awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence (goal)
Goal:
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
inst : Finite ι
E : ι → Type u_3
inst_1 : (i : ι) → NormedAddCommGroup (E i)
inst_2 : (i : ι) → MeasurableSpace (E i)
inst_3 : ∀ (i : ι), CompleteSpace.{u_3} (E i)
inst_4 : ∀ (i : ι), BorelSpace.{u_3} (E i)
inst_5 : ∀ (i : ι), SecondCountableTopology.{u_3} (E i)
inst_6 : (i : ι) → NormedSpace ℝ (E i)
X : (i : ι) → Ω → E i
hX1 : ∀ (i : ι), HasGaussianLaw (X i) P
hX2 : iIndepFun (β := E) X P
this_1 : IsProbabilityMeasure P
this : Fintype ι := Fintype.ofFinite ι
L : StrongDual ℝ ((x : ι) → E x)
⊢ Eq (α := ℂ)
    (∏ i : ι,
      charFunDual (mE := inst_2 i) (Measure.map (X i) P)
        (ContinuousLinearMap.comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i)))
    (∏ x : ι,
      cexp
        ((↑((L : (a : (x : ι) → E x) → ℝ)
                  (Pi.single (M := E) x (@integral _ _ _ _ (inst_2 x) (Measure.map (X x) P) fun (x : E x) ↦ x))) :
              ℂ) *
            I -
          (↑((((covarianceBilinDual (mE := inst_2 x) (Measure.map (X x) P) :
                          (a : StrongDual ℝ (E x)) → StrongDual ℝ (E x) →L[ℝ] ℝ)
                        (ContinuousLinearMap.comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E x)) :
                      StrongDual ℝ (E x) →L[ℝ] ℝ) :
                    (a : StrongDual ℝ (E x)) → ℝ)
                  (ContinuousLinearMap.comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E x))) :
              ℂ) /
            2))
have e_f : ∀ (i : ι),
  charFunDual (mE := inst_2 i) (Measure.map (X i) P) (ContinuousLinearMap.comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i)) =
    cexp
      ((↑((L : (a : (x : ι) → E x) → ℝ)
                (Pi.single (M := E) i (@integral _ _ _ _ (inst_2 i) (Measure.map (X i) P) fun (x : E i) ↦ x))) :
            ℂ) *
          I -
        (↑((((covarianceBilinDual (mE := inst_2 i) (Measure.map (X i) P) :
                        (a : StrongDual ℝ (E i)) → StrongDual ℝ (E i) →L[ℝ] ℝ)
                      (ContinuousLinearMap.comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i)) :
                    StrongDual ℝ (E i) →L[ℝ] ℝ) :
                  (a : StrongDual ℝ (E i)) → ℝ)
                (ContinuousLinearMap.comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i))) :
            ℂ) /
          2)
Tactic: congr with i
-- Lean 4 proof step from Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence (goal)
Goal:
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
inst : Finite ι
E : ι → Type u_3
inst_1 : (i : ι) → NormedAddCommGroup (E i)
inst_2 : (i : ι) → MeasurableSpace (E i)
inst_3 : ∀ (i : ι), CompleteSpace.{u_3} (E i)
inst_4 : ∀ (i : ι), BorelSpace.{u_3} (E i)
inst_5 : ∀ (i : ι), SecondCountableTopology.{u_3} (E i)
inst_6 : (i : ι) → NormedSpace ℝ (E i)
X : (i : ι) → Ω → E i
hX1 : ∀ (i : ι), HasGaussianLaw (X i) P
hX2 : iIndepFun (β := E) X P
this_1 : IsProbabilityMeasure P
this : Fintype ι := Fintype.ofFinite ι
L : StrongDual ℝ ((x : ι) → E x)
i : ι
⊢ charFunDual (mE := inst_2 i) (Measure.map (X i) P) (ContinuousLinearMap.comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i)) =
    cexp
      ((↑((L : (a : (x : ι) → E x) → ℝ)
                (Pi.single (M := E) i (@integral _ _ _ _ (inst_2 i) (Measure.map (X i) P) fun (x : E i) ↦ x))) :
            ℂ) *
          I -
        (↑((((covarianceBilinDual (mE := inst_2 i) (Measure.map (X i) P) :
                        (a : StrongDual ℝ (E i)) → StrongDual ℝ (E i) →L[ℝ] ℝ)
                      (ContinuousLinearMap.comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i)) :
                    StrongDual ℝ (E i) →L[ℝ] ℝ) :
                  (a : StrongDual ℝ (E i)) → ℝ)
                (ContinuousLinearMap.comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i))) :
            ℂ) /
          2)
have e_f : cexp
    ((↑((ContinuousLinearMap.comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i) : (a : E i) → ℝ)
              (@integral _ _ _ _ (inst_2 i) (Measure.map (X i) P) fun (x : E i) ↦ x)) :
          ℂ) *
        I -
      (↑(variance (mΩ := inst_2 i) (⇑(ContinuousLinearMap.comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i)) : (a : E i) → ℝ)
              (Measure.map (X i) P)) :
          ℂ) /
        2) =
  cexp
    ((↑((L : (a : (x : ι) → E x) → ℝ)
              (Pi.single (M := E) i (@integral _ _ _ _ (inst_2 i) (Measure.map (X i) P) fun (x : E i) ↦ x))) :
          ℂ) *
        I -
      (↑(variance (mΩ := inst_2 i) (⇑(ContinuousLinearMap.comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i)) : (a : E i) → ℝ)
              (Measure.map (X i) P)) :
          ℂ) /
        2)
Tactic: rw [(hX1 i).isGaussian_map.charFunDual_eq, integral_complex_ofReal, integral_comp_id_comm,
      covarianceBilinDual_self_eq_variance]
-- Lean 4 proof step from Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence (goal)
Goal:
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
inst : Finite ι
E : ι → Type u_3
inst_1 : (i : ι) → NormedAddCommGroup (E i)
inst_2 : (i : ι) → MeasurableSpace (E i)
inst_3 : ∀ (i : ι), CompleteSpace.{u_3} (E i)
inst_4 : ∀ (i : ι), BorelSpace.{u_3} (E i)
inst_5 : ∀ (i : ι), SecondCountableTopology.{u_3} (E i)
inst_6 : (i : ι) → NormedSpace ℝ (E i)
X : (i : ι) → Ω → E i
hX1 : ∀ (i : ι), HasGaussianLaw (X i) P
hX2 : iIndepFun (β := E) X P
this_1 : IsProbabilityMeasure P
this : Fintype ι := Fintype.ofFinite ι
L : StrongDual ℝ ((x : ι) → E x)
i : ι
e_f :
  cexp
      ((↑((ContinuousLinearMap.comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i) : (a : E i) → ℝ)
                (@integral _ _ _ _ (inst_2 i) (Measure.map (X i) P) fun (x : E i) ↦ x)) :
            ℂ) *
          I -
        (↑(variance (mΩ := inst_2 i)
                (⇑(ContinuousLinearMap.comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i)) : (a : E i) → ℝ)
                (Measure.map (X i) P)) :
            ℂ) /
          2) =
    cexp
      ((↑((L : (a : (x : ι) → E x) → ℝ)
                (Pi.single (M := E) i (@integral _ _ _ _ (inst_2 i) (Measure.map (X i) P) fun (x : E i) ↦ x))) :
            ℂ) *
          I -
        (↑(variance (mΩ := inst_2 i)
                (⇑(ContinuousLinearMap.comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i)) : (a : E i) → ℝ)
                (Measure.map (X i) P)) :
            ℂ) /
          2)
⊢ charFunDual (mE := inst_2 i) (Measure.map (X i) P) (ContinuousLinearMap.comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i)) =
    cexp
      ((↑((L : (a : (x : ι) → E x) → ℝ)
                (Pi.single (M := E) i (@integral _ _ _ _ (inst_2 i) (Measure.map (X i) P) fun (x : E i) ↦ x))) :
            ℂ) *
          I -
        (↑((((covarianceBilinDual (mE := inst_2 i) (Measure.map (X i) P) :
                        (a : StrongDual ℝ (E i)) → StrongDual ℝ (E i) →L[ℝ] ℝ)
                      (ContinuousLinearMap.comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i)) :
                    StrongDual ℝ (E i) →L[ℝ] ℝ) :
                  (a : StrongDual ℝ (E i)) → ℝ)
                (ContinuousLinearMap.comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i))) :
            ℂ) /
          2)
have e_f₁ : MemLp (m0 := inst_2 i) id 2 (Measure.map (X i) P)
Tactic: rw [(hX1 i).isGaussian_map.charFunDual_eq, integral_complex_ofReal, integral_comp_id_comm,
      covarianceBilinDual_self_eq_variance]
-- Lean 4 proof step from Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence (goal)
Goal:
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
inst : Finite ι
E : ι → Type u_3
inst_1 : (i : ι) → NormedAddCommGroup (E i)
inst_2 : (i : ι) → MeasurableSpace (E i)
inst_3 : ∀ (i : ι), CompleteSpace.{u_3} (E i)
inst_4 : ∀ (i : ι), BorelSpace.{u_3} (E i)
inst_5 : ∀ (i : ι), SecondCountableTopology.{u_3} (E i)
inst_6 : (i : ι) → NormedSpace ℝ (E i)
X : (i : ι) → Ω → E i
hX1 : ∀ (i : ι), HasGaussianLaw (X i) P
hX2 : iIndepFun (β := E) X P
this_1 : IsProbabilityMeasure P
this : Fintype ι := Fintype.ofFinite ι
L : StrongDual ℝ ((x : ι) → E x)
i : ι
e_f :
  cexp
      ((↑((ContinuousLinearMap.comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i) : (a : E i) → ℝ)
                (@integral _ _ _ _ (inst_2 i) (Measure.map (X i) P) fun (x : E i) ↦ x)) :
            ℂ) *
          I -
        (↑(variance (mΩ := inst_2 i)
                (⇑(ContinuousLinearMap.comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i)) : (a : E i) → ℝ)
                (Measure.map (X i) P)) :
            ℂ) /
          2) =
    cexp
      ((↑((L : (a : (x : ι) → E x) → ℝ)
                (Pi.single (M := E) i (@integral _ _ _ _ (inst_2 i) (Measure.map (X i) P) fun (x : E i) ↦ x))) :
            ℂ) *
          I -
        (↑(variance (mΩ := inst_2 i)
                (⇑(ContinuousLinearMap.comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i)) : (a : E i) → ℝ)
                (Measure.map (X i) P)) :
            ℂ) /
          2)
e_f₁ : MemLp (m0 := inst_2 i) id 2 (Measure.map (X i) P)
⊢ charFunDual (mE := inst_2 i) (Measure.map (X i) P) (ContinuousLinearMap.comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i)) =
    cexp
      ((↑((L : (a : (x : ι) → E x) → ℝ)
                (Pi.single (M := E) i (@integral _ _ _ _ (inst_2 i) (Measure.map (X i) P) fun (x : E i) ↦ x))) :
            ℂ) *
          I -
        (↑((((covarianceBilinDual (mE := inst_2 i) (Measure.map (X i) P) :
                        (a : StrongDual ℝ (E i)) → StrongDual ℝ (E i) →L[ℝ] ℝ)
                      (ContinuousLinearMap.comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i)) :
                    StrongDual ℝ (E i) →L[ℝ] ℝ) :
                  (a : StrongDual ℝ (E i)) → ℝ)
                (ContinuousLinearMap.comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i))) :
            ℂ) /
          2)
have e_f₂ : @Integrable _ _ _ _ (inst_2 i) id (Measure.map (X i) P)
Tactic: rw [(hX1 i).isGaussian_map.charFunDual_eq, integral_complex_ofReal, integral_comp_id_comm,
      covarianceBilinDual_self_eq_variance]
-- Lean 4 proof step from Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence (goal)
Goal:
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
inst : Finite ι
E : ι → Type u_3
inst_1 : (i : ι) → NormedAddCommGroup (E i)
inst_2 : (i : ι) → MeasurableSpace (E i)
inst_3 : ∀ (i : ι), CompleteSpace.{u_3} (E i)
inst_4 : ∀ (i : ι), BorelSpace.{u_3} (E i)
inst_5 : ∀ (i : ι), SecondCountableTopology.{u_3} (E i)
inst_6 : (i : ι) → NormedSpace ℝ (E i)
X : (i : ι) → Ω → E i
hX1 : ∀ (i : ι), HasGaussianLaw (X i) P
hX2 : iIndepFun (β := E) X P
this_1 : IsProbabilityMeasure P
this : Fintype ι := Fintype.ofFinite ι
⊢ ∃ (m : (x : ι) → E x) (f : StrongDual ℝ ((x : ι) → E x) →L[ℝ] StrongDual ℝ ((x : ι) → E x) →L[ℝ] ℝ),
    f.toBilinForm.IsPosSemidef ∧
      ∀ (L : StrongDual ℝ ((x : ι) → E x)),
        charFunDual (mE := MeasurableSpace.pi (X := E)) (Measure.map (fun (ω : Ω) (x : ι) ↦ X x ω) P) L =
          cexp
            ((↑((L : (a : (x : ι) → E x) → ℝ) m) : ℂ) * I -
              (↑((((f : (a : StrongDual ℝ ((x : ι) → E x)) → StrongDual ℝ ((x : ι) → E x) →L[ℝ] ℝ) L :
                          StrongDual ℝ ((x : ι) → E x) →L[ℝ] ℝ) :
                        (a : StrongDual ℝ ((x : ι) → E x)) → ℝ)
                      L) :
                  ℂ) /
                2)
have refine : ∀ (L : StrongDual ℝ ((x : ι) → E x)),
  charFunDual (mE := MeasurableSpace.pi (X := E)) (Measure.map (fun (ω : Ω) (x : ι) ↦ X x ω) P) L =
    cexp
      ((↑((L : (a : (x : ι) → E x) → ℝ) fun (i : ι) ↦
                @integral _ _ _ _ (inst_2 i) (Measure.map (X i) P) fun (x : E i) ↦ x) :
            ℂ) *
          I -
        (↑((((diagonalStrongDualPi (E := E) fun (i : ι) ↦ covarianceBilinDual (mE := inst_2 i) (Measure.map (X i) P) :
                        (a : StrongDual ℝ ((x : ι) → E x)) → StrongDual ℝ ((x : ι) → E x) →L[ℝ] ℝ)
                      L :
                    StrongDual ℝ ((x : ι) → E x) →L[ℝ] ℝ) :
                  (a : StrongDual ℝ ((x : ι) → E x)) → ℝ)
                L) :
            ℂ) /
          2)
Tactic: classical
    refine ⟨fun i ↦ ∫ x, x ∂P.map (X i),
      .diagonalStrongDualPi (fun i ↦ covarianceBilinDual (P.map (X i))),
      isPosSemidef_diagonalStrongDualPi (fun _ ↦ isPosSemidef_covarianceBilinDual), fun L ↦ ?_⟩
    rw [(iIndepFun_iff_charFunDual_pi (by fun_prop)).1 hX2]
    simp only [← LinearMap.sum_single_apply E (fun i ↦ ∫ x, x ∂P.map (X i)), map_sum, ofReal_sum,
      sum_mul, diagonalStrongDualPi_apply, sum_div, ← sum_sub_distrib, exp_sum]
    congr with i
    rw [(hX1 i).isGaussian_map.charFunDual_eq, integral_complex_ofReal, integral_comp_id_comm,
      covarianceBilinDual_self_eq_variance]
    · simp
    · exact (hX1 i).isGaussian_map.memLp_two_id
    · exact (hX1 i).isGaussian_map.integrable_id
-- Lean 4 proof step from Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence (hypothesis)
Goal:
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
inst : Finite ι
E : ι → Type u_3
inst_1 : (i : ι) → NormedAddCommGroup (E i)
inst_2 : (i : ι) → MeasurableSpace (E i)
inst_3 : ∀ (i : ι), CompleteSpace.{u_3} (E i)
inst_4 : ∀ (i : ι), BorelSpace.{u_3} (E i)
inst_5 : ∀ (i : ι), SecondCountableTopology.{u_3} (E i)
inst_6 : (i : ι) → NormedSpace ℝ (E i)
X : (i : ι) → Ω → E i
hX : HasGaussianLaw (E := (i : ι) → E i) (fun (ω : Ω) (i : ι) ↦ X i ω) P
h :
  ∀ (i j : ι),
    i ≠ j →
      ∀ (L₁ : StrongDual ℝ (E i)) (L₂ : StrongDual ℝ (E j)),
        cov[fun (x : Ω) ↦ (L₁ : (a : E i) → ℝ) (X i x), fun (x : Ω) ↦ (L₂ : (a : E j) → ℝ) (X j x); P] = 0
⊢ iIndepFun (β := E) X P
have this : IsProbabilityMeasure P
Tactic: have := hX.isProbabilityMeasure
-- Lean 4 proof step from Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence (goal)
Goal:
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
inst : Finite ι
E : ι → Type u_3
inst_1 : (i : ι) → NormedAddCommGroup (E i)
inst_2 : (i : ι) → MeasurableSpace (E i)
inst_3 : ∀ (i : ι), CompleteSpace.{u_3} (E i)
inst_4 : ∀ (i : ι), BorelSpace.{u_3} (E i)
inst_5 : ∀ (i : ι), SecondCountableTopology.{u_3} (E i)
inst_6 : (i : ι) → NormedSpace ℝ (E i)
X : (i : ι) → Ω → E i
hX : HasGaussianLaw (E := (i : ι) → E i) (fun (ω : Ω) (i : ι) ↦ X i ω) P
h :
  ∀ (i j : ι),
    i ≠ j →
      ∀ (L₁ : StrongDual ℝ (E i)) (L₂ : StrongDual ℝ (E j)),
        cov[fun (x : Ω) ↦ (L₁ : (a : E i) → ℝ) (X i x), fun (x : Ω) ↦ (L₂ : (a : E j) → ℝ) (X j x); P] = 0
this_1 : IsProbabilityMeasure P
this : Fintype ι := Fintype.ofFinite ι
⊢ iIndepFun (β := E) X P
have rw : ∀ (L : StrongDual ℝ ((x : ι) → E x)),
  charFunDual (mE := MeasurableSpace.pi (X := E)) (Measure.map (fun (ω : Ω) (x : ι) ↦ X x ω) P) L =
    ∏ i : ι,
      charFunDual.{u_3} (E := E i) (mE := inst_2 i) (Measure.map (fun (x : Ω) ↦ X i x) P)
        (comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i))
Tactic: rw [iIndepFun_iff_charFunDual_pi fun i ↦ hX.aemeasurable.eval i]
-- Lean 4 proof step from Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence (hypothesis)
Goal:
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
inst : Finite ι
E : ι → Type u_3
inst_1 : (i : ι) → NormedAddCommGroup (E i)
inst_2 : (i : ι) → MeasurableSpace (E i)
inst_3 : ∀ (i : ι), CompleteSpace.{u_3} (E i)
inst_4 : ∀ (i : ι), BorelSpace.{u_3} (E i)
inst_5 : ∀ (i : ι), SecondCountableTopology.{u_3} (E i)
inst_6 : (i : ι) → NormedSpace ℝ (E i)
X : (i : ι) → Ω → E i
hX : HasGaussianLaw (E := (i : ι) → E i) (fun (ω : Ω) (i : ι) ↦ X i ω) P
h :
  ∀ (i j : ι),
    i ≠ j →
      ∀ (L₁ : StrongDual ℝ (E i)) (L₂ : StrongDual ℝ (E j)),
        cov[fun (x : Ω) ↦ (L₁ : (a : E i) → ℝ) (X i x), fun (x : Ω) ↦ (L₂ : (a : E j) → ℝ) (X j x); P] = 0
this_1 : IsProbabilityMeasure P
this : Fintype ι := Fintype.ofFinite ι
L : StrongDual ℝ ((x : ι) → E x)
⊢ charFunDual (mE := MeasurableSpace.pi (X := E)) (Measure.map (fun (ω : Ω) (x : ι) ↦ X x ω) P) L =
    ∏ i : ι,
      charFunDual.{u_3} (E := E i) (mE := inst_2 i) (Measure.map (fun (x : Ω) ↦ X i x) P)
        (comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i))
have this₁ : ∀ (ω : Ω),
  Eq (α := ℝ) ((L : (a : (x : ι) → E x) → ℝ) fun (x : ι) ↦ X x ω : ℝ)
    (∑ i : ι, (comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i) : (a : E i) → ℝ) (X i ω) : ℝ)
Tactic: have this ω : L (X · ω) = ∑ i, (L ∘L (single ℝ E i)) (X i ω) := by
    simp [← map_sum, LinearMap.sum_single_apply]
-- Lean 4 proof step from Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence (goal)
Goal:
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
inst : Finite ι
E : ι → Type u_3
inst_1 : (i : ι) → NormedAddCommGroup (E i)
inst_2 : (i : ι) → MeasurableSpace (E i)
inst_3 : ∀ (i : ι), CompleteSpace.{u_3} (E i)
inst_4 : ∀ (i : ι), BorelSpace.{u_3} (E i)
inst_5 : ∀ (i : ι), SecondCountableTopology.{u_3} (E i)
inst_6 : (i : ι) → NormedSpace ℝ (E i)
X : (i : ι) → Ω → E i
hX : HasGaussianLaw (E := (i : ι) → E i) (fun (ω : Ω) (i : ι) ↦ X i ω) P
h :
  ∀ (i j : ι),
    i ≠ j →
      ∀ (L₁ : StrongDual ℝ (E i)) (L₂ : StrongDual ℝ (E j)),
        cov[fun (x : Ω) ↦ (L₁ : (a : E i) → ℝ) (X i x), fun (x : Ω) ↦ (L₂ : (a : E j) → ℝ) (X j x); P] = 0
this_1 : IsProbabilityMeasure P
this_2 : Fintype ι := Fintype.ofFinite ι
L : StrongDual ℝ ((x : ι) → E x)
this :
  ∀ (ω : Ω),
    Eq (α := ℝ) ((L : (a : (x : ι) → E x) → ℝ) fun (x : ι) ↦ X x ω : ℝ)
      (∑ i : ι, (comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i) : (a : E i) → ℝ) (X i ω) : ℝ)
⊢ charFunDual (mE := MeasurableSpace.pi (X := E)) (Measure.map (fun (ω : Ω) (x : ι) ↦ X x ω) P) L =
    ∏ i : ι,
      charFunDual.{u_3} (E := E i) (mE := inst_2 i) (Measure.map (fun (x : Ω) ↦ X i x) P)
        (comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i))
have simp_rw : cexp
    ((↑(∫ (ω : Ω), ∑ i : ι, (comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i) : (a : E i) → ℝ) (X i ω) ∂P) : ℂ) * I -
      (↑Var[fun (ω : Ω) ↦ ∑ i : ι, (comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i) : (a : E i) → ℝ) (X i ω); P] : ℂ) / 2) =
  cexp
    ((∑ i : ι, (↑(∫ (ω : Ω), (comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i) : (a : E i) → ℝ) (X i ω) ∂P) : ℂ)) * I -
      ∑ x : ι, (↑Var[fun (ω : Ω) ↦ (comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E x) : (a : E x) → ℝ) (X x ω); P] : ℂ) / 2)
Tactic: simp_rw [hX.charFunDual_map_eq_fun, fun i ↦ (hX.eval i).charFunDual_map_eq_fun, ← Complex.exp_sum,
    sum_sub_distrib, ← sum_mul, this]
-- Lean 4 proof step from Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence (goal)
Goal:
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
inst : Finite ι
E : ι → Type u_3
inst_1 : (i : ι) → NormedAddCommGroup (E i)
inst_2 : (i : ι) → MeasurableSpace (E i)
inst_3 : ∀ (i : ι), CompleteSpace.{u_3} (E i)
inst_4 : ∀ (i : ι), BorelSpace.{u_3} (E i)
inst_5 : ∀ (i : ι), SecondCountableTopology.{u_3} (E i)
inst_6 : (i : ι) → NormedSpace ℝ (E i)
X : (i : ι) → Ω → E i
hX : HasGaussianLaw (E := (i : ι) → E i) (fun (ω : Ω) (i : ι) ↦ X i ω) P
h :
  ∀ (i j : ι),
    i ≠ j →
      ∀ (L₁ : StrongDual ℝ (E i)) (L₂ : StrongDual ℝ (E j)),
        cov[fun (x : Ω) ↦ (L₁ : (a : E i) → ℝ) (X i x), fun (x : Ω) ↦ (L₂ : (a : E j) → ℝ) (X j x); P] = 0
this_1 : IsProbabilityMeasure P
this_2 : Fintype ι := Fintype.ofFinite ι
L : StrongDual ℝ ((x : ι) → E x)
this :
  ∀ (ω : Ω),
    Eq (α := ℝ) ((L : (a : (x : ι) → E x) → ℝ) fun (x : ι) ↦ X x ω : ℝ)
      (∑ i : ι, (comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i) : (a : E i) → ℝ) (X i ω) : ℝ)
⊢ cexp
      ((↑(∫ (ω : Ω), ∑ i : ι, (comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i) : (a : E i) → ℝ) (X i ω) ∂P) : ℂ) * I -
        (↑Var[fun (ω : Ω) ↦ ∑ i : ι, (comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i) : (a : E i) → ℝ) (X i ω); P] : ℂ) /
          2) =
    cexp
      ((∑ i : ι, (↑(∫ (ω : Ω), (comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i) : (a : E i) → ℝ) (X i ω) ∂P) : ℂ)) * I -
        ∑ x : ι, (↑Var[fun (ω : Ω) ↦ (comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E x) : (a : E x) → ℝ) (X x ω); P] : ℂ) / 2)
have e_z : (↑(∫ (ω : Ω), ∑ i : ι, (comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i) : (a : E i) → ℝ) (X i ω) ∂P) : ℂ) =
  ∑ i : ι, (↑(∫ (ω : Ω), (comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i) : (a : E i) → ℝ) (X i ω) ∂P) : ℂ)
Tactic: congr
-- Lean 4 proof step from Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence (goal)
Goal:
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
inst : Finite ι
E : ι → Type u_3
inst_1 : (i : ι) → NormedAddCommGroup (E i)
inst_2 : (i : ι) → MeasurableSpace (E i)
inst_3 : ∀ (i : ι), CompleteSpace.{u_3} (E i)
inst_4 : ∀ (i : ι), BorelSpace.{u_3} (E i)
inst_5 : ∀ (i : ι), SecondCountableTopology.{u_3} (E i)
inst_6 : (i : ι) → NormedSpace ℝ (E i)
X : (i : ι) → Ω → E i
hX : HasGaussianLaw (E := (i : ι) → E i) (fun (ω : Ω) (i : ι) ↦ X i ω) P
h :
  ∀ (i j : ι),
    i ≠ j →
      ∀ (L₁ : StrongDual ℝ (E i)) (L₂ : StrongDual ℝ (E j)),
        cov[fun (x : Ω) ↦ (L₁ : (a : E i) → ℝ) (X i x), fun (x : Ω) ↦ (L₂ : (a : E j) → ℝ) (X j x); P] = 0
this_1 : IsProbabilityMeasure P
this_2 : Fintype ι := Fintype.ofFinite ι
L : StrongDual ℝ ((x : ι) → E x)
this :
  ∀ (ω : Ω),
    Eq (α := ℝ) ((L : (a : (x : ι) → E x) → ℝ) fun (x : ι) ↦ X x ω : ℝ)
      (∑ i : ι, (comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i) : (a : E i) → ℝ) (X i ω) : ℝ)
e_z :
  (↑(∫ (ω : Ω), ∑ i : ι, (comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i) : (a : E i) → ℝ) (X i ω) ∂P) : ℂ) =
    ∑ i : ι, (↑(∫ (ω : Ω), (comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i) : (a : E i) → ℝ) (X i ω) ∂P) : ℂ)
⊢ cexp
      ((↑(∫ (ω : Ω), ∑ i : ι, (comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i) : (a : E i) → ℝ) (X i ω) ∂P) : ℂ) * I -
        (↑Var[fun (ω : Ω) ↦ ∑ i : ι, (comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i) : (a : E i) → ℝ) (X i ω); P] : ℂ) /
          2) =
    cexp
      ((∑ i : ι, (↑(∫ (ω : Ω), (comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i) : (a : E i) → ℝ) (X i ω) ∂P) : ℂ)) * I -
        ∑ x : ι, (↑Var[fun (ω : Ω) ↦ (comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E x) : (a : E x) → ℝ) (X x ω); P] : ℂ) / 2)
have e_z₁ : (↑Var[fun (ω : Ω) ↦ ∑ i : ι, (comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i) : (a : E i) → ℝ) (X i ω); P] : ℂ) / 2 =
  ∑ x : ι, (↑Var[fun (ω : Ω) ↦ (comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E x) : (a : E x) → ℝ) (X x ω); P] : ℂ) / 2
Tactic: congr
-- Lean 4 proof step from Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence (goal)
Goal:
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
inst : Finite ι
E : ι → Type u_3
inst_1 : (i : ι) → NormedAddCommGroup (E i)
inst_2 : (i : ι) → MeasurableSpace (E i)
inst_3 : ∀ (i : ι), CompleteSpace.{u_3} (E i)
inst_4 : ∀ (i : ι), BorelSpace.{u_3} (E i)
inst_5 : ∀ (i : ι), SecondCountableTopology.{u_3} (E i)
inst_6 : (i : ι) → NormedSpace ℝ (E i)
X : (i : ι) → Ω → E i
hX : HasGaussianLaw (E := (i : ι) → E i) (fun (ω : Ω) (i : ι) ↦ X i ω) P
h :
  ∀ (i j : ι),
    i ≠ j →
      ∀ (L₁ : StrongDual ℝ (E i)) (L₂ : StrongDual ℝ (E j)),
        cov[fun (x : Ω) ↦ (L₁ : (a : E i) → ℝ) (X i x), fun (x : Ω) ↦ (L₂ : (a : E j) → ℝ) (X j x); P] = 0
this_1 : IsProbabilityMeasure P
this_2 : Fintype ι := Fintype.ofFinite ι
L : StrongDual ℝ ((x : ι) → E x)
this :
  ∀ (ω : Ω),
    Eq (α := ℝ) ((L : (a : (x : ι) → E x) → ℝ) fun (x : ι) ↦ X x ω : ℝ)
      (∑ i : ι, (comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i) : (a : E i) → ℝ) (X i ω) : ℝ)
⊢ (↑(∫ (ω : Ω), ∑ i : ι, (comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i) : (a : E i) → ℝ) (X i ω) ∂P) : ℂ) =
    ∑ i : ι, (↑(∫ (ω : Ω), (comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i) : (a : E i) → ℝ) (X i ω) ∂P) : ℂ)
have e_z : (↑(∫ (ω : Ω), ∑ i : ι, (comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i) : (a : E i) → ℝ) (X i ω) ∂P) : ℂ) =
  (↑(∑ i : ι, ∫ (ω : Ω), (comp (σ₁₂ := RingHom.id ℝ) L (single ℝ E i) : (a : E i) → ℝ) (X i ω) ∂P) : ℂ)
Tactic: simp_rw [← Complex.ofReal_sum]