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

Goal state before:
pp: case h
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
hf : ∀ (x : X), 0 ≤ f x
K : Set X
hK : IsCompact K
hfK : ∀ x ∈ K, f x = 1
p : NNReal
hp : Λ f = ↑p
⊢ (∀ x ∈ { carrier := K, isCompact' := hK }, 1 ≤ f.nnrealPart x) ∧ (toNNRealLinear Λ) f.nnrealPart = p

Tactic:
nnrealPart_apply,

Goal state after:
case h
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
hf : ∀ (x : X), 0 ≤ f x
K : Set X
hK : IsCompact K
hfK : ∀ x ∈ K, f x = 1
p : NNReal
hp : Λ f = ↑p
⊢ (∀ x ∈ { carrier := K, isCompact' := hK }, 1 ≤ (f x).toNNReal) ∧ (toNNRealLinear Λ) f.nnrealPart = p
Module: Mathlib.MeasureTheory.MeasurableSpace.Card

Goal state before:
pp: case inl.inr
α : Type u
s : Set (Set α)
i : Ordinal.{u_1}
t✝ : Set α
p : Set α → Prop
hs : ∀ t ∈ s, p t
h0 : p ∅
hc : ∀ (u : Set α), p u → (∃ j < i, u ∈ generateMeasurableRec s j) → p uᶜ
hn : ∀ (f : ℕ → Set α), (∀ (n : ℕ), p (f n) ∧ ∃ j < i, f n ∈ generateMeasurableRec s j) → p (⋃ n, f n)
k✝ k : Ordinal.{u_1}
hk : k ≤ i
t : Set α
IH : ∀ j < k, ∀ t ∈ generateMeasurableRec s j, p t
ht : t ∈ compl '' ⋃ j, ⋃ (_ : j < k), generateMeasurableRec s j
⊢ p t

Tactic:
[

Goal state after:
case inl.inr
α : Type u
s : Set (Set α)
i : Ordinal.{u_1}
t✝ : Set α
p : Set α → Prop
hs : ∀ t ∈ s, p t
h0 : p ∅
hc : ∀ (u : Set α), p u → (∃ j < i, u ∈ generateMeasurableRec s j) → p uᶜ
hn : ∀ (f : ℕ → Set α), (∀ (n : ℕ), p (f n) ∧ ∃ j < i, f n ∈ generateMeasurableRec s j) → p (⋃ n, f n)
k✝ k : Ordinal.{u_1}
hk : k ≤ i
t : Set α
IH : ∀ j < k, ∀ t ∈ generateMeasurableRec s j, p t
ht : t ∈ compl '' ⋃ j, ⋃ (_ : j < k), generateMeasurableRec s j
⊢ p t
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: case h.refine_2
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ
a ε : ℝ
hε : 0 < ε
N : ℕ
hf : range ⇑f ⊆ Ioo a (a + ↑N * ε)
b : ℝ := ⋯
y : Fin N → ℝ := ⋯
hy : ∀ {n m : Fin N}, n < m → y n + ε ≤ y m
E : Fin N → Set X := ⋯
m : Fin N
a✝¹ : m ∈ univ
n : Fin N
a✝ : n ∈ univ
hmn : m ≠ n
⊢ Disjoint {a | a ∈ ⇑f ⁻¹' Ioc (y m - ε) (y m) ∧ a ∈ tsupport ⇑f} {a | a ∈ ⇑f ⁻¹' Ioc (y n - ε) (y n) ∧ a ∈ tsupport ⇑f}

Tactic:
[

Goal state after:
case h.refine_2
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ
a ε : ℝ
hε : 0 < ε
N : ℕ
hf : range ⇑f ⊆ Ioo a (a + ↑N * ε)
b : ℝ := ⋯
y : Fin N → ℝ := ⋯
hy : ∀ {n m : Fin N}, n < m → y n + ε ≤ y m
E : Fin N → Set X := ⋯
m : Fin N
a✝¹ : m ∈ univ
n : Fin N
a✝ : n ∈ univ
hmn : m ≠ n
⊢ Disjoint {a | a ∈ ⇑f ⁻¹' Ioc (y m - ε) (y m) ∧ a ∈ tsupport ⇑f} {a | a ∈ ⇑f ⁻¹' Ioc (y n - ε) (y n) ∧ a ∈ tsupport ⇑f}
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: case h.refine_2
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ
a ε : ℝ
hε : 0 < ε
N : ℕ
hf : range ⇑f ⊆ Ioo a (a + ↑N * ε)
b : ℝ := a + ↑N * ε
y : Fin N → ℝ := fun n ↦ a + ε * (↑↑n + 1)
hy : ∀ {n m : Fin N}, n < m → y n + ε ≤ y m
E : Fin N → Set X := fun n ↦ ⇑f ⁻¹' Ioc (y n - ε) (y n) ∩ tsupport ⇑f
m : Fin N
a✝¹ : m ∈ univ
n : Fin N
a✝ : n ∈ univ
hmn : m ≠ n
x : X
hx : y m - ε < f x ∧ f x ≤ y m ∧ x ∈ tsupport ⇑f
⊢ x ∉ {a | (y n - ε < f a ∧ f a ≤ y n) ∧ a ∈ tsupport ⇑f}

Tactic:
[

Goal state after:
case h.refine_2
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ
a ε : ℝ
hε : 0 < ε
N : ℕ
hf : range ⇑f ⊆ Ioo a (a + ↑N * ε)
b : ℝ := a + ↑N * ε
y : Fin N → ℝ := fun n ↦ a + ε * (↑↑n + 1)
hy : ∀ {n m : Fin N}, n < m → y n + ε ≤ y m
E : Fin N → Set X := fun n ↦ ⇑f ⁻¹' Ioc (y n - ε) (y n) ∩ tsupport ⇑f
m : Fin N
a✝¹ : m ∈ univ
n : Fin N
a✝ : n ∈ univ
hmn : m ≠ n
x : X
hx : y m - ε < f x ∧ f x ≤ y m ∧ x ∈ tsupport ⇑f
⊢ x ∉ {a | (y n - ε < f a ∧ f a ≤ y n) ∧ a ∈ tsupport ⇑f}
Module: Mathlib.MeasureTheory.MeasurableSpace.Card

Goal state before:
pp: case refine_4
α : Type u
s : Set (Set α)
t : Set α
ht : t ∈ generateMeasurableRec s (ω_ 1)
f : ℕ → Set α
H : ∀ (n : ℕ), (∃ i, ∃ (_ : i < ω_ 1), f n ∈ generateMeasurableRec s i) ∧ ∃ j < ω_ 1, f n ∈ generateMeasurableRec s j
I : ℕ → Ordinal.{v}
hI : ∀ (n : ℕ), ∃ (_ : I n < ω_ 1), f n ∈ generateMeasurableRec s (I n)
⊢ ∃ i, ∃ (_ : i < ω_ 1), ⋃ n, f n ∈ generateMeasurableRec s i

Tactic:
[

Goal state after:
case refine_4
α : Type u
s : Set (Set α)
t : Set α
ht : t ∈ generateMeasurableRec s (ω_ 1)
f : ℕ → Set α
H : ∀ (n : ℕ), (∃ i, ∃ (_ : i < ω_ 1), f n ∈ generateMeasurableRec s i) ∧ ∃ j < ω_ 1, f n ∈ generateMeasurableRec s j
I : ℕ → Ordinal.{v}
hI : ∀ (n : ℕ), ∃ (_ : I n < ω_ 1), f n ∈ generateMeasurableRec s (I n)
⊢ ∃ i, ∃ (_ : i < ω_ 1), ⋃ n, f n ∈ generateMeasurableRec s i
Module: Mathlib.MeasureTheory.MeasurableSpace.Card

Goal state before:
pp: case iUnion
α : Type u
s : Set (Set α)
t : Set α
f : ℕ → Set α
a✝ : ∀ (n : ℕ), GenerateMeasurable s (f n)
IH : ∀ (n : ℕ), f n ∈ generateMeasurableRec s (ω_ 1)
⊢ ⋃ i, f i ∈ generateMeasurableRec s (ω_ 1)

Tactic:
[

Goal state after:
case iUnion
α : Type u
s : Set (Set α)
t : Set α
f : ℕ → Set α
a✝ : ∀ (n : ℕ), GenerateMeasurable s (f n)
IH : ∀ (n : ℕ), f n ∈ generateMeasurableRec s (ω_ 1)
⊢ ⋃ i, f i ∈ generateMeasurableRec s (ω_ 1)
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
⊢ Λ f ≤ ∫ (x : X), f x ∂rieszMeasure Λ + ε

Tactic:
obtain ⟨N, hN, hε'⟩ := exists_nat_large (b - a) (2 * μ.real K + |a| + b) hε

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
hε' : (b - a) / ↑N * (2 * μ.real K + |a| + b + (b - a) / ↑N) ≤ ε
⊢ Λ f ≤ ∫ (x : X), f x ∂rieszMeasure Λ + ε
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
⊢ E n ⊆ ⋃ j, E j

Tactic:
exact subset_iUnion _ _

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : IsFiniteMeasure μ
n : ℕ
hint : MemLp id (↑n) μ
⊢ ContDiff ℝ (↑n) (charFun μ)

Tactic:
[

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : IsFiniteMeasure μ
n : ℕ
hint : MemLp id (↑n) μ
⊢ ContDiff ℝ (↑n) (charFun μ)
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.TaylorExpansion

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : IsFiniteMeasure μ
n : ℕ∞
hint : ∀ (k : ℕ), MemLp id (↑k) μ
⊢ ContDiff ℝ (↑n) (charFun μ)

Tactic:
[

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : IsFiniteMeasure μ
n : ℕ∞
hint : ∀ (k : ℕ), MemLp id (↑k) μ
⊢ ContDiff ℝ (↑n) (charFun μ)
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.TaylorExpansion

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : IsFiniteMeasure μ
n : ℕ
t : E
hint : MemLp id (↑n) μ
x : Fin n → E
h : innerₗ E = ContinuousLinearMap.toLinearMap₁₂ (innerSL ℝ)
hint' : ∀ (k : ℕ), ↑k ≤ ↑n → Integrable (fun x ↦ ‖x‖ ^ k * ‖1 x‖) μ
⊢ (iteratedFDeriv ℝ n (charFun μ) t) x = I ^ n * ∫ (y : E), ↑(∏ i, ⟪y, x i⟫) * cexp (↑⟪y, t⟫ * I) ∂μ

Tactic:
[

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : IsFiniteMeasure μ
n : ℕ
t : E
hint : MemLp id (↑n) μ
x : Fin n → E
h : innerₗ E = ContinuousLinearMap.toLinearMap₁₂ (innerSL ℝ)
hint' : ∀ (k : ℕ), ↑k ≤ ↑n → Integrable (fun x ↦ ‖x‖ ^ k * ‖1 x‖) μ
⊢ (iteratedFDeriv ℝ n (charFun μ) t) x = I ^ n * ∫ (y : E), ↑(∏ i, ⟪y, x i⟫) * cexp (↑⟪y, t⟫ * I) ∂μ
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.TaylorExpansion

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : IsFiniteMeasure μ
n : ℕ
t : E
hint : MemLp id (↑n) μ
x : Fin n → E
h : innerₗ E = ContinuousLinearMap.toLinearMap₁₂ (innerSL ℝ)
hint' : ∀ (k : ℕ), ↑k ≤ ↑n → Integrable (fun x ↦ ‖x‖ ^ k * ‖1 x‖) μ
⊢ (-((↑Real.pi)⁻¹ * 2⁻¹)) ^ n *
      ∫ (v : E),
        cexp (2 * ↑Real.pi * ((↑Real.pi)⁻¹ * 2⁻¹ * ↑⟪v, t⟫) * I) * ((-(2 * ↑Real.pi * I)) ^ n * ∏ i, ↑⟪v, x i⟫) ∂μ =
    I ^ n * ∫ (y : E), (∏ i, ↑⟪y, x i⟫) * cexp (↑⟪y, t⟫ * I) ∂μ

Tactic:
[

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : IsFiniteMeasure μ
n : ℕ
t : E
hint : MemLp id (↑n) μ
x : Fin n → E
h : innerₗ E = ContinuousLinearMap.toLinearMap₁₂ (innerSL ℝ)
hint' : ∀ (k : ℕ), ↑k ≤ ↑n → Integrable (fun x ↦ ‖x‖ ^ k * ‖1 x‖) μ
⊢ (-((↑Real.pi)⁻¹ * 2⁻¹)) ^ n *
      ∫ (v : E),
        cexp (2 * ↑Real.pi * ((↑Real.pi)⁻¹ * 2⁻¹ * ↑⟪v, t⟫) * I) * ((-(2 * ↑Real.pi * I)) ^ n * ∏ i, ↑⟪v, x i⟫) ∂μ =
    I ^ n * ∫ (y : E), (∏ i, ↑⟪y, x i⟫) * cexp (↑⟪y, t⟫ * I) ∂μ
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
⊢ taylorWithinEval (charFun μ) n univ 0 t = ∑ k ∈ Finset.range (n + 1), (↑k !)⁻¹ * (↑t * I) ^ k * ↑(∫ (x : ℝ), x ^ k ∂μ)

Tactic:
[

Goal state after:
μ : Measure ℝ
inst✝ : IsFiniteMeasure μ
n : ℕ
hint : MemLp id (↑n) μ
t : ℝ
⊢ taylorWithinEval (charFun μ) n univ 0 t = ∑ k ∈ Finset.range (n + 1), (↑k !)⁻¹ * (↑t * I) ^ k * ↑(∫ (x : ℝ), x ^ k ∂μ)
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.TaylorExpansion

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝ : IsProbabilityMeasure P
X : Ω → ℝ
hX : AEMeasurable X P
h0 : ∫ (x : Ω), X x ∂P = 0
h1 : ∫ (x : Ω), (X ^ 2) x ∂P = 1
⊢ (fun t ↦ charFun (Measure.map X P) t - (1 - ↑t ^ 2 / 2)) =o[𝓝 0] fun t ↦ t ^ 2

Tactic:
[

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝ : IsProbabilityMeasure P
X : Ω → ℝ
hX : AEMeasurable X P
h0 : ∫ (x : Ω), X x ∂P = 0
h1 : ∫ (x : Ω), (X ^ 2) x ∂P = 1
⊢ (fun t ↦ charFun (Measure.map X P) t - (1 - ↑t ^ 2 / 2)) =o[𝓝 0] fun t ↦ t ^ 2
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: case e_f
E : Type u_2
mE : MeasurableSpace E
μ : Measure E
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
t x : E
⊢ cexp (↑⟪x, t⟫ * I) = cexp (2 * ↑π * ((2 * ↑π)⁻¹ * ↑⟪x, t⟫) * I)

Tactic:
rw [← mul_assoc, mul_inv_cancel₀ (by simp [pi_ne_zero]), one_mul]

Goal state after:
Module: Mathlib.Topology.Separation.CompletelyRegular

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
⊢ CompletelyRegularSpace X = ?m.25

Tactic:
[

Goal state after:
X : Type u
inst✝ : TopologicalSpace X
⊢ CompletelyRegularSpace X = ?m.25
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:
[

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 (↑⟪r • x, t⟫ * I) ∂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
μ : Measure E
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : BorelSpace E
r t : E
⊢ charFun (Measure.map (fun x ↦ r + x) μ) t = charFun μ t * cexp (↑⟪r, t⟫ * I)

Tactic:
[

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
⊢ charFun (Measure.map (fun x ↦ r + x) μ) t = charFun μ t * cexp (↑⟪r, t⟫ * I)
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_3
inst✝⁷ : MeasurableSpace E
μ ν : Measure E
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : BorelSpace E
inst✝³ : SecondCountableTopology E
inst✝² : CompleteSpace E
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : charFun μ = charFun ν
⊢ μ = ν

Tactic:
[

Goal state after:
E : Type u_3
inst✝⁷ : MeasurableSpace E
μ ν : Measure E
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : BorelSpace E
inst✝³ : SecondCountableTopology E
inst✝² : CompleteSpace E
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : charFun μ = charFun ν
⊢ μ = ν
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_3
inst✝⁶ : MeasurableSpace E
μ ν : Measure E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : BorelSpace E
inst✝² : SecondCountableTopology E
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
t : E
⊢ charFun (μ ∗ ν) t = charFun μ t * charFun ν t

Tactic:
[

Goal state after:
E : Type u_3
inst✝⁶ : MeasurableSpace E
μ ν : Measure E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : BorelSpace E
inst✝² : SecondCountableTopology E
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
t : E
⊢ charFun (μ ∗ ν) t = charFun μ t * charFun ν t
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)
⊢ charFun (Measure.map (toLp 2) (μ.prod ν)) t = charFun μ t.ofLp.1 * charFun ν t.ofLp.2

Tactic:
[

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)
⊢ charFun (Measure.map (toLp 2) (μ.prod ν)) t = charFun μ t.ofLp.1 * charFun ν t.ofLp.2
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
⊢ charFun (Measure.map (toLp 2) (Measure.pi μ)) t = ∏ i, charFun (μ i) (t.ofLp i)

Tactic:
[

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
⊢ charFun (Measure.map (toLp 2) (Measure.pi μ)) t = ∏ i, charFun (μ i) (t.ofLp i)
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 : x ∈ K
⊢ ∃ f, Continuous[⨅ i, t i, _] f ∧ f x = 0 ∧ EqOn f 1 Kᶜ

Tactic:
[

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 : x ∈ K
⊢ ∃ f, Continuous[⨅ i, t i, _] f ∧ f x = 0 ∧ EqOn f 1 Kᶜ
Module: Mathlib.Topology.Separation.CompletelyRegular

Goal state before:
pp: case a
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompletelyRegularSpace X
x : X
⊢ comap stoneCechUnit (𝓝 (stoneCechUnit x)) ≤ 𝓝 x

Tactic:
[

Goal state after:
case a
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompletelyRegularSpace X
x : X
⊢ comap stoneCechUnit (𝓝 (stoneCechUnit x)) ≤ 𝓝 x
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
mE : MeasurableSpace E
μ : Measure E
inst✝ : BorelSpace E
r : E
L : StrongDual ℝ E
⊢ charFunDual (Measure.map (fun x ↦ r + x) μ) L = charFunDual μ L * cexp (↑(L r) * I)

Tactic:
[

Goal state after:
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
mE : MeasurableSpace E
μ : Measure E
inst✝ : BorelSpace E
r : E
L : StrongDual ℝ E
⊢ charFunDual (Measure.map (fun x ↦ r + x) μ) L = charFunDual μ L * cexp (↑(L r) * I)
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
ν : Measure F
inst✝¹ : SFinite μ
inst✝ : SFinite ν
L : StrongDual ℝ (E × F)
⊢ charFunDual (μ.prod ν) L =
    charFunDual μ (L ∘SL ContinuousLinearMap.inl ℝ E F) * charFunDual ν (L ∘SL ContinuousLinearMap.inr ℝ E F)

Tactic:
[

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
ν : Measure F
inst✝¹ : SFinite μ
inst✝ : SFinite ν
L : StrongDual ℝ (E × F)
⊢ charFunDual (μ.prod ν) L =
    charFunDual μ (L ∘SL ContinuousLinearMap.inl ℝ E F) * charFunDual ν (L ∘SL ContinuousLinearMap.inr ℝ E F)
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
ν : Measure F
inst✝¹ : SFinite μ
inst✝ : SFinite ν
L : StrongDual ℝ (E × F)
⊢ charFunDual (μ.prod ν) L =
    charFunDual μ (L ∘SL ContinuousLinearMap.inl ℝ E F) * charFunDual ν (L ∘SL ContinuousLinearMap.inr ℝ E F)

Tactic:
by
  simp_rw [charFunDual_apply, ← L.comp_inl_add_comp_inr, ofReal_add, add_mul,
    Complex.exp_add, ← integral_prod_mul]

Goal state after:
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))
⊢ charFunDual (Measure.map (toLp p) (μ.prod ν)) L =
    charFunDual μ (L ∘SL ↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inl ℝ E F) *
      charFunDual ν (L ∘SL ↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inr ℝ E F)

Tactic:
[

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))
⊢ charFunDual (Measure.map (toLp p) (μ.prod ν)) L =
    charFunDual μ (L ∘SL ↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inl ℝ E F) *
      charFunDual ν (L ∘SL ↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inr ℝ E F)
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: 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 ((i : ι) → E i)
⊢ charFunDual (Measure.pi μ) L = ∏ i, charFunDual (μ i) (L ∘SL ContinuousLinearMap.single ℝ E i)

Tactic:
[

Goal state after:
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 ((i : ι) → E i)
⊢ charFunDual (Measure.pi μ) L = ∏ i, charFunDual (μ i) (L ∘SL ContinuousLinearMap.single ℝ E i)
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)
⊢ charFunDual (Measure.map (toLp p) (Measure.pi μ)) L =
    ∏ i, charFunDual (μ i) (L ∘SL ↑(PiLp.continuousLinearEquiv p ℝ E).symm ∘SL ContinuousLinearMap.single ℝ E i)

Tactic:
[

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)
⊢ charFunDual (Measure.map (toLp p) (Measure.pi μ)) L =
    ∏ i, charFunDual (μ i) (L ∘SL ↑(PiLp.continuousLinearEquiv p ℝ E).symm ∘SL ContinuousLinearMap.single ℝ E i)