Theorem: GaussianFourier.integral_cexp_neg_mul_sq_norm_add_of_euclideanSpace
File: Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean

tactic: have := (EuclideanSpace.volume_preserving_measurableEquiv ι).symm
state_before: b : ℂ
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
⊢ ∫ (v : EuclideanSpace ℝ ι), cexp (-b * ↑‖v‖ ^ 2 + c * ↑⟪w, v⟫_ℝ) =
    (↑π / b) ^ (↑(Fintype.card ι) / 2) * cexp (c ^ 2 * ↑‖w‖ ^ 2 / (4 * b))
state_after: b : ℂ
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι).symm) volume volume
⊢ ∫ (v : EuclideanSpace ℝ ι), cexp (-b * ↑‖v‖ ^ 2 + c * ↑⟪w, v⟫_ℝ) =
    (↑π / b) ^ (↑(Fintype.card ι) / 2) * cexp (c ^ 2 * ↑‖w‖ ^ 2 / (4 * b))

tactic: rw [← this.integral_comp (MeasurableEquiv.measurableEmbedding _)]
state_before: b : ℂ
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι).symm) volume volume
⊢ ∫ (v : EuclideanSpace ℝ ι), cexp (-b * ↑‖v‖ ^ 2 + c * ↑⟪w, v⟫_ℝ) =
    (↑π / b) ^ (↑(Fintype.card ι) / 2) * cexp (c ^ 2 * ↑‖w‖ ^ 2 / (4 * b))
state_after: b : ℂ
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι).symm) volume volume
⊢ ∫ (x : ι → ℝ),
      cexp
        (-b * ↑‖(EuclideanSpace.measurableEquiv ι).symm x‖ ^ 2 +
          c * ↑⟪w, (EuclideanSpace.measurableEquiv ι).symm x⟫_ℝ) ∂volume =
    (↑π / b) ^ (↑(Fintype.card ι) / 2) * cexp (c ^ 2 * ↑‖w‖ ^ 2 / (4 * b))

tactic: simp only [neg_mul, Function.comp_def]
state_before: b : ℂ
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι).symm) volume volume
⊢ ∫ (x : ι → ℝ),
      cexp
        (-b * ↑‖(EuclideanSpace.measurableEquiv ι).symm x‖ ^ 2 +
          c * ↑⟪w, (EuclideanSpace.measurableEquiv ι).symm x⟫_ℝ) ∂volume =
    (↑π / b) ^ (↑(Fintype.card ι) / 2) * cexp (c ^ 2 * ↑‖w‖ ^ 2 / (4 * b))
state_after: b : ℂ
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι).symm) volume volume
⊢ ∫ (x : ι → ℝ),
      cexp
        (-(b * ↑‖(EuclideanSpace.measurableEquiv ι).symm x‖ ^ 2) +
          c * ↑⟪w, (EuclideanSpace.measurableEquiv ι).symm x⟫_ℝ) ∂volume =
    (↑π / b) ^ (↑(Fintype.card ι) / 2) * cexp (c ^ 2 * ↑‖w‖ ^ 2 / (4 * b))

tactic: convert integral_cexp_neg_mul_sum_add hb (fun i ↦ c * w i) using 5 with _x y
state_before: b : ℂ
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι).symm) volume volume
⊢ ∫ (x : ι → ℝ),
      cexp
        (-(b * ↑‖(EuclideanSpace.measurableEquiv ι).symm x‖ ^ 2) +
          c * ↑⟪w, (EuclideanSpace.measurableEquiv ι).symm x⟫_ℝ) ∂volume =
    (↑π / b) ^ (↑(Fintype.card ι) / 2) * cexp (c ^ 2 * ↑‖w‖ ^ 2 / (4 * b))
state_after: case h.e'_2.h.e'_7.h.h.e'_1.h.e'_5
b : ℂ
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι).symm) volume volume
y : ι → ℝ
⊢ -(b * ↑‖(EuclideanSpace.measurableEquiv ι).symm y‖ ^ 2) = -b * ∑ i : ι, ↑(y i) ^ 2

case h.e'_2.h.e'_7.h.h.e'_1.h.e'_6
b : ℂ
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι).symm) volume volume
y : ι → ℝ
⊢ c * ↑⟪w, (EuclideanSpace.measurableEquiv ι).symm y⟫_ℝ = ∑ i : ι, c * ↑(w i) * ↑(y i)

case h.e'_3.h.e'_6.h.e'_1.h.e'_5
b : ℂ
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι).symm) volume volume
⊢ c ^ 2 * ↑‖w‖ ^ 2 = ∑ i : ι, (c * ↑(w i)) ^ 2

tactic: simp only [EuclideanSpace.measurableEquiv, MeasurableEquiv.symm_mk, MeasurableEquiv.coe_mk,
  EuclideanSpace.norm_eq, WithLp.equiv_symm_pi_apply, Real.norm_eq_abs, sq_abs, neg_mul,
  neg_inj, mul_eq_mul_left_iff]
state_before: case h.e'_2.h.e'_7.h.h.e'_1.h.e'_5
b : ℂ
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι).symm) volume volume
y : ι → ℝ
⊢ -(b * ↑‖(EuclideanSpace.measurableEquiv ι).symm y‖ ^ 2) = -b * ∑ i : ι, ↑(y i) ^ 2
state_after: case h.e'_2.h.e'_7.h.h.e'_1.h.e'_5
b : ℂ
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι).symm) volume volume
y : ι → ℝ
⊢ ↑√(∑ x : ι, y x ^ 2) ^ 2 = ∑ i : ι, ↑(y i) ^ 2 ∨ b = 0

tactic: norm_cast
state_before: case h.e'_2.h.e'_7.h.h.e'_1.h.e'_5
b : ℂ
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι).symm) volume volume
y : ι → ℝ
⊢ ↑√(∑ x : ι, y x ^ 2) ^ 2 = ∑ i : ι, ↑(y i) ^ 2 ∨ b = 0
state_after: case h.e'_2.h.e'_7.h.h.e'_1.h.e'_5
b : ℂ
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι).symm) volume volume
y : ι → ℝ
⊢ √(∑ x : ι, y x ^ 2) ^ 2 = ∑ i : ι, y i ^ 2 ∨ b = 0

tactic: left
state_before: case h.e'_2.h.e'_7.h.h.e'_1.h.e'_5
b : ℂ
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι).symm) volume volume
y : ι → ℝ
⊢ √(∑ x : ι, y x ^ 2) ^ 2 = ∑ i : ι, y i ^ 2 ∨ b = 0
state_after: case h.e'_2.h.e'_7.h.h.e'_1.h.e'_5.h
b : ℂ
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι).symm) volume volume
y : ι → ℝ
⊢ √(∑ x : ι, y x ^ 2) ^ 2 = ∑ i : ι, y i ^ 2

tactic: rw [sq_sqrt]
state_before: case h.e'_2.h.e'_7.h.h.e'_1.h.e'_5.h
b : ℂ
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι).symm) volume volume
y : ι → ℝ
⊢ √(∑ x : ι, y x ^ 2) ^ 2 = ∑ i : ι, y i ^ 2
state_after: case h.e'_2.h.e'_7.h.h.e'_1.h.e'_5.h
b : ℂ
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι).symm) volume volume
y : ι → ℝ
⊢ 0 ≤ ∑ x : ι, y x ^ 2

tactic: exact Finset.sum_nonneg (fun i _hi ↦ by positivity)
state_before: case h.e'_2.h.e'_7.h.h.e'_1.h.e'_5.h
b : ℂ
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι).symm) volume volume
y : ι → ℝ
⊢ 0 ≤ ∑ x : ι, y x ^ 2
state_after: no goals

tactic: positivity
state_before: b : ℂ
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι).symm) volume volume
y : ι → ℝ
i : ι
_hi : i ∈ Finset.univ
⊢ 0 ≤ y i ^ 2
state_after: no goals

tactic: simp [PiLp.inner_apply, EuclideanSpace.measurableEquiv, Finset.mul_sum, mul_assoc]
state_before: case h.e'_2.h.e'_7.h.h.e'_1.h.e'_6
b : ℂ
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι).symm) volume volume
y : ι → ℝ
⊢ c * ↑⟪w, (EuclideanSpace.measurableEquiv ι).symm y⟫_ℝ = ∑ i : ι, c * ↑(w i) * ↑(y i)
state_after: no goals

tactic: simp only [EuclideanSpace.norm_eq, Real.norm_eq_abs, sq_abs, mul_pow, ← Finset.mul_sum]
state_before: case h.e'_3.h.e'_6.h.e'_1.h.e'_5
b : ℂ
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι).symm) volume volume
⊢ c ^ 2 * ↑‖w‖ ^ 2 = ∑ i : ι, (c * ↑(w i)) ^ 2
state_after: case h.e'_3.h.e'_6.h.e'_1.h.e'_5
b : ℂ
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι).symm) volume volume
⊢ c ^ 2 * ↑√(∑ x : ι, w x ^ 2) ^ 2 = c ^ 2 * ∑ i : ι, ↑(w i) ^ 2

tactic: congr
state_before: case h.e'_3.h.e'_6.h.e'_1.h.e'_5
b : ℂ
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι).symm) volume volume
⊢ c ^ 2 * ↑√(∑ x : ι, w x ^ 2) ^ 2 = c ^ 2 * ∑ i : ι, ↑(w i) ^ 2
state_after: case h.e'_3.h.e'_6.h.e'_1.h.e'_5.e_a
b : ℂ
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι).symm) volume volume
⊢ ↑√(∑ x : ι, w x ^ 2) ^ 2 = ∑ i : ι, ↑(w i) ^ 2

tactic: norm_cast
state_before: case h.e'_3.h.e'_6.h.e'_1.h.e'_5.e_a
b : ℂ
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι).symm) volume volume
⊢ ↑√(∑ x : ι, w x ^ 2) ^ 2 = ∑ i : ι, ↑(w i) ^ 2
state_after: case h.e'_3.h.e'_6.h.e'_1.h.e'_5.e_a
b : ℂ
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι).symm) volume volume
⊢ √(∑ x : ι, w x ^ 2) ^ 2 = ∑ i : ι, w i ^ 2

tactic: rw [sq_sqrt]
state_before: case h.e'_3.h.e'_6.h.e'_1.h.e'_5.e_a
b : ℂ
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι).symm) volume volume
⊢ √(∑ x : ι, w x ^ 2) ^ 2 = ∑ i : ι, w i ^ 2
state_after: case h.e'_3.h.e'_6.h.e'_1.h.e'_5.e_a
b : ℂ
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι).symm) volume volume
⊢ 0 ≤ ∑ x : ι, w x ^ 2

tactic: exact Finset.sum_nonneg (fun i _hi ↦ by positivity)
state_before: case h.e'_3.h.e'_6.h.e'_1.h.e'_5.e_a
b : ℂ
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι).symm) volume volume
⊢ 0 ≤ ∑ x : ι, w x ^ 2
state_after: no goals

tactic: positivity
state_before: b : ℂ
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι).symm) volume volume
i : ι
_hi : i ∈ Finset.univ
⊢ 0 ≤ w i ^ 2
state_after: no goals