ha
    lift b to I using Ioo_subset_Icc_self hb
    simpa [hψφ] using hφt a ha b hb

Goal state after:
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_1 ∈ Ioo 0 1, ∀ b_1 ∈ Ioo 0 1, φ (a_1, b_1) ∈ t
hω : ∀ a_1 ∈ Ioo 0 1, ∀ b_1 ∈ Ioo 0 1, (a_1, b_1) ∉ s → HasFDerivWithinAt ω (dω (φ (a_1, b_1))) t (φ (a_1, b_1))
hωc : ContinuousOn ω (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t)
hdω_symm :
  ∀ a_1 ∈ Ioo 0 1,
    ∀ b_1 ∈ Ioo 0 1,
      (a_1, b_1) ∉ s →
        ∀ u ∈ tangentConeAt ℝ t (φ (a_1, b_1)),
          ∀ v ∈ tangentConeAt ℝ t (φ (a_1, b_1)), ((dω (φ (a_1, b_1))) u) v = ((dω (φ (a_1, 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_1 b_1 : ↑I), ψ (↑a_1, ↑b_1) = φ (a_1, b_1)
hψ_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ψ
hψUt : MapsTo ψ U t
⊢ ∫ᶜ (x : E) in γ₁, ω x + ∫ᶜ (x : E) in φ.evalAt 1, ω x = ∫ᶜ (x : E) in γ₂, ω x + ∫ᶜ (x : E) in φ.evalAt 0, ω x
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ
ε : ℝ
hε : 0 < ε
E : Set X
μ : Content X
hμ : μ.outerMeasure E ≠ ∞
hμ' : MeasurableSet E
c : ℝ
hfE : ∀ x ∈ E, f x < c
⊢ ∃ V, E ⊆ ↑V ∧ (∀ x ∈ V, f x < c) ∧ μ.measure ↑V ≤ μ.measure E + ENNReal.ofReal ε

Tactic:
have hε' := ne_of_gt <| Real.toNNReal_pos.mpr hε
  obtain ⟨V₁ : Opens X, hV₁⟩ := Content.outerMeasure_exists_open μ hμ hε'
  let V₂ : Opens X := ⟨(f ⁻¹' Iio c), IsOpen.preimage f.1.2 isOpen_Iio⟩
  use V₁ ⊓ V₂
  refine ⟨subset_inter hV₁.1 hfE, ?_, ?_⟩
  · intro x hx
    suffices ∀ x ∈ V₂.carrier, f x < c from this x (mem_of_mem_inter_right hx)
    exact fun _ a ↦ a
  · calc
      _ ≤ μ.measure V₁ := by simp [measure_mono]
      _ = μ.outerMeasure V₁ := Content.measure_apply μ (V₁.2.measurableSet)
      _ ≤ μ.outerMeasure E + ε.toNNReal := hV₁.2
      _ = _ := by rw [Content.measure_apply μ hμ', ENNReal.ofNNReal_toNNReal]

Goal state after:
Module: Mathlib.MeasureTheory.MeasurableSpace.Card

Goal state before:
pp: α : Type u
s : Set (Set α)
i✝¹ i✝ i : Ordinal.{v}
IH : ∀ y < i, y ≤ ω_ 1 → #↑(generateMeasurableRec s y) ≤ max (#↑s) 2 ^ ℵ₀
hi : i ≤ ω_ 1
A : 𝔠 ≤ max (#↑s) 2 ^ ℵ₀
B : ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀
C : #↑(⋃ j, ⋃ (_ : j < i), generateMeasurableRec s j) ≤ max (#↑s) 2 ^ ℵ₀
⊢ #↑((s ∪ {∅} ∪ compl '' ⋃ j, ⋃ (_ : j < i), generateMeasurableRec s j) ∪ range fun f ↦ ⋃ n, ↑(f n)) ≤ max (#↑s) 2 ^ ℵ₀

Tactic:
apply_rules [(mk_union_le _ _).trans, add_le_of_le (aleph_one_le_continuum.trans A),
    mk_image_le.trans]

Goal state after:
case h1.h1.h1
α : Type u
s : Set (Set α)
i✝¹ i✝ i : Ordinal.{v}
IH : ∀ y < i, y ≤ ω_ 1 → #↑(generateMeasurableRec s y) ≤ max (#↑s) 2 ^ ℵ₀
hi : i ≤ ω_ 1
A : 𝔠 ≤ max (#↑s) 2 ^ ℵ₀
B : ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀
C : #↑(⋃ j, ⋃ (_ : j < i), generateMeasurableRec s j) ≤ max (#↑s) 2 ^ ℵ₀
⊢ #↑s ≤ max (#↑s) 2 ^ ℵ₀

case h1.h1.h2
α : Type u
s : Set (Set α)
i✝¹ i✝ i : Ordinal.{v}
IH : ∀ y < i, y ≤ ω_ 1 → #↑(generateMeasurableRec s y) ≤ max (#↑s) 2 ^ ℵ₀
hi : i ≤ ω_ 1
A : 𝔠 ≤ max (#↑s) 2 ^ ℵ₀
B : ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀
C : #↑(⋃ j, ⋃ (_ : j < i), generateMeasurableRec s j) ≤ max (#↑s) 2 ^ ℵ₀
⊢ #↑{∅} ≤ max (#↑s) 2 ^ ℵ₀

case h2
α : Type u
s : Set (Set α)
i✝¹ i✝ i : Ordinal.{v}
IH : ∀ y < i, y ≤ ω_ 1 → #↑(generateMeasurableRec s y) ≤ max (#↑s) 2 ^ ℵ₀
hi : i ≤ ω_ 1
A : 𝔠 ≤ max (#↑s) 2 ^ ℵ₀
B : ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀
C : #↑(⋃ j, ⋃ (_ : j < i), generateMeasurableRec s j) ≤ max (#↑s) 2 ^ ℵ₀
⊢ #↑(range fun f ↦ ⋃ n, ↑(f n)) ≤ max (#↑s) 2 ^ ℵ₀
Module: Mathlib.MeasureTheory.Measure.FiniteMeasureExt

Goal state before:
pp: E : Type u_1
𝕜 : Type u_2
inst✝⁷ : RCLike 𝕜
inst✝⁶ : MeasurableSpace E
inst✝⁵ : PseudoEMetricSpace E
inst✝⁴ : BorelSpace E
inst✝³ : CompleteSpace E
inst✝² : SecondCountableTopology E
P P' : Measure E
inst✝¹ : IsFiniteMeasure P
inst✝ : IsFiniteMeasure P'
A : StarSubalgebra 𝕜 (E →ᵇ 𝕜)
hA : (StarSubalgebra.map (toContinuousMapStarₐ 𝕜) A).SeparatesPoints
heq : ∀ g ∈ A, ∫ (x : E), g x ∂P = ∫ (x : E), g x ∂P'
⊢ P = P'

Tactic:
let A_toReal := (A.restrictScalars ℝ).comap
    (ofRealAm.compLeftContinuousBounded ℝ lipschitzWith_ofReal)
  --the real subalgebra separates points
  have hA_toReal : (A_toReal.map (toContinuousMapₐ ℝ)).SeparatesPoints := by
    rw [RCLike.restrict_toContinuousMap_eq_toContinuousMapStar_restrict]
    exact Subalgebra.SeparatesPoints.rclike_to_real hA
  --integrals of elements of the real subalgebra w.r.t. P, P', respectively, coincide
  have heq' : ∀ g ∈ A_toReal, ∫ x, (g : E → ℝ) x ∂P = ∫ x, (g : E → ℝ) x ∂P' := by
    intro g hgA_toReal
    rw [← @ofReal_inj 𝕜, ← integral_ofReal, ← integral_ofReal]
    exact heq _ hgA_toReal
  apply ext_of_forall_integral_eq_of_IsFiniteMeasure
  intro f
  have h0 : Tendsto (fun ε : ℝ => 6 * √ε) (𝓝[>] 0) (𝓝 0) := by
    nth_rewrite 3 [← mul_zero 6]
    apply tendsto_nhdsWithin_of_tendsto_nhds (Tendsto.const_mul 6 _)
    nth_rewrite 2 [← sqrt_zero]
    exact Continuous.tendsto continuous_sqrt 0
  have lim1 : Tendsto (fun ε => |∫ x, mulExpNegMulSq ε (f x) ∂P - ∫ x, mulExpNegMulSq ε (f x) ∂P'|)
      (𝓝[>] 0) (𝓝 0) := by
    apply squeeze_zero' (eventually_nhdsWithin_of_forall (fun x _ => abs_nonneg _))
      (eventually_nhdsWithin_of_forall _) h0
    exact fun ε hε => dist_integral_mulExpNegMulSq_comp_le f hA_toReal heq' hε
  have lim2 : Tendsto (fun ε => |∫ x, mulExpNegMulSq ε (f x) ∂P
      - ∫ x, mulExpNegMulSq ε (f x) ∂P'|) (𝓝[>] 0)
      (𝓝 |∫ x, f x ∂↑P - ∫ x, f x ∂↑P'|) :=
    Tendsto.abs (Tendsto.sub (tendsto_integral_mulExpNegMulSq_comp f)
      (tendsto_integral_mulExpNegMulSq_comp f))
  exact eq_of_abs_sub_eq_zero (tendsto_nhds_unique lim2 lim1)

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

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
mF : MeasurableSpace F
μ : Measure E
inst✝¹ : OpensMeasurableSpace E
inst✝ : BorelSpace F
L : E →L[ℝ] F
L' : StrongDual ℝ F
⊢ charFunDual (Measure.map (⇑L) μ) L' = charFunDual μ (L' ∘SL L)

Tactic:
charFunDual_eq_charFun_map_one,

Goal state after:
E : Type u_2
F : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
mF : MeasurableSpace F
μ : Measure E
inst✝¹ : OpensMeasurableSpace E
inst✝ : BorelSpace F
L : E →L[ℝ] F
L' : StrongDual ℝ F
⊢ charFun (Measure.map (⇑L') (Measure.map (⇑L) μ)) 1 = charFunDual μ (L' ∘SL L)
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
mF : MeasurableSpace F
μ : Measure E
inst✝¹ : OpensMeasurableSpace E
inst✝ : BorelSpace F
L : E →L[ℝ] F
L' : StrongDual ℝ F
⊢ charFun (Measure.map (⇑L') (Measure.map (⇑L) μ)) 1 = charFunDual μ (L' ∘SL L)

Tactic:
charFunDual_eq_charFun_map_one,

Goal state after:
E : Type u_2
F : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
mF : MeasurableSpace F
μ : Measure E
inst✝¹ : OpensMeasurableSpace E
inst✝ : BorelSpace F
L : E →L[ℝ] F
L' : StrongDual ℝ F
⊢ charFun (Measure.map (⇑L') (Measure.map (⇑L) μ)) 1 = charFun (Measure.map (⇑(L' ∘SL L)) μ) 1
Module: Mathlib.Topology.Separation.CompletelyRegular

Goal state before:
pp: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompletelyRegularSpace X
hX : #X < 𝔠
x : X
s : Set X
hxs : x ∈ s
hs : IsOpen[inst✝¹] s
f : X → ↑I
hfc : Continuous[inst✝¹, _] f
hf₀ : f x = 0
hf₁ : EqOn f 1 sᶜ
R : Set ↑I := range f
hR : #↑R < #↑I
r : ↑I
hr : r ∈ Rᶜ
⊢ ∃ v ∈ {s | IsClopen s}, x ∈ v ∧ v ⊆ s

Tactic:
have hr' : ∀ (x : X), f x ≠ r := by simpa [R] using hr

Goal state after:
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompletelyRegularSpace X
hX : #X < 𝔠
x : X
s : Set X
hxs : x ∈ s
hs : IsOpen[inst✝¹] s
f : X → ↑I
hfc : Continuous[inst✝¹, _] f
hf₀ : f x = 0
hf₁ : EqOn f 1 sᶜ
R : Set ↑I := range f
hR : #↑R < #↑I
r : ↑I
hr : r ∈ Rᶜ
hr' : ∀ (x : X), f x ≠ r
⊢ ∃ v ∈ {s | IsClopen s}, x ∈ v ∧ v ⊆ s