Physlib.SpaceAndTime.Space.IsDistBounded

Physlib
SpaceAndTime
Space
IsDistBounded

Imports

/-
Copyright (c) 2025 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
module

public import Physlib.SpaceAndTime.Space.Integrals.RadialAngularMeasure
public import Physlib.SpaceAndTime.Time.Basic
public import Physlib.Relativity.Tensors.RealTensor.Vector.Basic
public import Mathlib.Analysis.Distribution.SchwartzSpace.Deriv

Functions on `Space d` which can be made into distributions

i. Overview

In this module, for functions f : Space d → F, we define the property IsDistBounded f. Functions satisfying this property can be used to create distributions Space d →d[ℝ] F by integrating them against Schwartz maps.

The condition IsDistBounded f essentially says that f is bounded by a finite sum of terms of the form c * ‖x + g‖ ^ p for constants c, g and - (d - 1) ≤ p where d is the dimension of the space.

ii. Key results

IsDistBounded : The boundedness condition on functions Space d → F for them to form distributions.

IsDistBounded.integrable_space : If f satisfies IsDistBounded f, then fun x => η x • f x is integrable for any Schwartz map η : 𝓢(Space d, ℝ).

IsDistBounded.integrable_time_space : If f satisfies IsDistBounded f, then fun x => η x • f x.2 is integrable for any Schwartz map η : 𝓢(Time × Space d, ℝ).

IsDistBounded.mono : If f₁ satisfies IsDistBounded f₁ and ‖f₂ x‖ ≤ ‖f₁ x‖ for all x, then f₂ satisfies IsDistBounded f₂.

iii. Table of contents

A. The predicate IsDistBounded f

B. Integrability properties of functions satisfying IsDistBounded f

B.1. AEStronglyMeasurable conditions

B.2. Integrability with respect to Schwartz maps on space

B.3. Integrability with respect to Schwartz maps on time and space

B.4. Integrability with respect to inverse powers

C. Integral on Schwartz maps is bounded by seminorms

D. Construction rules for IsDistBounded f

D.1. Addition

D.2. Finite sums

D.3. Scalar multiplication

D.4. Components of functions

D.5. Compositions with additions and subtractions

D.6. Congruence with respect to the norm

D.7. Monotonicity with respect to the norm

D.8. Inner products

D.9. Scalar multiplication with constant

E. Specific functions that are IsDistBounded

E.1. Constant functions

E.2. Powers of norms

F. Multiplication by norms and components

iv. References

@[expose] public section

A. The predicate `IsDistBounded f`

The boundedness condition on a function Space d → F for it to form a distribution.

@[fun_prop]
def IsDistBounded {d : ℕ} (f : Space d → F) : Prop :=
  AEStronglyMeasurable (fun x => f x) volume ∧
  ∃ n, ∃ c : Fin n → ℝ, ∃ g : Fin n → Space d,
    ∃ p : Fin n → ℤ,
    (∀ i, 0 ≤ c i) ∧
    (∀ i, - (d - 1 : ℕ) ≤ p i) ∧
    ∀ x, ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p i

B. Integrability properties of functions satisfying `IsDistBounded f`

B.1. `AEStronglyMeasurable` conditions

              omit [NormedSpace ℝ F] in
@[fun_prop]
lemma aestronglyMeasurable {d : ℕ} {f : Space d → F} (hf : IsDistBounded f) :
    AEStronglyMeasurable (fun x => f x) volume := hf.1@[fun_prop]
lemma aeStronglyMeasurable_schwartzMap_smul {d : ℕ} {f : Space d → F}
    (hf : IsDistBounded f) (η : 𝓢(Space d, ℝ)) :
    AEStronglyMeasurable (fun x => η x • f x) := byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ AEStronglyMeasurable (fun x => η x • f x) volume
  fun_propAll goals completed! 🐙@[fun_prop]
lemma aeStronglyMeasurable_fderiv_schwartzMap_smul {d : ℕ} {f : Space d → F}
    (hf : IsDistBounded f) (η : 𝓢(Space d, ℝ)) (y : Space d) :
    AEStronglyMeasurable (fun x => fderiv ℝ η x y • f x) := byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)y:Space d⊢ AEStronglyMeasurable (fun x => (fderiv ℝ (⇑η) x) y • f x) volume
  fun_propAll goals completed! 🐙@[fun_prop]
lemma aeStronglyMeasurable_inv_pow {d r : ℕ} {f : Space d → F}
    (hf : IsDistBounded f) :
    AEStronglyMeasurable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) :=
  AEStronglyMeasurable.smul (AEMeasurable.aestronglyMeasurable (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕr:ℕf:Space d → Fhf:IsDistBounded f⊢ AEMeasurable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖) volume fun_propAll goals completed! 🐙)) (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕr:ℕf:Space d → Fhf:IsDistBounded f⊢ AEStronglyMeasurable f volume fun_propAll goals completed! 🐙)@[fun_prop]
lemma aeStronglyMeasurable_time_schwartzMap_smul {d : ℕ} {f : Space d → F}
    (hf : IsDistBounded f) (η : 𝓢(Time × Space d, ℝ)) :
    AEStronglyMeasurable (fun x => η x • f x.2) :=
  AEStronglyMeasurable.smul (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ AEStronglyMeasurable (⇑η) volume fun_propAll goals completed! 🐙) (AEStronglyMeasurable.comp_snd (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ AEStronglyMeasurable f volume fun_propAll goals completed! 🐙))

B.2. Integrability with respect to Schwartz maps on space

              

@[fun_prop]
lemma integrable_space {d : ℕ} {f : Space d → F} (hf : IsDistBounded f)
    (η : 𝓢(Space d, ℝ)) :
    Integrable (fun x : Space d => η x • f x) volume := byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ Integrable (fun x => η x • f x) volume
  /- Reducing the problem to `Integrable (fun x : Space d => η x * ‖x + c‖ ^ p)` -/
  suffices h2 : ∀ (p : ℤ) (hp : - (d - 1 : ℕ) ≤ p) (c : Space d) (η : 𝓢(Space d, ℝ)),
      Integrable (fun x : Space d => η x * ‖x + c‖ ^ p) volume byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volume⊢ Integrable (fun x => η x • f x) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volume
    obtain ⟨n, c, g, p, c_nonneg, p_bound, bound⟩ := hf.2F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p i⊢ Integrable (fun x => η x • f x) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volume
    apply Integrable.mono (g := fun x => ∑ i, (c i * (‖η x‖ * ‖x + g i‖ ^ p i))) _hfF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p i⊢ AEStronglyMeasurable (fun x => η x • f x) volumehF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p i⊢ ∀ᵐ (a : Space d), ‖η a • f a‖ ≤ ‖∑ i, c i * (‖η a‖ * ‖a + g i‖ ^ p i)‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p i⊢ Integrable (fun x => ∑ i, c i * (‖η x‖ * ‖x + g i‖ ^ p i)) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volume
    ·hfF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p i⊢ AEStronglyMeasurable (fun x => η x • f x) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volume fun_propAll goals completed! 🐙F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volume
    ·hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p i⊢ ∀ᵐ (a : Space d), ‖η a • f a‖ ≤ ‖∑ i, c i * (‖η a‖ * ‖a + g i‖ ^ p i)‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volume filter_upwards with xhF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Space d⊢ ‖η x • f x‖ ≤ ‖∑ i, c i * (‖η x‖ * ‖x + g i‖ ^ p i)‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volume
      rw [norm_smulhF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Space d⊢ ‖η x‖ * ‖f x‖ ≤ ‖∑ i, c i * (‖η x‖ * ‖x + g i‖ ^ p i)‖hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Space d⊢ ‖η x‖ * ‖f x‖ ≤ ‖∑ i, c i * (‖η x‖ * ‖x + g i‖ ^ p i)‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volume]hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Space d⊢ ‖η x‖ * ‖f x‖ ≤ ‖∑ i, c i * (‖η x‖ * ‖x + g i‖ ^ p i)‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volume
      refine (mul_le_mul_of_nonneg_left (bound x) (norm_nonneg (η x))).trans (le_of_eq ?_)hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Space d⊢ ‖η x‖ * ∑ i, c i * ‖x + g i‖ ^ p i = ‖∑ i, c i * (‖η x‖ * ‖x + g i‖ ^ p i)‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volume
      simp only [Real.norm_eq_abs]hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Space d⊢ |η x| * ∑ i, c i * ‖x + g i‖ ^ p i = |∑ x_1, c x_1 * (|η x| * ‖x + g x_1‖ ^ p x_1)|F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volume
      rw [Finset.abs_sum_of_nonneg (fun i _ => mul_nonneg (c_nonneg i) (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Space di:Fin nx✝:i ∈ Finset.univ⊢ 0 ≤ |η x| * ‖x + g i‖ ^ p ihF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Space d⊢ ∑ i, |η x| * (c i * ‖x + g i‖ ^ p i) = ∑ i, c i * (|η x| * ‖x + g i‖ ^ p i)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volume positivityAll goals completed! 🐙hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Space d⊢ ∑ i, |η x| * (c i * ‖x + g i‖ ^ p i) = ∑ i, c i * (|η x| * ‖x + g i‖ ^ p i)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volume)),
        Finset.mul_sumhF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Space d⊢ ∑ i, |η x| * (c i * ‖x + g i‖ ^ p i) = ∑ i, c i * (|η x| * ‖x + g i‖ ^ p i)hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Space d⊢ ∑ i, |η x| * (c i * ‖x + g i‖ ^ p i) = ∑ i, c i * (|η x| * ‖x + g i‖ ^ p i)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volume]hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Space d⊢ ∑ i, |η x| * (c i * ‖x + g i‖ ^ p i) = ∑ i, c i * (|η x| * ‖x + g i‖ ^ p i)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volume
      ring_nfAll goals completed! 🐙F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volume
    ·F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p i⊢ Integrable (fun x => ∑ i, c i * (‖η x‖ * ‖x + g i‖ ^ p i)) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volume refine MeasureTheory.integrable_finsetSum _ fun i _ => Integrable.const_mul ?_ _F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ii:Fin nx✝:i ∈ Finset.univ⊢ Integrable (fun x => ‖η x‖ * ‖x + g i‖ ^ p i) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volume
      simpa using (h2 (p i) (p_bound i) (g i) η).normF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volume
  /- Reducing the problem to `Integrable (fun x : Space d => η x * ‖x‖ ^ p)` -/
  suffices h0 : ∀ (p : ℤ) (hp : - (d - 1 : ℕ) ≤ p) (η : 𝓢(Space d, ℝ)),
      Integrable (fun x : Space d => η x * ‖x‖ ^ p) volume byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volume⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x + c‖ ^ p) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volume
    intro p hp c ηF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Space d, ℝ)⊢ Integrable (fun x => η x * ‖x + c‖ ^ p) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volume
    suffices h1 : Integrable (fun x => η ((x + c) - c) * ‖x + c‖ ^ p) volume byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Space d, ℝ)h1:Integrable (fun x => η (x + c - c) * ‖x + c‖ ^ p) volume⊢ Integrable (fun x => η x * ‖x + c‖ ^ p) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Space d, ℝ)⊢ Integrable (fun x => η (x + c - c) * ‖x + c‖ ^ p) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volume
      simpa using h1F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Space d, ℝ)⊢ Integrable (fun x => η (x + c - c) * ‖x + c‖ ^ p) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Space d, ℝ)⊢ Integrable (fun x => η (x + c - c) * ‖x + c‖ ^ p) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volume
    apply MeasureTheory.Integrable.comp_add_right (g := c) (f := fun x => η (x - c) * ‖x‖ ^ p)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Space d, ℝ)⊢ Integrable (fun x => η (x - c) * ‖x‖ ^ p) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volume
    apply h0 p hp (η.compCLM (𝕜 := ℝ) ?_ ?_)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Space d, ℝ)⊢ Function.HasTemperateGrowth fun x => x - cF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Space d, ℝ)⊢ ∃ k C, ∀ (x : Space d), ‖x‖ ≤ C * (1 + ‖x - c‖) ^ kF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volume
    ·F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Space d, ℝ)⊢ Function.HasTemperateGrowth fun x => x - cF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volume apply Function.HasTemperateGrowth.of_fderiv (k := 1) (C := 1 + ‖c‖)h'fF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Space d, ℝ)⊢ Function.HasTemperateGrowth (fderiv ℝ fun x => x - c)hfF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Space d, ℝ)⊢ Differentiable ℝ fun x => x - chF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Space d, ℝ)⊢ ∀ (x : Space d), ‖x - c‖ ≤ (1 + ‖c‖) * (1 + ‖x‖) ^ 1F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volume
      ·h'fF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Space d, ℝ)⊢ Function.HasTemperateGrowth (fderiv ℝ fun x => x - c)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volume convert Function.HasTemperateGrowth.const (ContinuousLinearMap.id ℝ (Space d))e'_7F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Space d, ℝ)x✝:Space d⊢ fderiv ℝ (fun x => x - c) x✝ = ContinuousLinearMap.id ℝ (Space d)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volume
        simp [fderiv_sub_const]All goals completed! 🐙F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volume
      ·hfF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Space d, ℝ)⊢ Differentiable ℝ fun x => x - cF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volume fun_propAll goals completed! 🐙F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volume
      ·hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Space d, ℝ)⊢ ∀ (x : Space d), ‖x - c‖ ≤ (1 + ‖c‖) * (1 + ‖x‖) ^ 1F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volume refine fun x => (norm_sub_le _ _).trans (le_of_sub_nonneg ?_)hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Space d, ℝ)x:Space d⊢ 0 ≤ (1 + ‖c‖) * (1 + ‖x‖) ^ 1 - (‖x‖ + ‖c‖)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volume
        ring_nfhF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Space d, ℝ)x:Space d⊢ 0 ≤ 1 + ‖c‖ * ‖x‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volume
        positivityAll goals completed! 🐙F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volume
    ·F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Space d, ℝ)⊢ ∃ k C, ∀ (x : Space d), ‖x‖ ≤ C * (1 + ‖x - c‖) ^ kF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volume refine ⟨1, (1 + ‖c‖), fun x => (norm_le_norm_add_norm_sub' x c).trans (le_of_sub_nonneg ?_)⟩F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Space d, ℝ)x:Space d⊢ 0 ≤ (1 + ‖c‖) * (1 + ‖x - c‖) ^ 1 - (‖c‖ + ‖x - c‖)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volume
      ring_nfF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Space d, ℝ)x:Space d⊢ 0 ≤ 1 + ‖c‖ * ‖x - c‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volume
      positivityF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Space d, ℝ)), Integrable (fun x => η x * ‖x‖ ^ p) volume
  /- Proving `Integrable (fun x : Space d => η x * ‖x + c‖ ^ p)` -/
  intro p hp ηF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Space d, ℝ)p:ℤhp:-↑(d - 1) ≤ pη:𝓢(Space d, ℝ)⊢ Integrable (fun x => η x * ‖x‖ ^ p) volume
  rw [← MeasureTheory.integrable_norm_iff (AEMeasurable.aestronglyMeasurable (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Space d, ℝ)p:ℤhp:-↑(d - 1) ≤ pη:𝓢(Space d, ℝ)⊢ AEMeasurable (fun x => η x * ‖x‖ ^ p) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Space d, ℝ)p:ℤhp:-↑(d - 1) ≤ pη:𝓢(Space d, ℝ)⊢ Integrable (fun a => ‖η a * ‖a‖ ^ p‖) volume fun_propAll goals completed! 🐙F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Space d, ℝ)p:ℤhp:-↑(d - 1) ≤ pη:𝓢(Space d, ℝ)⊢ Integrable (fun a => ‖η a * ‖a‖ ^ p‖) volume))]F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Space d, ℝ)p:ℤhp:-↑(d - 1) ≤ pη:𝓢(Space d, ℝ)⊢ Integrable (fun a => ‖η a * ‖a‖ ^ p‖) volume
  simp only [norm_mul, norm_zpow, norm_norm]F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Space d, ℝ)p:ℤhp:-↑(d - 1) ≤ pη:𝓢(Space d, ℝ)⊢ Integrable (fun a => ‖η a‖ * ‖a‖ ^ p) volume
  match d with
  | 0 =>F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕp:ℤf:Space 0 → Fhf:IsDistBounded fη✝:𝓢(Space 0, ℝ)hp:-↑(0 - 1) ≤ pη:𝓢(Space 0, ℝ)⊢ Integrable (fun a => ‖η a‖ * ‖a‖ ^ p) volume simp only [Real.norm_eq_abs, Integrable.of_finite]All goals completed! 🐙
  | d + 1 =>F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)⊢ Integrable (fun a => ‖η a‖ * ‖a‖ ^ p) volume
  by_cases hp' : p = 0posF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':p = 0⊢ Integrable (fun a => ‖η a‖ * ‖a‖ ^ p) volumenegF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0⊢ Integrable (fun a => ‖η a‖ * ‖a‖ ^ p) volume
  ·posF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':p = 0⊢ Integrable (fun a => ‖η a‖ * ‖a‖ ^ p) volume subst hp'posF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)η:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ 0⊢ Integrable (fun a => ‖η a‖ * ‖a‖ ^ 0) volume
    simpa using η.integrable.normAll goals completed! 🐙
  suffices h1 : Integrable (fun x => ‖η x‖ * ‖x‖ ^ (p + d)) (radialAngularMeasure (d := (d + 1))) byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0h1:Integrable (fun x => ‖η x‖ * ‖x‖ ^ (p + ↑d)) radialAngularMeasure⊢ Integrable (fun a => ‖η a‖ * ‖a‖ ^ p) volumenegF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x‖ ^ (p + ↑d)) radialAngularMeasure
    rw [integrable_radialAngularMeasure_iffF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0h1:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖η x‖ * ‖x‖ ^ (p + ↑d))) volume⊢ Integrable (fun a => ‖η a‖ * ‖a‖ ^ p) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0h1:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖η x‖ * ‖x‖ ^ (p + ↑d))) volume⊢ Integrable (fun a => ‖η a‖ * ‖a‖ ^ p) volumenegF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x‖ ^ (p + ↑d)) radialAngularMeasure] at h1F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0h1:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖η x‖ * ‖x‖ ^ (p + ↑d))) volume⊢ Integrable (fun a => ‖η a‖ * ‖a‖ ^ p) volumenegF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x‖ ^ (p + ↑d)) radialAngularMeasure
    convert h1 using 1F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0h1:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖η x‖ * ‖x‖ ^ (p + ↑d))) volume⊢ (fun a => ‖η a‖ * ‖a‖ ^ p) = fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖η x‖ * ‖x‖ ^ (p + ↑d))negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x‖ ^ (p + ↑d)) radialAngularMeasure
    funext xF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0h1:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖η x‖ * ‖x‖ ^ (p + ↑d))) volumex:Space (d + 1)⊢ ‖η x‖ * ‖x‖ ^ p = (1 / ‖x‖ ^ (d + 1 - 1)) • (‖η x‖ * ‖x‖ ^ (p + ↑d))negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x‖ ^ (p + ↑d)) radialAngularMeasure
    generalize ‖x‖ = rF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0h1:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖η x‖ * ‖x‖ ^ (p + ↑d))) volumex:Space (d + 1)r:ℝ⊢ ‖η x‖ * r ^ p = (1 / r ^ (d + 1 - 1)) • (‖η x‖ * r ^ (p + ↑d))negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x‖ ^ (p + ↑d)) radialAngularMeasure
    simp only [Real.norm_eq_abs, add_tsub_cancel_right, one_div, smul_eq_mul]F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0h1:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖η x‖ * ‖x‖ ^ (p + ↑d))) volumex:Space (d + 1)r:ℝ⊢ |η x| * r ^ p = (r ^ d)⁻¹ * (|η x| * r ^ (p + ↑d))negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x‖ ^ (p + ↑d)) radialAngularMeasure
    rw [mul_left_commF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0h1:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖η x‖ * ‖x‖ ^ (p + ↑d))) volumex:Space (d + 1)r:ℝ⊢ |η x| * r ^ p = |η x| * ((r ^ d)⁻¹ * r ^ (p + ↑d))F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0h1:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖η x‖ * ‖x‖ ^ (p + ↑d))) volumex:Space (d + 1)r:ℝ⊢ |η x| * r ^ p = |η x| * ((r ^ d)⁻¹ * r ^ (p + ↑d))negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x‖ ^ (p + ↑d)) radialAngularMeasure]F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0h1:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖η x‖ * ‖x‖ ^ (p + ↑d))) volumex:Space (d + 1)r:ℝ⊢ |η x| * r ^ p = |η x| * ((r ^ d)⁻¹ * r ^ (p + ↑d))negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x‖ ^ (p + ↑d)) radialAngularMeasure
    congr 1e_aF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0h1:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖η x‖ * ‖x‖ ^ (p + ↑d))) volumex:Space (d + 1)r:ℝ⊢ r ^ p = (r ^ d)⁻¹ * r ^ (p + ↑d)negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x‖ ^ (p + ↑d)) radialAngularMeasure
    by_cases hr : r = 0posF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0h1:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖η x‖ * ‖x‖ ^ (p + ↑d))) volumex:Space (d + 1)r:ℝhr:r = 0⊢ r ^ p = (r ^ d)⁻¹ * r ^ (p + ↑d)negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0h1:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖η x‖ * ‖x‖ ^ (p + ↑d))) volumex:Space (d + 1)r:ℝhr:¬r = 0⊢ r ^ p = (r ^ d)⁻¹ * r ^ (p + ↑d)negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x‖ ^ (p + ↑d)) radialAngularMeasure
    ·posF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0h1:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖η x‖ * ‖x‖ ^ (p + ↑d))) volumex:Space (d + 1)r:ℝhr:r = 0⊢ r ^ p = (r ^ d)⁻¹ * r ^ (p + ↑d)negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x‖ ^ (p + ↑d)) radialAngularMeasure subst hrposF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0h1:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖η x‖ * ‖x‖ ^ (p + ↑d))) volumex:Space (d + 1)⊢ 0 ^ p = (0 ^ d)⁻¹ * 0 ^ (p + ↑d)negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x‖ ^ (p + ↑d)) radialAngularMeasure
      simp [zero_pow_eq, zero_zpow_eq, hp']posF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0h1:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖η x‖ * ‖x‖ ^ (p + ↑d))) volumex:Space (d + 1)⊢ p + ↑d = 0 → ¬d = 0negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x‖ ^ (p + ↑d)) radialAngularMeasure
      omegaAll goals completed! 🐙negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x‖ ^ (p + ↑d)) radialAngularMeasure
    field_simpnegF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0h1:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖η x‖ * ‖x‖ ^ (p + ↑d))) volumex:Space (d + 1)r:ℝhr:¬r = 0⊢ r ^ p * r ^ d = r ^ (p + ↑d)negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x‖ ^ (p + ↑d)) radialAngularMeasure
    rw [zpow_add₀ hrnegF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0h1:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖η x‖ * ‖x‖ ^ (p + ↑d))) volumex:Space (d + 1)r:ℝhr:¬r = 0⊢ r ^ p * r ^ d = r ^ p * r ^ ↑dnegF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0h1:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖η x‖ * ‖x‖ ^ (p + ↑d))) volumex:Space (d + 1)r:ℝhr:¬r = 0⊢ r ^ p * r ^ d = r ^ p * r ^ ↑dnegF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x‖ ^ (p + ↑d)) radialAngularMeasure]negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0h1:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖η x‖ * ‖x‖ ^ (p + ↑d))) volumex:Space (d + 1)r:ℝhr:¬r = 0⊢ r ^ p * r ^ d = r ^ p * r ^ ↑dnegF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x‖ ^ (p + ↑d)) radialAngularMeasure
    rflnegF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x‖ ^ (p + ↑d)) radialAngularMeasurenegF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x‖ ^ (p + ↑d)) radialAngularMeasure
  convert integrable_pow_mul_iteratedFDeriv radialAngularMeasure η (p + d).toNat 0 using 1F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0⊢ (fun x => ‖η x‖ * ‖x‖ ^ (p + ↑d)) = fun x => ‖x‖ ^ (p + ↑d).toNat * ‖iteratedFDeriv ℝ 0 (⇑η) x‖
  funext xF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0x:Space (d + 1)⊢ ‖η x‖ * ‖x‖ ^ (p + ↑d) = ‖x‖ ^ (p + ↑d).toNat * ‖iteratedFDeriv ℝ 0 (⇑η) x‖
  simp only [Real.norm_eq_abs, norm_iteratedFDeriv_zero]F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0x:Space (d + 1)⊢ |η x| * ‖x‖ ^ (p + ↑d) = ‖x‖ ^ (p + ↑d).toNat * |η x|
  rw [mul_comm,F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0x:Space (d + 1)⊢ ‖x‖ ^ (p + ↑d) * |η x| = ‖x‖ ^ (p + ↑d).toNat * |η x|All goals completed! 🐙 ← zpow_natCast,F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0x:Space (d + 1)⊢ ‖x‖ ^ (p + ↑d) * |η x| = ‖x‖ ^ ↑(p + ↑d).toNat * |η x|All goals completed! 🐙 Int.toNat_of_nonneg (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤd:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Space (d + 1), ℝ)hp':¬p = 0x:Space (d + 1)⊢ 0 ≤ p + ↑dAll goals completed! 🐙 omegaAll goals completed! 🐙All goals completed! 🐙)]All goals completed! 🐙@[fun_prop]
lemma integrable_space_mul {d : ℕ} {f : Space d → ℝ} (hf : IsDistBounded f)
    (η : 𝓢(Space d, ℝ)) :
    Integrable (fun x : Space d => η x * f x) volume := byd:ℕf:Space d → ℝhf:IsDistBounded fη:𝓢(Space d, ℝ)⊢ Integrable (fun x => η x * f x) volume
  exact hf.integrable_space ηAll goals completed! 🐙@[fun_prop]
lemma integrable_space_fderiv {d : ℕ} {f : Space d → F} (hf : IsDistBounded f)
    (η : 𝓢(Space d, ℝ)) (y : Space d) :
    Integrable (fun x : Space d => fderiv ℝ η x y • f x) volume :=
  hf.integrable_space (LineDeriv.lineDerivOpCLM ℝ _ y η)@[fun_prop]
lemma integrable_space_fderiv_mul {d : ℕ} {f : Space d → ℝ} (hf : IsDistBounded f)
    (η : 𝓢(Space d, ℝ)) (y : Space d) :
    Integrable (fun x : Space d => fderiv ℝ η x y * f x) volume :=
  hf.integrable_space (LineDeriv.lineDerivOpCLM ℝ _ y η)

B.3. Integrability with respect to Schwartz maps on time and space



instance {D1 : Type} [NormedAddCommGroup D1] [MeasurableSpace D1]
    {D2 : Type} [NormedAddCommGroup D2] [MeasurableSpace D2]
    (μ1 : Measure D1) (μ2 : Measure D2)
    [Measure.HasTemperateGrowth μ1] [Measure.HasTemperateGrowth μ2]
    [OpensMeasurableSpace (D1 × D2)] :
    Measure.HasTemperateGrowth (μ1.prod μ2) where
  exists_integrable := by𝕜:TypeE:TypeF:TypeF':Typeinst✝¹³:RCLike 𝕜inst✝¹²:NormedAddCommGroup Einst✝¹¹:NormedAddCommGroup Finst✝¹⁰:NormedAddCommGroup F'inst✝⁹:NormedSpace ℝ Finst✝⁸:NormedSpace ℝ F'inst✝⁷:NormedSpace ℝ ED1:Typeinst✝⁶:NormedAddCommGroup D1inst✝⁵:MeasurableSpace D1D2:Typeinst✝⁴:NormedAddCommGroup D2inst✝³:MeasurableSpace D2μ1:Measure D1μ2:Measure D2inst✝²:μ1.HasTemperateGrowthinst✝¹:μ2.HasTemperateGrowthinst✝:OpensMeasurableSpace (D1 × D2)⊢ ∃ n, Integrable (fun x => (1 + ‖x‖) ^ (-↑n)) (μ1.prod μ2)
    obtain ⟨rt1, h1⟩ := Measure.HasTemperateGrowth.exists_integrable (μ := μ1)𝕜:TypeE:TypeF:TypeF':Typeinst✝¹³:RCLike 𝕜inst✝¹²:NormedAddCommGroup Einst✝¹¹:NormedAddCommGroup Finst✝¹⁰:NormedAddCommGroup F'inst✝⁹:NormedSpace ℝ Finst✝⁸:NormedSpace ℝ F'inst✝⁷:NormedSpace ℝ ED1:Typeinst✝⁶:NormedAddCommGroup D1inst✝⁵:MeasurableSpace D1D2:Typeinst✝⁴:NormedAddCommGroup D2inst✝³:MeasurableSpace D2μ1:Measure D1μ2:Measure D2inst✝²:μ1.HasTemperateGrowthinst✝¹:μ2.HasTemperateGrowthinst✝:OpensMeasurableSpace (D1 × D2)rt1:ℕh1:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt1)) μ1⊢ ∃ n, Integrable (fun x => (1 + ‖x‖) ^ (-↑n)) (μ1.prod μ2)
    obtain ⟨rt2, h2⟩ := Measure.HasTemperateGrowth.exists_integrable (μ := μ2)𝕜:TypeE:TypeF:TypeF':Typeinst✝¹³:RCLike 𝕜inst✝¹²:NormedAddCommGroup Einst✝¹¹:NormedAddCommGroup Finst✝¹⁰:NormedAddCommGroup F'inst✝⁹:NormedSpace ℝ Finst✝⁸:NormedSpace ℝ F'inst✝⁷:NormedSpace ℝ ED1:Typeinst✝⁶:NormedAddCommGroup D1inst✝⁵:MeasurableSpace D1D2:Typeinst✝⁴:NormedAddCommGroup D2inst✝³:MeasurableSpace D2μ1:Measure D1μ2:Measure D2inst✝²:μ1.HasTemperateGrowthinst✝¹:μ2.HasTemperateGrowthinst✝:OpensMeasurableSpace (D1 × D2)rt1:ℕh1:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt1)) μ1rt2:ℕh2:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt2)) μ2⊢ ∃ n, Integrable (fun x => (1 + ‖x‖) ^ (-↑n)) (μ1.prod μ2)
    use rt1 + rt2h𝕜:TypeE:TypeF:TypeF':Typeinst✝¹³:RCLike 𝕜inst✝¹²:NormedAddCommGroup Einst✝¹¹:NormedAddCommGroup Finst✝¹⁰:NormedAddCommGroup F'inst✝⁹:NormedSpace ℝ Finst✝⁸:NormedSpace ℝ F'inst✝⁷:NormedSpace ℝ ED1:Typeinst✝⁶:NormedAddCommGroup D1inst✝⁵:MeasurableSpace D1D2:Typeinst✝⁴:NormedAddCommGroup D2inst✝³:MeasurableSpace D2μ1:Measure D1μ2:Measure D2inst✝²:μ1.HasTemperateGrowthinst✝¹:μ2.HasTemperateGrowthinst✝:OpensMeasurableSpace (D1 × D2)rt1:ℕh1:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt1)) μ1rt2:ℕh2:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt2)) μ2⊢ Integrable (fun x => (1 + ‖x‖) ^ (-↑(rt1 + rt2))) (μ1.prod μ2)
    apply Integrable.mono' (h1.mul_prod h2)h.hf𝕜:TypeE:TypeF:TypeF':Typeinst✝¹³:RCLike 𝕜inst✝¹²:NormedAddCommGroup Einst✝¹¹:NormedAddCommGroup Finst✝¹⁰:NormedAddCommGroup F'inst✝⁹:NormedSpace ℝ Finst✝⁸:NormedSpace ℝ F'inst✝⁷:NormedSpace ℝ ED1:Typeinst✝⁶:NormedAddCommGroup D1inst✝⁵:MeasurableSpace D1D2:Typeinst✝⁴:NormedAddCommGroup D2inst✝³:MeasurableSpace D2μ1:Measure D1μ2:Measure D2inst✝²:μ1.HasTemperateGrowthinst✝¹:μ2.HasTemperateGrowthinst✝:OpensMeasurableSpace (D1 × D2)rt1:ℕh1:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt1)) μ1rt2:ℕh2:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt2)) μ2⊢ AEStronglyMeasurable (fun x => (1 + ‖x‖) ^ (-↑(rt1 + rt2))) (μ1.prod μ2)h.h𝕜:TypeE:TypeF:TypeF':Typeinst✝¹³:RCLike 𝕜inst✝¹²:NormedAddCommGroup Einst✝¹¹:NormedAddCommGroup Finst✝¹⁰:NormedAddCommGroup F'inst✝⁹:NormedSpace ℝ Finst✝⁸:NormedSpace ℝ F'inst✝⁷:NormedSpace ℝ ED1:Typeinst✝⁶:NormedAddCommGroup D1inst✝⁵:MeasurableSpace D1D2:Typeinst✝⁴:NormedAddCommGroup D2inst✝³:MeasurableSpace D2μ1:Measure D1μ2:Measure D2inst✝²:μ1.HasTemperateGrowthinst✝¹:μ2.HasTemperateGrowthinst✝:OpensMeasurableSpace (D1 × D2)rt1:ℕh1:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt1)) μ1rt2:ℕh2:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt2)) μ2⊢ ∀ᵐ (a : D1 × D2) ∂μ1.prod μ2, ‖(1 + ‖a‖) ^ (-↑(rt1 + rt2))‖ ≤ (1 + ‖a.1‖) ^ (-↑rt1) * (1 + ‖a.2‖) ^ (-↑rt2)
    ·h.hf𝕜:TypeE:TypeF:TypeF':Typeinst✝¹³:RCLike 𝕜inst✝¹²:NormedAddCommGroup Einst✝¹¹:NormedAddCommGroup Finst✝¹⁰:NormedAddCommGroup F'inst✝⁹:NormedSpace ℝ Finst✝⁸:NormedSpace ℝ F'inst✝⁷:NormedSpace ℝ ED1:Typeinst✝⁶:NormedAddCommGroup D1inst✝⁵:MeasurableSpace D1D2:Typeinst✝⁴:NormedAddCommGroup D2inst✝³:MeasurableSpace D2μ1:Measure D1μ2:Measure D2inst✝²:μ1.HasTemperateGrowthinst✝¹:μ2.HasTemperateGrowthinst✝:OpensMeasurableSpace (D1 × D2)rt1:ℕh1:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt1)) μ1rt2:ℕh2:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt2)) μ2⊢ AEStronglyMeasurable (fun x => (1 + ‖x‖) ^ (-↑(rt1 + rt2))) (μ1.prod μ2) apply AEMeasurable.aestronglyMeasurableh.hf𝕜:TypeE:TypeF:TypeF':Typeinst✝¹³:RCLike 𝕜inst✝¹²:NormedAddCommGroup Einst✝¹¹:NormedAddCommGroup Finst✝¹⁰:NormedAddCommGroup F'inst✝⁹:NormedSpace ℝ Finst✝⁸:NormedSpace ℝ F'inst✝⁷:NormedSpace ℝ ED1:Typeinst✝⁶:NormedAddCommGroup D1inst✝⁵:MeasurableSpace D1D2:Typeinst✝⁴:NormedAddCommGroup D2inst✝³:MeasurableSpace D2μ1:Measure D1μ2:Measure D2inst✝²:μ1.HasTemperateGrowthinst✝¹:μ2.HasTemperateGrowthinst✝:OpensMeasurableSpace (D1 × D2)rt1:ℕh1:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt1)) μ1rt2:ℕh2:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt2)) μ2⊢ AEMeasurable (fun x => (1 + ‖x‖) ^ (-↑(rt1 + rt2))) (μ1.prod μ2)
      fun_propAll goals completed! 🐙
    filter_upwards with xh.h𝕜:TypeE:TypeF:TypeF':Typeinst✝¹³:RCLike 𝕜inst✝¹²:NormedAddCommGroup Einst✝¹¹:NormedAddCommGroup Finst✝¹⁰:NormedAddCommGroup F'inst✝⁹:NormedSpace ℝ Finst✝⁸:NormedSpace ℝ F'inst✝⁷:NormedSpace ℝ ED1:Typeinst✝⁶:NormedAddCommGroup D1inst✝⁵:MeasurableSpace D1D2:Typeinst✝⁴:NormedAddCommGroup D2inst✝³:MeasurableSpace D2μ1:Measure D1μ2:Measure D2inst✝²:μ1.HasTemperateGrowthinst✝¹:μ2.HasTemperateGrowthinst✝:OpensMeasurableSpace (D1 × D2)rt1:ℕh1:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt1)) μ1rt2:ℕh2:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt2)) μ2x:D1 × D2⊢ ‖(1 + ‖x‖) ^ (-↑(rt1 + rt2))‖ ≤ (1 + ‖x.1‖) ^ (-↑rt1) * (1 + ‖x.2‖) ^ (-↑rt2)
    simp only [Nat.cast_add, neg_add_rev, Real.norm_eq_abs, Real.rpow_neg_natCast, zpow_neg,
      zpow_natCast]h.h𝕜:TypeE:TypeF:TypeF':Typeinst✝¹³:RCLike 𝕜inst✝¹²:NormedAddCommGroup Einst✝¹¹:NormedAddCommGroup Finst✝¹⁰:NormedAddCommGroup F'inst✝⁹:NormedSpace ℝ Finst✝⁸:NormedSpace ℝ F'inst✝⁷:NormedSpace ℝ ED1:Typeinst✝⁶:NormedAddCommGroup D1inst✝⁵:MeasurableSpace D1D2:Typeinst✝⁴:NormedAddCommGroup D2inst✝³:MeasurableSpace D2μ1:Measure D1μ2:Measure D2inst✝²:μ1.HasTemperateGrowthinst✝¹:μ2.HasTemperateGrowthinst✝:OpensMeasurableSpace (D1 × D2)rt1:ℕh1:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt1)) μ1rt2:ℕh2:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt2)) μ2x:D1 × D2⊢ |(1 + ‖x‖) ^ (-↑rt2 + -↑rt1)| ≤ ((1 + ‖x.1‖) ^ rt1)⁻¹ * ((1 + ‖x.2‖) ^ rt2)⁻¹
    calc _
      _ = |(1 + ‖x‖) ^ (-(rt1 : ℝ)) * (1 + ‖x‖) ^ (-(rt2 : ℝ))| := by𝕜:TypeE:TypeF:TypeF':Typeinst✝¹³:RCLike 𝕜inst✝¹²:NormedAddCommGroup Einst✝¹¹:NormedAddCommGroup Finst✝¹⁰:NormedAddCommGroup F'inst✝⁹:NormedSpace ℝ Finst✝⁸:NormedSpace ℝ F'inst✝⁷:NormedSpace ℝ ED1:Typeinst✝⁶:NormedAddCommGroup D1inst✝⁵:MeasurableSpace D1D2:Typeinst✝⁴:NormedAddCommGroup D2inst✝³:MeasurableSpace D2μ1:Measure D1μ2:Measure D2inst✝²:μ1.HasTemperateGrowthinst✝¹:μ2.HasTemperateGrowthinst✝:OpensMeasurableSpace (D1 × D2)rt1:ℕh1:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt1)) μ1rt2:ℕh2:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt2)) μ2x:D1 × D2⊢ |(1 + ‖x‖) ^ (-↑rt2 + -↑rt1)| = |(1 + ‖x‖) ^ (-↑rt1) * (1 + ‖x‖) ^ (-↑rt2)|
        rw [Real.rpow_add (by𝕜:TypeE:TypeF:TypeF':Typeinst✝¹³:RCLike 𝕜inst✝¹²:NormedAddCommGroup Einst✝¹¹:NormedAddCommGroup Finst✝¹⁰:NormedAddCommGroup F'inst✝⁹:NormedSpace ℝ Finst✝⁸:NormedSpace ℝ F'inst✝⁷:NormedSpace ℝ ED1:Typeinst✝⁶:NormedAddCommGroup D1inst✝⁵:MeasurableSpace D1D2:Typeinst✝⁴:NormedAddCommGroup D2inst✝³:MeasurableSpace D2μ1:Measure D1μ2:Measure D2inst✝²:μ1.HasTemperateGrowthinst✝¹:μ2.HasTemperateGrowthinst✝:OpensMeasurableSpace (D1 × D2)rt1:ℕh1:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt1)) μ1rt2:ℕh2:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt2)) μ2x:D1 × D2⊢ 0 < 1 + ‖x‖All goals completed! 🐙 positivityAll goals completed! 🐙All goals completed! 🐙), mul_comm𝕜:TypeE:TypeF:TypeF':Typeinst✝¹³:RCLike 𝕜inst✝¹²:NormedAddCommGroup Einst✝¹¹:NormedAddCommGroup Finst✝¹⁰:NormedAddCommGroup F'inst✝⁹:NormedSpace ℝ Finst✝⁸:NormedSpace ℝ F'inst✝⁷:NormedSpace ℝ ED1:Typeinst✝⁶:NormedAddCommGroup D1inst✝⁵:MeasurableSpace D1D2:Typeinst✝⁴:NormedAddCommGroup D2inst✝³:MeasurableSpace D2μ1:Measure D1μ2:Measure D2inst✝²:μ1.HasTemperateGrowthinst✝¹:μ2.HasTemperateGrowthinst✝:OpensMeasurableSpace (D1 × D2)rt1:ℕh1:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt1)) μ1rt2:ℕh2:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt2)) μ2x:D1 × D2⊢ |(1 + ‖x‖) ^ (-↑rt1) * (1 + ‖x‖) ^ (-↑rt2)| = |(1 + ‖x‖) ^ (-↑rt1) * (1 + ‖x‖) ^ (-↑rt2)|All goals completed! 🐙]All goals completed! 🐙
      _ = (1 + ‖x‖) ^ (-(rt1 : ℝ)) * (1 + ‖x‖) ^ (-(rt2 : ℝ)) := by𝕜:TypeE:TypeF:TypeF':Typeinst✝¹³:RCLike 𝕜inst✝¹²:NormedAddCommGroup Einst✝¹¹:NormedAddCommGroup Finst✝¹⁰:NormedAddCommGroup F'inst✝⁹:NormedSpace ℝ Finst✝⁸:NormedSpace ℝ F'inst✝⁷:NormedSpace ℝ ED1:Typeinst✝⁶:NormedAddCommGroup D1inst✝⁵:MeasurableSpace D1D2:Typeinst✝⁴:NormedAddCommGroup D2inst✝³:MeasurableSpace D2μ1:Measure D1μ2:Measure D2inst✝²:μ1.HasTemperateGrowthinst✝¹:μ2.HasTemperateGrowthinst✝:OpensMeasurableSpace (D1 × D2)rt1:ℕh1:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt1)) μ1rt2:ℕh2:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt2)) μ2x:D1 × D2⊢ |(1 + ‖x‖) ^ (-↑rt1) * (1 + ‖x‖) ^ (-↑rt2)| = (1 + ‖x‖) ^ (-↑rt1) * (1 + ‖x‖) ^ (-↑rt2)
        rw [abs_of_nonneg (by𝕜:TypeE:TypeF:TypeF':Typeinst✝¹³:RCLike 𝕜inst✝¹²:NormedAddCommGroup Einst✝¹¹:NormedAddCommGroup Finst✝¹⁰:NormedAddCommGroup F'inst✝⁹:NormedSpace ℝ Finst✝⁸:NormedSpace ℝ F'inst✝⁷:NormedSpace ℝ ED1:Typeinst✝⁶:NormedAddCommGroup D1inst✝⁵:MeasurableSpace D1D2:Typeinst✝⁴:NormedAddCommGroup D2inst✝³:MeasurableSpace D2μ1:Measure D1μ2:Measure D2inst✝²:μ1.HasTemperateGrowthinst✝¹:μ2.HasTemperateGrowthinst✝:OpensMeasurableSpace (D1 × D2)rt1:ℕh1:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt1)) μ1rt2:ℕh2:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt2)) μ2x:D1 × D2⊢ 0 ≤ (1 + ‖x‖) ^ (-↑rt1) * (1 + ‖x‖) ^ (-↑rt2)All goals completed! 🐙 positivityAll goals completed! 🐙All goals completed! 🐙)]All goals completed! 🐙
    simp only [Real.rpow_neg_natCast, zpow_neg, zpow_natCast]h.h.calc.step𝕜:TypeE:TypeF:TypeF':Typeinst✝¹³:RCLike 𝕜inst✝¹²:NormedAddCommGroup Einst✝¹¹:NormedAddCommGroup Finst✝¹⁰:NormedAddCommGroup F'inst✝⁹:NormedSpace ℝ Finst✝⁸:NormedSpace ℝ F'inst✝⁷:NormedSpace ℝ ED1:Typeinst✝⁶:NormedAddCommGroup D1inst✝⁵:MeasurableSpace D1D2:Typeinst✝⁴:NormedAddCommGroup D2inst✝³:MeasurableSpace D2μ1:Measure D1μ2:Measure D2inst✝²:μ1.HasTemperateGrowthinst✝¹:μ2.HasTemperateGrowthinst✝:OpensMeasurableSpace (D1 × D2)rt1:ℕh1:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt1)) μ1rt2:ℕh2:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt2)) μ2x:D1 × D2⊢ ((1 + ‖x‖) ^ rt1)⁻¹ * ((1 + ‖x‖) ^ rt2)⁻¹ ≤ ((1 + ‖x.1‖) ^ rt1)⁻¹ * ((1 + ‖x.2‖) ^ rt2)⁻¹
    apply mul_le_mul _ _ (by𝕜:TypeE:TypeF:TypeF':Typeinst✝¹³:RCLike 𝕜inst✝¹²:NormedAddCommGroup Einst✝¹¹:NormedAddCommGroup Finst✝¹⁰:NormedAddCommGroup F'inst✝⁹:NormedSpace ℝ Finst✝⁸:NormedSpace ℝ F'inst✝⁷:NormedSpace ℝ ED1:Typeinst✝⁶:NormedAddCommGroup D1inst✝⁵:MeasurableSpace D1D2:Typeinst✝⁴:NormedAddCommGroup D2inst✝³:MeasurableSpace D2μ1:Measure D1μ2:Measure D2inst✝²:μ1.HasTemperateGrowthinst✝¹:μ2.HasTemperateGrowthinst✝:OpensMeasurableSpace (D1 × D2)rt1:ℕh1:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt1)) μ1rt2:ℕh2:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt2)) μ2x:D1 × D2⊢ 0 ≤ ((1 + ‖x‖) ^ rt2)⁻¹ positivityAll goals completed! 🐙) (by𝕜:TypeE:TypeF:TypeF':Typeinst✝¹³:RCLike 𝕜inst✝¹²:NormedAddCommGroup Einst✝¹¹:NormedAddCommGroup Finst✝¹⁰:NormedAddCommGroup F'inst✝⁹:NormedSpace ℝ Finst✝⁸:NormedSpace ℝ F'inst✝⁷:NormedSpace ℝ ED1:Typeinst✝⁶:NormedAddCommGroup D1inst✝⁵:MeasurableSpace D1D2:Typeinst✝⁴:NormedAddCommGroup D2inst✝³:MeasurableSpace D2μ1:Measure D1μ2:Measure D2inst✝²:μ1.HasTemperateGrowthinst✝¹:μ2.HasTemperateGrowthinst✝:OpensMeasurableSpace (D1 × D2)rt1:ℕh1:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt1)) μ1rt2:ℕh2:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt2)) μ2x:D1 × D2⊢ 0 ≤ ((1 + ‖x.1‖) ^ rt1)⁻¹ positivityAll goals completed! 🐙)
    ·𝕜:TypeE:TypeF:TypeF':Typeinst✝¹³:RCLike 𝕜inst✝¹²:NormedAddCommGroup Einst✝¹¹:NormedAddCommGroup Finst✝¹⁰:NormedAddCommGroup F'inst✝⁹:NormedSpace ℝ Finst✝⁸:NormedSpace ℝ F'inst✝⁷:NormedSpace ℝ ED1:Typeinst✝⁶:NormedAddCommGroup D1inst✝⁵:MeasurableSpace D1D2:Typeinst✝⁴:NormedAddCommGroup D2inst✝³:MeasurableSpace D2μ1:Measure D1μ2:Measure D2inst✝²:μ1.HasTemperateGrowthinst✝¹:μ2.HasTemperateGrowthinst✝:OpensMeasurableSpace (D1 × D2)rt1:ℕh1:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt1)) μ1rt2:ℕh2:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt2)) μ2x:D1 × D2⊢ ((1 + ‖x‖) ^ rt1)⁻¹ ≤ ((1 + ‖x.1‖) ^ rt1)⁻¹ exact inv_anti₀ (by𝕜:TypeE:TypeF:TypeF':Typeinst✝¹³:RCLike 𝕜inst✝¹²:NormedAddCommGroup Einst✝¹¹:NormedAddCommGroup Finst✝¹⁰:NormedAddCommGroup F'inst✝⁹:NormedSpace ℝ Finst✝⁸:NormedSpace ℝ F'inst✝⁷:NormedSpace ℝ ED1:Typeinst✝⁶:NormedAddCommGroup D1inst✝⁵:MeasurableSpace D1D2:Typeinst✝⁴:NormedAddCommGroup D2inst✝³:MeasurableSpace D2μ1:Measure D1μ2:Measure D2inst✝²:μ1.HasTemperateGrowthinst✝¹:μ2.HasTemperateGrowthinst✝:OpensMeasurableSpace (D1 × D2)rt1:ℕh1:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt1)) μ1rt2:ℕh2:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt2)) μ2x:D1 × D2⊢ 0 < (1 + ‖x.1‖) ^ rt1 positivityAll goals completed! 🐙)
        (pow_le_pow_left₀ (by𝕜:TypeE:TypeF:TypeF':Typeinst✝¹³:RCLike 𝕜inst✝¹²:NormedAddCommGroup Einst✝¹¹:NormedAddCommGroup Finst✝¹⁰:NormedAddCommGroup F'inst✝⁹:NormedSpace ℝ Finst✝⁸:NormedSpace ℝ F'inst✝⁷:NormedSpace ℝ ED1:Typeinst✝⁶:NormedAddCommGroup D1inst✝⁵:MeasurableSpace D1D2:Typeinst✝⁴:NormedAddCommGroup D2inst✝³:MeasurableSpace D2μ1:Measure D1μ2:Measure D2inst✝²:μ1.HasTemperateGrowthinst✝¹:μ2.HasTemperateGrowthinst✝:OpensMeasurableSpace (D1 × D2)rt1:ℕh1:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt1)) μ1rt2:ℕh2:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt2)) μ2x:D1 × D2⊢ 0 ≤ 1 + ‖x.1‖ positivityAll goals completed! 🐙) (by𝕜:TypeE:TypeF:TypeF':Typeinst✝¹³:RCLike 𝕜inst✝¹²:NormedAddCommGroup Einst✝¹¹:NormedAddCommGroup Finst✝¹⁰:NormedAddCommGroup F'inst✝⁹:NormedSpace ℝ Finst✝⁸:NormedSpace ℝ F'inst✝⁷:NormedSpace ℝ ED1:Typeinst✝⁶:NormedAddCommGroup D1inst✝⁵:MeasurableSpace D1D2:Typeinst✝⁴:NormedAddCommGroup D2inst✝³:MeasurableSpace D2μ1:Measure D1μ2:Measure D2inst✝²:μ1.HasTemperateGrowthinst✝¹:μ2.HasTemperateGrowthinst✝:OpensMeasurableSpace (D1 × D2)rt1:ℕh1:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt1)) μ1rt2:ℕh2:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt2)) μ2x:D1 × D2⊢ 1 + ‖x.1‖ ≤ 1 + ‖x‖ simpa using norm_fst_le xAll goals completed! 🐙) rt1)
    ·𝕜:TypeE:TypeF:TypeF':Typeinst✝¹³:RCLike 𝕜inst✝¹²:NormedAddCommGroup Einst✝¹¹:NormedAddCommGroup Finst✝¹⁰:NormedAddCommGroup F'inst✝⁹:NormedSpace ℝ Finst✝⁸:NormedSpace ℝ F'inst✝⁷:NormedSpace ℝ ED1:Typeinst✝⁶:NormedAddCommGroup D1inst✝⁵:MeasurableSpace D1D2:Typeinst✝⁴:NormedAddCommGroup D2inst✝³:MeasurableSpace D2μ1:Measure D1μ2:Measure D2inst✝²:μ1.HasTemperateGrowthinst✝¹:μ2.HasTemperateGrowthinst✝:OpensMeasurableSpace (D1 × D2)rt1:ℕh1:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt1)) μ1rt2:ℕh2:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt2)) μ2x:D1 × D2⊢ ((1 + ‖x‖) ^ rt2)⁻¹ ≤ ((1 + ‖x.2‖) ^ rt2)⁻¹ exact inv_anti₀ (by𝕜:TypeE:TypeF:TypeF':Typeinst✝¹³:RCLike 𝕜inst✝¹²:NormedAddCommGroup Einst✝¹¹:NormedAddCommGroup Finst✝¹⁰:NormedAddCommGroup F'inst✝⁹:NormedSpace ℝ Finst✝⁸:NormedSpace ℝ F'inst✝⁷:NormedSpace ℝ ED1:Typeinst✝⁶:NormedAddCommGroup D1inst✝⁵:MeasurableSpace D1D2:Typeinst✝⁴:NormedAddCommGroup D2inst✝³:MeasurableSpace D2μ1:Measure D1μ2:Measure D2inst✝²:μ1.HasTemperateGrowthinst✝¹:μ2.HasTemperateGrowthinst✝:OpensMeasurableSpace (D1 × D2)rt1:ℕh1:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt1)) μ1rt2:ℕh2:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt2)) μ2x:D1 × D2⊢ 0 < (1 + ‖x.2‖) ^ rt2 positivityAll goals completed! 🐙)
        (pow_le_pow_left₀ (by𝕜:TypeE:TypeF:TypeF':Typeinst✝¹³:RCLike 𝕜inst✝¹²:NormedAddCommGroup Einst✝¹¹:NormedAddCommGroup Finst✝¹⁰:NormedAddCommGroup F'inst✝⁹:NormedSpace ℝ Finst✝⁸:NormedSpace ℝ F'inst✝⁷:NormedSpace ℝ ED1:Typeinst✝⁶:NormedAddCommGroup D1inst✝⁵:MeasurableSpace D1D2:Typeinst✝⁴:NormedAddCommGroup D2inst✝³:MeasurableSpace D2μ1:Measure D1μ2:Measure D2inst✝²:μ1.HasTemperateGrowthinst✝¹:μ2.HasTemperateGrowthinst✝:OpensMeasurableSpace (D1 × D2)rt1:ℕh1:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt1)) μ1rt2:ℕh2:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt2)) μ2x:D1 × D2⊢ 0 ≤ 1 + ‖x.2‖ positivityAll goals completed! 🐙) (by𝕜:TypeE:TypeF:TypeF':Typeinst✝¹³:RCLike 𝕜inst✝¹²:NormedAddCommGroup Einst✝¹¹:NormedAddCommGroup Finst✝¹⁰:NormedAddCommGroup F'inst✝⁹:NormedSpace ℝ Finst✝⁸:NormedSpace ℝ F'inst✝⁷:NormedSpace ℝ ED1:Typeinst✝⁶:NormedAddCommGroup D1inst✝⁵:MeasurableSpace D1D2:Typeinst✝⁴:NormedAddCommGroup D2inst✝³:MeasurableSpace D2μ1:Measure D1μ2:Measure D2inst✝²:μ1.HasTemperateGrowthinst✝¹:μ2.HasTemperateGrowthinst✝:OpensMeasurableSpace (D1 × D2)rt1:ℕh1:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt1)) μ1rt2:ℕh2:Integrable (fun x => (1 + ‖x‖) ^ (-↑rt2)) μ2x:D1 × D2⊢ 1 + ‖x.2‖ ≤ 1 + ‖x‖ simpa using norm_snd_le xAll goals completed! 🐙) rt2)



@[fun_prop]
lemma integrable_time_space {d : ℕ} {f : Space d → F} (hf : IsDistBounded f)
    (η : 𝓢(Time × Space d, ℝ)) :
    Integrable (fun x : Time × Space d => η x • f x.2) volume := byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ Integrable (fun x => η x • f x.2) volume
  /- Reducing the problem to `Integrable (fun x : Time × Space d => η x * ‖x.2 + c‖ ^ p)` -/
  suffices h2 : ∀ (p : ℤ) (hp : - (d - 1 : ℕ) ≤ p) (c : Space d) (η : 𝓢(Time × Space d, ℝ)),
      Integrable (fun x : Time × Space d => η x * ‖x.2 + c‖ ^ p) volume byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volume⊢ Integrable (fun x => η x • f x.2) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volume
    obtain ⟨n, c, g, p, c_nonneg, p_bound, bound⟩ := hf.2F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p i⊢ Integrable (fun x => η x • f x.2) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volume
    apply Integrable.mono (g := fun x => ∑ i, (c i * (‖η x‖ * ‖x.2 + g i‖ ^ p i))) _hfF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p i⊢ AEStronglyMeasurable (fun x => η x • f x.2) volumehF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p i⊢ ∀ᵐ (a : Time × Space d), ‖η a • f a.2‖ ≤ ‖∑ i, c i * (‖η a‖ * ‖a.2 + g i‖ ^ p i)‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p i⊢ Integrable (fun x => ∑ i, c i * (‖η x‖ * ‖x.2 + g i‖ ^ p i)) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volume
    ·hfF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p i⊢ AEStronglyMeasurable (fun x => η x • f x.2) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volume fun_propAll goals completed! 🐙F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volume
    ·hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p i⊢ ∀ᵐ (a : Time × Space d), ‖η a • f a.2‖ ≤ ‖∑ i, c i * (‖η a‖ * ‖a.2 + g i‖ ^ p i)‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volume filter_upwards with xhF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Time × Space d⊢ ‖η x • f x.2‖ ≤ ‖∑ i, c i * (‖η x‖ * ‖x.2 + g i‖ ^ p i)‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volume
      rw [norm_smulhF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Time × Space d⊢ ‖η x‖ * ‖f x.2‖ ≤ ‖∑ i, c i * (‖η x‖ * ‖x.2 + g i‖ ^ p i)‖hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Time × Space d⊢ ‖η x‖ * ‖f x.2‖ ≤ ‖∑ i, c i * (‖η x‖ * ‖x.2 + g i‖ ^ p i)‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volume]hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Time × Space d⊢ ‖η x‖ * ‖f x.2‖ ≤ ‖∑ i, c i * (‖η x‖ * ‖x.2 + g i‖ ^ p i)‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volume
      refine (mul_le_mul_of_nonneg_left (bound x.2) (norm_nonneg (η x))).trans (le_of_eq ?_)hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Time × Space d⊢ ‖η x‖ * ∑ i, c i * ‖x.2 + g i‖ ^ p i = ‖∑ i, c i * (‖η x‖ * ‖x.2 + g i‖ ^ p i)‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volume
      simp only [Real.norm_eq_abs]hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Time × Space d⊢ |η x| * ∑ i, c i * ‖x.2 + g i‖ ^ p i = |∑ x_1, c x_1 * (|η x| * ‖x.2 + g x_1‖ ^ p x_1)|F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volume
      rw [Finset.abs_sum_of_nonneg (fun i _ => mul_nonneg (c_nonneg i) (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Time × Space di:Fin nx✝:i ∈ Finset.univ⊢ 0 ≤ |η x| * ‖x.2 + g i‖ ^ p ihF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Time × Space d⊢ ∑ i, |η x| * (c i * ‖x.2 + g i‖ ^ p i) = ∑ i, c i * (|η x| * ‖x.2 + g i‖ ^ p i)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volume positivityAll goals completed! 🐙hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Time × Space d⊢ ∑ i, |η x| * (c i * ‖x.2 + g i‖ ^ p i) = ∑ i, c i * (|η x| * ‖x.2 + g i‖ ^ p i)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volume)),
        Finset.mul_sumhF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Time × Space d⊢ ∑ i, |η x| * (c i * ‖x.2 + g i‖ ^ p i) = ∑ i, c i * (|η x| * ‖x.2 + g i‖ ^ p i)hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Time × Space d⊢ ∑ i, |η x| * (c i * ‖x.2 + g i‖ ^ p i) = ∑ i, c i * (|η x| * ‖x.2 + g i‖ ^ p i)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volume]hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Time × Space d⊢ ∑ i, |η x| * (c i * ‖x.2 + g i‖ ^ p i) = ∑ i, c i * (|η x| * ‖x.2 + g i‖ ^ p i)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volume
      ring_nfAll goals completed! 🐙F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volume
    ·F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p i⊢ Integrable (fun x => ∑ i, c i * (‖η x‖ * ‖x.2 + g i‖ ^ p i)) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volume refine MeasureTheory.integrable_finsetSum _ fun i _ => Integrable.const_mul ?_ _F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)h2:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ii:Fin nx✝:i ∈ Finset.univ⊢ Integrable (fun x => ‖η x‖ * ‖x.2 + g i‖ ^ p i) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volume
      simpa using (h2 (p i) (p_bound i) (g i) η).normF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volume
  /- Reducing the problem to `Integrable (fun x : Space d => η x * ‖x‖ ^ p)` -/
  suffices h0 : ∀ (p : ℤ) (hp : - (d - 1 : ℕ) ≤ p) (η : 𝓢(Time × Space d, ℝ)),
      Integrable (fun x : Time × Space d => η x * ‖x.2‖ ^ p) volume byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volume⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d) (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volume
    intro p hp c ηF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Time × Space d, ℝ)⊢ Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volume
    suffices h1 : Integrable (fun (x : Time × Space d) =>
        η ((x + (0, c)) - (0, c)) * ‖(x + (0, c)).2‖ ^ p) (volume.prod volume) byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Time × Space d, ℝ)h1:Integrable (fun x => η (x + (0, c) - (0, c)) * ‖(x + (0, c)).2‖ ^ p) (volume.prod volume)⊢ Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Time × Space d, ℝ)⊢ Integrable (fun x => η (x + (0, c) - (0, c)) * ‖(x + (0, c)).2‖ ^ p) (volume.prod volume)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volume
      simp_all only [add_sub_cancel_right, Prod.snd_add]F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Time × Space d, ℝ)h1:Integrable (fun x => η x * ‖x.2 + c‖ ^ p) (volume.prod volume)⊢ Integrable (fun x => η x * ‖x.2 + c‖ ^ p) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Time × Space d, ℝ)⊢ Integrable (fun x => η (x + (0, c) - (0, c)) * ‖(x + (0, c)).2‖ ^ p) (volume.prod volume)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volume
      exact h1F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Time × Space d, ℝ)⊢ Integrable (fun x => η (x + (0, c) - (0, c)) * ‖(x + (0, c)).2‖ ^ p) (volume.prod volume)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Time × Space d, ℝ)⊢ Integrable (fun x => η (x + (0, c) - (0, c)) * ‖(x + (0, c)).2‖ ^ p) (volume.prod volume)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volume
    apply MeasureTheory.Integrable.comp_add_right (g := (0, c))
      (f := fun x => η (x - (0, c)) * ‖x.2‖ ^ p)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Time × Space d, ℝ)⊢ Integrable (fun x => η (x - (0, c)) * ‖x.2‖ ^ p) (volume.prod volume)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volume
    apply h0 p hp (η.compCLM (𝕜 := ℝ) ?_ ?_)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Time × Space d, ℝ)⊢ Function.HasTemperateGrowth fun x => x - (0, c)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Time × Space d, ℝ)⊢ ∃ k C, ∀ (x : Time × Space d), ‖x‖ ≤ C * (1 + ‖x - (0, c)‖) ^ kF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volume
    ·F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Time × Space d, ℝ)⊢ Function.HasTemperateGrowth fun x => x - (0, c)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volume apply Function.HasTemperateGrowth.of_fderiv (k := 1) (C := 1 + ‖c‖)h'fF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Time × Space d, ℝ)⊢ Function.HasTemperateGrowth (fderiv ℝ fun x => x - (0, c))hfF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Time × Space d, ℝ)⊢ Differentiable ℝ fun x => x - (0, c)hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Time × Space d, ℝ)⊢ ∀ (x : Time × Space d), ‖x - (0, c)‖ ≤ (1 + ‖c‖) * (1 + ‖x‖) ^ 1F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volume
      ·h'fF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Time × Space d, ℝ)⊢ Function.HasTemperateGrowth (fderiv ℝ fun x => x - (0, c))F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volume convert Function.HasTemperateGrowth.const (ContinuousLinearMap.id ℝ (Time × Space d))e'_7F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Time × Space d, ℝ)x✝:Time × Space d⊢ fderiv ℝ (fun x => x - (0, c)) x✝ = ContinuousLinearMap.id ℝ (Time × Space d)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volume
        simp [fderiv_sub_const]All goals completed! 🐙F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volume
      ·hfF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Time × Space d, ℝ)⊢ Differentiable ℝ fun x => x - (0, c)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volume fun_propAll goals completed! 🐙F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volume
      ·hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Time × Space d, ℝ)⊢ ∀ (x : Time × Space d), ‖x - (0, c)‖ ≤ (1 + ‖c‖) * (1 + ‖x‖) ^ 1F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volume refine fun x => (norm_sub_le _ _).trans (le_of_sub_nonneg ?_)hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Time × Space d, ℝ)x:Time × Space d⊢ 0 ≤ (1 + ‖c‖) * (1 + ‖x‖) ^ 1 - (‖x‖ + ‖(0, c)‖)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volume
        ring_nfhF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Time × Space d, ℝ)x:Time × Space d⊢ 0 ≤ 1 + ‖c‖ + ‖c‖ * ‖x‖ - ‖(0, c)‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volume
        simp only [Prod.norm_mk, norm_zero, norm_nonneg, sup_of_le_right]hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Time × Space d, ℝ)x:Time × Space d⊢ 0 ≤ 1 + ‖c‖ + ‖c‖ * ‖x‖ - ‖c‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volume
        ring_nfhF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Time × Space d, ℝ)x:Time × Space d⊢ 0 ≤ 1 + ‖c‖ * ‖x‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volume
        positivityAll goals completed! 🐙F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volume
    ·F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Time × Space d, ℝ)⊢ ∃ k C, ∀ (x : Time × Space d), ‖x‖ ≤ C * (1 + ‖x - (0, c)‖) ^ kF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volume refine ⟨1, (1 + ‖((0, c) : Time × Space d)‖),
        fun x => (norm_le_norm_add_norm_sub' x (0,c)).trans (le_of_sub_nonneg ?_)⟩F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Time × Space d, ℝ)x:Time × Space d⊢ 0 ≤ (1 + ‖(0, c)‖) * (1 + ‖x - (0, c)‖) ^ 1 - (‖(0, c)‖ + ‖x - (0, c)‖)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volume
      ring_nfF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)h0:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volumep:ℤhp:-↑(d - 1) ≤ pc:Space dη:𝓢(Time × Space d, ℝ)x:Time × Space d⊢ 0 ≤ 1 + ‖(0, c)‖ * ‖x - (0, c)‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volume
      positivityF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη:𝓢(Time × Space d, ℝ)⊢ ∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (η : 𝓢(Time × Space d, ℝ)), Integrable (fun x => η x * ‖x.2‖ ^ p) volume
  /- Proving `Integrable (fun x : Space d => η x * ‖x.2‖ ^ p)` -/
  intro p hp ηF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)⊢ Integrable (fun x => η x * ‖x.2‖ ^ p) volume
  rw [← MeasureTheory.integrable_norm_iff (AEMeasurable.aestronglyMeasurable (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)⊢ AEMeasurable (fun x => η x * ‖x.2‖ ^ p) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)⊢ Integrable (fun a => ‖η a * ‖a.2‖ ^ p‖) volume fun_propAll goals completed! 🐙F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)⊢ Integrable (fun a => ‖η a * ‖a.2‖ ^ p‖) volume))]F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)⊢ Integrable (fun a => ‖η a * ‖a.2‖ ^ p‖) volume
  simp only [norm_mul, norm_zpow, norm_norm]F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)⊢ Integrable (fun a => ‖η a‖ * ‖a.2‖ ^ p) volume
  by_cases hp : p = 0posF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:p = 0⊢ Integrable (fun a => ‖η a‖ * ‖a.2‖ ^ p) volumenegF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun a => ‖η a‖ * ‖a.2‖ ^ p) volume
  ·posF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:p = 0⊢ Integrable (fun a => ‖η a‖ * ‖a.2‖ ^ p) volume subst hpposF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)η:𝓢(Time × Space d, ℝ)hp:-↑(d - 1) ≤ 0⊢ Integrable (fun a => ‖η a‖ * ‖a.2‖ ^ 0) volume
    simp only [zpow_zero, mul_one]posF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)η:𝓢(Time × Space d, ℝ)hp:-↑(d - 1) ≤ 0⊢ Integrable (fun a => ‖η a‖) volume
    apply Integrable.normposF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)η:𝓢(Time × Space d, ℝ)hp:-↑(d - 1) ≤ 0⊢ Integrable (⇑η) volume
    change Integrable (⇑η) (volume.prod volume)posF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)η:𝓢(Time × Space d, ℝ)hp:-↑(d - 1) ≤ 0⊢ Integrable (⇑η) (volume.prod volume)
    exact η.integrableAll goals completed! 🐙
  suffices h1 : Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + (d - 1 : ℕ)))
      (volume.prod (radialAngularMeasure (d := d))) byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0h1:Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)⊢ Integrable (fun a => ‖η a‖ * ‖a.2‖ ^ p) volumenegF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)
    match d with
    | 0 =>F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕp:ℤhp✝:¬p = 0f:Space 0 → Fhf:IsDistBounded fη✝:𝓢(Time × Space 0, ℝ)hp:-↑(0 - 1) ≤ pη:𝓢(Time × Space 0, ℝ)h1:Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(0 - 1))) (volume.prod radialAngularMeasure)⊢ Integrable (fun a => ‖η a‖ * ‖a.2‖ ^ p) volumenegF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)
      simp_all only [zero_tsub, CharP.cast_eq_zero, neg_zero, Real.norm_eq_abs, add_zero,
        radialAngularMeasure_zero_eq_volume]F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕp:ℤhp✝:¬p = 0f:Space 0 → Fhf:IsDistBounded fη✝:𝓢(Time × Space 0, ℝ)η:𝓢(Time × Space 0, ℝ)hp:0 ≤ ph1:Integrable (fun x => |η x| * ‖x.2‖ ^ p) (volume.prod volume)⊢ Integrable (fun a => |η a| * ‖a.2‖ ^ p) volumenegF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)
      exact h1All goals completed! 🐙negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)
    | d + 1 =>F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1))) (volume.prod radialAngularMeasure)⊢ Integrable (fun a => ‖η a‖ * ‖a.2‖ ^ p) volumenegF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)
    rw [radialAngularMeasure,F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))
  (volume.prod (volume.withDensity fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))))⊢ Integrable (fun a => ‖η a‖ * ‖a.2‖ ^ p) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))
  ((volume.prod volume).withDensity fun z => ENNReal.ofReal (1 / ‖z.2‖ ^ (d + 1 - 1)))⊢ Integrable (fun a => ‖η a‖ * ‖a.2‖ ^ p) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))
  (volume.prod (volume.withDensity fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))))⊢ Measurable fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure) MeasureTheory.prod_withDensity_rightF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))
  ((volume.prod volume).withDensity fun z => ENNReal.ofReal (1 / ‖z.2‖ ^ (d + 1 - 1)))⊢ Integrable (fun a => ‖η a‖ * ‖a.2‖ ^ p) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))
  (volume.prod (volume.withDensity fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))))⊢ Measurable fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))
  ((volume.prod volume).withDensity fun z => ENNReal.ofReal (1 / ‖z.2‖ ^ (d + 1 - 1)))⊢ Integrable (fun a => ‖η a‖ * ‖a.2‖ ^ p) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))
  (volume.prod (volume.withDensity fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))))⊢ Measurable fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)] at h1F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))
  ((volume.prod volume).withDensity fun z => ENNReal.ofReal (1 / ‖z.2‖ ^ (d + 1 - 1)))⊢ Integrable (fun a => ‖η a‖ * ‖a.2‖ ^ p) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))
  (volume.prod (volume.withDensity fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))))⊢ Measurable fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)
    erw [integrable_withDensity_iff_integrable_smul₀ (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))
  ((volume.prod volume).withDensity fun z => ENNReal.ofReal (1 / ‖z.2‖ ^ (d + 1 - 1)))⊢ AEMeasurable (fun z => (1 / ‖z.2‖ ^ (d + 1 - 1)).toNNReal) (volume.prod volume)negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure) fun_propAll goals completed! 🐙negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure))]F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)⊢ Integrable (fun a => ‖η a‖ * ‖a.2‖ ^ p) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))
  (volume.prod (volume.withDensity fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))))⊢ Measurable fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure) at h1
    convert! h1 using 1e'_6F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)⊢ (fun a => ‖η a‖ * ‖a.2‖ ^ p) = fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))
  (volume.prod (volume.withDensity fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))))⊢ Measurable fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)
    funext xe'_6F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)⊢ ‖η x‖ * ‖x.2‖ ^ p = (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))
  (volume.prod (volume.withDensity fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))))⊢ Measurable fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)
    simp only [Real.norm_eq_abs, one_div]e'_6F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)⊢ |η x| * ‖x.2‖ ^ p = (‖x.2‖ ^ (d + 1 - 1))⁻¹.toNNReal • (|η x| * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))
  (volume.prod (volume.withDensity fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))))⊢ Measurable fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)
    rw [Real.toNNReal_of_nonneg,e'_6F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)⊢ |η x| * ‖x.2‖ ^ p = NNReal.mk (‖x.2‖ ^ (d + 1 - 1))⁻¹ ?e'_6✝ • (|η x| * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))e'_6F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)⊢ 0 ≤ (‖x.2‖ ^ (d + 1 - 1))⁻¹F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))
  (volume.prod (volume.withDensity fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))))⊢ Measurable fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))e'_6F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)⊢ |η x| * ‖x.2‖ ^ p = ↑(NNReal.mk (‖x.2‖ ^ (d + 1 - 1))⁻¹ ?e'_6✝) • (|η x| * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))e'_6F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)⊢ 0 ≤ (‖x.2‖ ^ (d + 1 - 1))⁻¹F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))
  (volume.prod (volume.withDensity fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))))⊢ Measurable fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure) NNReal.smul_defe'_6F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)⊢ |η x| * ‖x.2‖ ^ p = ↑(NNReal.mk (‖x.2‖ ^ (d + 1 - 1))⁻¹ ?e'_6✝) • (|η x| * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))e'_6F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)⊢ 0 ≤ (‖x.2‖ ^ (d + 1 - 1))⁻¹F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))
  (volume.prod (volume.withDensity fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))))⊢ Measurable fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))e'_6F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)⊢ |η x| * ‖x.2‖ ^ p = ↑(NNReal.mk (‖x.2‖ ^ (d + 1 - 1))⁻¹ ?e'_6✝) • (|η x| * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))e'_6F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)⊢ 0 ≤ (‖x.2‖ ^ (d + 1 - 1))⁻¹F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))
  (volume.prod (volume.withDensity fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))))⊢ Measurable fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)]e'_6F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)⊢ |η x| * ‖x.2‖ ^ p = ↑(NNReal.mk (‖x.2‖ ^ (d + 1 - 1))⁻¹ ?e'_6✝) • (|η x| * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))e'_6F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)⊢ 0 ≤ (‖x.2‖ ^ (d + 1 - 1))⁻¹F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))
  (volume.prod (volume.withDensity fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))))⊢ Measurable fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)
    simp only [inv_nonneg, norm_nonneg, pow_nonneg, coe_mk, smul_eq_mul]e'_6F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)⊢ |η x| * ‖x.2‖ ^ p = (‖x.2‖ ^ (d + 1 - 1))⁻¹ * (|η x| * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))e'_6F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)⊢ 0 ≤ (‖x.2‖ ^ (d + 1 - 1))⁻¹F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))
  (volume.prod (volume.withDensity fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))))⊢ Measurable fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)
    ring_nfe'_6F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)⊢ |η x| * ‖x.2‖ ^ p = |η x| * ‖x.2‖⁻¹ ^ (1 + d - 1) * ‖x.2‖ ^ (p + ↑(1 + d - 1))e'_6F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)⊢ 0 ≤ (‖x.2‖ ^ (d + 1 - 1))⁻¹F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))
  (volume.prod (volume.withDensity fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))))⊢ Measurable fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)
    rw [mul_assoce'_6F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)⊢ |η x| * ‖x.2‖ ^ p = |η x| * (‖x.2‖⁻¹ ^ (1 + d - 1) * ‖x.2‖ ^ (p + ↑(1 + d - 1)))e'_6F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)⊢ 0 ≤ (‖x.2‖ ^ (d + 1 - 1))⁻¹F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))
  (volume.prod (volume.withDensity fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))))⊢ Measurable fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))e'_6F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)⊢ |η x| * ‖x.2‖ ^ p = |η x| * (‖x.2‖⁻¹ ^ (1 + d - 1) * ‖x.2‖ ^ (p + ↑(1 + d - 1)))e'_6F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)⊢ 0 ≤ (‖x.2‖ ^ (d + 1 - 1))⁻¹F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))
  (volume.prod (volume.withDensity fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))))⊢ Measurable fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)]e'_6F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)⊢ |η x| * ‖x.2‖ ^ p = |η x| * (‖x.2‖⁻¹ ^ (1 + d - 1) * ‖x.2‖ ^ (p + ↑(1 + d - 1)))e'_6F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)⊢ 0 ≤ (‖x.2‖ ^ (d + 1 - 1))⁻¹F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))
  (volume.prod (volume.withDensity fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))))⊢ Measurable fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)
    congre'_6.e_aF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)⊢ ‖x.2‖ ^ p = ‖x.2‖⁻¹ ^ (1 + d - 1) * ‖x.2‖ ^ (p + ↑(1 + d - 1))e'_6F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)⊢ 0 ≤ (‖x.2‖ ^ (d + 1 - 1))⁻¹F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))
  (volume.prod (volume.withDensity fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))))⊢ Measurable fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)
    have hx : 0 ≤ ‖x.2‖ := norm_nonneg x.2e'_6.e_aF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)hx:0 ≤ ‖x.2‖⊢ ‖x.2‖ ^ p = ‖x.2‖⁻¹ ^ (1 + d - 1) * ‖x.2‖ ^ (p + ↑(1 + d - 1))e'_6F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)⊢ 0 ≤ (‖x.2‖ ^ (d + 1 - 1))⁻¹F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))
  (volume.prod (volume.withDensity fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))))⊢ Measurable fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)
    generalize ‖x.2‖ = r at *e'_6.e_aF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)r:ℝhx:0 ≤ r⊢ r ^ p = r⁻¹ ^ (1 + d - 1) * r ^ (p + ↑(1 + d - 1))e'_6F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)⊢ 0 ≤ (‖x.2‖ ^ (d + 1 - 1))⁻¹F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))
  (volume.prod (volume.withDensity fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))))⊢ Measurable fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)
    by_cases hr : r = 0posF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)r:ℝhx:0 ≤ rhr:r = 0⊢ r ^ p = r⁻¹ ^ (1 + d - 1) * r ^ (p + ↑(1 + d - 1))negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)r:ℝhx:0 ≤ rhr:¬r = 0⊢ r ^ p = r⁻¹ ^ (1 + d - 1) * r ^ (p + ↑(1 + d - 1))e'_6F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)⊢ 0 ≤ (‖x.2‖ ^ (d + 1 - 1))⁻¹F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))
  (volume.prod (volume.withDensity fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))))⊢ Measurable fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)
    ·posF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)r:ℝhx:0 ≤ rhr:r = 0⊢ r ^ p = r⁻¹ ^ (1 + d - 1) * r ^ (p + ↑(1 + d - 1))negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure) subst hrposF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)hx:0 ≤ 0⊢ 0 ^ p = 0⁻¹ ^ (1 + d - 1) * 0 ^ (p + ↑(1 + d - 1))negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)
      simp only [inv_zero]posF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)hx:0 ≤ 0⊢ 0 ^ p = 0 ^ (1 + d - 1) * 0 ^ (p + ↑(1 + d - 1))negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)
      rw [zero_pow_eq,posF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)hx:0 ≤ 0⊢ 0 ^ p = (if 1 + d - 1 = 0 then 1 else 0) * 0 ^ (p + ↑(1 + d - 1))posF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)hx:0 ≤ 0⊢ (if p = 0 then 1 else 0) = (if 1 + d - 1 = 0 then 1 else 0) * if p + ↑(1 + d - 1) = 0 then 1 else 0negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure) zero_zpow_eq,posF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)hx:0 ≤ 0⊢ (if p = 0 then 1 else 0) = (if 1 + d - 1 = 0 then 1 else 0) * 0 ^ (p + ↑(1 + d - 1))posF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)hx:0 ≤ 0⊢ (if p = 0 then 1 else 0) = (if 1 + d - 1 = 0 then 1 else 0) * if p + ↑(1 + d - 1) = 0 then 1 else 0negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure) zero_zpow_eqposF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)hx:0 ≤ 0⊢ (if p = 0 then 1 else 0) = (if 1 + d - 1 = 0 then 1 else 0) * if p + ↑(1 + d - 1) = 0 then 1 else 0posF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)hx:0 ≤ 0⊢ (if p = 0 then 1 else 0) = (if 1 + d - 1 = 0 then 1 else 0) * if p + ↑(1 + d - 1) = 0 then 1 else 0negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)]posF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)hx:0 ≤ 0⊢ (if p = 0 then 1 else 0) = (if 1 + d - 1 = 0 then 1 else 0) * if p + ↑(1 + d - 1) = 0 then 1 else 0negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)
      split_ifsposF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)hx:0 ≤ 0h✝¹:1 + d - 1 = 0h✝:p + ↑(1 + d - 1) = 0⊢ 0 = 1 * 1negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)hx:0 ≤ 0h✝¹:1 + d - 1 = 0h✝:¬p + ↑(1 + d - 1) = 0⊢ 0 = 1 * 0posF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)hx:0 ≤ 0h✝¹:¬1 + d - 1 = 0h✝:p + ↑(1 + d - 1) = 0⊢ 0 = 0 * 1negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)hx:0 ≤ 0h✝¹:¬1 + d - 1 = 0h✝:¬p + ↑(1 + d - 1) = 0⊢ 0 = 0 * 0negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure) <;>posF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)hx:0 ≤ 0h✝¹:1 + d - 1 = 0h✝:p + ↑(1 + d - 1) = 0⊢ 0 = 1 * 1negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)hx:0 ≤ 0h✝¹:1 + d - 1 = 0h✝:¬p + ↑(1 + d - 1) = 0⊢ 0 = 1 * 0posF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)hx:0 ≤ 0h✝¹:¬1 + d - 1 = 0h✝:p + ↑(1 + d - 1) = 0⊢ 0 = 0 * 1negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)hx:0 ≤ 0h✝¹:¬1 + d - 1 = 0h✝:¬p + ↑(1 + d - 1) = 0⊢ 0 = 0 * 0negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure) simpAll goals completed! 🐙negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)
      any_goals omegaAll goals completed! 🐙negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)
    ·negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)r:ℝhx:0 ≤ rhr:¬r = 0⊢ r ^ p = r⁻¹ ^ (1 + d - 1) * r ^ (p + ↑(1 + d - 1))negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure) simp only [inv_pow]negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)r:ℝhx:0 ≤ rhr:¬r = 0⊢ r ^ p = (r ^ (1 + d - 1))⁻¹ * r ^ (p + ↑(1 + d - 1))negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)
      field_simpnegF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)r:ℝhx:0 ≤ rhr:¬r = 0⊢ r ^ p * r ^ (1 + d - 1) = r ^ (p + ↑(1 + d - 1))negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)
      rw [zpow_add₀ hrnegF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)r:ℝhx:0 ≤ rhr:¬r = 0⊢ r ^ p * r ^ (1 + d - 1) = r ^ p * r ^ ↑(1 + d - 1)negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)r:ℝhx:0 ≤ rhr:¬r = 0⊢ r ^ p * r ^ (1 + d - 1) = r ^ p * r ^ ↑(1 + d - 1)negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)]negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)r:ℝhx:0 ≤ rhr:¬r = 0⊢ r ^ p * r ^ (1 + d - 1) = r ^ p * r ^ ↑(1 + d - 1)negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)
      simpAll goals completed! 🐙negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)
    ·e'_6F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => (1 / ‖x.2‖ ^ (d + 1 - 1)).toNNReal • (‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))) (volume.prod volume)x:Time × Space (d + 1)⊢ 0 ≤ (‖x.2‖ ^ (d + 1 - 1))⁻¹negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure) simpAll goals completed! 🐙negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)
    ·F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕp:ℤhp✝:¬p = 0d:ℕf:Space (d + 1) → Fhf:IsDistBounded fη✝:𝓢(Time × Space (d + 1), ℝ)hp:-↑(d + 1 - 1) ≤ pη:𝓢(Time × Space (d + 1), ℝ)h1:Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d + 1 - 1)))
  (volume.prod (volume.withDensity fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))))⊢ Measurable fun x => ENNReal.ofReal (1 / ‖x‖ ^ (d + 1 - 1))negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure) fun_propnegF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ Integrable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure)
  apply Integrable.mono' (integrable_pow_mul_iteratedFDeriv _ η (p + (d - 1 : ℕ)).toNat 0)
    (AEMeasurable.aestronglyMeasurable (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0⊢ AEMeasurable (fun x => ‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))) (volume.prod radialAngularMeasure) fun_propAll goals completed! 🐙))
  filter_upwards with xnegF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0x:Time × Space d⊢ ‖‖η x‖ * ‖x.2‖ ^ (p + ↑(d - 1))‖ ≤ ‖x‖ ^ (p + ↑(d - 1)).toNat * ‖iteratedFDeriv ℝ 0 (⇑η) x‖
  simp only [Real.norm_eq_abs, norm_iteratedFDeriv_zero]negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0x:Time × Space d⊢ ||η x| * ‖x.2‖ ^ (p + ↑(d - 1))| ≤ ‖x‖ ^ (p + ↑(d - 1)).toNat * |η x|
  rw [abs_of_nonneg (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0x:Time × Space d⊢ 0 ≤ |η x| * ‖x.2‖ ^ (p + ↑(d - 1))negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0x:Time × Space d⊢ ‖x.2‖ ^ (p + ↑(d - 1)) * |η x| ≤ ‖x‖ ^ (p + ↑(d - 1)) * |η x| positivityAll goals completed! 🐙negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0x:Time × Space d⊢ ‖x.2‖ ^ (p + ↑(d - 1)) * |η x| ≤ ‖x‖ ^ (p + ↑(d - 1)) * |η x|), mul_comm,negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0x:Time × Space d⊢ ‖x.2‖ ^ (p + ↑(d - 1)) * |η x| ≤ ‖x‖ ^ (p + ↑(d - 1)).toNat * |η x|negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0x:Time × Space d⊢ ‖x.2‖ ^ (p + ↑(d - 1)) * |η x| ≤ ‖x‖ ^ (p + ↑(d - 1)) * |η x| ← zpow_natCast,negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0x:Time × Space d⊢ ‖x.2‖ ^ (p + ↑(d - 1)) * |η x| ≤ ‖x‖ ^ ↑(p + ↑(d - 1)).toNat * |η x|negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0x:Time × Space d⊢ ‖x.2‖ ^ (p + ↑(d - 1)) * |η x| ≤ ‖x‖ ^ (p + ↑(d - 1)) * |η x| Int.toNat_of_nonneg (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0x:Time × Space d⊢ 0 ≤ p + ↑(d - 1)negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0x:Time × Space d⊢ ‖x.2‖ ^ (p + ↑(d - 1)) * |η x| ≤ ‖x‖ ^ (p + ↑(d - 1)) * |η x| omegaAll goals completed! 🐙negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0x:Time × Space d⊢ ‖x.2‖ ^ (p + ↑(d - 1)) * |η x| ≤ ‖x‖ ^ (p + ↑(d - 1)) * |η x|)]negF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0x:Time × Space d⊢ ‖x.2‖ ^ (p + ↑(d - 1)) * |η x| ≤ ‖x‖ ^ (p + ↑(d - 1)) * |η x|
  exact mul_le_mul_of_nonneg_right
    (zpow_le_zpow_left₀ (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fη✝:𝓢(Time × Space d, ℝ)p:ℤhp✝:-↑(d - 1) ≤ pη:𝓢(Time × Space d, ℝ)hp:¬p = 0x:Time × Space d⊢ 0 ≤ p + ↑(d - 1) omegaAll goals completed! 🐙) (norm_nonneg _) (norm_snd_le x)) (abs_nonneg _)

B.4. Integrability with respect to inverse powers



lemma integrable_mul_inv_pow {d : ℕ}
    {f : Space d → F} (hf : IsDistBounded f) :
    ∃ r, Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volume := byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∃ r, Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volume
  suffices h0 : ∀ pmax, ∃ r, ∀ (p : ℤ) (hp : - (d - 1 : ℕ) ≤ p) (c : Space d)
      (p_le : p ≤ pmax),
      Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fh0:∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume⊢ ∃ r, Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume
    obtain ⟨n, c, g, p, c_nonneg, p_bound, bound⟩ := hf.2F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fh0:∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p i⊢ ∃ r, Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume
    match n with
    | 0 =>F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fh0:∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumen:ℕc:Fin 0 → ℝg:Fin 0 → Space dp:Fin 0 → ℤc_nonneg:∀ (i : Fin 0), 0 ≤ c ip_bound:∀ (i : Fin 0), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p i⊢ ∃ r, Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume simp at boundF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fh0:∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumen:ℕc:Fin 0 → ℝg:Fin 0 → Space dp:Fin 0 → ℤc_nonneg:∀ (i : Fin 0), 0 ≤ c ip_bound:∀ (i : Fin 0), -↑(d - 1) ≤ p ibound:∀ (x : Space d), f x = 0⊢ ∃ r, Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume; simp [bound]All goals completed! 🐙F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume
    | n + 1 =>F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fh0:∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumen✝:ℕn:ℕc:Fin (n + 1) → ℝg:Fin (n + 1) → Space dp:Fin (n + 1) → ℤc_nonneg:∀ (i : Fin (n + 1)), 0 ≤ c ip_bound:∀ (i : Fin (n + 1)), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p i⊢ ∃ r, Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume
    let pMax := Finset.max' (Finset.image p Finset.univ) (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fh0:∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumen✝:ℕn:ℕc:Fin (n + 1) → ℝg:Fin (n + 1) → Space dp:Fin (n + 1) → ℤc_nonneg:∀ (i : Fin (n + 1)), 0 ≤ c ip_bound:∀ (i : Fin (n + 1)), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p i⊢ (Finset.image p Finset.univ).NonemptyF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fh0:∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumen✝:ℕn:ℕc:Fin (n + 1) → ℝg:Fin (n + 1) → Space dp:Fin (n + 1) → ℤc_nonneg:∀ (i : Fin (n + 1)), 0 ≤ c ip_bound:∀ (i : Fin (n + 1)), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ipMax:ℤ := (Finset.image p Finset.univ).max' ⋯⊢ ∃ r, Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume simpAll goals completed! 🐙F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fh0:∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumen✝:ℕn:ℕc:Fin (n + 1) → ℝg:Fin (n + 1) → Space dp:Fin (n + 1) → ℤc_nonneg:∀ (i : Fin (n + 1)), 0 ≤ c ip_bound:∀ (i : Fin (n + 1)), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ipMax:ℤ := (Finset.image p Finset.univ).max' ⋯⊢ ∃ r, Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fh0:∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumen✝:ℕn:ℕc:Fin (n + 1) → ℝg:Fin (n + 1) → Space dp:Fin (n + 1) → ℤc_nonneg:∀ (i : Fin (n + 1)), 0 ≤ c ip_bound:∀ (i : Fin (n + 1)), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ipMax:ℤ := (Finset.image p Finset.univ).max' ⋯⊢ ∃ r, Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume
    have pMax_max (i : Fin n.succ) : p i ≤ pMax :=
      Finset.le_max' _ _ (Finset.mem_image_of_mem p (Finset.mem_univ i))F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fh0:∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumen✝:ℕn:ℕc:Fin (n + 1) → ℝg:Fin (n + 1) → Space dp:Fin (n + 1) → ℤc_nonneg:∀ (i : Fin (n + 1)), 0 ≤ c ip_bound:∀ (i : Fin (n + 1)), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ipMax:ℤ := (Finset.image p Finset.univ).max' ⋯pMax_max:∀ (i : Fin n.succ), p i ≤ pMax⊢ ∃ r, Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume
    obtain ⟨r, hr⟩ := h0 pMaxF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fh0:∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumen✝:ℕn:ℕc:Fin (n + 1) → ℝg:Fin (n + 1) → Space dp:Fin (n + 1) → ℤc_nonneg:∀ (i : Fin (n + 1)), 0 ≤ c ip_bound:∀ (i : Fin (n + 1)), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ipMax:ℤ := (Finset.image p Finset.univ).max' ⋯pMax_max:∀ (i : Fin n.succ), p i ≤ pMaxr:ℕhr:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pMax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume⊢ ∃ r, Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume
    use rhF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fh0:∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumen✝:ℕn:ℕc:Fin (n + 1) → ℝg:Fin (n + 1) → Space dp:Fin (n + 1) → ℤc_nonneg:∀ (i : Fin (n + 1)), 0 ≤ c ip_bound:∀ (i : Fin (n + 1)), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ipMax:ℤ := (Finset.image p Finset.univ).max' ⋯pMax_max:∀ (i : Fin n.succ), p i ≤ pMaxr:ℕhr:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pMax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume⊢ Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume
    apply Integrable.mono (g := fun x => ∑ i, (c i * (‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + g i‖ ^ p i))) _h.hfF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fh0:∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumen✝:ℕn:ℕc:Fin (n + 1) → ℝg:Fin (n + 1) → Space dp:Fin (n + 1) → ℤc_nonneg:∀ (i : Fin (n + 1)), 0 ≤ c ip_bound:∀ (i : Fin (n + 1)), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ipMax:ℤ := (Finset.image p Finset.univ).max' ⋯pMax_max:∀ (i : Fin n.succ), p i ≤ pMaxr:ℕhr:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pMax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume⊢ AEStronglyMeasurable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeh.hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fh0:∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumen✝:ℕn:ℕc:Fin (n + 1) → ℝg:Fin (n + 1) → Space dp:Fin (n + 1) → ℤc_nonneg:∀ (i : Fin (n + 1)), 0 ≤ c ip_bound:∀ (i : Fin (n + 1)), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ipMax:ℤ := (Finset.image p Finset.univ).max' ⋯pMax_max:∀ (i : Fin n.succ), p i ≤ pMaxr:ℕhr:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pMax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume⊢ ∀ᵐ (a : Space d), ‖‖((1 + ‖a‖) ^ r)⁻¹‖ • f a‖ ≤ ‖∑ i, c i * (‖((1 + ‖a‖) ^ r)⁻¹‖ * ‖a + g i‖ ^ p i)‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fh0:∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumen✝:ℕn:ℕc:Fin (n + 1) → ℝg:Fin (n + 1) → Space dp:Fin (n + 1) → ℤc_nonneg:∀ (i : Fin (n + 1)), 0 ≤ c ip_bound:∀ (i : Fin (n + 1)), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ipMax:ℤ := (Finset.image p Finset.univ).max' ⋯pMax_max:∀ (i : Fin n.succ), p i ≤ pMaxr:ℕhr:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pMax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume⊢ Integrable (fun x => ∑ i, c i * (‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + g i‖ ^ p i)) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume
    ·h.hfF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fh0:∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumen✝:ℕn:ℕc:Fin (n + 1) → ℝg:Fin (n + 1) → Space dp:Fin (n + 1) → ℤc_nonneg:∀ (i : Fin (n + 1)), 0 ≤ c ip_bound:∀ (i : Fin (n + 1)), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ipMax:ℤ := (Finset.image p Finset.univ).max' ⋯pMax_max:∀ (i : Fin n.succ), p i ≤ pMaxr:ℕhr:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pMax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume⊢ AEStronglyMeasurable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume fun_propAll goals completed! 🐙F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume
    ·h.hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fh0:∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumen✝:ℕn:ℕc:Fin (n + 1) → ℝg:Fin (n + 1) → Space dp:Fin (n + 1) → ℤc_nonneg:∀ (i : Fin (n + 1)), 0 ≤ c ip_bound:∀ (i : Fin (n + 1)), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ipMax:ℤ := (Finset.image p Finset.univ).max' ⋯pMax_max:∀ (i : Fin n.succ), p i ≤ pMaxr:ℕhr:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pMax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume⊢ ∀ᵐ (a : Space d), ‖‖((1 + ‖a‖) ^ r)⁻¹‖ • f a‖ ≤ ‖∑ i, c i * (‖((1 + ‖a‖) ^ r)⁻¹‖ * ‖a + g i‖ ^ p i)‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume filter_upwards with xh.hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fh0:∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumen✝:ℕn:ℕc:Fin (n + 1) → ℝg:Fin (n + 1) → Space dp:Fin (n + 1) → ℤc_nonneg:∀ (i : Fin (n + 1)), 0 ≤ c ip_bound:∀ (i : Fin (n + 1)), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ipMax:ℤ := (Finset.image p Finset.univ).max' ⋯pMax_max:∀ (i : Fin n.succ), p i ≤ pMaxr:ℕhr:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pMax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumex:Space d⊢ ‖‖((1 + ‖x‖) ^ r)⁻¹‖ • f x‖ ≤ ‖∑ i, c i * (‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + g i‖ ^ p i)‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume
      rw [norm_smulh.hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fh0:∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumen✝:ℕn:ℕc:Fin (n + 1) → ℝg:Fin (n + 1) → Space dp:Fin (n + 1) → ℤc_nonneg:∀ (i : Fin (n + 1)), 0 ≤ c ip_bound:∀ (i : Fin (n + 1)), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ipMax:ℤ := (Finset.image p Finset.univ).max' ⋯pMax_max:∀ (i : Fin n.succ), p i ≤ pMaxr:ℕhr:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pMax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumex:Space d⊢ ‖‖((1 + ‖x‖) ^ r)⁻¹‖‖ * ‖f x‖ ≤ ‖∑ i, c i * (‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + g i‖ ^ p i)‖h.hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fh0:∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumen✝:ℕn:ℕc:Fin (n + 1) → ℝg:Fin (n + 1) → Space dp:Fin (n + 1) → ℤc_nonneg:∀ (i : Fin (n + 1)), 0 ≤ c ip_bound:∀ (i : Fin (n + 1)), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ipMax:ℤ := (Finset.image p Finset.univ).max' ⋯pMax_max:∀ (i : Fin n.succ), p i ≤ pMaxr:ℕhr:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pMax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumex:Space d⊢ ‖‖((1 + ‖x‖) ^ r)⁻¹‖‖ * ‖f x‖ ≤ ‖∑ i, c i * (‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + g i‖ ^ p i)‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume]h.hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fh0:∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumen✝:ℕn:ℕc:Fin (n + 1) → ℝg:Fin (n + 1) → Space dp:Fin (n + 1) → ℤc_nonneg:∀ (i : Fin (n + 1)), 0 ≤ c ip_bound:∀ (i : Fin (n + 1)), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ipMax:ℤ := (Finset.image p Finset.univ).max' ⋯pMax_max:∀ (i : Fin n.succ), p i ≤ pMaxr:ℕhr:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pMax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumex:Space d⊢ ‖‖((1 + ‖x‖) ^ r)⁻¹‖‖ * ‖f x‖ ≤ ‖∑ i, c i * (‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + g i‖ ^ p i)‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume
      refine (mul_le_mul_of_nonneg_left (bound x) (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fh0:∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumen✝:ℕn:ℕc:Fin (n + 1) → ℝg:Fin (n + 1) → Space dp:Fin (n + 1) → ℤc_nonneg:∀ (i : Fin (n + 1)), 0 ≤ c ip_bound:∀ (i : Fin (n + 1)), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ipMax:ℤ := (Finset.image p Finset.univ).max' ⋯pMax_max:∀ (i : Fin n.succ), p i ≤ pMaxr:ℕhr:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pMax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumex:Space d⊢ 0 ≤ ‖‖((1 + ‖x‖) ^ r)⁻¹‖‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume positivityAll goals completed! 🐙F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume)).trans (le_of_eq ?_)
      simp only [norm_inv, norm_pow, Real.norm_eq_abs, abs_abs]h.hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fh0:∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumen✝:ℕn:ℕc:Fin (n + 1) → ℝg:Fin (n + 1) → Space dp:Fin (n + 1) → ℤc_nonneg:∀ (i : Fin (n + 1)), 0 ≤ c ip_bound:∀ (i : Fin (n + 1)), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ipMax:ℤ := (Finset.image p Finset.univ).max' ⋯pMax_max:∀ (i : Fin n.succ), p i ≤ pMaxr:ℕhr:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pMax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumex:Space d⊢ (|1 + ‖x‖| ^ r)⁻¹ * ∑ i, c i * ‖x + g i‖ ^ p i = |∑ x_1, c x_1 * ((|1 + ‖x‖| ^ r)⁻¹ * ‖x + g x_1‖ ^ p x_1)|F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume
      rw [Finset.abs_sum_of_nonneg (fun i _ => mul_nonneg (c_nonneg i) (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fh0:∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumen✝:ℕn:ℕc:Fin (n + 1) → ℝg:Fin (n + 1) → Space dp:Fin (n + 1) → ℤc_nonneg:∀ (i : Fin (n + 1)), 0 ≤ c ip_bound:∀ (i : Fin (n + 1)), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ipMax:ℤ := (Finset.image p Finset.univ).max' ⋯pMax_max:∀ (i : Fin n.succ), p i ≤ pMaxr:ℕhr:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pMax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumex:Space di:Fin (n + 1)x✝:i ∈ Finset.univ⊢ 0 ≤ (|1 + ‖x‖| ^ r)⁻¹ * ‖x + g i‖ ^ p ih.hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fh0:∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumen✝:ℕn:ℕc:Fin (n + 1) → ℝg:Fin (n + 1) → Space dp:Fin (n + 1) → ℤc_nonneg:∀ (i : Fin (n + 1)), 0 ≤ c ip_bound:∀ (i : Fin (n + 1)), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ipMax:ℤ := (Finset.image p Finset.univ).max' ⋯pMax_max:∀ (i : Fin n.succ), p i ≤ pMaxr:ℕhr:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pMax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumex:Space d⊢ ∑ i, (|1 + ‖x‖| ^ r)⁻¹ * (c i * ‖x + g i‖ ^ p i) = ∑ i, c i * ((|1 + ‖x‖| ^ r)⁻¹ * ‖x + g i‖ ^ p i)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume positivityAll goals completed! 🐙h.hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fh0:∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumen✝:ℕn:ℕc:Fin (n + 1) → ℝg:Fin (n + 1) → Space dp:Fin (n + 1) → ℤc_nonneg:∀ (i : Fin (n + 1)), 0 ≤ c ip_bound:∀ (i : Fin (n + 1)), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ipMax:ℤ := (Finset.image p Finset.univ).max' ⋯pMax_max:∀ (i : Fin n.succ), p i ≤ pMaxr:ℕhr:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pMax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumex:Space d⊢ ∑ i, (|1 + ‖x‖| ^ r)⁻¹ * (c i * ‖x + g i‖ ^ p i) = ∑ i, c i * ((|1 + ‖x‖| ^ r)⁻¹ * ‖x + g i‖ ^ p i)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume)),
        Finset.mul_sumh.hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fh0:∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumen✝:ℕn:ℕc:Fin (n + 1) → ℝg:Fin (n + 1) → Space dp:Fin (n + 1) → ℤc_nonneg:∀ (i : Fin (n + 1)), 0 ≤ c ip_bound:∀ (i : Fin (n + 1)), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ipMax:ℤ := (Finset.image p Finset.univ).max' ⋯pMax_max:∀ (i : Fin n.succ), p i ≤ pMaxr:ℕhr:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pMax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumex:Space d⊢ ∑ i, (|1 + ‖x‖| ^ r)⁻¹ * (c i * ‖x + g i‖ ^ p i) = ∑ i, c i * ((|1 + ‖x‖| ^ r)⁻¹ * ‖x + g i‖ ^ p i)h.hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fh0:∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumen✝:ℕn:ℕc:Fin (n + 1) → ℝg:Fin (n + 1) → Space dp:Fin (n + 1) → ℤc_nonneg:∀ (i : Fin (n + 1)), 0 ≤ c ip_bound:∀ (i : Fin (n + 1)), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ipMax:ℤ := (Finset.image p Finset.univ).max' ⋯pMax_max:∀ (i : Fin n.succ), p i ≤ pMaxr:ℕhr:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pMax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumex:Space d⊢ ∑ i, (|1 + ‖x‖| ^ r)⁻¹ * (c i * ‖x + g i‖ ^ p i) = ∑ i, c i * ((|1 + ‖x‖| ^ r)⁻¹ * ‖x + g i‖ ^ p i)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume]h.hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fh0:∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumen✝:ℕn:ℕc:Fin (n + 1) → ℝg:Fin (n + 1) → Space dp:Fin (n + 1) → ℤc_nonneg:∀ (i : Fin (n + 1)), 0 ≤ c ip_bound:∀ (i : Fin (n + 1)), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ipMax:ℤ := (Finset.image p Finset.univ).max' ⋯pMax_max:∀ (i : Fin n.succ), p i ≤ pMaxr:ℕhr:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pMax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumex:Space d⊢ ∑ i, (|1 + ‖x‖| ^ r)⁻¹ * (c i * ‖x + g i‖ ^ p i) = ∑ i, c i * ((|1 + ‖x‖| ^ r)⁻¹ * ‖x + g i‖ ^ p i)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume
      ring_nfAll goals completed! 🐙F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume
    ·F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fh0:∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumen✝:ℕn:ℕc:Fin (n + 1) → ℝg:Fin (n + 1) → Space dp:Fin (n + 1) → ℤc_nonneg:∀ (i : Fin (n + 1)), 0 ≤ c ip_bound:∀ (i : Fin (n + 1)), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ipMax:ℤ := (Finset.image p Finset.univ).max' ⋯pMax_max:∀ (i : Fin n.succ), p i ≤ pMaxr:ℕhr:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pMax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume⊢ Integrable (fun x => ∑ i, c i * (‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + g i‖ ^ p i)) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume refine MeasureTheory.integrable_finsetSum _ fun i _ => Integrable.const_mul ?_ _F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fh0:∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumen✝:ℕn:ℕc:Fin (n + 1) → ℝg:Fin (n + 1) → Space dp:Fin (n + 1) → ℤc_nonneg:∀ (i : Fin (n + 1)), 0 ≤ c ip_bound:∀ (i : Fin (n + 1)), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ipMax:ℤ := (Finset.image p Finset.univ).max' ⋯pMax_max:∀ (i : Fin n.succ), p i ≤ pMaxr:ℕhr:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pMax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumei:Fin (n + 1)x✝:i ∈ Finset.univ⊢ Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + g i‖ ^ p i) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume
      refine (hr (p i) (p_bound i) (g i) (pMax_max i)).mono (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fh0:∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumen✝:ℕn:ℕc:Fin (n + 1) → ℝg:Fin (n + 1) → Space dp:Fin (n + 1) → ℤc_nonneg:∀ (i : Fin (n + 1)), 0 ≤ c ip_bound:∀ (i : Fin (n + 1)), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ipMax:ℤ := (Finset.image p Finset.univ).max' ⋯pMax_max:∀ (i : Fin n.succ), p i ≤ pMaxr:ℕhr:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pMax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumei:Fin (n + 1)x✝:i ∈ Finset.univ⊢ AEStronglyMeasurable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + g i‖ ^ p i) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume fun_propAll goals completed! 🐙F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume) ?_
      filter_upwards with xF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fh0:∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumen✝:ℕn:ℕc:Fin (n + 1) → ℝg:Fin (n + 1) → Space dp:Fin (n + 1) → ℤc_nonneg:∀ (i : Fin (n + 1)), 0 ≤ c ip_bound:∀ (i : Fin (n + 1)), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ipMax:ℤ := (Finset.image p Finset.univ).max' ⋯pMax_max:∀ (i : Fin n.succ), p i ≤ pMaxr:ℕhr:∀ (p : ℤ), -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pMax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumei:Fin (n + 1)x✝:i ∈ Finset.univx:Space d⊢ ‖‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + g i‖ ^ p i‖ ≤ ‖‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + g i‖ ^ p i‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume
      simpF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d - 1) ≤ p → ∀ (c : Space d), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume
  match d with
  | 0 =>F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space 0 → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(0 - 1) ≤ p → ∀ (c : Space 0), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume simpAll goals completed! 🐙
  | d + 1 =>F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded f⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d + 1 - 1) ≤ p →
        ∀ (c : Space (d + 1)), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volume
  suffices h0 : ∀ (q : ℕ) (c : Space (d + 1)), Integrable (fun x => ‖x + c‖ ^ (q - d : ℤ)
        * ‖((1 + ‖x‖) ^ (q + (radialAngularMeasure (d := d + 1)).integrablePower))⁻¹‖) volume byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fh0:∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volume⊢ ∀ (pmax : ℤ),
  ∃ r,
    ∀ (p : ℤ),
      -↑(d + 1 - 1) ≤ p →
        ∀ (c : Space (d + 1)), p ≤ pmax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded f⊢ ∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volume
    intro pMaxF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fh0:∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volumepMax:ℤ⊢ ∃ r,
  ∀ (p : ℤ),
    -↑(d + 1 - 1) ≤ p → ∀ (c : Space (d + 1)), p ≤ pMax → Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ * ‖x + c‖ ^ p) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded f⊢ ∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volume
    use (pMax + d).toNat + (radialAngularMeasure (d := d + 1)).integrablePowerhF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fh0:∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volumepMax:ℤ⊢ ∀ (p : ℤ),
  -↑(d + 1 - 1) ≤ p →
    ∀ (c : Space (d + 1)),
      p ≤ pMax →
        Integrable (fun x => ‖((1 + ‖x‖) ^ ((pMax + ↑d).toNat + radialAngularMeasure.integrablePower))⁻¹‖ * ‖x + c‖ ^ p)
          volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded f⊢ ∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volume
    intro p hp c p_lehF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fh0:∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volumepMax:ℤp:ℤhp:-↑(d + 1 - 1) ≤ pc:Space (d + 1)p_le:p ≤ pMax⊢ Integrable (fun x => ‖((1 + ‖x‖) ^ ((pMax + ↑d).toNat + radialAngularMeasure.integrablePower))⁻¹‖ * ‖x + c‖ ^ p) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded f⊢ ∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volume
    refine (h0 (p + d).toNat c).mono (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fh0:∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volumepMax:ℤp:ℤhp:-↑(d + 1 - 1) ≤ pc:Space (d + 1)p_le:p ≤ pMax⊢ AEStronglyMeasurable
  (fun x => ‖((1 + ‖x‖) ^ ((pMax + ↑d).toNat + radialAngularMeasure.integrablePower))⁻¹‖ * ‖x + c‖ ^ p) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded f⊢ ∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volume fun_propAll goals completed! 🐙F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded f⊢ ∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volume) ?_
    filter_upwards with xhF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fh0:∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volumepMax:ℤp:ℤhp:-↑(d + 1 - 1) ≤ pc:Space (d + 1)p_le:p ≤ pMaxx:Space (d + 1)⊢ ‖‖((1 + ‖x‖) ^ ((pMax + ↑d).toNat + radialAngularMeasure.integrablePower))⁻¹‖ * ‖x + c‖ ^ p‖ ≤
  ‖‖x + c‖ ^ (↑(p + ↑d).toNat - ↑d) * ‖((1 + ‖x‖) ^ ((p + ↑d).toNat + radialAngularMeasure.integrablePower))⁻¹‖‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded f⊢ ∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volume
    simp only [norm_inv, norm_pow, Real.norm_eq_abs, norm_mul, abs_abs, norm_zpow,
      Int.ofNat_toNat]hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fh0:∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volumepMax:ℤp:ℤhp:-↑(d + 1 - 1) ≤ pc:Space (d + 1)p_le:p ≤ pMaxx:Space (d + 1)⊢ (|1 + ‖x‖| ^ ((pMax + ↑d).toNat + radialAngularMeasure.integrablePower))⁻¹ * |‖x + c‖| ^ p ≤
  |‖x + c‖| ^ (max (p + ↑d) 0 - ↑d) * (|1 + ‖x‖| ^ ((p + ↑d).toNat + radialAngularMeasure.integrablePower))⁻¹F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded f⊢ ∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volume
    rw [mul_commhF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fh0:∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volumepMax:ℤp:ℤhp:-↑(d + 1 - 1) ≤ pc:Space (d + 1)p_le:p ≤ pMaxx:Space (d + 1)⊢ |‖x + c‖| ^ p * (|1 + ‖x‖| ^ ((pMax + ↑d).toNat + radialAngularMeasure.integrablePower))⁻¹ ≤
  |‖x + c‖| ^ (max (p + ↑d) 0 - ↑d) * (|1 + ‖x‖| ^ ((p + ↑d).toNat + radialAngularMeasure.integrablePower))⁻¹hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fh0:∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volumepMax:ℤp:ℤhp:-↑(d + 1 - 1) ≤ pc:Space (d + 1)p_le:p ≤ pMaxx:Space (d + 1)⊢ |‖x + c‖| ^ p * (|1 + ‖x‖| ^ ((pMax + ↑d).toNat + radialAngularMeasure.integrablePower))⁻¹ ≤
  |‖x + c‖| ^ (max (p + ↑d) 0 - ↑d) * (|1 + ‖x‖| ^ ((p + ↑d).toNat + radialAngularMeasure.integrablePower))⁻¹F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded f⊢ ∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volume]hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fh0:∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volumepMax:ℤp:ℤhp:-↑(d + 1 - 1) ≤ pc:Space (d + 1)p_le:p ≤ pMaxx:Space (d + 1)⊢ |‖x + c‖| ^ p * (|1 + ‖x‖| ^ ((pMax + ↑d).toNat + radialAngularMeasure.integrablePower))⁻¹ ≤
  |‖x + c‖| ^ (max (p + ↑d) 0 - ↑d) * (|1 + ‖x‖| ^ ((p + ↑d).toNat + radialAngularMeasure.integrablePower))⁻¹F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded f⊢ ∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volume
    refine mul_le_mul ?_ ?_ (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fh0:∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volumepMax:ℤp:ℤhp:-↑(d + 1 - 1) ≤ pc:Space (d + 1)p_le:p ≤ pMaxx:Space (d + 1)⊢ 0 ≤ (|1 + ‖x‖| ^ ((pMax + ↑d).toNat + radialAngularMeasure.integrablePower))⁻¹F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded f⊢ ∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volume positivityAll goals completed! 🐙F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded f⊢ ∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volume) (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fh0:∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volumepMax:ℤp:ℤhp:-↑(d + 1 - 1) ≤ pc:Space (d + 1)p_le:p ≤ pMaxx:Space (d + 1)⊢ 0 ≤ |‖x + c‖| ^ (max (p + ↑d) 0 - ↑d)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded f⊢ ∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volume positivityAll goals completed! 🐙F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded f⊢ ∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volume)
    ·h.refine_1F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fh0:∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volumepMax:ℤp:ℤhp:-↑(d + 1 - 1) ≤ pc:Space (d + 1)p_le:p ≤ pMaxx:Space (d + 1)⊢ |‖x + c‖| ^ p ≤ |‖x + c‖| ^ (max (p + ↑d) 0 - ↑d)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded f⊢ ∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volume rw [max_eq_left (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fh0:∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volumepMax:ℤp:ℤhp:-↑(d + 1 - 1) ≤ pc:Space (d + 1)p_le:p ≤ pMaxx:Space (d + 1)⊢ 0 ≤ p + ↑dh.refine_1F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fh0:∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volumepMax:ℤp:ℤhp:-↑(d + 1 - 1) ≤ pc:Space (d + 1)p_le:p ≤ pMaxx:Space (d + 1)⊢ |‖x + c‖| ^ p ≤ |‖x + c‖| ^ (p + ↑d - ↑d)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded f⊢ ∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volume omegaAll goals completed! 🐙h.refine_1F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fh0:∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volumepMax:ℤp:ℤhp:-↑(d + 1 - 1) ≤ pc:Space (d + 1)p_le:p ≤ pMaxx:Space (d + 1)⊢ |‖x + c‖| ^ p ≤ |‖x + c‖| ^ (p + ↑d - ↑d)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded f⊢ ∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volume)]h.refine_1F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fh0:∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volumepMax:ℤp:ℤhp:-↑(d + 1 - 1) ≤ pc:Space (d + 1)p_le:p ≤ pMaxx:Space (d + 1)⊢ |‖x + c‖| ^ p ≤ |‖x + c‖| ^ (p + ↑d - ↑d)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded f⊢ ∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volume
      simpAll goals completed! 🐙F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded f⊢ ∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volume
    ·h.refine_2F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fh0:∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volumepMax:ℤp:ℤhp:-↑(d + 1 - 1) ≤ pc:Space (d + 1)p_le:p ≤ pMaxx:Space (d + 1)⊢ (|1 + ‖x‖| ^ ((pMax + ↑d).toNat + radialAngularMeasure.integrablePower))⁻¹ ≤
  (|1 + ‖x‖| ^ ((p + ↑d).toNat + radialAngularMeasure.integrablePower))⁻¹F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded f⊢ ∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volume refine inv_pow_le_inv_pow_of_le ?_ ?_h.refine_2.refine_1F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fh0:∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volumepMax:ℤp:ℤhp:-↑(d + 1 - 1) ≤ pc:Space (d + 1)p_le:p ≤ pMaxx:Space (d + 1)⊢ 1 ≤ |1 + ‖x‖|h.refine_2.refine_2F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fh0:∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volumepMax:ℤp:ℤhp:-↑(d + 1 - 1) ≤ pc:Space (d + 1)p_le:p ≤ pMaxx:Space (d + 1)⊢ (p + ↑d).toNat + radialAngularMeasure.integrablePower ≤ (pMax + ↑d).toNat + radialAngularMeasure.integrablePowerF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded f⊢ ∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volume
      ·h.refine_2.refine_1F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fh0:∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volumepMax:ℤp:ℤhp:-↑(d + 1 - 1) ≤ pc:Space (d + 1)p_le:p ≤ pMaxx:Space (d + 1)⊢ 1 ≤ |1 + ‖x‖|F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded f⊢ ∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volume rw [abs_of_nonneg (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fh0:∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volumepMax:ℤp:ℤhp:-↑(d + 1 - 1) ≤ pc:Space (d + 1)p_le:p ≤ pMaxx:Space (d + 1)⊢ 0 ≤ 1 + ‖x‖h.refine_2.refine_1F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fh0:∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volumepMax:ℤp:ℤhp:-↑(d + 1 - 1) ≤ pc:Space (d + 1)p_le:p ≤ pMaxx:Space (d + 1)⊢ 1 ≤ 1 + ‖x‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded f⊢ ∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volume positivityAll goals completed! 🐙h.refine_2.refine_1F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fh0:∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volumepMax:ℤp:ℤhp:-↑(d + 1 - 1) ≤ pc:Space (d + 1)p_le:p ≤ pMaxx:Space (d + 1)⊢ 1 ≤ 1 + ‖x‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded f⊢ ∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volume)]h.refine_2.refine_1F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fh0:∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volumepMax:ℤp:ℤhp:-↑(d + 1 - 1) ≤ pc:Space (d + 1)p_le:p ≤ pMaxx:Space (d + 1)⊢ 1 ≤ 1 + ‖x‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded f⊢ ∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volume
        simpAll goals completed! 🐙F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded f⊢ ∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volume
      ·h.refine_2.refine_2F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fh0:∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volumepMax:ℤp:ℤhp:-↑(d + 1 - 1) ≤ pc:Space (d + 1)p_le:p ≤ pMaxx:Space (d + 1)⊢ (p + ↑d).toNat + radialAngularMeasure.integrablePower ≤ (pMax + ↑d).toNat + radialAngularMeasure.integrablePowerF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded f⊢ ∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volume simp_allF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded f⊢ ∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded f⊢ ∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volume
  let m := (radialAngularMeasure (d := (d + 1))).integrablePowerF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePower⊢ ∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volume
  suffices h0 : ∀ (q : ℕ) (c : Space (d + 1)),
      Integrable (fun x => ‖x‖ ^ (q - d : ℤ) * ‖((1 + ‖x - c‖) ^ (q + m))⁻¹‖) volume byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerh0:∀ (q : ℕ) (c : Space (d + 1)), Integrable (fun x => ‖x‖ ^ (↑q - ↑d) * ‖((1 + ‖x - c‖) ^ (q + m))⁻¹‖) volume⊢ ∀ (q : ℕ) (c : Space (d + 1)),
  Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePower⊢ ∀ (q : ℕ) (c : Space (d + 1)), Integrable (fun x => ‖x‖ ^ (↑q - ↑d) * ‖((1 + ‖x - c‖) ^ (q + m))⁻¹‖) volume
    intro q cF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerh0:∀ (q : ℕ) (c : Space (d + 1)), Integrable (fun x => ‖x‖ ^ (↑q - ↑d) * ‖((1 + ‖x - c‖) ^ (q + m))⁻¹‖) volumeq:ℕc:Space (d + 1)⊢ Integrable (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePower⊢ ∀ (q : ℕ) (c : Space (d + 1)), Integrable (fun x => ‖x‖ ^ (↑q - ↑d) * ‖((1 + ‖x - c‖) ^ (q + m))⁻¹‖) volume
    convert (h0 q c).comp_add_right c using 1F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerh0:∀ (q : ℕ) (c : Space (d + 1)), Integrable (fun x => ‖x‖ ^ (↑q - ↑d) * ‖((1 + ‖x - c‖) ^ (q + m))⁻¹‖) volumeq:ℕc:Space (d + 1)⊢ (fun x => ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖) = fun t =>
  ‖t + c‖ ^ (↑q - ↑d) * ‖((1 + ‖t + c - c‖) ^ (q + m))⁻¹‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePower⊢ ∀ (q : ℕ) (c : Space (d + 1)), Integrable (fun x => ‖x‖ ^ (↑q - ↑d) * ‖((1 + ‖x - c‖) ^ (q + m))⁻¹‖) volume
    funext xF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerh0:∀ (q : ℕ) (c : Space (d + 1)), Integrable (fun x => ‖x‖ ^ (↑q - ↑d) * ‖((1 + ‖x - c‖) ^ (q + m))⁻¹‖) volumeq:ℕc:Space (d + 1)x:Space (d + 1)⊢ ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x‖) ^ (q + radialAngularMeasure.integrablePower))⁻¹‖ =
  ‖x + c‖ ^ (↑q - ↑d) * ‖((1 + ‖x + c - c‖) ^ (q + m))⁻¹‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePower⊢ ∀ (q : ℕ) (c : Space (d + 1)), Integrable (fun x => ‖x‖ ^ (↑q - ↑d) * ‖((1 + ‖x - c‖) ^ (q + m))⁻¹‖) volume
    simp [m]F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePower⊢ ∀ (q : ℕ) (c : Space (d + 1)), Integrable (fun x => ‖x‖ ^ (↑q - ↑d) * ‖((1 + ‖x - c‖) ^ (q + m))⁻¹‖) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePower⊢ ∀ (q : ℕ) (c : Space (d + 1)), Integrable (fun x => ‖x‖ ^ (↑q - ↑d) * ‖((1 + ‖x - c‖) ^ (q + m))⁻¹‖) volume
  suffices h0 : ∀ (q : ℕ) (v : Space (d + 1)),
      Integrable (fun x => ‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖) radialAngularMeasure byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerh0:∀ (q : ℕ) (v : Space (d + 1)), Integrable (fun x => ‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖) radialAngularMeasure⊢ ∀ (q : ℕ) (c : Space (d + 1)), Integrable (fun x => ‖x‖ ^ (↑q - ↑d) * ‖((1 + ‖x - c‖) ^ (q + m))⁻¹‖) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePower⊢ ∀ (q : ℕ) (v : Space (d + 1)), Integrable (fun x => ‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖) radialAngularMeasure
    intro q vF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerh0:∀ (q : ℕ) (v : Space (d + 1)), Integrable (fun x => ‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖) radialAngularMeasureq:ℕv:Space (d + 1)⊢ Integrable (fun x => ‖x‖ ^ (↑q - ↑d) * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePower⊢ ∀ (q : ℕ) (v : Space (d + 1)), Integrable (fun x => ‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖) radialAngularMeasure
    specialize h0 q vF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)h0:Integrable (fun x => ‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖) radialAngularMeasure⊢ Integrable (fun x => ‖x‖ ^ (↑q - ↑d) * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePower⊢ ∀ (q : ℕ) (v : Space (d + 1)), Integrable (fun x => ‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖) radialAngularMeasure
    rw [integrable_radialAngularMeasure_iffF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)h0:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖)) volume⊢ Integrable (fun x => ‖x‖ ^ (↑q - ↑d) * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)h0:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖)) volume⊢ Integrable (fun x => ‖x‖ ^ (↑q - ↑d) * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePower⊢ ∀ (q : ℕ) (v : Space (d + 1)), Integrable (fun x => ‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖) radialAngularMeasure] at h0F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)h0:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖)) volume⊢ Integrable (fun x => ‖x‖ ^ (↑q - ↑d) * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePower⊢ ∀ (q : ℕ) (v : Space (d + 1)), Integrable (fun x => ‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖) radialAngularMeasure
    refine h0.congr (Filter.eventuallyEq_of_mem (compl_mem_ae_iff.mpr (measure_singleton 0))
      fun x hx => ?_)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)h0:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖)) volumex:Space (d + 1)hx:x ∈ {0}ᶜ⊢ (1 / ‖x‖ ^ (d + 1 - 1)) • (‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖) = ‖x‖ ^ (↑q - ↑d) * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePower⊢ ∀ (q : ℕ) (v : Space (d + 1)), Integrable (fun x => ‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖) radialAngularMeasure
    simp [← mul_assoc]F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)h0:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖)) volumex:Space (d + 1)hx:x ∈ {0}ᶜ⊢ (‖x‖ ^ d)⁻¹ * ‖x‖ ^ q = ‖x‖ ^ (↑q - ↑d) ∨ 1 + ‖x - v‖ = 0 ∧ (q = 0 → ¬m = 0)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePower⊢ ∀ (q : ℕ) (v : Space (d + 1)), Integrable (fun x => ‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖) radialAngularMeasure
    leftF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)h0:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖)) volumex:Space (d + 1)hx:x ∈ {0}ᶜ⊢ (‖x‖ ^ d)⁻¹ * ‖x‖ ^ q = ‖x‖ ^ (↑q - ↑d)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePower⊢ ∀ (q : ℕ) (v : Space (d + 1)), Integrable (fun x => ‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖) radialAngularMeasure
    rw [zpow_sub₀ (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)h0:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖)) volumex:Space (d + 1)hx:x ∈ {0}ᶜ⊢ ‖x‖ ≠ 0F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)h0:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖)) volumex:Space (d + 1)hx:x ∈ {0}ᶜ⊢ (‖x‖ ^ d)⁻¹ * ‖x‖ ^ q = ‖x‖ ^ q / ‖x‖ ^ dF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePower⊢ ∀ (q : ℕ) (v : Space (d + 1)), Integrable (fun x => ‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖) radialAngularMeasure simpa using hxAll goals completed! 🐙F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)h0:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖)) volumex:Space (d + 1)hx:x ∈ {0}ᶜ⊢ (‖x‖ ^ d)⁻¹ * ‖x‖ ^ q = ‖x‖ ^ q / ‖x‖ ^ dF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePower⊢ ∀ (q : ℕ) (v : Space (d + 1)), Integrable (fun x => ‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖) radialAngularMeasure), zpow_natCast,F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)h0:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖)) volumex:Space (d + 1)hx:x ∈ {0}ᶜ⊢ (‖x‖ ^ d)⁻¹ * ‖x‖ ^ q = ‖x‖ ^ q / ‖x‖ ^ ↑dF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)h0:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖)) volumex:Space (d + 1)hx:x ∈ {0}ᶜ⊢ (‖x‖ ^ d)⁻¹ * ‖x‖ ^ q = ‖x‖ ^ q / ‖x‖ ^ dF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePower⊢ ∀ (q : ℕ) (v : Space (d + 1)), Integrable (fun x => ‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖) radialAngularMeasure zpow_natCastF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)h0:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖)) volumex:Space (d + 1)hx:x ∈ {0}ᶜ⊢ (‖x‖ ^ d)⁻¹ * ‖x‖ ^ q = ‖x‖ ^ q / ‖x‖ ^ dF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)h0:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖)) volumex:Space (d + 1)hx:x ∈ {0}ᶜ⊢ (‖x‖ ^ d)⁻¹ * ‖x‖ ^ q = ‖x‖ ^ q / ‖x‖ ^ dF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePower⊢ ∀ (q : ℕ) (v : Space (d + 1)), Integrable (fun x => ‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖) radialAngularMeasure]F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)h0:Integrable (fun x => (1 / ‖x‖ ^ (d + 1 - 1)) • (‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖)) volumex:Space (d + 1)hx:x ∈ {0}ᶜ⊢ (‖x‖ ^ d)⁻¹ * ‖x‖ ^ q = ‖x‖ ^ q / ‖x‖ ^ dF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePower⊢ ∀ (q : ℕ) (v : Space (d + 1)), Integrable (fun x => ‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖) radialAngularMeasure
    field_simpF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePower⊢ ∀ (q : ℕ) (v : Space (d + 1)), Integrable (fun x => ‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖) radialAngularMeasureF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePower⊢ ∀ (q : ℕ) (v : Space (d + 1)), Integrable (fun x => ‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖) radialAngularMeasure
  intro q vF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)⊢ Integrable (fun x => ‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖) radialAngularMeasure
  have hr1 (x : Space (d + 1)) :
        ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ = ((1 + ‖x - v‖) ^ (q + m))⁻¹ := byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∃ r, Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)hr1:∀ (x : Space (d + 1)), ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ = ((1 + ‖x - v‖) ^ (q + m))⁻¹⊢ Integrable (fun x => ‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖) radialAngularMeasure
      simp only [norm_inv, norm_pow, Real.norm_eq_abs, inv_inj]F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)x:Space (d + 1)⊢ |1 + ‖x - v‖| ^ (q + m) = (1 + ‖x - v‖) ^ (q + m)F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)hr1:∀ (x : Space (d + 1)), ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ = ((1 + ‖x - v‖) ^ (q + m))⁻¹⊢ Integrable (fun x => ‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖) radialAngularMeasure
      rw [abs_of_nonneg (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)x:Space (d + 1)⊢ 0 ≤ 1 + ‖x - v‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)hr1:∀ (x : Space (d + 1)), ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ = ((1 + ‖x - v‖) ^ (q + m))⁻¹⊢ Integrable (fun x => ‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖) radialAngularMeasure positivityAll goals completed! 🐙F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)hr1:∀ (x : Space (d + 1)), ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ = ((1 + ‖x - v‖) ^ (q + m))⁻¹⊢ Integrable (fun x => ‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖) radialAngularMeasure)]F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)hr1:∀ (x : Space (d + 1)), ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ = ((1 + ‖x - v‖) ^ (q + m))⁻¹⊢ Integrable (fun x => ‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖) radialAngularMeasureF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)hr1:∀ (x : Space (d + 1)), ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ = ((1 + ‖x - v‖) ^ (q + m))⁻¹⊢ Integrable (fun x => ‖x‖ ^ q * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖) radialAngularMeasure
  apply integrable_of_le_of_pow_mul_le (C₁ := 1) (C₂ := (1 + ‖v‖) ^ (q + m))hfF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)hr1:∀ (x : Space (d + 1)), ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ = ((1 + ‖x - v‖) ^ (q + m))⁻¹⊢ ∀ (x : Space (d + 1)), ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ ≤ 1h'fF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)hr1:∀ (x : Space (d + 1)), ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ = ((1 + ‖x - v‖) ^ (q + m))⁻¹⊢ ∀ (x : Space (d + 1)),
  ‖x‖ ^ (q + radialAngularMeasure.integrablePower) * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ ≤ (1 + ‖v‖) ^ (q + m)h''fF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)hr1:∀ (x : Space (d + 1)), ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ = ((1 + ‖x - v‖) ^ (q + m))⁻¹⊢ AEStronglyMeasurable (fun x => ((1 + ‖x - v‖) ^ (q + m))⁻¹) radialAngularMeasure
  ·hfF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)hr1:∀ (x : Space (d + 1)), ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ = ((1 + ‖x - v‖) ^ (q + m))⁻¹⊢ ∀ (x : Space (d + 1)), ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ ≤ 1 intro xhfF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)hr1:∀ (x : Space (d + 1)), ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ = ((1 + ‖x - v‖) ^ (q + m))⁻¹x:Space (d + 1)⊢ ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ ≤ 1
    rw [hr1hfF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)hr1:∀ (x : Space (d + 1)), ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ = ((1 + ‖x - v‖) ^ (q + m))⁻¹x:Space (d + 1)⊢ ((1 + ‖x - v‖) ^ (q + m))⁻¹ ≤ 1hfF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)hr1:∀ (x : Space (d + 1)), ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ = ((1 + ‖x - v‖) ^ (q + m))⁻¹x:Space (d + 1)⊢ ((1 + ‖x - v‖) ^ (q + m))⁻¹ ≤ 1]hfF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)hr1:∀ (x : Space (d + 1)), ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ = ((1 + ‖x - v‖) ^ (q + m))⁻¹x:Space (d + 1)⊢ ((1 + ‖x - v‖) ^ (q + m))⁻¹ ≤ 1
    exact inv_le_one_of_one_le₀ (one_le_pow₀ (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)hr1:∀ (x : Space (d + 1)), ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ = ((1 + ‖x - v‖) ^ (q + m))⁻¹x:Space (d + 1)⊢ 1 ≤ 1 + ‖x - v‖ simpAll goals completed! 🐙))
  ·h'fF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)hr1:∀ (x : Space (d + 1)), ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ = ((1 + ‖x - v‖) ^ (q + m))⁻¹⊢ ∀ (x : Space (d + 1)),
  ‖x‖ ^ (q + radialAngularMeasure.integrablePower) * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ ≤ (1 + ‖v‖) ^ (q + m) intro xh'fF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)hr1:∀ (x : Space (d + 1)), ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ = ((1 + ‖x - v‖) ^ (q + m))⁻¹x:Space (d + 1)⊢ ‖x‖ ^ (q + radialAngularMeasure.integrablePower) * ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ ≤ (1 + ‖v‖) ^ (q + m)
    rw [hr1h'fF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)hr1:∀ (x : Space (d + 1)), ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ = ((1 + ‖x - v‖) ^ (q + m))⁻¹x:Space (d + 1)⊢ ‖x‖ ^ (q + radialAngularMeasure.integrablePower) * ((1 + ‖x - v‖) ^ (q + m))⁻¹ ≤ (1 + ‖v‖) ^ (q + m)h'fF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)hr1:∀ (x : Space (d + 1)), ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ = ((1 + ‖x - v‖) ^ (q + m))⁻¹x:Space (d + 1)⊢ ‖x‖ ^ (q + radialAngularMeasure.integrablePower) * ((1 + ‖x - v‖) ^ (q + m))⁻¹ ≤ (1 + ‖v‖) ^ (q + m)]h'fF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)hr1:∀ (x : Space (d + 1)), ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ = ((1 + ‖x - v‖) ^ (q + m))⁻¹x:Space (d + 1)⊢ ‖x‖ ^ (q + radialAngularMeasure.integrablePower) * ((1 + ‖x - v‖) ^ (q + m))⁻¹ ≤ (1 + ‖v‖) ^ (q + m)
    refine mul_inv_le_of_le_mul₀ ?_ (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)hr1:∀ (x : Space (d + 1)), ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ = ((1 + ‖x - v‖) ^ (q + m))⁻¹x:Space (d + 1)⊢ 0 ≤ (1 + ‖v‖) ^ (q + m) positivityAll goals completed! 🐙) ?_
    ·h'f.refine_1F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)hr1:∀ (x : Space (d + 1)), ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ = ((1 + ‖x - v‖) ^ (q + m))⁻¹x:Space (d + 1)⊢ 0 ≤ (1 + ‖x - v‖) ^ (q + m) positivityAll goals completed! 🐙
    change ‖x‖ ^ (q + m) ≤ _h'f.refine_2F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)hr1:∀ (x : Space (d + 1)), ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ = ((1 + ‖x - v‖) ^ (q + m))⁻¹x:Space (d + 1)⊢ ‖x‖ ^ (q + m) ≤ (1 + ‖v‖) ^ (q + m) * (1 + ‖x - v‖) ^ (q + m)
    calc ‖x‖ ^ (q + m) ≤ ((1 + ‖v‖) * (1 + ‖x - v‖)) ^ (q + m) := byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)hr1:∀ (x : Space (d + 1)), ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ = ((1 + ‖x - v‖) ^ (q + m))⁻¹x:Space (d + 1)⊢ ‖x‖ ^ (q + m) ≤ ((1 + ‖v‖) * (1 + ‖x - v‖)) ^ (q + m)
          refine pow_le_pow_left₀ (norm_nonneg x) ?_ _F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)hr1:∀ (x : Space (d + 1)), ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ = ((1 + ‖x - v‖) ^ (q + m))⁻¹x:Space (d + 1)⊢ ‖x‖ ≤ (1 + ‖v‖) * (1 + ‖x - v‖)
          nlinarith [norm_le_norm_add_norm_sub' x v, norm_nonneg (x - v), norm_nonneg v]All goals completed! 🐙
      _ = (1 + ‖v‖) ^ (q + m) * (1 + ‖x - v‖) ^ (q + m) := mul_pow _ _ _
  ·h''fF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)hr1:∀ (x : Space (d + 1)), ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ = ((1 + ‖x - v‖) ^ (q + m))⁻¹⊢ AEStronglyMeasurable (fun x => ((1 + ‖x - v‖) ^ (q + m))⁻¹) radialAngularMeasure exact Measurable.aestronglyMeasurable (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd✝:ℕd:ℕf:Space (d + 1) → Fhf:IsDistBounded fm:ℕ := radialAngularMeasure.integrablePowerq:ℕv:Space (d + 1)hr1:∀ (x : Space (d + 1)), ‖((1 + ‖x - v‖) ^ (q + m))⁻¹‖ = ((1 + ‖x - v‖) ^ (q + m))⁻¹⊢ Measurable fun x => ((1 + ‖x - v‖) ^ (q + m))⁻¹ fun_propAll goals completed! 🐙)

C. Integral on Schwartz maps is bounded by seminorms



lemma integral_mul_schwartzMap_bounded {d : ℕ} {f : Space d → F} (hf : IsDistBounded f) :
    ∃ (s : Finset (ℕ × ℕ)) (C : ℝ),
    0 ≤ C ∧ ∀ (η : 𝓢(Space d, ℝ)),
    ‖∫ (x : Space d), η x • f x‖ ≤ C * (s.sup (schwartzSeminormFamily ℝ (Space d) ℝ)) η := byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded f⊢ ∃ s C,
  0 ≤ C ∧ ∀ (η : 𝓢(Space d, ℝ)), ‖∫ (x : Space d), η x • f x‖ ≤ C * (s.sup (schwartzSeminormFamily ℝ (Space d) ℝ)) η
  obtain ⟨r, hr⟩ := hf.integrable_mul_inv_powF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volume⊢ ∃ s C,
  0 ≤ C ∧ ∀ (η : 𝓢(Space d, ℝ)), ‖∫ (x : Space d), η x • f x‖ ≤ C * (s.sup (schwartzSeminormFamily ℝ (Space d) ℝ)) η
  use Finset.Iic (r, 0), 2 ^ r * ∫ x, ‖f x‖ * ‖((1 + ‖x‖) ^ r)⁻¹‖hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volume⊢ 0 ≤ 2 ^ r * ∫ (x : Space d), ‖f x‖ * ‖((1 + ‖x‖) ^ r)⁻¹‖ ∧
  ∀ (η : 𝓢(Space d, ℝ)),
    ‖∫ (x : Space d), η x • f x‖ ≤
      (2 ^ r * ∫ (x : Space d), ‖f x‖ * ‖((1 + ‖x‖) ^ r)⁻¹‖) *
        ((Finset.Iic (r, 0)).sup (schwartzSeminormFamily ℝ (Space d) ℝ)) η
  refine ⟨byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volume⊢ 0 ≤ 2 ^ r * ∫ (x : Space d), ‖f x‖ * ‖((1 + ‖x‖) ^ r)⁻¹‖ positivityAll goals completed! 🐙, fun η ↦ (norm_integral_le_integral_norm _).trans ?_⟩
  rw [← integral_const_mul,hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)⊢ ∫ (a : Space d), ‖η a • f a‖ ≤
  (∫ (a : Space d), 2 ^ r * (‖f a‖ * ‖((1 + ‖a‖) ^ r)⁻¹‖)) *
    ((Finset.Iic (r, 0)).sup (schwartzSeminormFamily ℝ (Space d) ℝ)) ηhF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)⊢ ∫ (a : Space d), ‖η a • f a‖ ≤
  ∫ (a : Space d),
    2 ^ r * (‖f a‖ * ‖((1 + ‖a‖) ^ r)⁻¹‖) * ((Finset.Iic (r, 0)).sup (schwartzSeminormFamily ℝ (Space d) ℝ)) η ← integral_mul_consthF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)⊢ ∫ (a : Space d), ‖η a • f a‖ ≤
  ∫ (a : Space d),
    2 ^ r * (‖f a‖ * ‖((1 + ‖a‖) ^ r)⁻¹‖) * ((Finset.Iic (r, 0)).sup (schwartzSeminormFamily ℝ (Space d) ℝ)) ηhF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)⊢ ∫ (a : Space d), ‖η a • f a‖ ≤
  ∫ (a : Space d),
    2 ^ r * (‖f a‖ * ‖((1 + ‖a‖) ^ r)⁻¹‖) * ((Finset.Iic (r, 0)).sup (schwartzSeminormFamily ℝ (Space d) ℝ)) η]hF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)⊢ ∫ (a : Space d), ‖η a • f a‖ ≤
  ∫ (a : Space d),
    2 ^ r * (‖f a‖ * ‖((1 + ‖a‖) ^ r)⁻¹‖) * ((Finset.Iic (r, 0)).sup (schwartzSeminormFamily ℝ (Space d) ℝ)) η
  refine integral_mono_of_nonneg ?_ ?_ ?_h.refine_1F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)⊢ 0 ≤ᵐ[volume] fun a => ‖η a • f a‖h.refine_2F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)⊢ Integrable
  (fun a => 2 ^ r * (‖f a‖ * ‖((1 + ‖a‖) ^ r)⁻¹‖) * ((Finset.Iic (r, 0)).sup (schwartzSeminormFamily ℝ (Space d) ℝ)) η)
  volumeh.refine_3F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)⊢ (fun a => ‖η a • f a‖) ≤ᵐ[volume] fun a =>
  2 ^ r * (‖f a‖ * ‖((1 + ‖a‖) ^ r)⁻¹‖) * ((Finset.Iic (r, 0)).sup (schwartzSeminormFamily ℝ (Space d) ℝ)) η
  ·h.refine_1F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)⊢ 0 ≤ᵐ[volume] fun a => ‖η a • f a‖ filter_upwards with xh.refine_1F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)x:Space d⊢ 0 x ≤ ‖η x • f x‖
    positivityAll goals completed! 🐙
  ·h.refine_2F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)⊢ Integrable
  (fun a => 2 ^ r * (‖f a‖ * ‖((1 + ‖a‖) ^ r)⁻¹‖) * ((Finset.Iic (r, 0)).sup (schwartzSeminormFamily ℝ (Space d) ℝ)) η)
  volume refine (Integrable.congr' hr (AEStronglyMeasurable.mul (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)⊢ AEStronglyMeasurable (fun a => ‖f a‖) volume fun_propAll goals completed! 🐙)
      (AEMeasurable.aestronglyMeasurable (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)⊢ AEMeasurable (fun a => ‖((1 + ‖a‖) ^ r)⁻¹‖) volume fun_propAll goals completed! 🐙))) ?_).const_mul _ |>.mul_const _
    filter_upwards with xh.refine_2F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)x:Space d⊢ ‖‖((1 + ‖x‖) ^ r)⁻¹‖ • f x‖ = ‖((fun a => ‖f a‖) * fun a => ‖((1 + ‖a‖) ^ r)⁻¹‖) x‖
    simp [norm_smul, mul_comm]All goals completed! 🐙
  ·h.refine_3F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)⊢ (fun a => ‖η a • f a‖) ≤ᵐ[volume] fun a =>
  2 ^ r * (‖f a‖ * ‖((1 + ‖a‖) ^ r)⁻¹‖) * ((Finset.Iic (r, 0)).sup (schwartzSeminormFamily ℝ (Space d) ℝ)) η filter_upwards with xh.refine_3F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)x:Space d⊢ ‖η x • f x‖ ≤ 2 ^ r * (‖f x‖ * ‖((1 + ‖x‖) ^ r)⁻¹‖) * ((Finset.Iic (r, 0)).sup (schwartzSeminormFamily ℝ (Space d) ℝ)) η
    simp [norm_smul]h.refine_3F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)x:Space d⊢ |η x| * ‖f x‖ ≤ 2 ^ r * (‖f x‖ * (|1 + ‖x‖| ^ r)⁻¹) * ((Finset.Iic (r, 0)).sup (schwartzSeminormFamily ℝ (Space d) ℝ)) η
    trans (2 ^ r *
      ((Finset.Iic (r, 0)).sup (schwartzSeminormFamily ℝ (Space d) ℝ)) η
      *(|1 + ‖x‖| ^ r)⁻¹) * ‖f x‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)x:Space d⊢ |η x| * ‖f x‖ ≤ 2 ^ r * ((Finset.Iic (r, 0)).sup (schwartzSeminormFamily ℝ (Space d) ℝ)) η * (|1 + ‖x‖| ^ r)⁻¹ * ‖f x‖F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)x:Space d⊢ 2 ^ r * ((Finset.Iic (r, 0)).sup (schwartzSeminormFamily ℝ (Space d) ℝ)) η * (|1 + ‖x‖| ^ r)⁻¹ * ‖f x‖ ≤
  2 ^ r * (‖f x‖ * (|1 + ‖x‖| ^ r)⁻¹) * ((Finset.Iic (r, 0)).sup (schwartzSeminormFamily ℝ (Space d) ℝ)) η; swapF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)x:Space d⊢ 2 ^ r * ((Finset.Iic (r, 0)).sup (schwartzSeminormFamily ℝ (Space d) ℝ)) η * (|1 + ‖x‖| ^ r)⁻¹ * ‖f x‖ ≤
  2 ^ r * (‖f x‖ * (|1 + ‖x‖| ^ r)⁻¹) * ((Finset.Iic (r, 0)).sup (schwartzSeminormFamily ℝ (Space d) ℝ)) ηF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)x:Space d⊢ |η x| * ‖f x‖ ≤ 2 ^ r * ((Finset.Iic (r, 0)).sup (schwartzSeminormFamily ℝ (Space d) ℝ)) η * (|1 + ‖x‖| ^ r)⁻¹ * ‖f x‖
    ·F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)x:Space d⊢ 2 ^ r * ((Finset.Iic (r, 0)).sup (schwartzSeminormFamily ℝ (Space d) ℝ)) η * (|1 + ‖x‖| ^ r)⁻¹ * ‖f x‖ ≤
  2 ^ r * (‖f x‖ * (|1 + ‖x‖| ^ r)⁻¹) * ((Finset.Iic (r, 0)).sup (schwartzSeminormFamily ℝ (Space d) ℝ)) η apply le_of_eqF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)x:Space d⊢ 2 ^ r * ((Finset.Iic (r, 0)).sup (schwartzSeminormFamily ℝ (Space d) ℝ)) η * (|1 + ‖x‖| ^ r)⁻¹ * ‖f x‖ =
  2 ^ r * (‖f x‖ * (|1 + ‖x‖| ^ r)⁻¹) * ((Finset.Iic (r, 0)).sup (schwartzSeminormFamily ℝ (Space d) ℝ)) η
      ringAll goals completed! 🐙
    apply mul_le_mul_of_nonneg ?_ (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)x:Space d⊢ ‖f x‖ ≤ ‖f x‖ rflAll goals completed! 🐙) (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)x:Space d⊢ 0 ≤ |η x| positivityAll goals completed! 🐙) (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)x:Space d⊢ 0 ≤ ‖f x‖ positivityAll goals completed! 🐙)
    have h0 := one_add_le_sup_seminorm_apply (𝕜 := ℝ) (m := (r, 0))
      (k := r) (n := 0) le_rfl le_rfl η xF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)x:Space dh0:(1 + ‖x‖) ^ r * ‖iteratedFDeriv ℝ 0 (⇑η) x‖ ≤
  2 ^ (r, 0).1 * ((Finset.Iic (r, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ |η x| ≤ 2 ^ r * ((Finset.Iic (r, 0)).sup (schwartzSeminormFamily ℝ (Space d) ℝ)) η * (|1 + ‖x‖| ^ r)⁻¹
    rw [le_mul_inv_iff₀ (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)x:Space dh0:(1 + ‖x‖) ^ r * ‖iteratedFDeriv ℝ 0 (⇑η) x‖ ≤
  2 ^ (r, 0).1 * ((Finset.Iic (r, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ 0 < |1 + ‖x‖| ^ rF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)x:Space dh0:(1 + ‖x‖) ^ r * ‖iteratedFDeriv ℝ 0 (⇑η) x‖ ≤
  2 ^ (r, 0).1 * ((Finset.Iic (r, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ |η x| * |1 + ‖x‖| ^ r ≤ 2 ^ r * ((Finset.Iic (r, 0)).sup (schwartzSeminormFamily ℝ (Space d) ℝ)) η positivityAll goals completed! 🐙F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)x:Space dh0:(1 + ‖x‖) ^ r * ‖iteratedFDeriv ℝ 0 (⇑η) x‖ ≤
  2 ^ (r, 0).1 * ((Finset.Iic (r, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ |η x| * |1 + ‖x‖| ^ r ≤ 2 ^ r * ((Finset.Iic (r, 0)).sup (schwartzSeminormFamily ℝ (Space d) ℝ)) η)]F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)x:Space dh0:(1 + ‖x‖) ^ r * ‖iteratedFDeriv ℝ 0 (⇑η) x‖ ≤
  2 ^ (r, 0).1 * ((Finset.Iic (r, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ |η x| * |1 + ‖x‖| ^ r ≤ 2 ^ r * ((Finset.Iic (r, 0)).sup (schwartzSeminormFamily ℝ (Space d) ℝ)) η
    convert! h0 using 1e'_3F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)x:Space dh0:(1 + ‖x‖) ^ r * ‖iteratedFDeriv ℝ 0 (⇑η) x‖ ≤
  2 ^ (r, 0).1 * ((Finset.Iic (r, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ |η x| * |1 + ‖x‖| ^ r = (1 + ‖x‖) ^ r * ‖iteratedFDeriv ℝ 0 (⇑η) x‖
    simp only [norm_iteratedFDeriv_zero, Real.norm_eq_abs]e'_3F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)x:Space dh0:(1 + ‖x‖) ^ r * ‖iteratedFDeriv ℝ 0 (⇑η) x‖ ≤
  2 ^ (r, 0).1 * ((Finset.Iic (r, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ |η x| * |1 + ‖x‖| ^ r = (1 + ‖x‖) ^ r * |η x|
    ring_nfe'_3F:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)x:Space dh0:(1 + ‖x‖) ^ r * ‖iteratedFDeriv ℝ 0 (⇑η) x‖ ≤
  2 ^ (r, 0).1 * ((Finset.Iic (r, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ |η x| * |1 + ‖x‖| ^ r = |η x| * (1 + ‖x‖) ^ r
    congre'_3.e_a.e_aF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)x:Space dh0:(1 + ‖x‖) ^ r * ‖iteratedFDeriv ℝ 0 (⇑η) x‖ ≤
  2 ^ (r, 0).1 * ((Finset.Iic (r, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ |1 + ‖x‖| = 1 + ‖x‖
    rw [abs_of_nonneg (byF:Typeinst✝¹:NormedAddCommGroup Finst✝:NormedSpace ℝ Fd:ℕf:Space d → Fhf:IsDistBounded fr:ℕhr:Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖ • f x) volumeη:𝓢(Space d, ℝ)x:Space dh0:(1 + ‖x‖) ^ r * ‖iteratedFDeriv ℝ 0 (⇑η) x‖ ≤
  2 ^ (r, 0).1 * ((Finset.Iic (r, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ 0 ≤ 1 + ‖x‖All goals completed! 🐙 positivityAll goals completed! 🐙All goals completed! 🐙)]All goals completed! 🐙

D. Construction rules for `IsDistBounded f`

@[fun_prop]
lemma zero {d} : IsDistBounded (0 : Space d → F) :=
  ⟨byF:Typeinst✝:NormedAddCommGroup Fd:ℕ⊢ AEStronglyMeasurable (fun x => 0 x) volume fun_propAll goals completed! 🐙, 1, fun _ => 0, fun _ => 0, fun _ => 0, byF:Typeinst✝:NormedAddCommGroup Fd:ℕ⊢ (∀ (i : Fin 1), 0 ≤ (fun x => 0) i) ∧
  (∀ (i : Fin 1), -↑(d - 1) ≤ (fun x => 0) i) ∧
    ∀ (x : Space d), ‖0 x‖ ≤ ∑ i, (fun x => 0) i * ‖x + (fun x => 0) i‖ ^ (fun x => 0) i simpAll goals completed! 🐙⟩

D.1. Addition

@[fun_prop]
lemma add {d : ℕ} {f g : Space d → F}
    (hf : IsDistBounded f) (hg : IsDistBounded g) : IsDistBounded (f + g) := byF:Typeinst✝:NormedAddCommGroup Fd:ℕf:Space d → Fg:Space d → Fhf:IsDistBounded fhg:IsDistBounded g⊢ IsDistBounded (f + g)
  rcases hf with ⟨hae1, ⟨n1, c1, g1, p1, c1_nonneg, p1_bound, bound1⟩⟩F:Typeinst✝:NormedAddCommGroup Fd:ℕf:Space d → Fg:Space d → Fhg:IsDistBounded ghae1:AEStronglyMeasurable (fun x => f x) volumen1:ℕc1:Fin n1 → ℝg1:Fin n1 → Space dp1:Fin n1 → ℤc1_nonneg:∀ (i : Fin n1), 0 ≤ c1 ip1_bound:∀ (i : Fin n1), -↑(d - 1) ≤ p1 ibound1:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c1 i * ‖x + g1 i‖ ^ p1 i⊢ IsDistBounded (f + g)
  rcases hg with ⟨hae2, ⟨n2, c2, g2, p2, c2_nonneg, p2_bound, bound2⟩⟩F:Typeinst✝:NormedAddCommGroup Fd:ℕf:Space d → Fg:Space d → Fhae1:AEStronglyMeasurable (fun x => f x) volumen1:ℕc1:Fin n1 → ℝg1:Fin n1 → Space dp1:Fin n1 → ℤc1_nonneg:∀ (i : Fin n1), 0 ≤ c1 ip1_bound:∀ (i : Fin n1), -↑(d - 1) ≤ p1 ibound1:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c1 i * ‖x + g1 i‖ ^ p1 ihae2:AEStronglyMeasurable (fun x => g x) volumen2:ℕc2:Fin n2 → ℝg2:Fin n2 → Space dp2:Fin n2 → ℤc2_nonneg:∀ (i : Fin n2), 0 ≤ c2 ip2_bound:∀ (i : Fin n2), -↑(d - 1) ≤ p2 ibound2:∀ (x : Space d), ‖g x‖ ≤ ∑ i, c2 i * ‖x + g2 i‖ ^ p2 i⊢ IsDistBounded (f + g)
  refine ⟨byF:Typeinst✝:NormedAddCommGroup Fd:ℕf:Space d → Fg:Space d → Fhae1:AEStronglyMeasurable (fun x => f x) volumen1:ℕc1:Fin n1 → ℝg1:Fin n1 → Space dp1:Fin n1 → ℤc1_nonneg:∀ (i : Fin n1), 0 ≤ c1 ip1_bound:∀ (i : Fin n1), -↑(d - 1) ≤ p1 ibound1:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c1 i * ‖x + g1 i‖ ^ p1 ihae2:AEStronglyMeasurable (fun x => g x) volumen2:ℕc2:Fin n2 → ℝg2:Fin n2 → Space dp2:Fin n2 → ℤc2_nonneg:∀ (i : Fin n2), 0 ≤ c2 ip2_bound:∀ (i : Fin n2), -↑(d - 1) ≤ p2 ibound2:∀ (x : Space d), ‖g x‖ ≤ ∑ i, c2 i * ‖x + g2 i‖ ^ p2 i⊢ AEStronglyMeasurable (fun x => (f + g) x) volume fun_propAll goals completed! 🐙, n1 + n2, Fin.append c1 c2, Fin.append g1 g2, Fin.append p1 p2, ?_, ?_, ?_⟩
  ·refine_1F:Typeinst✝:NormedAddCommGroup Fd:ℕf:Space d → Fg:Space d → Fhae1:AEStronglyMeasurable (fun x => f x) volumen1:ℕc1:Fin n1 → ℝg1:Fin n1 → Space dp1:Fin n1 → ℤc1_nonneg:∀ (i : Fin n1), 0 ≤ c1 ip1_bound:∀ (i : Fin n1), -↑(d - 1) ≤ p1 ibound1:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c1 i * ‖x + g1 i‖ ^ p1 ihae2:AEStronglyMeasurable (fun x => g x) volumen2:ℕc2:Fin n2 → ℝg2:Fin n2 → Space dp2:Fin n2 → ℤc2_nonneg:∀ (i : Fin n2), 0 ≤ c2 ip2_bound:∀ (i : Fin n2), -↑(d - 1) ≤ p2 ibound2:∀ (x : Space d), ‖g x‖ ≤ ∑ i, c2 i * ‖x + g2 i‖ ^ p2 i⊢ ∀ (i : Fin (n1 + n2)), 0 ≤ Fin.append c1 c2 i intro irefine_1F:Typeinst✝:NormedAddCommGroup Fd:ℕf:Space d → Fg:Space d → Fhae1:AEStronglyMeasurable (fun x => f x) volumen1:ℕc1:Fin n1 → ℝg1:Fin n1 → Space dp1:Fin n1 → ℤc1_nonneg:∀ (i : Fin n1), 0 ≤ c1 ip1_bound:∀ (i : Fin n1), -↑(d - 1) ≤ p1 ibound1:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c1 i * ‖x + g1 i‖ ^ p1 ihae2:AEStronglyMeasurable (fun x => g x) volumen2:ℕc2:Fin n2 → ℝg2:Fin n2 → Space dp2:Fin n2 → ℤc2_nonneg:∀ (i : Fin n2), 0 ≤ c2 ip2_bound:∀ (i : Fin n2), -↑(d - 1) ≤ p2 ibound2:∀ (x : Space d), ‖g x‖ ≤ ∑ i, c2 i * ‖x + g2 i‖ ^ p2 ii:Fin (n1 + n2)⊢ 0 ≤ Fin.append c1 c2 i
    induction i using Fin.addCases with
    | left i =>refine_1.leftF:Typeinst✝:NormedAddCommGroup Fd:ℕf:Space d → Fg:Space d → Fhae1:AEStronglyMeasurable (fun x => f x) volumen1:ℕc1:Fin n1 → ℝg1:Fin n1 → Space dp1:Fin n1 → ℤc1_nonneg:∀ (i : Fin n1), 0 ≤ c1 ip1_bound:∀ (i : Fin n1), -↑(d - 1) ≤ p1 ibound1:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c1 i * ‖x + g1 i‖ ^ p1 ihae2:AEStronglyMeasurable (fun x => g x) volumen2:ℕc2:Fin n2 → ℝg2:Fin n2 → Space dp2:Fin n2 → ℤc2_nonneg:∀ (i : Fin n2), 0 ≤ c2 ip2_bound:∀ (i : Fin n2), -↑(d - 1) ≤ p2 ibound2:∀ (x : Space d), ‖g x‖ ≤ ∑ i, c2 i * ‖x + g2 i‖ ^ p2 ii:Fin n1⊢ 0 ≤ Fin.append c1 c2 (Fin.castAdd n2 i) simpa using c1_nonneg iAll goals completed! 🐙
    | right i =>refine_1.rightF:Typeinst✝:NormedAddCommGroup Fd:ℕf:Space d → Fg:Space d → Fhae1:AEStronglyMeasurable (fun x => f x) volumen1:ℕc1:Fin n1 → ℝg1:Fin n1 → Space dp1:Fin n1 → ℤc1_nonneg:∀ (i : Fin n1), 0 ≤ c1 ip1_bound:∀ (i : Fin n1), -↑(d - 1) ≤ p1 ibound1:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c1 i * ‖x + g1 i‖ ^ p1 ihae2:AEStronglyMeasurable (fun x => g x) volumen2:ℕc2:Fin n2 → ℝg2:Fin n2 → Space dp2:Fin n2 → ℤc2_nonneg:∀ (i : Fin n2), 0 ≤ c2 ip2_bound:∀ (i : Fin n2), -↑(d - 1) ≤ p2 ibound2:∀ (x : Space d), ‖g x‖ ≤ ∑ i, c2 i * ‖x + g2 i‖ ^ p2 ii:Fin n2⊢ 0 ≤ Fin.append c1 c2 (Fin.natAdd n1 i) simpa using c2_nonneg iAll goals completed! 🐙
  ·refine_2F:Typeinst✝:NormedAddCommGroup Fd:ℕf:Space d → Fg:Space d → Fhae1:AEStronglyMeasurable (fun x => f x) volumen1:ℕc1:Fin n1 → ℝg1:Fin n1 → Space dp1:Fin n1 → ℤc1_nonneg:∀ (i : Fin n1), 0 ≤ c1 ip1_bound:∀ (i : Fin n1), -↑(d - 1) ≤ p1 ibound1:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c1 i * ‖x + g1 i‖ ^ p1 ihae2:AEStronglyMeasurable (fun x => g x) volumen2:ℕc2:Fin n2 → ℝg2:Fin n2 → Space dp2:Fin n2 → ℤc2_nonneg:∀ (i : Fin n2), 0 ≤ c2 ip2_bound:∀ (i : Fin n2), -↑(d - 1) ≤ p2 ibound2:∀ (x : Space d), ‖g x‖ ≤ ∑ i, c2 i * ‖x + g2 i‖ ^ p2 i⊢ ∀ (i : Fin (n1 + n2)), -↑(d - 1) ≤ Fin.append p1 p2 i intro irefine_2F:Typeinst✝:NormedAddCommGroup Fd:ℕf:Space d → Fg:Space d → Fhae1:AEStronglyMeasurable (fun x => f x) volumen1:ℕc1:Fin n1 → ℝg1:Fin n1 → Space dp1:Fin n1 → ℤc1_nonneg:∀ (i : Fin n1), 0 ≤ c1 ip1_bound:∀ (i : Fin n1), -↑(d - 1) ≤ p1 ibound1:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c1 i * ‖x + g1 i‖ ^ p1 ihae2:AEStronglyMeasurable (fun x => g x) volumen2:ℕc2:Fin n2 → ℝg2:Fin n2 → Space dp2:Fin n2 → ℤc2_nonneg:∀ (i : Fin n2), 0 ≤ c2 ip2_bound:∀ (i : Fin n2), -↑(d - 1) ≤ p2 ibound2:∀ (x : Space d), ‖g x‖ ≤ ∑ i, c2 i * ‖x + g2 i‖ ^ p2 ii:Fin (n1 + n2)⊢ -↑(d - 1) ≤ Fin.append p1 p2 i
    induction i using Fin.addCases with
    | left i =>refine_2.leftF:Typeinst✝:NormedAddCommGroup Fd:ℕf:Space d → Fg:Space d → Fhae1:AEStronglyMeasurable (fun x => f x) volumen1:ℕc1:Fin n1 → ℝg1:Fin n1 → Space dp1:Fin n1 → ℤc1_nonneg:∀ (i : Fin n1), 0 ≤ c1 ip1_bound:∀ (i : Fin n1), -↑(d - 1) ≤ p1 ibound1:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c1 i * ‖x + g1 i‖ ^ p1 ihae2:AEStronglyMeasurable (fun x => g x) volumen2:ℕc2:Fin n2 → ℝg2:Fin n2 → Space dp2:Fin n2 → ℤc2_nonneg:∀ (i : Fin n2), 0 ≤ c2 ip2_bound:∀ (i : Fin n2), -↑(d - 1) ≤ p2 ibound2:∀ (x : Space d), ‖g x‖ ≤ ∑ i, c2 i * ‖x + g2 i‖ ^ p2 ii:Fin n1⊢ -↑(d - 1) ≤ Fin.append p1 p2 (Fin.castAdd n2 i) simpa using p1_bound iAll goals completed! 🐙
    | right i =>refine_2.rightF:Typeinst✝:NormedAddCommGroup Fd:ℕf:Space d → Fg:Space d → Fhae1:AEStronglyMeasurable (fun x => f x) volumen1:ℕc1:Fin n1 → ℝg1:Fin n1 → Space dp1:Fin n1 → ℤc1_nonneg:∀ (i : Fin n1), 0 ≤ c1 ip1_bound:∀ (i : Fin n1), -↑(d - 1) ≤ p1 ibound1:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c1 i * ‖x + g1 i‖ ^ p1 ihae2:AEStronglyMeasurable (fun x => g x) volumen2:ℕc2:Fin n2 → ℝg2:Fin n2 → Space dp2:Fin n2 → ℤc2_nonneg:∀ (i : Fin n2), 0 ≤ c2 ip2_bound:∀ (i : Fin n2), -↑(d - 1) ≤ p2 ibound2:∀ (x : Space d), ‖g x‖ ≤ ∑ i, c2 i * ‖x + g2 i‖ ^ p2 ii:Fin n2⊢ -↑(d - 1) ≤ Fin.append p1 p2 (Fin.natAdd n1 i) simpa using p2_bound iAll goals completed! 🐙
  ·refine_3F:Typeinst✝:NormedAddCommGroup Fd:ℕf:Space d → Fg:Space d → Fhae1:AEStronglyMeasurable (fun x => f x) volumen1:ℕc1:Fin n1 → ℝg1:Fin n1 → Space dp1:Fin n1 → ℤc1_nonneg:∀ (i : Fin n1), 0 ≤ c1 ip1_bound:∀ (i : Fin n1), -↑(d - 1) ≤ p1 ibound1:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c1 i * ‖x + g1 i‖ ^ p1 ihae2:AEStronglyMeasurable (fun x => g x) volumen2:ℕc2:Fin n2 → ℝg2:Fin n2 → Space dp2:Fin n2 → ℤc2_nonneg:∀ (i : Fin n2), 0 ≤ c2 ip2_bound:∀ (i : Fin n2), -↑(d - 1) ≤ p2 ibound2:∀ (x : Space d), ‖g x‖ ≤ ∑ i, c2 i * ‖x + g2 i‖ ^ p2 i⊢ ∀ (x : Space d), ‖(f + g) x‖ ≤ ∑ i, Fin.append c1 c2 i * ‖x + Fin.append g1 g2 i‖ ^ Fin.append p1 p2 i intro xrefine_3F:Typeinst✝:NormedAddCommGroup Fd:ℕf:Space d → Fg:Space d → Fhae1:AEStronglyMeasurable (fun x => f x) volumen1:ℕc1:Fin n1 → ℝg1:Fin n1 → Space dp1:Fin n1 → ℤc1_nonneg:∀ (i : Fin n1), 0 ≤ c1 ip1_bound:∀ (i : Fin n1), -↑(d - 1) ≤ p1 ibound1:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c1 i * ‖x + g1 i‖ ^ p1 ihae2:AEStronglyMeasurable (fun x => g x) volumen2:ℕc2:Fin n2 → ℝg2:Fin n2 → Space dp2:Fin n2 → ℤc2_nonneg:∀ (i : Fin n2), 0 ≤ c2 ip2_bound:∀ (i : Fin n2), -↑(d - 1) ≤ p2 ibound2:∀ (x : Space d), ‖g x‖ ≤ ∑ i, c2 i * ‖x + g2 i‖ ^ p2 ix:Space d⊢ ‖(f + g) x‖ ≤ ∑ i, Fin.append c1 c2 i * ‖x + Fin.append g1 g2 i‖ ^ Fin.append p1 p2 i
    refine ((norm_add_le _ _).trans (add_le_add (bound1 x) (bound2 x))).trans_eq ?_refine_3F:Typeinst✝:NormedAddCommGroup Fd:ℕf:Space d → Fg:Space d → Fhae1:AEStronglyMeasurable (fun x => f x) volumen1:ℕc1:Fin n1 → ℝg1:Fin n1 → Space dp1:Fin n1 → ℤc1_nonneg:∀ (i : Fin n1), 0 ≤ c1 ip1_bound:∀ (i : Fin n1), -↑(d - 1) ≤ p1 ibound1:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c1 i * ‖x + g1 i‖ ^ p1 ihae2:AEStronglyMeasurable (fun x => g x) volumen2:ℕc2:Fin n2 → ℝg2:Fin n2 → Space dp2:Fin n2 → ℤc2_nonneg:∀ (i : Fin n2), 0 ≤ c2 ip2_bound:∀ (i : Fin n2), -↑(d - 1) ≤ p2 ibound2:∀ (x : Space d), ‖g x‖ ≤ ∑ i, c2 i * ‖x + g2 i‖ ^ p2 ix:Space d⊢ ∑ i, c1 i * ‖x + g1 i‖ ^ p1 i + ∑ i, c2 i * ‖x + g2 i‖ ^ p2 i =
  ∑ i, Fin.append c1 c2 i * ‖x + Fin.append g1 g2 i‖ ^ Fin.append p1 p2 i
    simp [Fin.sum_univ_add]All goals completed! 🐙

@[fun_prop]
lemma fun_add {d : ℕ} {f g : Space d → F}
    (hf : IsDistBounded f) (hg : IsDistBounded g) : IsDistBounded (fun x => f x + g x) := byF:Typeinst✝:NormedAddCommGroup Fd:ℕf:Space d → Fg:Space d → Fhf:IsDistBounded fhg:IsDistBounded g⊢ IsDistBounded fun x => f x + g x
  exact hf.add hgAll goals completed! 🐙

D.2. Finite sums



lemma sum {ι : Type*} {s : Finset ι} {d : ℕ} {f : ι → Space d → F}
    (hf : ∀ i ∈ s, IsDistBounded (f i)) : IsDistBounded (∑ i ∈ s, f i) := byF:Typeinst✝:NormedAddCommGroup Fι:Type u_1s:Finset ιd:ℕf:ι → Space d → Fhf:∀ i ∈ s, IsDistBounded (f i)⊢ IsDistBounded (∑ i ∈ s, f i)
  classical
  induction s using Finset.induction with
  | empty =>emptyF:Typeinst✝:NormedAddCommGroup Fι:Type u_1d:ℕf:ι → Space d → Fhf:∀ i ∈ ∅, IsDistBounded (f i)⊢ IsDistBounded (∑ i ∈ ∅, f i) simpa using zeroAll goals completed! 🐙
  | insert i s hi ih =>insertF:Typeinst✝:NormedAddCommGroup Fι:Type u_1d:ℕf:ι → Space d → Fi:ιs:Finset ιhi:i ∉ sih:(∀ i ∈ s, IsDistBounded (f i)) → IsDistBounded (∑ i ∈ s, f i)hf:∀ i_1 ∈ insert i s, IsDistBounded (f i_1)⊢ IsDistBounded (∑ i ∈ insert i s, f i)
    rw [Finset.sum_insert hiinsertF:Typeinst✝:NormedAddCommGroup Fι:Type u_1d:ℕf:ι → Space d → Fi:ιs:Finset ιhi:i ∉ sih:(∀ i ∈ s, IsDistBounded (f i)) → IsDistBounded (∑ i ∈ s, f i)hf:∀ i_1 ∈ insert i s, IsDistBounded (f i_1)⊢ IsDistBounded (f i + ∑ x ∈ s, f x)insertF:Typeinst✝:NormedAddCommGroup Fι:Type u_1d:ℕf:ι → Space d → Fi:ιs:Finset ιhi:i ∉ sih:(∀ i ∈ s, IsDistBounded (f i)) → IsDistBounded (∑ i ∈ s, f i)hf:∀ i_1 ∈ insert i s, IsDistBounded (f i_1)⊢ IsDistBounded (f i + ∑ x ∈ s, f x)]insertF:Typeinst✝:NormedAddCommGroup Fι:Type u_1d:ℕf:ι → Space d → Fi:ιs:Finset ιhi:i ∉ sih:(∀ i ∈ s, IsDistBounded (f i)) → IsDistBounded (∑ i ∈ s, f i)hf:∀ i_1 ∈ insert i s, IsDistBounded (f i_1)⊢ IsDistBounded (f i + ∑ x ∈ s, f x)
    exact (hf i (s.mem_insert_self i)).add (ih fun j hj => hf j (s.mem_insert_of_mem hj))All goals completed! 🐙

lemma sum_fun {ι : Type*} {s : Finset ι} {d : ℕ} {f : ι → Space d → F}
    (hf : ∀ i ∈ s, IsDistBounded (f i)) : IsDistBounded (fun x => ∑ i ∈ s, f i x) :=
  Finset.sum_fn s f ▸ sum hf

D.3. Scalar multiplication

              @[fun_prop]
lemma const_smul {d : ℕ} [NormedSpace ℝ F] {f : Space d → F}
    (hf : IsDistBounded f) (c : ℝ) : IsDistBounded (c • f) := byF:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhf:IsDistBounded fc:ℝ⊢ IsDistBounded (c • f)
  rcases hf with ⟨hae1, ⟨n1, c1, g1, p1, c1_nonneg, p1_bound, bound1⟩⟩F:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fc:ℝhae1:AEStronglyMeasurable (fun x => f x) volumen1:ℕc1:Fin n1 → ℝg1:Fin n1 → Space dp1:Fin n1 → ℤc1_nonneg:∀ (i : Fin n1), 0 ≤ c1 ip1_bound:∀ (i : Fin n1), -↑(d - 1) ≤ p1 ibound1:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c1 i * ‖x + g1 i‖ ^ p1 i⊢ IsDistBounded (c • f)
  refine ⟨byF:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fc:ℝhae1:AEStronglyMeasurable (fun x => f x) volumen1:ℕc1:Fin n1 → ℝg1:Fin n1 → Space dp1:Fin n1 → ℤc1_nonneg:∀ (i : Fin n1), 0 ≤ c1 ip1_bound:∀ (i : Fin n1), -↑(d - 1) ≤ p1 ibound1:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c1 i * ‖x + g1 i‖ ^ p1 i⊢ AEStronglyMeasurable (fun x => (c • f) x) volume fun_propAll goals completed! 🐙, n1, ‖c‖ • c1, g1, p1,
    fun i => mul_nonneg (norm_nonneg c) (c1_nonneg i), p1_bound, fun x => ?_⟩
  simp only [Pi.smul_apply, norm_smul, smul_eq_mul, mul_assoc, ← Finset.mul_sum]F:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fc:ℝhae1:AEStronglyMeasurable (fun x => f x) volumen1:ℕc1:Fin n1 → ℝg1:Fin n1 → Space dp1:Fin n1 → ℤc1_nonneg:∀ (i : Fin n1), 0 ≤ c1 ip1_bound:∀ (i : Fin n1), -↑(d - 1) ≤ p1 ibound1:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c1 i * ‖x + g1 i‖ ^ p1 ix:Space d⊢ ‖c‖ * ‖f x‖ ≤ ‖c‖ * ∑ i, c1 i * ‖x + g1 i‖ ^ p1 i
  exact mul_le_mul_of_nonneg_left (bound1 x) (norm_nonneg c)All goals completed! 🐙@[fun_prop]
lemma neg {d : ℕ} [NormedSpace ℝ F] {f : Space d → F}
    (hf : IsDistBounded f) : IsDistBounded (fun x => - f x) := byF:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhf:IsDistBounded f⊢ IsDistBounded fun x => -f x
  simpa [Pi.neg_def] using hf.const_smul (-1)All goals completed! 🐙@[fun_prop]
lemma const_fun_smul {d : ℕ} [NormedSpace ℝ F] {f : Space d → F}
    (hf : IsDistBounded f) (c : ℝ) : IsDistBounded (fun x => c • f x) := hf.const_smul c@[fun_prop]
lemma const_mul_fun {d : ℕ}
    {f : Space d → ℝ}
    (hf : IsDistBounded f) (c : ℝ) : IsDistBounded (fun x => c * f x) := hf.const_smul c@[fun_prop]
lemma mul_const_fun {d : ℕ}
    {f : Space d → ℝ}
    (hf : IsDistBounded f) (c : ℝ) : IsDistBounded (fun x => f x * c) := byd:ℕf:Space d → ℝhf:IsDistBounded fc:ℝ⊢ IsDistBounded fun x => f x * c
  simpa [Pi.smul_def, mul_comm] using hf.const_smul cAll goals completed! 🐙

D.4. Components of functions

@[fun_prop]
lemma pi_comp {d n : ℕ}
    {f : Space d → EuclideanSpace ℝ (Fin n)}
    (hf : IsDistBounded f) (j : Fin n) : IsDistBounded (fun x => f x j) := byd:ℕn:ℕf:Space d → EuclideanSpace ℝ (Fin n)hf:IsDistBounded fj:Fin n⊢ IsDistBounded fun x => (f x).ofLp j
  rcases hf with ⟨hae1, ⟨n1, c1, g1, p1, c1_nonneg, p1_bound, bound1⟩⟩d:ℕn:ℕf:Space d → EuclideanSpace ℝ (Fin n)j:Fin nhae1:AEStronglyMeasurable (fun x => f x) volumen1:ℕc1:Fin n1 → ℝg1:Fin n1 → Space dp1:Fin n1 → ℤc1_nonneg:∀ (i : Fin n1), 0 ≤ c1 ip1_bound:∀ (i : Fin n1), -↑(d - 1) ≤ p1 ibound1:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c1 i * ‖x + g1 i‖ ^ p1 i⊢ IsDistBounded fun x => (f x).ofLp j
  exact ⟨byd:ℕn:ℕf:Space d → EuclideanSpace ℝ (Fin n)j:Fin nhae1:AEStronglyMeasurable (fun x => f x) volumen1:ℕc1:Fin n1 → ℝg1:Fin n1 → Space dp1:Fin n1 → ℤc1_nonneg:∀ (i : Fin n1), 0 ≤ c1 ip1_bound:∀ (i : Fin n1), -↑(d - 1) ≤ p1 ibound1:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c1 i * ‖x + g1 i‖ ^ p1 i⊢ AEStronglyMeasurable (fun x => (fun x => (f x).ofLp j) x) volume fun_propAll goals completed! 🐙, n1, c1, g1, p1, c1_nonneg, p1_bound,
    fun x => (PiLp.norm_apply_le (f x) j).trans (bound1 x)⟩

lemma vector_component {d n : ℕ} {f : Space d → Lorentz.Vector n}
    (hf : IsDistBounded f) (j : Fin 1 ⊕ Fin n) : IsDistBounded (fun x => f x j) := byd:ℕn:ℕf:Space d → Lorentz.Vector nhf:IsDistBounded fj:Fin 1 ⊕ Fin n⊢ IsDistBounded fun x => f x j
  rcases hf with ⟨hae1, ⟨n1, c1, g1, p1, c1_nonneg, p1_bound, bound1⟩⟩d:ℕn:ℕf:Space d → Lorentz.Vector nj:Fin 1 ⊕ Fin nhae1:AEStronglyMeasurable (fun x => f x) volumen1:ℕc1:Fin n1 → ℝg1:Fin n1 → Space dp1:Fin n1 → ℤc1_nonneg:∀ (i : Fin n1), 0 ≤ c1 ip1_bound:∀ (i : Fin n1), -↑(d - 1) ≤ p1 ibound1:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c1 i * ‖x + g1 i‖ ^ p1 i⊢ IsDistBounded fun x => f x j
  refine ⟨byd:ℕn:ℕf:Space d → Lorentz.Vector nj:Fin 1 ⊕ Fin nhae1:AEStronglyMeasurable (fun x => f x) volumen1:ℕc1:Fin n1 → ℝg1:Fin n1 → Space dp1:Fin n1 → ℤc1_nonneg:∀ (i : Fin n1), 0 ≤ c1 ip1_bound:∀ (i : Fin n1), -↑(d - 1) ≤ p1 ibound1:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c1 i * ‖x + g1 i‖ ^ p1 i⊢ AEStronglyMeasurable (fun x => (fun x => f x j) x) volume fun_propAll goals completed! 🐙, n1, c1, g1, p1, c1_nonneg, p1_bound, fun x => le_trans ?_ (bound1 x)⟩
  simp [Real.norm_eq_abs]All goals completed! 🐙

D.5. Compositions with additions and subtractions



lemma comp_add_right {d : ℕ} {f : Space d → F}
    (hf : IsDistBounded f) (c : Space d) :
    IsDistBounded (fun x => f (x + c)) := byF:Typeinst✝:NormedAddCommGroup Fd:ℕf:Space d → Fhf:IsDistBounded fc:Space d⊢ IsDistBounded fun x => f (x + c)
  rcases hf with ⟨hae1, ⟨n1, c1, g1, p1, c1_nonneg, p1_bound, bound1⟩⟩F:Typeinst✝:NormedAddCommGroup Fd:ℕf:Space d → Fc:Space dhae1:AEStronglyMeasurable (fun x => f x) volumen1:ℕc1:Fin n1 → ℝg1:Fin n1 → Space dp1:Fin n1 → ℤc1_nonneg:∀ (i : Fin n1), 0 ≤ c1 ip1_bound:∀ (i : Fin n1), -↑(d - 1) ≤ p1 ibound1:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c1 i * ‖x + g1 i‖ ^ p1 i⊢ IsDistBounded fun x => f (x + c)
  refine ⟨hae1.comp_measurePreserving (measurePreserving_add_right volume c),
    n1, c1, fun i => g1 i + c, p1, c1_nonneg, p1_bound, fun x => ?_⟩F:Typeinst✝:NormedAddCommGroup Fd:ℕf:Space d → Fc:Space dhae1:AEStronglyMeasurable (fun x => f x) volumen1:ℕc1:Fin n1 → ℝg1:Fin n1 → Space dp1:Fin n1 → ℤc1_nonneg:∀ (i : Fin n1), 0 ≤ c1 ip1_bound:∀ (i : Fin n1), -↑(d - 1) ≤ p1 ibound1:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c1 i * ‖x + g1 i‖ ^ p1 ix:Space d⊢ ‖(fun x => f (x + c)) x‖ ≤ ∑ i, c1 i * ‖x + (fun i => g1 i + c) i‖ ^ p1 i
  refine (bound1 (x + c)).trans_eq (Finset.sum_congr rfl fun i _ => ?_)F:Typeinst✝:NormedAddCommGroup Fd:ℕf:Space d → Fc:Space dhae1:AEStronglyMeasurable (fun x => f x) volumen1:ℕc1:Fin n1 → ℝg1:Fin n1 → Space dp1:Fin n1 → ℤc1_nonneg:∀ (i : Fin n1), 0 ≤ c1 ip1_bound:∀ (i : Fin n1), -↑(d - 1) ≤ p1 ibound1:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c1 i * ‖x + g1 i‖ ^ p1 ix:Space di:Fin n1x✝:i ∈ Finset.univ⊢ c1 i * ‖x + c + g1 i‖ ^ p1 i = c1 i * ‖x + (fun i => g1 i + c) i‖ ^ p1 i
  rw [add_right_comm,F:Typeinst✝:NormedAddCommGroup Fd:ℕf:Space d → Fc:Space dhae1:AEStronglyMeasurable (fun x => f x) volumen1:ℕc1:Fin n1 → ℝg1:Fin n1 → Space dp1:Fin n1 → ℤc1_nonneg:∀ (i : Fin n1), 0 ≤ c1 ip1_bound:∀ (i : Fin n1), -↑(d - 1) ≤ p1 ibound1:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c1 i * ‖x + g1 i‖ ^ p1 ix:Space di:Fin n1x✝:i ∈ Finset.univ⊢ c1 i * ‖x + g1 i + c‖ ^ p1 i = c1 i * ‖x + (fun i => g1 i + c) i‖ ^ p1 iAll goals completed! 🐙 add_assocF:Typeinst✝:NormedAddCommGroup Fd:ℕf:Space d → Fc:Space dhae1:AEStronglyMeasurable (fun x => f x) volumen1:ℕc1:Fin n1 → ℝg1:Fin n1 → Space dp1:Fin n1 → ℤc1_nonneg:∀ (i : Fin n1), 0 ≤ c1 ip1_bound:∀ (i : Fin n1), -↑(d - 1) ≤ p1 ibound1:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c1 i * ‖x + g1 i‖ ^ p1 ix:Space di:Fin n1x✝:i ∈ Finset.univ⊢ c1 i * ‖x + (g1 i + c)‖ ^ p1 i = c1 i * ‖x + (fun i => g1 i + c) i‖ ^ p1 iAll goals completed! 🐙]All goals completed! 🐙

lemma comp_sub_right {d : ℕ} {f : Space d → F}
    (hf : IsDistBounded f) (c : Space d) :
    IsDistBounded (fun x => f (x - c)) := hf.comp_add_right (- c)

D.6. Congruence with respect to the norm

omit [NormedSpace ℝ F'] in
lemma congr {d : ℕ} {f : Space d → F}
    {g : Space d → F'}
    (hf : IsDistBounded f) (hae : AEStronglyMeasurable g) (hfg : ∀ x, ‖g x‖ = ‖f x‖) :
      IsDistBounded g := byF:TypeF':Typeinst✝¹:NormedAddCommGroup Finst✝:NormedAddCommGroup F'd:ℕf:Space d → Fg:Space d → F'hf:IsDistBounded fhae:AEStronglyMeasurable g volumehfg:∀ (x : Space d), ‖g x‖ = ‖f x‖⊢ IsDistBounded g
  rcases hf with ⟨hae1, ⟨n1, c1, g1, p1, c1_nonneg, p1_bound, bound1⟩⟩F:TypeF':Typeinst✝¹:NormedAddCommGroup Finst✝:NormedAddCommGroup F'd:ℕf:Space d → Fg:Space d → F'hae:AEStronglyMeasurable g volumehfg:∀ (x : Space d), ‖g x‖ = ‖f x‖hae1:AEStronglyMeasurable (fun x => f x) volumen1:ℕc1:Fin n1 → ℝg1:Fin n1 → Space dp1:Fin n1 → ℤc1_nonneg:∀ (i : Fin n1), 0 ≤ c1 ip1_bound:∀ (i : Fin n1), -↑(d - 1) ≤ p1 ibound1:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c1 i * ‖x + g1 i‖ ^ p1 i⊢ IsDistBounded g
  exact ⟨hae, n1, c1, g1, p1, c1_nonneg, p1_bound, fun x => (hfg x).le.trans (bound1 x)⟩All goals completed! 🐙

D.7. Monotonicity with respect to the norm

omit [NormedSpace ℝ F'] in
lemma mono {d : ℕ} {f : Space d → F}
    {g : Space d → F'}
    (hf : IsDistBounded f) (hae : AEStronglyMeasurable g)
    (hfg : ∀ x, ‖g x‖ ≤ ‖f x‖) : IsDistBounded g := byF:TypeF':Typeinst✝¹:NormedAddCommGroup Finst✝:NormedAddCommGroup F'd:ℕf:Space d → Fg:Space d → F'hf:IsDistBounded fhae:AEStronglyMeasurable g volumehfg:∀ (x : Space d), ‖g x‖ ≤ ‖f x‖⊢ IsDistBounded g
  rcases hf with ⟨hae1, ⟨n1, c1, g1, p1, c1_nonneg, p1_bound, bound1⟩⟩F:TypeF':Typeinst✝¹:NormedAddCommGroup Finst✝:NormedAddCommGroup F'd:ℕf:Space d → Fg:Space d → F'hae:AEStronglyMeasurable g volumehfg:∀ (x : Space d), ‖g x‖ ≤ ‖f x‖hae1:AEStronglyMeasurable (fun x => f x) volumen1:ℕc1:Fin n1 → ℝg1:Fin n1 → Space dp1:Fin n1 → ℤc1_nonneg:∀ (i : Fin n1), 0 ≤ c1 ip1_bound:∀ (i : Fin n1), -↑(d - 1) ≤ p1 ibound1:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c1 i * ‖x + g1 i‖ ^ p1 i⊢ IsDistBounded g
  exact ⟨hae, n1, c1, g1, p1, c1_nonneg, p1_bound, fun x => (hfg x).trans (bound1 x)⟩All goals completed! 🐙

D.8. Inner products

@[fun_prop]
lemma inner_left {d n : ℕ}
    {f : Space d → EuclideanSpace ℝ (Fin n) }
    (hf : IsDistBounded f) (y : EuclideanSpace ℝ (Fin n)) :
    IsDistBounded (fun x => ⟪f x, y⟫_ℝ) := byd:ℕn:ℕf:Space d → EuclideanSpace ℝ (Fin n)hf:IsDistBounded fy:EuclideanSpace ℝ (Fin n)⊢ IsDistBounded fun x => ⟪f x, y⟫_ℝ
  rcases hf with ⟨hae1, ⟨n1, c1, g1, p1, c1_nonneg, p1_bound, bound1⟩⟩d:ℕn:ℕf:Space d → EuclideanSpace ℝ (Fin n)y:EuclideanSpace ℝ (Fin n)hae1:AEStronglyMeasurable (fun x => f x) volumen1:ℕc1:Fin n1 → ℝg1:Fin n1 → Space dp1:Fin n1 → ℤc1_nonneg:∀ (i : Fin n1), 0 ≤ c1 ip1_bound:∀ (i : Fin n1), -↑(d - 1) ≤ p1 ibound1:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c1 i * ‖x + g1 i‖ ^ p1 i⊢ IsDistBounded fun x => ⟪f x, y⟫_ℝ
  refine ⟨byd:ℕn:ℕf:Space d → EuclideanSpace ℝ (Fin n)y:EuclideanSpace ℝ (Fin n)hae1:AEStronglyMeasurable (fun x => f x) volumen1:ℕc1:Fin n1 → ℝg1:Fin n1 → Space dp1:Fin n1 → ℤc1_nonneg:∀ (i : Fin n1), 0 ≤ c1 ip1_bound:∀ (i : Fin n1), -↑(d - 1) ≤ p1 ibound1:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c1 i * ‖x + g1 i‖ ^ p1 i⊢ AEStronglyMeasurable (fun x => (fun x => ⟪f x, y⟫_ℝ) x) volume fun_propAll goals completed! 🐙, n1, fun i => ‖y‖ * c1 i, g1, p1,
    fun i => mul_nonneg (norm_nonneg y) (c1_nonneg i), p1_bound, fun x => ?_⟩
  simp only [mul_assoc, ← Finset.mul_sum]d:ℕn:ℕf:Space d → EuclideanSpace ℝ (Fin n)y:EuclideanSpace ℝ (Fin n)hae1:AEStronglyMeasurable (fun x => f x) volumen1:ℕc1:Fin n1 → ℝg1:Fin n1 → Space dp1:Fin n1 → ℤc1_nonneg:∀ (i : Fin n1), 0 ≤ c1 ip1_bound:∀ (i : Fin n1), -↑(d - 1) ≤ p1 ibound1:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c1 i * ‖x + g1 i‖ ^ p1 ix:Space d⊢ ‖⟪f x, y⟫_ℝ‖ ≤ ‖y‖ * ∑ i, c1 i * ‖x + g1 i‖ ^ p1 i
  exact ((norm_inner_le_norm (f x) y).trans_eq (mul_comm _ _)).trans
    (mul_le_mul_of_nonneg_left (bound1 x) (norm_nonneg y))All goals completed! 🐙

D.9. Scalar multiplication with constant

@[fun_prop]
lemma smul_const {d : ℕ} [NormedSpace ℝ F] {c : Space d → ℝ}
    (hc : IsDistBounded c) (f : F) : IsDistBounded (fun x => c x • f) :=
  (hc.mul_const_fun ‖f‖).congr (byF:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Fc:Space d → ℝhc:IsDistBounded cf:F⊢ AEStronglyMeasurable (fun x => c x • f) volume fun_propAll goals completed! 🐙) fun x => byF:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Fc:Space d → ℝhc:IsDistBounded cf:Fx:Space d⊢ ‖c x • f‖ = ‖c x * ‖f‖‖ simp [norm_smul]All goals completed! 🐙

E. Specific functions that are `IsDistBounded`

E.1. Constant functions

@[fun_prop]
lemma const {d : ℕ} (f : F) :
    IsDistBounded (d := d) (fun _ : Space d => f) :=
  ⟨byF:Typeinst✝:NormedAddCommGroup Fd:ℕf:F⊢ AEStronglyMeasurable (fun x => (fun x => f) x) volume fun_propAll goals completed! 🐙, 1, fun _ => ‖f‖, fun _ => 0, fun _ => 0, byF:Typeinst✝:NormedAddCommGroup Fd:ℕf:F⊢ (∀ (i : Fin 1), 0 ≤ (fun x => ‖f‖) i) ∧
  (∀ (i : Fin 1), -↑(d - 1) ≤ (fun x => 0) i) ∧
    ∀ (x : Space d), ‖(fun x => f) x‖ ≤ ∑ i, (fun x => ‖f‖) i * ‖x + (fun x => 0) i‖ ^ (fun x => 0) i simpAll goals completed! 🐙⟩

E.2. Powers of norms

              @[fun_prop]
lemma pow {d : ℕ} (n : ℤ) (hn : - (d - 1 : ℕ) ≤ n) :
    IsDistBounded (d := d) (fun x => ‖x‖ ^ n) :=
  ⟨AEMeasurable.aestronglyMeasurable (byd:ℕn:ℤhn:-↑(d - 1) ≤ n⊢ AEMeasurable (fun x => (fun x => ‖x‖ ^ n) x) volume fun_propAll goals completed! 🐙), 1, fun _ => 1, fun _ => 0, fun _ => n,
    fun _ => zero_le_one, fun _ => hn, fun x => byd:ℕn:ℤhn:-↑(d - 1) ≤ nx:Space d⊢ ‖(fun x => ‖x‖ ^ n) x‖ ≤ ∑ i, (fun x => 1) i * ‖x + (fun x => 0) i‖ ^ (fun x => n) i simpAll goals completed! 🐙⟩@[fun_prop]
lemma pow_shift {d : ℕ} (n : ℤ)
    (g : Space d) (hn : - (d - 1 : ℕ) ≤ n) :
    IsDistBounded (d := d) (fun x => ‖x - g‖ ^ n) :=
  ⟨AEMeasurable.aestronglyMeasurable (byd:ℕn:ℤg:Space dhn:-↑(d - 1) ≤ n⊢ AEMeasurable (fun x => (fun x => ‖x - g‖ ^ n) x) volume fun_propAll goals completed! 🐙), 1, fun _ => 1, fun _ => - g, fun _ => n,
    fun _ => zero_le_one, fun _ => hn, fun x => byd:ℕn:ℤg:Space dhn:-↑(d - 1) ≤ nx:Space d⊢ ‖(fun x => ‖x - g‖ ^ n) x‖ ≤ ∑ i, (fun x => 1) i * ‖x + (fun x => -g) i‖ ^ (fun x => n) i simp [sub_eq_add_neg]All goals completed! 🐙⟩@[fun_prop]
lemma inv_shift {d : ℕ} (g : Space d) (hd : 2 ≤ d := by omega) :
    IsDistBounded (d := d) (fun x => ‖x - g‖⁻¹) := byd:ℕg:Space dhd:2 ≤ d⊢ IsDistBounded fun x => ‖x - g‖⁻¹
  simpa using IsDistBounded.pow_shift (d := d) (-1) g (byd:ℕg:Space dhd:2 ≤ d⊢ -↑(d - 1) ≤ -1 omegaAll goals completed! 🐙)@[fun_prop]
lemma nat_pow {d : ℕ} (n : ℕ) :
    IsDistBounded (d := d) (fun x => ‖x‖ ^ n) := byd:ℕn:ℕ⊢ IsDistBounded fun x => ‖x‖ ^ n
  exact IsDistBounded.pow (d := d) (n : ℤ) (byd:ℕn:ℕ⊢ -↑(d - 1) ≤ ↑n omegaAll goals completed! 🐙)@[fun_prop]
lemma norm_add_nat_pow {d : ℕ} (n : ℕ) (a : ℝ) :
    IsDistBounded (d := d) (fun x => (‖x‖ + a) ^ n) := byd:ℕn:ℕa:ℝ⊢ IsDistBounded fun x => (‖x‖ + a) ^ n
  simp only [add_pow]d:ℕn:ℕa:ℝ⊢ IsDistBounded fun x => ∑ m ∈ Finset.range (n + 1), ‖x‖ ^ m * a ^ (n - m) * ↑(n.choose m)
  exact sum_fun fun i _ => byd:ℕn:ℕa:ℝi:ℕx✝:i ∈ Finset.range (n + 1)⊢ IsDistBounded fun x => ‖x‖ ^ i * a ^ (n - i) * ↑(n.choose i) fun_propAll goals completed! 🐙@[fun_prop]
lemma norm_add_pos_nat_zpow {d : ℕ} (n : ℤ) (a : ℝ) (ha : 0 < a) :
    IsDistBounded (d := d) (fun x => (‖x‖ + a) ^ n) := byd:ℕn:ℤa:ℝha:0 < a⊢ IsDistBounded fun x => (‖x‖ + a) ^ n
  match n with
  | Int.ofNat n =>d:ℕn✝:ℤa:ℝha:0 < an:ℕ⊢ IsDistBounded fun x => (‖x‖ + a) ^ Int.ofNat n fun_propAll goals completed! 🐙
  | Int.negSucc n =>d:ℕn✝:ℤa:ℝha:0 < an:ℕ⊢ IsDistBounded fun x => (‖x‖ + a) ^ Int.negSucc n
    refine IsDistBounded.mono (f := fun x => (a ^ ((n + 1)))⁻¹) (byd:ℕn✝:ℤa:ℝha:0 < an:ℕ⊢ IsDistBounded fun x => (a ^ (n + 1))⁻¹ fun_propAll goals completed! 🐙)
      (AEMeasurable.aestronglyMeasurable (byd:ℕn✝:ℤa:ℝha:0 < an:ℕ⊢ AEMeasurable (fun x => (‖x‖ + a) ^ Int.negSucc n) volume fun_propAll goals completed! 🐙)) fun x => ?_
    simp only [zpow_negSucc, norm_inv, norm_pow, Real.norm_eq_abs, abs_of_nonneg ha.le,
      abs_of_nonneg (show (0:ℝ) ≤ ‖x‖ + a by positivity)]d:ℕn✝:ℤa:ℝha:0 < an:ℕx:Space d⊢ ((‖x‖ + a) ^ (n + 1))⁻¹ ≤ (a ^ (n + 1))⁻¹
    exact inv_anti₀ (byd:ℕn✝:ℤa:ℝha:0 < an:ℕx:Space d⊢ 0 < a ^ (n + 1) positivityAll goals completed! 🐙) (pow_le_pow_left₀ ha.le (byd:ℕn✝:ℤa:ℝha:0 < an:ℕx:Space d⊢ a ≤ ‖x‖ + a simpAll goals completed! 🐙) _)@[fun_prop]
lemma nat_pow_shift {d : ℕ} (n : ℕ)
    (g : Space d) :
    IsDistBounded (d := d) (fun x => ‖x - g‖ ^ n) :=
  IsDistBounded.pow_shift (d := d) (n : ℤ) g (byd:ℕn:ℕg:Space d⊢ -↑(d - 1) ≤ ↑n omegaAll goals completed! 🐙)@[fun_prop]
lemma norm_sub {d : ℕ} (g : Space d) :
    IsDistBounded (d := d) (fun x => ‖x - g‖) := byd:ℕg:Space d⊢ IsDistBounded fun x => ‖x - g‖
  simpa using IsDistBounded.nat_pow_shift (d := d) 1 gAll goals completed! 🐙@[fun_prop]
lemma norm_add {d : ℕ} (g : Space d) :
    IsDistBounded (d := d) (fun x => ‖x + g‖) := byd:ℕg:Space d⊢ IsDistBounded fun x => ‖x + g‖
  simpa using IsDistBounded.nat_pow_shift (d := d) 1 (- g)All goals completed! 🐙@[fun_prop]
lemma inv {d : ℕ} (hd: 2 ≤ d := by omega):
    IsDistBounded (d := d) (fun x => ‖x‖⁻¹) := byd:ℕhd:2 ≤ d⊢ IsDistBounded fun x => ‖x‖⁻¹
  simpa using IsDistBounded.pow (d := d) (-1) (byd:ℕhd:2 ≤ d⊢ -↑(d - 1) ≤ -1 omegaAll goals completed! 🐙)@[fun_prop]
lemma norm {d : ℕ} : IsDistBounded (d := d) (fun x => ‖x‖) := byd:ℕ⊢ IsDistBounded fun x => ‖x‖
  simpa using IsDistBounded.nat_pow (d := d) 1All goals completed! 🐙

@[fun_prop]
lemma log_norm {d : ℕ} (hd : 2 ≤ d := by omega) :
    IsDistBounded (d := d) (fun x => Real.log ‖x‖) := byd:ℕhd:2 ≤ d⊢ IsDistBounded fun x => Real.log ‖x‖
  refine IsDistBounded.mono (f := fun x => ‖x‖⁻¹ + ‖x‖) (byd:ℕhd:2 ≤ d⊢ IsDistBounded fun x => ‖x‖⁻¹ + ‖x‖ fun_propAll goals completed! 🐙)
    (AEMeasurable.aestronglyMeasurable (byd:ℕhd:2 ≤ d⊢ AEMeasurable (fun x => Real.log ‖x‖) volume fun_propAll goals completed! 🐙)) fun x => ?_
  have h1 := Real.neg_inv_le_log (x := ‖x‖) (norm_nonneg x)d:ℕhd:2 ≤ dx:Space dh1:-‖x‖⁻¹ ≤ Real.log ‖x‖⊢ ‖Real.log ‖x‖‖ ≤ ‖‖x‖⁻¹ + ‖x‖‖
  have h2 := Real.log_le_rpow_div (x := ‖x‖) (norm_nonneg x) one_posd:ℕhd:2 ≤ dx:Space dh1:-‖x‖⁻¹ ≤ Real.log ‖x‖h2:Real.log ‖x‖ ≤ ‖x‖ ^ 1 / 1⊢ ‖Real.log ‖x‖‖ ≤ ‖‖x‖⁻¹ + ‖x‖‖
  simp only [Real.rpow_one, div_one] at h2d:ℕhd:2 ≤ dx:Space dh1:-‖x‖⁻¹ ≤ Real.log ‖x‖h2:Real.log ‖x‖ ≤ ‖x‖⊢ ‖Real.log ‖x‖‖ ≤ ‖‖x‖⁻¹ + ‖x‖‖
  rw [Real.norm_eq_abs,d:ℕhd:2 ≤ dx:Space dh1:-‖x‖⁻¹ ≤ Real.log ‖x‖h2:Real.log ‖x‖ ≤ ‖x‖⊢ |Real.log ‖x‖| ≤ ‖‖x‖⁻¹ + ‖x‖‖d:ℕhd:2 ≤ dx:Space dh1:-‖x‖⁻¹ ≤ Real.log ‖x‖h2:Real.log ‖x‖ ≤ ‖x‖⊢ Real.log ‖x‖ ≤ ‖x‖⁻¹ + ‖x‖ ∧ -Real.log ‖x‖ ≤ ‖x‖⁻¹ + ‖x‖ Real.norm_eq_abs,d:ℕhd:2 ≤ dx:Space dh1:-‖x‖⁻¹ ≤ Real.log ‖x‖h2:Real.log ‖x‖ ≤ ‖x‖⊢ |Real.log ‖x‖| ≤ |‖x‖⁻¹ + ‖x‖|d:ℕhd:2 ≤ dx:Space dh1:-‖x‖⁻¹ ≤ Real.log ‖x‖h2:Real.log ‖x‖ ≤ ‖x‖⊢ Real.log ‖x‖ ≤ ‖x‖⁻¹ + ‖x‖ ∧ -Real.log ‖x‖ ≤ ‖x‖⁻¹ + ‖x‖
    abs_of_nonneg (show (0:ℝ) ≤ ‖x‖⁻¹ + ‖x‖ byd:ℕhd:2 ≤ d⊢ IsDistBounded fun x => Real.log ‖x‖d:ℕhd:2 ≤ dx:Space dh1:-‖x‖⁻¹ ≤ Real.log ‖x‖h2:Real.log ‖x‖ ≤ ‖x‖⊢ Real.log ‖x‖ ≤ ‖x‖⁻¹ + ‖x‖ ∧ -Real.log ‖x‖ ≤ ‖x‖⁻¹ + ‖x‖ positivityAll goals completed! 🐙d:ℕhd:2 ≤ dx:Space dh1:-‖x‖⁻¹ ≤ Real.log ‖x‖h2:Real.log ‖x‖ ≤ ‖x‖⊢ Real.log ‖x‖ ≤ ‖x‖⁻¹ + ‖x‖ ∧ -Real.log ‖x‖ ≤ ‖x‖⁻¹ + ‖x‖), abs_le'd:ℕhd:2 ≤ dx:Space dh1:-‖x‖⁻¹ ≤ Real.log ‖x‖h2:Real.log ‖x‖ ≤ ‖x‖⊢ Real.log ‖x‖ ≤ ‖x‖⁻¹ + ‖x‖ ∧ -Real.log ‖x‖ ≤ ‖x‖⁻¹ + ‖x‖d:ℕhd:2 ≤ dx:Space dh1:-‖x‖⁻¹ ≤ Real.log ‖x‖h2:Real.log ‖x‖ ≤ ‖x‖⊢ Real.log ‖x‖ ≤ ‖x‖⁻¹ + ‖x‖ ∧ -Real.log ‖x‖ ≤ ‖x‖⁻¹ + ‖x‖]d:ℕhd:2 ≤ dx:Space dh1:-‖x‖⁻¹ ≤ Real.log ‖x‖h2:Real.log ‖x‖ ≤ ‖x‖⊢ Real.log ‖x‖ ≤ ‖x‖⁻¹ + ‖x‖ ∧ -Real.log ‖x‖ ≤ ‖x‖⁻¹ + ‖x‖
  constructorleftd:ℕhd:2 ≤ dx:Space dh1:-‖x‖⁻¹ ≤ Real.log ‖x‖h2:Real.log ‖x‖ ≤ ‖x‖⊢ Real.log ‖x‖ ≤ ‖x‖⁻¹ + ‖x‖rightd:ℕhd:2 ≤ dx:Space dh1:-‖x‖⁻¹ ≤ Real.log ‖x‖h2:Real.log ‖x‖ ≤ ‖x‖⊢ -Real.log ‖x‖ ≤ ‖x‖⁻¹ + ‖x‖
  ·leftd:ℕhd:2 ≤ dx:Space dh1:-‖x‖⁻¹ ≤ Real.log ‖x‖h2:Real.log ‖x‖ ≤ ‖x‖⊢ Real.log ‖x‖ ≤ ‖x‖⁻¹ + ‖x‖ exact h2.trans (byd:ℕhd:2 ≤ dx:Space dh1:-‖x‖⁻¹ ≤ Real.log ‖x‖h2:Real.log ‖x‖ ≤ ‖x‖⊢ ‖x‖ ≤ ‖x‖⁻¹ + ‖x‖ simpAll goals completed! 🐙)
  ·rightd:ℕhd:2 ≤ dx:Space dh1:-‖x‖⁻¹ ≤ Real.log ‖x‖h2:Real.log ‖x‖ ≤ ‖x‖⊢ -Real.log ‖x‖ ≤ ‖x‖⁻¹ + ‖x‖ linarith [norm_nonneg x]All goals completed! 🐙lemma zpow_smul_self {d : ℕ} (n : ℤ) (hn : - (d - 1 : ℕ) - 1 ≤ n) :
    IsDistBounded (d := d) (fun x => ‖x‖ ^ n • x) := byd:ℕn:ℤhn:-↑(d - 1) - 1 ≤ n⊢ IsDistBounded fun x => ‖x‖ ^ n • x
  by_cases hzero : n = -1posd:ℕn:ℤhn:-↑(d - 1) - 1 ≤ nhzero:n = -1⊢ IsDistBounded fun x => ‖x‖ ^ n • xnegd:ℕn:ℤhn:-↑(d - 1) - 1 ≤ nhzero:¬n = -1⊢ IsDistBounded fun x => ‖x‖ ^ n • x
  ·posd:ℕn:ℤhn:-↑(d - 1) - 1 ≤ nhzero:n = -1⊢ IsDistBounded fun x => ‖x‖ ^ n • x subst hzeroposd:ℕhn:-↑(d - 1) - 1 ≤ -1⊢ IsDistBounded fun x => ‖x‖ ^ (-1) • x
    refine IsDistBounded.mono (f := fun x => (1 : ℝ)) (byd:ℕhn:-↑(d - 1) - 1 ≤ -1⊢ IsDistBounded fun x => 1 fun_propAll goals completed! 🐙)
      (AEMeasurable.aestronglyMeasurable (byd:ℕhn:-↑(d - 1) - 1 ≤ -1⊢ AEMeasurable (fun x => ‖x‖ ^ (-1) • x) volume fun_propAll goals completed! 🐙)) fun x => ?_
    simpa [norm_smul, inv_mul_eq_div] using div_self_le_one ‖x‖All goals completed! 🐙
  refine IsDistBounded.congr (f := fun x => ‖x‖ ^ (n + 1)) (pow _ (byd:ℕn:ℤhn:-↑(d - 1) - 1 ≤ nhzero:¬n = -1⊢ -↑(d - 1) ≤ n + 1 omegaAll goals completed! 🐙))
    (AEMeasurable.aestronglyMeasurable (byd:ℕn:ℤhn:-↑(d - 1) - 1 ≤ nhzero:¬n = -1⊢ AEMeasurable (fun x => ‖x‖ ^ n • x) volume fun_propAll goals completed! 🐙)) fun x => ?_
  rcases eq_or_ne x 0 with rfl | hxneg.inld:ℕn:ℤhn:-↑(d - 1) - 1 ≤ nhzero:¬n = -1⊢ ‖‖0‖ ^ n • 0‖ = ‖‖0‖ ^ (n + 1)‖neg.inrd:ℕn:ℤhn:-↑(d - 1) - 1 ≤ nhzero:¬n = -1x:Space dhx:x ≠ 0⊢ ‖‖x‖ ^ n • x‖ = ‖‖x‖ ^ (n + 1)‖
  ·neg.inld:ℕn:ℤhn:-↑(d - 1) - 1 ≤ nhzero:¬n = -1⊢ ‖‖0‖ ^ n • 0‖ = ‖‖0‖ ^ (n + 1)‖ simp [zero_zpow (n + 1) (by omega)]All goals completed! 🐙
  ·neg.inrd:ℕn:ℤhn:-↑(d - 1) - 1 ≤ nhzero:¬n = -1x:Space dhx:x ≠ 0⊢ ‖‖x‖ ^ n • x‖ = ‖‖x‖ ^ (n + 1)‖ simp [norm_smul, zpow_add₀ (norm_ne_zero_iff.mpr hx), mul_comm]All goals completed! 🐙lemma zpow_smul_repr_self {d : ℕ} (n : ℤ) (hn : - (d - 1 : ℕ) - 1 ≤ n) :
    IsDistBounded (d := d) (fun x => ‖x‖ ^ n • basis.repr x) :=
  (zpow_smul_self n hn).congr (AEMeasurable.aestronglyMeasurable (byd:ℕn:ℤhn:-↑(d - 1) - 1 ≤ n⊢ AEMeasurable (fun x => ‖x‖ ^ n • basis.repr x) volume fun_propAll goals completed! 🐙))
    fun x => byd:ℕn:ℤhn:-↑(d - 1) - 1 ≤ nx:Space d⊢ ‖‖x‖ ^ n • basis.repr x‖ = ‖‖x‖ ^ n • x‖ simp [norm_smul]All goals completed! 🐙lemma zpow_smul_repr_self_sub {d : ℕ} (n : ℤ) (hn : - (d - 1 : ℕ) - 1 ≤ n)
    (y : Space d) :
    IsDistBounded (d := d) (fun x => ‖x - y‖ ^ n • basis.repr (x - y)) :=
  (zpow_smul_repr_self n hn).comp_sub_right ylemma inv_pow_smul_self {d : ℕ} (n : ℕ) (hn : - (d - 1 : ℕ) - 1 ≤ (- n : ℤ)) :
    IsDistBounded (d := d) (fun x => ‖x‖⁻¹ ^ n • x) := byd:ℕn:ℕhn:-↑(d - 1) - 1 ≤ -↑n⊢ IsDistBounded fun x => ‖x‖⁻¹ ^ n • x
  simpa using zpow_smul_self (n := - (n : ℤ)) (byd:ℕn:ℕhn:-↑(d - 1) - 1 ≤ -↑n⊢ -↑(d - 1) - 1 ≤ -↑n omegaAll goals completed! 🐙)lemma inv_pow_smul_repr_self {d : ℕ} (n : ℕ) (hn : - (d - 1 : ℕ) - 1 ≤ (- n : ℤ)) :
    IsDistBounded (d := d) (fun x => ‖x‖⁻¹ ^ n • basis.repr x) := byd:ℕn:ℕhn:-↑(d - 1) - 1 ≤ -↑n⊢ IsDistBounded fun x => ‖x‖⁻¹ ^ n • basis.repr x
  simpa using zpow_smul_repr_self (n := - (n : ℤ)) (byd:ℕn:ℕhn:-↑(d - 1) - 1 ≤ -↑n⊢ -↑(d - 1) - 1 ≤ -↑n omegaAll goals completed! 🐙)

F. Multiplication by norms and components

              

lemma norm_smul_nat_pow {d} (p : ℕ) (c : Space d) :
    IsDistBounded (fun x => ‖x‖ * ‖x + c‖ ^ p) := byd:ℕp:ℕc:Space d⊢ IsDistBounded fun x => ‖x‖ * ‖x + c‖ ^ p
  refine IsDistBounded.mono (f := fun x => ‖x‖ * (‖x‖ + ‖c‖) ^ p) ?_
    (AEMeasurable.aestronglyMeasurable (byd:ℕp:ℕc:Space d⊢ AEMeasurable (fun x => ‖x‖ * ‖x + c‖ ^ p) volume fun_propAll goals completed! 🐙)) fun x => ?_
  ·refine_1d:ℕp:ℕc:Space d⊢ IsDistBounded fun x => ‖x‖ * (‖x‖ + ‖c‖) ^ p simp only [add_pow, Finset.mul_sum, ← mul_assoc]refine_1d:ℕp:ℕc:Space d⊢ IsDistBounded fun x => ∑ x_1 ∈ Finset.range (p + 1), ‖x‖ * ‖x‖ ^ x_1 * ‖c‖ ^ (p - x_1) * ↑(p.choose x_1)
    refine IsDistBounded.sum_fun fun i _ => mul_const_fun (mul_const_fun ?_ _) _refine_1d:ℕp:ℕc:Space di:ℕx✝:i ∈ Finset.range (p + 1)⊢ IsDistBounded fun x => ‖x‖ * ‖x‖ ^ i
    simpa [pow_succ'] using IsDistBounded.nat_pow (d := d) (i + 1)All goals completed! 🐙
  ·refine_2d:ℕp:ℕc:Space dx:Space d⊢ ‖‖x‖ * ‖x + c‖ ^ p‖ ≤ ‖‖x‖ * (‖x‖ + ‖c‖) ^ p‖ simp [norm_mul, norm_pow, Real.norm_eq_abs]refine_2d:ℕp:ℕc:Space dx:Space d⊢ ‖x‖ * ‖x + c‖ ^ p ≤ ‖x‖ * |‖x‖ + ‖c‖| ^ p
    rw [abs_of_nonneg (byd:ℕp:ℕc:Space dx:Space d⊢ 0 ≤ ‖x‖ + ‖c‖refine_2d:ℕp:ℕc:Space dx:Space d⊢ ‖x‖ * ‖x + c‖ ^ p ≤ ‖x‖ * (‖x‖ + ‖c‖) ^ p positivityAll goals completed! 🐙refine_2d:ℕp:ℕc:Space dx:Space d⊢ ‖x‖ * ‖x + c‖ ^ p ≤ ‖x‖ * (‖x‖ + ‖c‖) ^ p)]refine_2d:ℕp:ℕc:Space dx:Space d⊢ ‖x‖ * ‖x + c‖ ^ p ≤ ‖x‖ * (‖x‖ + ‖c‖) ^ p
    gcongrhabd:ℕp:ℕc:Space dx:Space d⊢ ‖x + c‖ ≤ ‖x‖ + ‖c‖
    exact norm_add_le x cAll goals completed! 🐙

lemma norm_smul_zpow {d} (p : ℤ) (c : Space d) (hn : - (d - 1 : ℕ) ≤ p) :
    IsDistBounded (fun x => ‖x‖ * ‖x + c‖ ^ p) := byd:ℕp:ℤc:Space dhn:-↑(d - 1) ≤ p⊢ IsDistBounded fun x => ‖x‖ * ‖x + c‖ ^ p
  match p with
  | Int.ofNat p =>d:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.ofNat p⊢ IsDistBounded fun x => ‖x‖ * ‖x + c‖ ^ Int.ofNat p exact norm_smul_nat_pow p cAll goals completed! 🐙
  | Int.negSucc p =>d:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc p⊢ IsDistBounded fun x => ‖x‖ * ‖x + c‖ ^ Int.negSucc p
    suffices h0 : IsDistBounded (fun x => ‖x - c‖ * (‖x‖ ^ (p + 1))⁻¹) byd:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc ph0:IsDistBounded fun x => ‖x - c‖ * (‖x‖ ^ (p + 1))⁻¹⊢ IsDistBounded fun x => ‖x‖ * ‖x + c‖ ^ Int.negSucc pd:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc p⊢ IsDistBounded fun x => ‖x - c‖ * (‖x‖ ^ (p + 1))⁻¹
      simpa using h0.comp_sub_right (- c)d:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc p⊢ IsDistBounded fun x => ‖x - c‖ * (‖x‖ ^ (p + 1))⁻¹d:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc p⊢ IsDistBounded fun x => ‖x - c‖ * (‖x‖ ^ (p + 1))⁻¹
    suffices h0 : IsDistBounded (fun x => (‖x‖ + ‖c‖) * (‖x‖ ^ (p + 1))⁻¹) byd:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc ph0:IsDistBounded fun x => (‖x‖ + ‖c‖) * (‖x‖ ^ (p + 1))⁻¹⊢ IsDistBounded fun x => ‖x - c‖ * (‖x‖ ^ (p + 1))⁻¹d:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc p⊢ IsDistBounded fun x => (‖x‖ + ‖c‖) * (‖x‖ ^ (p + 1))⁻¹
      refine h0.mono (byd:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc ph0:IsDistBounded fun x => (‖x‖ + ‖c‖) * (‖x‖ ^ (p + 1))⁻¹⊢ AEStronglyMeasurable (fun x => ‖x - c‖ * (‖x‖ ^ (p + 1))⁻¹) volumed:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc p⊢ IsDistBounded fun x => (‖x‖ + ‖c‖) * (‖x‖ ^ (p + 1))⁻¹ fun_propAll goals completed! 🐙d:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc p⊢ IsDistBounded fun x => (‖x‖ + ‖c‖) * (‖x‖ ^ (p + 1))⁻¹) fun x => ?_
      simp [norm_mul, norm_inv, norm_pow, Real.norm_eq_abs]d:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc ph0:IsDistBounded fun x => (‖x‖ + ‖c‖) * (‖x‖ ^ (p + 1))⁻¹x:Space d⊢ ‖x - c‖ * (‖x‖ ^ (p + 1))⁻¹ ≤ |‖x‖ + ‖c‖| * (‖x‖ ^ (p + 1))⁻¹d:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc p⊢ IsDistBounded fun x => (‖x‖ + ‖c‖) * (‖x‖ ^ (p + 1))⁻¹
      rw [abs_of_nonneg (byd:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc ph0:IsDistBounded fun x => (‖x‖ + ‖c‖) * (‖x‖ ^ (p + 1))⁻¹x:Space d⊢ 0 ≤ ‖x‖ + ‖c‖d:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc ph0:IsDistBounded fun x => (‖x‖ + ‖c‖) * (‖x‖ ^ (p + 1))⁻¹x:Space d⊢ ‖x - c‖ * (‖x‖ ^ (p + 1))⁻¹ ≤ (‖x‖ + ‖c‖) * (‖x‖ ^ (p + 1))⁻¹d:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc p⊢ IsDistBounded fun x => (‖x‖ + ‖c‖) * (‖x‖ ^ (p + 1))⁻¹ positivityAll goals completed! 🐙d:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc ph0:IsDistBounded fun x => (‖x‖ + ‖c‖) * (‖x‖ ^ (p + 1))⁻¹x:Space d⊢ ‖x - c‖ * (‖x‖ ^ (p + 1))⁻¹ ≤ (‖x‖ + ‖c‖) * (‖x‖ ^ (p + 1))⁻¹d:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc p⊢ IsDistBounded fun x => (‖x‖ + ‖c‖) * (‖x‖ ^ (p + 1))⁻¹)]d:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc ph0:IsDistBounded fun x => (‖x‖ + ‖c‖) * (‖x‖ ^ (p + 1))⁻¹x:Space d⊢ ‖x - c‖ * (‖x‖ ^ (p + 1))⁻¹ ≤ (‖x‖ + ‖c‖) * (‖x‖ ^ (p + 1))⁻¹d:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc p⊢ IsDistBounded fun x => (‖x‖ + ‖c‖) * (‖x‖ ^ (p + 1))⁻¹
      gcongrhbcd:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc ph0:IsDistBounded fun x => (‖x‖ + ‖c‖) * (‖x‖ ^ (p + 1))⁻¹x:Space d⊢ ‖x - c‖ ≤ ‖x‖ + ‖c‖d:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc p⊢ IsDistBounded fun x => (‖x‖ + ‖c‖) * (‖x‖ ^ (p + 1))⁻¹
      exact norm_sub_le x cd:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc p⊢ IsDistBounded fun x => (‖x‖ + ‖c‖) * (‖x‖ ^ (p + 1))⁻¹d:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc p⊢ IsDistBounded fun x => (‖x‖ + ‖c‖) * (‖x‖ ^ (p + 1))⁻¹
    suffices h0 : IsDistBounded (fun x => ‖x‖ * (‖x‖ ^ (p + 1))⁻¹ + ‖c‖ * (‖x‖ ^ (p + 1))⁻¹) byd:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc ph0:IsDistBounded fun x => ‖x‖ * (‖x‖ ^ (p + 1))⁻¹ + ‖c‖ * (‖x‖ ^ (p + 1))⁻¹⊢ IsDistBounded fun x => (‖x‖ + ‖c‖) * (‖x‖ ^ (p + 1))⁻¹d:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc p⊢ IsDistBounded fun x => ‖x‖ * (‖x‖ ^ (p + 1))⁻¹ + ‖c‖ * (‖x‖ ^ (p + 1))⁻¹
      simpa [add_mul] using h0d:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc p⊢ IsDistBounded fun x => ‖x‖ * (‖x‖ ^ (p + 1))⁻¹ + ‖c‖ * (‖x‖ ^ (p + 1))⁻¹d:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc p⊢ IsDistBounded fun x => ‖x‖ * (‖x‖ ^ (p + 1))⁻¹ + ‖c‖ * (‖x‖ ^ (p + 1))⁻¹
    suffices h0 : IsDistBounded (fun x => ‖x‖ * (‖x‖ ^ (p + 1))⁻¹) byd:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc ph0:IsDistBounded fun x => ‖x‖ * (‖x‖ ^ (p + 1))⁻¹⊢ IsDistBounded fun x => ‖x‖ * (‖x‖ ^ (p + 1))⁻¹ + ‖c‖ * (‖x‖ ^ (p + 1))⁻¹d:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc p⊢ IsDistBounded fun x => ‖x‖ * (‖x‖ ^ (p + 1))⁻¹
      refine h0.add (const_mul_fun ?_ ‖c‖)d:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc ph0:IsDistBounded fun x => ‖x‖ * (‖x‖ ^ (p + 1))⁻¹⊢ IsDistBounded fun x => (‖x‖ ^ (p + 1))⁻¹d:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc p⊢ IsDistBounded fun x => ‖x‖ * (‖x‖ ^ (p + 1))⁻¹
      exact IsDistBounded.pow (d := d) (n := -(p + 1)) (byd:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc ph0:IsDistBounded fun x => ‖x‖ * (‖x‖ ^ (p + 1))⁻¹⊢ -↑(d - 1) ≤ -(↑p + 1)d:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc p⊢ IsDistBounded fun x => ‖x‖ * (‖x‖ ^ (p + 1))⁻¹ grindAll goals completed! 🐙d:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc p⊢ IsDistBounded fun x => ‖x‖ * (‖x‖ ^ (p + 1))⁻¹)d:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc p⊢ IsDistBounded fun x => ‖x‖ * (‖x‖ ^ (p + 1))⁻¹
    by_cases hp : p = 0posd:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc php:p = 0⊢ IsDistBounded fun x => ‖x‖ * (‖x‖ ^ (p + 1))⁻¹negd:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc php:¬p = 0⊢ IsDistBounded fun x => ‖x‖ * (‖x‖ ^ (p + 1))⁻¹
    ·posd:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc php:p = 0⊢ IsDistBounded fun x => ‖x‖ * (‖x‖ ^ (p + 1))⁻¹ subst hpposd:ℕp:ℤc:Space dhn:-↑(d - 1) ≤ Int.negSucc 0⊢ IsDistBounded fun x => ‖x‖ * (‖x‖ ^ (0 + 1))⁻¹
      simp only [zero_add, pow_one]posd:ℕp:ℤc:Space dhn:-↑(d - 1) ≤ Int.negSucc 0⊢ IsDistBounded fun x => ‖x‖ * ‖x‖⁻¹
      refine IsDistBounded.mono (f := fun x => (1 : ℝ)) (byd:ℕp:ℤc:Space dhn:-↑(d - 1) ≤ Int.negSucc 0⊢ IsDistBounded fun x => 1 fun_propAll goals completed! 🐙)
        (AEMeasurable.aestronglyMeasurable (byd:ℕp:ℤc:Space dhn:-↑(d - 1) ≤ Int.negSucc 0⊢ AEMeasurable (fun x => ‖x‖ * ‖x‖⁻¹) volume fun_propAll goals completed! 🐙)) fun x => ?_
      simpa [← div_eq_mul_inv] using div_self_le_one ‖x‖All goals completed! 🐙
    convert IsDistBounded.pow (d := d) (n := - p) (byd:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc php:¬p = 0⊢ -↑(d - 1) ≤ -↑p grindAll goals completed! 🐙) using 1
    funext xd:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc php:¬p = 0x:Space d⊢ ‖x‖ * (‖x‖ ^ (p + 1))⁻¹ = ‖x‖ ^ (-↑p)
    rw [zpow_neg,d:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc php:¬p = 0x:Space d⊢ ‖x‖ * (‖x‖ ^ (p + 1))⁻¹ = (‖x‖ ^ ↑p)⁻¹d:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc php:¬p = 0x:Space d⊢ ‖x‖ * (‖x‖ ^ (p + 1))⁻¹ = (‖x‖ ^ p)⁻¹ zpow_natCastd:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc php:¬p = 0x:Space d⊢ ‖x‖ * (‖x‖ ^ (p + 1))⁻¹ = (‖x‖ ^ p)⁻¹d:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc php:¬p = 0x:Space d⊢ ‖x‖ * (‖x‖ ^ (p + 1))⁻¹ = (‖x‖ ^ p)⁻¹]d:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc php:¬p = 0x:Space d⊢ ‖x‖ * (‖x‖ ^ (p + 1))⁻¹ = (‖x‖ ^ p)⁻¹
    rcases eq_or_ne ‖x‖ 0 with hx | hxinld:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc php:¬p = 0x:Space dhx:‖x‖ = 0⊢ ‖x‖ * (‖x‖ ^ (p + 1))⁻¹ = (‖x‖ ^ p)⁻¹inrd:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc php:¬p = 0x:Space dhx:‖x‖ ≠ 0⊢ ‖x‖ * (‖x‖ ^ (p + 1))⁻¹ = (‖x‖ ^ p)⁻¹
    ·inld:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc php:¬p = 0x:Space dhx:‖x‖ = 0⊢ ‖x‖ * (‖x‖ ^ (p + 1))⁻¹ = (‖x‖ ^ p)⁻¹ simp [hx, zero_pow hp]All goals completed! 🐙
    ·inrd:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc php:¬p = 0x:Space dhx:‖x‖ ≠ 0⊢ ‖x‖ * (‖x‖ ^ (p + 1))⁻¹ = (‖x‖ ^ p)⁻¹ field_simpinrd:ℕp✝:ℤc:Space dp:ℕhn:-↑(d - 1) ≤ Int.negSucc php:¬p = 0x:Space dhx:‖x‖ ≠ 0⊢ ‖x‖ * ‖x‖ ^ p = ‖x‖ ^ (p + 1)
      ringAll goals completed! 🐙

@[fun_prop]
lemma norm_smul_isDistBounded {d : ℕ} [NormedSpace ℝ F] {f : Space d → F}
    (hf : IsDistBounded f) :
    IsDistBounded (fun x => ‖x‖ • f x) := byF:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhf:IsDistBounded f⊢ IsDistBounded fun x => ‖x‖ • f x
  obtain ⟨hae, ⟨n, c, g, p, c_nonneg, p_bound, bound⟩⟩ := hfF:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhae:AEStronglyMeasurable (fun x => f x) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p i⊢ IsDistBounded fun x => ‖x‖ • f x
  refine IsDistBounded.mono (f := fun x => ‖x‖ * ∑ i, (c i * ‖x + g i‖ ^ (p i)))
    (IsDistBounded.congr (f := fun x => ∑ i, (c i * (‖x‖ * ‖x + g i‖ ^ (p i))))
      (sum_fun fun i _ => const_mul_fun (norm_smul_zpow (p i) (g i) (p_bound i)) (c i))
      (byF:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhae:AEStronglyMeasurable (fun x => f x) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p i⊢ AEStronglyMeasurable (fun x => ‖x‖ * ∑ i, c i * ‖x + g i‖ ^ p i) volume fun_propAll goals completed! 🐙) fun x => ?_) (byF:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhae:AEStronglyMeasurable (fun x => f x) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p i⊢ AEStronglyMeasurable (fun x => ‖x‖ • f x) volume fun_propAll goals completed! 🐙) fun x => ?_
  ·refine_1F:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhae:AEStronglyMeasurable (fun x => f x) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Space d⊢ ‖‖x‖ * ∑ i, c i * ‖x + g i‖ ^ p i‖ = ‖∑ i, c i * (‖x‖ * ‖x + g i‖ ^ p i)‖ rw [Finset.mul_sumrefine_1F:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhae:AEStronglyMeasurable (fun x => f x) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Space d⊢ ‖∑ i, ‖x‖ * (c i * ‖x + g i‖ ^ p i)‖ = ‖∑ i, c i * (‖x‖ * ‖x + g i‖ ^ p i)‖refine_1F:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhae:AEStronglyMeasurable (fun x => f x) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Space d⊢ ‖∑ i, ‖x‖ * (c i * ‖x + g i‖ ^ p i)‖ = ‖∑ i, c i * (‖x‖ * ‖x + g i‖ ^ p i)‖]refine_1F:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhae:AEStronglyMeasurable (fun x => f x) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Space d⊢ ‖∑ i, ‖x‖ * (c i * ‖x + g i‖ ^ p i)‖ = ‖∑ i, c i * (‖x‖ * ‖x + g i‖ ^ p i)‖
    congr 1refine_1F:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhae:AEStronglyMeasurable (fun x => f x) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Space d⊢ ∑ i, ‖x‖ * (c i * ‖x + g i‖ ^ p i) = ∑ i, c i * (‖x‖ * ‖x + g i‖ ^ p i)
    exact Finset.sum_congr rfl fun i _ => mul_left_comm _ _ _All goals completed! 🐙
  ·refine_2F:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhae:AEStronglyMeasurable (fun x => f x) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Space d⊢ ‖‖x‖ • f x‖ ≤ ‖‖x‖ * ∑ i, c i * ‖x + g i‖ ^ p i‖ have h : (0:ℝ) ≤ ∑ i, c i * ‖x + g i‖ ^ p i :=
      Finset.sum_nonneg fun i _ => mul_nonneg (c_nonneg i) (byF:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhae:AEStronglyMeasurable (fun x => f x) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Space di:Fin nx✝:i ∈ Finset.univ⊢ 0 ≤ ‖x + g i‖ ^ p irefine_2F:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhae:AEStronglyMeasurable (fun x => f x) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Space dh:0 ≤ ∑ i, c i * ‖x + g i‖ ^ p i⊢ ‖‖x‖ • f x‖ ≤ ‖‖x‖ * ∑ i, c i * ‖x + g i‖ ^ p i‖ positivityAll goals completed! 🐙refine_2F:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhae:AEStronglyMeasurable (fun x => f x) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Space dh:0 ≤ ∑ i, c i * ‖x + g i‖ ^ p i⊢ ‖‖x‖ • f x‖ ≤ ‖‖x‖ * ∑ i, c i * ‖x + g i‖ ^ p i‖)refine_2F:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhae:AEStronglyMeasurable (fun x => f x) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Space dh:0 ≤ ∑ i, c i * ‖x + g i‖ ^ p i⊢ ‖‖x‖ • f x‖ ≤ ‖‖x‖ * ∑ i, c i * ‖x + g i‖ ^ p i‖
    simp only [_root_.norm_smul, Real.norm_eq_abs, abs_mul, abs_norm, abs_of_nonneg h]refine_2F:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhae:AEStronglyMeasurable (fun x => f x) volumen:ℕc:Fin n → ℝg:Fin n → Space dp:Fin n → ℤc_nonneg:∀ (i : Fin n), 0 ≤ c ip_bound:∀ (i : Fin n), -↑(d - 1) ≤ p ibound:∀ (x : Space d), ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p ix:Space dh:0 ≤ ∑ i, c i * ‖x + g i‖ ^ p i⊢ ‖x‖ * ‖f x‖ ≤ ‖x‖ * ∑ i, c i * ‖x + g i‖ ^ p i
    exact mul_le_mul_of_nonneg_left (bound x) (norm_nonneg x)All goals completed! 🐙@[fun_prop]
lemma norm_mul_isDistBounded {d : ℕ} {f : Space d → ℝ}
    (hf : IsDistBounded f) :
    IsDistBounded (fun x => ‖x‖ * f x) := hf.norm_smul_isDistBounded@[fun_prop]
lemma component_smul_isDistBounded {d : ℕ} [NormedSpace ℝ F] {f : Space d → F}
    (hf : IsDistBounded f) (i : Fin d) :
    IsDistBounded (fun x => x i • f x) := byF:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhf:IsDistBounded fi:Fin d⊢ IsDistBounded fun x => x.val i • f x
  refine IsDistBounded.mono (f := fun x => ‖x‖ • f x) (byF:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhf:IsDistBounded fi:Fin d⊢ IsDistBounded fun x => ‖x‖ • f x fun_propAll goals completed! 🐙)
    (AEStronglyMeasurable.smul ?_ (byF:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhf:IsDistBounded fi:Fin d⊢ AEStronglyMeasurable f volume fun_propAll goals completed! 🐙)) fun x => ?_
  ·refine_1F:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhf:IsDistBounded fi:Fin d⊢ AEStronglyMeasurable (fun x => x.val i) volume simpa [coordCLM_apply, coord_apply] using
      (byF:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhf:IsDistBounded fi:Fin d⊢ AEStronglyMeasurable (fun x => (coordCLM i) x) volume fun_propAll goals completed! 🐙 : AEStronglyMeasurable (fun x => Space.coordCLM i x))
  ·refine_2F:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhf:IsDistBounded fi:Fin dx:Space d⊢ ‖x.val i • f x‖ ≤ ‖‖x‖ • f x‖ simp [norm_smul]refine_2F:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhf:IsDistBounded fi:Fin dx:Space d⊢ |x.val i| * ‖f x‖ ≤ ‖x‖ * ‖f x‖
    exact mul_le_mul_of_nonneg_right (abs_eval_le_norm x i) (byF:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhf:IsDistBounded fi:Fin dx:Space d⊢ 0 ≤ ‖f x‖ positivityAll goals completed! 🐙)@[fun_prop]
lemma component_mul_isDistBounded {d : ℕ} {f : Space d → ℝ}
    (hf : IsDistBounded f) (i : Fin d) :
    IsDistBounded (fun x => x i * f x) := hf.component_smul_isDistBounded i@[fun_prop]
lemma isDistBounded_smul_self {d : ℕ} {f : Space d → ℝ}
    (hf : IsDistBounded f) : IsDistBounded (fun x => f x • x) := byd:ℕf:Space d → ℝhf:IsDistBounded f⊢ IsDistBounded fun x => f x • x
  refine IsDistBounded.congr (f := fun x => ‖x‖ * f x) (byd:ℕf:Space d → ℝhf:IsDistBounded f⊢ IsDistBounded fun x => ‖x‖ * f x fun_propAll goals completed! 🐙)
    (AEStronglyMeasurable.smul (f := f) (g := fun x => x) (byd:ℕf:Space d → ℝhf:IsDistBounded f⊢ AEStronglyMeasurable f volume fun_propAll goals completed! 🐙) (byd:ℕf:Space d → ℝhf:IsDistBounded f⊢ AEStronglyMeasurable (fun x => x) volume fun_propAll goals completed! 🐙)) fun x => ?_
  simp [norm_smul, mul_comm]All goals completed! 🐙@[fun_prop]
lemma isDistBounded_smul_self_repr {d : ℕ} {f : Space d → ℝ}
    (hf : IsDistBounded f) : IsDistBounded (fun x => f x • basis.repr x) := byd:ℕf:Space d → ℝhf:IsDistBounded f⊢ IsDistBounded fun x => f x • basis.repr x
  refine IsDistBounded.congr (f := fun x => ‖x‖ * f x) (byd:ℕf:Space d → ℝhf:IsDistBounded f⊢ IsDistBounded fun x => ‖x‖ * f x fun_propAll goals completed! 🐙)
    (AEStronglyMeasurable.smul (f := f) (g := fun x => basis.repr x) (byd:ℕf:Space d → ℝhf:IsDistBounded f⊢ AEStronglyMeasurable f volume fun_propAll goals completed! 🐙) (byd:ℕf:Space d → ℝhf:IsDistBounded f⊢ AEStronglyMeasurable (fun x => basis.repr x) volume fun_propAll goals completed! 🐙))
      fun x => ?_
  simp [norm_smul, mul_comm]All goals completed! 🐙

@[fun_prop]
lemma isDistBounded_smul_inner {d : ℕ} [NormedSpace ℝ F] {f : Space d → F}
    (hf : IsDistBounded f) (y : Space d) : IsDistBounded (fun x => ⟪y, x⟫_ℝ • f x) := byF:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhf:IsDistBounded fy:Space d⊢ IsDistBounded fun x => ⟪y, x⟫_ℝ • f x
  have h1 (x : Space d) : ⟪y, x⟫_ℝ • f x = ∑ i, (y i * x i) • f x := by
    rw [inner_eq_sum,F:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhf:IsDistBounded fy:Space dx:Space d⊢ (∑ i, y.val i * x.val i) • f x = ∑ i, (y.val i * x.val i) • f xF:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhf:IsDistBounded fy:Space dh1:∀ (x : Space d), ⟪y, x⟫_ℝ • f x = ∑ i, (y.val i * x.val i) • f x⊢ IsDistBounded fun x => ⟪y, x⟫_ℝ • f x ← Finset.sum_smulF:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhf:IsDistBounded fy:Space dx:Space d⊢ (∑ i, y.val i * x.val i) • f x = (∑ i, y.val i * x.val i) • f xF:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhf:IsDistBounded fy:Space dh1:∀ (x : Space d), ⟪y, x⟫_ℝ • f x = ∑ i, (y.val i * x.val i) • f x⊢ IsDistBounded fun x => ⟪y, x⟫_ℝ • f x]F:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhf:IsDistBounded fy:Space dh1:∀ (x : Space d), ⟪y, x⟫_ℝ • f x = ∑ i, (y.val i * x.val i) • f x⊢ IsDistBounded fun x => ⟪y, x⟫_ℝ • f xF:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhf:IsDistBounded fy:Space dh1:∀ (x : Space d), ⟪y, x⟫_ℝ • f x = ∑ i, (y.val i * x.val i) • f x⊢ IsDistBounded fun x => ⟪y, x⟫_ℝ • f x
  simp only [h1, ← smul_smul]F:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhf:IsDistBounded fy:Space dh1:∀ (x : Space d), ⟪y, x⟫_ℝ • f x = ∑ i, (y.val i * x.val i) • f x⊢ IsDistBounded fun x => ∑ x_1, y.val x_1 • x.val x_1 • f x
  exact sum_fun fun i _ => const_fun_smul (byF:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhf:IsDistBounded fy:Space dh1:∀ (x : Space d), ⟪y, x⟫_ℝ • f x = ∑ i, (y.val i * x.val i) • f xi:Fin dx✝:i ∈ Finset.univ⊢ IsDistBounded fun x => x.val i • f x fun_propAll goals completed! 🐙) (y i)

lemma isDistBounded_smul_inner_of_smul_norm {d : ℕ} [NormedSpace ℝ F] {f : Space d → F}
    (hf : IsDistBounded (fun x => ‖x‖ • f x)) (hae : AEStronglyMeasurable f) (y : Space d) :
    IsDistBounded (fun x => ⟪y, x⟫_ℝ • f x) := byF:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhf:IsDistBounded fun x => ‖x‖ • f xhae:AEStronglyMeasurable f volumey:Space d⊢ IsDistBounded fun x => ⟪y, x⟫_ℝ • f x
  have h1 (x : Space d) : ⟪y, x⟫_ℝ • f x = ∑ i, (y i * x i) • f x := by
    rw [inner_eq_sum,F:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhf:IsDistBounded fun x => ‖x‖ • f xhae:AEStronglyMeasurable f volumey:Space dx:Space d⊢ (∑ i, y.val i * x.val i) • f x = ∑ i, (y.val i * x.val i) • f xF:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhf:IsDistBounded fun x => ‖x‖ • f xhae:AEStronglyMeasurable f volumey:Space dh1:∀ (x : Space d), ⟪y, x⟫_ℝ • f x = ∑ i, (y.val i * x.val i) • f x⊢ IsDistBounded fun x => ⟪y, x⟫_ℝ • f x ← Finset.sum_smulF:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhf:IsDistBounded fun x => ‖x‖ • f xhae:AEStronglyMeasurable f volumey:Space dx:Space d⊢ (∑ i, y.val i * x.val i) • f x = (∑ i, y.val i * x.val i) • f xF:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhf:IsDistBounded fun x => ‖x‖ • f xhae:AEStronglyMeasurable f volumey:Space dh1:∀ (x : Space d), ⟪y, x⟫_ℝ • f x = ∑ i, (y.val i * x.val i) • f x⊢ IsDistBounded fun x => ⟪y, x⟫_ℝ • f x]F:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhf:IsDistBounded fun x => ‖x‖ • f xhae:AEStronglyMeasurable f volumey:Space dh1:∀ (x : Space d), ⟪y, x⟫_ℝ • f x = ∑ i, (y.val i * x.val i) • f x⊢ IsDistBounded fun x => ⟪y, x⟫_ℝ • f xF:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhf:IsDistBounded fun x => ‖x‖ • f xhae:AEStronglyMeasurable f volumey:Space dh1:∀ (x : Space d), ⟪y, x⟫_ℝ • f x = ∑ i, (y.val i * x.val i) • f x⊢ IsDistBounded fun x => ⟪y, x⟫_ℝ • f x
  simp only [h1, ← smul_smul]F:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhf:IsDistBounded fun x => ‖x‖ • f xhae:AEStronglyMeasurable f volumey:Space dh1:∀ (x : Space d), ⟪y, x⟫_ℝ • f x = ∑ i, (y.val i * x.val i) • f x⊢ IsDistBounded fun x => ∑ x_1, y.val x_1 • x.val x_1 • f x
  refine sum_fun fun i _ => const_fun_smul (hf.mono (byF:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhf:IsDistBounded fun x => ‖x‖ • f xhae:AEStronglyMeasurable f volumey:Space dh1:∀ (x : Space d), ⟪y, x⟫_ℝ • f x = ∑ i, (y.val i * x.val i) • f xi:Fin dx✝:i ∈ Finset.univ⊢ AEStronglyMeasurable (fun x => x.val i • f x) volume fun_propAll goals completed! 🐙) fun x => ?_) (y i)
  simp [norm_smul]F:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhf:IsDistBounded fun x => ‖x‖ • f xhae:AEStronglyMeasurable f volumey:Space dh1:∀ (x : Space d), ⟪y, x⟫_ℝ • f x = ∑ i, (y.val i * x.val i) • f xi:Fin dx✝:i ∈ Finset.univx:Space d⊢ |x.val i| * ‖f x‖ ≤ ‖x‖ * ‖f x‖
  exact mul_le_mul_of_nonneg_right (abs_eval_le_norm x i) (byF:Typeinst✝¹:NormedAddCommGroup Fd:ℕinst✝:NormedSpace ℝ Ff:Space d → Fhf:IsDistBounded fun x => ‖x‖ • f xhae:AEStronglyMeasurable f volumey:Space dh1:∀ (x : Space d), ⟪y, x⟫_ℝ • f x = ∑ i, (y.val i * x.val i) • f xi:Fin dx✝:i ∈ Finset.univx:Space d⊢ 0 ≤ ‖f x‖ positivityAll goals completed! 🐙)@[fun_prop]
lemma isDistBounded_mul_inner {d : ℕ} {f : Space d → ℝ}
    (hf : IsDistBounded f) (y : Space d) : IsDistBounded (fun x => ⟪y, x⟫_ℝ * f x) :=
  hf.isDistBounded_smul_inner ylemma isDistBounded_mul_inner' {d : ℕ} {f : Space d → ℝ}
    (hf : IsDistBounded f) (y : Space d) : IsDistBounded (fun x => ⟪x, y⟫_ℝ * f x) := byd:ℕf:Space d → ℝhf:IsDistBounded fy:Space d⊢ IsDistBounded fun x => ⟪x, y⟫_ℝ * f x
  simpa only [smul_eq_mul, real_inner_comm y] using hf.isDistBounded_smul_inner yAll goals completed! 🐙lemma isDistBounded_mul_inner_of_smul_norm {d : ℕ} {f : Space d → ℝ}
    (hf : IsDistBounded (fun x => ‖x‖ * f x)) (hae : AEStronglyMeasurable f) (y : Space d) :
    IsDistBounded (fun x => ⟪y, x⟫_ℝ * f x) := hf.isDistBounded_smul_inner_of_smul_norm hae y@[fun_prop]
lemma mul_inner_pow_neg_two {d : ℕ} (y : Space d) (hd : 2 ≤ d := by omega) :
    IsDistBounded (fun x => ⟪y, x⟫_ℝ * ‖x‖ ^ (- 2 : ℤ)) := byd:ℕy:Space dhd:2 ≤ d⊢ IsDistBounded fun x => ⟪y, x⟫_ℝ * ‖x‖ ^ (-2)
  refine IsDistBounded.mono (f := fun x => (‖y‖ * ‖x‖) * ‖x‖ ^ (- 2 : ℤ)) ?_
    (AEMeasurable.aestronglyMeasurable (byd:ℕy:Space dhd:2 ≤ d⊢ AEMeasurable (fun x => ⟪y, x⟫_ℝ * ‖x‖ ^ (-2)) volume fun_propAll goals completed! 🐙)) fun x => ?_
  ·refine_1d:ℕy:Space dhd:2 ≤ d⊢ IsDistBounded fun x => ‖y‖ * ‖x‖ * ‖x‖ ^ (-2) simp only [mul_assoc]refine_1d:ℕy:Space dhd:2 ≤ d⊢ IsDistBounded fun x => ‖y‖ * (‖x‖ * ‖x‖ ^ (-2))
    refine IsDistBounded.const_mul_fun (IsDistBounded.congr (f := fun x => ‖x‖ ^ (- 1 : ℤ))
      (IsDistBounded.pow (d := d) (-1) (byd:ℕy:Space dhd:2 ≤ d⊢ -↑(d - 1) ≤ -1 omegaAll goals completed! 🐙))
      (AEMeasurable.aestronglyMeasurable (byd:ℕy:Space dhd:2 ≤ d⊢ AEMeasurable (fun x => ‖x‖ * ‖x‖ ^ (-2)) volume fun_propAll goals completed! 🐙)) fun x => ?_) ‖y‖
    simp only [norm_mul, norm_norm, norm_inv, norm_zpow, Int.reduceNeg, zpow_neg, zpow_one]refine_1d:ℕy:Space dhd:2 ≤ dx:Space d⊢ ‖x‖ * (‖x‖ ^ 2)⁻¹ = ‖x‖⁻¹
    rcases eq_or_ne x 0 with rfl | hxrefine_1.inld:ℕy:Space dhd:2 ≤ d⊢ ‖0‖ * (‖0‖ ^ 2)⁻¹ = ‖0‖⁻¹refine_1.inrd:ℕy:Space dhd:2 ≤ dx:Space dhx:x ≠ 0⊢ ‖x‖ * (‖x‖ ^ 2)⁻¹ = ‖x‖⁻¹
    ·refine_1.inld:ℕy:Space dhd:2 ≤ d⊢ ‖0‖ * (‖0‖ ^ 2)⁻¹ = ‖0‖⁻¹ simpAll goals completed! 🐙
    ·refine_1.inrd:ℕy:Space dhd:2 ≤ dx:Space dhx:x ≠ 0⊢ ‖x‖ * (‖x‖ ^ 2)⁻¹ = ‖x‖⁻¹ field_simp [norm_ne_zero_iff.mpr hx]All goals completed! 🐙
  ·refine_2d:ℕy:Space dhd:2 ≤ dx:Space d⊢ ‖⟪y, x⟫_ℝ * ‖x‖ ^ (-2)‖ ≤ ‖‖y‖ * ‖x‖ * ‖x‖ ^ (-2)‖ simprefine_2d:ℕy:Space dhd:2 ≤ dx:Space d⊢ |⟪y, x⟫_ℝ| * (‖x‖ ^ 2)⁻¹ ≤ ‖y‖ * ‖x‖ * (‖x‖ ^ 2)⁻¹
    exact mul_le_mul_of_nonneg (abs_real_inner_le_norm y x) (byd:ℕy:Space dhd:2 ≤ dx:Space d⊢ (‖x‖ ^ 2)⁻¹ ≤ (‖x‖ ^ 2)⁻¹ rflAll goals completed! 🐙) (byd:ℕy:Space dhd:2 ≤ dx:Space d⊢ 0 ≤ |⟪y, x⟫_ℝ| positivityAll goals completed! 🐙) (byd:ℕy:Space dhd:2 ≤ dx:Space d⊢ 0 ≤ (‖x‖ ^ 2)⁻¹ positivityAll goals completed! 🐙)

Functions on Space d which can be made into distributions

i. Overview

ii. Key results

iii. Table of contents

iv. References

A. The predicate IsDistBounded f

B. Integrability properties of functions satisfying IsDistBounded f

B.1. AEStronglyMeasurable conditions

B.2. Integrability with respect to Schwartz maps on space

B.3. Integrability with respect to Schwartz maps on time and space

B.4. Integrability with respect to inverse powers

C. Integral on Schwartz maps is bounded by seminorms

D. Construction rules for IsDistBounded f

D.1. Addition

D.2. Finite sums

D.3. Scalar multiplication

D.4. Components of functions

D.5. Compositions with additions and subtractions

D.6. Congruence with respect to the norm

D.7. Monotonicity with respect to the norm

D.8. Inner products

D.9. Scalar multiplication with constant

E. Specific functions that are IsDistBounded

E.1. Constant functions

E.2. Powers of norms

F. Multiplication by norms and components

Functions on `Space d` which can be made into distributions

A. The predicate `IsDistBounded f`

B. Integrability properties of functions satisfying `IsDistBounded f`

B.1. `AEStronglyMeasurable` conditions

D. Construction rules for `IsDistBounded f`

E. Specific functions that are `IsDistBounded`