mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Group.FoelnerFilter

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝³ : MeasurableSpace X
μ : Measure X
inst✝² : Group G
inst✝¹ : MulAction G X
ι : Type u_3
u : Ultrafilter ι
F : ι → Set X
inst✝ : SMulInvariantMeasure G X μ
hfoel : IsFoelner G μ (↑u) F
g✝ h✝ : G
s : Set X
g h : G
i : ι
hi : μ (F i) ≠ 0
⊢ μ (g • s ∩ F i) / μ (F i) ≤ μ (h • s ∩ F i) / μ (F i) + μ ((g⁻¹ • F i) ∆ (h⁻¹ • F i)) / μ (F i)

Tactic:
← tsub_le_iff_left,

Goal state after:
G : Type u_1
X : Type u_2
inst✝³ : MeasurableSpace X
μ : Measure X
inst✝² : Group G
inst✝¹ : MulAction G X
ι : Type u_3
u : Ultrafilter ι
F : ι → Set X
inst✝ : SMulInvariantMeasure G X μ
hfoel : IsFoelner G μ (↑u) F
g✝ h✝ : G
s : Set X
g h : G
i : ι
hi : μ (F i) ≠ 0
⊢ μ (g • s ∩ F i) / μ (F i) - μ (h • s ∩ F i) / μ (F i) ≤ μ ((g⁻¹ • F i) ∆ (h⁻¹ • F i)) / μ (F i)
Module: Mathlib.MeasureTheory.Integral.CircleTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
z w : ℂ
f : ℂ → E
⊢ Periodic (circleTransformDeriv R z w f) (2 * π)

Tactic:
simp [circleTransformDeriv, periodic_circleMap z R _, periodic_circleMap 0 R _]

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

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
s : Set α
f g : α → ℝ
μ : Measure α
hs_conn : IsConnected s
hs_meas : MeasurableSet s
hf : ContinuousOn f s
hg : IntegrableOn g s μ
hfg : IntegrableOn (fun x ↦ f x * g x) s μ
hg0 : ∀ᵐ (x : α) ∂μ.restrict s, 0 ≤ g x
ρ : α → ENNReal := fun x ↦ ENNReal.ofReal (g x)
ν : Measure α := μ.withDensity ρ
hρ_ae : AEMeasurable ρ (μ.restrict s)
hρ_top : ∀ᵐ (x : α) ∂μ.restrict s, ρ x < ⊤
h_toReal_ae : (fun x ↦ (ρ x).toReal) =ᵐ[μ.restrict s] g
heq : ∫ (x : α) in s, f x * g x ∂μ = ∫ (x : α) in s, f x ∂ν
⊢ ∃ c ∈ s, ∫ (x : α) in s, f x * g x ∂μ = f c * ∫ (x : α) in s, g x ∂μ

Tactic:
have hg1 : ∫ x in s, g x ∂μ = ∫ x in s, (1 : ℝ) ∂ν := by
    have h := setIntegral_withDensity_eq_setIntegral_toReal_smul₀
      hρ_ae hρ_top (fun _ ↦ (1 : ℝ)) hs_meas
    calc
      _ = ∫ x in s, (ρ x).toReal ∂μ := by rw [integral_congr_ae h_toReal_ae]
      _ = _ := by simp [ν, h]

Goal state after:
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
s : Set α
f g : α → ℝ
μ : Measure α
hs_conn : IsConnected s
hs_meas : MeasurableSet s
hf : ContinuousOn f s
hg : IntegrableOn g s μ
hfg : IntegrableOn (fun x ↦ f x * g x) s μ
hg0 : ∀ᵐ (x : α) ∂μ.restrict s, 0 ≤ g x
ρ : α → ENNReal := fun x ↦ ENNReal.ofReal (g x)
ν : Measure α := μ.withDensity ρ
hρ_ae : AEMeasurable ρ (μ.restrict s)
hρ_top : ∀ᵐ (x : α) ∂μ.restrict s, ρ x < ⊤
h_toReal_ae : (fun x ↦ (ρ x).toReal) =ᵐ[μ.restrict s] g
heq : ∫ (x : α) in s, f x * g x ∂μ = ∫ (x : α) in s, f x ∂ν
hg1 : ∫ (x : α) in s, g x ∂μ = ∫ (x : α) in s, 1 ∂ν
⊢ ∃ c ∈ s, ∫ (x : α) in s, f x * g x ∂μ = f c * ∫ (x : α) in s, g x ∂μ
Module: Mathlib.MeasureTheory.Integral.MeanValue

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
s : Set α
f g : α → ℝ
μ : Measure α
hs_conn : IsConnected s
hs_meas : MeasurableSet s
hf : ContinuousOn f s
hg : IntegrableOn g s μ
hfg : IntegrableOn (fun x ↦ f x * g x) s μ
hg0 : ∀ᵐ (x : α) ∂μ.restrict s, 0 ≤ g x
ρ : α → ENNReal := fun x ↦ ENNReal.ofReal (g x)
ν : Measure α := μ.withDensity ρ
hρ_ae : AEMeasurable ρ (μ.restrict s)
hρ_top : ∀ᵐ (x : α) ∂μ.restrict s, ρ x < ⊤
h_toReal_ae : (fun x ↦ (ρ x).toReal) =ᵐ[μ.restrict s] g
heq : ∫ (x : α) in s, f x * g x ∂μ = ∫ (x : α) in s, f x ∂ν
hg1 : g =ᵐ[μ.restrict s] 0
hν0 : ∫ (x : α) in s, 1 ∂ν = 0
c : α
hc : c ∈ s
x : α
hx : g x = 0 x
⊢ (fun x ↦ f x * g x) x = 0 x

Tactic:
simp [hx]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.MeanValue

Goal state before:
pp: a✝ b✝ : ℝ
f g : ℝ → ℝ
μ : Measure ℝ
a b : ℝ
hf : ContinuousOn f [[a, b]]
hg : IntervalIntegrable g μ a b
hg0 : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), 0 ≤ g x
h✝ : ¬a = b
hab : a < b
s : Set ℝ := Ι a b
hs : s = Ioc a b
hs' : s ⊆ [[a, b]]
hs_conn : IsConnected s
hfg : IntegrableOn (fun x ↦ f x * g x) s μ
c : ℝ
hc : c ∈ s
h : ∫ (x : ℝ) in s, f x * g x ∂μ = f c * ∫ (x : ℝ) in s, g x ∂μ
⊢ ∫ (x : ℝ) in a..b, f x * g x ∂μ = f c * ∫ (x : ℝ) in a..b, g x ∂μ

Tactic:
simpa [intervalIntegral.integral_of_le hab.le, hs] using h

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : SecondCountableTopology E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
f : α → ℝ≥0
hf : Measurable f
g g' : α → E
g'meas : Measurable g'
hg' : g =ᵐ[μ.withDensity fun x ↦ ↑(f x)] g'
A : MeasurableSet {x | f x ≠ 0}
x : α
hx : f x = 0
⊢ ↑(f x) • g x = ↑(f x) • g' x

Tactic:
simp [hx]

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

Goal state before:
pp: case a
X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : T2Space X
U : Set X
hU : IsOpen U
μ : Measure X
inst✝² : IsFiniteMeasure μ
inst✝¹ : μ.InnerRegularCompactLTTop
inst✝ : NormalSpace X
f : X → ℝ
f_cont : Continuous f
fU : EqOn f 0 Uᶜ
f_nonneg : 0 ≤ f
⊢ f ≤ 1 → ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ μ U

Tactic:
f_le

Goal state after:
case a
X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
inst✝³ : T2Space X
U : Set X
hU : IsOpen U
μ : Measure X
inst✝² : IsFiniteMeasure μ
inst✝¹ : μ.InnerRegularCompactLTTop
inst✝ : NormalSpace X
f : X → ℝ
f_cont : Continuous f
fU : EqOn f 0 Uᶜ
f_nonneg : 0 ≤ f
f_le : f ≤ 1
⊢ ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ μ U
Module: Mathlib.Order.Interval.Set.Union

Goal state before:
pp: X : Type u_1
inst✝ : LinearOrder X
N : ℕ
a : ℕ → X
⊢ Ioc (a 0) (a N) ⊆ ⋃ i ∈ Finset.range N, Ioc (a i) (a (i + 1))

Tactic:
induction N with

Goal state after:
Module: Mathlib.Order.Interval.Set.Union

Goal state before:
pp: X : Type u_1
inst✝ : LinearOrder X
N : ℕ
a : ℕ → X
⊢ Ico (a 0) (a N) ⊆ ⋃ i ∈ Finset.range N, Ico (a i) (a (i + 1))

Tactic:
induction N with

Goal state after:
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
f₁ f₂ : X →C_c ℝ≥0
h : f₁ ≤ f₂
⊢ Λ f₁ ≤ Λ f₂

Tactic:
obtain ⟨g, hg⟩ := CompactlySupportedContinuousMap.exists_add_of_le h

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
f₁ f₂ : X →C_c ℝ≥0
h : f₁ ≤ f₂
g : X →C_c ℝ≥0
hg : f₁ + g = f₂
⊢ Λ f₁ ≤ Λ f₂
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

Goal state before:
pp: case h.refine_2
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
⊢ (f 0) x = 1

Tactic:
have : (f 1) x = 0 := by
      rw [← notMem_support]
      have : s₁ ⊆ (tsupport (f 1))ᶜ := by
        apply subset_trans _ (compl_subset_compl.mpr (f_supp_in_so 1))
        rw [hso]
        simp only [Fin.isValue, one_ne_zero, ↓reduceIte, compl_compl, subset_refl]
      apply notMem_of_mem_compl
      exact mem_of_subset_of_mem (subset_trans this (compl_subset_compl_of_subset subset_closure))
        hx

Goal state after:
case h.refine_2
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
this : (f 1) x = 0
⊢ (f 0) x = 1
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

Goal state before:
pp: case neg
X : Type u_1
inst✝² : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
K₁ K₂ : Compacts X
disj : Disjoint ↑K₁ ↑K₂
b : ℝ≥0
f : X →C_c ℝ≥0
hf : f ∈ {f | ∀ x ∈ K₁ ⊔ K₂, 1 ≤ f x}
Λf_eq_b : Λ f = b
hsuppf : ∀ x ∈ K₁ ⊔ K₂, x ∈ support ⇑f
hsubsuppf : ↑K₁ ∪ ↑K₂ ⊆ tsupport ⇑f
g₁ g₂ : X →C_c ℝ≥0
hg₁ : EqOn (⇑g₁) 1 K₁.carrier
hg₂ : EqOn (⇑g₂) 1 K₂.carrier
sum_g : EqOn (⇑g₁ + ⇑g₂) 1 (tsupport ⇑f)
x : X
h : ¬f x = 0
⊢ ↑(f x) = ↑((⇑g₁ + ⇑g₂) x) * ↑(f x)

Tactic:
sum_g (mem_of_subset_of_mem subset_closure (mem_support.mpr h))

Goal state after:
case neg
X : Type u_1
inst✝² : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
K₁ K₂ : Compacts X
disj : Disjoint ↑K₁ ↑K₂
b : ℝ≥0
f : X →C_c ℝ≥0
hf : f ∈ {f | ∀ x ∈ K₁ ⊔ K₂, 1 ≤ f x}
Λf_eq_b : Λ f = b
hsuppf : ∀ x ∈ K₁ ⊔ K₂, x ∈ support ⇑f
hsubsuppf : ↑K₁ ∪ ↑K₂ ⊆ tsupport ⇑f
g₁ g₂ : X →C_c ℝ≥0
hg₁ : EqOn (⇑g₁) 1 K₁.carrier
hg₂ : EqOn (⇑g₂) 1 K₂.carrier
sum_g : EqOn (⇑g₁ + ⇑g₂) 1 (tsupport ⇑f)
x : X
h : ¬f x = 0
⊢ ↑(f x) = ↑(1 x) * ↑(f x)
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
K : Compacts X
⊢ (rieszContent Λ) K ≠ ∞

Tactic:
simp [rieszContent, ne_eq, not_false_eq_true]

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

Goal state before:
pp: μ : Measure ℝ
inst✝ : IsFiniteMeasure μ
n : ℕ
hint : MemLp id (↑n) μ
t : ℝ
k : ℕ
hkn : k ∈ Finset.range (n + 1)
⊢ (↑k !)⁻¹ * ↑t ^ k * iteratedDerivWithin k (charFun μ) univ 0 = (↑k !)⁻¹ * (↑t * I) ^ k * ↑(∫ (x : ℝ), x ^ k ∂μ)

Tactic:
have hint' : MemLp id k μ := hint.mono_exponent (by simp_all)

Goal state after:
μ : Measure ℝ
inst✝ : IsFiniteMeasure μ
n : ℕ
hint : MemLp id (↑n) μ
t : ℝ
k : ℕ
hkn : k ∈ Finset.range (n + 1)
hint' : MemLp id (↑k) μ
⊢ (↑k !)⁻¹ * ↑t ^ k * iteratedDerivWithin k (charFun μ) univ 0 = (↑k !)⁻¹ * (↑t * I) ^ k * ↑(∫ (x : ℝ), x ^ k ∂μ)
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_2
mE : MeasurableSpace E
μ : Measure E
inst✝² : SeminormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : BorelSpace E
r : ℝ
t : E
⊢ ∫ (x : E), cexp (↑⟪r • x, t⟫ * I) ∂μ = ∫ (x : E), cexp (↑⟪x, r • t⟫ * I) ∂μ

Tactic:
simp_rw [inner_smul_right, ← real_inner_smul_left]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
Λ : (X →C_c ℝ) →ₚ[ℝ] ℝ
inst✝ : LocallyCompactSpace X
f : X →C_c ℝ
μ : Measure X := rieszMeasure Λ
K : Set X := tsupport ⇑f
ε : ℝ
hε : 0 < ε
a b : ℝ
hab : a < b ∧ range ⇑f ⊆ Ioo a b
N : ℕ
hN : 0 < N
ε' : ℝ := (b - a) / ↑N
hε' : 0 < ε' ∧ ε' * (2 * μ.real K + |a| + b + ε') ≤ ε
E : Fin N → Set X
hE :
  tsupport ⇑f = ⋃ j, E j ∧
    univ.PairwiseDisjoint E ∧
      (∀ (n : Fin N), ∀ x ∈ E n, a + ε' * ↑↑n < f x ∧ f x ≤ a + ε' * (↑↑n + 1)) ∧ ∀ (n : Fin N), MeasurableSet (E n)
y : Fin N → ℝ := fun n ↦ a + ε' * (↑↑n + 1)
hE' : ∀ (n : Fin N), μ (E n) ≠ ∞
V : Fin N → Opens X
hV : ∀ (n : Fin N), E n ⊆ ↑(V n) ∧ (∀ x ∈ V n, f x < y n + ε') ∧ μ ↑(V n) ≤ μ (E n) + ENNReal.ofReal (ε' / ↑N)
g : Fin N → X →C_c ℝ
hg :
  (∀ (n : Fin N), tsupport ⇑(g n) ⊆ (V n).carrier) ∧
    EqOn (∑ n, ⇑(g n)) 1 (tsupport f.toFun) ∧
      (∀ (n : Fin N) (x : X), (g n) x ∈ Icc 0 1) ∧ ∀ (n : Fin N), HasCompactSupport ⇑(g n)
this : ∀ (n : Fin N), (|a| + y n + ε') * μ.real (E n) = (|a| + 2 * ε') * μ.real (E n) + (y n - ε') * μ.real (E n)
h : μ K = ∑ i, μ (E i)
⊢ ∑ x, (μ (E x)).toReal = (∑ i, μ (E i)).toReal

Tactic:
exact Eq.symm <| ENNReal.toReal_sum <| fun n _ ↦ hE' n

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

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
β : Type u_2
inst✝ : MeasurableSpace β
ν : FiniteMeasure β
⊢ (prod 0 ν).mass = 0

Tactic:
mass_prod,

Goal state after:
α : Type u_1
inst✝¹ : MeasurableSpace α
β : Type u_2
inst✝ : MeasurableSpace β
ν : FiniteMeasure β
⊢ mass 0 * ν.mass = 0
Module: Mathlib.MeasureTheory.Measure.FiniteMeasureProd

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
β : Type u_2
inst✝ : MeasurableSpace β
μ : FiniteMeasure α
⊢ (μ.prod 0).mass = 0

Tactic:
mass_prod,

Goal state after:
α : Type u_1
inst✝¹ : MeasurableSpace α
β : Type u_2
inst✝ : MeasurableSpace β
μ : FiniteMeasure α
⊢ μ.mass * mass 0 = 0
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: case bound_integrable
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
⊢ Integrable 1 (volume.restrict A)

Tactic:
have aux : IsFiniteMeasure (volume.restrict A) := ⟨by simp [lt_top_iff_ne_top, A_finmeas]⟩
    apply integrable_const

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.Extension

Goal state before:
pp: case inr
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 : ∀ (c : C), ((H.pushforward μA) { toContinuousMap := f, hasCompactSupport' := hf2 }) c ≤ (μA {1}).toReal
hC : μC Set.univ < ⊤
⊢ ENNReal.ofReal (∫ (c : C), ((H.pushforward μA) { toContinuousMap := f, hasCompactSupport' := hf2 }) c ∂μC) ≤
    μC Set.univ * μA {1}

Tactic:
have : IsFiniteMeasure μC := ⟨hC⟩

Goal state after:
case inr
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 : ∀ (c : C), ((H.pushforward μA) { toContinuousMap := f, hasCompactSupport' := hf2 }) c ≤ (μA {1}).toReal
hC : μC Set.univ < ⊤
this : IsFiniteMeasure μC
⊢ ENNReal.ofReal (∫ (c : C), ((H.pushforward μA) { toContinuousMap := f, hasCompactSupport' := hf2 }) c ∂μC) ≤
    μC Set.univ * μA {1}
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: case inl
E : Type u_1
p : ℝ
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hp : 0 < p
hE : Subsingleton E
⊢ μ {0} = ENNReal.ofReal (μ.real {0} • Real.exp (-‖0‖ ^ p))

Tactic:
norm_zero,

Goal state after:
case inl
E : Type u_1
p : ℝ
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hp : 0 < p
hE : Subsingleton E
⊢ μ {0} = ENNReal.ofReal (μ.real {0} • Real.exp (-0 ^ p))
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

Goal state before:
pp: case hbc.refine_3
μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
x : ℝ
hx : x ∈ {x | 2 < |2 * r⁻¹ * x|}
⊢ 2⁻¹ ≤ 1 - sinc (2 * r⁻¹ * x)

Tactic:
· have hx_ne : 2 * r⁻¹ * x ≠ 0 := by
        intro hx0
        simp only [hx0, Set.mem_ofPred_eq, abs_zero] at hx
        linarith
      rw [le_sub_iff_add_le, ← le_sub_iff_add_le']
      norm_num
      rw [one_div]
      refine (sinc_le_inv_abs hx_ne).trans ?_
      exact (inv_le_inv₀ (by positivity) (by positivity)).mpr (le_of_lt hx)

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

Goal state before:
pp: case e'_3
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : ∀ (x : ι → ℂ), 0 ≤ ∑ i, ‖x i‖ ^ p
h₂ : ∀ (x : ι → ℂ), 0 ≤ (∑ i, ‖x i‖ ^ p) ^ (1 / p)
hr : 0 < r
⊢ ENNReal.ofReal (r ^ (2 * card ι)) = ENNReal.ofReal (r ^ finrank ℝ (ι → ℂ))

Tactic:
finrank_pi_fintype ℝ,

Goal state after:
case e'_3
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : ∀ (x : ι → ℂ), 0 ≤ ∑ i, ‖x i‖ ^ p
h₂ : ∀ (x : ι → ℂ), 0 ≤ (∑ i, ‖x i‖ ^ p) ^ (1 / p)
hr : 0 < r
⊢ ENNReal.ofReal (r ^ (2 * card ι)) = ENNReal.ofReal (r ^ ∑ i, finrank ℝ ℂ)
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)
⊢ IsTightMeasureSet (Set.range μ)

Tactic:
by
  refine isTightMeasureSet_range_of_tendsto_limsup_inner 𝕜 fun y ↦ ?_
  by_cases hy : y = 0
  · simp only [hy, inner_zero_left]
    refine (tendsto_congr' ?_).mpr tendsto_const_nhds
    filter_upwards [eventually_ge_atTop 0] with r hr
    simp [not_lt.mpr hr]
  have h' : Tendsto (fun r : ℝ ↦ limsup (fun n ↦ μ n {x | ‖y‖⁻¹ * r < ‖⟪(‖y‖⁻¹ : 𝕜) • y, x⟫_𝕜‖})
      atTop) atTop (𝓝 0) := by
    specialize h ((‖y‖⁻¹ : 𝕜) • y) ?_
    · simp only [norm_smul, norm_inv, norm_algebraMap', Real.norm_eq_abs, abs_norm]
      rw [inv_mul_cancel₀ (by positivity)]
    exact h.comp <| (tendsto_const_mul_atTop_of_pos (by positivity)).mpr tendsto_id
  convert! h' using 7 with r n x
  rw [inner_smul_left]
  simp only [map_inv₀, RCLike.conj_ofReal, norm_mul, norm_inv, norm_algebraMap', norm_norm]
  rw [mul_lt_mul_iff_right₀]
  positivity

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

Goal state before:
pp: case refine_1
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
⊢ ∀ (n : ℕ), AEStronglyMeasurable (fun t ↦ 1 - charFun (μ n) (t • z)) (volume.restrict (Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹)))

Tactic:
exact fun _ ↦ Measurable.aestronglyMeasurable <| by fun_prop

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

Goal state before:
pp: X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
s : ℕ → Set X
hs : ∀ (i : ℕ), MeasurableSet (s i)
hs' : Pairwise (Disjoint on s)
n : ℕ
hn : ¬n = 0
ε' : ℝ≥0
hε' : 0 < ε'
hsnetop : preVariationFun f (⋃ i, s i) < ∞
ε : ℝ≥0 := ε' / ↑n
hε : 0 < ε
hs'' : ∀ (i : ℕ), preVariationFun f (s i) ≠ ∞
P : (i : ℕ) → Finpartition ⟨s i, ⋯⟩
hP : ∀ (i : ℕ), preVariationFun f (s i) ≤ ∑ p ∈ (P i).parts, f ↑p + ↑ε
⊢ ∑ i ∈ Finset.range n, ∑ p ∈ (P i).parts, f ↑p + ↑ε' ≤ preVariationFun f (⋃ i, s i) + ↑ε'

Tactic:
by
      gcongr; exact sum_le_preVariationFun_iUnion' f hs hs' P n

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

Goal state before:
pp: X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
hf : IsSigmaSubadditiveSetFun f
hf' : f ∅ = 0
s : ℕ → Set X
hs : ∀ (i : ℕ), MeasurableSet (s i)
hs' : Pairwise (Disjoint on s)
b : ℝ≥0∞
Q : Finpartition ⟨⋃ i, s i, ⋯⟩
hQ : b < ∑ p ∈ Q.parts, f ↑p
s' : ℕ → Subtype MeasurableSet := fun i ↦ ⟨s i, ⋯⟩
P : (i : ℕ) → Finpartition (s' i) := fun i ↦ Q.restrict ⋯
⊢ ∃ I, b < ∑ i ∈ I, preVariationFun f (s i)

Tactic:
have splitting : ∑ q ∈ Q.parts, f q ≤ ∑' i, ∑ p ∈ (P i).parts, f p := by
    calc ∑ q ∈ Q.parts, f q
      _ ≤ ∑ q ∈ Q.parts, ∑' i, f (q ⊓ s' i) := by
          apply Finset.sum_le_sum fun q hq => ?_
          have hq_eq : q.val = ⋃ i, q.val ∩ s i := by
            rw [← Set.inter_iUnion]; exact (Set.inter_eq_left.mpr (Q.le hq)).symm
          let t (i : ℕ) : Subtype MeasurableSet := ⟨q.val ∩ s i, q.2.inter (hs i)⟩
          have ht_disj : Pairwise (Disjoint on (Subtype.val ∘ t)) :=
            fun i j hij => (hs' hij).mono Set.inter_subset_right Set.inter_subset_right
          calc f q
            _ = f (⋃ i, q.val ∩ s i) := congrArg f hq_eq
            _ = f (⋃ i, (t i).val) := rfl
            _ ≤ ∑' i, f (t i) := hf t ht_disj
            _ = ∑' i, f (q ⊓ s' i) := rfl
      _ = ∑' i, ∑ q ∈ Q.parts, f (q ⊓ s' i) :=
          (Summable.tsum_finsetSum (fun _ _ ↦ ENNReal.summable)).symm
      _ = ∑' i, ∑ p ∈ (P i).parts, f p := by
          congr 1; funext i
          exact (Q.sum_restrict _ (fun p => f p) hf').symm

Goal state after:
X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
hf : IsSigmaSubadditiveSetFun f
hf' : f ∅ = 0
s : ℕ → Set X
hs : ∀ (i : ℕ), MeasurableSet (s i)
hs' : Pairwise (Disjoint on s)
b : ℝ≥0∞
Q : Finpartition ⟨⋃ i, s i, ⋯⟩
hQ : b < ∑ p ∈ Q.parts, f ↑p
s' : ℕ → Subtype MeasurableSet := fun i ↦ ⟨s i, ⋯⟩
P : (i : ℕ) → Finpartition (s' i) := fun i ↦ Q.restrict ⋯
splitting : ∑ q ∈ Q.parts, f ↑q ≤ ∑' (i : ℕ), ∑ p ∈ (P i).parts, f ↑p
⊢ ∃ I, b < ∑ i ∈ I, preVariationFun f (s i)
Module: Mathlib.MeasureTheory.Measure.Support

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : MeasurableSpace X
μ : Measure X
inst✝ : OpensMeasurableSpace X
s : Set X
x : X
hxs : x ∈ interior s
hxμ : x ∈ μ.support
⊢ x ∈ (μ.restrict s).support

Tactic:
rw [mem_support_restrict, (nhdsWithin_basis_open x s).frequently_smallSets pos_mono]

Goal state after:
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : MeasurableSpace X
μ : Measure X
inst✝ : OpensMeasurableSpace X
s : Set X
x : X
hxs : x ∈ interior s
hxμ : x ∈ μ.support
⊢ ∀ (i : Set X), x ∈ i ∧ IsOpen[inst✝²] i → 0 < μ (i ∩ s)
Module: Mathlib.MeasureTheory.Measure.ResolventTransform

Goal state before:
pp: case h
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : MeasurableSpace 𝕜
inst✝² : CompleteSpace 𝕜
inst✝¹ : NormedDivisionRing A
inst✝ : NormedAlgebra 𝕜 A
μ : Measure 𝕜
a : A
hz : a ∉ ⇑(algebraMap 𝕜 A) '' μ.support
x : 𝕜
hx : x ∈ μ.support
this : 0 < infDist a (⇑(algebraMap 𝕜 A) '' μ.support)
⊢ (algebraMap 𝕜 A) x ∈ ⇑(algebraMap 𝕜 A) '' μ.support

Tactic:
simp [hx]

Goal state after: