mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
s₀ s₁ t : Set X
s₀_compact : IsCompact s₀
s₁_compact : IsCompact s₁
t_compact : IsCompact t
disj : Disjoint s₀ s₁
hst : s₀ ∪ s₁ ⊆ t
so : Fin 2 → Set X := ⋯
hso : so = fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
soopen : ∀ (j : Fin 2), IsOpen (so j)
hsot : t ⊆ ⋃ j, so j
f : Fin 2 → C(X, ℝ)
f_supp_in_so : ∀ (i : Fin 2), tsupport ⇑(f i) ⊆ so i
sum_f_one_on_t : EqOn (∑ i, ⇑(f i)) 1 t
f_in_icc : ∀ (i : Fin 2) (x : X), (f i) x ∈ Icc 0 1
f_hcs : ∀ (i : Fin 2), HasCompactSupport ⇑(f i)
sum_one_x : ∀ x ∈ t, (f 0) x + (f 1) x = 1
x : X
hx : x ∈ s₀
⊢ s₀ ⊆ (so 0)ᶜ

Tactic:
hso

Goal state after:
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
s₀ s₁ t : Set X
s₀_compact : IsCompact s₀
s₁_compact : IsCompact s₁
t_compact : IsCompact t
disj : Disjoint s₀ s₁
hst : s₀ ∪ s₁ ⊆ t
so : Fin 2 → Set X := fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
hso : so = fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
soopen : ∀ (j : Fin 2), IsOpen (so j)
hsot : t ⊆ ⋃ j, so j
f : Fin 2 → C(X, ℝ)
f_supp_in_so : ∀ (i : Fin 2), tsupport ⇑(f i) ⊆ so i
sum_f_one_on_t : EqOn (∑ i, ⇑(f i)) 1 t
f_in_icc : ∀ (i : Fin 2) (x : X), (f i) x ∈ Icc 0 1
f_hcs : ∀ (i : Fin 2), HasCompactSupport ⇑(f i)
sum_one_x : ∀ x ∈ t, (f 0) x + (f 1) x = 1
x : X
hx : x ∈ s₀
⊢ s₀ ⊆ ((fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ) 0)ᶜ
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
s₀ s₁ t : Set X
s₀_compact : IsCompact s₀
s₁_compact : IsCompact s₁
t_compact : IsCompact t
disj : Disjoint s₀ s₁
hst : s₀ ∪ s₁ ⊆ t
so : Fin 2 → Set X := ⋯
hso : so = fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
soopen : ∀ (j : Fin 2), IsOpen (so j)
hsot : t ⊆ ⋃ j, so j
f : Fin 2 → C(X, ℝ)
f_supp_in_so : ∀ (i : Fin 2), tsupport ⇑(f i) ⊆ so i
sum_f_one_on_t : EqOn (∑ i, ⇑(f i)) 1 t
f_in_icc : ∀ (i : Fin 2) (x : X), (f i) x ∈ Icc 0 1
f_hcs : ∀ (i : Fin 2), HasCompactSupport ⇑(f i)
sum_one_x : ∀ x ∈ t, (f 0) x + (f 1) x = 1
x : X
hx : x ∈ s₁
⊢ s₁ ⊆ (so 1)ᶜ

Tactic:
hso

Goal state after:
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
s₀ s₁ t : Set X
s₀_compact : IsCompact s₀
s₁_compact : IsCompact s₁
t_compact : IsCompact t
disj : Disjoint s₀ s₁
hst : s₀ ∪ s₁ ⊆ t
so : Fin 2 → Set X := fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
hso : so = fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
soopen : ∀ (j : Fin 2), IsOpen (so j)
hsot : t ⊆ ⋃ j, so j
f : Fin 2 → C(X, ℝ)
f_supp_in_so : ∀ (i : Fin 2), tsupport ⇑(f i) ⊆ so i
sum_f_one_on_t : EqOn (∑ i, ⇑(f i)) 1 t
f_in_icc : ∀ (i : Fin 2) (x : X), (f i) x ∈ Icc 0 1
f_hcs : ∀ (i : Fin 2), HasCompactSupport ⇑(f i)
sum_one_x : ∀ x ∈ t, (f 0) x + (f 1) x = 1
x : X
hx : x ∈ s₁
⊢ s₁ ⊆ ((fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ) 1)ᶜ
Module: Mathlib.MeasureTheory.MeasurableSpace.Card

Goal state before:
pp: α : Type u
s : Set (Set α)
i : Ordinal.{u_1}
⊢ s ⊆ generateMeasurableRec s i

Tactic:
by
  unfold generateMeasurableRec
  apply_rules [subset_union_of_subset_left]
  exact subset_rfl

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

Goal state before:
pp: E : Type u_4
F : Type u_5
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ E
inst✝² : InnerProductSpace ℝ F
mE : MeasurableSpace E
mF : MeasurableSpace F
μ : Measure E
ν : Measure F
inst✝¹ : SFinite μ
inst✝ : SFinite ν
t : WithLp 2 (E × F)
⊢ ∫ (x : WithLp 2 (E × F)), cexp (↑(⟪x.ofLp.1, t.ofLp.1⟫ + ⟪x.ofLp.2, t.ofLp.2⟫) * I) ∂Measure.map (toLp 2) (μ.prod ν) =
    (∫ (x : E), cexp (↑⟪x, t.ofLp.1⟫ * I) ∂μ) * ∫ (x : F), cexp (↑⟪x, t.ofLp.2⟫ * I) ∂ν

Tactic:
← MeasurableEquiv.coe_toLp,

Goal state after:
E : Type u_4
F : Type u_5
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ E
inst✝² : InnerProductSpace ℝ F
mE : MeasurableSpace E
mF : MeasurableSpace F
μ : Measure E
ν : Measure F
inst✝¹ : SFinite μ
inst✝ : SFinite ν
t : WithLp 2 (E × F)
⊢ ∫ (x : WithLp 2 (E × F)),
      cexp
        (↑(⟪x.ofLp.1, t.ofLp.1⟫ + ⟪x.ofLp.2, t.ofLp.2⟫) *
          I) ∂Measure.map (⇑(MeasurableEquiv.toLp 2 (E × F))) (μ.prod ν) =
    (∫ (x : E), cexp (↑⟪x, t.ofLp.1⟫ * I) ∂μ) * ∫ (x : F), cexp (↑⟪x, t.ofLp.2⟫ * I) ∂ν
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_4
F : Type u_5
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ E
inst✝² : InnerProductSpace ℝ F
mE : MeasurableSpace E
mF : MeasurableSpace F
μ : Measure E
ν : Measure F
inst✝¹ : SFinite μ
inst✝ : SFinite ν
t : WithLp 2 (E × F)
⊢ ∫ (x : WithLp 2 (E × F)),
      cexp
        (↑(⟪x.ofLp.1, t.ofLp.1⟫ + ⟪x.ofLp.2, t.ofLp.2⟫) *
          I) ∂Measure.map (⇑(MeasurableEquiv.toLp 2 (E × F))) (μ.prod ν) =
    (∫ (x : E), cexp (↑⟪x, t.ofLp.1⟫ * I) ∂μ) * ∫ (x : F), cexp (↑⟪x, t.ofLp.2⟫ * I) ∂ν

Tactic:
← integral_prod_mul,

Goal state after:
E : Type u_4
F : Type u_5
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ E
inst✝² : InnerProductSpace ℝ F
mE : MeasurableSpace E
mF : MeasurableSpace F
μ : Measure E
ν : Measure F
inst✝¹ : SFinite μ
inst✝ : SFinite ν
t : WithLp 2 (E × F)
⊢ ∫ (x : WithLp 2 (E × F)),
      cexp
        (↑(⟪x.ofLp.1, t.ofLp.1⟫ + ⟪x.ofLp.2, t.ofLp.2⟫) *
          I) ∂Measure.map (⇑(MeasurableEquiv.toLp 2 (E × F))) (μ.prod ν) =
    ∫ (z : E × F), cexp (↑⟪z.1, t.ofLp.1⟫ * I) * cexp (↑⟪z.2, t.ofLp.2⟫ * I) ∂μ.prod ν
Module: Mathlib.Topology.Separation.CompletelyRegular

Goal state before:
pp: ι : Type u_1
X : Type u_2
t : ι → TopologicalSpace X
ht : ∀ (i : ι), CompletelyRegularSpace X
this : TopologicalSpace X := ⨅ i, t i
x : X
K : Set X
hK : IsOpen[⨅ i, t i] K
hxK : K ∈ 𝓝 x
⊢ ∃ f, Continuous[⨅ i, t i, _] f ∧ f x = 0 ∧ EqOn f 1 Kᶜ

Tactic:
nhds_iInf,

Goal state after:
ι : Type u_1
X : Type u_2
t : ι → TopologicalSpace X
ht : ∀ (i : ι), CompletelyRegularSpace X
this : TopologicalSpace X := ⨅ i, t i
x : X
K : Set X
hK : IsOpen[⨅ i, t i] K
hxK : K ∈ ⨅ i, 𝓝 x
⊢ ∃ f, Continuous[⨅ i, t i, _] f ∧ f x = 0 ∧ EqOn f 1 Kᶜ
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: ι : Type u_6
inst✝³ : Fintype ι
E : ι → Type u_7
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → InnerProductSpace ℝ (E i)
mE : (i : ι) → MeasurableSpace (E i)
μ : (i : ι) → Measure (E i)
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
t : PiLp 2 E
⊢ ∫ (x : WithLp 2 ((i : ι) → E i)), cexp (↑(∑ i, ⟪x.ofLp i, t.ofLp i⟫) * I) ∂Measure.map (toLp 2) (Measure.pi μ) =
    ∏ i, ∫ (x : E i), cexp (↑⟪x, t.ofLp i⟫ * I) ∂μ i

Tactic:
← MeasurableEquiv.coe_toLp,

Goal state after:
ι : Type u_6
inst✝³ : Fintype ι
E : ι → Type u_7
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → InnerProductSpace ℝ (E i)
mE : (i : ι) → MeasurableSpace (E i)
μ : (i : ι) → Measure (E i)
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
t : PiLp 2 E
⊢ ∫ (x : WithLp 2 ((i : ι) → E i)),
      cexp (↑(∑ i, ⟪x.ofLp i, t.ofLp i⟫) * I) ∂Measure.map (⇑(MeasurableEquiv.toLp 2 ((i : ι) → E i))) (Measure.pi μ) =
    ∏ i, ∫ (x : E i), cexp (↑⟪x, t.ofLp i⟫ * I) ∂μ i
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
mF : MeasurableSpace F
μ : Measure E
ν : Measure F
p : ℝ≥0∞
inst✝² : Fact (1 ≤ p)
inst✝¹ : SFinite μ
inst✝ : SFinite ν
L : StrongDual ℝ (WithLp p (E × F))
⊢ ∫ (v : WithLp p (E × F)), cexp (↑(L v) * I) ∂Measure.map (toLp p) (μ.prod ν) =
    (∫ (v : E),
        cexp (↑((L ∘SL ↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inl ℝ E F) v) * I) ∂μ) *
      ∫ (v : F), cexp (↑((L ∘SL ↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inr ℝ E F) v) * I) ∂ν

Tactic:
← integral_prod_mul,

Goal state after:
E : Type u_2
F : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
mF : MeasurableSpace F
μ : Measure E
ν : Measure F
p : ℝ≥0∞
inst✝² : Fact (1 ≤ p)
inst✝¹ : SFinite μ
inst✝ : SFinite ν
L : StrongDual ℝ (WithLp p (E × F))
⊢ ∫ (v : WithLp p (E × F)), cexp (↑(L v) * I) ∂Measure.map (toLp p) (μ.prod ν) =
    ∫ (z : E × F),
      cexp (↑((L ∘SL ↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inl ℝ E F) z.1) * I) *
        cexp (↑((L ∘SL ↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inr ℝ E F) z.2) * I) ∂μ.prod ν
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
mF : MeasurableSpace F
μ : Measure E
ν : Measure F
p : ℝ≥0∞
inst✝² : Fact (1 ≤ p)
inst✝¹ : SFinite μ
inst✝ : SFinite ν
L : StrongDual ℝ (WithLp p (E × F))
⊢ ∫ (v : WithLp p (E × F)), cexp (↑(L v) * I) ∂Measure.map (toLp p) (μ.prod ν) =
    ∫ (z : E × F), cexp (↑(L (↑(prodContinuousLinearEquiv p ℝ E F).symm z)) * I) ∂μ.prod ν

Tactic:
← MeasurableEquiv.coe_toLp,

Goal state after:
E : Type u_2
F : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
mF : MeasurableSpace F
μ : Measure E
ν : Measure F
p : ℝ≥0∞
inst✝² : Fact (1 ≤ p)
inst✝¹ : SFinite μ
inst✝ : SFinite ν
L : StrongDual ℝ (WithLp p (E × F))
⊢ ∫ (v : WithLp p (E × F)), cexp (↑(L v) * I) ∂Measure.map (⇑(MeasurableEquiv.toLp p (E × F))) (μ.prod ν) =
    ∫ (z : E × F), cexp (↑(L (↑(prodContinuousLinearEquiv p ℝ E F).symm z)) * I) ∂μ.prod ν
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: p : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ p)
ι : Type u_4
inst✝⁴ : Fintype ι
inst✝³ : DecidableEq ι
E : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace ℝ (E i)
mE : (i : ι) → MeasurableSpace (E i)
μ : (i : ι) → Measure (E i)
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
L : StrongDual ℝ (PiLp p E)
⊢ ∫ (v : WithLp p ((i : ι) → E i)), cexp (↑(L v) * I) ∂Measure.map (toLp p) (Measure.pi μ) =
    ∫ (x : (i : ι) → E i), cexp (↑(L (↑(PiLp.continuousLinearEquiv p ℝ E).symm x)) * I) ∂Measure.pi μ

Tactic:
← MeasurableEquiv.coe_toLp,

Goal state after:
p : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ p)
ι : Type u_4
inst✝⁴ : Fintype ι
inst✝³ : DecidableEq ι
E : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace ℝ (E i)
mE : (i : ι) → MeasurableSpace (E i)
μ : (i : ι) → Measure (E i)
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
L : StrongDual ℝ (PiLp p E)
⊢ ∫ (v : WithLp p ((i : ι) → E i)),
      cexp (↑(L v) * I) ∂Measure.map (⇑(MeasurableEquiv.toLp p ((i : ι) → E i))) (Measure.pi μ) =
    ∫ (x : (i : ι) → E i), cexp (↑(L (↑(PiLp.continuousLinearEquiv p ℝ E).symm x)) * I) ∂Measure.pi μ
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : PseudoEMetricSpace Ω
μ ν : Measure Ω
δ : ℝ≥0∞
h :
  ∀ (ε : ℝ≥0∞) (B : Set Ω),
    0 < ε →
      ε < ∞ →
        MeasurableSet B → μ B ≤ ν (thickening (δ + ε).toReal B) + δ + ε ∧ ν B ≤ μ (thickening (δ + ε).toReal B) + δ + ε
⊢ levyProkhorovEDist μ ν ≤ δ

Tactic:
by
  apply ENNReal.le_of_forall_pos_le_add
  intro ε hε _
  by_cases ε_top : ε = ∞
  · simp only [ε_top, add_top, le_top]
  apply sInf_le
  intro B B_mble
  simpa only [add_assoc] using h ε B (by positivity) coe_lt_top B_mble

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

Goal state before:
pp: Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoEMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ ν : Measure Ω
hLP : levyProkhorovEDist μ ν = 0
s : Set Ω
s_closed : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s
hμs : ∃ δ > 0, μ (thickening δ s) ≠ ∞
hνs : ∃ δ > 0, ν (thickening δ s) ≠ ∞
⊢ μ s = ν s

Tactic:
by
  apply le_antisymm
  · exact measure_le_measure_closure_of_levyProkhorovEDist_eq_zero
      hLP s_closed.measurableSet hνs |>.trans <|
      le_of_eq (congr_arg _ s_closed.closure_eq)
  · exact measure_le_measure_closure_of_levyProkhorovEDist_eq_zero
      (levyProkhorovEDist_comm μ ν ▸ hLP) s_closed.measurableSet hμs |>.trans <|
      le_of_eq (congr_arg _ s_closed.closure_eq)

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

Goal state before:
pp: case inr
Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
ε : ℝ
ε_pos : 0 < ε
h✝ : Nonempty Ω
⊢ ∃ As,
    (∀ (n : ℕ), MeasurableSet (As n)) ∧
      (∀ (n : ℕ), Bornology.IsBounded (As n)) ∧
        (∀ (n : ℕ), diam (As n) ≤ ε) ∧ ⋃ n, As n = univ ∧ Pairwise fun n m ↦ Disjoint (As n) (As m)

Tactic:
obtain ⟨xs, xs_dense⟩ := exists_dense_seq Ω

Goal state after:
case inr
Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
ε : ℝ
ε_pos : 0 < ε
h✝ : Nonempty Ω
xs : ℕ → Ω
xs_dense : DenseRange xs
⊢ ∃ As,
    (∀ (n : ℕ), MeasurableSet (As n)) ∧
      (∀ (n : ℕ), Bornology.IsBounded (As n)) ∧
        (∀ (n : ℕ), diam (As n) ≤ ε) ∧ ⋃ n, As n = univ ∧ Pairwise fun n m ↦ Disjoint (As n) (As m)
Module: Mathlib.MeasureTheory.Measure.Haar.Extension

Goal state before:
pp: A : Type u_1
B : Type u_2
C : Type u_3
inst✝¹⁷ : Group A
inst✝¹⁶ : Group B
inst✝¹⁵ : Group C
inst✝¹⁴ : TopologicalSpace A
inst✝¹³ : TopologicalSpace B
inst✝¹² : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst✝¹¹ : IsTopologicalGroup A
inst✝¹⁰ : IsTopologicalGroup B
inst✝⁹ : MeasurableSpace A
inst✝⁸ : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst✝⁷ : IsTopologicalGroup C
inst✝⁶ : LocallyCompactSpace B
inst✝⁵ : MeasurableSpace C
inst✝⁴ : BorelSpace C
μC : Measure C
hμC : μC.IsHaarMeasure
inst✝³ : T2Space B
inst✝² : MeasurableSpace B
inst✝¹ : BorelSpace B
U : Set B
hU : IsOpen U
inst✝ : DiscreteTopology A
h✝ : μC Set.univ * μA {1} < (H.inducedMeasure μA μC) U
ho : 0 < μA {1}
ht : μA {1} < ⊤
K : Set B
hKU : K ⊆ U
hK : IsCompact K
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 :
  μC Set.univ * μA {1} <
    ENNReal.ofReal (∫ (c : C), ((H.pushforward μA) { toContinuousMap := f, hasCompactSupport' := hf2 }) c ∂μC)
hc : Set.InjOn (⇑ψ) U
b : B
⊢ ∫ (a : A), f (b * φ a) ∂μA ≤ (μA {1}).toReal

Tactic:
← setIntegral_support

Goal state after:
A : Type u_1
B : Type u_2
C : Type u_3
inst✝¹⁷ : Group A
inst✝¹⁶ : Group B
inst✝¹⁵ : Group C
inst✝¹⁴ : TopologicalSpace A
inst✝¹³ : TopologicalSpace B
inst✝¹² : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst✝¹¹ : IsTopologicalGroup A
inst✝¹⁰ : IsTopologicalGroup B
inst✝⁹ : MeasurableSpace A
inst✝⁸ : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst✝⁷ : IsTopologicalGroup C
inst✝⁶ : LocallyCompactSpace B
inst✝⁵ : MeasurableSpace C
inst✝⁴ : BorelSpace C
μC : Measure C
hμC : μC.IsHaarMeasure
inst✝³ : T2Space B
inst✝² : MeasurableSpace B
inst✝¹ : BorelSpace B
U : Set B
hU : IsOpen U
inst✝ : DiscreteTopology A
h✝ : μC Set.univ * μA {1} < (H.inducedMeasure μA μC) U
ho : 0 < μA {1}
ht : μA {1} < ⊤
K : Set B
hKU : K ⊆ U
hK : IsCompact K
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 :
  μC Set.univ * μA {1} <
    ENNReal.ofReal (∫ (c : C), ((H.pushforward μA) { toContinuousMap := f, hasCompactSupport' := hf2 }) c ∂μC)
hc : Set.InjOn (⇑ψ) U
b : B
⊢ ∫ (x : A) in Function.support fun a ↦ f (b * φ a), f (b * φ x) ∂μA ≤ (μA {1}).toReal
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
⊢ ∫ᶜ (x : E) in Path.segment a b, ω x + ∫ᶜ (x : E) in Path.segment b c, ω x = ∫ᶜ (x : E) in Path.segment a c, ω x

Tactic:
set φ := ContinuousMap.Homotopy.affine (Path.segment a b : C(I, E)) (Path.segment a c)
  have hφs : range φ ⊆ s := by
    rw [range_subset_iff]
    intro x
    simp [φ, ha, hb, hc, hs.lineMap_mem]
  have := φ.curveIntegral_add_curveIntegral_eq_of_hasFDerivWithinAt (t := range φ) (ω := ω)
    (dω := dω) ?_ ?_ ?_ ?_ ?_
  · convert! this using 2
    · dsimp [φ]
      rw [← Path.cast_segment (lineMap_apply_one a b) (lineMap_apply_one a c), curveIntegral_cast]
    · dsimp [φ]
      rw [← Path.cast_segment (lineMap_apply_zero a b) (lineMap_apply_zero a c)]
      simp
  · intros
    apply mem_range_self
  · exact fun x hx ↦ (hω x (hφs hx)).mono hφs
  · rw [(isCompact_range <| map_continuous _).isClosed.closure_eq]
    exact fun x hx ↦ (hω x <| hφs hx).continuousWithinAt.mono hφs
  · intro x hx u hu v hv
    apply hdω <;> grw [← hφs] <;> assumption
  · have : EqOn (fun x : ℝ × ℝ ↦ IccExtend zero_le_one (φ.extend x.1) x.2)
        (fun x ↦ lineMap (lineMap a b x.2) (lineMap a c x.2) x.1) (Icc 0 1) := by
      rw [Icc_prod_eq]
      rintro ⟨x, y⟩ ⟨hx, hy⟩
      lift x to I using hx
      lift y to I using hy
      simp [φ]
    exact .congr (by fun_prop) this

Goal state after:
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)
h : ∀ (y : E), ‖y‖ = 1 → Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
y : E
hy : ¬y = 0
⊢ ‖(↑‖y‖)⁻¹ • y‖ = 1

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)
y : E
h : Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪(↑‖y‖)⁻¹ • y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
hy : ¬y = 0
⊢ Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | ‖y‖⁻¹ * r < ‖⟪(↑‖y‖)⁻¹ • y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)

Tactic:
· simp only [norm_smul, norm_inv, norm_algebraMap', Real.norm_eq_abs, abs_norm]
      rw [inv_mul_cancel₀ (by positivity)]

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)
y : E
h : Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪(↑‖y‖)⁻¹ • y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
hy : ¬y = 0
⊢ Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | ‖y‖⁻¹ * r < ‖⟪(↑‖y‖)⁻¹ • y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
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)
h : ∀ (y : E), ‖y‖ = 1 → Tendsto (fun r ↦ limsup (fun n ↦ (μ n).real {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
C : ℝ≥0
hμ : ∀ᶠ (n : ℕ) in atTop, (μ n) Set.univ ≤ ↑C
⊢ IsTightMeasureSet (Set.range μ)

Tactic:
refine isTightMeasureSet_range_of_tendsto_limsup_inner_of_norm_eq_one 𝕜 fun z hz ↦ ?_

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 : ∀ (y : E), ‖y‖ = 1 → Tendsto (fun r ↦ limsup (fun n ↦ (μ n).real {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
C : ℝ≥0
hμ : ∀ᶠ (n : ℕ) in atTop, (μ n) Set.univ ≤ ↑C
z : E
hz : ‖z‖ = 1
⊢ 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: case refine_3
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)‖
r : ℝ
hr : 0 < r
hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹
⊢ ∀ᵐ (a : ℝ) ∂volume.restrict (Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹)),
    Tendsto (fun n ↦ 1 - charFun (μ n) (a • z)) atTop (𝓝 (1 - f (a • z)))

Tactic:
· exact ae_of_all _ fun _ ↦ tendsto_const_nhds.sub (h _)

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

Goal state before:
pp: case refine_1
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
ν : ℕ → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n)).InnerRegular
νK : ∀ (n : ℕ), (ν n) ((partialSups K) n)ᶜ = 0
hν : ∀ (n : ℕ), Tendsto (fun ρ ↦ ρ.restrict (disjointed K n)) (↑f) (𝓝 (ν n))
B : (Measure.sum fun n ↦ ↑(ν n)) univ ≤ ↑C
μ : FiniteMeasure E := ⟨Measure.sum fun n ↦ ↑(ν n), ⋯⟩
L : Tendsto id (↑f) (𝓝 μ)
⊢ ((Measure.sum fun n ↦ ↑(ν n)) univ).toNNReal ≤ C

Tactic:
show C = (C : ℝ≥0∞).toNNReal by simp

Goal state after:
case refine_1
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
ν : ℕ → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n)).InnerRegular
νK : ∀ (n : ℕ), (ν n) ((partialSups K) n)ᶜ = 0
hν : ∀ (n : ℕ), Tendsto (fun ρ ↦ ρ.restrict (disjointed K n)) (↑f) (𝓝 (ν n))
B : (Measure.sum fun n ↦ ↑(ν n)) univ ≤ ↑C
μ : FiniteMeasure E := ⟨Measure.sum fun n ↦ ↑(ν n), ⋯⟩
L : Tendsto id (↑f) (𝓝 μ)
⊢ ((Measure.sum fun n ↦ ↑(ν n)) univ).toNNReal ≤ (↑C).toNNReal
Module: Mathlib.MeasureTheory.Measure.PreVariation

Goal state before:
pp: X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
s : Set X
hs : MeasurableSet s
ε : ℝ≥0
hε : 0 < ε
h : preVariationFun f s ≠ ∞
⊢ ∃ P, preVariationFun f s ≤ ∑ p ∈ P.parts, f ↑p + ↑ε

Tactic:
by
  let ε' := min ε (preVariationFun f s).toNNReal
  have hε' : ε' ≤ preVariationFun f s := by simp_all [ε']
  have : ε' ≤ ε := by simp_all [ε']
  obtain hw | hw : preVariationFun f s ≠ 0 ∨ preVariationFun f s = 0 := ne_or_eq _ _
  · have : 0 < ε' := by
      simp only [lt_inf_iff, ε']
      exact ⟨hε, toNNReal_pos hw h⟩
    let a := preVariationFun f s - ε'
    have ha : a < preVariationFun f s := ENNReal.sub_lt_self h hw (by positivity)
    obtain ⟨P, hP⟩ := exists_Finpartition_sum_gt f hs ha
    use P
    calc preVariationFun f s
      _ = a + ε' := (tsub_add_cancel_of_le hε').symm
      _ ≤ ∑ p ∈ P.parts, f p + ε' := by
        exact (ENNReal.add_le_add_iff_right coe_ne_top).mpr (le_of_lt hP)
      _ ≤ ∑ p ∈ P.parts, f p + ε := by gcongr
  · simp [*]

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

Goal state before:
pp: X : Type u_1
inst✝ : MeasurableSpace X
⊢ IsSigmaSubadditiveSetFun 0

Tactic:
intro; simp

Goal state after: