mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Basic

Goal state before:
pp: Ω : Type u_1
E : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
X : Ω → E
inst✝ : NormedSpace ℝ E
L : StrongDual ℝ E
hX : HasGaussianLaw X P
⊢ cexp (↑(∫ (x : Ω), (⇑L ∘ X) x ∂P) * I - ↑Var[⇑L ∘ X; P] / 2) =
    cexp (↑(∫ (ω : Ω), L (X ω) ∂P) * I - ↑Var[fun ω ↦ L (X ω); P] / 2)

Tactic:
Function.comp_def

Goal state after:
Ω : Type u_1
E : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
X : Ω → E
inst✝ : NormedSpace ℝ E
L : StrongDual ℝ E
hX : HasGaussianLaw X P
⊢ cexp (↑(∫ (x : Ω), (fun x ↦ L (X x)) x ∂P) * I - ↑Var[fun x ↦ L (X x); P] / 2) =
    cexp (↑(∫ (ω : Ω), L (X ω) ∂P) * I - ↑Var[fun ω ↦ L (X ω); P] / 2)
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Basic

Goal state before:
pp: Ω : Type u_1
E : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
X : Ω → E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
hX : HasGaussianLaw X P
p : ℝ≥0∞
hp : p ≠ ∞
⊢ MemLp id p (Measure.map X P)

pp: case hg
Ω : Type u_1
E : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
X : Ω → E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
hX : HasGaussianLaw X P
p : ℝ≥0∞
hp : p ≠ ∞
⊢ AEStronglyMeasurable id (Measure.map X P)

pp: case hf
Ω : Type u_1
E : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
X : Ω → E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
hX : HasGaussianLaw X P
p : ℝ≥0∞
hp : p ≠ ∞
⊢ AEMeasurable X P

pp: Ω : Type u_1
E : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
X : Ω → E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
hX : HasGaussianLaw X P
p : ℝ≥0∞
hp : p ≠ ∞
⊢ MeasurableSpace E

Tactic:
· exact hX.isGaussian_map.memLp_id _ p hp

Goal state after:
case hg
Ω : Type u_1
E : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
X : Ω → E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
hX : HasGaussianLaw X P
p : ℝ≥0∞
hp : p ≠ ∞
⊢ AEStronglyMeasurable id (Measure.map X P)

case hf
Ω : Type u_1
E : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
X : Ω → E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
hX : HasGaussianLaw X P
p : ℝ≥0∞
hp : p ≠ ∞
⊢ AEMeasurable X P
Module: Mathlib.Probability.Independence.Process.HasIndepIncrements.Basic

Goal state before:
pp: T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
hX : HasIndepIncrements X P
t : ℕ → T
ht : Monotone t
x✝ : Fin (0 + 1)
⊢ ∀ ⦃b : Fin (0 + 1)⦄, x✝ ≤ b → (fun x ↦ t 0) x✝ ≤ (fun x ↦ t 0) b

Tactic:
grind

Goal state after:
Module: Mathlib.Probability.Independence.Process.HasIndepIncrements.Basic

Goal state before:
pp: T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
h : ∀ (t : ℕ → T), Monotone t → EventuallyConst t atTop → iIndepFun (fun i ω ↦ X (t (i + 1)) ω - X (t i) ω) P
n : ℕ
t : Fin (n + 1) → T
ht : Monotone t
k : ℕ
⊢ min n k < n + 1

Tactic:
grind

Goal state after:
Module: Mathlib.Probability.Independence.Process.HasIndepIncrements.Basic

Goal state before:
pp: case e'_6.e'_5.e'_2
T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
h : ∀ (t : ℕ → T), Monotone t → EventuallyConst t atTop → iIndepFun (fun i ω ↦ X (t (i + 1)) ω - X (t i) ω) P
n : ℕ
t : Fin (n + 1) → T
ht : Monotone t
t' : ℕ → T := fun k ↦ t ⟨min n k, ⋯⟩
i : Fin n
ω : Ω
⊢ ↑i.succ = min n (↑i + 1)

Tactic:
grind

Goal state after:
Module: Mathlib.Probability.Independence.Process.HasIndepIncrements.Basic

Goal state before:
pp: case e'_6.e'_6.e'_2
T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
h : ∀ (t : ℕ → T), Monotone t → EventuallyConst t atTop → iIndepFun (fun i ω ↦ X (t (i + 1)) ω - X (t i) ω) P
n : ℕ
t : Fin (n + 1) → T
ht : Monotone t
t' : ℕ → T := fun k ↦ t ⟨min n k, ⋯⟩
i : Fin n
ω : Ω
⊢ ↑i.castSucc = min n ↑i

Tactic:
grind

Goal state after:
Module: Mathlib.Probability.Independence.Process.HasIndepIncrements.Basic

Goal state before:
pp: T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
h : ∀ (t : ℕ → T), Monotone t → EventuallyConst t atTop → iIndepFun (fun i ω ↦ X (t (i + 1)) ω - X (t i) ω) P
n : ℕ
t : Fin (n + 1) → T
ht : Monotone t
t' : ℕ → T := fun k ↦ t ⟨min n k, ⋯⟩
a b : ℕ
hab : a ≤ b
⊢ ⟨min n a, ⋯⟩ ≤ ⟨min n b, ⋯⟩

Tactic:
grind

Goal state after:
Module: Mathlib.Probability.Independence.Process.HasIndepIncrements.Basic

Goal state before:
pp: T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
h : ∀ (t : ℕ → T), Monotone t → EventuallyConst t atTop → iIndepFun (fun i ω ↦ X (t (i + 1)) ω - X (t i) ω) P
n : ℕ
t : Fin (n + 1) → T
ht : Monotone t
t' : ℕ → T := fun k ↦ t ⟨min n k, ⋯⟩
⊢ ∀ (j : ℕ), n ≤ j → t' j = t' n

Tactic:
grind

Goal state after:
Module: Mathlib.Probability.Independence.Process.HasIndepIncrements.Basic

Goal state before:
pp: T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
hX : HasIndepIncrements X P
r s t : T
hrs : r ≤ s
hst : s ≤ t
τ : ℕ → T :=
  fun x ↦
    match x with
    | 0 => r
    | 1 => s
    | x => t
x✝ : ℕ
⊢ ∀ ⦃b : ℕ⦄, x✝ ≤ b → τ x✝ ≤ τ b

Tactic:
grind

Goal state after:
Module: Mathlib.Probability.Independence.Process.HasIndepIncrements.Basic

Goal state before:
pp: T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
hX : HasIndepIncrements X P
r s t : T
hrs : r ≤ s
hst : s ≤ t
τ : ℕ → T :=
  fun x ↦
    match x with
    | 0 => r
    | 1 => s
    | x => t
⊢ 0 ≠ 1

Tactic:
grind

Goal state after:
Module: Mathlib.Probability.Independence.CharacteristicFunction

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝¹⁰ : IsFiniteMeasure P
E : Type u_2
F : Type u_3
mE : MeasurableSpace E
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : BorelSpace E
inst✝⁷ : SecondCountableTopology E
mF : MeasurableSpace F
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : CompleteSpace F
inst✝⁴ : BorelSpace F
inst✝³ : SecondCountableTopology F
X : Ω → E
Y : Ω → F
inst✝² : InnerProductSpace ℝ E
inst✝¹ : InnerProductSpace ℝ F
inst✝ : CompleteSpace E
hX : AEMeasurable X P
hY : AEMeasurable Y P
⊢ (∀ (t : WithLp 2 (E × F)),
      charFun (Measure.map (toLp 2 ∘ fun ω ↦ (X ω, Y ω)) P) t =
        charFun (Measure.map X P) t.ofLp.1 * charFun (Measure.map Y P) t.ofLp.2) ↔
    ∀ (t : WithLp 2 (E × F)),
      charFun (Measure.map (fun ω ↦ toLp 2 (X ω, Y ω)) P) t =
        charFun (Measure.map X P) t.ofLp.1 * charFun (Measure.map Y P) t.ofLp.2

Tactic:
Function.comp_def

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝¹⁰ : IsFiniteMeasure P
E : Type u_2
F : Type u_3
mE : MeasurableSpace E
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : BorelSpace E
inst✝⁷ : SecondCountableTopology E
mF : MeasurableSpace F
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : CompleteSpace F
inst✝⁴ : BorelSpace F
inst✝³ : SecondCountableTopology F
X : Ω → E
Y : Ω → F
inst✝² : InnerProductSpace ℝ E
inst✝¹ : InnerProductSpace ℝ F
inst✝ : CompleteSpace E
hX : AEMeasurable X P
hY : AEMeasurable Y P
⊢ (∀ (t : WithLp 2 (E × F)),
      charFun (Measure.map (fun x ↦ toLp 2 (X x, Y x)) P) t =
        charFun (Measure.map X P) t.ofLp.1 * charFun (Measure.map Y P) t.ofLp.2) ↔
    ∀ (t : WithLp 2 (E × F)),
      charFun (Measure.map (fun ω ↦ toLp 2 (X ω, Y ω)) P) t =
        charFun (Measure.map X P) t.ofLp.1 * charFun (Measure.map Y P) t.ofLp.2
Module: Mathlib.Probability.Process.FiniteDimensionalLaws

Goal state before:
pp: T : Type u_1
Ω : Type u_2
𝓧 : T → Type u_3
mΩ : MeasurableSpace Ω
mα : (t : T) → MeasurableSpace (𝓧 t)
X : (t : T) → Ω → 𝓧 t
P : Measure Ω
hX : AEMeasurable (fun ω x ↦ X x ω) P
I : Finset T
⊢ Measure.map (I.restrict ∘ fun ω x ↦ X x ω) P = (fun I ↦ Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P) I

Tactic:
Function.comp_def

Goal state after:
T : Type u_1
Ω : Type u_2
𝓧 : T → Type u_3
mΩ : MeasurableSpace Ω
mα : (t : T) → MeasurableSpace (𝓧 t)
X : (t : T) → Ω → 𝓧 t
P : Measure Ω
hX : AEMeasurable (fun ω x ↦ X x ω) P
I : Finset T
⊢ Measure.map (fun x ↦ I.restrict fun x_1 ↦ X x_1 x) P = (fun I ↦ Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P) I
Module: Mathlib.Probability.Process.FiniteDimensionalLaws

Goal state before:
pp: T : Type u_1
Ω : Type u_2
𝓧 : T → Type u_3
mΩ : MeasurableSpace Ω
mα : (t : T) → MeasurableSpace (𝓧 t)
X Y : (t : T) → Ω → 𝓧 t
P : Measure Ω
inst✝ : IsFiniteMeasure P
hX : AEMeasurable (fun ω x ↦ X x ω) P
hY : AEMeasurable (fun ω x ↦ Y x ω) P
h : Measure.map (fun ω x ↦ X x ω) P = Measure.map (fun ω x ↦ Y x ω) P
I : Finset T
⊢ Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P = Measure.map (I.restrict ∘ fun ω x ↦ X x ω) P

Tactic:
Function.comp_def

Goal state after:
T : Type u_1
Ω : Type u_2
𝓧 : T → Type u_3
mΩ : MeasurableSpace Ω
mα : (t : T) → MeasurableSpace (𝓧 t)
X Y : (t : T) → Ω → 𝓧 t
P : Measure Ω
inst✝ : IsFiniteMeasure P
hX : AEMeasurable (fun ω x ↦ X x ω) P
hY : AEMeasurable (fun ω x ↦ Y x ω) P
h : Measure.map (fun ω x ↦ X x ω) P = Measure.map (fun ω x ↦ Y x ω) P
I : Finset T
⊢ Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P = Measure.map (fun x ↦ I.restrict fun x_1 ↦ X x_1 x) P
Module: Mathlib.Probability.Process.FiniteDimensionalLaws

Goal state before:
pp: T : Type u_1
Ω : Type u_2
𝓧 : T → Type u_3
mΩ : MeasurableSpace Ω
mα : (t : T) → MeasurableSpace (𝓧 t)
X Y : (t : T) → Ω → 𝓧 t
P : Measure Ω
inst✝ : IsFiniteMeasure P
hX : AEMeasurable (fun ω x ↦ X x ω) P
hY : AEMeasurable (fun ω x ↦ Y x ω) P
h : Measure.map (fun ω x ↦ X x ω) P = Measure.map (fun ω x ↦ Y x ω) P
I : Finset T
hX' : Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P = Measure.map I.restrict (Measure.map (fun ω x ↦ X x ω) P)
⊢ Measure.map (fun ω ↦ I.restrict fun x ↦ Y x ω) P = Measure.map (I.restrict ∘ fun ω x ↦ Y x ω) P

Tactic:
Function.comp_def

Goal state after:
T : Type u_1
Ω : Type u_2
𝓧 : T → Type u_3
mΩ : MeasurableSpace Ω
mα : (t : T) → MeasurableSpace (𝓧 t)
X Y : (t : T) → Ω → 𝓧 t
P : Measure Ω
inst✝ : IsFiniteMeasure P
hX : AEMeasurable (fun ω x ↦ X x ω) P
hY : AEMeasurable (fun ω x ↦ Y x ω) P
h : Measure.map (fun ω x ↦ X x ω) P = Measure.map (fun ω x ↦ Y x ω) P
I : Finset T
hX' : Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P = Measure.map I.restrict (Measure.map (fun ω x ↦ X x ω) P)
⊢ Measure.map (fun ω ↦ I.restrict fun x ↦ Y x ω) P = Measure.map (fun x ↦ I.restrict fun x_1 ↦ Y x_1 x) P
Module: Mathlib.Probability.Independence.CharacteristicFunction

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
p : ℝ≥0∞
inst✝¹¹ : Fact (1 ≤ p)
inst✝¹⁰ : IsFiniteMeasure P
E : Type u_2
F : Type u_3
mE : MeasurableSpace E
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : BorelSpace E
inst✝⁷ : SecondCountableTopology E
mF : MeasurableSpace F
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : CompleteSpace F
inst✝⁴ : BorelSpace F
inst✝³ : SecondCountableTopology F
X : Ω → E
Y : Ω → F
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace E
hX : AEMeasurable X P
hY : AEMeasurable Y P
⊢ (∀ (L : StrongDual ℝ (WithLp p (E × F))),
      charFunDual (Measure.map (toLp p ∘ fun ω ↦ (X ω, Y ω)) P) L =
        charFunDual (Measure.map X P)
            (L ∘SL ↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inl ℝ E F) *
          charFunDual (Measure.map Y P)
            (L ∘SL ↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inr ℝ E F)) ↔
    ∀ (L : StrongDual ℝ (WithLp p (E × F))),
      charFunDual (Measure.map (fun ω ↦ toLp p (X ω, Y ω)) P) L =
        charFunDual (Measure.map X P)
            (L ∘SL ↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inl ℝ E F) *
          charFunDual (Measure.map Y P)
            (L ∘SL ↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inr ℝ E F)

Tactic:
Function.comp_def

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
p : ℝ≥0∞
inst✝¹¹ : Fact (1 ≤ p)
inst✝¹⁰ : IsFiniteMeasure P
E : Type u_2
F : Type u_3
mE : MeasurableSpace E
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : BorelSpace E
inst✝⁷ : SecondCountableTopology E
mF : MeasurableSpace F
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : CompleteSpace F
inst✝⁴ : BorelSpace F
inst✝³ : SecondCountableTopology F
X : Ω → E
Y : Ω → F
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace E
hX : AEMeasurable X P
hY : AEMeasurable Y P
⊢ (∀ (L : StrongDual ℝ (WithLp p (E × F))),
      charFunDual (Measure.map (fun x ↦ toLp p (X x, Y x)) P) L =
        charFunDual (Measure.map X P)
            (L ∘SL ↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inl ℝ E F) *
          charFunDual (Measure.map Y P)
            (L ∘SL ↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inr ℝ E F)) ↔
    ∀ (L : StrongDual ℝ (WithLp p (E × F))),
      charFunDual (Measure.map (fun ω ↦ toLp p (X ω, Y ω)) P) L =
        charFunDual (Measure.map X P)
            (L ∘SL ↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inl ℝ E F) *
          charFunDual (Measure.map Y P)
            (L ∘SL ↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inr ℝ E F)
Module: Mathlib.Probability.Process.FiniteDimensionalLaws

Goal state before:
pp: T : Type u_1
Ω : Type u_2
𝓧 : T → Type u_3
mΩ : MeasurableSpace Ω
mα : (t : T) → MeasurableSpace (𝓧 t)
X Y : (t : T) → Ω → 𝓧 t
P : Measure Ω
inst✝ : IsFiniteMeasure P
hX : AEMeasurable (fun ω x ↦ X x ω) P
hY : AEMeasurable (fun ω x ↦ Y x ω) P
h : ∀ (t : T), X t =ᵐ[P] Y t
⊢ ∀ (I : Finset T), Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P = Measure.map (fun ω ↦ I.restrict fun x ↦ Y x ω) P

Tactic:
exact fun I ↦ map_restrict_eq_of_forall_ae_eq h I

Goal state after:
Module: Mathlib.Probability.Independence.CharacteristicFunction

Goal state before:
pp: case convert_9
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
s✝ : Finset ι
E : Type u_3
inst✝⁴ : MeasurableSpace E
inst✝³ : NormedAddCommGroup E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
X : ι → Ω → E
inst✝ : NormedSpace ℝ E
this : IsProbabilityMeasure P
i : ι
s : Finset ι
hi : i ∉ s
hs :
  (∀ i ∈ s, AEMeasurable (X i) P) →
    iIndepFun (s.restrict X) P → charFunDual (Measure.map (∑ i ∈ s, X i) P) = ∏ i ∈ s, charFunDual (Measure.map (X i) P)
mX : ∀ i_1 ∈ insert i s, AEMeasurable (X i_1) P
hX : iIndepFun ((insert i s).restrict X) P
e : ↥(insert i s) → ι := Subtype.val
x✝ : ↥(insert i s)
a✝ : x✝ ∈ ↑{x | ↑x ∈ s}
⊢ ↑x✝ ∈ ↑s

Tactic:
grind

Goal state after:
Module: Mathlib.Probability.Independence.CharacteristicFunction

Goal state before:
pp: case convert_10
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
s✝ : Finset ι
E : Type u_3
inst✝⁴ : MeasurableSpace E
inst✝³ : NormedAddCommGroup E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
X : ι → Ω → E
inst✝ : NormedSpace ℝ E
this : IsProbabilityMeasure P
i : ι
s : Finset ι
hi : i ∉ s
hs :
  (∀ i ∈ s, AEMeasurable (X i) P) →
    iIndepFun (s.restrict X) P → charFunDual (Measure.map (∑ i ∈ s, X i) P) = ∏ i ∈ s, charFunDual (Measure.map (X i) P)
mX : ∀ i_1 ∈ insert i s, AEMeasurable (X i_1) P
hX : iIndepFun ((insert i s).restrict X) P
e : ↥(insert i s) → ι := Subtype.val
⊢ ∀ i_1 ∈ s, (i_1 = i ∨ i_1 ∈ s → i_1 ∈ s → ¬i_1 = i ∧ i_1 ∉ s) → X i_1 = 0

Tactic:
grind

Goal state after:
Module: Mathlib.Probability.Independence.CharacteristicFunction

Goal state before:
pp: case convert_11
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
s✝ : Finset ι
E : Type u_3
inst✝⁴ : MeasurableSpace E
inst✝³ : NormedAddCommGroup E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
X : ι → Ω → E
inst✝ : NormedSpace ℝ E
this : IsProbabilityMeasure P
i : ι
s : Finset ι
hi : i ∉ s
hs :
  (∀ i ∈ s, AEMeasurable (X i) P) →
    iIndepFun (s.restrict X) P → charFunDual (Measure.map (∑ i ∈ s, X i) P) = ∏ i ∈ s, charFunDual (Measure.map (X i) P)
mX : ∀ i_1 ∈ insert i s, AEMeasurable (X i_1) P
hX : iIndepFun ((insert i s).restrict X) P
e : ↥(insert i s) → ι := Subtype.val
⊢ ∀ i_1 ∈ {x | ↑x ∈ s}, X ↑i_1 = X ↑i_1

Tactic:
grind

Goal state after:
Module: Mathlib.Probability.Independence.CharacteristicFunction

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
inst✝⁶ : Fintype ι
inst✝⁵ : IsProbabilityMeasure P
E : ι → Type u_3
mE : (i : ι) → MeasurableSpace (E i)
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : ∀ (i : ι), CompleteSpace (E i)
inst✝² : ∀ (i : ι), BorelSpace (E i)
inst✝¹ : ∀ (i : ι), SecondCountableTopology (E i)
X : (i : ι) → Ω → E i
inst✝ : (i : ι) → InnerProductSpace ℝ (E i)
hX : ∀ (i : ι), AEMeasurable (X i) P
⊢ (∀ (t : WithLp 2 ((i : ι) → E i)),
      charFun (Measure.map (toLp 2 ∘ fun ω i ↦ X i ω) P) t = ∏ i, charFun (Measure.map (X i) P) (t.ofLp i)) ↔
    ∀ (t : WithLp 2 ((x : ι) → E x)),
      charFun (Measure.map (fun ω ↦ toLp 2 fun x ↦ X x ω) P) t = ∏ i, charFun (Measure.map (X i) P) (t.ofLp i)

Tactic:
Function.comp_def

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
inst✝⁶ : Fintype ι
inst✝⁵ : IsProbabilityMeasure P
E : ι → Type u_3
mE : (i : ι) → MeasurableSpace (E i)
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : ∀ (i : ι), CompleteSpace (E i)
inst✝² : ∀ (i : ι), BorelSpace (E i)
inst✝¹ : ∀ (i : ι), SecondCountableTopology (E i)
X : (i : ι) → Ω → E i
inst✝ : (i : ι) → InnerProductSpace ℝ (E i)
hX : ∀ (i : ι), AEMeasurable (X i) P
⊢ (∀ (t : WithLp 2 ((i : ι) → E i)),
      charFun (Measure.map (fun x ↦ toLp 2 fun i ↦ X i x) P) t = ∏ i, charFun (Measure.map (X i) P) (t.ofLp i)) ↔
    ∀ (t : WithLp 2 ((x : ι) → E x)),
      charFun (Measure.map (fun ω ↦ toLp 2 fun x ↦ X x ω) P) t = ∏ i, charFun (Measure.map (X i) P) (t.ofLp i)
Module: Mathlib.Probability.Independence.CharacteristicFunction

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
p : ℝ≥0∞
inst✝⁸ : Fact (1 ≤ p)
ι : Type u_2
inst✝⁷ : Fintype ι
inst✝⁶ : IsProbabilityMeasure P
E : ι → Type u_3
mE : (i : ι) → MeasurableSpace (E i)
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : ∀ (i : ι), CompleteSpace (E i)
inst✝³ : ∀ (i : ι), BorelSpace (E i)
inst✝² : ∀ (i : ι), SecondCountableTopology (E i)
X : (i : ι) → Ω → E i
inst✝¹ : (i : ι) → NormedSpace ℝ (E i)
inst✝ : DecidableEq ι
hX : ∀ (i : ι), AEMeasurable (X i) P
⊢ (∀ (L : StrongDual ℝ (WithLp p ((i : ι) → E i))),
      charFunDual (Measure.map (toLp p ∘ fun ω i ↦ X i ω) P) L =
        ∏ i,
          charFunDual (Measure.map (X i) P)
            (L ∘SL ↑(PiLp.continuousLinearEquiv p ℝ E).symm ∘SL ContinuousLinearMap.single ℝ E i)) ↔
    ∀ (L : StrongDual ℝ (WithLp p ((x : ι) → E x))),
      charFunDual (Measure.map (fun ω ↦ toLp p fun x ↦ X x ω) P) L =
        ∏ i,
          charFunDual (Measure.map (X i) P)
            (L ∘SL ↑(PiLp.continuousLinearEquiv p ℝ E).symm ∘SL ContinuousLinearMap.single ℝ E i)

Tactic:
Function.comp_def

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
p : ℝ≥0∞
inst✝⁸ : Fact (1 ≤ p)
ι : Type u_2
inst✝⁷ : Fintype ι
inst✝⁶ : IsProbabilityMeasure P
E : ι → Type u_3
mE : (i : ι) → MeasurableSpace (E i)
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : ∀ (i : ι), CompleteSpace (E i)
inst✝³ : ∀ (i : ι), BorelSpace (E i)
inst✝² : ∀ (i : ι), SecondCountableTopology (E i)
X : (i : ι) → Ω → E i
inst✝¹ : (i : ι) → NormedSpace ℝ (E i)
inst✝ : DecidableEq ι
hX : ∀ (i : ι), AEMeasurable (X i) P
⊢ (∀ (L : StrongDual ℝ (WithLp p ((i : ι) → E i))),
      charFunDual (Measure.map (fun x ↦ toLp p fun i ↦ X i x) P) L =
        ∏ i,
          charFunDual (Measure.map (X i) P)
            (L ∘SL ↑(PiLp.continuousLinearEquiv p ℝ E).symm ∘SL ContinuousLinearMap.single ℝ E i)) ↔
    ∀ (L : StrongDual ℝ (WithLp p ((x : ι) → E x))),
      charFunDual (Measure.map (fun ω ↦ toLp p fun x ↦ X x ω) P) L =
        ∏ i,
          charFunDual (Measure.map (X i) P)
            (L ∘SL ↑(PiLp.continuousLinearEquiv p ℝ E).symm ∘SL ContinuousLinearMap.single ℝ E i)
Module: Mathlib.Probability.Independence.Process.Basic

Goal state before:
pp: S : Type u_1
Ω : Type u_3
mΩ : MeasurableSpace Ω
α : Type u_4
mα : MeasurableSpace α
κ : Kernel α Ω
P : Measure α
𝓧 : S → Type u_5
𝓨 : Type u_6
inst✝² : (i : S) → MeasurableSpace (𝓧 i)
inst✝¹ : MeasurableSpace 𝓨
X : (i : S) → Ω → 𝓧 i
Y : Ω → 𝓨
hX : ∀ (i : S), Measurable (X i)
hY : Measurable Y
h : ∀ (I : Finset S), IndepFun (fun ω i ↦ X (↑i) ω) Y κ P
inst✝ : IsZeroOrMarkovKernel κ
πX : Set (Set Ω) := {s | ∃ t ∈ squareCylinders fun i ↦ {s | MeasurableSet s}, (fun ω i ↦ X i ω) ⁻¹' t = s}
πX_pi : IsPiSystem πX
⊢ generateFrom ((preimage fun ω i ↦ X i ω) '' squareCylinders fun i ↦ {s | MeasurableSet s}) = generateFrom πX

Tactic:
rfl

Goal state after:
Module: Mathlib.Probability.Distributions.Fernique

Goal state before:
pp: E : Type u_1
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : SecondCountableTopology E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
h_meas_Ioo : ∀ (a : ℝ), 0 < a → 2⁻¹ < μ {x | ‖x‖ ≤ a} → 1 ≤ μ {x | ‖x‖ ≤ a}
h_ne : ∀ (b : ℝ), μ {x | ‖x‖ ≤ b} ≠ 1
a a' : ℝ
ha : 0 < a'
h_lt : ¬μ {x | ‖x‖ ≤ a'} ≤ 2⁻¹
⊢ False

Tactic:
refine h_ne a' ?_

Goal state after:
E : Type u_1
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : SecondCountableTopology E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
h_meas_Ioo : ∀ (a : ℝ), 0 < a → 2⁻¹ < μ {x | ‖x‖ ≤ a} → 1 ≤ μ {x | ‖x‖ ≤ a}
h_ne : ∀ (b : ℝ), μ {x | ‖x‖ ≤ b} ≠ 1
a a' : ℝ
ha : 0 < a'
h_lt : ¬μ {x | ‖x‖ ≤ a'} ≤ 2⁻¹
⊢ μ {x | ‖x‖ ≤ a'} = 1