Imports
The norm on space
i. Overview
The main content of this file is defining Space.normPowerSeries, a power series which is
differentiable everywhere, and which tends to the norm in the limit as n → ∞.
We use properties of this power series to prove various results about distributions involving norms.
ii. Key results
normPowerSeries : A power series which is differentiable everywhere, and in the limit
as n → ∞ tends to ‖x‖.
normPowerSeries_differentiable : The power series is differentiable everywhere.
normPowerSeries_tendsto : The power series tends to the norm in the limit as n → ∞.
distGrad_distOfFunction_norm_zpow : The gradient of the distribution defined by a power of the
norm.
distGrad_distOfFunction_log_norm : The gradient of the distribution defined by the logarithm
of the norm.
distDiv_norm_zpow_smul_repr_self_eq_smul : The divergence of the distribution defined by
x ↦ ‖x‖ ^ q • x.
distLaplacian_distOfFunction_norm_zpow : The Laplacian of the distribution defined by a power
of the norm.
distDiv_inv_pow_eq_dim : The divergence of x ↦ ‖x‖ ^ (-d) • x equals d * volume (ball 0 1)
times the Dirac delta at the origin.
distLaplacian_fundamentalSolution_norm_zpow : The Laplacian of the power-form fundamental
solution ‖x‖ ^ (2 - d), in every dimension (trivial at d = 0, 2).
distLaplacian_fundamentalSolution_log_norm : The Laplacian of the two-dimensional logarithmic
fundamental solution Real.log ‖x‖.
iii. Table of contents
A. The norm as a power series
A.1. Differentiability of the norm power series
A.2. The limit of the norm power series
A.3. The derivative of the norm power series
A.4. Limits of the derivative of the power series
A.5. The power series is AEStronglyMeasurable
A.6. Bounds on the norm power series
A.7. The IsDistBounded property of the norm power series
A.8. Differentiability of functions
A.9. Derivatives of functions
A.10. Gradients of distributions based on powers
A.11. Gradients of distributions based on logs
B. Distributions involving norms
B.1. The gradient of distributions based on powers
B.2. The gradient of distributions based on logs
B.3. Divergence of radial norm-power distributions
B.4. The Laplacian of distributions based on powers
B.5. Divergence equal dirac delta
B.6. The Laplacian of the fundamental solution
iv. References
@[ expose ] public section
A. The norm as a power series
A power series which is differentiable everywhere, and in the limit
as n → ∞ tends to ‖x‖.
def normPowerSeries { d } : ℕ → Space d → ℝ := fun n x =>
√ ( ‖ x ‖ ^ 2 + 1 / ( n + 1 ) )
lemma normPowerSeries_eq ( n : ℕ ) :
normPowerSeries ( d := d ) n = fun x => √ ( ‖ x ‖ ^ 2 + 1 / ( n + 1 ) ) := rfl lemma normPowerSeries_eq_rpow { d } ( n : ℕ ) :
normPowerSeries ( d := d ) n = fun x => ( ( ‖ x ‖ ^ 2 + 1 / ( n + 1 ) ) ) ^ ( 1 / 2 : ℝ ) :=
funext fun _ => Real.sqrt_eq_rpow _
A.1. Differentiability of the norm power series
A.2. The limit of the norm power series
A.3. The derivative of the norm power series
lemma deriv_normPowerSeries { d } ( n : ℕ ) ( x : Space d ) ( i : Fin d ) :
∂[ i ] ( normPowerSeries n ) x = x i * ( normPowerSeries n x ) ⁻¹ := by d : ℕ n : ℕ x : Space d i : Fin d ⊢ deriv i ( normPowerSeries n ) x = x . val i * ( normPowerSeries n x ) ⁻¹
rw [ deriv_eq_fderiv_basis , d : ℕ n : ℕ x : Space d i : Fin d ⊢ ( fderiv ℝ ( normPowerSeries n ) x ) ( basis i ) = x . val i * ( normPowerSeries n x ) ⁻¹ d : ℕ n : ℕ x : Space d i : Fin d ⊢ ( ( 1 / ( 2 * √ ( ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) ) ) • fderiv ℝ (fun x => ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) x ) ( basis i ) =
x . val i * ( (fun x => √ ( ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) ) x ) ⁻¹ hf d : ℕ n : ℕ x : Space d i : Fin d ⊢ DifferentiableAt ℝ (fun x => ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) x hx d : ℕ n : ℕ x : Space d i : Fin d ⊢ ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ≠ 0 normPowerSeries_eq , d : ℕ n : ℕ x : Space d i : Fin d ⊢ ( fderiv ℝ (fun x => √ ( ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) ) x ) ( basis i ) = x . val i * ( (fun x => √ ( ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) ) x ) ⁻¹ d : ℕ n : ℕ x : Space d i : Fin d ⊢ ( ( 1 / ( 2 * √ ( ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) ) ) • fderiv ℝ (fun x => ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) x ) ( basis i ) =
x . val i * ( (fun x => √ ( ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) ) x ) ⁻¹ hf d : ℕ n : ℕ x : Space d i : Fin d ⊢ DifferentiableAt ℝ (fun x => ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) x hx d : ℕ n : ℕ x : Space d i : Fin d ⊢ ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ≠ 0 fderiv_sqrt d : ℕ n : ℕ x : Space d i : Fin d ⊢ ( ( 1 / ( 2 * √ ( ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) ) ) • fderiv ℝ (fun x => ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) x ) ( basis i ) =
x . val i * ( (fun x => √ ( ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) ) x ) ⁻¹ hf d : ℕ n : ℕ x : Space d i : Fin d ⊢ DifferentiableAt ℝ (fun x => ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) x hx d : ℕ n : ℕ x : Space d i : Fin d ⊢ ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ≠ 0 d : ℕ n : ℕ x : Space d i : Fin d ⊢ ( ( 1 / ( 2 * √ ( ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) ) ) • fderiv ℝ (fun x => ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) x ) ( basis i ) =
x . val i * ( (fun x => √ ( ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) ) x ) ⁻¹ hf d : ℕ n : ℕ x : Space d i : Fin d ⊢ DifferentiableAt ℝ (fun x => ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) x hx d : ℕ n : ℕ x : Space d i : Fin d ⊢ ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ≠ 0 ] d : ℕ n : ℕ x : Space d i : Fin d ⊢ ( ( 1 / ( 2 * √ ( ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) ) ) • fderiv ℝ (fun x => ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) x ) ( basis i ) =
x . val i * ( (fun x => √ ( ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) ) x ) ⁻¹ hf d : ℕ n : ℕ x : Space d i : Fin d ⊢ DifferentiableAt ℝ (fun x => ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) x hx d : ℕ n : ℕ x : Space d i : Fin d ⊢ ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ≠ 0
simp only [ one_div , mul_inv_rev , fderiv_add_const , FunLike.coe_smul , Pi.smul_apply ,
smul_eq_mul ] d : ℕ n : ℕ x : Space d i : Fin d ⊢ ( √ ( ‖ x ‖ ^ 2 + ( ↑ n + 1 ) ⁻¹ ) ) ⁻¹ * 2 ⁻¹ * ( fderiv ℝ (fun y => ‖ y ‖ ^ 2 ) x ) ( basis i ) = x . val i * ( √ ( ‖ x ‖ ^ 2 + ( ↑ n + 1 ) ⁻¹ ) ) ⁻¹ hf d : ℕ n : ℕ x : Space d i : Fin d ⊢ DifferentiableAt ℝ (fun x => ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) x hx d : ℕ n : ℕ x : Space d i : Fin d ⊢ ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ≠ 0
rw [ ← deriv_eq_fderiv_basis , d : ℕ n : ℕ x : Space d i : Fin d ⊢ ( √ ( ‖ x ‖ ^ 2 + ( ↑ n + 1 ) ⁻¹ ) ) ⁻¹ * 2 ⁻¹ * deriv i (fun y => ‖ y ‖ ^ 2 ) x = x . val i * ( √ ( ‖ x ‖ ^ 2 + ( ↑ n + 1 ) ⁻¹ ) ) ⁻¹ hf d : ℕ n : ℕ x : Space d i : Fin d ⊢ DifferentiableAt ℝ (fun x => ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) x hx d : ℕ n : ℕ x : Space d i : Fin d ⊢ ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ≠ 0 d : ℕ n : ℕ x : Space d i : Fin d ⊢ ( √ ( ‖ x ‖ ^ 2 + ( ↑ n + 1 ) ⁻¹ ) ) ⁻¹ * 2 ⁻¹ * ( 2 * x . val i ) = x . val i * ( √ ( ‖ x ‖ ^ 2 + ( ↑ n + 1 ) ⁻¹ ) ) ⁻¹ hf d : ℕ n : ℕ x : Space d i : Fin d ⊢ DifferentiableAt ℝ (fun x => ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) x hx d : ℕ n : ℕ x : Space d i : Fin d ⊢ ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ≠ 0 deriv_norm_sq d : ℕ n : ℕ x : Space d i : Fin d ⊢ ( √ ( ‖ x ‖ ^ 2 + ( ↑ n + 1 ) ⁻¹ ) ) ⁻¹ * 2 ⁻¹ * ( 2 * x . val i ) = x . val i * ( √ ( ‖ x ‖ ^ 2 + ( ↑ n + 1 ) ⁻¹ ) ) ⁻¹ hf d : ℕ n : ℕ x : Space d i : Fin d ⊢ DifferentiableAt ℝ (fun x => ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) x hx d : ℕ n : ℕ x : Space d i : Fin d ⊢ ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ≠ 0 d : ℕ n : ℕ x : Space d i : Fin d ⊢ ( √ ( ‖ x ‖ ^ 2 + ( ↑ n + 1 ) ⁻¹ ) ) ⁻¹ * 2 ⁻¹ * ( 2 * x . val i ) = x . val i * ( √ ( ‖ x ‖ ^ 2 + ( ↑ n + 1 ) ⁻¹ ) ) ⁻¹ hf d : ℕ n : ℕ x : Space d i : Fin d ⊢ DifferentiableAt ℝ (fun x => ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) x hx d : ℕ n : ℕ x : Space d i : Fin d ⊢ ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ≠ 0 ] d : ℕ n : ℕ x : Space d i : Fin d ⊢ ( √ ( ‖ x ‖ ^ 2 + ( ↑ n + 1 ) ⁻¹ ) ) ⁻¹ * 2 ⁻¹ * ( 2 * x . val i ) = x . val i * ( √ ( ‖ x ‖ ^ 2 + ( ↑ n + 1 ) ⁻¹ ) ) ⁻¹ hf d : ℕ n : ℕ x : Space d i : Fin d ⊢ DifferentiableAt ℝ (fun x => ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) x hx d : ℕ n : ℕ x : Space d i : Fin d ⊢ ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ≠ 0
ring hf d : ℕ n : ℕ x : Space d i : Fin d ⊢ DifferentiableAt ℝ (fun x => ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) x hx d : ℕ n : ℕ x : Space d i : Fin d ⊢ ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ≠ 0
· hf d : ℕ n : ℕ x : Space d i : Fin d ⊢ DifferentiableAt ℝ (fun x => ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) x exact ( ( differentiable_id . norm_sq ℝ ) . add_const _ ) . differentiableAt All goals completed! 🐙
· hx d : ℕ n : ℕ x : Space d i : Fin d ⊢ ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ≠ 0 positivity All goals completed! 🐙
lemma fderiv_normPowerSeries { d } ( n : ℕ ) ( x y : Space d ) :
fderiv ℝ ( fun ( x : Space d ) => normPowerSeries n x ) x y =
⟪ y , x ⟫_ ℝ * ( normPowerSeries n x ) ⁻¹ := by d : ℕ n : ℕ x : Space d y : Space d ⊢ ( fderiv ℝ (fun x => normPowerSeries n x ) x ) y = ⟪ y , x ⟫_ ℝ * ( normPowerSeries n x ) ⁻¹
rw [ fderiv_eq_sum_deriv , d : ℕ n : ℕ x : Space d y : Space d ⊢ ∑ i , y . val i • deriv i (fun x => normPowerSeries n x ) x = ⟪ y , x ⟫_ ℝ * ( normPowerSeries n x ) ⁻¹ d : ℕ n : ℕ x : Space d y : Space d ⊢ ∑ i , y . val i • deriv i (fun x => normPowerSeries n x ) x = ∑ i , y . val i * x . val i * ( normPowerSeries n x ) ⁻¹ inner_eq_sum , d : ℕ n : ℕ x : Space d y : Space d ⊢ ∑ i , y . val i • deriv i (fun x => normPowerSeries n x ) x = (∑ i , y . val i * x . val i ) * ( normPowerSeries n x ) ⁻¹ d : ℕ n : ℕ x : Space d y : Space d ⊢ ∑ i , y . val i • deriv i (fun x => normPowerSeries n x ) x = ∑ i , y . val i * x . val i * ( normPowerSeries n x ) ⁻¹ Finset.sum_mul d : ℕ n : ℕ x : Space d y : Space d ⊢ ∑ i , y . val i • deriv i (fun x => normPowerSeries n x ) x = ∑ i , y . val i * x . val i * ( normPowerSeries n x ) ⁻¹ d : ℕ n : ℕ x : Space d y : Space d ⊢ ∑ i , y . val i • deriv i (fun x => normPowerSeries n x ) x = ∑ i , y . val i * x . val i * ( normPowerSeries n x ) ⁻¹ ] d : ℕ n : ℕ x : Space d y : Space d ⊢ ∑ i , y . val i • deriv i (fun x => normPowerSeries n x ) x = ∑ i , y . val i * x . val i * ( normPowerSeries n x ) ⁻¹
exact Finset.sum_congr rfl fun i _ => by d : ℕ n : ℕ x : Space d y : Space d i : Fin d x✝ : i ∈ Finset.univ ⊢ y . val i • deriv i (fun x => normPowerSeries n x ) x = y . val i * x . val i * ( normPowerSeries n x ) ⁻¹ simp [ deriv_normPowerSeries , mul_assoc ] All goals completed! 🐙
A.4. Limits of the derivative of the power series
A.5. The power series is AEStronglyMeasurable
A.6. Bounds on the norm power series
A.7. The IsDistBounded property of the norm power series
@[ fun_prop ]
lemma IsDistBounded.normPowerSeries_zpow { d : ℕ } { n : ℕ } ( m : ℤ ) :
IsDistBounded ( d := d ) ( fun x => ( normPowerSeries n x ) ^ m ) := by d : ℕ n : ℕ m : ℤ ⊢ IsDistBounded fun x => normPowerSeries n x ^ m
match m with
| .ofNat m => d : ℕ n : ℕ m✝ : ℤ m : ℕ ⊢ IsDistBounded fun x => normPowerSeries n x ^ Int.ofNat m
simp only [ Int.ofNat_eq_natCast , zpow_natCast ] d : ℕ n : ℕ m✝ : ℤ m : ℕ ⊢ IsDistBounded fun x => normPowerSeries n x ^ m
apply IsDistBounded.mono ( f := fun ( x : Space d ) => ( ‖ x ‖ + 1 ) ^ m ) hf d : ℕ n : ℕ m✝ : ℤ m : ℕ ⊢ IsDistBounded fun x => ( ‖ x ‖ + 1 ) ^ m hae d : ℕ n : ℕ m✝ : ℤ m : ℕ ⊢ AEStronglyMeasurable (fun x => normPowerSeries n x ^ m ) volume hfg d : ℕ n : ℕ m✝ : ℤ m : ℕ ⊢ ∀ ( x : Space d ), ‖ normPowerSeries n x ^ m ‖ ≤ ‖ ( ‖ x ‖ + 1 ) ^ m ‖
· hf d : ℕ n : ℕ m✝ : ℤ m : ℕ ⊢ IsDistBounded fun x => ( ‖ x ‖ + 1 ) ^ m fun_prop All goals completed! 🐙
· hae d : ℕ n : ℕ m✝ : ℤ m : ℕ ⊢ AEStronglyMeasurable (fun x => normPowerSeries n x ^ m ) volume fun_prop All goals completed! 🐙
intro x hfg d : ℕ n : ℕ m✝ : ℤ m : ℕ x : Space d ⊢ ‖ normPowerSeries n x ^ m ‖ ≤ ‖ ( ‖ x ‖ + 1 ) ^ m ‖
simp only [ norm_pow , Real.norm_eq_abs ] hfg d : ℕ n : ℕ m✝ : ℤ m : ℕ x : Space d ⊢ | normPowerSeries n x | ^ m ≤ | ‖ x ‖ + 1 | ^ m
refine pow_le_pow_left₀ ( by d : ℕ n : ℕ m✝ : ℤ m : ℕ x : Space d ⊢ 0 ≤ | normPowerSeries n x | positivity All goals completed! 🐙 ) ? _ m
rw [ abs_of_nonneg ( by d : ℕ n : ℕ m✝ : ℤ m : ℕ x : Space d ⊢ 0 ≤ normPowerSeries n x hfg d : ℕ n : ℕ m✝ : ℤ m : ℕ x : Space d ⊢ normPowerSeries n x ≤ ‖ x ‖ + 1 simp All goals completed! 🐙 hfg d : ℕ n : ℕ m✝ : ℤ m : ℕ x : Space d ⊢ normPowerSeries n x ≤ ‖ x ‖ + 1 ) , abs_of_nonneg ( by d : ℕ n : ℕ m✝ : ℤ m : ℕ x : Space d ⊢ 0 ≤ ‖ x ‖ + 1 hfg d : ℕ n : ℕ m✝ : ℤ m : ℕ x : Space d ⊢ normPowerSeries n x ≤ ‖ x ‖ + 1 positivity All goals completed! 🐙 hfg d : ℕ n : ℕ m✝ : ℤ m : ℕ x : Space d ⊢ normPowerSeries n x ≤ ‖ x ‖ + 1 ) ] hfg d : ℕ n : ℕ m✝ : ℤ m : ℕ x : Space d ⊢ normPowerSeries n x ≤ ‖ x ‖ + 1
exact normPowerSeries_le_norm_sq_add_one n x All goals completed! 🐙
| .negSucc m => d : ℕ n : ℕ m✝ : ℤ m : ℕ ⊢ IsDistBounded fun x => normPowerSeries n x ^ Int.negSucc m
simp only [ zpow_negSucc ] d : ℕ n : ℕ m✝ : ℤ m : ℕ ⊢ IsDistBounded fun x => ( normPowerSeries n x ^ ( m + 1 ) ) ⁻¹
apply IsDistBounded.mono ( f := fun ( x : Space d ) => ( ( √ ( 1 / ( n + 1 ) ) : ℝ ) ^ ( m + 1 ) ) ⁻¹ ) hf d : ℕ n : ℕ m✝ : ℤ m : ℕ ⊢ IsDistBounded fun x => ( √ ( 1 / ( ↑ n + 1 ) ) ^ ( m + 1 ) ) ⁻¹ hae d : ℕ n : ℕ m✝ : ℤ m : ℕ ⊢ AEStronglyMeasurable (fun x => ( normPowerSeries n x ^ ( m + 1 ) ) ⁻¹ ) volume hfg d : ℕ n : ℕ m✝ : ℤ m : ℕ ⊢ ∀ ( x : Space d ), ‖ ( normPowerSeries n x ^ ( m + 1 ) ) ⁻¹ ‖ ≤ ‖ ( √ ( 1 / ( ↑ n + 1 ) ) ^ ( m + 1 ) ) ⁻¹ ‖
· hf d : ℕ n : ℕ m✝ : ℤ m : ℕ ⊢ IsDistBounded fun x => ( √ ( 1 / ( ↑ n + 1 ) ) ^ ( m + 1 ) ) ⁻¹ fun_prop All goals completed! 🐙
· hae d : ℕ n : ℕ m✝ : ℤ m : ℕ ⊢ AEStronglyMeasurable (fun x => ( normPowerSeries n x ^ ( m + 1 ) ) ⁻¹ ) volume exact ( ( ( normPowerSeries_differentiable n ) . continuous . pow _ ) . inv₀
fun x => pow_ne_zero _ ( normPowerSeries_ne_zero n x ) ) . aestronglyMeasurable All goals completed! 🐙
· hfg d : ℕ n : ℕ m✝ : ℤ m : ℕ ⊢ ∀ ( x : Space d ), ‖ ( normPowerSeries n x ^ ( m + 1 ) ) ⁻¹ ‖ ≤ ‖ ( √ ( 1 / ( ↑ n + 1 ) ) ^ ( m + 1 ) ) ⁻¹ ‖ intro x hfg d : ℕ n : ℕ m✝ : ℤ m : ℕ x : Space d ⊢ ‖ ( normPowerSeries n x ^ ( m + 1 ) ) ⁻¹ ‖ ≤ ‖ ( √ ( 1 / ( ↑ n + 1 ) ) ^ ( m + 1 ) ) ⁻¹ ‖
simp only [ norm_inv , norm_pow , Real.norm_eq_abs , one_div ] hfg d : ℕ n : ℕ m✝ : ℤ m : ℕ x : Space d ⊢ ( | normPowerSeries n x | ^ ( m + 1 ) ) ⁻¹ ≤ ( | √ ( ↑ n + 1 ) ⁻¹ | ^ ( m + 1 ) ) ⁻¹
refine inv_anti₀ ( by d : ℕ n : ℕ m✝ : ℤ m : ℕ x : Space d ⊢ 0 < | √ ( ↑ n + 1 ) ⁻¹ | ^ ( m + 1 ) positivity All goals completed! 🐙 ) ( pow_le_pow_left₀ ( abs_nonneg _ ) ? _ _ )
rw [ abs_of_nonneg ( by d : ℕ n : ℕ m✝ : ℤ m : ℕ x : Space d ⊢ 0 ≤ √ ( ↑ n + 1 ) ⁻¹ hfg d : ℕ n : ℕ m✝ : ℤ m : ℕ x : Space d ⊢ √ ( ↑ n + 1 ) ⁻¹ ≤ (fun x => √ ( ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) ) x positivity All goals completed! 🐙 hfg d : ℕ n : ℕ m✝ : ℤ m : ℕ x : Space d ⊢ √ ( ↑ n + 1 ) ⁻¹ ≤ (fun x => √ ( ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) ) x ) , abs_of_nonneg ( by d : ℕ n : ℕ m✝ : ℤ m : ℕ x : Space d ⊢ 0 ≤ normPowerSeries n x hfg d : ℕ n : ℕ m✝ : ℤ m : ℕ x : Space d ⊢ √ ( ↑ n + 1 ) ⁻¹ ≤ (fun x => √ ( ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) ) x simp All goals completed! 🐙 hfg d : ℕ n : ℕ m✝ : ℤ m : ℕ x : Space d ⊢ √ ( ↑ n + 1 ) ⁻¹ ≤ (fun x => √ ( ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) ) x ) , normPowerSeries_eq hfg d : ℕ n : ℕ m✝ : ℤ m : ℕ x : Space d ⊢ √ ( ↑ n + 1 ) ⁻¹ ≤ (fun x => √ ( ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) ) x hfg d : ℕ n : ℕ m✝ : ℤ m : ℕ x : Space d ⊢ √ ( ↑ n + 1 ) ⁻¹ ≤ (fun x => √ ( ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) ) x ] hfg d : ℕ n : ℕ m✝ : ℤ m : ℕ x : Space d ⊢ √ ( ↑ n + 1 ) ⁻¹ ≤ (fun x => √ ( ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) ) ) x
exact Real.sqrt_le_sqrt ( by d : ℕ n : ℕ m✝ : ℤ m : ℕ x : Space d ⊢ ( ↑ n + 1 ) ⁻¹ ≤ ‖ x ‖ ^ 2 + 1 / ( ↑ n + 1 ) simp All goals completed! 🐙 ) @[ fun_prop ]
lemma IsDistBounded.normPowerSeries_single { d : ℕ } { n : ℕ } :
IsDistBounded ( d := d ) ( fun x => ( normPowerSeries n x ) ) := by d : ℕ n : ℕ ⊢ IsDistBounded fun x => normPowerSeries n x
simpa using IsDistBounded.normPowerSeries_zpow ( n := n ) ( m := 1 ) All goals completed! 🐙 @[ fun_prop ]
lemma IsDistBounded.normPowerSeries_inv { d : ℕ } { n : ℕ } :
IsDistBounded ( d := d ) ( fun x => ( normPowerSeries n x ) ⁻¹ ) := by d : ℕ n : ℕ ⊢ IsDistBounded fun x => ( normPowerSeries n x ) ⁻¹
simpa using normPowerSeries_zpow ( n := n ) ( - 1 ) All goals completed! 🐙 @[ fun_prop ]
lemma IsDistBounded.normPowerSeries_deriv { d : ℕ } ( n : ℕ ) ( i : Fin d ) :
IsDistBounded ( d := d ) ( fun x => ∂[ i ] ( normPowerSeries n ) x ) := by d : ℕ n : ℕ i : Fin d ⊢ IsDistBounded fun x => deriv i ( normPowerSeries n ) x
simp only [ deriv_normPowerSeries ] d : ℕ n : ℕ i : Fin d ⊢ IsDistBounded fun x => x . val i * ( normPowerSeries n x ) ⁻¹
fun_prop All goals completed! 🐙 @[ fun_prop ]
lemma IsDistBounded.normPowerSeries_fderiv { d : ℕ } ( n : ℕ ) ( y : Space d ) :
IsDistBounded ( d := d ) ( fun x => fderiv ℝ ( fun ( x : Space d ) => normPowerSeries n x ) x y ) := by d : ℕ n : ℕ y : Space d ⊢ IsDistBounded fun x => ( fderiv ℝ (fun x => normPowerSeries n x ) x ) y
simp only [ fderiv_eq_sum_deriv ] d : ℕ n : ℕ y : Space d ⊢ IsDistBounded fun x => ∑ i , y . val i • deriv i (fun x => normPowerSeries n x ) x
exact IsDistBounded.sum_fun ( by d : ℕ n : ℕ y : Space d ⊢ ∀ i ∈ Finset.univ , IsDistBounded fun x => y . val i • deriv i (fun x => normPowerSeries n x ) x fun_prop All goals completed! 🐙 ) @[ fun_prop ]
lemma IsDistBounded.normPowerSeries_log { d : ℕ } ( n : ℕ ) :
IsDistBounded ( d := d ) ( fun x => Real.log ( normPowerSeries n x ) ) := by d : ℕ n : ℕ ⊢ IsDistBounded fun x => Real.log ( normPowerSeries n x )
apply IsDistBounded.mono ( f := fun x => ( normPowerSeries n x ) ⁻¹ + ( normPowerSeries n x ) ) hf d : ℕ n : ℕ ⊢ IsDistBounded fun x => ( normPowerSeries n x ) ⁻¹ + normPowerSeries n x hae d : ℕ n : ℕ ⊢ AEStronglyMeasurable (fun x => Real.log ( normPowerSeries n x ) ) volume hfg d : ℕ n : ℕ ⊢ ∀ ( x : Space d ), ‖ Real.log ( normPowerSeries n x ) ‖ ≤ ‖ ( normPowerSeries n x ) ⁻¹ + normPowerSeries n x ‖
· hf d : ℕ n : ℕ ⊢ IsDistBounded fun x => ( normPowerSeries n x ) ⁻¹ + normPowerSeries n x fun_prop All goals completed! 🐙
· hae d : ℕ n : ℕ ⊢ AEStronglyMeasurable (fun x => Real.log ( normPowerSeries n x ) ) volume exact ( ( normPowerSeries_differentiable n ) . continuous . log
( normPowerSeries_ne_zero n ) ) . aestronglyMeasurable All goals completed! 🐙
· hfg d : ℕ n : ℕ ⊢ ∀ ( x : Space d ), ‖ Real.log ( normPowerSeries n x ) ‖ ≤ ‖ ( normPowerSeries n x ) ⁻¹ + normPowerSeries n x ‖ exact fun x => ( normPowerSeries_log_le_normPowerSeries n x ) . trans ( le_abs_self _ ) All goals completed! 🐙
A.8. Differentiability of functions
A.9. Derivatives of functions
lemma deriv_normPowerSeries_zpow { d : ℕ } { n : ℕ } ( m : ℤ ) ( x : Space d ) ( i : Fin d ) :
∂[ i ] ( fun x : Space d => ( normPowerSeries n x ) ^ m ) x =
m * x i * ( normPowerSeries n x ) ^ ( m - 2 ) := by d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ deriv i (fun x => normPowerSeries n x ^ m ) x = ↑ m * x . val i * normPowerSeries n x ^ ( m - 2 )
rw [ deriv_eq_fderiv_basis d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ ( fderiv ℝ (fun x => normPowerSeries n x ^ m ) x ) ( basis i ) = ↑ m * x . val i * normPowerSeries n x ^ ( m - 2 ) d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ ( fderiv ℝ (fun x => normPowerSeries n x ^ m ) x ) ( basis i ) = ↑ m * x . val i * normPowerSeries n x ^ ( m - 2 ) ] d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ ( fderiv ℝ (fun x => normPowerSeries n x ^ m ) x ) ( basis i ) = ↑ m * x . val i * normPowerSeries n x ^ ( m - 2 )
change ( fderiv ℝ ( ( fun x => x ^ m ) ∘ normPowerSeries n ) x ) ( basis i ) = _ d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ ( fderiv ℝ ( (fun x => x ^ m ) ∘ normPowerSeries n ) x ) ( basis i ) = ↑ m * x . val i * normPowerSeries n x ^ ( m - 2 )
rw [ show m - 2 = m - 1 - 1 by d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ deriv i (fun x => normPowerSeries n x ^ m ) x = ↑ m * x . val i * normPowerSeries n x ^ ( m - 2 ) d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ ( fderiv ℝ (fun x => x ^ m ) ( normPowerSeries n x ) ∘SL fderiv ℝ ( normPowerSeries n ) x ) ( basis i ) =
↑ m * x . val i * ( normPowerSeries n x ^ ( m - 1 ) * ( normPowerSeries n x ) ⁻¹ ) hg d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ DifferentiableAt ℝ (fun x => x ^ m ) ( normPowerSeries n x ) hf d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ DifferentiableAt ℝ ( normPowerSeries n ) x ring All goals completed! 🐙 d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ ( fderiv ℝ (fun x => x ^ m ) ( normPowerSeries n x ) ∘SL fderiv ℝ ( normPowerSeries n ) x ) ( basis i ) =
↑ m * x . val i * ( normPowerSeries n x ^ ( m - 1 ) * ( normPowerSeries n x ) ⁻¹ ) hg d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ DifferentiableAt ℝ (fun x => x ^ m ) ( normPowerSeries n x ) hf d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ DifferentiableAt ℝ ( normPowerSeries n ) x , zpow_sub_one₀ ( normPowerSeries_ne_zero n x ) , d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ ( fderiv ℝ ( (fun x => x ^ m ) ∘ normPowerSeries n ) x ) ( basis i ) =
↑ m * x . val i * ( normPowerSeries n x ^ ( m - 1 ) * ( normPowerSeries n x ) ⁻¹ ) d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ ( fderiv ℝ (fun x => x ^ m ) ( normPowerSeries n x ) ∘SL fderiv ℝ ( normPowerSeries n ) x ) ( basis i ) =
↑ m * x . val i * ( normPowerSeries n x ^ ( m - 1 ) * ( normPowerSeries n x ) ⁻¹ ) hg d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ DifferentiableAt ℝ (fun x => x ^ m ) ( normPowerSeries n x ) hf d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ DifferentiableAt ℝ ( normPowerSeries n ) x fderiv_comp d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ ( fderiv ℝ (fun x => x ^ m ) ( normPowerSeries n x ) ∘SL fderiv ℝ ( normPowerSeries n ) x ) ( basis i ) =
↑ m * x . val i * ( normPowerSeries n x ^ ( m - 1 ) * ( normPowerSeries n x ) ⁻¹ ) hg d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ DifferentiableAt ℝ (fun x => x ^ m ) ( normPowerSeries n x ) hf d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ DifferentiableAt ℝ ( normPowerSeries n ) x d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ ( fderiv ℝ (fun x => x ^ m ) ( normPowerSeries n x ) ∘SL fderiv ℝ ( normPowerSeries n ) x ) ( basis i ) =
↑ m * x . val i * ( normPowerSeries n x ^ ( m - 1 ) * ( normPowerSeries n x ) ⁻¹ ) hg d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ DifferentiableAt ℝ (fun x => x ^ m ) ( normPowerSeries n x ) hf d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ DifferentiableAt ℝ ( normPowerSeries n ) x ] d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ ( fderiv ℝ (fun x => x ^ m ) ( normPowerSeries n x ) ∘SL fderiv ℝ ( normPowerSeries n ) x ) ( basis i ) =
↑ m * x . val i * ( normPowerSeries n x ^ ( m - 1 ) * ( normPowerSeries n x ) ⁻¹ ) hg d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ DifferentiableAt ℝ (fun x => x ^ m ) ( normPowerSeries n x ) hf d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ DifferentiableAt ℝ ( normPowerSeries n ) x
simp only [ ContinuousLinearMap.coe_comp , Function.comp_apply , fderiv_eq_smul_deriv , deriv_zpow' ,
smul_eq_mul ] d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ ( fderiv ℝ ( normPowerSeries n ) x ) ( basis i ) * ( ↑ m * normPowerSeries n x ^ ( m - 1 ) ) =
↑ m * x . val i * ( normPowerSeries n x ^ ( m - 1 ) * ( normPowerSeries n x ) ⁻¹ ) hg d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ DifferentiableAt ℝ (fun x => x ^ m ) ( normPowerSeries n x ) hf d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ DifferentiableAt ℝ ( normPowerSeries n ) x
rw [ fderiv_normPowerSeries , d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ ⟪ basis i , x ⟫_ ℝ * ( normPowerSeries n x ) ⁻¹ * ( ↑ m * normPowerSeries n x ^ ( m - 1 ) ) =
↑ m * x . val i * ( normPowerSeries n x ^ ( m - 1 ) * ( normPowerSeries n x ) ⁻¹ ) hg d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ DifferentiableAt ℝ (fun x => x ^ m ) ( normPowerSeries n x ) hf d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ DifferentiableAt ℝ ( normPowerSeries n ) x d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ x . val i * ( normPowerSeries n x ) ⁻¹ * ( ↑ m * normPowerSeries n x ^ ( m - 1 ) ) =
↑ m * x . val i * ( normPowerSeries n x ^ ( m - 1 ) * ( normPowerSeries n x ) ⁻¹ ) hg d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ DifferentiableAt ℝ (fun x => x ^ m ) ( normPowerSeries n x ) hf d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ DifferentiableAt ℝ ( normPowerSeries n ) x basis_inner d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ x . val i * ( normPowerSeries n x ) ⁻¹ * ( ↑ m * normPowerSeries n x ^ ( m - 1 ) ) =
↑ m * x . val i * ( normPowerSeries n x ^ ( m - 1 ) * ( normPowerSeries n x ) ⁻¹ ) hg d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ DifferentiableAt ℝ (fun x => x ^ m ) ( normPowerSeries n x ) hf d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ DifferentiableAt ℝ ( normPowerSeries n ) x d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ x . val i * ( normPowerSeries n x ) ⁻¹ * ( ↑ m * normPowerSeries n x ^ ( m - 1 ) ) =
↑ m * x . val i * ( normPowerSeries n x ^ ( m - 1 ) * ( normPowerSeries n x ) ⁻¹ ) hg d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ DifferentiableAt ℝ (fun x => x ^ m ) ( normPowerSeries n x ) hf d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ DifferentiableAt ℝ ( normPowerSeries n ) x ] d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ x . val i * ( normPowerSeries n x ) ⁻¹ * ( ↑ m * normPowerSeries n x ^ ( m - 1 ) ) =
↑ m * x . val i * ( normPowerSeries n x ^ ( m - 1 ) * ( normPowerSeries n x ) ⁻¹ ) hg d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ DifferentiableAt ℝ (fun x => x ^ m ) ( normPowerSeries n x ) hf d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ DifferentiableAt ℝ ( normPowerSeries n ) x
ring hg d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ DifferentiableAt ℝ (fun x => x ^ m ) ( normPowerSeries n x ) hf d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ DifferentiableAt ℝ ( normPowerSeries n ) x
· hg d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ DifferentiableAt ℝ (fun x => x ^ m ) ( normPowerSeries n x ) exact differentiableAt_zpow . mpr ( .inl ( normPowerSeries_ne_zero n x ) ) All goals completed! 🐙
· hf d : ℕ n : ℕ m : ℤ x : Space d i : Fin d ⊢ DifferentiableAt ℝ ( normPowerSeries n ) x fun_prop All goals completed! 🐙
lemma fderiv_normPowerSeries_zpow { d : ℕ } { n : ℕ } ( m : ℤ ) ( x y : Space d ) :
fderiv ℝ ( fun x : Space d => ( normPowerSeries n x ) ^ m ) x y =
m * ⟪ y , x ⟫_ ℝ * ( normPowerSeries n x ) ^ ( m - 2 ) := by d : ℕ n : ℕ m : ℤ x : Space d y : Space d ⊢ ( fderiv ℝ (fun x => normPowerSeries n x ^ m ) x ) y = ↑ m * ⟪ y , x ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 )
rw [ fderiv_eq_sum_deriv , d : ℕ n : ℕ m : ℤ x : Space d y : Space d ⊢ ∑ i , y . val i • deriv i (fun x => normPowerSeries n x ^ m ) x = ↑ m * ⟪ y , x ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) d : ℕ n : ℕ m : ℤ x : Space d y : Space d ⊢ ∑ i , y . val i • deriv i (fun x => normPowerSeries n x ^ m ) x =
∑ i , ↑ m * ( y . val i * x . val i ) * normPowerSeries n x ^ ( m - 2 ) inner_eq_sum , d : ℕ n : ℕ m : ℤ x : Space d y : Space d ⊢ ∑ i , y . val i • deriv i (fun x => normPowerSeries n x ^ m ) x =
( ↑ m * ∑ i , y . val i * x . val i ) * normPowerSeries n x ^ ( m - 2 ) d : ℕ n : ℕ m : ℤ x : Space d y : Space d ⊢ ∑ i , y . val i • deriv i (fun x => normPowerSeries n x ^ m ) x =
∑ i , ↑ m * ( y . val i * x . val i ) * normPowerSeries n x ^ ( m - 2 ) Finset.mul_sum , d : ℕ n : ℕ m : ℤ x : Space d y : Space d ⊢ ∑ i , y . val i • deriv i (fun x => normPowerSeries n x ^ m ) x =
(∑ i , ↑ m * ( y . val i * x . val i ) ) * normPowerSeries n x ^ ( m - 2 ) d : ℕ n : ℕ m : ℤ x : Space d y : Space d ⊢ ∑ i , y . val i • deriv i (fun x => normPowerSeries n x ^ m ) x =
∑ i , ↑ m * ( y . val i * x . val i ) * normPowerSeries n x ^ ( m - 2 ) Finset.sum_mul d : ℕ n : ℕ m : ℤ x : Space d y : Space d ⊢ ∑ i , y . val i • deriv i (fun x => normPowerSeries n x ^ m ) x =
∑ i , ↑ m * ( y . val i * x . val i ) * normPowerSeries n x ^ ( m - 2 ) d : ℕ n : ℕ m : ℤ x : Space d y : Space d ⊢ ∑ i , y . val i • deriv i (fun x => normPowerSeries n x ^ m ) x =
∑ i , ↑ m * ( y . val i * x . val i ) * normPowerSeries n x ^ ( m - 2 ) ] d : ℕ n : ℕ m : ℤ x : Space d y : Space d ⊢ ∑ i , y . val i • deriv i (fun x => normPowerSeries n x ^ m ) x =
∑ i , ↑ m * ( y . val i * x . val i ) * normPowerSeries n x ^ ( m - 2 )
exact Finset.sum_congr rfl fun i _ => by d : ℕ n : ℕ m : ℤ x : Space d y : Space d i : Fin d x✝ : i ∈ Finset.univ ⊢ y . val i • deriv i (fun x => normPowerSeries n x ^ m ) x = ↑ m * ( y . val i * x . val i ) * normPowerSeries n x ^ ( m - 2 )
simp [ deriv_normPowerSeries_zpow , mul_assoc , mul_comm , mul_left_comm ] All goals completed! 🐙
lemma deriv_log_normPowerSeries { d : ℕ } { n : ℕ } ( x : Space d ) ( i : Fin d ) :
∂[ i ] ( fun x : Space d => Real.log ( normPowerSeries n x ) ) x =
x i * ( normPowerSeries n x ) ^ ( - 2 : ℤ ) := by d : ℕ n : ℕ x : Space d i : Fin d ⊢ deriv i (fun x => Real.log ( normPowerSeries n x ) ) x = x . val i * normPowerSeries n x ^ (- 2 )
rw [ deriv_eq_fderiv_basis d : ℕ n : ℕ x : Space d i : Fin d ⊢ ( fderiv ℝ (fun x => Real.log ( normPowerSeries n x ) ) x ) ( basis i ) = x . val i * normPowerSeries n x ^ (- 2 ) d : ℕ n : ℕ x : Space d i : Fin d ⊢ ( fderiv ℝ (fun x => Real.log ( normPowerSeries n x ) ) x ) ( basis i ) = x . val i * normPowerSeries n x ^ (- 2 ) ] d : ℕ n : ℕ x : Space d i : Fin d ⊢ ( fderiv ℝ (fun x => Real.log ( normPowerSeries n x ) ) x ) ( basis i ) = x . val i * normPowerSeries n x ^ (- 2 )
change ( fderiv ℝ ( Real.log ∘ normPowerSeries n ) x ) ( basis i ) = _ d : ℕ n : ℕ x : Space d i : Fin d ⊢ ( fderiv ℝ ( Real.log ∘ normPowerSeries n ) x ) ( basis i ) = x . val i * normPowerSeries n x ^ (- 2 )
rw [ fderiv_comp d : ℕ n : ℕ x : Space d i : Fin d ⊢ ( fderiv ℝ Real.log ( normPowerSeries n x ) ∘SL fderiv ℝ ( normPowerSeries n ) x ) ( basis i ) =
x . val i * normPowerSeries n x ^ (- 2 ) hg d : ℕ n : ℕ x : Space d i : Fin d ⊢ DifferentiableAt ℝ Real.log ( normPowerSeries n x ) hf d : ℕ n : ℕ x : Space d i : Fin d ⊢ DifferentiableAt ℝ ( normPowerSeries n ) x d : ℕ n : ℕ x : Space d i : Fin d ⊢ ( fderiv ℝ Real.log ( normPowerSeries n x ) ∘SL fderiv ℝ ( normPowerSeries n ) x ) ( basis i ) =
x . val i * normPowerSeries n x ^ (- 2 ) hg d : ℕ n : ℕ x : Space d i : Fin d ⊢ DifferentiableAt ℝ Real.log ( normPowerSeries n x ) hf d : ℕ n : ℕ x : Space d i : Fin d ⊢ DifferentiableAt ℝ ( normPowerSeries n ) x ] d : ℕ n : ℕ x : Space d i : Fin d ⊢ ( fderiv ℝ Real.log ( normPowerSeries n x ) ∘SL fderiv ℝ ( normPowerSeries n ) x ) ( basis i ) =
x . val i * normPowerSeries n x ^ (- 2 ) hg d : ℕ n : ℕ x : Space d i : Fin d ⊢ DifferentiableAt ℝ Real.log ( normPowerSeries n x ) hf d : ℕ n : ℕ x : Space d i : Fin d ⊢ DifferentiableAt ℝ ( normPowerSeries n ) x
simp only [ ContinuousLinearMap.coe_comp , Function.comp_apply , fderiv_eq_smul_deriv ,
Real.deriv_log' , smul_eq_mul , Int.reduceNeg , zpow_neg ] d : ℕ n : ℕ x : Space d i : Fin d ⊢ ( fderiv ℝ ( normPowerSeries n ) x ) ( basis i ) * ( normPowerSeries n x ) ⁻¹ = x . val i * ( normPowerSeries n x ^ 2 ) ⁻¹ hg d : ℕ n : ℕ x : Space d i : Fin d ⊢ DifferentiableAt ℝ Real.log ( normPowerSeries n x ) hf d : ℕ n : ℕ x : Space d i : Fin d ⊢ DifferentiableAt ℝ ( normPowerSeries n ) x
simp [ fderiv_normPowerSeries , zpow_ofNat , sq ] d : ℕ n : ℕ x : Space d i : Fin d ⊢ x . val i * ( normPowerSeries n x ) ⁻¹ * ( normPowerSeries n x ) ⁻¹ =
x . val i * ( ( normPowerSeries n x ) ⁻¹ * ( normPowerSeries n x ) ⁻¹ ) hg d : ℕ n : ℕ x : Space d i : Fin d ⊢ DifferentiableAt ℝ Real.log ( normPowerSeries n x ) hf d : ℕ n : ℕ x : Space d i : Fin d ⊢ DifferentiableAt ℝ ( normPowerSeries n ) x
ring hg d : ℕ n : ℕ x : Space d i : Fin d ⊢ DifferentiableAt ℝ Real.log ( normPowerSeries n x ) hf d : ℕ n : ℕ x : Space d i : Fin d ⊢ DifferentiableAt ℝ ( normPowerSeries n ) x
· hg d : ℕ n : ℕ x : Space d i : Fin d ⊢ DifferentiableAt ℝ Real.log ( normPowerSeries n x ) exact Real.differentiableAt_log ( normPowerSeries_ne_zero n x ) All goals completed! 🐙
· hf d : ℕ n : ℕ x : Space d i : Fin d ⊢ DifferentiableAt ℝ ( normPowerSeries n ) x fun_prop All goals completed! 🐙
lemma fderiv_log_normPowerSeries { d : ℕ } { n : ℕ } ( x y : Space d ) :
fderiv ℝ ( fun x : Space d => Real.log ( normPowerSeries n x ) ) x y =
⟪ y , x ⟫_ ℝ * ( normPowerSeries n x ) ^ ( - 2 : ℤ ) := by d : ℕ n : ℕ x : Space d y : Space d ⊢ ( fderiv ℝ (fun x => Real.log ( normPowerSeries n x ) ) x ) y = ⟪ y , x ⟫_ ℝ * normPowerSeries n x ^ (- 2 )
rw [ fderiv_eq_sum_deriv , d : ℕ n : ℕ x : Space d y : Space d ⊢ ∑ i , y . val i • deriv i (fun x => Real.log ( normPowerSeries n x ) ) x = ⟪ y , x ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) d : ℕ n : ℕ x : Space d y : Space d ⊢ ∑ i , y . val i • deriv i (fun x => Real.log ( normPowerSeries n x ) ) x = ∑ i , y . val i * x . val i * normPowerSeries n x ^ (- 2 ) inner_eq_sum , d : ℕ n : ℕ x : Space d y : Space d ⊢ ∑ i , y . val i • deriv i (fun x => Real.log ( normPowerSeries n x ) ) x =
(∑ i , y . val i * x . val i ) * normPowerSeries n x ^ (- 2 ) d : ℕ n : ℕ x : Space d y : Space d ⊢ ∑ i , y . val i • deriv i (fun x => Real.log ( normPowerSeries n x ) ) x = ∑ i , y . val i * x . val i * normPowerSeries n x ^ (- 2 ) Finset.sum_mul d : ℕ n : ℕ x : Space d y : Space d ⊢ ∑ i , y . val i • deriv i (fun x => Real.log ( normPowerSeries n x ) ) x = ∑ i , y . val i * x . val i * normPowerSeries n x ^ (- 2 ) d : ℕ n : ℕ x : Space d y : Space d ⊢ ∑ i , y . val i • deriv i (fun x => Real.log ( normPowerSeries n x ) ) x = ∑ i , y . val i * x . val i * normPowerSeries n x ^ (- 2 ) ] d : ℕ n : ℕ x : Space d y : Space d ⊢ ∑ i , y . val i • deriv i (fun x => Real.log ( normPowerSeries n x ) ) x = ∑ i , y . val i * x . val i * normPowerSeries n x ^ (- 2 )
exact Finset.sum_congr rfl fun i _ => by d : ℕ n : ℕ x : Space d y : Space d i : Fin d x✝ : i ∈ Finset.univ ⊢ y . val i • deriv i (fun x => Real.log ( normPowerSeries n x ) ) x = y . val i * x . val i * normPowerSeries n x ^ (- 2 ) simp [ deriv_log_normPowerSeries , mul_assoc ] All goals completed! 🐙
A.10. Gradients of distributions based on powers
lemma gradient_dist_normPowerSeries_zpow { d : ℕ } { n : ℕ } ( m : ℤ ) :
∇ᵈ ( distOfFunction ( fun x : Space d => ( normPowerSeries n x ) ^ m ) ( by 𝕜 : Type E : Type F : Type F' : Type inst✝⁵ : RCLike 𝕜 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedAddCommGroup F inst✝² : NormedAddCommGroup F' inst✝¹ : NormedSpace ℝ E inst✝ : NormedSpace ℝ F d : ℕ n : ℕ m : ℤ ⊢ IsDistBounded fun x => normPowerSeries n x ^ m fun_prop All goals completed! 🐙 ) ) =
distOfFunction ( fun x : Space d => ( m * ( normPowerSeries n x ) ^ ( m - 2 ) ) • basis . repr x )
( by 𝕜 : Type E : Type F : Type F' : Type inst✝⁵ : RCLike 𝕜 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedAddCommGroup F inst✝² : NormedAddCommGroup F' inst✝¹ : NormedSpace ℝ E inst✝ : NormedSpace ℝ F d : ℕ n : ℕ m : ℤ ⊢ IsDistBounded fun x => ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x fun_prop All goals completed! 🐙 ) := by d : ℕ n : ℕ m : ℤ ⊢ ∇ᵈ ( distOfFunction (fun x => normPowerSeries n x ^ m ) ⋯ ) =
distOfFunction (fun x => ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x ) ⋯
ext1 η d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) ⊢ ( ∇ᵈ ( distOfFunction (fun x => normPowerSeries n x ^ m ) ⋯ ) ) η =
( distOfFunction (fun x => ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x ) ⋯ ) η
refine ext_inner_right ℝ fun y => ? _ d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ ⟪ ( ∇ᵈ ( distOfFunction (fun x => normPowerSeries n x ^ m ) ⋯ ) ) η , y ⟫_ ℝ =
⟪ ( distOfFunction (fun x => ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x ) ⋯ ) η , y ⟫_ ℝ
simp [ distGrad_inner_eq ] d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ ( ( ( Distribution.fderivD ℝ ) ( distOfFunction (fun x => normPowerSeries n x ^ m ) ⋯ ) ) η ) ( basis . repr . symm y ) =
⟪ ( distOfFunction (fun x => ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x ) ⋯ ) η , y ⟫_ ℝ
rw [ Distribution.fderivD_apply , d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ - ( distOfFunction (fun x => normPowerSeries n x ^ m ) ⋯ )
( ( SchwartzMap.evalCLM ℝ ( Space d ) ℝ ( basis . repr . symm y ) ) ( ( fderivCLM ℝ ( Space d ) ℝ ) η ) ) =
⟪ ( distOfFunction (fun x => ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x ) ⋯ ) η , y ⟫_ ℝ d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ - ∫ ( x : Space d ) ,
( ( SchwartzMap.evalCLM ℝ ( Space d ) ℝ ( basis . repr . symm y ) ) ( ( fderivCLM ℝ ( Space d ) ℝ ) η ) ) x •
normPowerSeries n x ^ m =
∫ ( x : Space d ) , η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ distOfFunction_apply , d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ - ∫ ( x : Space d ) ,
( ( SchwartzMap.evalCLM ℝ ( Space d ) ℝ ( basis . repr . symm y ) ) ( ( fderivCLM ℝ ( Space d ) ℝ ) η ) ) x •
normPowerSeries n x ^ m =
⟪ ( distOfFunction (fun x => ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x ) ⋯ ) η , y ⟫_ ℝ d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ - ∫ ( x : Space d ) ,
( ( SchwartzMap.evalCLM ℝ ( Space d ) ℝ ( basis . repr . symm y ) ) ( ( fderivCLM ℝ ( Space d ) ℝ ) η ) ) x •
normPowerSeries n x ^ m =
∫ ( x : Space d ) , η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ distOfFunction_inner d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ - ∫ ( x : Space d ) ,
( ( SchwartzMap.evalCLM ℝ ( Space d ) ℝ ( basis . repr . symm y ) ) ( ( fderivCLM ℝ ( Space d ) ℝ ) η ) ) x •
normPowerSeries n x ^ m =
∫ ( x : Space d ) , η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ - ∫ ( x : Space d ) ,
( ( SchwartzMap.evalCLM ℝ ( Space d ) ℝ ( basis . repr . symm y ) ) ( ( fderivCLM ℝ ( Space d ) ℝ ) η ) ) x •
normPowerSeries n x ^ m =
∫ ( x : Space d ) , η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ ] d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ - ∫ ( x : Space d ) ,
( ( SchwartzMap.evalCLM ℝ ( Space d ) ℝ ( basis . repr . symm y ) ) ( ( fderivCLM ℝ ( Space d ) ℝ ) η ) ) x •
normPowerSeries n x ^ m =
∫ ( x : Space d ) , η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ
calc _
_ = - ∫ ( x : Space d ) , fderiv ℝ η x ( basis . repr . symm y ) * normPowerSeries n x ^ m := by d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ - ∫ ( x : Space d ) ,
( ( SchwartzMap.evalCLM ℝ ( Space d ) ℝ ( basis . repr . symm y ) ) ( ( fderivCLM ℝ ( Space d ) ℝ ) η ) ) x •
normPowerSeries n x ^ m =
- ∫ ( x : Space d ) , ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * normPowerSeries n x ^ m
rfl All goals completed! 🐙
_ = ∫ ( x : Space d ) , η x * fderiv ℝ ( normPowerSeries n · ^ m ) x ( basis . repr . symm y ) := by d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ - ∫ ( x : Space d ) , ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * normPowerSeries n x ^ m =
∫ ( x : Space d ) , η x * ( fderiv ℝ (fun x => normPowerSeries n x ^ m ) x ) ( basis . repr . symm y )
rw [ integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ - ∫ ( x : Space d ) , ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * normPowerSeries n x ^ m =
- ∫ ( x : Space d ) , ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * normPowerSeries n x ^ m hf'g d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Integrable (fun x => ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * normPowerSeries n x ^ m ) volume hfg' d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Integrable (fun x => η x * ( fderiv ℝ (fun x => normPowerSeries n x ^ m ) x ) ( basis . repr . symm y ) ) volume hfg d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Integrable (fun x => η x * normPowerSeries n x ^ m ) volume hf d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ ∀ x ∈ tsupport fun x => normPowerSeries n x ^ m , DifferentiableAt ℝ (⇑ η ) x hg d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ ∀ x ∈ tsupport ⇑ η , DifferentiableAt ℝ (fun x => normPowerSeries n x ^ m ) x hf'g d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Integrable (fun x => ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * normPowerSeries n x ^ m ) volume hfg' d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Integrable (fun x => η x * ( fderiv ℝ (fun x => normPowerSeries n x ^ m ) x ) ( basis . repr . symm y ) ) volume hfg d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Integrable (fun x => η x * normPowerSeries n x ^ m ) volume hf d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ ∀ x ∈ tsupport fun x => normPowerSeries n x ^ m , DifferentiableAt ℝ (⇑ η ) x hg d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ ∀ x ∈ tsupport ⇑ η , DifferentiableAt ℝ (fun x => normPowerSeries n x ^ m ) x ] hf'g d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Integrable (fun x => ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * normPowerSeries n x ^ m ) volume hfg' d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Integrable (fun x => η x * ( fderiv ℝ (fun x => normPowerSeries n x ^ m ) x ) ( basis . repr . symm y ) ) volume hfg d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Integrable (fun x => η x * normPowerSeries n x ^ m ) volume hf d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ ∀ x ∈ tsupport fun x => normPowerSeries n x ^ m , DifferentiableAt ℝ (⇑ η ) x hg d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ ∀ x ∈ tsupport ⇑ η , DifferentiableAt ℝ (fun x => normPowerSeries n x ^ m ) x
· hf'g d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Integrable (fun x => ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * normPowerSeries n x ^ m ) volume fun_prop All goals completed! 🐙
· hfg' d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Integrable (fun x => η x * ( fderiv ℝ (fun x => normPowerSeries n x ^ m ) x ) ( basis . repr . symm y ) ) volume refine IsDistBounded.integrable_space_mul ? _ η hfg' d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ IsDistBounded fun x => ( fderiv ℝ (fun x => normPowerSeries n x ^ m ) x ) ( basis . repr . symm y )
simp only [ fderiv_normPowerSeries_zpow , mul_assoc ] hfg' d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ IsDistBounded fun x => ↑ m * ( ⟪ basis . repr . symm y , x ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) )
fun_prop All goals completed! 🐙
· hfg d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Integrable (fun x => η x * normPowerSeries n x ^ m ) volume fun_prop All goals completed! 🐙
· hf d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ ∀ x ∈ tsupport fun x => normPowerSeries n x ^ m , DifferentiableAt ℝ (⇑ η ) x fun_prop All goals completed! 🐙
· hg d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ ∀ x ∈ tsupport ⇑ η , DifferentiableAt ℝ (fun x => normPowerSeries n x ^ m ) x exact fun _ _ => ( differentiable_normPowerSeries_zpow m ) . differentiableAt All goals completed! 🐙
_ = ∫ ( x : Space d ) , η x *
( m * ⟪ ( basis . repr . symm y ) , x ⟫_ ℝ * ( normPowerSeries n x ) ^ ( m - 2 ) ) := by d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ ∫ ( x : Space d ) , η x * ( fderiv ℝ (fun x => normPowerSeries n x ^ m ) x ) ( basis . repr . symm y ) =
∫ ( x : Space d ) , η x * ( ↑ m * ⟪ basis . repr . symm y , x ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) )
simp only [ fderiv_normPowerSeries_zpow ] All goals completed! 🐙
congr calc.step.e_f d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ (fun x => η x * ( ↑ m * ⟪ basis . repr . symm y , x ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) = fun x =>
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ
funext x calc.step.e_f d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) x : Space d ⊢ η x * ( ↑ m * ⟪ basis . repr . symm y , x ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) =
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ
simp [ inner_smul_left_eq_smul ] calc.step.e_f d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) x : Space d ⊢ ↑ m * ⟪ basis . repr . symm y , x ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) =
↑ m * normPowerSeries n x ^ ( m - 2 ) * ⟪ basis . repr x , y ⟫_ ℝ ∨
η x = 0
left calc.step.e_f d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) x : Space d ⊢ ↑ m * ⟪ basis . repr . symm y , x ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) = ↑ m * normPowerSeries n x ^ ( m - 2 ) * ⟪ basis . repr x , y ⟫_ ℝ
rw [ real_inner_comm , calc.step.e_f d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) x : Space d ⊢ ↑ m * ⟪ x , basis . repr . symm y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) = ↑ m * normPowerSeries n x ^ ( m - 2 ) * ⟪ basis . repr x , y ⟫_ ℝ calc.step.e_f d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) x : Space d ⊢ ↑ m * ⟪ x , basis . repr . symm y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) =
↑ m * normPowerSeries n x ^ ( m - 2 ) * ⟪ x , basis . repr . symm y ⟫_ ℝ basis_repr_inner_eq calc.step.e_f d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) x : Space d ⊢ ↑ m * ⟪ x , basis . repr . symm y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) =
↑ m * normPowerSeries n x ^ ( m - 2 ) * ⟪ x , basis . repr . symm y ⟫_ ℝ calc.step.e_f d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) x : Space d ⊢ ↑ m * ⟪ x , basis . repr . symm y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) =
↑ m * normPowerSeries n x ^ ( m - 2 ) * ⟪ x , basis . repr . symm y ⟫_ ℝ ] calc.step.e_f d : ℕ n : ℕ m : ℤ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) x : Space d ⊢ ↑ m * ⟪ x , basis . repr . symm y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) =
↑ m * normPowerSeries n x ^ ( m - 2 ) * ⟪ x , basis . repr . symm y ⟫_ ℝ
ring All goals completed! 🐙
A.10.1. The limits of gradients of distributions based on powers
lemma gradient_dist_normPowerSeries_zpow_tendsTo_distGrad_norm { d : ℕ } [ NeZero d ] ( m : ℤ )
( hm : - ( d - 1 : ℕ ) ≤ m ) ( η : 𝓢( Space d , ℝ ) )
( y : EuclideanSpace ℝ ( Fin d ) ) :
Filter.Tendsto ( fun n =>
⟪ ( ∇ᵈ ( distOfFunction
( fun x : Space d => ( normPowerSeries n x ) ^ m ) ( by 𝕜 : Type E : Type F : Type F' : Type inst✝⁶ : RCLike 𝕜 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedAddCommGroup F' inst✝² : NormedSpace ℝ E inst✝¹ : NormedSpace ℝ F d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) n : ℕ ⊢ IsDistBounded fun x => normPowerSeries n x ^ m fun_prop All goals completed! 🐙 ) ) ) η , y ⟫_ ℝ )
Filter.atTop
( 𝓝 ( ⟪ ∇ᵈ ( distOfFunction ( fun x : Space d => ‖ x ‖ ^ m )
( IsDistBounded.pow m hm ) ) η , y ⟫_ ℝ ) ) := by d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Filter.Tendsto (fun n => ⟪ ( ∇ᵈ ( distOfFunction (fun x => normPowerSeries n x ^ m ) ⋯ ) ) η , y ⟫_ ℝ ) Filter.atTop
( 𝓝 ⟪ ( ∇ᵈ ( distOfFunction (fun x => ‖ x ‖ ^ m ) ⋯ ) ) η , y ⟫_ ℝ )
simp only [ distGrad_inner_eq , Distribution.fderivD_apply , distOfFunction_apply ] d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Filter.Tendsto
(fun n =>
- ∫ ( x : Space d ) ,
( ( SchwartzMap.evalCLM ℝ ( Space d ) ℝ ( basis . repr . symm y ) ) ( ( fderivCLM ℝ ( Space d ) ℝ ) η ) ) x •
normPowerSeries n x ^ m )
Filter.atTop
( 𝓝
( - ∫ ( x : Space d ) ,
( ( SchwartzMap.evalCLM ℝ ( Space d ) ℝ ( basis . repr . symm y ) ) ( ( fderivCLM ℝ ( Space d ) ℝ ) η ) ) x • ‖ x ‖ ^ m ) )
change Filter.Tendsto ( fun n => - ∫ ( x : Space d ) ,
fderiv ℝ η x ( basis . repr . symm y ) * normPowerSeries n x ^ m )
Filter.atTop ( 𝓝 ( - ∫ ( x : Space d ) , fderiv ℝ η x ( basis . repr . symm y ) * ‖ x ‖ ^ m ) ) d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Filter.Tendsto (fun n => - ∫ ( x : Space d ) , ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * normPowerSeries n x ^ m ) Filter.atTop
( 𝓝 ( - ∫ ( x : Space d ) , ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * ‖ x ‖ ^ m ) )
apply Filter.Tendsto.neg d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Filter.Tendsto (fun x => ∫ ( x_1 : Space d ) , ( fderiv ℝ (⇑ η ) x_1 ) ( basis . repr . symm y ) * normPowerSeries x x_1 ^ m )
Filter.atTop ( 𝓝 ( ∫ ( x : Space d ) , ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * ‖ x ‖ ^ m ) )
apply MeasureTheory.tendsto_integral_of_dominated_convergence
( bound := fun x => | fderiv ℝ η x ( basis . repr . symm y ) | * ( ( ‖ x ‖ + 1 ) ^ m + ‖ x ‖ ^ m ) ) F_measurable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ ∀ ( n : ℕ ), AEStronglyMeasurable (fun a => ( fderiv ℝ (⇑ η ) a ) ( basis . repr . symm y ) * normPowerSeries n a ^ m ) volume bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Integrable (fun x => | ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) | * ( ( ‖ x ‖ + 1 ) ^ m + ‖ x ‖ ^ m ) ) volume h_bound d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ ∀ ( n : ℕ ),
∀ᵐ ( a : Space d ) ,
‖ ( fderiv ℝ (⇑ η ) a ) ( basis . repr . symm y ) * normPowerSeries n a ^ m ‖ ≤
| ( fderiv ℝ (⇑ η ) a ) ( basis . repr . symm y ) | * ( ( ‖ a ‖ + 1 ) ^ m + ‖ a ‖ ^ m ) h_lim d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ ∀ᵐ ( a : Space d ) ,
Filter.Tendsto (fun n => ( fderiv ℝ (⇑ η ) a ) ( basis . repr . symm y ) * normPowerSeries n a ^ m ) Filter.atTop
( 𝓝 ( ( fderiv ℝ (⇑ η ) a ) ( basis . repr . symm y ) * ‖ a ‖ ^ m ) )
· F_measurable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ ∀ ( n : ℕ ), AEStronglyMeasurable (fun a => ( fderiv ℝ (⇑ η ) a ) ( basis . repr . symm y ) * normPowerSeries n a ^ m ) volume intro n F_measurable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) n : ℕ ⊢ AEStronglyMeasurable (fun a => ( fderiv ℝ (⇑ η ) a ) ( basis . repr . symm y ) * normPowerSeries n a ^ m ) volume
exact IsDistBounded.aeStronglyMeasurable_fderiv_schwartzMap_smul ( F := ℝ ) ( by d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) n : ℕ ⊢ IsDistBounded fun a => normPowerSeries n a ^ m fun_prop All goals completed! 🐙 ) η _
· bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Integrable (fun x => | ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) | * ( ( ‖ x ‖ + 1 ) ^ m + ‖ x ‖ ^ m ) ) volume have h1 : Integrable ( fun x =>
( fderiv ℝ ( ⇑ η ) x ) ( basis . repr . symm y ) * ( ( ‖ x ‖ + 1 ) ^ m + ‖ x ‖ ^ m ) ) volume := by d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Filter.Tendsto (fun n => ⟪ ( ∇ᵈ ( distOfFunction (fun x => normPowerSeries n x ^ m ) ⋯ ) ) η , y ⟫_ ℝ ) Filter.atTop
( 𝓝 ⟪ ( ∇ᵈ ( distOfFunction (fun x => ‖ x ‖ ^ m ) ⋯ ) ) η , y ⟫_ ℝ ) bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : Integrable (fun x => ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * ( ( ‖ x ‖ + 1 ) ^ m + ‖ x ‖ ^ m ) ) volume ⊢ Integrable (fun x => | ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) | * ( ( ‖ x ‖ + 1 ) ^ m + ‖ x ‖ ^ m ) ) volume
apply IsDistBounded.integrable_space_fderiv
( ( IsDistBounded.norm_add_pos_nat_zpow m 1 one_pos ) . add ( IsDistBounded.pow m hm ) ) bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : Integrable (fun x => ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * ( ( ‖ x ‖ + 1 ) ^ m + ‖ x ‖ ^ m ) ) volume ⊢ Integrable (fun x => | ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) | * ( ( ‖ x ‖ + 1 ) ^ m + ‖ x ‖ ^ m ) ) volume bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : Integrable (fun x => ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * ( ( ‖ x ‖ + 1 ) ^ m + ‖ x ‖ ^ m ) ) volume ⊢ Integrable (fun x => | ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) | * ( ( ‖ x ‖ + 1 ) ^ m + ‖ x ‖ ^ m ) ) volume
refine h1 . abs . congr ( ae_of_all _ fun x => ? _ ) bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : Integrable (fun x => ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * ( ( ‖ x ‖ + 1 ) ^ m + ‖ x ‖ ^ m ) ) volume x : Space d ⊢ (fun a => | ( fderiv ℝ (⇑ η ) a ) ( basis . repr . symm y ) * ( ( ‖ a ‖ + 1 ) ^ m + ‖ a ‖ ^ m ) | ) x =
(fun x => | ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) | * ( ( ‖ x ‖ + 1 ) ^ m + ‖ x ‖ ^ m ) ) x
simp only [ abs_mul ] bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : Integrable (fun x => ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * ( ( ‖ x ‖ + 1 ) ^ m + ‖ x ‖ ^ m ) ) volume x : Space d ⊢ | ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) | * | ( ‖ x ‖ + 1 ) ^ m + ‖ x ‖ ^ m | =
| ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) | * ( ( ‖ x ‖ + 1 ) ^ m + ‖ x ‖ ^ m )
congr 1 bound_integrable.e_a d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : Integrable (fun x => ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * ( ( ‖ x ‖ + 1 ) ^ m + ‖ x ‖ ^ m ) ) volume x : Space d ⊢ | ( ‖ x ‖ + 1 ) ^ m + ‖ x ‖ ^ m | = ( ‖ x ‖ + 1 ) ^ m + ‖ x ‖ ^ m
exact abs_of_nonneg ( by d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : Integrable (fun x => ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * ( ( ‖ x ‖ + 1 ) ^ m + ‖ x ‖ ^ m ) ) volume x : Space d ⊢ 0 ≤ ( ‖ x ‖ + 1 ) ^ m + ‖ x ‖ ^ m positivity All goals completed! 🐙 )
· h_bound d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ ∀ ( n : ℕ ),
∀ᵐ ( a : Space d ) ,
‖ ( fderiv ℝ (⇑ η ) a ) ( basis . repr . symm y ) * normPowerSeries n a ^ m ‖ ≤
| ( fderiv ℝ (⇑ η ) a ) ( basis . repr . symm y ) | * ( ( ‖ a ‖ + 1 ) ^ m + ‖ a ‖ ^ m ) intro n h_bound d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) n : ℕ ⊢ ∀ᵐ ( a : Space d ) ,
‖ ( fderiv ℝ (⇑ η ) a ) ( basis . repr . symm y ) * normPowerSeries n a ^ m ‖ ≤
| ( fderiv ℝ (⇑ η ) a ) ( basis . repr . symm y ) | * ( ( ‖ a ‖ + 1 ) ^ m + ‖ a ‖ ^ m )
filter_upwards [ Measure.ae_ne volume 0 ] with x hx d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) n : ℕ x : Space d hx : x ≠ 0 ⊢ ‖ ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * normPowerSeries n x ^ m ‖ ≤
| ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) | * ( ( ‖ x ‖ + 1 ) ^ m + ‖ x ‖ ^ m )
simp [ abs_of_nonneg ( normPowerSeries_nonneg n x ) ] d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) n : ℕ x : Space d hx : x ≠ 0 ⊢ | ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) | * normPowerSeries n x ^ m ≤
| ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) | * ( ( ‖ x ‖ + 1 ) ^ m + ‖ x ‖ ^ m )
exact mul_le_mul_of_nonneg_left
( normPowerSeries_zpow_le_norm_sq_add_one n m x hx ) ( abs_nonneg _ ) All goals completed! 🐙
· h_lim d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ ∀ᵐ ( a : Space d ) ,
Filter.Tendsto (fun n => ( fderiv ℝ (⇑ η ) a ) ( basis . repr . symm y ) * normPowerSeries n a ^ m ) Filter.atTop
( 𝓝 ( ( fderiv ℝ (⇑ η ) a ) ( basis . repr . symm y ) * ‖ a ‖ ^ m ) ) filter_upwards [ Measure.ae_ne volume 0 ] with x hx d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) x : Space d hx : x ≠ 0 ⊢ Filter.Tendsto (fun n => ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * normPowerSeries n x ^ m ) Filter.atTop
( 𝓝 ( ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * ‖ x ‖ ^ m ) )
exact tendsto_const_nhds . mul
( ( normPowerSeries_tendsto x hx ) . zpow₀ m ( .inl ( norm_ne_zero_iff . mpr hx ) ) ) All goals completed! 🐙
lemma gradient_dist_normPowerSeries_zpow_tendsTo { d : ℕ } [ NeZero d ] ( m : ℤ )
( hm : - ( d - 1 : ℕ ) + 1 ≤ m )
( η : 𝓢( Space d , ℝ ) ) ( y : EuclideanSpace ℝ ( Fin d ) ) :
Filter.Tendsto ( fun n =>
⟪ ( ∇ᵈ ( distOfFunction ( fun x : Space d => ( normPowerSeries n x ) ^ m )
( by 𝕜 : Type E : Type F : Type F' : Type inst✝⁶ : RCLike 𝕜 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedAddCommGroup F' inst✝² : NormedSpace ℝ E inst✝¹ : NormedSpace ℝ F d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) n : ℕ ⊢ IsDistBounded fun x => normPowerSeries n x ^ m fun_prop All goals completed! 🐙 ) ) ) η , y ⟫_ ℝ )
Filter.atTop
( 𝓝 ( ⟪ distOfFunction ( fun x : Space d => ( m * ‖ x ‖ ^ ( m - 2 ) ) • basis . repr x ) ( by 𝕜 : Type E : Type F : Type F' : Type inst✝⁶ : RCLike 𝕜 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedAddCommGroup F' inst✝² : NormedSpace ℝ E inst✝¹ : NormedSpace ℝ F d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ IsDistBounded fun x => ( ↑ m * ‖ x ‖ ^ ( m - 2 ) ) • basis . repr x
simp [ ← smul_smul ] 𝕜 : Type E : Type F : Type F' : Type inst✝⁶ : RCLike 𝕜 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedAddCommGroup F' inst✝² : NormedSpace ℝ E inst✝¹ : NormedSpace ℝ F d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ IsDistBounded fun x => ↑ m • ‖ x ‖ ^ ( m - 2 ) • basis . repr x
refine IsDistBounded.const_fun_smul ? _ ↑ m 𝕜 : Type E : Type F : Type F' : Type inst✝⁶ : RCLike 𝕜 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedAddCommGroup F' inst✝² : NormedSpace ℝ E inst✝¹ : NormedSpace ℝ F d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ IsDistBounded fun x => ‖ x ‖ ^ ( m - 2 ) • basis . repr x
apply IsDistBounded.zpow_smul_repr_self 𝕜 : Type E : Type F : Type F' : Type inst✝⁶ : RCLike 𝕜 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedAddCommGroup F' inst✝² : NormedSpace ℝ E inst✝¹ : NormedSpace ℝ F d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ - ↑ ( d - 1 ) - 1 ≤ m - 2
omega All goals completed! 🐙 ) η , y ⟫_ ℝ ) ) := by d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Filter.Tendsto (fun n => ⟪ ( ∇ᵈ ( distOfFunction (fun x => normPowerSeries n x ^ m ) ⋯ ) ) η , y ⟫_ ℝ ) Filter.atTop
( 𝓝 ⟪ ( distOfFunction (fun x => ( ↑ m * ‖ x ‖ ^ ( m - 2 ) ) • basis . repr x ) ⋯ ) η , y ⟫_ ℝ )
simp only [ gradient_dist_normPowerSeries_zpow ] d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Filter.Tendsto (fun n => ⟪ ( distOfFunction (fun x => ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x ) ⋯ ) η , y ⟫_ ℝ )
Filter.atTop ( 𝓝 ⟪ ( distOfFunction (fun x => ( ↑ m * ‖ x ‖ ^ ( m - 2 ) ) • basis . repr x ) ⋯ ) η , y ⟫_ ℝ )
simp [ distOfFunction_inner ] d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Filter.Tendsto (fun n => ∫ ( x : Space d ) , η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ ) Filter.atTop
( 𝓝 ( ∫ ( x : Space d ) , η x * ⟪ ( ↑ m * ‖ x ‖ ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ ) )
have h1 ( n : ℕ ) ( x : Space d ) :
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , ( y ) ⟫_ ℝ =
η x * ( m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( normPowerSeries n x ) ^ ( m - 2 ) ) ) := by d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Filter.Tendsto (fun n => ⟪ ( ∇ᵈ ( distOfFunction (fun x => normPowerSeries n x ^ m ) ⋯ ) ) η , y ⟫_ ℝ ) Filter.atTop
( 𝓝 ⟪ ( distOfFunction (fun x => ( ↑ m * ‖ x ‖ ^ ( m - 2 ) ) • basis . repr x ) ⋯ ) η , y ⟫_ ℝ ) d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ Filter.Tendsto (fun n => ∫ ( x : Space d ) , η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ ) Filter.atTop
( 𝓝 ( ∫ ( x : Space d ) , η x * ⟪ ( ↑ m * ‖ x ‖ ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ ) )
rw [ real_inner_smul_left d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) n : ℕ x : Space d ⊢ η x * ( ↑ m * normPowerSeries n x ^ ( m - 2 ) * ⟪ basis . repr x , y ⟫_ ℝ ) =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) n : ℕ x : Space d ⊢ η x * ( ↑ m * normPowerSeries n x ^ ( m - 2 ) * ⟪ basis . repr x , y ⟫_ ℝ ) =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ Filter.Tendsto (fun n => ∫ ( x : Space d ) , η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ ) Filter.atTop
( 𝓝 ( ∫ ( x : Space d ) , η x * ⟪ ( ↑ m * ‖ x ‖ ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ ) ) ] d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) n : ℕ x : Space d ⊢ η x * ( ↑ m * normPowerSeries n x ^ ( m - 2 ) * ⟪ basis . repr x , y ⟫_ ℝ ) =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ Filter.Tendsto (fun n => ∫ ( x : Space d ) , η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ ) Filter.atTop
( 𝓝 ( ∫ ( x : Space d ) , η x * ⟪ ( ↑ m * ‖ x ‖ ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ ) )
ring d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ Filter.Tendsto (fun n => ∫ ( x : Space d ) , η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ ) Filter.atTop
( 𝓝 ( ∫ ( x : Space d ) , η x * ⟪ ( ↑ m * ‖ x ‖ ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ ) ) d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ Filter.Tendsto (fun n => ∫ ( x : Space d ) , η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ ) Filter.atTop
( 𝓝 ( ∫ ( x : Space d ) , η x * ⟪ ( ↑ m * ‖ x ‖ ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ ) )
simp only [ h1 ] d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ Filter.Tendsto (fun n => ∫ ( x : Space d ) , η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) )
Filter.atTop ( 𝓝 ( ∫ ( x : Space d ) , η x * ⟪ ( ↑ m * ‖ x ‖ ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ ) )
apply MeasureTheory.tendsto_integral_of_dominated_convergence
( bound := fun x => | η x | * | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) F_measurable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ ∀ ( n : ℕ ), AEStronglyMeasurable (fun a => η a * ( ↑ m * ( ⟪ basis . repr a , y ⟫_ ℝ * normPowerSeries n a ^ ( m - 2 ) ) ) ) volume bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume h_bound d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ ∀ ( n : ℕ ),
∀ᵐ ( a : Space d ) ,
‖ η a * ( ↑ m * ( ⟪ basis . repr a , y ⟫_ ℝ * normPowerSeries n a ^ ( m - 2 ) ) ) ‖ ≤
| η a | * ↑ | m | * | ⟪ basis . repr a , y ⟫_ ℝ | * ( ( ‖ a ‖ + 1 ) ^ ( m - 2 ) + ‖ a ‖ ^ ( m - 2 ) ) h_lim d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ ∀ᵐ ( a : Space d ) ,
Filter.Tendsto (fun n => η a * ( ↑ m * ( ⟪ basis . repr a , y ⟫_ ℝ * normPowerSeries n a ^ ( m - 2 ) ) ) ) Filter.atTop
( 𝓝 ( η a * ⟪ ( ↑ m * ‖ a ‖ ^ ( m - 2 ) ) • basis . repr a , y ⟫_ ℝ ) )
· F_measurable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ ∀ ( n : ℕ ), AEStronglyMeasurable (fun a => η a * ( ↑ m * ( ⟪ basis . repr a , y ⟫_ ℝ * normPowerSeries n a ^ ( m - 2 ) ) ) ) volume intro n F_measurable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) n : ℕ ⊢ AEStronglyMeasurable (fun a => η a * ( ↑ m * ( ⟪ basis . repr a , y ⟫_ ℝ * normPowerSeries n a ^ ( m - 2 ) ) ) ) volume
apply IsDistBounded.aeStronglyMeasurable_schwartzMap_smul ( F := ℝ ) ? _ η d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) n : ℕ ⊢ IsDistBounded fun x => ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) )
apply IsDistBounded.const_mul_fun d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) n : ℕ ⊢ IsDistBounded fun x => ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 )
simp [ basis_repr_inner_eq ] d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) n : ℕ ⊢ IsDistBounded fun x => ⟪ x , basis . repr . symm y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 )
exact IsDistBounded.isDistBounded_mul_inner' ( by d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) n : ℕ ⊢ IsDistBounded fun x => normPowerSeries n x ^ ( m - 2 ) fun_prop All goals completed! 🐙 ) _
· bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume have h1 : Integrable ( fun x =>
η x * ( m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume := by d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Filter.Tendsto (fun n => ⟪ ( ∇ᵈ ( distOfFunction (fun x => normPowerSeries n x ^ m ) ⋯ ) ) η , y ⟫_ ℝ ) Filter.atTop
( 𝓝 ⟪ ( distOfFunction (fun x => ( ↑ m * ‖ x ‖ ^ ( m - 2 ) ) • basis . repr x ) ⋯ ) η , y ⟫_ ℝ ) bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume
apply IsDistBounded.integrable_space_mul ? _ η d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ IsDistBounded fun x => ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume
apply IsDistBounded.const_mul_fun d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ IsDistBounded fun x => ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume
simp [ mul_add ] d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ IsDistBounded fun x => ⟪ basis . repr x , y ⟫_ ℝ * ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ⟪ basis . repr x , y ⟫_ ℝ * ‖ x ‖ ^ ( m - 2 ) bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume
apply IsDistBounded.add hf d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ IsDistBounded fun x => ⟪ basis . repr x , y ⟫_ ℝ * ( ‖ x ‖ + 1 ) ^ ( m - 2 ) hg d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ IsDistBounded fun x => ⟪ basis . repr x , y ⟫_ ℝ * ‖ x ‖ ^ ( m - 2 ) bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume
· hf d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ IsDistBounded fun x => ⟪ basis . repr x , y ⟫_ ℝ * ( ‖ x ‖ + 1 ) ^ ( m - 2 ) bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume simp [ basis_repr_inner_eq ] hf d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ IsDistBounded fun x => ⟪ x , basis . repr . symm y ⟫_ ℝ * ( ‖ x ‖ + 1 ) ^ ( m - 2 ) bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume
exact IsDistBounded.isDistBounded_mul_inner'
( IsDistBounded.norm_add_pos_nat_zpow ( m - 2 ) 1 one_pos ) _ All goals completed! 🐙 bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume
· hg d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ IsDistBounded fun x => ⟪ basis . repr x , y ⟫_ ℝ * ‖ x ‖ ^ ( m - 2 ) bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume simp [ basis_repr_inner_eq ] hg d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ IsDistBounded fun x => ⟪ x , basis . repr . symm y ⟫_ ℝ * ‖ x ‖ ^ ( m - 2 ) bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume
conv => d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) | IsDistBounded fun x => ⟪ x , basis . repr . symm y ⟫_ ℝ * ‖ x ‖ ^ ( m - 2 ) bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume
enter [ 1 , x ] d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) x : Space d | ⟪ x , basis . repr . symm y ⟫_ ℝ * ‖ x ‖ ^ ( m - 2 ) bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume
rw [ real_inner_comm ] d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) x : Space d | ⟪ basis . repr . symm y , x ⟫_ ℝ * ‖ x ‖ ^ ( m - 2 ) bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume
apply IsDistBounded.isDistBounded_mul_inner_of_smul_norm hg.hf d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ IsDistBounded fun x => ‖ x ‖ * ‖ x ‖ ^ ( m - 2 ) hg.hae d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ AEStronglyMeasurable (fun x => ‖ x ‖ ^ ( m - 2 ) ) volume bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume
· hg.hf d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ IsDistBounded fun x => ‖ x ‖ * ‖ x ‖ ^ ( m - 2 ) bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume apply IsDistBounded.mono ( f := fun x => ‖ x ‖ ^ ( m - 1 ) + 1 ) hg.hf.hf d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ IsDistBounded fun x => ‖ x ‖ ^ ( m - 1 ) + 1 hg.hf.hae d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ AEStronglyMeasurable (fun x => ‖ x ‖ * ‖ x ‖ ^ ( m - 2 ) ) volume hg.hf.hfg d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ ∀ ( x : Space d ), ‖ ‖ x ‖ * ‖ x ‖ ^ ( m - 2 ) ‖ ≤ ‖ ‖ x ‖ ^ ( m - 1 ) + 1 ‖ bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume
· hg.hf.hf d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ IsDistBounded fun x => ‖ x ‖ ^ ( m - 1 ) + 1 bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume exact ( IsDistBounded.pow ( m - 1 ) ( by d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ - ↑ ( d - 1 ) ≤ m - 1 bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume omega All goals completed! 🐙 bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume ) ) . add ( by d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ IsDistBounded fun x => 1 bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume fun_prop All goals completed! 🐙 bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume )
· hg.hf.hae d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ AEStronglyMeasurable (fun x => ‖ x ‖ * ‖ x ‖ ^ ( m - 2 ) ) volume bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume exact AEMeasurable.aestronglyMeasurable ( by d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ AEMeasurable (fun x => ‖ x ‖ * ‖ x ‖ ^ ( m - 2 ) ) volume bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume fun_prop All goals completed! 🐙 bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume )
· hg.hf.hfg d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ ∀ ( x : Space d ), ‖ ‖ x ‖ * ‖ x ‖ ^ ( m - 2 ) ‖ ≤ ‖ ‖ x ‖ ^ ( m - 1 ) + 1 ‖ bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume intro x hg.hf.hfg d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) x : Space d ⊢ ‖ ‖ x ‖ * ‖ x ‖ ^ ( m - 2 ) ‖ ≤ ‖ ‖ x ‖ ^ ( m - 1 ) + 1 ‖ bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume
simp only [ norm_mul , Real.norm_eq_abs , abs_norm , norm_zpow ] hg.hf.hfg d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) x : Space d ⊢ ‖ x ‖ * ‖ x ‖ ^ ( m - 2 ) ≤ | ‖ x ‖ ^ ( m - 1 ) + 1 | bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume
rw [ abs_of_nonneg ( by d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) x : Space d ⊢ 0 ≤ ‖ x ‖ ^ ( m - 1 ) + 1 hg.hf.hfg d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) x : Space d ⊢ ‖ x ‖ * ‖ x ‖ ^ ( m - 2 ) ≤ ‖ x ‖ ^ ( m - 1 ) + 1 bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume positivity All goals completed! 🐙 hg.hf.hfg d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) x : Space d ⊢ ‖ x ‖ * ‖ x ‖ ^ ( m - 2 ) ≤ ‖ x ‖ ^ ( m - 1 ) + 1 bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume ) ] hg.hf.hfg d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) x : Space d ⊢ ‖ x ‖ * ‖ x ‖ ^ ( m - 2 ) ≤ ‖ x ‖ ^ ( m - 1 ) + 1 bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume
by_cases hx : x = 0 pos d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) x : Space d hx : x = 0 ⊢ ‖ x ‖ * ‖ x ‖ ^ ( m - 2 ) ≤ ‖ x ‖ ^ ( m - 1 ) + 1 neg d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) x : Space d hx : ¬ x = 0 ⊢ ‖ x ‖ * ‖ x ‖ ^ ( m - 2 ) ≤ ‖ x ‖ ^ ( m - 1 ) + 1 bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume
· pos d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) x : Space d hx : x = 0 ⊢ ‖ x ‖ * ‖ x ‖ ^ ( m - 2 ) ≤ ‖ x ‖ ^ ( m - 1 ) + 1 bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume subst hx pos d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ ‖ 0 ‖ * ‖ 0 ‖ ^ ( m - 2 ) ≤ ‖ 0 ‖ ^ ( m - 1 ) + 1 bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume
simp [ zero_zpow_eq ] pos d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ 0 ≤ ( if m - 1 = 0 then 1 else 0 ) + 1 bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume
split_ifs pos d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h✝ : m - 1 = 0 ⊢ 0 ≤ 1 + 1 neg d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h✝ : ¬ m - 1 = 0 ⊢ 0 ≤ 0 + 1 bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume <;> pos d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h✝ : m - 1 = 0 ⊢ 0 ≤ 1 + 1 neg d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h✝ : ¬ m - 1 = 0 ⊢ 0 ≤ 0 + 1 bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume grind All goals completed! 🐙 bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume
· neg d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) x : Space d hx : ¬ x = 0 ⊢ ‖ x ‖ * ‖ x ‖ ^ ( m - 2 ) ≤ ‖ x ‖ ^ ( m - 1 ) + 1 bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume rw [ mul_comm , neg d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) x : Space d hx : ¬ x = 0 ⊢ ‖ x ‖ ^ ( m - 2 ) * ‖ x ‖ ≤ ‖ x ‖ ^ ( m - 1 ) + 1 neg d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) x : Space d hx : ¬ x = 0 ⊢ ‖ x ‖ ^ ( m - 1 ) ≤ ‖ x ‖ ^ ( m - 1 ) + 1 bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume ← zpow_add_one₀ ( norm_ne_zero_iff . mpr hx ) , neg d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) x : Space d hx : ¬ x = 0 ⊢ ‖ x ‖ ^ ( m - 2 + 1 ) ≤ ‖ x ‖ ^ ( m - 1 ) + 1 neg d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) x : Space d hx : ¬ x = 0 ⊢ ‖ x ‖ ^ ( m - 1 ) ≤ ‖ x ‖ ^ ( m - 1 ) + 1 bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume
show m - 2 + 1 = m - 1 by d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Filter.Tendsto (fun n => ⟪ ( ∇ᵈ ( distOfFunction (fun x => normPowerSeries n x ^ m ) ⋯ ) ) η , y ⟫_ ℝ ) Filter.atTop
( 𝓝 ⟪ ( distOfFunction (fun x => ( ↑ m * ‖ x ‖ ^ ( m - 2 ) ) • basis . repr x ) ⋯ ) η , y ⟫_ ℝ ) neg d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) x : Space d hx : ¬ x = 0 ⊢ ‖ x ‖ ^ ( m - 1 ) ≤ ‖ x ‖ ^ ( m - 1 ) + 1 bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume ring All goals completed! 🐙 neg d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) x : Space d hx : ¬ x = 0 ⊢ ‖ x ‖ ^ ( m - 1 ) ≤ ‖ x ‖ ^ ( m - 1 ) + 1 bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume ] neg d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) x : Space d hx : ¬ x = 0 ⊢ ‖ x ‖ ^ ( m - 1 ) ≤ ‖ x ‖ ^ ( m - 1 ) + 1 bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume
simp All goals completed! 🐙 bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume
· hg.hae d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ AEStronglyMeasurable (fun x => ‖ x ‖ ^ ( m - 2 ) ) volume bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume exact AEMeasurable.aestronglyMeasurable ( by d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ AEMeasurable (fun x => ‖ x ‖ ^ ( m - 2 ) ) volume bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume fun_prop All goals completed! 🐙 bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume ) bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume ⊢ Integrable (fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) volume
refine h1 . abs . congr ( ae_of_all _ fun x => ? _ ) bound_integrable d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) h1 : Integrable (fun x => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) ) ) volume x : Space d ⊢ (fun a => | η a * ( ↑ m * ( ⟪ basis . repr a , y ⟫_ ℝ * ( ( ‖ a ‖ + 1 ) ^ ( m - 2 ) + ‖ a ‖ ^ ( m - 2 ) ) ) ) | ) x =
(fun x => | η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) x
simp only [ abs_mul , mul_assoc , Int.cast_abs ,
abs_of_nonneg ( show ( 0 : ℝ ) ≤ ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) by positivity ) ] All goals completed! 🐙
· h_bound d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ ∀ ( n : ℕ ),
∀ᵐ ( a : Space d ) ,
‖ η a * ( ↑ m * ( ⟪ basis . repr a , y ⟫_ ℝ * normPowerSeries n a ^ ( m - 2 ) ) ) ‖ ≤
| η a | * ↑ | m | * | ⟪ basis . repr a , y ⟫_ ℝ | * ( ( ‖ a ‖ + 1 ) ^ ( m - 2 ) + ‖ a ‖ ^ ( m - 2 ) ) intro n h_bound d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) n : ℕ ⊢ ∀ᵐ ( a : Space d ) ,
‖ η a * ( ↑ m * ( ⟪ basis . repr a , y ⟫_ ℝ * normPowerSeries n a ^ ( m - 2 ) ) ) ‖ ≤
| η a | * ↑ | m | * | ⟪ basis . repr a , y ⟫_ ℝ | * ( ( ‖ a ‖ + 1 ) ^ ( m - 2 ) + ‖ a ‖ ^ ( m - 2 ) )
filter_upwards [ Measure.ae_ne volume 0 ] with x hx d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) n : ℕ x : Space d hx : x ≠ 0 ⊢ ‖ η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ‖ ≤
| η x | * ↑ | m | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) )
simp [ mul_assoc ] d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) n : ℕ x : Space d hx : x ≠ 0 ⊢ | η x | * ( | ↑ m | * ( | ⟪ basis . repr x , y ⟫_ ℝ | * | normPowerSeries n x | ^ ( m - 2 ) ) ) ≤
| η x | * ( | ↑ m | * ( | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ) )
gcongr hbc d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) n : ℕ x : Space d hx : x ≠ 0 ⊢ | normPowerSeries n x | ^ ( m - 2 ) ≤ ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 )
rw [ abs_of_nonneg ( by d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) n : ℕ x : Space d hx : x ≠ 0 ⊢ 0 ≤ normPowerSeries n x hbc d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) n : ℕ x : Space d hx : x ≠ 0 ⊢ normPowerSeries n x ^ ( m - 2 ) ≤ ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) simp All goals completed! 🐙 hbc d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) n : ℕ x : Space d hx : x ≠ 0 ⊢ normPowerSeries n x ^ ( m - 2 ) ≤ ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 ) ) ] hbc d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) n : ℕ x : Space d hx : x ≠ 0 ⊢ normPowerSeries n x ^ ( m - 2 ) ≤ ( ‖ x ‖ + 1 ) ^ ( m - 2 ) + ‖ x ‖ ^ ( m - 2 )
exact normPowerSeries_zpow_le_norm_sq_add_one n ( m - 2 ) x hx All goals completed! 🐙
· h_lim d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ⊢ ∀ᵐ ( a : Space d ) ,
Filter.Tendsto (fun n => η a * ( ↑ m * ( ⟪ basis . repr a , y ⟫_ ℝ * normPowerSeries n a ^ ( m - 2 ) ) ) ) Filter.atTop
( 𝓝 ( η a * ⟪ ( ↑ m * ‖ a ‖ ^ ( m - 2 ) ) • basis . repr a , y ⟫_ ℝ ) ) filter_upwards [ Measure.ae_ne volume 0 ] with x hx d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) x : Space d hx : x ≠ 0 ⊢ Filter.Tendsto (fun n => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ) Filter.atTop
( 𝓝 ( η x * ⟪ ( ↑ m * ‖ x ‖ ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ ) )
have h2 : ⟪ ( ( m : ℝ ) * ‖ x ‖ ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
( m : ℝ ) * ( ⟪ basis . repr x , y ⟫_ ℝ * ‖ x ‖ ^ ( m - 2 ) ) := by d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Filter.Tendsto (fun n => ⟪ ( ∇ᵈ ( distOfFunction (fun x => normPowerSeries n x ^ m ) ⋯ ) ) η , y ⟫_ ℝ ) Filter.atTop
( 𝓝 ⟪ ( distOfFunction (fun x => ( ↑ m * ‖ x ‖ ^ ( m - 2 ) ) • basis . repr x ) ⋯ ) η , y ⟫_ ℝ ) d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) x : Space d hx : x ≠ 0 h2 : ⟪ ( ↑ m * ‖ x ‖ ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ = ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ‖ x ‖ ^ ( m - 2 ) ) ⊢ Filter.Tendsto (fun n => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ) Filter.atTop
( 𝓝 ( η x * ⟪ ( ↑ m * ‖ x ‖ ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ ) )
rw [ real_inner_smul_left d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) x : Space d hx : x ≠ 0 ⊢ ↑ m * ‖ x ‖ ^ ( m - 2 ) * ⟪ basis . repr x , y ⟫_ ℝ = ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ‖ x ‖ ^ ( m - 2 ) ) d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) x : Space d hx : x ≠ 0 ⊢ ↑ m * ‖ x ‖ ^ ( m - 2 ) * ⟪ basis . repr x , y ⟫_ ℝ = ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ‖ x ‖ ^ ( m - 2 ) ) d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) x : Space d hx : x ≠ 0 h2 : ⟪ ( ↑ m * ‖ x ‖ ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ = ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ‖ x ‖ ^ ( m - 2 ) ) ⊢ Filter.Tendsto (fun n => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ) Filter.atTop
( 𝓝 ( η x * ⟪ ( ↑ m * ‖ x ‖ ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ ) ) ] d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) x : Space d hx : x ≠ 0 ⊢ ↑ m * ‖ x ‖ ^ ( m - 2 ) * ⟪ basis . repr x , y ⟫_ ℝ = ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ‖ x ‖ ^ ( m - 2 ) ) d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) x : Space d hx : x ≠ 0 h2 : ⟪ ( ↑ m * ‖ x ‖ ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ = ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ‖ x ‖ ^ ( m - 2 ) ) ⊢ Filter.Tendsto (fun n => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ) Filter.atTop
( 𝓝 ( η x * ⟪ ( ↑ m * ‖ x ‖ ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ ) )
ring d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) x : Space d hx : x ≠ 0 h2 : ⟪ ( ↑ m * ‖ x ‖ ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ = ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ‖ x ‖ ^ ( m - 2 ) ) ⊢ Filter.Tendsto (fun n => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ) Filter.atTop
( 𝓝 ( η x * ⟪ ( ↑ m * ‖ x ‖ ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ ) ) d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) x : Space d hx : x ≠ 0 h2 : ⟪ ( ↑ m * ‖ x ‖ ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ = ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ‖ x ‖ ^ ( m - 2 ) ) ⊢ Filter.Tendsto (fun n => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ) Filter.atTop
( 𝓝 ( η x * ⟪ ( ↑ m * ‖ x ‖ ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ ) )
rw [ h2 d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) x : Space d hx : x ≠ 0 h2 : ⟪ ( ↑ m * ‖ x ‖ ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ = ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ‖ x ‖ ^ ( m - 2 ) ) ⊢ Filter.Tendsto (fun n => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ) Filter.atTop
( 𝓝 ( η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ‖ x ‖ ^ ( m - 2 ) ) ) ) ) d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) x : Space d hx : x ≠ 0 h2 : ⟪ ( ↑ m * ‖ x ‖ ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ = ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ‖ x ‖ ^ ( m - 2 ) ) ⊢ Filter.Tendsto (fun n => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ) Filter.atTop
( 𝓝 ( η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ‖ x ‖ ^ ( m - 2 ) ) ) ) ) ] d : ℕ inst✝ : NeZero d m : ℤ hm : - ↑ ( d - 1 ) + 1 ≤ m η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ ( ↑ m * normPowerSeries n x ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) x : Space d hx : x ≠ 0 h2 : ⟪ ( ↑ m * ‖ x ‖ ^ ( m - 2 ) ) • basis . repr x , y ⟫_ ℝ = ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ‖ x ‖ ^ ( m - 2 ) ) ⊢ Filter.Tendsto (fun n => η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ ( m - 2 ) ) ) ) Filter.atTop
( 𝓝 ( η x * ( ↑ m * ( ⟪ basis . repr x , y ⟫_ ℝ * ‖ x ‖ ^ ( m - 2 ) ) ) ) )
exact tendsto_const_nhds . mul ( tendsto_const_nhds . mul ( tendsto_const_nhds . mul
( ( normPowerSeries_tendsto x hx ) . zpow₀ _ ( .inl ( norm_ne_zero_iff . mpr hx ) ) ) ) ) All goals completed! 🐙
A.11. Gradients of distributions based on logs
lemma gradient_dist_normPowerSeries_log { d : ℕ } { n : ℕ } :
∇ᵈ ( distOfFunction ( fun x : Space d => Real.log ( normPowerSeries n x ) ) ( by 𝕜 : Type E : Type F : Type F' : Type inst✝⁵ : RCLike 𝕜 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedAddCommGroup F inst✝² : NormedAddCommGroup F' inst✝¹ : NormedSpace ℝ E inst✝ : NormedSpace ℝ F d : ℕ n : ℕ ⊢ IsDistBounded fun x => Real.log ( normPowerSeries n x ) fun_prop All goals completed! 🐙 ) ) =
distOfFunction ( fun x : Space d => ( ( normPowerSeries n x ) ^ ( - 2 : ℤ ) ) • basis . repr x )
( by 𝕜 : Type E : Type F : Type F' : Type inst✝⁵ : RCLike 𝕜 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedAddCommGroup F inst✝² : NormedAddCommGroup F' inst✝¹ : NormedSpace ℝ E inst✝ : NormedSpace ℝ F d : ℕ n : ℕ ⊢ IsDistBounded fun x => normPowerSeries n x ^ (- 2 ) • basis . repr x fun_prop All goals completed! 🐙 ) := by d : ℕ n : ℕ ⊢ ∇ᵈ ( distOfFunction (fun x => Real.log ( normPowerSeries n x ) ) ⋯ ) =
distOfFunction (fun x => normPowerSeries n x ^ (- 2 ) • basis . repr x ) ⋯
ext1 η d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) ⊢ ( ∇ᵈ ( distOfFunction (fun x => Real.log ( normPowerSeries n x ) ) ⋯ ) ) η =
( distOfFunction (fun x => normPowerSeries n x ^ (- 2 ) • basis . repr x ) ⋯ ) η
refine ext_inner_right ℝ fun y => ? _ d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ ⟪ ( ∇ᵈ ( distOfFunction (fun x => Real.log ( normPowerSeries n x ) ) ⋯ ) ) η , y ⟫_ ℝ =
⟪ ( distOfFunction (fun x => normPowerSeries n x ^ (- 2 ) • basis . repr x ) ⋯ ) η , y ⟫_ ℝ
simp [ distGrad_inner_eq ] d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ ( ( ( Distribution.fderivD ℝ ) ( distOfFunction (fun x => Real.log ( normPowerSeries n x ) ) ⋯ ) ) η ) ( basis . repr . symm y ) =
⟪ ( distOfFunction (fun x => ( normPowerSeries n x ^ 2 ) ⁻¹ • basis . repr x ) ⋯ ) η , y ⟫_ ℝ
rw [ Distribution.fderivD_apply , d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ - ( distOfFunction (fun x => Real.log ( normPowerSeries n x ) ) ⋯ )
( ( SchwartzMap.evalCLM ℝ ( Space d ) ℝ ( basis . repr . symm y ) ) ( ( fderivCLM ℝ ( Space d ) ℝ ) η ) ) =
⟪ ( distOfFunction (fun x => ( normPowerSeries n x ^ 2 ) ⁻¹ • basis . repr x ) ⋯ ) η , y ⟫_ ℝ d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ - ∫ ( x : Space d ) ,
( ( SchwartzMap.evalCLM ℝ ( Space d ) ℝ ( basis . repr . symm y ) ) ( ( fderivCLM ℝ ( Space d ) ℝ ) η ) ) x •
Real.log ( normPowerSeries n x ) =
∫ ( x : Space d ) , η x * ⟪ ( normPowerSeries n x ^ 2 ) ⁻¹ • basis . repr x , y ⟫_ ℝ distOfFunction_apply , d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ - ∫ ( x : Space d ) ,
( ( SchwartzMap.evalCLM ℝ ( Space d ) ℝ ( basis . repr . symm y ) ) ( ( fderivCLM ℝ ( Space d ) ℝ ) η ) ) x •
Real.log ( normPowerSeries n x ) =
⟪ ( distOfFunction (fun x => ( normPowerSeries n x ^ 2 ) ⁻¹ • basis . repr x ) ⋯ ) η , y ⟫_ ℝ d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ - ∫ ( x : Space d ) ,
( ( SchwartzMap.evalCLM ℝ ( Space d ) ℝ ( basis . repr . symm y ) ) ( ( fderivCLM ℝ ( Space d ) ℝ ) η ) ) x •
Real.log ( normPowerSeries n x ) =
∫ ( x : Space d ) , η x * ⟪ ( normPowerSeries n x ^ 2 ) ⁻¹ • basis . repr x , y ⟫_ ℝ distOfFunction_inner d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ - ∫ ( x : Space d ) ,
( ( SchwartzMap.evalCLM ℝ ( Space d ) ℝ ( basis . repr . symm y ) ) ( ( fderivCLM ℝ ( Space d ) ℝ ) η ) ) x •
Real.log ( normPowerSeries n x ) =
∫ ( x : Space d ) , η x * ⟪ ( normPowerSeries n x ^ 2 ) ⁻¹ • basis . repr x , y ⟫_ ℝ d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ - ∫ ( x : Space d ) ,
( ( SchwartzMap.evalCLM ℝ ( Space d ) ℝ ( basis . repr . symm y ) ) ( ( fderivCLM ℝ ( Space d ) ℝ ) η ) ) x •
Real.log ( normPowerSeries n x ) =
∫ ( x : Space d ) , η x * ⟪ ( normPowerSeries n x ^ 2 ) ⁻¹ • basis . repr x , y ⟫_ ℝ ] d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ - ∫ ( x : Space d ) ,
( ( SchwartzMap.evalCLM ℝ ( Space d ) ℝ ( basis . repr . symm y ) ) ( ( fderivCLM ℝ ( Space d ) ℝ ) η ) ) x •
Real.log ( normPowerSeries n x ) =
∫ ( x : Space d ) , η x * ⟪ ( normPowerSeries n x ^ 2 ) ⁻¹ • basis . repr x , y ⟫_ ℝ
calc _
_ = - ∫ ( x : Space d ) , fderiv ℝ η x ( basis . repr . symm y ) * Real.log ( normPowerSeries n x ) := by d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ - ∫ ( x : Space d ) ,
( ( SchwartzMap.evalCLM ℝ ( Space d ) ℝ ( basis . repr . symm y ) ) ( ( fderivCLM ℝ ( Space d ) ℝ ) η ) ) x •
Real.log ( normPowerSeries n x ) =
- ∫ ( x : Space d ) , ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * Real.log ( normPowerSeries n x )
rfl All goals completed! 🐙
_ = ∫ ( x : Space d ) , η x *
fderiv ℝ ( fun x => Real.log ( normPowerSeries n x ) ) x ( basis . repr . symm y ) := by d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ - ∫ ( x : Space d ) , ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * Real.log ( normPowerSeries n x ) =
∫ ( x : Space d ) , η x * ( fderiv ℝ (fun x => Real.log ( normPowerSeries n x ) ) x ) ( basis . repr . symm y )
rw [ integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ - ∫ ( x : Space d ) , ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * Real.log ( normPowerSeries n x ) =
- ∫ ( x : Space d ) , ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * Real.log ( normPowerSeries n x ) hf'g d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Integrable (fun x => ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * Real.log ( normPowerSeries n x ) ) volume hfg' d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Integrable (fun x => η x * ( fderiv ℝ (fun x => Real.log ( normPowerSeries n x ) ) x ) ( basis . repr . symm y ) ) volume hfg d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Integrable (fun x => η x * Real.log ( normPowerSeries n x ) ) volume hf d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ ∀ x ∈ tsupport fun x => Real.log ( normPowerSeries n x ) , DifferentiableAt ℝ (⇑ η ) x hg d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ ∀ x ∈ tsupport ⇑ η , DifferentiableAt ℝ (fun x => Real.log ( normPowerSeries n x ) ) x hf'g d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Integrable (fun x => ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * Real.log ( normPowerSeries n x ) ) volume hfg' d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Integrable (fun x => η x * ( fderiv ℝ (fun x => Real.log ( normPowerSeries n x ) ) x ) ( basis . repr . symm y ) ) volume hfg d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Integrable (fun x => η x * Real.log ( normPowerSeries n x ) ) volume hf d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ ∀ x ∈ tsupport fun x => Real.log ( normPowerSeries n x ) , DifferentiableAt ℝ (⇑ η ) x hg d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ ∀ x ∈ tsupport ⇑ η , DifferentiableAt ℝ (fun x => Real.log ( normPowerSeries n x ) ) x ] hf'g d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Integrable (fun x => ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * Real.log ( normPowerSeries n x ) ) volume hfg' d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Integrable (fun x => η x * ( fderiv ℝ (fun x => Real.log ( normPowerSeries n x ) ) x ) ( basis . repr . symm y ) ) volume hfg d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Integrable (fun x => η x * Real.log ( normPowerSeries n x ) ) volume hf d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ ∀ x ∈ tsupport fun x => Real.log ( normPowerSeries n x ) , DifferentiableAt ℝ (⇑ η ) x hg d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ ∀ x ∈ tsupport ⇑ η , DifferentiableAt ℝ (fun x => Real.log ( normPowerSeries n x ) ) x
· hf'g d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Integrable (fun x => ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * Real.log ( normPowerSeries n x ) ) volume fun_prop All goals completed! 🐙
· hfg' d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Integrable (fun x => η x * ( fderiv ℝ (fun x => Real.log ( normPowerSeries n x ) ) x ) ( basis . repr . symm y ) ) volume refine IsDistBounded.integrable_space_mul ? _ η hfg' d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ IsDistBounded fun x => ( fderiv ℝ (fun x => Real.log ( normPowerSeries n x ) ) x ) ( basis . repr . symm y )
simp only [ fderiv_log_normPowerSeries ] hfg' d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ IsDistBounded fun x => ⟪ basis . repr . symm y , x ⟫_ ℝ * normPowerSeries n x ^ (- 2 )
fun_prop All goals completed! 🐙
· hfg d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Integrable (fun x => η x * Real.log ( normPowerSeries n x ) ) volume fun_prop All goals completed! 🐙
· hf d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ ∀ x ∈ tsupport fun x => Real.log ( normPowerSeries n x ) , DifferentiableAt ℝ (⇑ η ) x fun_prop All goals completed! 🐙
· hg d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ ∀ x ∈ tsupport ⇑ η , DifferentiableAt ℝ (fun x => Real.log ( normPowerSeries n x ) ) x exact fun _ _ => Differentiable.differentiableAt ( by d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) x✝¹ : Space d x✝ : x✝¹ ∈ tsupport ⇑ η ⊢ Differentiable ℝ fun x => Real.log ( normPowerSeries n x ) fun_prop All goals completed! 🐙 )
_ = ∫ ( x : Space d ) , η x * ( ⟪ basis . repr . symm y , x ⟫_ ℝ * ( normPowerSeries n x ) ^ ( - 2 : ℤ ) ) := by d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ ∫ ( x : Space d ) , η x * ( fderiv ℝ (fun x => Real.log ( normPowerSeries n x ) ) x ) ( basis . repr . symm y ) =
∫ ( x : Space d ) , η x * ( ⟪ basis . repr . symm y , x ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) )
simp only [ fderiv_log_normPowerSeries ] All goals completed! 🐙
congr calc.step.e_f d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ (fun x => η x * ( ⟪ basis . repr . symm y , x ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ) = fun x =>
η x * ⟪ ( normPowerSeries n x ^ 2 ) ⁻¹ • basis . repr x , y ⟫_ ℝ
funext x calc.step.e_f d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) x : Space d ⊢ η x * ( ⟪ basis . repr . symm y , x ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) = η x * ⟪ ( normPowerSeries n x ^ 2 ) ⁻¹ • basis . repr x , y ⟫_ ℝ
simp [ inner_smul_left_eq_smul ] calc.step.e_f d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) x : Space d ⊢ ⟪ basis . repr . symm y , x ⟫_ ℝ * ( normPowerSeries n x ^ 2 ) ⁻¹ = ( normPowerSeries n x ^ 2 ) ⁻¹ * ⟪ basis . repr x , y ⟫_ ℝ ∨ η x = 0
left calc.step.e_f d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) x : Space d ⊢ ⟪ basis . repr . symm y , x ⟫_ ℝ * ( normPowerSeries n x ^ 2 ) ⁻¹ = ( normPowerSeries n x ^ 2 ) ⁻¹ * ⟪ basis . repr x , y ⟫_ ℝ
rw [ real_inner_comm , calc.step.e_f d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) x : Space d ⊢ ⟪ x , basis . repr . symm y ⟫_ ℝ * ( normPowerSeries n x ^ 2 ) ⁻¹ = ( normPowerSeries n x ^ 2 ) ⁻¹ * ⟪ basis . repr x , y ⟫_ ℝ calc.step.e_f d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) x : Space d ⊢ ⟪ x , basis . repr . symm y ⟫_ ℝ * ( normPowerSeries n x ^ 2 ) ⁻¹ = ( normPowerSeries n x ^ 2 ) ⁻¹ * ⟪ x , basis . repr . symm y ⟫_ ℝ basis_repr_inner_eq calc.step.e_f d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) x : Space d ⊢ ⟪ x , basis . repr . symm y ⟫_ ℝ * ( normPowerSeries n x ^ 2 ) ⁻¹ = ( normPowerSeries n x ^ 2 ) ⁻¹ * ⟪ x , basis . repr . symm y ⟫_ ℝ calc.step.e_f d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) x : Space d ⊢ ⟪ x , basis . repr . symm y ⟫_ ℝ * ( normPowerSeries n x ^ 2 ) ⁻¹ = ( normPowerSeries n x ^ 2 ) ⁻¹ * ⟪ x , basis . repr . symm y ⟫_ ℝ ] calc.step.e_f d : ℕ n : ℕ η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) x : Space d ⊢ ⟪ x , basis . repr . symm y ⟫_ ℝ * ( normPowerSeries n x ^ 2 ) ⁻¹ = ( normPowerSeries n x ^ 2 ) ⁻¹ * ⟪ x , basis . repr . symm y ⟫_ ℝ
ring All goals completed! 🐙
A.11.1. The limits of gradients of distributions based on logs
lemma gradient_dist_normPowerSeries_log_tendsTo_distGrad_norm { d : ℕ } ( hd : 2 ≤ d )
( η : 𝓢( Space d , ℝ ) ) ( y : EuclideanSpace ℝ ( Fin d ) ) :
Filter.Tendsto ( fun n =>
⟪ ( ∇ᵈ ( distOfFunction
( fun x : Space d => Real.log ( normPowerSeries n x ) ) ( by 𝕜 : Type E : Type F : Type F' : Type inst✝⁵ : RCLike 𝕜 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedAddCommGroup F inst✝² : NormedAddCommGroup F' inst✝¹ : NormedSpace ℝ E inst✝ : NormedSpace ℝ F d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) n : ℕ ⊢ IsDistBounded fun x => Real.log ( normPowerSeries n x ) fun_prop All goals completed! 🐙 ) ) ) η , y ⟫_ ℝ )
Filter.atTop
( 𝓝 ( ⟪ ∇ᵈ ( distOfFunction ( fun x : Space d => Real.log ‖ x ‖ )
( IsDistBounded.log_norm ) ) η , y ⟫_ ℝ ) ) := by d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Filter.Tendsto (fun n => ⟪ ( ∇ᵈ ( distOfFunction (fun x => Real.log ( normPowerSeries n x ) ) ⋯ ) ) η , y ⟫_ ℝ ) Filter.atTop
( 𝓝 ⟪ ( ∇ᵈ ( distOfFunction (fun x => Real.log ‖ x ‖ ) ⋯ ) ) η , y ⟫_ ℝ )
haveI : NeZero d := ⟨ by d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ d ≠ 0 omega All goals completed! 🐙 ⟩
simp only [ distGrad_inner_eq , Distribution.fderivD_apply , distOfFunction_apply ] d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d ⊢ Filter.Tendsto
(fun n =>
- ∫ ( x : Space d ) ,
( ( SchwartzMap.evalCLM ℝ ( Space d ) ℝ ( basis . repr . symm y ) ) ( ( fderivCLM ℝ ( Space d ) ℝ ) η ) ) x •
Real.log ( normPowerSeries n x ) )
Filter.atTop
( 𝓝
( - ∫ ( x : Space d ) ,
( ( SchwartzMap.evalCLM ℝ ( Space d ) ℝ ( basis . repr . symm y ) ) ( ( fderivCLM ℝ ( Space d ) ℝ ) η ) ) x • Real.log ‖ x ‖ ) )
change Filter.Tendsto ( fun n => -
∫ ( x : Space d ) , fderiv ℝ η x ( basis . repr . symm y ) * Real.log ( normPowerSeries n x ) )
Filter.atTop ( 𝓝 ( - ∫ ( x : Space d ) , fderiv ℝ η x ( basis . repr . symm y ) * Real.log ‖ x ‖ ) ) d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d ⊢ Filter.Tendsto (fun n => - ∫ ( x : Space d ) , ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * Real.log ( normPowerSeries n x ) )
Filter.atTop ( 𝓝 ( - ∫ ( x : Space d ) , ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * Real.log ‖ x ‖ ) )
apply Filter.Tendsto.neg d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d ⊢ Filter.Tendsto (fun x => ∫ ( x_1 : Space d ) , ( fderiv ℝ (⇑ η ) x_1 ) ( basis . repr . symm y ) * Real.log ( normPowerSeries x x_1 ) )
Filter.atTop ( 𝓝 ( ∫ ( x : Space d ) , ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * Real.log ‖ x ‖ ) )
apply MeasureTheory.tendsto_integral_of_dominated_convergence
( bound := fun x => | fderiv ℝ η x ( basis . repr . symm y ) | * ( ‖ x ‖ ⁻¹ + ( ‖ x ‖ + 1 ) ) ) F_measurable d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d ⊢ ∀ ( n : ℕ ), AEStronglyMeasurable (fun a => ( fderiv ℝ (⇑ η ) a ) ( basis . repr . symm y ) * Real.log ( normPowerSeries n a ) ) volume bound_integrable d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d ⊢ Integrable (fun x => | ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) | * ( ‖ x ‖ ⁻¹ + ( ‖ x ‖ + 1 ) ) ) volume h_bound d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d ⊢ ∀ ( n : ℕ ),
∀ᵐ ( a : Space d ) ,
‖ ( fderiv ℝ (⇑ η ) a ) ( basis . repr . symm y ) * Real.log ( normPowerSeries n a ) ‖ ≤
| ( fderiv ℝ (⇑ η ) a ) ( basis . repr . symm y ) | * ( ‖ a ‖ ⁻¹ + ( ‖ a ‖ + 1 ) ) h_lim d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d ⊢ ∀ᵐ ( a : Space d ) ,
Filter.Tendsto (fun n => ( fderiv ℝ (⇑ η ) a ) ( basis . repr . symm y ) * Real.log ( normPowerSeries n a ) ) Filter.atTop
( 𝓝 ( ( fderiv ℝ (⇑ η ) a ) ( basis . repr . symm y ) * Real.log ‖ a ‖ ) )
· F_measurable d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d ⊢ ∀ ( n : ℕ ), AEStronglyMeasurable (fun a => ( fderiv ℝ (⇑ η ) a ) ( basis . repr . symm y ) * Real.log ( normPowerSeries n a ) ) volume intro n F_measurable d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d n : ℕ ⊢ AEStronglyMeasurable (fun a => ( fderiv ℝ (⇑ η ) a ) ( basis . repr . symm y ) * Real.log ( normPowerSeries n a ) ) volume
exact IsDistBounded.aeStronglyMeasurable_fderiv_schwartzMap_smul ( F := ℝ ) ( by d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d n : ℕ ⊢ IsDistBounded fun a => Real.log ( normPowerSeries n a ) fun_prop All goals completed! 🐙 ) η _
· bound_integrable d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d ⊢ Integrable (fun x => | ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) | * ( ‖ x ‖ ⁻¹ + ( ‖ x ‖ + 1 ) ) ) volume have h1 : Integrable ( fun x => ( fderiv ℝ ( ⇑ η ) x ) ( basis . repr . symm y ) *
( ‖ x ‖ ⁻¹ + ( ‖ x ‖ + 1 ) ) ) volume := by d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Filter.Tendsto (fun n => ⟪ ( ∇ᵈ ( distOfFunction (fun x => Real.log ( normPowerSeries n x ) ) ⋯ ) ) η , y ⟫_ ℝ ) Filter.atTop
( 𝓝 ⟪ ( ∇ᵈ ( distOfFunction (fun x => Real.log ‖ x ‖ ) ⋯ ) ) η , y ⟫_ ℝ ) bound_integrable d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : Integrable (fun x => ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * ( ‖ x ‖ ⁻¹ + ( ‖ x ‖ + 1 ) ) ) volume ⊢ Integrable (fun x => | ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) | * ( ‖ x ‖ ⁻¹ + ( ‖ x ‖ + 1 ) ) ) volume
apply IsDistBounded.integrable_space_fderiv
( IsDistBounded.add IsDistBounded.inv ( by d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d ⊢ IsDistBounded fun x => ‖ x ‖ + 1 bound_integrable d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : Integrable (fun x => ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * ( ‖ x ‖ ⁻¹ + ( ‖ x ‖ + 1 ) ) ) volume ⊢ Integrable (fun x => | ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) | * ( ‖ x ‖ ⁻¹ + ( ‖ x ‖ + 1 ) ) ) volume fun_prop All goals completed! 🐙 bound_integrable d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : Integrable (fun x => ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * ( ‖ x ‖ ⁻¹ + ( ‖ x ‖ + 1 ) ) ) volume ⊢ Integrable (fun x => | ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) | * ( ‖ x ‖ ⁻¹ + ( ‖ x ‖ + 1 ) ) ) volume ) ) bound_integrable d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : Integrable (fun x => ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * ( ‖ x ‖ ⁻¹ + ( ‖ x ‖ + 1 ) ) ) volume ⊢ Integrable (fun x => | ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) | * ( ‖ x ‖ ⁻¹ + ( ‖ x ‖ + 1 ) ) ) volume
refine h1 . abs . congr ( ae_of_all _ fun x => ? _ ) bound_integrable d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : Integrable (fun x => ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * ( ‖ x ‖ ⁻¹ + ( ‖ x ‖ + 1 ) ) ) volume x : Space d ⊢ (fun a => | ( fderiv ℝ (⇑ η ) a ) ( basis . repr . symm y ) * ( ‖ a ‖ ⁻¹ + ( ‖ a ‖ + 1 ) ) | ) x =
(fun x => | ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) | * ( ‖ x ‖ ⁻¹ + ( ‖ x ‖ + 1 ) ) ) x
simp only [ abs_mul ] bound_integrable d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : Integrable (fun x => ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * ( ‖ x ‖ ⁻¹ + ( ‖ x ‖ + 1 ) ) ) volume x : Space d ⊢ | ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) | * | ‖ x ‖ ⁻¹ + ( ‖ x ‖ + 1 ) | =
| ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) | * ( ‖ x ‖ ⁻¹ + ( ‖ x ‖ + 1 ) )
congr 1 bound_integrable.e_a d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : Integrable (fun x => ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * ( ‖ x ‖ ⁻¹ + ( ‖ x ‖ + 1 ) ) ) volume x : Space d ⊢ | ‖ x ‖ ⁻¹ + ( ‖ x ‖ + 1 ) | = ‖ x ‖ ⁻¹ + ( ‖ x ‖ + 1 )
exact abs_of_nonneg ( by d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : Integrable (fun x => ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * ( ‖ x ‖ ⁻¹ + ( ‖ x ‖ + 1 ) ) ) volume x : Space d ⊢ 0 ≤ ‖ x ‖ ⁻¹ + ( ‖ x ‖ + 1 ) positivity All goals completed! 🐙 )
· h_bound d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d ⊢ ∀ ( n : ℕ ),
∀ᵐ ( a : Space d ) ,
‖ ( fderiv ℝ (⇑ η ) a ) ( basis . repr . symm y ) * Real.log ( normPowerSeries n a ) ‖ ≤
| ( fderiv ℝ (⇑ η ) a ) ( basis . repr . symm y ) | * ( ‖ a ‖ ⁻¹ + ( ‖ a ‖ + 1 ) ) intro n h_bound d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d n : ℕ ⊢ ∀ᵐ ( a : Space d ) ,
‖ ( fderiv ℝ (⇑ η ) a ) ( basis . repr . symm y ) * Real.log ( normPowerSeries n a ) ‖ ≤
| ( fderiv ℝ (⇑ η ) a ) ( basis . repr . symm y ) | * ( ‖ a ‖ ⁻¹ + ( ‖ a ‖ + 1 ) )
filter_upwards [ Measure.ae_ne volume 0 ] with x hx d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d n : ℕ x : Space d hx : x ≠ 0 ⊢ ‖ ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * Real.log ( normPowerSeries n x ) ‖ ≤
| ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) | * ( ‖ x ‖ ⁻¹ + ( ‖ x ‖ + 1 ) )
simp only [ norm_mul , Real.norm_eq_abs ] d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d n : ℕ x : Space d hx : x ≠ 0 ⊢ | ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) | * | Real.log ( normPowerSeries n x ) | ≤
| ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) | * ( ‖ x ‖ ⁻¹ + ( ‖ x ‖ + 1 ) )
exact mul_le_mul_of_nonneg_left ( normPowerSeries_log_le n x hx ) ( abs_nonneg _ ) All goals completed! 🐙
· h_lim d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d ⊢ ∀ᵐ ( a : Space d ) ,
Filter.Tendsto (fun n => ( fderiv ℝ (⇑ η ) a ) ( basis . repr . symm y ) * Real.log ( normPowerSeries n a ) ) Filter.atTop
( 𝓝 ( ( fderiv ℝ (⇑ η ) a ) ( basis . repr . symm y ) * Real.log ‖ a ‖ ) ) filter_upwards [ Measure.ae_ne volume 0 ] with x hx d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d x : Space d hx : x ≠ 0 ⊢ Filter.Tendsto (fun n => ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * Real.log ( normPowerSeries n x ) ) Filter.atTop
( 𝓝 ( ( fderiv ℝ (⇑ η ) x ) ( basis . repr . symm y ) * Real.log ‖ x ‖ ) )
exact tendsto_const_nhds . mul
( ( normPowerSeries_tendsto x hx ) . log ( norm_ne_zero_iff . mpr hx ) ) All goals completed! 🐙
lemma gradient_dist_normPowerSeries_log_tendsTo { d : ℕ } ( hd : 2 ≤ d )
( η : 𝓢( Space d , ℝ ) ) ( y : EuclideanSpace ℝ ( Fin d ) ) :
Filter.Tendsto ( fun n =>
⟪ ( ∇ᵈ ( distOfFunction ( fun x : Space d => Real.log ( normPowerSeries n x ) )
( by 𝕜 : Type E : Type F : Type F' : Type inst✝⁵ : RCLike 𝕜 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedAddCommGroup F inst✝² : NormedAddCommGroup F' inst✝¹ : NormedSpace ℝ E inst✝ : NormedSpace ℝ F d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) n : ℕ ⊢ IsDistBounded fun x => Real.log ( normPowerSeries n x ) fun_prop All goals completed! 🐙 ) ) ) η , y ⟫_ ℝ )
Filter.atTop
( 𝓝 ( ⟪ distOfFunction ( fun x : Space d => ( ‖ x ‖ ^ ( - 2 : ℤ ) ) • basis . repr x ) ( by 𝕜 : Type E : Type F : Type F' : Type inst✝⁵ : RCLike 𝕜 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedAddCommGroup F inst✝² : NormedAddCommGroup F' inst✝¹ : NormedSpace ℝ E inst✝ : NormedSpace ℝ F d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ IsDistBounded fun x => ‖ x ‖ ^ (- 2 ) • basis . repr x
refine ( IsDistBounded.zpow_smul_repr_self _ ? _ ) 𝕜 : Type E : Type F : Type F' : Type inst✝⁵ : RCLike 𝕜 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedAddCommGroup F inst✝² : NormedAddCommGroup F' inst✝¹ : NormedSpace ℝ E inst✝ : NormedSpace ℝ F d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ - ↑ ( d - 1 ) - 1 ≤ - 2
omega All goals completed! 🐙 ) η , y ⟫_ ℝ ) ) := by d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Filter.Tendsto (fun n => ⟪ ( ∇ᵈ ( distOfFunction (fun x => Real.log ( normPowerSeries n x ) ) ⋯ ) ) η , y ⟫_ ℝ ) Filter.atTop
( 𝓝 ⟪ ( distOfFunction (fun x => ‖ x ‖ ^ (- 2 ) • basis . repr x ) ⋯ ) η , y ⟫_ ℝ )
haveI : NeZero d := ⟨ by d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ d ≠ 0 omega All goals completed! 🐙 ⟩
simp only [ gradient_dist_normPowerSeries_log , distOfFunction_inner ] d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d ⊢ Filter.Tendsto (fun n => ∫ ( x : Space d ) , η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ ) Filter.atTop
( 𝓝 ( ∫ ( x : Space d ) , η x * ⟪ ‖ x ‖ ^ (- 2 ) • basis . repr x , y ⟫_ ℝ ) )
have h1 ( n : ℕ ) ( x : Space d ) :
η x * ⟪ ( normPowerSeries n x ^ ( - 2 : ℤ ) ) • basis . repr x , y ⟫_ ℝ =
η x * ( ( ⟪ basis . repr x , y ⟫_ ℝ * ( normPowerSeries n x ) ^ ( - 2 : ℤ ) ) ) := by d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Filter.Tendsto (fun n => ⟪ ( ∇ᵈ ( distOfFunction (fun x => Real.log ( normPowerSeries n x ) ) ⋯ ) ) η , y ⟫_ ℝ ) Filter.atTop
( 𝓝 ⟪ ( distOfFunction (fun x => ‖ x ‖ ^ (- 2 ) • basis . repr x ) ⋯ ) η , y ⟫_ ℝ ) d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ⊢ Filter.Tendsto (fun n => ∫ ( x : Space d ) , η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ ) Filter.atTop
( 𝓝 ( ∫ ( x : Space d ) , η x * ⟪ ‖ x ‖ ^ (- 2 ) • basis . repr x , y ⟫_ ℝ ) )
rw [ real_inner_smul_left d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d n : ℕ x : Space d ⊢ η x * ( normPowerSeries n x ^ (- 2 ) * ⟪ basis . repr x , y ⟫_ ℝ ) = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d n : ℕ x : Space d ⊢ η x * ( normPowerSeries n x ^ (- 2 ) * ⟪ basis . repr x , y ⟫_ ℝ ) = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ⊢ Filter.Tendsto (fun n => ∫ ( x : Space d ) , η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ ) Filter.atTop
( 𝓝 ( ∫ ( x : Space d ) , η x * ⟪ ‖ x ‖ ^ (- 2 ) • basis . repr x , y ⟫_ ℝ ) ) ] d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d n : ℕ x : Space d ⊢ η x * ( normPowerSeries n x ^ (- 2 ) * ⟪ basis . repr x , y ⟫_ ℝ ) = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ⊢ Filter.Tendsto (fun n => ∫ ( x : Space d ) , η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ ) Filter.atTop
( 𝓝 ( ∫ ( x : Space d ) , η x * ⟪ ‖ x ‖ ^ (- 2 ) • basis . repr x , y ⟫_ ℝ ) )
ring d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ⊢ Filter.Tendsto (fun n => ∫ ( x : Space d ) , η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ ) Filter.atTop
( 𝓝 ( ∫ ( x : Space d ) , η x * ⟪ ‖ x ‖ ^ (- 2 ) • basis . repr x , y ⟫_ ℝ ) ) d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ⊢ Filter.Tendsto (fun n => ∫ ( x : Space d ) , η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ ) Filter.atTop
( 𝓝 ( ∫ ( x : Space d ) , η x * ⟪ ‖ x ‖ ^ (- 2 ) • basis . repr x , y ⟫_ ℝ ) )
simp only [ h1 ] d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ⊢ Filter.Tendsto (fun n => ∫ ( x : Space d ) , η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ) Filter.atTop
( 𝓝 ( ∫ ( x : Space d ) , η x * ⟪ ‖ x ‖ ^ (- 2 ) • basis . repr x , y ⟫_ ℝ ) )
apply MeasureTheory.tendsto_integral_of_dominated_convergence
( bound := fun x => | η x | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ ( - 2 : ℤ ) + ‖ x ‖ ^ ( - 2 : ℤ ) ) ) F_measurable d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ⊢ ∀ ( n : ℕ ), AEStronglyMeasurable (fun a => η a * ( ⟪ basis . repr a , y ⟫_ ℝ * normPowerSeries n a ^ (- 2 ) ) ) volume bound_integrable d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ⊢ Integrable (fun x => | η x | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ (- 2 ) + ‖ x ‖ ^ (- 2 ) ) ) volume h_bound d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ⊢ ∀ ( n : ℕ ),
∀ᵐ ( a : Space d ) ,
‖ η a * ( ⟪ basis . repr a , y ⟫_ ℝ * normPowerSeries n a ^ (- 2 ) ) ‖ ≤
| η a | * | ⟪ basis . repr a , y ⟫_ ℝ | * ( ( ‖ a ‖ + 1 ) ^ (- 2 ) + ‖ a ‖ ^ (- 2 ) ) h_lim d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ⊢ ∀ᵐ ( a : Space d ) ,
Filter.Tendsto (fun n => η a * ( ⟪ basis . repr a , y ⟫_ ℝ * normPowerSeries n a ^ (- 2 ) ) ) Filter.atTop
( 𝓝 ( η a * ⟪ ‖ a ‖ ^ (- 2 ) • basis . repr a , y ⟫_ ℝ ) )
· F_measurable d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ⊢ ∀ ( n : ℕ ), AEStronglyMeasurable (fun a => η a * ( ⟪ basis . repr a , y ⟫_ ℝ * normPowerSeries n a ^ (- 2 ) ) ) volume intro n F_measurable d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) n : ℕ ⊢ AEStronglyMeasurable (fun a => η a * ( ⟪ basis . repr a , y ⟫_ ℝ * normPowerSeries n a ^ (- 2 ) ) ) volume
refine IsDistBounded.aeStronglyMeasurable_schwartzMap_smul ( F := ℝ ) ? _ η F_measurable d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) n : ℕ ⊢ IsDistBounded fun a => ⟪ basis . repr a , y ⟫_ ℝ * normPowerSeries n a ^ (- 2 )
simp only [ basis_repr_inner_eq ] F_measurable d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) n : ℕ ⊢ IsDistBounded fun a => ⟪ a , basis . repr . symm y ⟫_ ℝ * normPowerSeries n a ^ (- 2 )
exact IsDistBounded.isDistBounded_mul_inner' ( by d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) n : ℕ ⊢ IsDistBounded fun a => normPowerSeries n a ^ (- 2 ) fun_prop All goals completed! 🐙 ) _
· bound_integrable d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ⊢ Integrable (fun x => | η x | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ (- 2 ) + ‖ x ‖ ^ (- 2 ) ) ) volume have h1 : Integrable ( fun x =>
η x * ( ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ ( - 2 : ℤ ) + ‖ x ‖ ^ ( - 2 : ℤ ) ) ) ) ) volume := by d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Filter.Tendsto (fun n => ⟪ ( ∇ᵈ ( distOfFunction (fun x => Real.log ( normPowerSeries n x ) ) ⋯ ) ) η , y ⟫_ ℝ ) Filter.atTop
( 𝓝 ⟪ ( distOfFunction (fun x => ‖ x ‖ ^ (- 2 ) • basis . repr x ) ⋯ ) η , y ⟫_ ℝ ) bound_integrable d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) h1 : Integrable (fun x => η x * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ (- 2 ) + ‖ x ‖ ^ (- 2 ) ) ) ) volume ⊢ Integrable (fun x => | η x | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ (- 2 ) + ‖ x ‖ ^ (- 2 ) ) ) volume
apply IsDistBounded.integrable_space_mul ? _ η d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ⊢ IsDistBounded fun x => ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ (- 2 ) + ‖ x ‖ ^ (- 2 ) ) bound_integrable d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) h1 : Integrable (fun x => η x * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ (- 2 ) + ‖ x ‖ ^ (- 2 ) ) ) ) volume ⊢ Integrable (fun x => | η x | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ (- 2 ) + ‖ x ‖ ^ (- 2 ) ) ) volume
simp [ mul_add ] d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ⊢ IsDistBounded fun x => ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ 2 ) ⁻¹ + ⟪ basis . repr x , y ⟫_ ℝ * ( ‖ x ‖ ^ 2 ) ⁻¹ bound_integrable d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) h1 : Integrable (fun x => η x * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ (- 2 ) + ‖ x ‖ ^ (- 2 ) ) ) ) volume ⊢ Integrable (fun x => | η x | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ (- 2 ) + ‖ x ‖ ^ (- 2 ) ) ) volume
apply IsDistBounded.add hf d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ⊢ IsDistBounded fun x => ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ 2 ) ⁻¹ hg d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ⊢ IsDistBounded fun x => ⟪ basis . repr x , y ⟫_ ℝ * ( ‖ x ‖ ^ 2 ) ⁻¹ bound_integrable d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) h1 : Integrable (fun x => η x * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ (- 2 ) + ‖ x ‖ ^ (- 2 ) ) ) ) volume ⊢ Integrable (fun x => | η x | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ (- 2 ) + ‖ x ‖ ^ (- 2 ) ) ) volume
· hf d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ⊢ IsDistBounded fun x => ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ 2 ) ⁻¹ bound_integrable d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) h1 : Integrable (fun x => η x * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ (- 2 ) + ‖ x ‖ ^ (- 2 ) ) ) ) volume ⊢ Integrable (fun x => | η x | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ (- 2 ) + ‖ x ‖ ^ (- 2 ) ) ) volume simp only [ basis_repr_inner_eq ] hf d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ⊢ IsDistBounded fun x => ⟪ x , basis . repr . symm y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ 2 ) ⁻¹ bound_integrable d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) h1 : Integrable (fun x => η x * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ (- 2 ) + ‖ x ‖ ^ (- 2 ) ) ) ) volume ⊢ Integrable (fun x => | η x | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ (- 2 ) + ‖ x ‖ ^ (- 2 ) ) ) volume
exact IsDistBounded.isDistBounded_mul_inner'
( IsDistBounded.norm_add_pos_nat_zpow ( - 2 ) 1 one_pos ) _ All goals completed! 🐙 bound_integrable d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) h1 : Integrable (fun x => η x * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ (- 2 ) + ‖ x ‖ ^ (- 2 ) ) ) ) volume ⊢ Integrable (fun x => | η x | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ (- 2 ) + ‖ x ‖ ^ (- 2 ) ) ) volume
· hg d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ⊢ IsDistBounded fun x => ⟪ basis . repr x , y ⟫_ ℝ * ( ‖ x ‖ ^ 2 ) ⁻¹ bound_integrable d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) h1 : Integrable (fun x => η x * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ (- 2 ) + ‖ x ‖ ^ (- 2 ) ) ) ) volume ⊢ Integrable (fun x => | η x | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ (- 2 ) + ‖ x ‖ ^ (- 2 ) ) ) volume simp only [ basis_repr_inner_eq ] hg d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ⊢ IsDistBounded fun x => ⟪ x , basis . repr . symm y ⟫_ ℝ * ( ‖ x ‖ ^ 2 ) ⁻¹ bound_integrable d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) h1 : Integrable (fun x => η x * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ (- 2 ) + ‖ x ‖ ^ (- 2 ) ) ) ) volume ⊢ Integrable (fun x => | η x | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ (- 2 ) + ‖ x ‖ ^ (- 2 ) ) ) volume
convert IsDistBounded.mul_inner_pow_neg_two ( basis . repr . symm y ) using 1 d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ⊢ (fun x => ⟪ x , basis . repr . symm y ⟫_ ℝ * ( ‖ x ‖ ^ 2 ) ⁻¹ ) = fun x => ⟪ basis . repr . symm y , x ⟫_ ℝ * ‖ x ‖ ^ (- 2 ) bound_integrable d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) h1 : Integrable (fun x => η x * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ (- 2 ) + ‖ x ‖ ^ (- 2 ) ) ) ) volume ⊢ Integrable (fun x => | η x | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ (- 2 ) + ‖ x ‖ ^ (- 2 ) ) ) volume
funext x d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) x : Space d ⊢ ⟪ x , basis . repr . symm y ⟫_ ℝ * ( ‖ x ‖ ^ 2 ) ⁻¹ = ⟪ basis . repr . symm y , x ⟫_ ℝ * ‖ x ‖ ^ (- 2 ) bound_integrable d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) h1 : Integrable (fun x => η x * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ (- 2 ) + ‖ x ‖ ^ (- 2 ) ) ) ) volume ⊢ Integrable (fun x => | η x | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ (- 2 ) + ‖ x ‖ ^ (- 2 ) ) ) volume
simp [ real_inner_comm ] bound_integrable d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) h1 : Integrable (fun x => η x * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ (- 2 ) + ‖ x ‖ ^ (- 2 ) ) ) ) volume ⊢ Integrable (fun x => | η x | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ (- 2 ) + ‖ x ‖ ^ (- 2 ) ) ) volume bound_integrable d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) h1 : Integrable (fun x => η x * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ (- 2 ) + ‖ x ‖ ^ (- 2 ) ) ) ) volume ⊢ Integrable (fun x => | η x | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ (- 2 ) + ‖ x ‖ ^ (- 2 ) ) ) volume
refine h1 . abs . congr ( ae_of_all _ fun x => ? _ ) bound_integrable d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1✝ : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) h1 : Integrable (fun x => η x * ( ⟪ basis . repr x , y ⟫_ ℝ * ( ( ‖ x ‖ + 1 ) ^ (- 2 ) + ‖ x ‖ ^ (- 2 ) ) ) ) volume x : Space d ⊢ (fun a => | η a * ( ⟪ basis . repr a , y ⟫_ ℝ * ( ( ‖ a ‖ + 1 ) ^ (- 2 ) + ‖ a ‖ ^ (- 2 ) ) ) | ) x =
(fun x => | η x | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ (- 2 ) + ‖ x ‖ ^ (- 2 ) ) ) x
simp only [ abs_mul , mul_assoc ,
abs_of_nonneg ( show ( 0 : ℝ ) ≤ ( ‖ x ‖ + 1 ) ^ ( - 2 : ℤ ) + ‖ x ‖ ^ ( - 2 : ℤ ) by positivity ) ] All goals completed! 🐙
· h_bound d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ⊢ ∀ ( n : ℕ ),
∀ᵐ ( a : Space d ) ,
‖ η a * ( ⟪ basis . repr a , y ⟫_ ℝ * normPowerSeries n a ^ (- 2 ) ) ‖ ≤
| η a | * | ⟪ basis . repr a , y ⟫_ ℝ | * ( ( ‖ a ‖ + 1 ) ^ (- 2 ) + ‖ a ‖ ^ (- 2 ) ) intro n h_bound d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) n : ℕ ⊢ ∀ᵐ ( a : Space d ) ,
‖ η a * ( ⟪ basis . repr a , y ⟫_ ℝ * normPowerSeries n a ^ (- 2 ) ) ‖ ≤
| η a | * | ⟪ basis . repr a , y ⟫_ ℝ | * ( ( ‖ a ‖ + 1 ) ^ (- 2 ) + ‖ a ‖ ^ (- 2 ) )
filter_upwards [ Measure.ae_ne volume 0 ] with x hx d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) n : ℕ x : Space d hx : x ≠ 0 ⊢ ‖ η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ‖ ≤
| η x | * | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ‖ x ‖ + 1 ) ^ (- 2 ) + ‖ x ‖ ^ (- 2 ) )
simp [ mul_assoc ] d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) n : ℕ x : Space d hx : x ≠ 0 ⊢ | η x | * ( | ⟪ basis . repr x , y ⟫_ ℝ | * ( | normPowerSeries n x | ^ 2 ) ⁻¹ ) ≤
| η x | * ( | ⟪ basis . repr x , y ⟫_ ℝ | * ( ( ( ‖ x ‖ + 1 ) ^ 2 ) ⁻¹ + ( ‖ x ‖ ^ 2 ) ⁻¹ ) )
gcongr hbc d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) n : ℕ x : Space d hx : x ≠ 0 ⊢ ( | normPowerSeries n x | ^ 2 ) ⁻¹ ≤ ( ( ‖ x ‖ + 1 ) ^ 2 ) ⁻¹ + ( ‖ x ‖ ^ 2 ) ⁻¹
rw [ abs_of_nonneg ( by d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) n : ℕ x : Space d hx : x ≠ 0 ⊢ 0 ≤ normPowerSeries n x hbc d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) n : ℕ x : Space d hx : x ≠ 0 ⊢ ( normPowerSeries n x ^ 2 ) ⁻¹ ≤ ( ( ‖ x ‖ + 1 ) ^ 2 ) ⁻¹ + ( ‖ x ‖ ^ 2 ) ⁻¹ simp All goals completed! 🐙 hbc d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) n : ℕ x : Space d hx : x ≠ 0 ⊢ ( normPowerSeries n x ^ 2 ) ⁻¹ ≤ ( ( ‖ x ‖ + 1 ) ^ 2 ) ⁻¹ + ( ‖ x ‖ ^ 2 ) ⁻¹ ) ] hbc d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) n : ℕ x : Space d hx : x ≠ 0 ⊢ ( normPowerSeries n x ^ 2 ) ⁻¹ ≤ ( ( ‖ x ‖ + 1 ) ^ 2 ) ⁻¹ + ( ‖ x ‖ ^ 2 ) ⁻¹
exact normPowerSeries_zpow_le_norm_sq_add_one n ( - 2 : ℤ ) x hx All goals completed! 🐙
· h_lim d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ⊢ ∀ᵐ ( a : Space d ) ,
Filter.Tendsto (fun n => η a * ( ⟪ basis . repr a , y ⟫_ ℝ * normPowerSeries n a ^ (- 2 ) ) ) Filter.atTop
( 𝓝 ( η a * ⟪ ‖ a ‖ ^ (- 2 ) • basis . repr a , y ⟫_ ℝ ) ) filter_upwards [ Measure.ae_ne volume 0 ] with x hx d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) x : Space d hx : x ≠ 0 ⊢ Filter.Tendsto (fun n => η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ) Filter.atTop
( 𝓝 ( η x * ⟪ ‖ x ‖ ^ (- 2 ) • basis . repr x , y ⟫_ ℝ ) )
have h2 : ⟪ ( ‖ x ‖ ^ ( - 2 : ℤ ) ) • basis . repr x , y ⟫_ ℝ =
⟪ basis . repr x , y ⟫_ ℝ * ‖ x ‖ ^ ( - 2 : ℤ ) := by d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) ⊢ Filter.Tendsto (fun n => ⟪ ( ∇ᵈ ( distOfFunction (fun x => Real.log ( normPowerSeries n x ) ) ⋯ ) ) η , y ⟫_ ℝ ) Filter.atTop
( 𝓝 ⟪ ( distOfFunction (fun x => ‖ x ‖ ^ (- 2 ) • basis . repr x ) ⋯ ) η , y ⟫_ ℝ ) d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) x : Space d hx : x ≠ 0 h2 : ⟪ ‖ x ‖ ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = ⟪ basis . repr x , y ⟫_ ℝ * ‖ x ‖ ^ (- 2 ) ⊢ Filter.Tendsto (fun n => η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ) Filter.atTop
( 𝓝 ( η x * ⟪ ‖ x ‖ ^ (- 2 ) • basis . repr x , y ⟫_ ℝ ) )
rw [ real_inner_smul_left d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) x : Space d hx : x ≠ 0 ⊢ ‖ x ‖ ^ (- 2 ) * ⟪ basis . repr x , y ⟫_ ℝ = ⟪ basis . repr x , y ⟫_ ℝ * ‖ x ‖ ^ (- 2 ) d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) x : Space d hx : x ≠ 0 ⊢ ‖ x ‖ ^ (- 2 ) * ⟪ basis . repr x , y ⟫_ ℝ = ⟪ basis . repr x , y ⟫_ ℝ * ‖ x ‖ ^ (- 2 ) d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) x : Space d hx : x ≠ 0 h2 : ⟪ ‖ x ‖ ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = ⟪ basis . repr x , y ⟫_ ℝ * ‖ x ‖ ^ (- 2 ) ⊢ Filter.Tendsto (fun n => η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ) Filter.atTop
( 𝓝 ( η x * ⟪ ‖ x ‖ ^ (- 2 ) • basis . repr x , y ⟫_ ℝ ) ) ] d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) x : Space d hx : x ≠ 0 ⊢ ‖ x ‖ ^ (- 2 ) * ⟪ basis . repr x , y ⟫_ ℝ = ⟪ basis . repr x , y ⟫_ ℝ * ‖ x ‖ ^ (- 2 ) d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) x : Space d hx : x ≠ 0 h2 : ⟪ ‖ x ‖ ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = ⟪ basis . repr x , y ⟫_ ℝ * ‖ x ‖ ^ (- 2 ) ⊢ Filter.Tendsto (fun n => η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ) Filter.atTop
( 𝓝 ( η x * ⟪ ‖ x ‖ ^ (- 2 ) • basis . repr x , y ⟫_ ℝ ) )
ring d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) x : Space d hx : x ≠ 0 h2 : ⟪ ‖ x ‖ ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = ⟪ basis . repr x , y ⟫_ ℝ * ‖ x ‖ ^ (- 2 ) ⊢ Filter.Tendsto (fun n => η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ) Filter.atTop
( 𝓝 ( η x * ⟪ ‖ x ‖ ^ (- 2 ) • basis . repr x , y ⟫_ ℝ ) ) d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) x : Space d hx : x ≠ 0 h2 : ⟪ ‖ x ‖ ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = ⟪ basis . repr x , y ⟫_ ℝ * ‖ x ‖ ^ (- 2 ) ⊢ Filter.Tendsto (fun n => η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ) Filter.atTop
( 𝓝 ( η x * ⟪ ‖ x ‖ ^ (- 2 ) • basis . repr x , y ⟫_ ℝ ) )
rw [ h2 d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) x : Space d hx : x ≠ 0 h2 : ⟪ ‖ x ‖ ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = ⟪ basis . repr x , y ⟫_ ℝ * ‖ x ‖ ^ (- 2 ) ⊢ Filter.Tendsto (fun n => η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ) Filter.atTop
( 𝓝 ( η x * ( ⟪ basis . repr x , y ⟫_ ℝ * ‖ x ‖ ^ (- 2 ) ) ) ) d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) x : Space d hx : x ≠ 0 h2 : ⟪ ‖ x ‖ ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = ⟪ basis . repr x , y ⟫_ ℝ * ‖ x ‖ ^ (- 2 ) ⊢ Filter.Tendsto (fun n => η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ) Filter.atTop
( 𝓝 ( η x * ( ⟪ basis . repr x , y ⟫_ ℝ * ‖ x ‖ ^ (- 2 ) ) ) ) ] d : ℕ hd : 2 ≤ d η : 𝓢( Space d , ℝ ) y : EuclideanSpace ℝ ( Fin d ) this : NeZero d h1 : ∀ ( n : ℕ ) ( x : Space d ),
η x * ⟪ normPowerSeries n x ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) x : Space d hx : x ≠ 0 h2 : ⟪ ‖ x ‖ ^ (- 2 ) • basis . repr x , y ⟫_ ℝ = ⟪ basis . repr x , y ⟫_ ℝ * ‖ x ‖ ^ (- 2 ) ⊢ Filter.Tendsto (fun n => η x * ( ⟪ basis . repr x , y ⟫_ ℝ * normPowerSeries n x ^ (- 2 ) ) ) Filter.atTop
( 𝓝 ( η x * ( ⟪ basis . repr x , y ⟫_ ℝ * ‖ x ‖ ^ (- 2 ) ) ) )
exact tendsto_const_nhds . mul ( tendsto_const_nhds . mul
( ( normPowerSeries_tendsto x hx ) . zpow₀ _ ( .inl ( norm_ne_zero_iff . mpr hx ) ) ) ) All goals completed! 🐙
B. Distributions involving norms
B.1. The gradient of distributions based on powers
B.2. The gradient of distributions based on logs
B.3. Divergence of radial norm-power distributions
private lemma integrable_real_pow_mul_schwartz
( ψ : 𝓢( ℝ , ℝ ) ) ( k : ℕ ) :
Integrable ( fun x : ℝ => x ^ k * ψ x ) volume := by ψ : 𝓢( ℝ , ℝ ) k : ℕ ⊢ Integrable (fun x => x ^ k * ψ x ) volume
refine ( ψ . integrable_pow_mul volume k ) . mono' ( by ψ : 𝓢( ℝ , ℝ ) k : ℕ ⊢ AEStronglyMeasurable (fun x => x ^ k * ψ x ) volume fun_prop All goals completed! 🐙 )
( ae_of_all _ fun x => by ψ : 𝓢( ℝ , ℝ ) k : ℕ x : ℝ ⊢ ‖ x ^ k * ψ x ‖ ≤ ‖ x ‖ ^ k * ‖ ψ x ‖ simp [ norm_mul , norm_pow ] All goals completed! 🐙 )
private lemma radial_power_deriv_integral_by_parts
{ d : ℕ } ( η : 𝓢( Space d , ℝ ) )
( n : ↑ ( Metric.sphere ( 0 : Space d ) 1 ) )
( p : ℕ ) ( hp : 0 < p ) :
- ∫ ( r : Set.Ioi ( 0 : ℝ ) ) ,
r . 1 ^ p * ( _root_.deriv ( fun a => η ( a • n . 1 ) ) r . 1 )
∂ ( .comap Subtype.val volume )
=
( p : ℝ ) * ∫ ( r : Set.Ioi ( 0 : ℝ ) ) ,
r . 1 ^ ( p - 1 ) * η ( r . 1 • n . 1 )
∂ ( .comap Subtype.val volume ) := by d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume
let η' : 𝓢( ℝ , ℝ ) := SchwartzMap.compCLM ( g := fun a => a • n . 1 ) ℝ ( by d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p ⊢ Function.HasTemperateGrowth fun a => a • ↑ n d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume
apply And.intro left d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p ⊢ ContDiff ℝ ↑ ⊤ fun a => a • ↑ n right d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p ⊢ ∀ ( n_1 : ℕ ), ∃ k C , ∀ ( x : ℝ ), ‖ iteratedFDeriv ℝ n_1 (fun a => a • ↑ n ) x ‖ ≤ C * ( 1 + ‖ x ‖ ) ^ k d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume
· left d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p ⊢ ContDiff ℝ ↑ ⊤ fun a => a • ↑ n d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume fun_prop All goals completed! 🐙 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume
· right d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p ⊢ ∀ ( n_1 : ℕ ), ∃ k C , ∀ ( x : ℝ ), ‖ iteratedFDeriv ℝ n_1 (fun a => a • ↑ n ) x ‖ ≤ C * ( 1 + ‖ x ‖ ) ^ k d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume intro n' right d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p n' : ℕ ⊢ ∃ k C , ∀ ( x : ℝ ), ‖ iteratedFDeriv ℝ n' (fun a => a • ↑ n ) x ‖ ≤ C * ( 1 + ‖ x ‖ ) ^ k d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume
match n' with
| 0 => d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p n' : ℕ ⊢ ∃ k C , ∀ ( x : ℝ ), ‖ iteratedFDeriv ℝ 0 (fun a => a • ↑ n ) x ‖ ≤ C * ( 1 + ‖ x ‖ ) ^ k d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume
use 1 , 1 h d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p n' : ℕ ⊢ ∀ ( x : ℝ ), ‖ iteratedFDeriv ℝ 0 (fun a => a • ↑ n ) x ‖ ≤ 1 * ( 1 + ‖ x ‖ ) ^ 1 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume
simp [ norm_smul ] All goals completed! 🐙 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume
| 1 => d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p n' : ℕ ⊢ ∃ k C , ∀ ( x : ℝ ), ‖ iteratedFDeriv ℝ 1 (fun a => a • ↑ n ) x ‖ ≤ C * ( 1 + ‖ x ‖ ) ^ k d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume
use 0 , 1 h d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p n' : ℕ ⊢ ∀ ( x : ℝ ), ‖ iteratedFDeriv ℝ 1 (fun a => a • ↑ n ) x ‖ ≤ 1 * ( 1 + ‖ x ‖ ) ^ 0 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume
intro x h d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p n' : ℕ x : ℝ ⊢ ‖ iteratedFDeriv ℝ 1 (fun a => a • ↑ n ) x ‖ ≤ 1 * ( 1 + ‖ x ‖ ) ^ 0 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume
simp [ fderiv_smul_const ] All goals completed! 🐙 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume
| n' + 1 + 1 => d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p n'✝ : ℕ n' : ℕ ⊢ ∃ k C , ∀ ( x : ℝ ), ‖ iteratedFDeriv ℝ ( n' + 1 + 1 ) (fun a => a • ↑ n ) x ‖ ≤ C * ( 1 + ‖ x ‖ ) ^ k d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume
use 0 , 0 h d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p n'✝ : ℕ n' : ℕ ⊢ ∀ ( x : ℝ ), ‖ iteratedFDeriv ℝ ( n' + 1 + 1 ) (fun a => a • ↑ n ) x ‖ ≤ 0 * ( 1 + ‖ x ‖ ) ^ 0 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume
intro x h d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p n'✝ : ℕ n' : ℕ x : ℝ ⊢ ‖ iteratedFDeriv ℝ ( n' + 1 + 1 ) (fun a => a • ↑ n ) x ‖ ≤ 0 * ( 1 + ‖ x ‖ ) ^ 0 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume
simp only [ Real.norm_eq_abs , pow_zero , mul_one , norm_le_zero_iff ] h d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p n'✝ : ℕ n' : ℕ x : ℝ ⊢ iteratedFDeriv ℝ ( n' + 1 + 1 ) (fun a => a • ↑ n ) x = 0 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume
rw [ iteratedFDeriv_succ_eq_comp_right h d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p n'✝ : ℕ n' : ℕ x : ℝ ⊢ ( ⇑ ( continuousMultilinearCurryRightEquiv' ℝ ( n' + 1 ) ℝ ( Space d ) ) . symm ∘
iteratedFDeriv ℝ ( n' + 1 ) fun y => fderiv ℝ (fun a => a • ↑ n ) y )
x =
0 h d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p n'✝ : ℕ n' : ℕ x : ℝ ⊢ ( ⇑ ( continuousMultilinearCurryRightEquiv' ℝ ( n' + 1 ) ℝ ( Space d ) ) . symm ∘
iteratedFDeriv ℝ ( n' + 1 ) fun y => fderiv ℝ (fun a => a • ↑ n ) y )
x =
0 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume ] h d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p n'✝ : ℕ n' : ℕ x : ℝ ⊢ ( ⇑ ( continuousMultilinearCurryRightEquiv' ℝ ( n' + 1 ) ℝ ( Space d ) ) . symm ∘
iteratedFDeriv ℝ ( n' + 1 ) fun y => fderiv ℝ (fun a => a • ↑ n ) y )
x =
0 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume
conv_lhs =>
enter [ 2 , 3 , y ] d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p n'✝ : ℕ n' : ℕ x : ℝ y : ℝ | fderiv ℝ (fun a => a • ↑ n ) y d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume
simp [ fderiv_smul_const ] d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p n'✝ : ℕ n' : ℕ x : ℝ y : ℝ | ( ContinuousLinearMap.id ℝ ℝ ) . smulRight ↑ n d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume
rw [ iteratedFDeriv_succ_const h d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p n'✝ : ℕ n' : ℕ x : ℝ ⊢ ( ⇑ ( continuousMultilinearCurryRightEquiv' ℝ ( n' + 1 ) ℝ ( Space d ) ) . symm ∘ 0 ) x = 0 h d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p n'✝ : ℕ n' : ℕ x : ℝ ⊢ ( ⇑ ( continuousMultilinearCurryRightEquiv' ℝ ( n' + 1 ) ℝ ( Space d ) ) . symm ∘ 0 ) x = 0 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume ] h d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p n'✝ : ℕ n' : ℕ x : ℝ ⊢ ( ⇑ ( continuousMultilinearCurryRightEquiv' ℝ ( n' + 1 ) ℝ ( Space d ) ) . symm ∘ 0 ) x = 0 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume
rfl All goals completed! 🐙 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume ) ( by d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p ⊢ ∃ k C , ∀ ( x : ℝ ), ‖ x ‖ ≤ C * ( 1 + ‖ x • ↑ n ‖ ) ^ k d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume use 1 , 1 h d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p ⊢ ∀ ( x : ℝ ), ‖ x ‖ ≤ 1 * ( 1 + ‖ x • ↑ n ‖ ) ^ 1 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume ; simp [ norm_smul ] All goals completed! 🐙 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume ) η d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume
have hη'_apply ( x : ℝ ) : η' x = η ( x • n . 1 ) := by
simp [ η' ] d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume
have hmul_iter_apply :
∀ k x , ( ( Physlib.Distribution.powOneMul ℝ ) ^[ k ] η' ) x = x ^ k * η' x := by
intro k d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) k : ℕ ⊢ ∀ ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume
induction k with
| zero => zero d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) ⊢ ∀ ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ 0 ] η' ) x = x ^ 0 * η' x d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume simp All goals completed! 🐙 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume
| succ k ih => succ d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) k : ℕ ih : ∀ ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x ⊢ ∀ ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k + 1 ] η' ) x = x ^ ( k + 1 ) * η' x d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume
intro x succ d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) k : ℕ ih : ∀ ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x x : ℝ ⊢ ( (⇑ ( powOneMul ℝ ) ) ^[ k + 1 ] η' ) x = x ^ ( k + 1 ) * η' x d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume
rw [ Function.iterate_succ_apply' , succ d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) k : ℕ ih : ∀ ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x x : ℝ ⊢ ( ( powOneMul ℝ ) ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) ) x = x ^ ( k + 1 ) * η' x succ d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) k : ℕ ih : ∀ ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x x : ℝ ⊢ ↑ x * ( x ^ k * η' x ) = x ^ k * x * η' x d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume Physlib.Distribution.powOneMul_apply , succ d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) k : ℕ ih : ∀ ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x x : ℝ ⊢ ↑ x * ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ ( k + 1 ) * η' x succ d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) k : ℕ ih : ∀ ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x x : ℝ ⊢ ↑ x * ( x ^ k * η' x ) = x ^ k * x * η' x d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume ih , succ d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) k : ℕ ih : ∀ ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x x : ℝ ⊢ ↑ x * ( x ^ k * η' x ) = x ^ ( k + 1 ) * η' x succ d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) k : ℕ ih : ∀ ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x x : ℝ ⊢ ↑ x * ( x ^ k * η' x ) = x ^ k * x * η' x d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume pow_succ succ d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) k : ℕ ih : ∀ ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x x : ℝ ⊢ ↑ x * ( x ^ k * η' x ) = x ^ k * x * η' x succ d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) k : ℕ ih : ∀ ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x x : ℝ ⊢ ↑ x * ( x ^ k * η' x ) = x ^ k * x * η' x d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume ] succ d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) k : ℕ ih : ∀ ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x x : ℝ ⊢ ↑ x * ( x ^ k * η' x ) = x ^ k * x * η' x d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume
change x * ( x ^ k * η' x ) = x ^ k * x * η' x succ d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) k : ℕ ih : ∀ ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x x : ℝ ⊢ x * ( x ^ k * η' x ) = x ^ k * x * η' x d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume
ring d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume
have hleft_subtype :
∫ ( r : Set.Ioi ( 0 : ℝ ) ) ,
r . 1 ^ p * _root_.deriv ( fun a => η ( a • n . 1 ) ) r . 1
∂ ( .comap Subtype.val volume )
=
∫ ( x : ℝ ) in Set.Ioi ( 0 : ℝ ) ,
x ^ p * _root_.deriv ( fun a => η ( a • n . 1 ) ) x :=
MeasureTheory.integral_subtype_comap measurableSet_Ioi
fun x : ℝ => x ^ p * _root_.deriv ( fun a => η ( a • n . 1 ) ) x d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume
have hright_subtype :
∫ ( r : Set.Ioi ( 0 : ℝ ) ) ,
r . 1 ^ ( p - 1 ) * η ( r . 1 • n . 1 )
∂ ( .comap Subtype.val volume )
=
∫ ( x : ℝ ) in Set.Ioi ( 0 : ℝ ) ,
x ^ ( p - 1 ) * η ( x • n . 1 ) :=
MeasureTheory.integral_subtype_comap measurableSet_Ioi fun x : ℝ => x ^ ( p - 1 ) * η ( x • n . 1 ) d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume
rw [ hleft_subtype , d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hright_subtype d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) ] d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n )
have hIBP :
∫ ( x : ℝ ) in Set.Ioi ( 0 : ℝ ) ,
x ^ p * _root_.deriv ( fun a => η ( a • n . 1 ) ) x
=
( 0 : ℝ ) - ( 0 : ℝ ) -
∫ ( x : ℝ ) in Set.Ioi ( 0 : ℝ ) ,
( ( p : ℝ ) * x ^ ( p - 1 ) ) * η ( x • n . 1 ) := by d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n )
refine MeasureTheory.integral_Ioi_mul_deriv_eq_deriv_mul
( a := ( 0 : ℝ ) )
( u := fun x : ℝ => x ^ p )
( u' := fun x : ℝ => ( p : ℝ ) * x ^ ( p - 1 ) )
( v := fun x : ℝ => η ( x • n . 1 ) )
( v' := fun x : ℝ => _root_.deriv ( fun a => η ( a • n . 1 ) ) x )
( a' := ( 0 : ℝ ) ) ( b' := ( 0 : ℝ ) ) ? _ ? _ ? _ ? _ ? _ ? _ refine_1 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ ∀ x ∈ Set.Ioi 0 , HasDerivAt (fun x => x ^ p ) ( ↑ p * x ^ ( p - 1 ) ) x refine_2 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ ∀ x ∈ Set.Ioi 0 , HasDerivAt (fun x => η ( x • ↑ n ) ) ( _root_.deriv (fun a => η ( a • ↑ n ) ) x ) x refine_3 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ IntegrableOn ( (fun x => x ^ p ) * fun x => _root_.deriv (fun a => η ( a • ↑ n ) ) x ) ( Set.Ioi 0 ) volume refine_4 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ IntegrableOn ( (fun x => ↑ p * x ^ ( p - 1 ) ) * fun x => η ( x • ↑ n ) ) ( Set.Ioi 0 ) volume refine_5 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ Filter.Tendsto ( (fun x => x ^ p ) * fun x => η ( x • ↑ n ) ) ( 𝓝[>] 0 ) ( 𝓝 0 ) refine_6 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ Filter.Tendsto ( (fun x => x ^ p ) * fun x => η ( x • ↑ n ) ) Filter.atTop ( 𝓝 0 ) d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n )
· refine_1 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ ∀ x ∈ Set.Ioi 0 , HasDerivAt (fun x => x ^ p ) ( ↑ p * x ^ ( p - 1 ) ) x d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) exact fun x _ => by d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) x : ℝ x✝ : x ∈ Set.Ioi 0 ⊢ HasDerivAt (fun x => x ^ p ) ( ↑ p * x ^ ( p - 1 ) ) x d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) simpa using hasDerivAt_pow p x All goals completed! 🐙 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n )
· refine_2 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ ∀ x ∈ Set.Ioi 0 , HasDerivAt (fun x => η ( x • ↑ n ) ) ( _root_.deriv (fun a => η ( a • ↑ n ) ) x ) x d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) exact fun x _ => DifferentiableAt.hasDerivAt ( by d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) x : ℝ x✝ : x ∈ Set.Ioi 0 ⊢ DifferentiableAt ℝ (fun x => η ( x • ↑ n ) ) x d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) fun_prop All goals completed! 🐙 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) )
· refine_3 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ IntegrableOn ( (fun x => x ^ p ) * fun x => _root_.deriv (fun a => η ( a • ↑ n ) ) x ) ( Set.Ioi 0 ) volume d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) refine ( integrable_real_pow_mul_schwartz ( ( SchwartzMap.derivCLM ℝ ℝ ) η' )
p ) . integrableOn . congr_fun ( fun x _ => ? _ ) measurableSet_Ioi refine_3 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) x : ℝ x✝ : x ∈ Set.Ioi 0 ⊢ (fun x => x ^ p * ( ( derivCLM ℝ ℝ ) η' ) x ) x = ( (fun x => x ^ p ) * fun x => _root_.deriv (fun a => η ( a • ↑ n ) ) x ) x d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n )
have hderiv_eq : _root_.deriv η' x = _root_.deriv ( fun a => η ( a • n . 1 ) ) x := by d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume refine_3 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) x : ℝ x✝ : x ∈ Set.Ioi 0 hderiv_eq : _root_.deriv (⇑ η' ) x = _root_.deriv (fun a => η ( a • ↑ n ) ) x ⊢ (fun x => x ^ p * ( ( derivCLM ℝ ℝ ) η' ) x ) x = ( (fun x => x ^ p ) * fun x => _root_.deriv (fun a => η ( a • ↑ n ) ) x ) x d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) congr 1 refine_3 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) x : ℝ x✝ : x ∈ Set.Ioi 0 hderiv_eq : _root_.deriv (⇑ η' ) x = _root_.deriv (fun a => η ( a • ↑ n ) ) x ⊢ (fun x => x ^ p * ( ( derivCLM ℝ ℝ ) η' ) x ) x = ( (fun x => x ^ p ) * fun x => _root_.deriv (fun a => η ( a • ↑ n ) ) x ) x d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) refine_3 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) x : ℝ x✝ : x ∈ Set.Ioi 0 hderiv_eq : _root_.deriv (⇑ η' ) x = _root_.deriv (fun a => η ( a • ↑ n ) ) x ⊢ (fun x => x ^ p * ( ( derivCLM ℝ ℝ ) η' ) x ) x = ( (fun x => x ^ p ) * fun x => _root_.deriv (fun a => η ( a • ↑ n ) ) x ) x d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n )
simp [ SchwartzMap.derivCLM_apply , hderiv_eq ] All goals completed! 🐙 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n )
· refine_4 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ IntegrableOn ( (fun x => ↑ p * x ^ ( p - 1 ) ) * fun x => η ( x • ↑ n ) ) ( Set.Ioi 0 ) volume d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) refine ( ( integrable_real_pow_mul_schwartz η' ( p - 1 ) ) . const_mul
( p : ℝ ) ) . integrableOn . congr_fun ( fun x _ => ? _ ) measurableSet_Ioi refine_4 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) x : ℝ x✝ : x ∈ Set.Ioi 0 ⊢ (fun x => ↑ p * ( x ^ ( p - 1 ) * η' x ) ) x = ( (fun x => ↑ p * x ^ ( p - 1 ) ) * fun x => η ( x • ↑ n ) ) x d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n )
ring_nf refine_4 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) x : ℝ x✝ : x ∈ Set.Ioi 0 ⊢ ↑ p * x ^ ( p - 1 ) * η' x = ( (fun x => ↑ p * x ^ ( p - 1 ) ) * fun x => η ( x • ↑ n ) ) x d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n )
simp [ hη'_apply , mul_assoc ] All goals completed! 🐙 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n )
· refine_5 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ Filter.Tendsto ( (fun x => x ^ p ) * fun x => η ( x • ↑ n ) ) ( 𝓝[>] 0 ) ( 𝓝 0 ) d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) have hcont : ContinuousAt ( fun x : ℝ => x ^ p * η ( x • n . 1 ) ) ( 0 : ℝ ) := by d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume refine_5 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hcont : ContinuousAt (fun x => x ^ p * η ( x • ↑ n ) ) 0 ⊢ Filter.Tendsto ( (fun x => x ^ p ) * fun x => η ( x • ↑ n ) ) ( 𝓝[>] 0 ) ( 𝓝 0 ) d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) fun_prop refine_5 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hcont : ContinuousAt (fun x => x ^ p * η ( x • ↑ n ) ) 0 ⊢ Filter.Tendsto ( (fun x => x ^ p ) * fun x => η ( x • ↑ n ) ) ( 𝓝[>] 0 ) ( 𝓝 0 ) d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) refine_5 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hcont : ContinuousAt (fun x => x ^ p * η ( x • ↑ n ) ) 0 ⊢ Filter.Tendsto ( (fun x => x ^ p ) * fun x => η ( x • ↑ n ) ) ( 𝓝[>] 0 ) ( 𝓝 0 ) d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n )
have hlim := tendsto_nhdsWithin_of_tendsto_nhds ( s := Set.Ioi ( 0 : ℝ ) ) hcont . tendsto refine_5 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hcont : ContinuousAt (fun x => x ^ p * η ( x • ↑ n ) ) 0 hlim : Filter.Tendsto (fun x => x ^ p * η ( x • ↑ n ) ) ( 𝓝[>] 0 ) ( 𝓝 ( 0 ^ p * η ( 0 • ↑ n ) ) ) ⊢ Filter.Tendsto ( (fun x => x ^ p ) * fun x => η ( x • ↑ n ) ) ( 𝓝[>] 0 ) ( 𝓝 0 ) d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n )
simp only [ ne_eq , hp . ne' , not_false_eq_true , zero_pow , zero_smul , zero_mul ] at hlim refine_5 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hcont : ContinuousAt (fun x => x ^ p * η ( x • ↑ n ) ) 0 hlim : Filter.Tendsto (fun x => x ^ p * η ( x • ↑ n ) ) ( 𝓝[>] 0 ) ( 𝓝 0 ) ⊢ Filter.Tendsto ( (fun x => x ^ p ) * fun x => η ( x • ↑ n ) ) ( 𝓝[>] 0 ) ( 𝓝 0 ) d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n )
exact hlim All goals completed! 🐙 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n )
· refine_6 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ Filter.Tendsto ( (fun x => x ^ p ) * fun x => η ( x • ↑ n ) ) Filter.atTop ( 𝓝 0 ) d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) have hsch : Filter.Tendsto ( fun x : ℝ => ( ( Physlib.Distribution.powOneMul ℝ ) ^[ p ] η' ) x )
Filter.atTop ( 𝓝 ( 0 : ℝ ) ) :=
Filter.Tendsto.mono_left
( ( ( Physlib.Distribution.powOneMul ℝ ) ^[ p ] η' ) . toZeroAtInfty . zero_at_infty' )
atTop_le_cocompact refine_6 d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hsch : Filter.Tendsto (fun x => ( (⇑ ( powOneMul ℝ ) ) ^[ p ] η' ) x ) Filter.atTop ( 𝓝 0 ) ⊢ Filter.Tendsto ( (fun x => x ^ p ) * fun x => η ( x • ↑ n ) ) Filter.atTop ( 𝓝 0 ) d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n )
exact hsch . congr' ( Filter.Eventually.of_forall ( hmul_iter_apply p ) ) d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n )
calc
- ∫ ( x : ℝ ) in Set.Ioi ( 0 : ℝ ) ,
x ^ p * _root_.deriv ( fun a => η ( a • n . 1 ) ) x
= ∫ ( x : ℝ ) in Set.Ioi ( 0 : ℝ ) ,
( ( p : ℝ ) * x ^ ( p - 1 ) ) * η ( x • n . 1 ) := by d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n )
rw [ hIBP d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ( 0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ) = ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ( 0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ) = ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ] d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ - ( 0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ) = ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n )
ring All goals completed! 🐙
_ = ( p : ℝ ) * ∫ ( x : ℝ ) in Set.Ioi ( 0 : ℝ ) ,
x ^ ( p - 1 ) * η ( x • n . 1 ) := by d : ℕ η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) p : ℕ hp : 0 < p η' : 𝓢( ℝ , ℝ ) := ( compCLM ℝ ⋯ ⋯ ) η hη'_apply : ∀ ( x : ℝ ), η' x = η ( x • ↑ n ) hmul_iter_apply : ∀ ( k : ℕ ) ( x : ℝ ), ( (⇑ ( powOneMul ℝ ) ) ^[ k ] η' ) x = x ^ k * η' x hleft_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x hright_subtype : ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume =
∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n ) hIBP : ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) x =
0 - 0 - ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) ⊢ ∫ ( x : ℝ ) in Set.Ioi 0 , ↑ p * x ^ ( p - 1 ) * η ( x • ↑ n ) = ↑ p * ∫ ( x : ℝ ) in Set.Ioi 0 , x ^ ( p - 1 ) * η ( x • ↑ n )
simp only [ mul_assoc , integral_const_mul ] All goals completed! 🐙
private lemma distDiv_norm_zpow_smul_repr_self_apply_eq_radial_deriv
{ d p : ℕ } [ NeZero d ] ( q : ℤ ) ( hq : 0 < q + ( d : ℤ ) )
( hp_int : ( p : ℤ ) = q + ( d : ℤ ) )
( η : 𝓢( Space d , ℝ ) ) :
( ∇ᵈ ⬝ ( distOfFunction ( fun x : Space d => ‖ x ‖ ^ q • basis . repr x )
( IsDistBounded.zpow_smul_repr_self q ( by 𝕜 : Type E : Type F : Type F' : Type inst✝⁶ : RCLike 𝕜 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedAddCommGroup F' inst✝² : NormedSpace ℝ E inst✝¹ : NormedSpace ℝ F d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) ⊢ - ↑ ( d - 1 ) - 1 ≤ q omega All goals completed! 🐙 ) ) ) ) η =
- ∫ n , ( ∫ ( r : Set.Ioi ( 0 : ℝ ) ) ,
r . 1 ^ p * ( _root_.deriv ( fun a => η ( a • n . 1 ) ) r . 1 )
∂ ( .comap Subtype.val volume ) )
∂ ( volume ( α := Space d ) . toSphere ) := by d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) ⊢ ( distDiv ( distOfFunction (fun x => ‖ x ‖ ^ q • basis . repr x ) ⋯ ) ) η =
- ∫ ( n : ↑ ( Metric.sphere 0 1 ) ) ,
∫ ( r : ↑ ( Set.Ioi 0 ) ) ,
↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume ∂ volume . toSphere
let F : Space d → ℝ := fun x =>
inner ℝ ( ‖ x ‖ ^ q • basis . repr x ) ( grad η x ) d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ ⊢ ( distDiv ( distOfFunction (fun x => ‖ x ‖ ^ q • basis . repr x ) ⋯ ) ) η =
- ∫ ( n : ↑ ( Metric.sphere 0 1 ) ) ,
∫ ( r : ↑ ( Set.Ioi 0 ) ) ,
↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume ∂ volume . toSphere
calc
( ∇ᵈ ⬝ ( distOfFunction ( fun x : Space d => ‖ x ‖ ^ q • basis . repr x )
( IsDistBounded.zpow_smul_repr_self q ( by d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ ⊢ - ↑ ( d - 1 ) - 1 ≤ q omega All goals completed! 🐙 ) ) ) ) η
= - ∫ x , F x := by d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ ⊢ ( distDiv ( distOfFunction (fun x => ‖ x ‖ ^ q • basis . repr x ) ⋯ ) ) η = - ∫ ( x : Space d ) , F x
rw [ distDiv_ofFunction d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ ⊢ - ∫ ( x : Space d ) , ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ = - ∫ ( x : Space d ) , F x All goals completed! 🐙 ] All goals completed! 🐙
_ = - ∫ r , F ( r . 2 . 1 • r . 1 . 1 )
∂ ( volume ( α := Space d ) . toSphere . prod
( Measure.volumeIoiPow ( Module.finrank ℝ ( Space d ) - 1 ) ) ) := by d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ ⊢ - ∫ ( x : Space d ) , F x =
- ∫ ( r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) ) ,
F ( ↑ r .2 • ↑ r .1 ) ∂ volume . toSphere . prod ( Measure.volumeIoiPow ( Module.finrank ℝ ( Space d ) - 1 ) )
rw [ integral_volume_eq_spherical d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ ⊢ - ∫ ( x : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) ) ,
F ( ↑ x .2 • ↑ x .1 ) ∂ volume . toSphere . prod ( Measure.volumeIoiPow ( Module.finrank ℝ ( Space d ) - 1 ) ) =
- ∫ ( r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) ) ,
F ( ↑ r .2 • ↑ r .1 ) ∂ volume . toSphere . prod ( Measure.volumeIoiPow ( Module.finrank ℝ ( Space d ) - 1 ) ) All goals completed! 🐙 ] All goals completed! 🐙
_ = - ∫ n , ( ∫ r , F ( r . 1 • n . 1 )
∂ ( Measure.volumeIoiPow ( Module.finrank ℝ ( Space d ) - 1 ) ) )
∂ ( volume ( α := Space d ) . toSphere ) := by d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ ⊢ - ∫ ( r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) ) ,
F ( ↑ r .2 • ↑ r .1 ) ∂ volume . toSphere . prod ( Measure.volumeIoiPow ( Module.finrank ℝ ( Space d ) - 1 ) ) =
- ∫ ( n : ↑ ( Metric.sphere 0 1 ) ) ,
∫ ( r : ↑ ( Set.Ioi 0 ) ) , F ( ↑ r • ↑ n ) ∂ Measure.volumeIoiPow ( Module.finrank ℝ ( Space d ) - 1 ) ∂ volume . toSphere
rw [ MeasureTheory.integral_prod d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ ⊢ - ∫ ( x : ↑ ( Metric.sphere 0 1 ) ) ,
∫ ( y : ↑ ( Set.Ioi 0 ) ) ,
F ( ↑ ( x , y ) .2 • ↑ ( x , y ) .1 ) ∂ Measure.volumeIoiPow ( Module.finrank ℝ ( Space d ) - 1 ) ∂ volume . toSphere =
- ∫ ( n : ↑ ( Metric.sphere 0 1 ) ) ,
∫ ( r : ↑ ( Set.Ioi 0 ) ) , F ( ↑ r • ↑ n ) ∂ Measure.volumeIoiPow ( Module.finrank ℝ ( Space d ) - 1 ) ∂ volume . toSphere hf d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ ⊢ Integrable (fun r => F ( ↑ r .2 • ↑ r .1 ) ) ( volume . toSphere . prod ( Measure.volumeIoiPow ( Module.finrank ℝ ( Space d ) - 1 ) ) ) hf d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ ⊢ Integrable (fun r => F ( ↑ r .2 • ↑ r .1 ) ) ( volume . toSphere . prod ( Measure.volumeIoiPow ( Module.finrank ℝ ( Space d ) - 1 ) ) ) ] hf d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ ⊢ Integrable (fun r => F ( ↑ r .2 • ↑ r .1 ) ) ( volume . toSphere . prod ( Measure.volumeIoiPow ( Module.finrank ℝ ( Space d ) - 1 ) ) )
exact integrable_isDistBounded_inner_grad_schwartzMap_spherical
( IsDistBounded.zpow_smul_repr_self q ( by d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ ⊢ - ↑ ( d - 1 ) - 1 ≤ q omega All goals completed! 🐙 ) ) η
_ = - ∫ n , ( ∫ ( r : Set.Ioi ( 0 : ℝ ) ) ,
r . 1 ^ p * ( _root_.deriv ( fun a => η ( a • n . 1 ) ) r . 1 )
∂ ( .comap Subtype.val volume ) )
∂ ( volume ( α := Space d ) . toSphere ) := by d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ ⊢ - ∫ ( n : ↑ ( Metric.sphere 0 1 ) ) ,
∫ ( r : ↑ ( Set.Ioi 0 ) ) , F ( ↑ r • ↑ n ) ∂ Measure.volumeIoiPow ( Module.finrank ℝ ( Space d ) - 1 ) ∂ volume . toSphere =
- ∫ ( n : ↑ ( Metric.sphere 0 1 ) ) ,
∫ ( r : ↑ ( Set.Ioi 0 ) ) ,
↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume ∂ volume . toSphere
congr e_f d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ ⊢ (fun n => ∫ ( r : ↑ ( Set.Ioi 0 ) ) , F ( ↑ r • ↑ n ) ∂ Measure.volumeIoiPow ( Module.finrank ℝ ( Space d ) - 1 ) ) = fun n =>
∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume
funext n e_f d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ n : ↑ ( Metric.sphere 0 1 ) ⊢ ∫ ( r : ↑ ( Set.Ioi 0 ) ) , F ( ↑ r • ↑ n ) ∂ Measure.volumeIoiPow ( Module.finrank ℝ ( Space d ) - 1 ) =
∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume
simp [ F , Measure.volumeIoiPow ] e_f d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ n : ↑ ( Metric.sphere 0 1 ) ⊢ ( ∫ ( r : ↑ ( Set.Ioi 0 ) ) ,
⟪ ‖ ↑ r • ↑ n ‖ ^ q • ↑ r • basis . repr ↑ n ,
∇ (⇑ η ) ( ↑ r • ↑ n ) ⟫_ ℝ ∂ ( Measure.comap Subtype.val volume ) . withDensity fun r => ENNReal.ofReal ( ↑ r ^ ( d - 1 ) ) ) =
∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume
erw [ integral_withDensity_eq_integral_smul ( by d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ n : ↑ ( Metric.sphere 0 1 ) ⊢ Measurable fun r => ( ↑ r ^ ( d - 1 ) ) . toNNReal fun_prop All goals completed! 🐙 ) ] e_f d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ n : ↑ ( Metric.sphere 0 1 ) ⊢ ∫ ( x : ↑ ( Set.Ioi 0 ) ) ,
( ↑ x ^ ( d - 1 ) ) . toNNReal •
⟪ ‖ ↑ x • ↑ n ‖ ^ q • ↑ x • basis . repr ↑ n , ∇ (⇑ η ) ( ↑ x • ↑ n ) ⟫_ ℝ ∂ Measure.comap Subtype.val volume =
∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume
· e_f d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ n : ↑ ( Metric.sphere 0 1 ) ⊢ ∫ ( x : ↑ ( Set.Ioi 0 ) ) ,
( ↑ x ^ ( d - 1 ) ) . toNNReal •
⟪ ‖ ↑ x • ↑ n ‖ ^ q • ↑ x • basis . repr ↑ n , ∇ (⇑ η ) ( ↑ x • ↑ n ) ⟫_ ℝ ∂ Measure.comap Subtype.val volume =
∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume congr e_f.e_f d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ n : ↑ ( Metric.sphere 0 1 ) ⊢ (fun x => ( ↑ x ^ ( d - 1 ) ) . toNNReal • ⟪ ‖ ↑ x • ↑ n ‖ ^ q • ↑ x • basis . repr ↑ n , ∇ (⇑ η ) ( ↑ x • ↑ n ) ⟫_ ℝ ) = fun r =>
↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r
funext r e_f.e_f d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ n : ↑ ( Metric.sphere 0 1 ) r : ↑ ( Set.Ioi 0 ) ⊢ ( ↑ r ^ ( d - 1 ) ) . toNNReal • ⟪ ‖ ↑ r • ↑ n ‖ ^ q • ↑ r • basis . repr ↑ n , ∇ (⇑ η ) ( ↑ r • ↑ n ) ⟫_ ℝ =
↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r
have hr : 0 < ( r : ℝ ) := r . 2 e_f.e_f d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ n : ↑ ( Metric.sphere 0 1 ) r : ↑ ( Set.Ioi 0 ) hr : 0 < ↑ r ⊢ ( ↑ r ^ ( d - 1 ) ) . toNNReal • ⟪ ‖ ↑ r • ↑ n ‖ ^ q • ↑ r • basis . repr ↑ n , ∇ (⇑ η ) ( ↑ r • ↑ n ) ⟫_ ℝ =
↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r
have hnorm := norm_smul_sphere n ( le_of_lt hr ) e_f.e_f d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ n : ↑ ( Metric.sphere 0 1 ) r : ↑ ( Set.Ioi 0 ) hr : 0 < ↑ r hnorm : ‖ ↑ r • ↑ n ‖ = ↑ r ⊢ ( ↑ r ^ ( d - 1 ) ) . toNNReal • ⟪ ‖ ↑ r • ↑ n ‖ ^ q • ↑ r • basis . repr ↑ n , ∇ (⇑ η ) ( ↑ r • ↑ n ) ⟫_ ℝ =
↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r
rw [ NNReal.smul_def e_f.e_f d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ n : ↑ ( Metric.sphere 0 1 ) r : ↑ ( Set.Ioi 0 ) hr : 0 < ↑ r hnorm : ‖ ↑ r • ↑ n ‖ = ↑ r ⊢ ↑ ( ↑ r ^ ( d - 1 ) ) . toNNReal • ⟪ ‖ ↑ r • ↑ n ‖ ^ q • ↑ r • basis . repr ↑ n , ∇ (⇑ η ) ( ↑ r • ↑ n ) ⟫_ ℝ =
↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r e_f.e_f d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ n : ↑ ( Metric.sphere 0 1 ) r : ↑ ( Set.Ioi 0 ) hr : 0 < ↑ r hnorm : ‖ ↑ r • ↑ n ‖ = ↑ r ⊢ ↑ ( ↑ r ^ ( d - 1 ) ) . toNNReal • ⟪ ‖ ↑ r • ↑ n ‖ ^ q • ↑ r • basis . repr ↑ n , ∇ (⇑ η ) ( ↑ r • ↑ n ) ⟫_ ℝ =
↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ] e_f.e_f d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ n : ↑ ( Metric.sphere 0 1 ) r : ↑ ( Set.Ioi 0 ) hr : 0 < ↑ r hnorm : ‖ ↑ r • ↑ n ‖ = ↑ r ⊢ ↑ ( ↑ r ^ ( d - 1 ) ) . toNNReal • ⟪ ‖ ↑ r • ↑ n ‖ ^ q • ↑ r • basis . repr ↑ n , ∇ (⇑ η ) ( ↑ r • ↑ n ) ⟫_ ℝ =
↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r
rw [ Real.coe_toNNReal _ ( pow_nonneg ( le_of_lt hr ) ( d - 1 ) ) e_f.e_f d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ n : ↑ ( Metric.sphere 0 1 ) r : ↑ ( Set.Ioi 0 ) hr : 0 < ↑ r hnorm : ‖ ↑ r • ↑ n ‖ = ↑ r ⊢ ↑ r ^ ( d - 1 ) • ⟪ ‖ ↑ r • ↑ n ‖ ^ q • ↑ r • basis . repr ↑ n , ∇ (⇑ η ) ( ↑ r • ↑ n ) ⟫_ ℝ = ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r e_f.e_f d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ n : ↑ ( Metric.sphere 0 1 ) r : ↑ ( Set.Ioi 0 ) hr : 0 < ↑ r hnorm : ‖ ↑ r • ↑ n ‖ = ↑ r ⊢ ↑ r ^ ( d - 1 ) • ⟪ ‖ ↑ r • ↑ n ‖ ^ q • ↑ r • basis . repr ↑ n , ∇ (⇑ η ) ( ↑ r • ↑ n ) ⟫_ ℝ = ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ] e_f.e_f d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ n : ↑ ( Metric.sphere 0 1 ) r : ↑ ( Set.Ioi 0 ) hr : 0 < ↑ r hnorm : ‖ ↑ r • ↑ n ‖ = ↑ r ⊢ ↑ r ^ ( d - 1 ) • ⟪ ‖ ↑ r • ↑ n ‖ ^ q • ↑ r • basis . repr ↑ n , ∇ (⇑ η ) ( ↑ r • ↑ n ) ⟫_ ℝ = ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r
· e_f.e_f d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ n : ↑ ( Metric.sphere 0 1 ) r : ↑ ( Set.Ioi 0 ) hr : 0 < ↑ r hnorm : ‖ ↑ r • ↑ n ‖ = ↑ r ⊢ ↑ r ^ ( d - 1 ) • ⟪ ‖ ↑ r • ↑ n ‖ ^ q • ↑ r • basis . repr ↑ n , ∇ (⇑ η ) ( ↑ r • ↑ n ) ⟫_ ℝ = ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r simp only [ smul_eq_mul ] e_f.e_f d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ n : ↑ ( Metric.sphere 0 1 ) r : ↑ ( Set.Ioi 0 ) hr : 0 < ↑ r hnorm : ‖ ↑ r • ↑ n ‖ = ↑ r ⊢ ↑ r ^ ( d - 1 ) * ⟪ ‖ ↑ r • ↑ n ‖ ^ q • ↑ r • basis . repr ↑ n , ∇ (⇑ η ) ( ↑ r • ↑ n ) ⟫_ ℝ = ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r
rw [ hnorm , e_f.e_f d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ n : ↑ ( Metric.sphere 0 1 ) r : ↑ ( Set.Ioi 0 ) hr : 0 < ↑ r hnorm : ‖ ↑ r • ↑ n ‖ = ↑ r ⊢ ↑ r ^ ( d - 1 ) * ⟪ ↑ r ^ q • ↑ r • basis . repr ↑ n , ∇ (⇑ η ) ( ↑ r • ↑ n ) ⟫_ ℝ = ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r e_f.e_f d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ n : ↑ ( Metric.sphere 0 1 ) r : ↑ ( Set.Ioi 0 ) hr : 0 < ↑ r hnorm : ‖ ↑ r • ↑ n ‖ = ↑ r ⊢ ↑ r ^ ( d - 1 ) * ⟪ ∇ (⇑ η ) ( ↑ r • ↑ n ) , ↑ r ^ q • ↑ r • basis . repr ↑ n ⟫_ ℝ = ↑ r ^ p * ⟪ ∇ (⇑ η ) ( ↑ r • ↑ n ) , basis . repr ↑ n ⟫_ ℝ ← grad_smul_inner_space ( n : Space d ) ( ⇑ η )
( SchwartzMap.differentiable η ) ( r : ℝ ) hr , e_f.e_f d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ n : ↑ ( Metric.sphere 0 1 ) r : ↑ ( Set.Ioi 0 ) hr : 0 < ↑ r hnorm : ‖ ↑ r • ↑ n ‖ = ↑ r ⊢ ↑ r ^ ( d - 1 ) * ⟪ ↑ r ^ q • ↑ r • basis . repr ↑ n , ∇ (⇑ η ) ( ↑ r • ↑ n ) ⟫_ ℝ = ↑ r ^ p * ⟪ ∇ (⇑ η ) ( ↑ r • ↑ n ) , basis . repr ↑ n ⟫_ ℝ e_f.e_f d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ n : ↑ ( Metric.sphere 0 1 ) r : ↑ ( Set.Ioi 0 ) hr : 0 < ↑ r hnorm : ‖ ↑ r • ↑ n ‖ = ↑ r ⊢ ↑ r ^ ( d - 1 ) * ⟪ ∇ (⇑ η ) ( ↑ r • ↑ n ) , ↑ r ^ q • ↑ r • basis . repr ↑ n ⟫_ ℝ = ↑ r ^ p * ⟪ ∇ (⇑ η ) ( ↑ r • ↑ n ) , basis . repr ↑ n ⟫_ ℝ real_inner_comm e_f.e_f d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ n : ↑ ( Metric.sphere 0 1 ) r : ↑ ( Set.Ioi 0 ) hr : 0 < ↑ r hnorm : ‖ ↑ r • ↑ n ‖ = ↑ r ⊢ ↑ r ^ ( d - 1 ) * ⟪ ∇ (⇑ η ) ( ↑ r • ↑ n ) , ↑ r ^ q • ↑ r • basis . repr ↑ n ⟫_ ℝ = ↑ r ^ p * ⟪ ∇ (⇑ η ) ( ↑ r • ↑ n ) , basis . repr ↑ n ⟫_ ℝ e_f.e_f d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ n : ↑ ( Metric.sphere 0 1 ) r : ↑ ( Set.Ioi 0 ) hr : 0 < ↑ r hnorm : ‖ ↑ r • ↑ n ‖ = ↑ r ⊢ ↑ r ^ ( d - 1 ) * ⟪ ∇ (⇑ η ) ( ↑ r • ↑ n ) , ↑ r ^ q • ↑ r • basis . repr ↑ n ⟫_ ℝ = ↑ r ^ p * ⟪ ∇ (⇑ η ) ( ↑ r • ↑ n ) , basis . repr ↑ n ⟫_ ℝ ] e_f.e_f d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ n : ↑ ( Metric.sphere 0 1 ) r : ↑ ( Set.Ioi 0 ) hr : 0 < ↑ r hnorm : ‖ ↑ r • ↑ n ‖ = ↑ r ⊢ ↑ r ^ ( d - 1 ) * ⟪ ∇ (⇑ η ) ( ↑ r • ↑ n ) , ↑ r ^ q • ↑ r • basis . repr ↑ n ⟫_ ℝ = ↑ r ^ p * ⟪ ∇ (⇑ η ) ( ↑ r • ↑ n ) , basis . repr ↑ n ⟫_ ℝ
simp only [ inner_smul_right ] e_f.e_f d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ n : ↑ ( Metric.sphere 0 1 ) r : ↑ ( Set.Ioi 0 ) hr : 0 < ↑ r hnorm : ‖ ↑ r • ↑ n ‖ = ↑ r ⊢ ↑ r ^ ( d - 1 ) * ( ↑ r ^ q * ( ↑ r * ⟪ ∇ (⇑ η ) ( ↑ r • ↑ n ) , basis . repr ↑ n ⟫_ ℝ ) ) = ↑ r ^ p * ⟪ ∇ (⇑ η ) ( ↑ r • ↑ n ) , basis . repr ↑ n ⟫_ ℝ
rw [ ← radial_jacobian_zpow_mul_self hp_int hr e_f.e_f d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ n : ↑ ( Metric.sphere 0 1 ) r : ↑ ( Set.Ioi 0 ) hr : 0 < ↑ r hnorm : ‖ ↑ r • ↑ n ‖ = ↑ r ⊢ ↑ r ^ ( d - 1 ) * ( ↑ r ^ q * ( ↑ r * ⟪ ∇ (⇑ η ) ( ↑ r • ↑ n ) , basis . repr ↑ n ⟫_ ℝ ) ) =
↑ r ^ ( d - 1 ) * ( ↑ r ^ q * ↑ r ) * ⟪ ∇ (⇑ η ) ( ↑ r • ↑ n ) , basis . repr ↑ n ⟫_ ℝ e_f.e_f d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ n : ↑ ( Metric.sphere 0 1 ) r : ↑ ( Set.Ioi 0 ) hr : 0 < ↑ r hnorm : ‖ ↑ r • ↑ n ‖ = ↑ r ⊢ ↑ r ^ ( d - 1 ) * ( ↑ r ^ q * ( ↑ r * ⟪ ∇ (⇑ η ) ( ↑ r • ↑ n ) , basis . repr ↑ n ⟫_ ℝ ) ) =
↑ r ^ ( d - 1 ) * ( ↑ r ^ q * ↑ r ) * ⟪ ∇ (⇑ η ) ( ↑ r • ↑ n ) , basis . repr ↑ n ⟫_ ℝ ] e_f.e_f d : ℕ p : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d hp_int : ↑ p = q + ↑ d η : 𝓢( Space d , ℝ ) F : Space d → ℝ := fun x => ⟪ ‖ x ‖ ^ q • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ n : ↑ ( Metric.sphere 0 1 ) r : ↑ ( Set.Ioi 0 ) hr : 0 < ↑ r hnorm : ‖ ↑ r • ↑ n ‖ = ↑ r ⊢ ↑ r ^ ( d - 1 ) * ( ↑ r ^ q * ( ↑ r * ⟪ ∇ (⇑ η ) ( ↑ r • ↑ n ) , basis . repr ↑ n ⟫_ ℝ ) ) =
↑ r ^ ( d - 1 ) * ( ↑ r ^ q * ↑ r ) * ⟪ ∇ (⇑ η ) ( ↑ r • ↑ n ) , basis . repr ↑ n ⟫_ ℝ
ring All goals completed! 🐙
lemma distDiv_norm_zpow_smul_repr_self_eq_smul
{ d : ℕ } [ NeZero d ] ( q : ℤ ) ( hq : 0 < q + ( d : ℤ ) ) :
∇ᵈ ⬝ ( distOfFunction ( fun x : Space d => ‖ x ‖ ^ q • basis . repr x )
( IsDistBounded.zpow_smul_repr_self q ( by 𝕜 : Type E : Type F : Type F' : Type inst✝⁶ : RCLike 𝕜 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedAddCommGroup F' inst✝² : NormedSpace ℝ E inst✝¹ : NormedSpace ℝ F d : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d ⊢ - ↑ ( d - 1 ) - 1 ≤ q omega All goals completed! 🐙 ) ) ) =
( ( ( q + d : ℤ ) : ℝ ) •
distOfFunction ( fun x : Space d => ‖ x ‖ ^ q )
( IsDistBounded.pow q ( by 𝕜 : Type E : Type F : Type F' : Type inst✝⁶ : RCLike 𝕜 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedAddCommGroup F' inst✝² : NormedSpace ℝ E inst✝¹ : NormedSpace ℝ F d : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d ⊢ - ↑ ( d - 1 ) ≤ q omega All goals completed! 🐙 ) ) ) := by d : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d ⊢ distDiv ( distOfFunction (fun x => ‖ x ‖ ^ q • basis . repr x ) ⋯ ) = ↑ ( q + ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ q ) ⋯
ext η d : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d η : 𝓢( Space d , ℝ ) ⊢ ( distDiv ( distOfFunction (fun x => ‖ x ‖ ^ q • basis . repr x ) ⋯ ) ) η = ( ↑ ( q + ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ q ) ⋯ ) η
let p : ℕ := Int.toNat ( q + ( d : ℤ ) ) d : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d η : 𝓢( Space d , ℝ ) p : ℕ := ( q + ↑ d ) . toNat ⊢ ( distDiv ( distOfFunction (fun x => ‖ x ‖ ^ q • basis . repr x ) ⋯ ) ) η = ( ↑ ( q + ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ q ) ⋯ ) η
have hp_int : ( p : ℤ ) = q + ( d : ℤ ) := by d : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d ⊢ distDiv ( distOfFunction (fun x => ‖ x ‖ ^ q • basis . repr x ) ⋯ ) = ↑ ( q + ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ q ) ⋯ d : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d η : 𝓢( Space d , ℝ ) p : ℕ := ( q + ↑ d ) . toNat hp_int : ↑ p = q + ↑ d ⊢ ( distDiv ( distOfFunction (fun x => ‖ x ‖ ^ q • basis . repr x ) ⋯ ) ) η = ( ↑ ( q + ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ q ) ⋯ ) η simpa [ p ] using Int.toNat_of_nonneg ( le_of_lt hq ) d : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d η : 𝓢( Space d , ℝ ) p : ℕ := ( q + ↑ d ) . toNat hp_int : ↑ p = q + ↑ d ⊢ ( distDiv ( distOfFunction (fun x => ‖ x ‖ ^ q • basis . repr x ) ⋯ ) ) η = ( ↑ ( q + ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ q ) ⋯ ) η d : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d η : 𝓢( Space d , ℝ ) p : ℕ := ( q + ↑ d ) . toNat hp_int : ↑ p = q + ↑ d ⊢ ( distDiv ( distOfFunction (fun x => ‖ x ‖ ^ q • basis . repr x ) ⋯ ) ) η = ( ↑ ( q + ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ q ) ⋯ ) η
have hp_pos : 0 < p := by d : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d ⊢ distDiv ( distOfFunction (fun x => ‖ x ‖ ^ q • basis . repr x ) ⋯ ) = ↑ ( q + ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ q ) ⋯ d : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d η : 𝓢( Space d , ℝ ) p : ℕ := ( q + ↑ d ) . toNat hp_int : ↑ p = q + ↑ d hp_pos : 0 < p ⊢ ( distDiv ( distOfFunction (fun x => ‖ x ‖ ^ q • basis . repr x ) ⋯ ) ) η = ( ↑ ( q + ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ q ) ⋯ ) η omega d : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d η : 𝓢( Space d , ℝ ) p : ℕ := ( q + ↑ d ) . toNat hp_int : ↑ p = q + ↑ d hp_pos : 0 < p ⊢ ( distDiv ( distOfFunction (fun x => ‖ x ‖ ^ q • basis . repr x ) ⋯ ) ) η = ( ↑ ( q + ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ q ) ⋯ ) η d : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d η : 𝓢( Space d , ℝ ) p : ℕ := ( q + ↑ d ) . toNat hp_int : ↑ p = q + ↑ d hp_pos : 0 < p ⊢ ( distDiv ( distOfFunction (fun x => ‖ x ‖ ^ q • basis . repr x ) ⋯ ) ) η = ( ↑ ( q + ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ q ) ⋯ ) η
have hcoef : ( ( ( q + d : ℤ ) : ℝ ) ) = ( p : ℝ ) := by d : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d ⊢ distDiv ( distOfFunction (fun x => ‖ x ‖ ^ q • basis . repr x ) ⋯ ) = ↑ ( q + ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ q ) ⋯ d : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d η : 𝓢( Space d , ℝ ) p : ℕ := ( q + ↑ d ) . toNat hp_int : ↑ p = q + ↑ d hp_pos : 0 < p hcoef : ↑ ( q + ↑ d ) = ↑ p ⊢ ( distDiv ( distOfFunction (fun x => ‖ x ‖ ^ q • basis . repr x ) ⋯ ) ) η = ( ↑ ( q + ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ q ) ⋯ ) η
exact_mod_cast hp_int . symm d : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d η : 𝓢( Space d , ℝ ) p : ℕ := ( q + ↑ d ) . toNat hp_int : ↑ p = q + ↑ d hp_pos : 0 < p hcoef : ↑ ( q + ↑ d ) = ↑ p ⊢ ( distDiv ( distOfFunction (fun x => ‖ x ‖ ^ q • basis . repr x ) ⋯ ) ) η = ( ↑ ( q + ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ q ) ⋯ ) η d : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d η : 𝓢( Space d , ℝ ) p : ℕ := ( q + ↑ d ) . toNat hp_int : ↑ p = q + ↑ d hp_pos : 0 < p hcoef : ↑ ( q + ↑ d ) = ↑ p ⊢ ( distDiv ( distOfFunction (fun x => ‖ x ‖ ^ q • basis . repr x ) ⋯ ) ) η = ( ↑ ( q + ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ q ) ⋯ ) η
calc
( ∇ᵈ ⬝ ( distOfFunction ( fun x : Space d => ‖ x ‖ ^ q • basis . repr x )
( IsDistBounded.zpow_smul_repr_self q ( by d : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d η : 𝓢( Space d , ℝ ) p : ℕ := ( q + ↑ d ) . toNat hp_int : ↑ p = q + ↑ d hp_pos : 0 < p hcoef : ↑ ( q + ↑ d ) = ↑ p ⊢ - ↑ ( d - 1 ) - 1 ≤ q omega All goals completed! 🐙 ) ) ) ) η
= - ∫ n , ( ∫ ( r : Set.Ioi ( 0 : ℝ ) ) ,
r . 1 ^ p * ( _root_.deriv ( fun a => η ( a • n . 1 ) ) r . 1 )
∂ ( .comap Subtype.val volume ) )
∂ ( volume ( α := Space d ) . toSphere ) := by d : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d η : 𝓢( Space d , ℝ ) p : ℕ := ( q + ↑ d ) . toNat hp_int : ↑ p = q + ↑ d hp_pos : 0 < p hcoef : ↑ ( q + ↑ d ) = ↑ p ⊢ ( distDiv ( distOfFunction (fun x => ‖ x ‖ ^ q • basis . repr x ) ⋯ ) ) η =
- ∫ ( n : ↑ ( Metric.sphere 0 1 ) ) ,
∫ ( r : ↑ ( Set.Ioi 0 ) ) ,
↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume ∂ volume . toSphere
exact distDiv_norm_zpow_smul_repr_self_apply_eq_radial_deriv q hq hp_int η All goals completed! 🐙
_ = ∫ n , ( p : ℝ ) * ∫ ( r : Set.Ioi ( 0 : ℝ ) ) ,
r . 1 ^ ( p - 1 ) * η ( r . 1 • n . 1 )
∂ ( .comap Subtype.val volume )
∂ ( volume ( α := Space d ) . toSphere ) := by d : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d η : 𝓢( Space d , ℝ ) p : ℕ := ( q + ↑ d ) . toNat hp_int : ↑ p = q + ↑ d hp_pos : 0 < p hcoef : ↑ ( q + ↑ d ) = ↑ p ⊢ - ∫ ( n : ↑ ( Metric.sphere 0 1 ) ) ,
∫ ( r : ↑ ( Set.Ioi 0 ) ) ,
↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume ∂ volume . toSphere =
∫ ( n : ↑ ( Metric.sphere 0 1 ) ) ,
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume ∂ volume . toSphere
rw [ ← integral_neg d : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d η : 𝓢( Space d , ℝ ) p : ℕ := ( q + ↑ d ) . toNat hp_int : ↑ p = q + ↑ d hp_pos : 0 < p hcoef : ↑ ( q + ↑ d ) = ↑ p ⊢ ∫ ( a : ↑ ( Metric.sphere 0 1 ) ) ,
- ∫ ( r : ↑ ( Set.Ioi 0 ) ) ,
↑ r ^ p * _root_.deriv (fun a_1 => η ( a_1 • ↑ a ) ) ↑ r ∂ Measure.comap Subtype.val volume ∂ volume . toSphere =
∫ ( n : ↑ ( Metric.sphere 0 1 ) ) ,
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume ∂ volume . toSphere d : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d η : 𝓢( Space d , ℝ ) p : ℕ := ( q + ↑ d ) . toNat hp_int : ↑ p = q + ↑ d hp_pos : 0 < p hcoef : ↑ ( q + ↑ d ) = ↑ p ⊢ ∫ ( a : ↑ ( Metric.sphere 0 1 ) ) ,
- ∫ ( r : ↑ ( Set.Ioi 0 ) ) ,
↑ r ^ p * _root_.deriv (fun a_1 => η ( a_1 • ↑ a ) ) ↑ r ∂ Measure.comap Subtype.val volume ∂ volume . toSphere =
∫ ( n : ↑ ( Metric.sphere 0 1 ) ) ,
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume ∂ volume . toSphere ] d : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d η : 𝓢( Space d , ℝ ) p : ℕ := ( q + ↑ d ) . toNat hp_int : ↑ p = q + ↑ d hp_pos : 0 < p hcoef : ↑ ( q + ↑ d ) = ↑ p ⊢ ∫ ( a : ↑ ( Metric.sphere 0 1 ) ) ,
- ∫ ( r : ↑ ( Set.Ioi 0 ) ) ,
↑ r ^ p * _root_.deriv (fun a_1 => η ( a_1 • ↑ a ) ) ↑ r ∂ Measure.comap Subtype.val volume ∂ volume . toSphere =
∫ ( n : ↑ ( Metric.sphere 0 1 ) ) ,
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume ∂ volume . toSphere
congr e_f d : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d η : 𝓢( Space d , ℝ ) p : ℕ := ( q + ↑ d ) . toNat hp_int : ↑ p = q + ↑ d hp_pos : 0 < p hcoef : ↑ ( q + ↑ d ) = ↑ p ⊢ (fun a => - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a_1 => η ( a_1 • ↑ a ) ) ↑ r ∂ Measure.comap Subtype.val volume ) =
fun n => ↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume
funext n e_f d : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d η : 𝓢( Space d , ℝ ) p : ℕ := ( q + ↑ d ) . toNat hp_int : ↑ p = q + ↑ d hp_pos : 0 < p hcoef : ↑ ( q + ↑ d ) = ↑ p n : ↑ ( Metric.sphere 0 1 ) ⊢ - ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ p * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume =
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume
exact radial_power_deriv_integral_by_parts η n p hp_pos All goals completed! 🐙
_ = ( p : ℝ ) * ∫ n : ↑ ( Metric.sphere ( 0 : Space d ) 1 ) ,
∫ ( r : Set.Ioi ( 0 : ℝ ) ) ,
r . 1 ^ ( p - 1 ) * η ( r . 1 • n . 1 )
∂ ( .comap Subtype.val volume )
∂ ( volume ( α := Space d ) . toSphere ) := by d : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d η : 𝓢( Space d , ℝ ) p : ℕ := ( q + ↑ d ) . toNat hp_int : ↑ p = q + ↑ d hp_pos : 0 < p hcoef : ↑ ( q + ↑ d ) = ↑ p ⊢ ∫ ( n : ↑ ( Metric.sphere 0 1 ) ) ,
↑ p * ∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume ∂ volume . toSphere =
↑ p *
∫ ( n : ↑ ( Metric.sphere 0 1 ) ) ,
∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume ∂ volume . toSphere
rw [ integral_const_mul d : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d η : 𝓢( Space d , ℝ ) p : ℕ := ( q + ↑ d ) . toNat hp_int : ↑ p = q + ↑ d hp_pos : 0 < p hcoef : ↑ ( q + ↑ d ) = ↑ p ⊢ ↑ p *
∫ ( a : ↑ ( Metric.sphere 0 1 ) ) ,
∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ a ) ∂ Measure.comap Subtype.val volume ∂ volume . toSphere =
↑ p *
∫ ( n : ↑ ( Metric.sphere 0 1 ) ) ,
∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume ∂ volume . toSphere All goals completed! 🐙 ] All goals completed! 🐙
_ = ( p : ℝ ) * ∫ x : Space d , η x * ‖ x ‖ ^ q := by d : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d η : 𝓢( Space d , ℝ ) p : ℕ := ( q + ↑ d ) . toNat hp_int : ↑ p = q + ↑ d hp_pos : 0 < p hcoef : ↑ ( q + ↑ d ) = ↑ p ⊢ ↑ p *
∫ ( n : ↑ ( Metric.sphere 0 1 ) ) ,
∫ ( r : ↑ ( Set.Ioi 0 ) ) , ↑ r ^ ( p - 1 ) * η ( ↑ r • ↑ n ) ∂ Measure.comap Subtype.val volume ∂ volume . toSphere =
↑ p * ∫ ( x : Space d ) , η x * ‖ x ‖ ^ q
rw [ ← radial_norm_power_spherical_integral_eq_space_integral hp_int hp_pos η d : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d η : 𝓢( Space d , ℝ ) p : ℕ := ( q + ↑ d ) . toNat hp_int : ↑ p = q + ↑ d hp_pos : 0 < p hcoef : ↑ ( q + ↑ d ) = ↑ p ⊢ ↑ p * ∫ ( x : Space d ) , η x * ‖ x ‖ ^ q = ↑ p * ∫ ( x : Space d ) , η x * ‖ x ‖ ^ q All goals completed! 🐙 ] All goals completed! 🐙
_ = ( ( ( q + ( d : ℤ ) : ℤ ) : ℝ ) •
distOfFunction ( fun x : Space d => ‖ x ‖ ^ q )
( IsDistBounded.pow q ( by d : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d η : 𝓢( Space d , ℝ ) p : ℕ := ( q + ↑ d ) . toNat hp_int : ↑ p = q + ↑ d hp_pos : 0 < p hcoef : ↑ ( q + ↑ d ) = ↑ p ⊢ - ↑ ( d - 1 ) ≤ q omega All goals completed! 🐙 ) ) ) η := by d : ℕ inst✝ : NeZero d q : ℤ hq : 0 < q + ↑ d η : 𝓢( Space d , ℝ ) p : ℕ := ( q + ↑ d ) . toNat hp_int : ↑ p = q + ↑ d hp_pos : 0 < p hcoef : ↑ ( q + ↑ d ) = ↑ p ⊢ ↑ p * ∫ ( x : Space d ) , η x * ‖ x ‖ ^ q = ( ↑ ( q + ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ q ) ⋯ ) η
simp [ distOfFunction_apply , hcoef ] All goals completed! 🐙
B.4. The Laplacian of distributions based on powers
B.5. Divergence equal dirac delta
We show that the divergence of x ↦ ‖x‖ ^ (- d) • x is equal to a multiple of the Dirac delta
at 0.
The distributional divergence of the radial field x ↦ ‖x‖ ^ (-d) • x (i.e. x / ‖x‖ ^ d)
equals d * volume (Metric.ball 0 1) — the surface area of the unit sphere S^{d-1} — times the
Dirac delta at the origin. This is the Gauss-law identity underlying the fundamental solution of
the Laplacian: away from 0 the field is divergence-free, and all of its flux concentrates at the
origin.
lemma distDiv_inv_pow_eq_dim { d : ℕ } [ NeZero d ] :
∇ᵈ ⬝ ( distOfFunction ( fun x : Space d => ‖ x ‖ ^ ( - d : ℤ ) • basis . repr x )
( IsDistBounded.zpow_smul_repr_self ( - d : ℤ ) ( by 𝕜 : Type E : Type F : Type F' : Type inst✝⁶ : RCLike 𝕜 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedAddCommGroup F' inst✝² : NormedSpace ℝ E inst✝¹ : NormedSpace ℝ F d : ℕ inst✝ : NeZero d ⊢ - ↑ ( d - 1 ) - 1 ≤ - ↑ d omega All goals completed! 🐙 ) ) ) =
( d * ( volume ( α := Space d ) ) . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0 := by d : ℕ inst✝ : NeZero d ⊢ distDiv ( distOfFunction (fun x => ‖ x ‖ ^ ( - ↑ d ) • basis . repr x ) ⋯ ) = ( ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0
ext η d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) ⊢ ( distDiv ( distOfFunction (fun x => ‖ x ‖ ^ ( - ↑ d ) • basis . repr x ) ⋯ ) ) η =
( ( ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0 ) η
calc _
_ = - ∫ x , ⟪ ‖ x ‖ ⁻¹ ^ d • basis . repr x , Space.grad η x ⟫_ ℝ := by d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) ⊢ ( distDiv ( distOfFunction (fun x => ‖ x ‖ ^ ( - ↑ d ) • basis . repr x ) ⋯ ) ) η =
- ∫ ( x : Space d ) , ⟪ ‖ x ‖ ⁻¹ ^ d • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ
simp only [ zpow_neg , zpow_natCast , distDiv_ofFunction , inv_pow ] All goals completed! 🐙
_ = - ∫ x , ‖ x ‖ ⁻¹ ^ ( d - 1 ) * ⟪ ‖ x ‖ ⁻¹ • basis . repr x , Space.grad η x ⟫_ ℝ := by d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) ⊢ - ∫ ( x : Space d ) , ⟪ ‖ x ‖ ⁻¹ ^ d • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ =
- ∫ ( x : Space d ) , ‖ x ‖ ⁻¹ ^ ( d - 1 ) * ⟪ ‖ x ‖ ⁻¹ • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ
simp only [ ← pow_sub_one_mul ( NeZero.ne d ) , inv_pow , inner_smul_left , conj_trivial ,
map_inv₀ , neg_inj ] d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) ⊢ ∫ ( x : Space d ) , ( ‖ x ‖ ^ ( d - 1 ) ) ⁻¹ * ‖ x ‖ ⁻¹ * ⟪ basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ =
∫ ( x : Space d ) , ( ‖ x ‖ ^ ( d - 1 ) ) ⁻¹ * ( ‖ x ‖ ⁻¹ * ⟪ basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ )
ring_nf All goals completed! 🐙
_ = - ∫ x , ‖ x ‖ ⁻¹ ^ ( d - 1 ) * ( _root_.deriv ( fun a => η ( a • ‖ x ‖ ⁻¹ • x ) ) ‖ x ‖ ) := by d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) ⊢ - ∫ ( x : Space d ) , ‖ x ‖ ⁻¹ ^ ( d - 1 ) * ⟪ ‖ x ‖ ⁻¹ • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ =
- ∫ ( x : Space d ) , ‖ x ‖ ⁻¹ ^ ( d - 1 ) * _root_.deriv (fun a => η ( a • ‖ x ‖ ⁻¹ • x ) ) ‖ x ‖
simp only [ real_inner_comm ,
← grad_inner_space_unit_vector _ _ ( SchwartzMap.differentiable η ) ] All goals completed! 🐙
_ = - ∫ r , ‖ r . 2 . 1 ‖ ⁻¹ ^ ( d - 1 ) * ( _root_.deriv ( fun a => η ( a • r . 1 ) ) ‖ r . 2 . 1 ‖ )
∂ ( volume ( α := Space d ) . toSphere . prod
( Measure.volumeIoiPow ( Module.finrank ℝ ( Space d ) - 1 ) ) ) := by d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) ⊢ - ∫ ( x : Space d ) , ‖ x ‖ ⁻¹ ^ ( d - 1 ) * _root_.deriv (fun a => η ( a • ‖ x ‖ ⁻¹ • x ) ) ‖ x ‖ =
- ∫ ( r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) ) ,
‖ ↑ r .2 ‖ ⁻¹ ^ ( d - 1 ) *
_root_.deriv (fun a => η ( a • ↑ r .1 ) )
‖ ↑ r .2 ‖ ∂ volume . toSphere . prod ( Measure.volumeIoiPow ( Module.finrank ℝ ( Space d ) - 1 ) )
rw [ ← MeasureTheory.MeasurePreserving.integral_comp ( f := homeomorphUnitSphereProd _ )
( MeasureTheory.Measure.measurePreserving_homeomorphUnitSphereProd
( volume ( α := Space d ) ) )
( Homeomorph.measurableEmbedding ( homeomorphUnitSphereProd ( Space d ) ) ) d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) ⊢ - ∫ ( x : Space d ) , ‖ x ‖ ⁻¹ ^ ( d - 1 ) * _root_.deriv (fun a => η ( a • ‖ x ‖ ⁻¹ • x ) ) ‖ x ‖ =
- ∫ ( x : ↑ { 0 } ᶜ ) ,
‖ ↑ ( ( homeomorphUnitSphereProd ( Space d ) ) x ) .2 ‖ ⁻¹ ^ ( d - 1 ) *
_root_.deriv (fun a => η ( a • ↑ ( ( homeomorphUnitSphereProd ( Space d ) ) x ) .1 ) )
‖ ↑ ( ( homeomorphUnitSphereProd ( Space d ) ) x ) .2 ‖ ∂ Measure.comap Subtype.val volume d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) ⊢ - ∫ ( x : Space d ) , ‖ x ‖ ⁻¹ ^ ( d - 1 ) * _root_.deriv (fun a => η ( a • ‖ x ‖ ⁻¹ • x ) ) ‖ x ‖ =
- ∫ ( x : ↑ { 0 } ᶜ ) ,
‖ ↑ ( ( homeomorphUnitSphereProd ( Space d ) ) x ) .2 ‖ ⁻¹ ^ ( d - 1 ) *
_root_.deriv (fun a => η ( a • ↑ ( ( homeomorphUnitSphereProd ( Space d ) ) x ) .1 ) )
‖ ↑ ( ( homeomorphUnitSphereProd ( Space d ) ) x ) .2 ‖ ∂ Measure.comap Subtype.val volume ] d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) ⊢ - ∫ ( x : Space d ) , ‖ x ‖ ⁻¹ ^ ( d - 1 ) * _root_.deriv (fun a => η ( a • ‖ x ‖ ⁻¹ • x ) ) ‖ x ‖ =
- ∫ ( x : ↑ { 0 } ᶜ ) ,
‖ ↑ ( ( homeomorphUnitSphereProd ( Space d ) ) x ) .2 ‖ ⁻¹ ^ ( d - 1 ) *
_root_.deriv (fun a => η ( a • ↑ ( ( homeomorphUnitSphereProd ( Space d ) ) x ) .1 ) )
‖ ↑ ( ( homeomorphUnitSphereProd ( Space d ) ) x ) .2 ‖ ∂ Measure.comap Subtype.val volume
congr 1 d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) ⊢ ∫ ( x : Space d ) , ‖ x ‖ ⁻¹ ^ ( d - 1 ) * _root_.deriv (fun a => η ( a • ‖ x ‖ ⁻¹ • x ) ) ‖ x ‖ =
∫ ( x : ↑ { 0 } ᶜ ) ,
‖ ↑ ( ( homeomorphUnitSphereProd ( Space d ) ) x ) .2 ‖ ⁻¹ ^ ( d - 1 ) *
_root_.deriv (fun a => η ( a • ↑ ( ( homeomorphUnitSphereProd ( Space d ) ) x ) .1 ) )
‖ ↑ ( ( homeomorphUnitSphereProd ( Space d ) ) x ) .2 ‖ ∂ Measure.comap Subtype.val volume
simp only [ inv_pow , homeomorphUnitSphereProd_apply_snd_coe , norm_norm ,
homeomorphUnitSphereProd_apply_fst_coe ] d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) ⊢ ∫ ( x : Space d ) , ( ‖ x ‖ ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ‖ x ‖ ⁻¹ • x ) ) ‖ x ‖ =
∫ ( x : ↑ { 0 } ᶜ ) , ( ‖ ↑ x ‖ ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ‖ ↑ x ‖ ⁻¹ • ↑ x ) ) ‖ ↑ x ‖ ∂ Measure.comap Subtype.val volume
let f ( x : Space d ) : ℝ :=
( ‖ ↑ x ‖ ^ ( d - 1 ) ) ⁻¹ * _root_.deriv ( fun a => η ( a • ‖ ↑ x ‖ ⁻¹ • ↑ x ) ) ‖ ↑ x ‖ d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) f : Space d → ℝ := fun x => ( ‖ x ‖ ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ‖ x ‖ ⁻¹ • x ) ) ‖ x ‖ ⊢ ∫ ( x : Space d ) , ( ‖ x ‖ ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ‖ x ‖ ⁻¹ • x ) ) ‖ x ‖ =
∫ ( x : ↑ { 0 } ᶜ ) , ( ‖ ↑ x ‖ ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ‖ ↑ x ‖ ⁻¹ • ↑ x ) ) ‖ ↑ x ‖ ∂ Measure.comap Subtype.val volume
conv_rhs =>
enter [ 2 , x ] d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) f : Space d → ℝ := fun x => ( ‖ x ‖ ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ‖ x ‖ ⁻¹ • x ) ) ‖ x ‖ x : ↑ { 0 } ᶜ | ( ‖ ↑ x ‖ ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ‖ ↑ x ‖ ⁻¹ • ↑ x ) ) ‖ ↑ x ‖
change f x . 1 d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) f : Space d → ℝ := fun x => ( ‖ x ‖ ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ‖ x ‖ ⁻¹ • x ) ) ‖ x ‖ x : ↑ { 0 } ᶜ | f ↑ x
rw [ MeasureTheory.integral_subtype_comap ( by d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) f : Space d → ℝ := fun x => ( ‖ x ‖ ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ‖ x ‖ ⁻¹ • x ) ) ‖ x ‖ ⊢ MeasurableSet { 0 } ᶜ d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) f : Space d → ℝ := fun x => ( ‖ x ‖ ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ‖ x ‖ ⁻¹ • x ) ) ‖ x ‖ ⊢ ∫ ( x : Space d ) in Set.univ , ( ‖ x ‖ ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ‖ x ‖ ⁻¹ • x ) ) ‖ x ‖ =
∫ ( x : Space d ) in { 0 } ᶜ , f x simp All goals completed! 🐙 d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) f : Space d → ℝ := fun x => ( ‖ x ‖ ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ‖ x ‖ ⁻¹ • x ) ) ‖ x ‖ ⊢ ∫ ( x : Space d ) in Set.univ , ( ‖ x ‖ ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ‖ x ‖ ⁻¹ • x ) ) ‖ x ‖ =
∫ ( x : Space d ) in { 0 } ᶜ , f x ) , ← setIntegral_univ d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) f : Space d → ℝ := fun x => ( ‖ x ‖ ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ‖ x ‖ ⁻¹ • x ) ) ‖ x ‖ ⊢ ∫ ( x : Space d ) in Set.univ , ( ‖ x ‖ ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ‖ x ‖ ⁻¹ • x ) ) ‖ x ‖ =
∫ ( x : Space d ) in { 0 } ᶜ , f x d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) f : Space d → ℝ := fun x => ( ‖ x ‖ ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ‖ x ‖ ⁻¹ • x ) ) ‖ x ‖ ⊢ ∫ ( x : Space d ) in Set.univ , ( ‖ x ‖ ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ‖ x ‖ ⁻¹ • x ) ) ‖ x ‖ =
∫ ( x : Space d ) in { 0 } ᶜ , f x ] d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) f : Space d → ℝ := fun x => ( ‖ x ‖ ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ‖ x ‖ ⁻¹ • x ) ) ‖ x ‖ ⊢ ∫ ( x : Space d ) in Set.univ , ( ‖ x ‖ ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ‖ x ‖ ⁻¹ • x ) ) ‖ x ‖ =
∫ ( x : Space d ) in { 0 } ᶜ , f x
change ∫ x in Set.univ , f x = ∫ ( x : Space d ) in _ , f x d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) f : Space d → ℝ := fun x => ( ‖ x ‖ ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ‖ x ‖ ⁻¹ • x ) ) ‖ x ‖ ⊢ ∫ ( x : Space d ) in Set.univ , f x = ∫ ( x : Space d ) in { 0 } ᶜ , f x
exact setIntegral_congr_set ( MeasureTheory.ae_eq_univ . mpr ( by d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) f : Space d → ℝ := fun x => ( ‖ x ‖ ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ‖ x ‖ ⁻¹ • x ) ) ‖ x ‖ ⊢ volume { 0 } ᶜ ᶜ = 0 simp All goals completed! 🐙 ) ) . symm
_ = - ∫ n , ( ∫ r , ‖ r . 1 ‖ ⁻¹ ^ ( d - 1 ) *
( _root_.deriv ( fun a => η ( a • n ) ) ‖ r . 1 ‖ )
∂ ( ( Measure.volumeIoiPow ( Module.finrank ℝ ( Space d ) - 1 ) ) ) )
∂ ( volume ( α := Space d ) . toSphere ) := by d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) ⊢ - ∫ ( r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) ) ,
‖ ↑ r .2 ‖ ⁻¹ ^ ( d - 1 ) *
_root_.deriv (fun a => η ( a • ↑ r .1 ) )
‖ ↑ r .2 ‖ ∂ volume . toSphere . prod ( Measure.volumeIoiPow ( Module.finrank ℝ ( Space d ) - 1 ) ) =
- ∫ ( n : ↑ ( Metric.sphere 0 1 ) ) ,
∫ ( r : ↑ ( Set.Ioi 0 ) ) ,
‖ ↑ r ‖ ⁻¹ ^ ( d - 1 ) *
_root_.deriv (fun a => η ( a • ↑ n ) )
‖ ↑ r ‖ ∂ Measure.volumeIoiPow ( Module.finrank ℝ ( Space d ) - 1 ) ∂ volume . toSphere
rw [ MeasureTheory.integral_prod d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) ⊢ - ∫ ( x : ↑ ( Metric.sphere 0 1 ) ) ,
∫ ( y : ↑ ( Set.Ioi 0 ) ) ,
‖ ↑ ( x , y ) .2 ‖ ⁻¹ ^ ( d - 1 ) *
_root_.deriv (fun a => η ( a • ↑ ( x , y ) .1 ) )
‖ ↑ ( x , y ) .2 ‖ ∂ Measure.volumeIoiPow ( Module.finrank ℝ ( Space d ) - 1 ) ∂ volume . toSphere =
- ∫ ( n : ↑ ( Metric.sphere 0 1 ) ) ,
∫ ( r : ↑ ( Set.Ioi 0 ) ) ,
‖ ↑ r ‖ ⁻¹ ^ ( d - 1 ) *
_root_.deriv (fun a => η ( a • ↑ n ) )
‖ ↑ r ‖ ∂ Measure.volumeIoiPow ( Module.finrank ℝ ( Space d ) - 1 ) ∂ volume . toSphere hf d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) ⊢ Integrable (fun r => ‖ ↑ r .2 ‖ ⁻¹ ^ ( d - 1 ) * _root_.deriv (fun a => η ( a • ↑ r .1 ) ) ‖ ↑ r .2 ‖ )
( volume . toSphere . prod ( Measure.volumeIoiPow ( Module.finrank ℝ ( Space d ) - 1 ) ) ) hf d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) ⊢ Integrable (fun r => ‖ ↑ r .2 ‖ ⁻¹ ^ ( d - 1 ) * _root_.deriv (fun a => η ( a • ↑ r .1 ) ) ‖ ↑ r .2 ‖ )
( volume . toSphere . prod ( Measure.volumeIoiPow ( Module.finrank ℝ ( Space d ) - 1 ) ) ) ] hf d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) ⊢ Integrable (fun r => ‖ ↑ r .2 ‖ ⁻¹ ^ ( d - 1 ) * _root_.deriv (fun a => η ( a • ↑ r .1 ) ) ‖ ↑ r .2 ‖ )
( volume . toSphere . prod ( Measure.volumeIoiPow ( Module.finrank ℝ ( Space d ) - 1 ) ) )
convert integrable_isDistBounded_inner_grad_schwartzMap_spherical
( IsDistBounded.inv_pow_smul_repr_self ( d ) ( by d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) ⊢ - ↑ ( d - 1 ) - 1 ≤ - ↑ d omega All goals completed! 🐙 ) ) η
rename_i r d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) ⊢ ‖ ↑ r .2 ‖ ⁻¹ ^ ( d - 1 ) * _root_.deriv (fun a => η ( a • ↑ r .1 ) ) ‖ ↑ r .2 ‖ =
( (fun x => ⟪ ‖ ↑ x ‖ ⁻¹ ^ d • basis . repr ↑ x , ∇ ⇑ η ↑ x ⟫_ ℝ ) ∘ ⇑ ( homeomorphUnitSphereProd ( Space d ) ) . symm ) r
simp only [ Real.norm_eq_abs , inv_pow , Function.comp_apply ,
homeomorphUnitSphereProd_symm_apply_coe , map_smul ] d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) ⊢ ( | ↑ r .2 | ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ↑ r .1 ) ) | ↑ r .2 | =
⟪ ( ‖ ↑ r .2 • ↑ r .1 ‖ ^ d ) ⁻¹ • ↑ r .2 • basis . repr ↑ r .1 , ∇ (⇑ η ) ( ↑ r .2 • ↑ r .1 ) ⟫_ ℝ
let x : Space d := r . 2 . 1 • r . 1 . 1 d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 ⊢ ( | ↑ r .2 | ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ↑ r .1 ) ) | ↑ r .2 | =
⟪ ( ‖ ↑ r .2 • ↑ r .1 ‖ ^ d ) ⁻¹ • ↑ r .2 • basis . repr ↑ r .1 , ∇ (⇑ η ) ( ↑ r .2 • ↑ r .1 ) ⟫_ ℝ
have hr : ( 0 : ℝ ) < r . 2 . 1 := r . 2 . 2 d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ ( | ↑ r .2 | ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ↑ r .1 ) ) | ↑ r .2 | =
⟪ ( ‖ ↑ r .2 • ↑ r .1 ‖ ^ d ) ⁻¹ • ↑ r .2 • basis . repr ↑ r .1 , ∇ (⇑ η ) ( ↑ r .2 • ↑ r .1 ) ⟫_ ℝ
rw [ abs_of_nonneg ( le_of_lt hr ) d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ ( ↑ r .2 ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ↑ r .1 ) ) ↑ r .2 =
⟪ ( ‖ ↑ r .2 • ↑ r .1 ‖ ^ d ) ⁻¹ • ↑ r .2 • basis . repr ↑ r .1 , ∇ (⇑ η ) ( ↑ r .2 • ↑ r .1 ) ⟫_ ℝ d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ ( ↑ r .2 ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ↑ r .1 ) ) ↑ r .2 =
⟪ ( ‖ ↑ r .2 • ↑ r .1 ‖ ^ d ) ⁻¹ • ↑ r .2 • basis . repr ↑ r .1 , ∇ (⇑ η ) ( ↑ r .2 • ↑ r .1 ) ⟫_ ℝ ] d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ ( ↑ r .2 ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ↑ r .1 ) ) ↑ r .2 =
⟪ ( ‖ ↑ r .2 • ↑ r .1 ‖ ^ d ) ⁻¹ • ↑ r .2 • basis . repr ↑ r .1 , ∇ (⇑ η ) ( ↑ r .2 • ↑ r .1 ) ⟫_ ℝ
trans ( r . 2 . 1 ^ ( d - 1 ) ) ⁻¹ * _root_.deriv ( fun a => η ( a • ‖ ↑ x ‖ ⁻¹ • ↑ x ) ) ‖ x ‖ d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ ( ↑ r .2 ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ↑ r .1 ) ) ↑ r .2 =
( ↑ r .2 ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ‖ x ‖ ⁻¹ • x ) ) ‖ x ‖ d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ ( ↑ r .2 ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ‖ x ‖ ⁻¹ • x ) ) ‖ x ‖ =
⟪ ( ‖ ↑ r .2 • ↑ r .1 ‖ ^ d ) ⁻¹ • ↑ r .2 • basis . repr ↑ r .1 , ∇ (⇑ η ) ( ↑ r .2 • ↑ r .1 ) ⟫_ ℝ
· d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ ( ↑ r .2 ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ↑ r .1 ) ) ↑ r .2 =
( ↑ r .2 ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ‖ x ‖ ⁻¹ • x ) ) ‖ x ‖ simp [ x , norm_smul ] d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ _root_.deriv (fun a => η ( a • ↑ r .1 ) ) ↑ r .2 = _root_.deriv (fun a => η ( a • | ↑ r .2 | ⁻¹ • ↑ r .2 • ↑ r .1 ) ) | ↑ r .2 | ∨
↑ r .2 = 0 ∧ ¬ d - 1 = 0
left d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ _root_.deriv (fun a => η ( a • ↑ r .1 ) ) ↑ r .2 = _root_.deriv (fun a => η ( a • | ↑ r .2 | ⁻¹ • ↑ r .2 • ↑ r .1 ) ) | ↑ r .2 |
congr e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ (fun a => η ( a • ↑ r .1 ) ) = fun a => η ( a • | ↑ r .2 | ⁻¹ • ↑ r .2 • ↑ r .1 ) e_x d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ ↑ r .2 = | ↑ r .2 |
funext a e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 a : ℝ ⊢ η ( a • ↑ r .1 ) = η ( a • | ↑ r .2 | ⁻¹ • ↑ r .2 • ↑ r .1 ) e_x d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ ↑ r .2 = | ↑ r .2 |
congr e_f.e_6.e_a d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 a : ℝ ⊢ ↑ r .1 = | ↑ r .2 | ⁻¹ • ↑ r .2 • ↑ r .1 e_x d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ ↑ r .2 = | ↑ r .2 |
simp [ smul_smul ] e_f.e_6.e_a d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 a : ℝ ⊢ ↑ r .1 = ( | ↑ r .2 | ⁻¹ * ↑ r .2 ) • ↑ r .1 e_x d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ ↑ r .2 = | ↑ r .2 |
rw [ abs_of_nonneg ( le_of_lt hr ) e_f.e_6.e_a d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 a : ℝ ⊢ ↑ r .1 = ( (↑ r .2 ) ⁻¹ * ↑ r .2 ) • ↑ r .1 e_x d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ ↑ r .2 = | ↑ r .2 | e_f.e_6.e_a d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 a : ℝ ⊢ ↑ r .1 = ( (↑ r .2 ) ⁻¹ * ↑ r .2 ) • ↑ r .1 e_x d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ ↑ r .2 = | ↑ r .2 | ] e_f.e_6.e_a d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 a : ℝ ⊢ ↑ r .1 = ( (↑ r .2 ) ⁻¹ * ↑ r .2 ) • ↑ r .1 e_x d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ ↑ r .2 = | ↑ r .2 |
field_simp e_f.e_6.e_a d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 a : ℝ ⊢ ↑ r .1 = 1 • ↑ r .1 e_x d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ ↑ r .2 = | ↑ r .2 |
simp only [ one_smul ] e_x d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ ↑ r .2 = | ↑ r .2 |
rw [ abs_of_nonneg ( le_of_lt hr ) e_x d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ ↑ r .2 = ↑ r .2 All goals completed! 🐙 ] All goals completed! 🐙
rw [ ← grad_inner_space_unit_vector , d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ ( ↑ r .2 ^ ( d - 1 ) ) ⁻¹ * ⟪ ∇ (⇑ η ) x , ‖ x ‖ ⁻¹ • basis . repr x ⟫_ ℝ =
⟪ ( ‖ ↑ r .2 • ↑ r .1 ‖ ^ d ) ⁻¹ • ↑ r .2 • basis . repr ↑ r .1 , ∇ (⇑ η ) ( ↑ r .2 • ↑ r .1 ) ⟫_ ℝ hd d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ Differentiable ℝ ⇑ η d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ ( ↑ r .2 ^ ( d - 1 ) ) ⁻¹ * ⟪ ‖ x ‖ ⁻¹ • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ =
⟪ ( ‖ ↑ r .2 • ↑ r .1 ‖ ^ ( d - 1 ) * ‖ ↑ r .2 • ↑ r .1 ‖ ) ⁻¹ • ↑ r .2 • basis . repr ↑ r .1 , ∇ (⇑ η ) ( ↑ r .2 • ↑ r .1 ) ⟫_ ℝ hd d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ Differentiable ℝ ⇑ η real_inner_comm , d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ ( ↑ r .2 ^ ( d - 1 ) ) ⁻¹ * ⟪ ‖ x ‖ ⁻¹ • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ =
⟪ ( ‖ ↑ r .2 • ↑ r .1 ‖ ^ d ) ⁻¹ • ↑ r .2 • basis . repr ↑ r .1 , ∇ (⇑ η ) ( ↑ r .2 • ↑ r .1 ) ⟫_ ℝ hd d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ Differentiable ℝ ⇑ η d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ ( ↑ r .2 ^ ( d - 1 ) ) ⁻¹ * ⟪ ‖ x ‖ ⁻¹ • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ =
⟪ ( ‖ ↑ r .2 • ↑ r .1 ‖ ^ ( d - 1 ) * ‖ ↑ r .2 • ↑ r .1 ‖ ) ⁻¹ • ↑ r .2 • basis . repr ↑ r .1 , ∇ (⇑ η ) ( ↑ r .2 • ↑ r .1 ) ⟫_ ℝ hd d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ Differentiable ℝ ⇑ η ← pow_sub_one_mul ( NeZero.ne d ) d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ ( ↑ r .2 ^ ( d - 1 ) ) ⁻¹ * ⟪ ‖ x ‖ ⁻¹ • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ =
⟪ ( ‖ ↑ r .2 • ↑ r .1 ‖ ^ ( d - 1 ) * ‖ ↑ r .2 • ↑ r .1 ‖ ) ⁻¹ • ↑ r .2 • basis . repr ↑ r .1 , ∇ (⇑ η ) ( ↑ r .2 • ↑ r .1 ) ⟫_ ℝ hd d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ Differentiable ℝ ⇑ η d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ ( ↑ r .2 ^ ( d - 1 ) ) ⁻¹ * ⟪ ‖ x ‖ ⁻¹ • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ =
⟪ ( ‖ ↑ r .2 • ↑ r .1 ‖ ^ ( d - 1 ) * ‖ ↑ r .2 • ↑ r .1 ‖ ) ⁻¹ • ↑ r .2 • basis . repr ↑ r .1 , ∇ (⇑ η ) ( ↑ r .2 • ↑ r .1 ) ⟫_ ℝ hd d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ Differentiable ℝ ⇑ η ] d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ ( ↑ r .2 ^ ( d - 1 ) ) ⁻¹ * ⟪ ‖ x ‖ ⁻¹ • basis . repr x , ∇ (⇑ η ) x ⟫_ ℝ =
⟪ ( ‖ ↑ r .2 • ↑ r .1 ‖ ^ ( d - 1 ) * ‖ ↑ r .2 • ↑ r .1 ‖ ) ⁻¹ • ↑ r .2 • basis . repr ↑ r .1 , ∇ (⇑ η ) ( ↑ r .2 • ↑ r .1 ) ⟫_ ℝ hd d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ Differentiable ℝ ⇑ η
simp only [ norm_smul , Real.norm_eq_abs , abs_of_nonneg ( le_of_lt hr ) ,
norm_eq_of_mem_sphere , mul_one , map_smul , inner_smul_left , map_inv₀ , conj_trivial ,
mul_inv_rev , x ] d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ ( ↑ r .2 ^ ( d - 1 ) ) ⁻¹ * ( (↑ r .2 ) ⁻¹ * ( ↑ r .2 * ⟪ basis . repr ↑ r .1 , ∇ (⇑ η ) ( ↑ r .2 • ↑ r .1 ) ⟫_ ℝ ) ) =
(↑ r .2 ) ⁻¹ * ( ↑ r .2 ^ ( d - 1 ) ) ⁻¹ * ( ↑ r .2 * ⟪ basis . repr ↑ r .1 , ∇ (⇑ η ) ( ↑ r .2 • ↑ r .1 ) ⟫_ ℝ ) hd d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ Differentiable ℝ ⇑ η
field_simp hd d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) r : ↑ ( Metric.sphere 0 1 ) × ↑ ( Set.Ioi 0 ) x : Space d := ↑ r .2 • ↑ r .1 hr : 0 < ↑ r .2 ⊢ Differentiable ℝ ⇑ η
exact SchwartzMap.differentiable η All goals completed! 🐙
_ = - ∫ n , ( ∫ ( r : Set.Ioi ( 0 : ℝ ) ) ,
( _root_.deriv ( fun a => η ( a • n ) ) r . 1 ) ∂ ( .comap Subtype.val volume ) )
∂ ( volume ( α := Space d ) . toSphere ) := by d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) ⊢ - ∫ ( n : ↑ ( Metric.sphere 0 1 ) ) ,
∫ ( r : ↑ ( Set.Ioi 0 ) ) ,
‖ ↑ r ‖ ⁻¹ ^ ( d - 1 ) *
_root_.deriv (fun a => η ( a • ↑ n ) )
‖ ↑ r ‖ ∂ Measure.volumeIoiPow ( Module.finrank ℝ ( Space d ) - 1 ) ∂ volume . toSphere =
- ∫ ( n : ↑ ( Metric.sphere 0 1 ) ) ,
∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume ∂ volume . toSphere
congr e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) ⊢ (fun n =>
∫ ( r : ↑ ( Set.Ioi 0 ) ) ,
‖ ↑ r ‖ ⁻¹ ^ ( d - 1 ) *
_root_.deriv (fun a => η ( a • ↑ n ) ) ‖ ↑ r ‖ ∂ Measure.volumeIoiPow ( Module.finrank ℝ ( Space d ) - 1 ) ) =
fun n => ∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume
funext n e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) ⊢ ∫ ( r : ↑ ( Set.Ioi 0 ) ) ,
‖ ↑ r ‖ ⁻¹ ^ ( d - 1 ) * _root_.deriv (fun a => η ( a • ↑ n ) ) ‖ ↑ r ‖ ∂ Measure.volumeIoiPow ( Module.finrank ℝ ( Space d ) - 1 ) =
∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume
simp [ Measure.volumeIoiPow ] e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) ⊢ ( ∫ ( r : ↑ ( Set.Ioi 0 ) ) ,
( | ↑ r | ^ ( d - 1 ) ) ⁻¹ *
_root_.deriv (fun a => η ( a • ↑ n ) )
| ↑ r | ∂ ( Measure.comap Subtype.val volume ) . withDensity fun r => ENNReal.ofReal ( ↑ r ^ ( d - 1 ) ) ) =
∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume
erw [ integral_withDensity_eq_integral_smul e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) ⊢ ∫ ( x : ↑ ( Set.Ioi 0 ) ) ,
( ↑ x ^ ( d - 1 ) ) . toNNReal •
( ( | ↑ x | ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ↑ n ) ) | ↑ x | ) ∂ Measure.comap Subtype.val volume =
∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume e_f.f_meas d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) ⊢ Measurable fun r => ( ↑ r ^ ( d - 1 ) ) . toNNReal ] e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) ⊢ ∫ ( x : ↑ ( Set.Ioi 0 ) ) ,
( ↑ x ^ ( d - 1 ) ) . toNNReal •
( ( | ↑ x | ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ↑ n ) ) | ↑ x | ) ∂ Measure.comap Subtype.val volume =
∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume e_f.f_meas d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) ⊢ Measurable fun r => ( ↑ r ^ ( d - 1 ) ) . toNNReal
congr e_f.e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) ⊢ (fun x => ( ↑ x ^ ( d - 1 ) ) . toNNReal • ( ( | ↑ x | ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ↑ n ) ) | ↑ x | ) ) = fun r =>
_root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r e_f.f_meas d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) ⊢ Measurable fun r => ( ↑ r ^ ( d - 1 ) ) . toNNReal
funext r e_f.e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) r : ↑ ( Set.Ioi 0 ) ⊢ ( ↑ r ^ ( d - 1 ) ) . toNNReal • ( ( | ↑ r | ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ↑ n ) ) | ↑ r | ) =
_root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r e_f.f_meas d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) ⊢ Measurable fun r => ( ↑ r ^ ( d - 1 ) ) . toNNReal
have hr : ( 0 : ℝ ) < r . 1 := r . 2 e_f.e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) r : ↑ ( Set.Ioi 0 ) hr : 0 < ↑ r ⊢ ( ↑ r ^ ( d - 1 ) ) . toNNReal • ( ( | ↑ r | ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ↑ n ) ) | ↑ r | ) =
_root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r e_f.f_meas d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) ⊢ Measurable fun r => ( ↑ r ^ ( d - 1 ) ) . toNNReal
rw [ abs_of_nonneg hr . le , e_f.e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) r : ↑ ( Set.Ioi 0 ) hr : 0 < ↑ r ⊢ ( ↑ r ^ ( d - 1 ) ) . toNNReal • ( ( ↑ r ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ) =
_root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r e_f.f_meas d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) ⊢ Measurable fun r => ( ↑ r ^ ( d - 1 ) ) . toNNReal e_f.f_meas d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) ⊢ Measurable fun r => ( ↑ r ^ ( d - 1 ) ) . toNNReal NNReal.smul_def , e_f.e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) r : ↑ ( Set.Ioi 0 ) hr : 0 < ↑ r ⊢ ↑ ( ↑ r ^ ( d - 1 ) ) . toNNReal • ( ( ↑ r ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ) =
_root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r e_f.f_meas d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) ⊢ Measurable fun r => ( ↑ r ^ ( d - 1 ) ) . toNNReal e_f.f_meas d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) ⊢ Measurable fun r => ( ↑ r ^ ( d - 1 ) ) . toNNReal Real.coe_toNNReal _ ( by d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) r : ↑ ( Set.Ioi 0 ) hr : 0 < ↑ r ⊢ 0 ≤ ↑ r ^ ( d - 1 ) e_f.f_meas d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) ⊢ Measurable fun r => ( ↑ r ^ ( d - 1 ) ) . toNNReal positivity All goals completed! 🐙 e_f.f_meas d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) ⊢ Measurable fun r => ( ↑ r ^ ( d - 1 ) ) . toNNReal ) ,
smul_eq_mul , e_f.e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) r : ↑ ( Set.Ioi 0 ) hr : 0 < ↑ r ⊢ ↑ r ^ ( d - 1 ) * ( ( ↑ r ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ) = _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r e_f.f_meas d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) ⊢ Measurable fun r => ( ↑ r ^ ( d - 1 ) ) . toNNReal e_f.f_meas d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) ⊢ Measurable fun r => ( ↑ r ^ ( d - 1 ) ) . toNNReal ← mul_assoc , e_f.e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) r : ↑ ( Set.Ioi 0 ) hr : 0 < ↑ r ⊢ ↑ r ^ ( d - 1 ) * ( ↑ r ^ ( d - 1 ) ) ⁻¹ * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r = _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r e_f.f_meas d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) ⊢ Measurable fun r => ( ↑ r ^ ( d - 1 ) ) . toNNReal e_f.f_meas d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) ⊢ Measurable fun r => ( ↑ r ^ ( d - 1 ) ) . toNNReal mul_inv_cancel₀ ( pow_ne_zero ( d - 1 ) hr . ne' ) , e_f.e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) r : ↑ ( Set.Ioi 0 ) hr : 0 < ↑ r ⊢ 1 * _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r = _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r e_f.f_meas d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) ⊢ Measurable fun r => ( ↑ r ^ ( d - 1 ) ) . toNNReal e_f.f_meas d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) ⊢ Measurable fun r => ( ↑ r ^ ( d - 1 ) ) . toNNReal one_mul e_f.e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) r : ↑ ( Set.Ioi 0 ) hr : 0 < ↑ r ⊢ _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r = _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r e_f.f_meas d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) ⊢ Measurable fun r => ( ↑ r ^ ( d - 1 ) ) . toNNReal e_f.f_meas d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) ⊢ Measurable fun r => ( ↑ r ^ ( d - 1 ) ) . toNNReal ] e_f.f_meas d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) ⊢ Measurable fun r => ( ↑ r ^ ( d - 1 ) ) . toNNReal
fun_prop All goals completed! 🐙
_ = - ∫ n , ( - η 0 ) ∂ ( volume ( α := Space d ) . toSphere ) := by d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) ⊢ - ∫ ( n : ↑ ( Metric.sphere 0 1 ) ) ,
∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume ∂ volume . toSphere =
- ∫ ( n : ↑ ( Metric.sphere 0 1 ) ) , - η 0 ∂ volume . toSphere
congr e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) ⊢ (fun n => ∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume ) = fun n => - η 0
funext n e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) ⊢ ∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume = - η 0
let η' ( n : ↑ ( Metric.sphere 0 1 ) ) : 𝓢( ℝ , ℝ ) := compCLM ( g := fun a => a • n . 1 ) ℝ ( by d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n✝ : ↑ ( Metric.sphere 0 1 ) n : ↑ ( Metric.sphere 0 1 ) ⊢ Function.HasTemperateGrowth fun a => a • ↑ n e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume = - η 0
apply And.intro left d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n✝ : ↑ ( Metric.sphere 0 1 ) n : ↑ ( Metric.sphere 0 1 ) ⊢ ContDiff ℝ ↑ ⊤ fun a => a • ↑ n right d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n✝ : ↑ ( Metric.sphere 0 1 ) n : ↑ ( Metric.sphere 0 1 ) ⊢ ∀ ( n_1 : ℕ ), ∃ k C , ∀ ( x : ℝ ), ‖ iteratedFDeriv ℝ n_1 (fun a => a • ↑ n ) x ‖ ≤ C * ( 1 + ‖ x ‖ ) ^ k e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume = - η 0
· left d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n✝ : ↑ ( Metric.sphere 0 1 ) n : ↑ ( Metric.sphere 0 1 ) ⊢ ContDiff ℝ ↑ ⊤ fun a => a • ↑ n e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume = - η 0 fun_prop All goals completed! 🐙 e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume = - η 0
· right d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n✝ : ↑ ( Metric.sphere 0 1 ) n : ↑ ( Metric.sphere 0 1 ) ⊢ ∀ ( n_1 : ℕ ), ∃ k C , ∀ ( x : ℝ ), ‖ iteratedFDeriv ℝ n_1 (fun a => a • ↑ n ) x ‖ ≤ C * ( 1 + ‖ x ‖ ) ^ k e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume = - η 0 intro n' right d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n✝ : ↑ ( Metric.sphere 0 1 ) n : ↑ ( Metric.sphere 0 1 ) n' : ℕ ⊢ ∃ k C , ∀ ( x : ℝ ), ‖ iteratedFDeriv ℝ n' (fun a => a • ↑ n ) x ‖ ≤ C * ( 1 + ‖ x ‖ ) ^ k e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume = - η 0
match n' with
| 0 => d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n✝ : ↑ ( Metric.sphere 0 1 ) n : ↑ ( Metric.sphere 0 1 ) n' : ℕ ⊢ ∃ k C , ∀ ( x : ℝ ), ‖ iteratedFDeriv ℝ 0 (fun a => a • ↑ n ) x ‖ ≤ C * ( 1 + ‖ x ‖ ) ^ k e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume = - η 0
use 1 , 1 h d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n✝ : ↑ ( Metric.sphere 0 1 ) n : ↑ ( Metric.sphere 0 1 ) n' : ℕ ⊢ ∀ ( x : ℝ ), ‖ iteratedFDeriv ℝ 0 (fun a => a • ↑ n ) x ‖ ≤ 1 * ( 1 + ‖ x ‖ ) ^ 1 e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume = - η 0
simp [ norm_smul ] All goals completed! 🐙 e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume = - η 0
| 1 => d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n✝ : ↑ ( Metric.sphere 0 1 ) n : ↑ ( Metric.sphere 0 1 ) n' : ℕ ⊢ ∃ k C , ∀ ( x : ℝ ), ‖ iteratedFDeriv ℝ 1 (fun a => a • ↑ n ) x ‖ ≤ C * ( 1 + ‖ x ‖ ) ^ k e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume = - η 0
use 0 , 1 h d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n✝ : ↑ ( Metric.sphere 0 1 ) n : ↑ ( Metric.sphere 0 1 ) n' : ℕ ⊢ ∀ ( x : ℝ ), ‖ iteratedFDeriv ℝ 1 (fun a => a • ↑ n ) x ‖ ≤ 1 * ( 1 + ‖ x ‖ ) ^ 0 e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume = - η 0
intro x h d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n✝ : ↑ ( Metric.sphere 0 1 ) n : ↑ ( Metric.sphere 0 1 ) n' : ℕ x : ℝ ⊢ ‖ iteratedFDeriv ℝ 1 (fun a => a • ↑ n ) x ‖ ≤ 1 * ( 1 + ‖ x ‖ ) ^ 0 e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume = - η 0
simp [ fderiv_smul_const ] All goals completed! 🐙 e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume = - η 0
| n' + 1 + 1 => d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n✝ : ↑ ( Metric.sphere 0 1 ) n : ↑ ( Metric.sphere 0 1 ) n'✝ : ℕ n' : ℕ ⊢ ∃ k C , ∀ ( x : ℝ ), ‖ iteratedFDeriv ℝ ( n' + 1 + 1 ) (fun a => a • ↑ n ) x ‖ ≤ C * ( 1 + ‖ x ‖ ) ^ k e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume = - η 0
use 0 , 0 h d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n✝ : ↑ ( Metric.sphere 0 1 ) n : ↑ ( Metric.sphere 0 1 ) n'✝ : ℕ n' : ℕ ⊢ ∀ ( x : ℝ ), ‖ iteratedFDeriv ℝ ( n' + 1 + 1 ) (fun a => a • ↑ n ) x ‖ ≤ 0 * ( 1 + ‖ x ‖ ) ^ 0 e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume = - η 0
intro x h d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n✝ : ↑ ( Metric.sphere 0 1 ) n : ↑ ( Metric.sphere 0 1 ) n'✝ : ℕ n' : ℕ x : ℝ ⊢ ‖ iteratedFDeriv ℝ ( n' + 1 + 1 ) (fun a => a • ↑ n ) x ‖ ≤ 0 * ( 1 + ‖ x ‖ ) ^ 0 e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume = - η 0
simp only [ Real.norm_eq_abs , pow_zero , mul_one , norm_le_zero_iff ] h d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n✝ : ↑ ( Metric.sphere 0 1 ) n : ↑ ( Metric.sphere 0 1 ) n'✝ : ℕ n' : ℕ x : ℝ ⊢ iteratedFDeriv ℝ ( n' + 1 + 1 ) (fun a => a • ↑ n ) x = 0 e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume = - η 0
rw [ iteratedFDeriv_succ_eq_comp_right h d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n✝ : ↑ ( Metric.sphere 0 1 ) n : ↑ ( Metric.sphere 0 1 ) n'✝ : ℕ n' : ℕ x : ℝ ⊢ ( ⇑ ( continuousMultilinearCurryRightEquiv' ℝ ( n' + 1 ) ℝ ( Space d ) ) . symm ∘
iteratedFDeriv ℝ ( n' + 1 ) fun y => fderiv ℝ (fun a => a • ↑ n ) y )
x =
0 h d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n✝ : ↑ ( Metric.sphere 0 1 ) n : ↑ ( Metric.sphere 0 1 ) n'✝ : ℕ n' : ℕ x : ℝ ⊢ ( ⇑ ( continuousMultilinearCurryRightEquiv' ℝ ( n' + 1 ) ℝ ( Space d ) ) . symm ∘
iteratedFDeriv ℝ ( n' + 1 ) fun y => fderiv ℝ (fun a => a • ↑ n ) y )
x =
0 e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume = - η 0 ] h d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n✝ : ↑ ( Metric.sphere 0 1 ) n : ↑ ( Metric.sphere 0 1 ) n'✝ : ℕ n' : ℕ x : ℝ ⊢ ( ⇑ ( continuousMultilinearCurryRightEquiv' ℝ ( n' + 1 ) ℝ ( Space d ) ) . symm ∘
iteratedFDeriv ℝ ( n' + 1 ) fun y => fderiv ℝ (fun a => a • ↑ n ) y )
x =
0 e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume = - η 0
conv_lhs =>
enter [ 2 , 3 , y ] d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n✝ : ↑ ( Metric.sphere 0 1 ) n : ↑ ( Metric.sphere 0 1 ) n'✝ : ℕ n' : ℕ x : ℝ y : ℝ | fderiv ℝ (fun a => a • ↑ n ) y e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume = - η 0
simp [ fderiv_smul_const ] d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n✝ : ↑ ( Metric.sphere 0 1 ) n : ↑ ( Metric.sphere 0 1 ) n'✝ : ℕ n' : ℕ x : ℝ y : ℝ | ( ContinuousLinearMap.id ℝ ℝ ) . smulRight ↑ n e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume = - η 0
rw [ iteratedFDeriv_succ_const h d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n✝ : ↑ ( Metric.sphere 0 1 ) n : ↑ ( Metric.sphere 0 1 ) n'✝ : ℕ n' : ℕ x : ℝ ⊢ ( ⇑ ( continuousMultilinearCurryRightEquiv' ℝ ( n' + 1 ) ℝ ( Space d ) ) . symm ∘ 0 ) x = 0 h d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n✝ : ↑ ( Metric.sphere 0 1 ) n : ↑ ( Metric.sphere 0 1 ) n'✝ : ℕ n' : ℕ x : ℝ ⊢ ( ⇑ ( continuousMultilinearCurryRightEquiv' ℝ ( n' + 1 ) ℝ ( Space d ) ) . symm ∘ 0 ) x = 0 e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume = - η 0 ] h d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n✝ : ↑ ( Metric.sphere 0 1 ) n : ↑ ( Metric.sphere 0 1 ) n'✝ : ℕ n' : ℕ x : ℝ ⊢ ( ⇑ ( continuousMultilinearCurryRightEquiv' ℝ ( n' + 1 ) ℝ ( Space d ) ) . symm ∘ 0 ) x = 0 e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume = - η 0
rfl All goals completed! 🐙 e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume = - η 0 ) ( by d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n✝ : ↑ ( Metric.sphere 0 1 ) n : ↑ ( Metric.sphere 0 1 ) ⊢ ∃ k C , ∀ ( x : ℝ ), ‖ x ‖ ≤ C * ( 1 + ‖ x • ↑ n ‖ ) ^ k e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume = - η 0 use 1 , 1 h d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n✝ : ↑ ( Metric.sphere 0 1 ) n : ↑ ( Metric.sphere 0 1 ) ⊢ ∀ ( x : ℝ ), ‖ x ‖ ≤ 1 * ( 1 + ‖ x • ↑ n ‖ ) ^ 1 e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume = - η 0 ; simp [ norm_smul ] All goals completed! 🐙 e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume = - η 0 ) η e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∫ ( r : ↑ ( Set.Ioi 0 ) ) , _root_.deriv (fun a => η ( a • ↑ n ) ) ↑ r ∂ Measure.comap Subtype.val volume = - η 0
rw [ MeasureTheory.integral_subtype_comap ( by d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ MeasurableSet ( Set.Ioi 0 ) e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ 0 - η ( 0 • ↑ n ) = - η 0 e_f.hcont d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ContinuousWithinAt (fun a => η ( a • ↑ n ) ) ( Set.Ici 0 ) 0 e_f.hderiv d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∀ x ∈ Set.Ioi 0 , HasDerivAt (fun a => η ( a • ↑ n ) ) ( _root_.deriv (fun a => η ( a • ↑ n ) ) x ) x e_f.f'int d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ IntegrableOn ( _root_.deriv fun a => η ( a • ↑ n ) ) ( Set.Ioi 0 ) volume e_f.hf d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ Filter.Tendsto (fun a => η ( a • ↑ n ) ) Filter.atTop ( 𝓝 0 ) simp All goals completed! 🐙 e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ 0 - η ( 0 • ↑ n ) = - η 0 e_f.hcont d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ContinuousWithinAt (fun a => η ( a • ↑ n ) ) ( Set.Ici 0 ) 0 e_f.hderiv d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∀ x ∈ Set.Ioi 0 , HasDerivAt (fun a => η ( a • ↑ n ) ) ( _root_.deriv (fun a => η ( a • ↑ n ) ) x ) x e_f.f'int d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ IntegrableOn ( _root_.deriv fun a => η ( a • ↑ n ) ) ( Set.Ioi 0 ) volume e_f.hf d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ Filter.Tendsto (fun a => η ( a • ↑ n ) ) Filter.atTop ( 𝓝 0 ) ) ,
MeasureTheory.integral_Ioi_of_hasDerivAt_of_tendsto ( f := fun a => η ( a • n ) ) ( m := 0 ) e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ 0 - η ( 0 • ↑ n ) = - η 0 e_f.hcont d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ContinuousWithinAt (fun a => η ( a • ↑ n ) ) ( Set.Ici 0 ) 0 e_f.hderiv d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∀ x ∈ Set.Ioi 0 , HasDerivAt (fun a => η ( a • ↑ n ) ) ( _root_.deriv (fun a => η ( a • ↑ n ) ) x ) x e_f.f'int d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ IntegrableOn ( _root_.deriv fun a => η ( a • ↑ n ) ) ( Set.Ioi 0 ) volume e_f.hf d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ Filter.Tendsto (fun a => η ( a • ↑ n ) ) Filter.atTop ( 𝓝 0 ) e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ 0 - η ( 0 • ↑ n ) = - η 0 e_f.hcont d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ContinuousWithinAt (fun a => η ( a • ↑ n ) ) ( Set.Ici 0 ) 0 e_f.hderiv d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∀ x ∈ Set.Ioi 0 , HasDerivAt (fun a => η ( a • ↑ n ) ) ( _root_.deriv (fun a => η ( a • ↑ n ) ) x ) x e_f.f'int d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ IntegrableOn ( _root_.deriv fun a => η ( a • ↑ n ) ) ( Set.Ioi 0 ) volume e_f.hf d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ Filter.Tendsto (fun a => η ( a • ↑ n ) ) Filter.atTop ( 𝓝 0 ) ] e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ 0 - η ( 0 • ↑ n ) = - η 0 e_f.hcont d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ContinuousWithinAt (fun a => η ( a • ↑ n ) ) ( Set.Ici 0 ) 0 e_f.hderiv d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∀ x ∈ Set.Ioi 0 , HasDerivAt (fun a => η ( a • ↑ n ) ) ( _root_.deriv (fun a => η ( a • ↑ n ) ) x ) x e_f.f'int d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ IntegrableOn ( _root_.deriv fun a => η ( a • ↑ n ) ) ( Set.Ioi 0 ) volume e_f.hf d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ Filter.Tendsto (fun a => η ( a • ↑ n ) ) Filter.atTop ( 𝓝 0 )
· e_f d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ 0 - η ( 0 • ↑ n ) = - η 0 simp All goals completed! 🐙
· e_f.hcont d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ContinuousWithinAt (fun a => η ( a • ↑ n ) ) ( Set.Ici 0 ) 0 exact ContinuousAt.continuousWithinAt ( by d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ContinuousAt (fun a => η ( a • ↑ n ) ) 0 fun_prop All goals completed! 🐙 )
· e_f.hderiv d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ ∀ x ∈ Set.Ioi 0 , HasDerivAt (fun a => η ( a • ↑ n ) ) ( _root_.deriv (fun a => η ( a • ↑ n ) ) x ) x exact fun x _ => DifferentiableAt.hasDerivAt ( by d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η x : ℝ x✝ : x ∈ Set.Ioi 0 ⊢ DifferentiableAt ℝ (fun a => η ( a • ↑ n ) ) x fun_prop All goals completed! 🐙 )
· e_f.f'int d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ IntegrableOn ( _root_.deriv fun a => η ( a • ↑ n ) ) ( Set.Ioi 0 ) volume exact ( integrable ( ( derivCLM ℝ ℝ ) ( η' n ) ) ) . integrableOn All goals completed! 🐙
· e_f.hf d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) n : ↑ ( Metric.sphere 0 1 ) η' : ↑ ( Metric.sphere 0 1 ) → 𝓢( ℝ , ℝ ) := fun n => ( compCLM ℝ ⋯ ⋯ ) η ⊢ Filter.Tendsto (fun a => η ( a • ↑ n ) ) Filter.atTop ( 𝓝 0 ) exact Filter.Tendsto.mono_left ( η' n ) . toZeroAtInfty . zero_at_infty' atTop_le_cocompact All goals completed! 🐙
_ = η 0 * ( d * ( volume ( α := Space d ) ) . real ( Metric.ball 0 1 ) ) := by d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) ⊢ - ∫ ( n : ↑ ( Metric.sphere 0 1 ) ) , - η 0 ∂ volume . toSphere = η 0 * ( ↑ d * volume . real ( Metric.ball 0 1 ) )
simp only [ integral_const , Measure.toSphere_real_apply_univ , finrank_eq_dim , smul_eq_mul ,
mul_neg , neg_neg ] d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) ⊢ ↑ d * volume . real ( Metric.ball 0 1 ) * η 0 = η 0 * ( ↑ d * volume . real ( Metric.ball 0 1 ) )
ring All goals completed! 🐙
simp only [ _root_.smul_apply , diracDelta_apply , smul_eq_mul ] calc.step d : ℕ inst✝ : NeZero d η : 𝓢( Space d , ℝ ) ⊢ η 0 * ( ↑ d * volume . real ( Metric.ball 0 1 ) ) = ↑ d * volume . real ( Metric.ball 0 1 ) * η 0
ring All goals completed! 🐙
B.6. The Laplacian of the fundamental solution
The distributional Laplacian of ‖x‖ ^ (2 - d) is (2 - d) * d * volume (Metric.ball 0 1)
times the Dirac delta at the origin. For d ≥ 3 this ‖x‖ ^ (2 - d) is the (singular)
fundamental solution of the Laplacian, and for d = 1 it is ‖x‖. When d = 2 the exponent
vanishes, so the identity collapses to the trivial Δᵈ 1 = 0; the genuine two-dimensional
fundamental solution is the logarithm, proved in distLaplacian_fundamentalSolution_log_norm.
The statement also holds vacuously for d = 0, where the space is trivial.
lemma distLaplacian_fundamentalSolution_norm_zpow { d : ℕ } :
Δᵈ ( distOfFunction ( fun x : Space d => ‖ x ‖ ^ ( - ( ( d : ℤ ) - 2 ) ) )
( IsDistBounded.pow _ ( by 𝕜 : Type E : Type F : Type F' : Type inst✝⁵ : RCLike 𝕜 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedAddCommGroup F inst✝² : NormedAddCommGroup F' inst✝¹ : NormedSpace ℝ E inst✝ : NormedSpace ℝ F d : ℕ ⊢ - ↑ ( d - 1 ) ≤ - ( ↑ d - 2 ) omega All goals completed! 🐙 ) ) ) =
( ( - ( ( d : ℝ ) - 2 ) ) * d *
( volume ( α := Space d ) ) . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0 := by d : ℕ ⊢ Δᵈ ( distOfFunction (fun x => ‖ x ‖ ^ ( - ( ↑ d - 2 ) ) ) ⋯ ) = ( - ( ↑ d - 2 ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0
by_cases h : d = 0 pos d : ℕ h : d = 0 ⊢ Δᵈ ( distOfFunction (fun x => ‖ x ‖ ^ ( - ( ↑ d - 2 ) ) ) ⋯ ) = ( - ( ↑ d - 2 ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0 neg d : ℕ h : ¬ d = 0 ⊢ Δᵈ ( distOfFunction (fun x => ‖ x ‖ ^ ( - ( ↑ d - 2 ) ) ) ⋯ ) = ( - ( ↑ d - 2 ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0
· pos d : ℕ h : d = 0 ⊢ Δᵈ ( distOfFunction (fun x => ‖ x ‖ ^ ( - ( ↑ d - 2 ) ) ) ⋯ ) = ( - ( ↑ d - 2 ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0 subst h pos ⊢ Δᵈ ( distOfFunction (fun x => ‖ x ‖ ^ ( - ( ↑ 0 - 2 ) ) ) ⋯ ) = ( - ( ↑ 0 - 2 ) * ↑ 0 * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0
have hzero :
distOfFunction ( fun x : Space 0 => ‖ x ‖ ^ 2 )
( IsDistBounded.pow _ ( by ⊢ - ↑ ( 0 - 1 ) ≤ 2 pos hzero : distOfFunction (fun x => ‖ x ‖ ^ 2 ) ⋯ = 0 ⊢ Δᵈ ( distOfFunction (fun x => ‖ x ‖ ^ ( - ( ↑ 0 - 2 ) ) ) ⋯ ) = ( - ( ↑ 0 - 2 ) * ↑ 0 * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0 omega All goals completed! 🐙 pos hzero : distOfFunction (fun x => ‖ x ‖ ^ 2 ) ⋯ = 0 ⊢ Δᵈ ( distOfFunction (fun x => ‖ x ‖ ^ ( - ( ↑ 0 - 2 ) ) ) ⋯ ) = ( - ( ↑ 0 - 2 ) * ↑ 0 * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0 ) ) = 0 := by d : ℕ ⊢ Δᵈ ( distOfFunction (fun x => ‖ x ‖ ^ ( - ( ↑ d - 2 ) ) ) ⋯ ) = ( - ( ↑ d - 2 ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0 pos hzero : distOfFunction (fun x => ‖ x ‖ ^ 2 ) ⋯ = 0 ⊢ Δᵈ ( distOfFunction (fun x => ‖ x ‖ ^ ( - ( ↑ 0 - 2 ) ) ) ⋯ ) = ( - ( ↑ 0 - 2 ) * ↑ 0 * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0
ext η η : 𝓢( Space 0 , ℝ ) ⊢ ( distOfFunction (fun x => ‖ x ‖ ^ 2 ) ⋯ ) η = 0 η pos hzero : distOfFunction (fun x => ‖ x ‖ ^ 2 ) ⋯ = 0 ⊢ Δᵈ ( distOfFunction (fun x => ‖ x ‖ ^ ( - ( ↑ 0 - 2 ) ) ) ⋯ ) = ( - ( ↑ 0 - 2 ) * ↑ 0 * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0
rw [ distOfFunction_apply η : 𝓢( Space 0 , ℝ ) ⊢ ∫ ( x : Space 0 ) , η x • ‖ x ‖ ^ 2 = 0 η η : 𝓢( Space 0 , ℝ ) ⊢ ∫ ( x : Space 0 ) , η x • ‖ x ‖ ^ 2 = 0 η pos hzero : distOfFunction (fun x => ‖ x ‖ ^ 2 ) ⋯ = 0 ⊢ Δᵈ ( distOfFunction (fun x => ‖ x ‖ ^ ( - ( ↑ 0 - 2 ) ) ) ⋯ ) = ( - ( ↑ 0 - 2 ) * ↑ 0 * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0 ] η : 𝓢( Space 0 , ℝ ) ⊢ ∫ ( x : Space 0 ) , η x • ‖ x ‖ ^ 2 = 0 η pos hzero : distOfFunction (fun x => ‖ x ‖ ^ 2 ) ⋯ = 0 ⊢ Δᵈ ( distOfFunction (fun x => ‖ x ‖ ^ ( - ( ↑ 0 - 2 ) ) ) ⋯ ) = ( - ( ↑ 0 - 2 ) * ↑ 0 * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0
refine integral_eq_zero_of_ae ( ae_of_all _ fun x => ? _ ) η : 𝓢( Space 0 , ℝ ) x : Space 0 ⊢ (fun x => η x • ‖ x ‖ ^ 2 ) x = 0 x pos hzero : distOfFunction (fun x => ‖ x ‖ ^ 2 ) ⋯ = 0 ⊢ Δᵈ ( distOfFunction (fun x => ‖ x ‖ ^ ( - ( ↑ 0 - 2 ) ) ) ⋯ ) = ( - ( ↑ 0 - 2 ) * ↑ 0 * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0
rw [ Subsingleton.elim x 0 η : 𝓢( Space 0 , ℝ ) x : Space 0 ⊢ (fun x => η x • ‖ x ‖ ^ 2 ) 0 = 0 0 η : 𝓢( Space 0 , ℝ ) x : Space 0 ⊢ (fun x => η x • ‖ x ‖ ^ 2 ) 0 = 0 0 pos hzero : distOfFunction (fun x => ‖ x ‖ ^ 2 ) ⋯ = 0 ⊢ Δᵈ ( distOfFunction (fun x => ‖ x ‖ ^ ( - ( ↑ 0 - 2 ) ) ) ⋯ ) = ( - ( ↑ 0 - 2 ) * ↑ 0 * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0 ] η : 𝓢( Space 0 , ℝ ) x : Space 0 ⊢ (fun x => η x • ‖ x ‖ ^ 2 ) 0 = 0 0 pos hzero : distOfFunction (fun x => ‖ x ‖ ^ 2 ) ⋯ = 0 ⊢ Δᵈ ( distOfFunction (fun x => ‖ x ‖ ^ ( - ( ↑ 0 - 2 ) ) ) ⋯ ) = ( - ( ↑ 0 - 2 ) * ↑ 0 * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0
simp [ zero_zpow_eq ] pos hzero : distOfFunction (fun x => ‖ x ‖ ^ 2 ) ⋯ = 0 ⊢ Δᵈ ( distOfFunction (fun x => ‖ x ‖ ^ ( - ( ↑ 0 - 2 ) ) ) ⋯ ) = ( - ( ↑ 0 - 2 ) * ↑ 0 * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0 pos hzero : distOfFunction (fun x => ‖ x ‖ ^ 2 ) ⋯ = 0 ⊢ Δᵈ ( distOfFunction (fun x => ‖ x ‖ ^ ( - ( ↑ 0 - 2 ) ) ) ⋯ ) = ( - ( ↑ 0 - 2 ) * ↑ 0 * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0
simp [ hzero ] All goals completed! 🐙
· neg d : ℕ h : ¬ d = 0 ⊢ Δᵈ ( distOfFunction (fun x => ‖ x ‖ ^ ( - ( ↑ d - 2 ) ) ) ⋯ ) = ( - ( ↑ d - 2 ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0 haveI : NeZero d := ⟨ by d : ℕ h : ¬ d = 0 ⊢ d ≠ 0 omega All goals completed! 🐙 ⟩
rw [ distLaplacian neg d : ℕ h : ¬ d = 0 this : NeZero d ⊢ ( distDiv ∘ₗ ∇ᵈ ) ( distOfFunction (fun x => ‖ x ‖ ^ ( - ( ↑ d - 2 ) ) ) ⋯ ) =
( - ( ↑ d - 2 ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0 neg d : ℕ h : ¬ d = 0 this : NeZero d ⊢ ( distDiv ∘ₗ ∇ᵈ ) ( distOfFunction (fun x => ‖ x ‖ ^ ( - ( ↑ d - 2 ) ) ) ⋯ ) =
( - ( ↑ d - 2 ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0 ] neg d : ℕ h : ¬ d = 0 this : NeZero d ⊢ ( distDiv ∘ₗ ∇ᵈ ) ( distOfFunction (fun x => ‖ x ‖ ^ ( - ( ↑ d - 2 ) ) ) ⋯ ) =
( - ( ↑ d - 2 ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0
change ∇ᵈ ⬝ ( ∇ᵈ ( distOfFunction
( fun x : Space d => ‖ x ‖ ^ ( - ( ( d : ℤ ) - 2 ) ) )
( IsDistBounded.pow ( - ( ( d : ℤ ) - 2 ) ) ( by d : ℕ h : ¬ d = 0 this : NeZero d ⊢ - ↑ ( d - 1 ) ≤ - ( ↑ d - 2 ) omega All goals completed! 🐙 ) ) ) ) = _
rw [ distGrad_distOfFunction_norm_zpow ( - ( ( d : ℤ ) - 2 ) ) ( by d : ℕ h : ¬ d = 0 this : NeZero d ⊢ - ↑ ( d - 1 ) + 1 ≤ - ( ↑ d - 2 ) neg d : ℕ h : ¬ d = 0 this : NeZero d ⊢ distDiv ( distOfFunction (fun x => ( ↑ ( - ( ↑ d - 2 ) ) * ‖ x ‖ ^ ( - ( ↑ d - 2 ) - 2 ) ) • basis . repr x ) ⋯ ) =
( - ( ↑ d - 2 ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0 omega All goals completed! 🐙 neg d : ℕ h : ¬ d = 0 this : NeZero d ⊢ distDiv ( distOfFunction (fun x => ( ↑ ( - ( ↑ d - 2 ) ) * ‖ x ‖ ^ ( - ( ↑ d - 2 ) - 2 ) ) • basis . repr x ) ⋯ ) =
( - ( ↑ d - 2 ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0 ) ] neg d : ℕ h : ¬ d = 0 this : NeZero d ⊢ distDiv ( distOfFunction (fun x => ( ↑ ( - ( ↑ d - 2 ) ) * ‖ x ‖ ^ ( - ( ↑ d - 2 ) - 2 ) ) • basis . repr x ) ⋯ ) =
( - ( ↑ d - 2 ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0
simp only [ neg_sub , Int.cast_sub , Int.cast_ofNat , Int.cast_natCast , sub_sub_cancel_left ] neg d : ℕ h : ¬ d = 0 this : NeZero d ⊢ distDiv ( distOfFunction (fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x ) ⋯ ) =
( ( 2 - ↑ d ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0
have hdist :
distOfFunction
( fun x : Space d =>
( ( 2 - ( d : ℝ ) ) * ‖ x ‖ ^ ( - ( d : ℤ ) ) ) • basis . repr x )
( by d : ℕ h : ¬ d = 0 this : NeZero d ⊢ IsDistBounded fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x neg d : ℕ h : ¬ d = 0 this : NeZero d hdist : distOfFunction (fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x ) ⋯ =
( 2 - ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ ( - ↑ d ) • basis . repr x ) ⋯ ⊢ distDiv ( distOfFunction (fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x ) ⋯ ) =
( ( 2 - ↑ d ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0
simpa [ smul_smul ] using
( IsDistBounded.const_fun_smul
( F := EuclideanSpace ℝ ( Fin d ) )
( IsDistBounded.zpow_smul_repr_self ( - ( d : ℤ ) ) ( by d : ℕ h : ¬ d = 0 this : NeZero d ⊢ - ↑ ( d - 1 ) - 1 ≤ - ↑ d neg d : ℕ h : ¬ d = 0 this : NeZero d hdist : distOfFunction (fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x ) ⋯ =
( 2 - ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ ( - ↑ d ) • basis . repr x ) ⋯ ⊢ distDiv ( distOfFunction (fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x ) ⋯ ) =
( ( 2 - ↑ d ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0 omega All goals completed! 🐙 neg d : ℕ h : ¬ d = 0 this : NeZero d hdist : distOfFunction (fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x ) ⋯ =
( 2 - ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ ( - ↑ d ) • basis . repr x ) ⋯ ⊢ distDiv ( distOfFunction (fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x ) ⋯ ) =
( ( 2 - ↑ d ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0 ) )
( 2 - ( d : ℝ ) ) ) ) =
( 2 - ( d : ℝ ) ) • distOfFunction
( fun x : Space d =>
‖ x ‖ ^ ( - ( d : ℤ ) ) • basis . repr x )
( IsDistBounded.zpow_smul_repr_self ( - ( d : ℤ ) ) ( by d : ℕ h : ¬ d = 0 this : NeZero d ⊢ - ↑ ( d - 1 ) - 1 ≤ - ↑ d neg d : ℕ h : ¬ d = 0 this : NeZero d hdist : distOfFunction (fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x ) ⋯ =
( 2 - ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ ( - ↑ d ) • basis . repr x ) ⋯ ⊢ distDiv ( distOfFunction (fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x ) ⋯ ) =
( ( 2 - ↑ d ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0 omega All goals completed! 🐙 neg d : ℕ h : ¬ d = 0 this : NeZero d hdist : distOfFunction (fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x ) ⋯ =
( 2 - ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ ( - ↑ d ) • basis . repr x ) ⋯ ⊢ distDiv ( distOfFunction (fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x ) ⋯ ) =
( ( 2 - ↑ d ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0 ) ) := by d : ℕ ⊢ Δᵈ ( distOfFunction (fun x => ‖ x ‖ ^ ( - ( ↑ d - 2 ) ) ) ⋯ ) = ( - ( ↑ d - 2 ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0 neg d : ℕ h : ¬ d = 0 this : NeZero d hdist : distOfFunction (fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x ) ⋯ =
( 2 - ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ ( - ↑ d ) • basis . repr x ) ⋯ ⊢ distDiv ( distOfFunction (fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x ) ⋯ ) =
( ( 2 - ↑ d ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0
convert distOfFunction_smul_fun
( fun x : Space d =>
‖ x ‖ ^ ( - ( d : ℤ ) ) • basis . repr x )
( IsDistBounded.zpow_smul_repr_self ( - ( d : ℤ ) ) ( by d : ℕ h : ¬ d = 0 this : NeZero d ⊢ - ↑ ( d - 1 ) - 1 ≤ - ↑ d neg d : ℕ h : ¬ d = 0 this : NeZero d hdist : distOfFunction (fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x ) ⋯ =
( 2 - ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ ( - ↑ d ) • basis . repr x ) ⋯ ⊢ distDiv ( distOfFunction (fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x ) ⋯ ) =
( ( 2 - ↑ d ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0 omega All goals completed! 🐙 neg d : ℕ h : ¬ d = 0 this : NeZero d hdist : distOfFunction (fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x ) ⋯ =
( 2 - ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ ( - ↑ d ) • basis . repr x ) ⋯ ⊢ distDiv ( distOfFunction (fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x ) ⋯ ) =
( ( 2 - ↑ d ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0 ) )
( 2 - ( d : ℝ ) ) using 1
ext x d : ℕ h : ¬ d = 0 this : NeZero d x : 𝓢( Space d , ℝ ) i✝ : Fin d ⊢ ( ( distOfFunction (fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x ) ⋯ ) x ) . ofLp i✝ =
( ( distOfFunction (fun x => ( 2 - ↑ d ) • ‖ x ‖ ^ ( - ↑ d ) • basis . repr x ) ⋯ ) x ) . ofLp i✝ neg d : ℕ h : ¬ d = 0 this : NeZero d hdist : distOfFunction (fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x ) ⋯ =
( 2 - ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ ( - ↑ d ) • basis . repr x ) ⋯ ⊢ distDiv ( distOfFunction (fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x ) ⋯ ) =
( ( 2 - ↑ d ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0
simp [ smul_smul ] neg d : ℕ h : ¬ d = 0 this : NeZero d hdist : distOfFunction (fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x ) ⋯ =
( 2 - ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ ( - ↑ d ) • basis . repr x ) ⋯ ⊢ distDiv ( distOfFunction (fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x ) ⋯ ) =
( ( 2 - ↑ d ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0 neg d : ℕ h : ¬ d = 0 this : NeZero d hdist : distOfFunction (fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x ) ⋯ =
( 2 - ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ ( - ↑ d ) • basis . repr x ) ⋯ ⊢ distDiv ( distOfFunction (fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x ) ⋯ ) =
( ( 2 - ↑ d ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0
rw [ hdist , neg d : ℕ h : ¬ d = 0 this : NeZero d hdist : distOfFunction (fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x ) ⋯ =
( 2 - ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ ( - ↑ d ) • basis . repr x ) ⋯ ⊢ distDiv ( ( 2 - ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ ( - ↑ d ) • basis . repr x ) ⋯ ) =
( ( 2 - ↑ d ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0 neg d : ℕ h : ¬ d = 0 this : NeZero d hdist : distOfFunction (fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x ) ⋯ =
( 2 - ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ ( - ↑ d ) • basis . repr x ) ⋯ ⊢ ( ( 2 - ↑ d ) * ( ↑ d * volume . real ( Metric.ball 0 1 ) ) ) • diracDelta ℝ 0 =
( ( 2 - ↑ d ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0 map_smul , neg d : ℕ h : ¬ d = 0 this : NeZero d hdist : distOfFunction (fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x ) ⋯ =
( 2 - ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ ( - ↑ d ) • basis . repr x ) ⋯ ⊢ ( 2 - ↑ d ) • distDiv ( distOfFunction (fun x => ‖ x ‖ ^ ( - ↑ d ) • basis . repr x ) ⋯ ) =
( ( 2 - ↑ d ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0 neg d : ℕ h : ¬ d = 0 this : NeZero d hdist : distOfFunction (fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x ) ⋯ =
( 2 - ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ ( - ↑ d ) • basis . repr x ) ⋯ ⊢ ( ( 2 - ↑ d ) * ( ↑ d * volume . real ( Metric.ball 0 1 ) ) ) • diracDelta ℝ 0 =
( ( 2 - ↑ d ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0 distDiv_inv_pow_eq_dim , neg d : ℕ h : ¬ d = 0 this : NeZero d hdist : distOfFunction (fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x ) ⋯ =
( 2 - ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ ( - ↑ d ) • basis . repr x ) ⋯ ⊢ ( 2 - ↑ d ) • ( ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0 =
( ( 2 - ↑ d ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0 neg d : ℕ h : ¬ d = 0 this : NeZero d hdist : distOfFunction (fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x ) ⋯ =
( 2 - ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ ( - ↑ d ) • basis . repr x ) ⋯ ⊢ ( ( 2 - ↑ d ) * ( ↑ d * volume . real ( Metric.ball 0 1 ) ) ) • diracDelta ℝ 0 =
( ( 2 - ↑ d ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0 smul_smul neg d : ℕ h : ¬ d = 0 this : NeZero d hdist : distOfFunction (fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x ) ⋯ =
( 2 - ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ ( - ↑ d ) • basis . repr x ) ⋯ ⊢ ( ( 2 - ↑ d ) * ( ↑ d * volume . real ( Metric.ball 0 1 ) ) ) • diracDelta ℝ 0 =
( ( 2 - ↑ d ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0 neg d : ℕ h : ¬ d = 0 this : NeZero d hdist : distOfFunction (fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x ) ⋯ =
( 2 - ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ ( - ↑ d ) • basis . repr x ) ⋯ ⊢ ( ( 2 - ↑ d ) * ( ↑ d * volume . real ( Metric.ball 0 1 ) ) ) • diracDelta ℝ 0 =
( ( 2 - ↑ d ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0 ] neg d : ℕ h : ¬ d = 0 this : NeZero d hdist : distOfFunction (fun x => ( ( 2 - ↑ d ) * ‖ x ‖ ^ ( - ↑ d ) ) • basis . repr x ) ⋯ =
( 2 - ↑ d ) • distOfFunction (fun x => ‖ x ‖ ^ ( - ↑ d ) • basis . repr x ) ⋯ ⊢ ( ( 2 - ↑ d ) * ( ↑ d * volume . real ( Metric.ball 0 1 ) ) ) • diracDelta ℝ 0 =
( ( 2 - ↑ d ) * ↑ d * volume . real ( Metric.ball 0 1 ) ) • diracDelta ℝ 0
ring_nf All goals completed! 🐙
In dimension two the fundamental solution of the Laplacian is the logarithm: the
distributional Laplacian of Real.log ‖x‖ is 2 * volume (Metric.ball 0 1) times the Dirac
delta at the origin.