mathlib-initiative/mathlib-tactics huggingface dataset:
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)
n : Fin N
h : E n ⊆ tsupport ⇑f
⊢ (rieszContent (toNNRealLinear Λ)).measure (tsupport ⇑f.toContinuousMap) < ∞

Tactic:
← ContinuousMap.toFun_eq_coe,

Goal state after:
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)
n : Fin N
h : E n ⊆ tsupport ⇑f
⊢ (rieszContent (toNNRealLinear Λ)).measure (tsupport f.toFun) < ∞
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_2
mE : MeasurableSpace E
μ : Measure E
t : E
inst✝² : SeminormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : IsProbabilityMeasure μ
⊢ 1 + 1 = 2

Tactic:
norm_num

Goal state after:
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
⊢ charFun (Measure.map (fun x ↦ r • x) μ) t = charFun μ (r • t)

Tactic:
charFun_apply,

Goal state after:
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 (↑⟪x, t⟫ * I) ∂Measure.map (fun x ↦ r • x) μ = charFun μ (r • t)
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 (↑⟪x, t⟫ * I) ∂Measure.map (fun x ↦ r • x) μ = charFun μ (r • t)

Tactic:
charFun_apply,

Goal state after:
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 (↑⟪x, t⟫ * I) ∂Measure.map (fun x ↦ r • x) μ = ∫ (x : E), cexp (↑⟪x, r • t⟫ * I) ∂μ
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_3
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : OpensMeasurableSpace E
x t : E
⊢ charFun (Measure.dirac x) t = cexp (↑⟪x, t⟫ * I)

Tactic:
charFun_apply,

Goal state after:
E : Type u_3
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : OpensMeasurableSpace E
x t : E
⊢ ∫ (x : E), cexp (↑⟪x, t⟫ * I) ∂Measure.dirac x = cexp (↑⟪x, t⟫ * I)
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_3
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : BorelSpace E
r t : E
⊢ charFun (Measure.map (fun x ↦ x + r) μ) t = charFun μ t * cexp (↑⟪r, t⟫ * I)

Tactic:
charFun_apply,

Goal state after:
E : Type u_3
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : BorelSpace E
r t : E
⊢ ∫ (x : E), cexp (↑⟪x, t⟫ * I) ∂Measure.map (fun x ↦ x + r) μ = charFun μ t * cexp (↑⟪r, t⟫ * I)
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_3
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : BorelSpace E
r t : E
⊢ ∫ (x : E), cexp (↑⟪x, t⟫ * I) ∂Measure.map (fun x ↦ x + r) μ = charFun μ t * cexp (↑⟪r, t⟫ * I)

Tactic:
charFun_apply,

Goal state after:
E : Type u_3
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : BorelSpace E
r t : E
⊢ ∫ (x : E), cexp (↑⟪x, t⟫ * I) ∂Measure.map (fun x ↦ x + r) μ = (∫ (x : E), cexp (↑⟪x, t⟫ * I) ∂μ) * cexp (↑⟪r, t⟫ * I)
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_4
F : Type u_5
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : InnerProductSpace ℝ E
inst✝⁹ : InnerProductSpace ℝ F
mE : MeasurableSpace E
mF : MeasurableSpace F
inst✝⁸ : CompleteSpace E
inst✝⁷ : CompleteSpace F
inst✝⁶ : SecondCountableTopology E
inst✝⁵ : SecondCountableTopology F
inst✝⁴ : BorelSpace E
inst✝³ : BorelSpace F
μ : Measure E
ν : Measure F
ξ : Measure (E × F)
inst✝² : IsFiniteMeasure μ
inst✝¹ : IsFiniteMeasure ν
inst✝ : IsFiniteMeasure ξ
h : ∀ (t : WithLp 2 (E × F)), charFun (Measure.map (toLp 2) ξ) t = charFun μ t.ofLp.1 * charFun ν t.ofLp.2
t : WithLp 2 (E × F)
⊢ charFun μ t.ofLp.1 * charFun ν t.ofLp.2 = charFun (Measure.map (toLp 2) (μ.prod ν)) t

Tactic:
charFun_prod

Goal state after:
E : Type u_4
F : Type u_5
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : InnerProductSpace ℝ E
inst✝⁹ : InnerProductSpace ℝ F
mE : MeasurableSpace E
mF : MeasurableSpace F
inst✝⁸ : CompleteSpace E
inst✝⁷ : CompleteSpace F
inst✝⁶ : SecondCountableTopology E
inst✝⁵ : SecondCountableTopology F
inst✝⁴ : BorelSpace E
inst✝³ : BorelSpace F
μ : Measure E
ν : Measure F
ξ : Measure (E × F)
inst✝² : IsFiniteMeasure μ
inst✝¹ : IsFiniteMeasure ν
inst✝ : IsFiniteMeasure ξ
h : ∀ (t : WithLp 2 (E × F)), charFun (Measure.map (toLp 2) ξ) t = charFun μ t.ofLp.1 * charFun ν t.ofLp.2
t : WithLp 2 (E × F)
⊢ charFun μ t.ofLp.1 * charFun ν t.ofLp.2 = charFun μ t.ofLp.1 * charFun ν t.ofLp.2
Module: Mathlib.Topology.Separation.CompletelyRegular

Goal state before:
pp: case h.right.right
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
I' : Finset ι
V U : ↥I' → Set X
hUV : ∀ (i : ↥I'), U i ⊆ V i
fs : ↥I' → X → ↑I
hfs : ∀ (i : ↥I'), Continuous[t ↑i, _] (fs i)
hxfs : ∀ (i : ↥I'), fs i x✝ = 0
hfsU : ∀ (i : ↥I') ⦃x : X⦄, x ∈ (U i)ᶜ → fs i x = ⊤
x : X
i : ↥I'
hxi : x ∈ (V i)ᶜ
⊢ ∃ b, fs b x = ⊤

Tactic:
specialize hfsU i (by tauto_set)

Goal state after:
case h.right.right
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
I' : Finset ι
V U : ↥I' → Set X
hUV : ∀ (i : ↥I'), U i ⊆ V i
fs : ↥I' → X → ↑I
hfs : ∀ (i : ↥I'), Continuous[t ↑i, _] (fs i)
hxfs : ∀ (i : ↥I'), fs i x✝ = 0
x : X
i : ↥I'
hfsU : fs i x = ⊤
hxi : x ∈ (V i)ᶜ
⊢ ∃ b, fs b x = ⊤
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: case refine_1
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : SecondCountableTopology E
inst✝² : CompleteSpace E
μ ν : Measure E
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : charFunDual μ = charFunDual ν
⊢ ∀ (v : E), v ≠ 0 → (ContinuousLinearMap.toLinearMap₁₂ ⋯.toContinuousLinearMap) v ≠ 0

pp: case refine_2
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : SecondCountableTopology E
inst✝² : CompleteSpace E
μ ν : Measure E
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : charFunDual μ = charFunDual ν
⊢ Continuous fun p ↦ ((ContinuousLinearMap.toLinearMap₁₂ ⋯.toContinuousLinearMap) p.1) p.2

Tactic:
· exact fun v hv ↦ DFunLike.ne_iff.mpr <| SeparatingDual.exists_ne_zero hv

Goal state after:
case refine_2
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : SecondCountableTopology E
inst✝² : CompleteSpace E
μ ν : Measure E
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : charFunDual μ = charFunDual ν
⊢ Continuous fun p ↦ ((ContinuousLinearMap.toLinearMap₁₂ ⋯.toContinuousLinearMap) p.1) p.2
Module: Mathlib.MeasureTheory.Measure.Haar.DistribChar

Goal state before:
pp: G : Type u_1
A : Type u_2
inst✝¹⁰ : Group G
inst✝⁹ : AddCommGroup A
inst✝⁸ : DistribMulAction G A
inst✝⁷ : TopologicalSpace A
inst✝⁶ : IsTopologicalAddGroup A
inst✝⁵ : LocallyCompactSpace A
inst✝⁴ : ContinuousConstSMul G A
inst✝³ : MeasurableSpace A
inst✝² : BorelSpace A
μ : Measure A
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : μ.Regular
s : Set A
hs₀ : μ s ≠ 0
hs : μ s ≠ ∞
g : G
⊢ ↑((distribHaarChar A) g) = μ (g • s) / μ s

Tactic:
← distribHaarChar_mul,

Goal state after:
G : Type u_1
A : Type u_2
inst✝¹⁰ : Group G
inst✝⁹ : AddCommGroup A
inst✝⁸ : DistribMulAction G A
inst✝⁷ : TopologicalSpace A
inst✝⁶ : IsTopologicalAddGroup A
inst✝⁵ : LocallyCompactSpace A
inst✝⁴ : ContinuousConstSMul G A
inst✝³ : MeasurableSpace A
inst✝² : BorelSpace A
μ : Measure A
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : μ.Regular
s : Set A
hs₀ : μ s ≠ 0
hs : μ s ≠ ∞
g : G
⊢ ↑((distribHaarChar A) g) = ↑((distribHaarChar A) g) * μ s / μ s
Module: Mathlib.MeasureTheory.Measure.Haar.Extension

Goal state before:
pp: A : Type u_1
B : Type u_2
C : Type u_3
E : Type u_4
inst✝¹³ : Group A
inst✝¹² : Group B
inst✝¹¹ : Group C
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : TopologicalSpace B
inst✝⁸ : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst✝⁷ : IsTopologicalGroup A
inst✝⁶ : IsTopologicalGroup B
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace A
inst✝³ : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst✝² : NormedSpace ℝ E
inst✝¹ : IsTopologicalGroup C
inst✝ : LocallyCompactSpace B
f : CompactlySupportedContinuousMap B E
this : UniformSpace B := IsTopologicalGroup.rightUniformSpace B
ε : ℝ
hε : ε > 0
U₀ : Set B
hU₀ : IsCompact U₀
K : Set B
hK : IsCompact K
hf₀ : ∀ x ∉ K, f x = 0
S : Set A := ⇑φ ⁻¹' (U₀⁻¹ * K)
hSc : IsCompact S
δ : ℝ
hδ0 : δ > 0
hδ : ENNReal.ofReal δ * μA S < ENNReal.ofReal ε
hS : ∀ {x : B}, x ∈ U₀ → ∀ {y : A}, y ∉ S → (H.pullback f x) y = 0
ha✝ : ∀ r ∈ uniformity E, ∀ᶠ (x : B × B) in uniformity B, (f x.1, f x.2) ∈ r
U : Set B
hU : U ∈ nhds 1
hf : (fun p ↦ p.2 * p.1⁻¹) ⁻¹' U ⊆ {x | (fun x ↦ (f x.1, f x.2) ∈ {p | dist p.1 p.2 < δ}) x}
t : B
ht : t ∈ U⁻¹
b : B
hb : U₀ ∈ nhds b
htb : (fun x1 x2 ↦ x1 * x2) t b ∈ U₀
a : A
ha : a ∈ S
⊢ edist ((H.pullback f (t * b)) a) ((H.pullback f b) a) ≤ ENNReal.ofReal δ

Tactic:
edist_dist

Goal state after:
A : Type u_1
B : Type u_2
C : Type u_3
E : Type u_4
inst✝¹³ : Group A
inst✝¹² : Group B
inst✝¹¹ : Group C
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : TopologicalSpace B
inst✝⁸ : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst✝⁷ : IsTopologicalGroup A
inst✝⁶ : IsTopologicalGroup B
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace A
inst✝³ : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst✝² : NormedSpace ℝ E
inst✝¹ : IsTopologicalGroup C
inst✝ : LocallyCompactSpace B
f : CompactlySupportedContinuousMap B E
this : UniformSpace B := IsTopologicalGroup.rightUniformSpace B
ε : ℝ
hε : ε > 0
U₀ : Set B
hU₀ : IsCompact U₀
K : Set B
hK : IsCompact K
hf₀ : ∀ x ∉ K, f x = 0
S : Set A := ⇑φ ⁻¹' (U₀⁻¹ * K)
hSc : IsCompact S
δ : ℝ
hδ0 : δ > 0
hδ : ENNReal.ofReal δ * μA S < ENNReal.ofReal ε
hS : ∀ {x : B}, x ∈ U₀ → ∀ {y : A}, y ∉ S → (H.pullback f x) y = 0
ha✝ : ∀ r ∈ uniformity E, ∀ᶠ (x : B × B) in uniformity B, (f x.1, f x.2) ∈ r
U : Set B
hU : U ∈ nhds 1
hf : (fun p ↦ p.2 * p.1⁻¹) ⁻¹' U ⊆ {x | (fun x ↦ (f x.1, f x.2) ∈ {p | dist p.1 p.2 < δ}) x}
t : B
ht : t ∈ U⁻¹
b : B
hb : U₀ ∈ nhds b
htb : (fun x1 x2 ↦ x1 * x2) t b ∈ U₀
a : A
ha : a ∈ S
⊢ ENNReal.ofReal (dist ((H.pullback f (t * b)) a) ((H.pullback f b) a)) ≤ ENNReal.ofReal δ
Module: Mathlib.MeasureTheory.Measure.Haar.Extension

Goal state before:
pp: A : Type u_1
B : Type u_2
C : Type u_3
E : Type u_4
inst✝¹⁸ : Group A
inst✝¹⁷ : Group B
inst✝¹⁶ : Group C
inst✝¹⁵ : TopologicalSpace A
inst✝¹⁴ : TopologicalSpace B
inst✝¹³ : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst✝¹² : IsTopologicalGroup A
inst✝¹¹ : IsTopologicalGroup B
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace A
inst✝⁸ : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : IsTopologicalGroup C
inst✝⁵ : LocallyCompactSpace B
inst✝⁴ : MeasurableSpace C
inst✝³ : BorelSpace C
μC : Measure C
hμC : μC.IsHaarMeasure
inst✝² : T2Space B
inst✝¹ : MeasurableSpace B
inst✝ : BorelSpace B
K : Set B
hK : IsCompact K
⊢ (H.inducedMeasure μA μC) K < ⊤

Tactic:
by
    obtain ⟨f, hf1, hf2, hf3, hf4⟩ :=
      exists_continuousMap_one_of_isCompact_subset_isOpen hK isOpen_univ K.subset_univ
    exact lt_of_le_of_lt (RealRMK.rieszMeasure_le_of_eq_one (f := ⟨f, hf2⟩) _
      (fun x ↦ (hf4 x).1) hK (fun x hx ↦ hf1 hx)) ENNReal.ofReal_lt_top

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

Goal state before:
pp: A : Type u_1
B : Type u_2
C : Type u_3
E : Type u_4
inst✝¹⁸ : Group A
inst✝¹⁷ : Group B
inst✝¹⁶ : Group C
inst✝¹⁵ : TopologicalSpace A
inst✝¹⁴ : TopologicalSpace B
inst✝¹³ : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst✝¹² : IsTopologicalGroup A
inst✝¹¹ : IsTopologicalGroup B
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace A
inst✝⁸ : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : IsTopologicalGroup C
inst✝⁵ : LocallyCompactSpace B
inst✝⁴ : MeasurableSpace C
inst✝³ : BorelSpace C
μC : Measure C
hμC : μC.IsHaarMeasure
inst✝² : T2Space B
inst✝¹ : MeasurableSpace B
inst✝ : BorelSpace B
b : B
this : (map (fun x ↦ b * x) (H.inducedMeasure μA μC)).Regular
f : CompactlySupportedContinuousMap B ℝ
h : ∀ (x : B), f (b * x) = (f.comp (Homeomorph.mulLeft b).toCocompactMap) x
⊢ ∫ (c : C), ((H.pushforward μA) (f.comp (Homeomorph.mulLeft b).toCocompactMap)) c ∂μC =
    ∫ (c : C), ((H.pushforward μA) f) c ∂μC

Tactic:
rw [← integral_mul_left_eq_self _ (ψ b)⁻¹]

Goal state after:
A : Type u_1
B : Type u_2
C : Type u_3
E : Type u_4
inst✝¹⁸ : Group A
inst✝¹⁷ : Group B
inst✝¹⁶ : Group C
inst✝¹⁵ : TopologicalSpace A
inst✝¹⁴ : TopologicalSpace B
inst✝¹³ : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst✝¹² : IsTopologicalGroup A
inst✝¹¹ : IsTopologicalGroup B
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace A
inst✝⁸ : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : IsTopologicalGroup C
inst✝⁵ : LocallyCompactSpace B
inst✝⁴ : MeasurableSpace C
inst✝³ : BorelSpace C
μC : Measure C
hμC : μC.IsHaarMeasure
inst✝² : T2Space B
inst✝¹ : MeasurableSpace B
inst✝ : BorelSpace B
b : B
this : (map (fun x ↦ b * x) (H.inducedMeasure μA μC)).Regular
f : CompactlySupportedContinuousMap B ℝ
h : ∀ (x : B), f (b * x) = (f.comp (Homeomorph.mulLeft b).toCocompactMap) x
⊢ ∫ (x : C), ((H.pushforward μA) (f.comp (Homeomorph.mulLeft b).toCocompactMap)) ((ψ b)⁻¹ * x) ∂μC =
    ∫ (c : C), ((H.pushforward μA) f) c ∂μC
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|}
hx_ne : 2 * r⁻¹ * x ≠ 0
⊢ sinc (2 * r⁻¹ * x) ≤ 1 - 2⁻¹

Tactic:
norm_num

Goal state after:
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|}
hx_ne : 2 * r⁻¹ * x ≠ 0
⊢ sinc (2 * r⁻¹ * x) ≤ 1 / 2
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: case refine_1
E : Type u_1
inst✝⁴ : MeasurableSpace E
inst✝³ : TopologicalSpace E
inst✝² : T2Space E
inst✝¹ : BorelSpace E
inst✝ : CompactSpace E
C : ℝ≥0
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C} ∈ f
Λ : (E →C_c ℝ) → ℝ
h₀Λ : ∀ (g : E →C_c ℝ), Λ g ∈ Icc (-↑C * ‖g.toBoundedContinuousFunction‖) (↑C * ‖g.toBoundedContinuousFunction‖)
hΛ : ∀ (g : E →C_c ℝ), Tendsto (fun μ ↦ ∫ (x : E), g x ∂↑μ) (↑f) (𝓝 (Λ g))
Λ' : (E →C_c ℝ) →ₚ[ℝ] ℝ := { toFun := Λ, map_add' := ⋯, map_smul' := ⋯, monotone' := ⋯ }
μlim : Measure E := RealRMK.rieszMeasure Λ'
μlim_le : μlim univ ≤ ENNReal.ofReal ↑C
μlim' : FiniteMeasure E := ⟨μlim, ⋯⟩
⊢ μlim' ∈ {μ | μ.mass ≤ C}

pp: case refine_2
E : Type u_1
inst✝⁴ : MeasurableSpace E
inst✝³ : TopologicalSpace E
inst✝² : T2Space E
inst✝¹ : BorelSpace E
inst✝ : CompactSpace E
C : ℝ≥0
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C} ∈ f
Λ : (E →C_c ℝ) → ℝ
h₀Λ : ∀ (g : E →C_c ℝ), Λ g ∈ Icc (-↑C * ‖g.toBoundedContinuousFunction‖) (↑C * ‖g.toBoundedContinuousFunction‖)
hΛ : ∀ (g : E →C_c ℝ), Tendsto (fun μ ↦ ∫ (x : E), g x ∂↑μ) (↑f) (𝓝 (Λ g))
Λ' : (E →C_c ℝ) →ₚ[ℝ] ℝ := { toFun := Λ, map_add' := ⋯, map_smul' := ⋯, monotone' := ⋯ }
μlim : Measure E := RealRMK.rieszMeasure Λ'
μlim_le : μlim univ ≤ ENNReal.ofReal ↑C
μlim' : FiniteMeasure E := ⟨μlim, ⋯⟩
⊢ ↑f ≤ 𝓝 μlim'

Tactic:
· simp only [mem_ofPred_eq, FiniteMeasure.mk_apply, μlim', FiniteMeasure.mass]
    rw [show C = (ENNReal.ofReal ↑C).toNNReal by simp]
    exact ENNReal.toNNReal_mono (by simp) μlim_le

Goal state after:
case refine_2
E : Type u_1
inst✝⁴ : MeasurableSpace E
inst✝³ : TopologicalSpace E
inst✝² : T2Space E
inst✝¹ : BorelSpace E
inst✝ : CompactSpace E
C : ℝ≥0
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C} ∈ f
Λ : (E →C_c ℝ) → ℝ
h₀Λ : ∀ (g : E →C_c ℝ), Λ g ∈ Icc (-↑C * ‖g.toBoundedContinuousFunction‖) (↑C * ‖g.toBoundedContinuousFunction‖)
hΛ : ∀ (g : E →C_c ℝ), Tendsto (fun μ ↦ ∫ (x : E), g x ∂↑μ) (↑f) (𝓝 (Λ g))
Λ' : (E →C_c ℝ) →ₚ[ℝ] ℝ := { toFun := Λ, map_add' := ⋯, map_smul' := ⋯, monotone' := ⋯ }
μlim : Measure E := RealRMK.rieszMeasure Λ'
μlim_le : μlim univ ≤ ENNReal.ofReal ↑C
μlim' : FiniteMeasure E := ⟨μlim, ⋯⟩
⊢ ↑f ≤ 𝓝 μlim'
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
k : ℕ
hk : finrank ℝ E = 2 * k + 1
x : E
r : ℝ
this : Nontrivial E
⊢ ENNReal.ofReal r ^ (2 * k + 1) * ENNReal.ofReal ((√π ^ 2) ^ k * √π / Gamma (↑(2 * k + 1) / 2 + 1)) =
    ENNReal.ofReal r ^ (2 * k + 1) * ENNReal.ofReal (π ^ k * 2 ^ (k + 1) / ↑(2 * k + 1)‼)

Tactic:
sq_sqrt pi_nonneg,

Goal state after:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
k : ℕ
hk : finrank ℝ E = 2 * k + 1
x : E
r : ℝ
this : Nontrivial E
⊢ ENNReal.ofReal r ^ (2 * k + 1) * ENNReal.ofReal (π ^ k * √π / Gamma (↑(2 * k + 1) / 2 + 1)) =
    ENNReal.ofReal r ^ (2 * k + 1) * ENNReal.ofReal (π ^ k * 2 ^ (k + 1) / ↑(2 * k + 1)‼)
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝ : PseudoMetricSpace E
hS : IsTightMeasureSet S
x : E
this : Tendsto ((⨆ μ ∈ S, ⇑μ) ∘ fun r ↦ (Metric.closedBall x r)ᶜ) atTop (𝓝 0)
⊢ Tendsto (fun r ↦ ⨆ μ ∈ S, μ (Metric.closedBall x r)ᶜ) atTop (𝓝 0)

Tactic:
convert! this with r

Goal state after:
E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝ : PseudoMetricSpace E
hS : IsTightMeasureSet S
x : E
this : Tendsto ((⨆ μ ∈ S, ⇑μ) ∘ fun r ↦ (Metric.closedBall x r)ᶜ) atTop (𝓝 0)
r : ℝ
⊢ ⨆ μ ∈ S, μ (Metric.closedBall x r)ᶜ = ((⨆ μ ∈ S, ⇑μ) ∘ fun r ↦ (Metric.closedBall x r)ᶜ) r
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝¹ : PseudoMetricSpace E
inst✝ : ProperSpace E
x : E
h✝ : ∀ ε > 0, ∃ N, ∀ n ≥ N, ⨆ μ ∈ S, μ (Metric.closedBall x n)ᶜ ≤ ε
ε : ℝ≥0∞
hε : 0 < ε
r : ℝ
h : ∀ n ≥ r, ⨆ μ ∈ S, μ (Metric.closedBall x n)ᶜ ≤ ε
⊢ ∀ μ ∈ S, μ (Metric.closedBall x r)ᶜ ≤ ε

Tactic:
simpa using h r le_rfl

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun n ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
n : ℕ
r : ℝ
hr : 0 < r
hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹
⊢ 2 * (1 + 1) ≤ 4

Tactic:
norm_num

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), Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
⊢ IsTightMeasureSet (Set.range μ)

Tactic:
refine isTightMeasureSet_of_inner_tendsto 𝕜 fun z ↦ ?_

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), Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
z : E
⊢ Tendsto (fun r ↦ ⨆ μ_1 ∈ Set.range μ, μ_1 {x | r < ‖⟪z, x⟫_𝕜‖}) atTop (𝓝 0)
Module: Mathlib.MeasureTheory.Measure.MeasuredSets

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : IsSetSemiring C
h'C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0
h : mα = generateFrom C
s : Set α
hs : MeasurableSet s
ε : ℝ≥0∞
hε : 0 < ε
⊢ ∃ t ∈ supClosure C, μ (t ∆ s) < ε

Tactic:
apply exists_measure_symmDiff_lt_of_generateFrom_isSetRing hC.isSetRing_supClosure ?_ ?_ hs hε

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : IsSetSemiring C
h'C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0
h : mα = generateFrom C
s : Set α
hs : MeasurableSet s
ε : ℝ≥0∞
hε : 0 < ε
⊢ ∃ D, D.Countable ∧ D ⊆ supClosure C ∧ μ (⋃₀ D)ᶜ = 0

α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : IsSetSemiring C
h'C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0
h : mα = generateFrom C
s : Set α
hs : MeasurableSet s
ε : ℝ≥0∞
hε : 0 < ε
⊢ mα = generateFrom (supClosure C)
Module: Mathlib.MeasureTheory.Measure.LevyConvergence

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

Tactic:
norm_num

Goal state after: